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I<br />
I<br />
I<br />
I
ORDER OUT OF CHAOS
ORDER OUT OF CHAOS:<br />
MAN'S NEW DIALOGUE WITH NATURE<br />
A Bantam Book I April 1984<br />
New Age and the accompanying figure design as well as the<br />
statement "a search for meaning, growth and change" are<br />
trademarks of Bantam Books, Inc.<br />
All rights reserved.<br />
Copyright © 1984 by llya Prigogine and Isabelle Stengers.<br />
The foreword "Science and Change" copyright© 1984 by<br />
Alvin Tofjler.<br />
Book design by Barbara N. Cohen<br />
This book may not be reproduced in whole or in part, by<br />
mimeograph or tiny other means, without permission.<br />
For information address: Bantam Books, Inc.<br />
Library of Congress Cataloging in Publication Data<br />
Prigogine. I. (llya)<br />
Order out of chaos.<br />
1<br />
Based on the authors' Ia nouvelle alliance.<br />
Includes bibliographical references and index.<br />
I. Science-Philosophy. 2. Physics-Philosophy.<br />
3. Thermodynamics. 4. Irreversible processes.<br />
I. Stengers, Isabelle. II. Prigogine, I. (Ilya)<br />
La nouvelle alliance. Ill. Title.<br />
QI75.P8823 1984 50 83-21 403<br />
ISBN 0-553-34082-4<br />
Published simultaneously in the United States and Canada<br />
Bantam Books are published by Bantam Books.Inc.lts trademark.<br />
consisting of the words "Bantam Books" and the portrayal<br />
of a rooster, is Registered in the United States Patent<br />
and Trademark Office and in other countries. Marca Registrada.<br />
Bantam Books, Inc., 666 Fifth Avenue, New York, New<br />
York 10103.<br />
PRINTED IN THE UNITED STATES OF AMERICA<br />
FG 0987654321
ORDER OUT OF CHAOS<br />
MAN'S NEW DIALOGUE<br />
WITH NATURE<br />
llya Prigogine<br />
and<br />
Isabelle Stengers<br />
Foreword by<br />
Alvin Toffler<br />
BANTAM BOOKS<br />
TORONTO· NEW YORK • LONDON • SYDNEY
This book is dedicated to the memory of<br />
Erich Jantsch<br />
Aharon Katchalsky<br />
Pierre Resibois<br />
Leon Rosenfeld
TABLE OF CONTENTS<br />
FOREWORD: Science and Change by Alvin Toffier xi<br />
PREFACE: Man's New Dialogue with Nature<br />
xxvii<br />
INTRODUCTION: The Challenge to Science 1<br />
Book One: The Delusion of the Universal<br />
CHAPTER I: The Triumph of Reason 27<br />
1. The New Moses 27<br />
2. A Dehumanized World 30<br />
3. The Newtonian Synthesis 37<br />
4. The Experimental Dialogue 41<br />
5. The Myth at the Origin of Science 44<br />
6. The Limits of Classical Science 51<br />
CHAPTER n: The Identification of the Real 57<br />
1. Newton's Laws 57<br />
2. Motion and Change 62<br />
3. The Language of Dynamics 68<br />
4. Laplace's Demon 75<br />
CHAPTER m: The 1Wo Cultures 79<br />
1. Diderot and the Discourse of the Living 79<br />
2. Kant's Critical Ratification 86<br />
3. A Philosophy of Nature? Hegel and Bergson 89<br />
4. Process and Reality: Whitehead 93<br />
5. "Ignoramus, lgnoramibus":<br />
The Positivist's Strain 96<br />
6. A New Start 98<br />
Book Two: The Science of Complexity<br />
CHAPTER IV: Energy and the Industrial Age 103
1. Heat, the Rival of Gravitation 1(}3<br />
2. The Principle of the Conservation of Energy 107<br />
3. Heat Engines and the Arrow of Time 111<br />
4. From Technology to Cosmology 115<br />
5. The Birth of Entropy 117<br />
6. Boltzmann's Order Principle 122<br />
7. Carnot and Darwin 127<br />
CHAPTER v: The Three Stages of Thermodynamics 131<br />
1. Flux and Force 131<br />
2. Linear Thermodynamics 137<br />
3. Far from Equilibrium 140<br />
4. Beyond the Threshold of Chemical Instability 146<br />
5. The Encounter with Molecular Biology 153<br />
6. Bifurcations and Symmetry-Breaking 160<br />
7. Cascading Bifurcations and<br />
the Transitions to Chaos 167<br />
8. From Euclid to Aristotle 171<br />
CHAPTER v1: Order Through Fluctuations 177<br />
1. Fluctuations and Chemistry 177<br />
2. Fluctuations and Correlations 179<br />
3. The Amplification of Fluctuations 181<br />
4. Structural Stability 189<br />
5. Logistic Evolution 192<br />
6. Evolutionary Feedback 196<br />
7. Modelizations of Complexity 203<br />
8. An Open World 207<br />
Book Three: From Being to Becoming<br />
CHAPTER vn: Rediscovering Time 213<br />
1. A Change of Emphasis 213<br />
2. The End of Universality 217<br />
3. The Rise of Quantum Mechanics 218<br />
- 4. Heisenberg's Uncertainty Relation 222<br />
5. The Temporal Evolution of Quantum Systems 226<br />
6. A Nonequilibrium Universe 229
CHAPTER vm: The Clash of Doctrines 233<br />
1. Probability and Irreversibility 233<br />
2. Boltzmann's Breakthrough 240<br />
3. Questioning Boltzmann's Interpretation 243<br />
4. Dynamics and Thermodynamics: 1Wo Separate<br />
Worlds 247<br />
5. Boltzmann and the Arrow of Time 253<br />
CHAPTER IX: Irreversibility-the Entropy Barrier 257<br />
1. Entropy and the Arrow of Time 257<br />
2. Irreversibility as a Symmetry-Breaking Process 260<br />
3. The Limits of Classical Concepts 261<br />
4. The Renewal of Dynamics 264<br />
5. From Randomness to Irreversibility 272<br />
6. The Entropy Barrier 277<br />
7. The Dynamics of Correlations 280<br />
8. Entropy as a Selection Principle 285<br />
9. Active Matter 286<br />
coNcLusioNs: From Earth to Heaventhe<br />
Reenchantment of Nature 291<br />
1. An Open Science 291<br />
2. Time and Times 293<br />
3. The Entropy Barrier 295<br />
4. The Evolutionary Paradigm 297<br />
5. Actors and Spectators 298<br />
6. A Whirlwind in a Tu rbulent Nature 301<br />
7. Beyond Tautology 305<br />
8. The Creative Course of Time 307<br />
9. The Human Condition 311<br />
10. The Renewal of Nature 312<br />
NOTES 315<br />
INDEX 335
FOREWORD<br />
SCIENCE AND<br />
CHANGE<br />
by Alvin Toffler<br />
One of the most highly developed skills in contemporary We stern<br />
civilization is dissection: the split-up of problems into their<br />
smallest possible components. We are good at it. So good, we<br />
often f<strong>org</strong>et to put the pieces back together again.<br />
This skill is perhaps most finely honed in science. There we<br />
not only routinely break problems down into bite-sized chunks<br />
and mini-chunks, we then very often isolate each one from its<br />
environment by means of a useful trick. We say ceteris paribus-all<br />
other things being equal. In this way we can ignore<br />
the complex interactions between our problem and the rest of<br />
the universe.<br />
llya Prigogine, who won the Nobel Prize in 1977 for his<br />
work on the thermodynamics of nonequilibrium systems, is<br />
net satisfied, however, with merely taking things apart. He has<br />
spent the better part of a lifetime trying to "put the pieces<br />
back together again"-the pieces in this case being biology<br />
and physics, necessity and chance, science and humanity.<br />
Born in Russia in 1917 and raised in Belgium since the age<br />
of ten, Prigogine is a compact man with gray hair, cleanly chiseled<br />
features, and a laserlike intensity. Deeply interested in<br />
archaeology, art, and history, he brings to science a remarkable<br />
polymathic mind. He lives with his engineer-wife, Marina,<br />
and his son, Pascal, in Brussels, where a crossdisciplinary<br />
team is busy exploring the implications of his<br />
ideas in fields as disparate as the social behavior of ant colonies,<br />
diffusion reactions in chemical systems and dissipative<br />
processes in quantum field theory.<br />
He spends part of each year at the Ilya Prigogine Center for<br />
Statistical Mechanics and Thermodynamics of the University<br />
of Texas in Austin. To his evident delight and surprise, he was<br />
xi
ORDER OUT OF CHAOS<br />
xii<br />
awarded the Nobel· Prize for his work on "dissipative structures"<br />
arising out of nonlinear processes in nonequilibrium<br />
systems. The coauthor of this volume, Isabelle Stengers, is a<br />
philosopher, chemist, and historian of science who served for<br />
a time as part of Prigogine 's Brussels team. She now lives in<br />
Paris and is associated with the Musee de Ia Villette.<br />
In Order Out of Chaos they have given us a landmark-a<br />
work that is contentious and mind-energizing, a book filled<br />
with flashing insights that subvert many of our most basic assumptions<br />
and suggest fresh ways to think about them.<br />
Under the title La nouvelle alliance, its appearance in<br />
France in 1979 triggered a marvelous scientific free-for-all<br />
among prestigious intellectuals in fields as diverse as entomology<br />
and literary criticism.<br />
It is a measure of America's insularity and cultural arrogance<br />
that this book, which is either published or about to<br />
be published in twelve languages, has taken so long to cross<br />
the Atlantic. The delay carries with it a silver lining, however,<br />
in that this edition includes Prigogine 's newest findings, particularly<br />
with respect to the Second Law of thermodynamics,<br />
which he sets into a fresh perspective.<br />
For all these reasons, Order Out of Chaos is more than just<br />
another book: It is a lever for changing science itself, for compelling<br />
us to reexamine its goals, its methods, its epistemology-its<br />
world view. Indeed, this book can serve as a symbol<br />
of today's historic transformation in science-one that no informed<br />
person can afford to ignore.<br />
Some scholars picture science as driven by its own internal<br />
logic, developing according to its own laws in splendid isolation<br />
from the world around it. Yet many scientific hypotheses,<br />
theories, metaphors, and models (not to mention the choices<br />
made by scientists either to study or to ignore various problems)<br />
are shaped by economic, cultural, and political forces<br />
· operating outside the laboratory.<br />
I do not mean to suggest too neat a parallel between the nature<br />
of society and the reigning scientific world view or "paradigm."<br />
Still less would I relegate science to some "superstructure , .<br />
mounted atop a socioeconomic "base," as Marxists are wont to<br />
do. But science is not an "independent variable.'' It is an open<br />
system embedded in society and linked to it by very dense feedback<br />
loops. It is powerfully influenced by its external environ-
xiii<br />
FOREWORD: SCIENCE AND CHANGE<br />
ment, and, in a general way, its development is shaped by cultural<br />
receptivity to its dominant ideas.<br />
Take that body of ideas that came together in the seventeenth<br />
and eighteenth centuries under the heading of "classical<br />
science" or "Newtonianism." They pictured a world in<br />
which every event was determined by initial conditions that<br />
were, at least in principle, determinable with precision. It was<br />
a world in which chance played no part, in which all the pieces<br />
came together like cogs in a cosmic machine.<br />
The acceptance of this mechanistic view coincided with the<br />
rise of a factory civilization. And divine dice-shooting seems<br />
hardly enough to account for the fact that the Age of the Machine<br />
enthusiastically embraced scientific theories that pictured<br />
the entire universe as a machine.<br />
This view of the world led Laplace to his famous claim that,<br />
given enough facts, we could not merely predict the future but<br />
retrodict the past. And this image of a simple, uniform; mechanical<br />
universe not only shaped the development of science,<br />
it also spilled over into many other fields. It influenced the<br />
framers of the American Constitution to create a machine for<br />
governing, its checks and balances clicking like parts of a<br />
clock. Metternich, when he rode forth to create his balance of<br />
power in Europe, carried a copy of Laplace's writings in his<br />
baggage. And the dramatic spread of factory civilization, with<br />
its vast clanking machines, its heroic engineering breakthroughs,<br />
the rise of the railroad, and new industries such as<br />
steel, textile, and auto, seemed merely to confirm the image of<br />
the universe as an engineer's Tinkertoy.<br />
Today, however, the Age of the Machine is screeching to a<br />
halt, if ages can screech-and ours certainly seems to. And<br />
the decline of the industrial age forces us to confront the painful<br />
limitations of the machine model of reality.<br />
Of course, most of these limitations are not freshly discovered.<br />
The notion that the world is a clockwork, the planets<br />
timelessly orbiting, all systems operating deterministically in<br />
equilibrium, all subject to universal laws that an outside observer<br />
could discover-this model has come under withering<br />
fire ever since it first arose.<br />
In the early nineteenth century, thermodynamics challenged<br />
the timelessness implied in the mechanistic image of<br />
the universe. If the world was a big machine, the thermodynamicists<br />
declared, it was running down, its useful en-
ORDER OUT OF CHAOS<br />
xiv<br />
ergy leaking out. It could not go on forever, and time,<br />
therefore, took on a new meaning. Darwin's followers soon<br />
introduced a contradictory thought: The world-machine might<br />
be running down, losing energy and <strong>org</strong>anization, but biological<br />
systems, at least, were running up, becoming more, not<br />
less, <strong>org</strong>anized.<br />
By the early twentieth century, Einstein had come along to<br />
put the observer back into the system: The machine looked<br />
diffe rent-indeed, for all practical purposes it was differentdepending<br />
upon where you stood within it. But it was still a<br />
deterministic machine, and God did not throw dice. Next, the<br />
quantum people and the uncertainty folks attacked the model<br />
with pickaxes, sledgehammers, and sticks of dynamite.<br />
Nevertheless, despite all the ifs, ands, and buts, it remains<br />
fair to say, as Prigogine and Stengers do, that the machine paradigm<br />
is still the "reference point" for physics and the core<br />
model of science in general. Indeed, so powerful is its continuing<br />
influence that much of social science, and especially<br />
economics, remains under its spell.<br />
The importance of this book is not simply that it uses original<br />
arguments to challenge the Newtonian model, but also<br />
that it shows how the still valid, though much limited, claims<br />
of Newtonianism might fit compatibly into a larger scientific<br />
image of reality. It argues that the old "universal laws" are not<br />
universal at all, but apply only to local regions of reality. And<br />
these happen to be the regions to which science has devoted<br />
the most effort.<br />
Thus, in broad-stroke terms, Prigogine and Stengers argue<br />
that traditional science in the Age of the Machine tended to<br />
emphasize stability, order, uniformity, and equilibrium. It concerned<br />
itself mostly with closed systems and linear relationships<br />
in which small inputs uniformly yield small results.<br />
With the transition from an industrial society based on<br />
heavy inputs of energy, capital, and labor to a high-technology<br />
society in which information and innovation are the critical<br />
resources, it is not surprising that new scientific world models<br />
should appear.<br />
What makes the Prigoginian paradigm especially interesting<br />
is that it shifts attention to those aspects of reality that characterize<br />
today's accelerated social change: disorder, instability,<br />
diversity, disequilibrium, nonlinear relationships (in which
xv<br />
FOREWORD: SCIENCE AND CHANGE<br />
small inputs can trigger massive consequences), and temporality-a<br />
heightened sensitivity to the flows of time.<br />
The work of Ilya Prigogine and his colleagues in the socalled<br />
"Brussels school" may well represent the next revolution<br />
in science as it enters into a new dialogue not merely with<br />
nature, but with society itself.<br />
The ideas of the Brussels school , based heavily on Prigogine's<br />
work, add up to a novel, comprehensive theory of<br />
change.<br />
Summed up and simplified, they hold that while some parts of<br />
the universe may operate like machines, these are closed systems,<br />
and closed systems, at best, form only a small part of the physical<br />
universe. Most phenomena of interest to us are, in fact, open<br />
systems, exchanging energy or matter (and, one might add, information)<br />
with their environment. Surely biological and social systems<br />
are open, which means that the attempt to understand them<br />
in mechanistic terms is doomed to failure.<br />
This suggests, moreover, that most of reality, instead of<br />
being orderly, stable, and equilibria!, is seething and bubbling<br />
with change, disorder, and process.<br />
In Prigoginian terms, all systems contain subsystems,<br />
which are continually "fluctuating." At times, a single fluctuation<br />
or a combination of them may become so powerful, as a<br />
result of positive feedback, that it shatters the preexisting <strong>org</strong>anization.<br />
At this revolutionary moment-the authors call it<br />
a "singular moment" or a "bifurcation point"-it is inherently<br />
impossible to determine in advance which direction change<br />
will take: whether the system will disintegrate into "chaos" or<br />
leap to a new, more differentiated, higher level of "order" or<br />
<strong>org</strong>anization, which they call a "dissipative structure." (Such<br />
physical or chemical structures are termed dissipative because,<br />
compared with the simpler structures they replace, they<br />
require more energy to sustain them.)<br />
One of the key controversies surrounding this concept has<br />
to do with Prigogine's insistence that order and <strong>org</strong>anization<br />
can actually arise "spontaneously" out of disorder and chaos<br />
through a process of "self-<strong>org</strong>anization."<br />
To grasp this extremely powerful idea. we first need to make<br />
a distinction between systems that are in "equilibrium," sys-
ORDER OUT OF CHAOS<br />
xvi<br />
terns that are "near equilibrium," and systems that are "far<br />
from equilibrium."<br />
Imagine a primitive tribe. If its birthrate and death rate are<br />
equal, the size of the population remains stable. Assuming adequate<br />
food and other resources, the tribe forms part of a local<br />
system in ecological equilibrium.<br />
Now increase the birthrate. A few additional births (without<br />
an equivalent number of deaths) might have little effect. The<br />
system may move to a near-equilibria! state. Nothing much<br />
happens. It takes a big jolt to produce big consequences in<br />
systems that are in equilibria] or near-equilibria] states.<br />
But if the birthrate should suddenly soar, the system is<br />
pushed into a far-from-equilibrium condition, and here nonlinear<br />
relationships prevail. In this state, systems do strange<br />
things. They become inordinately sensitive to external influences.<br />
Small inputs yield huge, startling effects. The entire<br />
system may re<strong>org</strong>anize itself in ways that strike us as bizarre.<br />
Examples of such self-re<strong>org</strong>anization abound in Order Out<br />
of Chaos. Heat moving evenly through a liquid suddenly, at a<br />
certain threshold, converts into a convection current that radically<br />
re<strong>org</strong>anizes the liquid, and millions of molecules, as if on<br />
cue, suddenly form themselves into hexagonal cells.<br />
Even more spectacular are the "chemical clocks" described<br />
by Prigogine and Stengers. Imagine a million white ping-pong<br />
balls mixed at random with a million black ones, bouncing<br />
around chaotically in a tank with a glass window in it. Most of<br />
the time, the mass seen through the window would appear to<br />
be gray, but now and then, at irregular moments, the sample<br />
seen through the glass might seem black or white, depending<br />
on the distribution of the balls at that moment in the vicinity of<br />
the window.<br />
Now imagine that suddenly the window goes all white, then<br />
all black, then all white again, and on and on, changing its<br />
color completely at fixed intervals-like a clock ticking.<br />
Why do all the white balls and all the black ones suddenly<br />
<strong>org</strong>anize themselves to change color in time with one another?<br />
By all the traditional rules, this should not happen at all. Yet, if<br />
we leave ping-pong behind and look at molecules in certain chemical<br />
reactions, we find that precisely such a self-<strong>org</strong>anization or<br />
ordering can and does occur--despite what classical physics and<br />
the probability theories of Boltzmann tell us.<br />
In far-from-equilibrium situations other seemingly spon-
xvii<br />
FOREWORD: SCIENCE AND CHANGE<br />
taneous, often dramatic re<strong>org</strong>anizations of matter within time<br />
and space also take place. And if we begin thinking in terms of<br />
two or three dimensions, the number and variety of such pos·<br />
sible structures become very great.<br />
Now add to this an additional discovery. Imagine a situation<br />
in which a chemical or other reaction produces an enzyme<br />
whose presence then encourages further production of the<br />
same enzyme. This is an example of what computer scientists<br />
would call a positive-feedback loop. In chemistry it is called<br />
"auto-catalysis." Such situations are rare in in<strong>org</strong>anic chemis·<br />
try. But in recent decades the molecular biologists have found<br />
that such loops (along with inhibitory or "negative" feedback<br />
and more complicated "cross-catalytic" processes) are the<br />
very stuff of life itself. Such processes help explain how we go<br />
from little lumps of DNA to complex living <strong>org</strong>anisms.<br />
More generally, therefore, in far-from-equilibrium conditions<br />
we find that very small perturbations or fluctuations can<br />
become amplified into gigantic, structure-breaking waves.<br />
And this sheds light on all sorts of "qualitative" or "revolu·<br />
tionary" change processes. When one combines the new insights<br />
gained from studying far-from-equilibrium states and<br />
nonlinear processes, along with these complicated feedback<br />
systems, a whole new approach is opened that makes it possible<br />
to relate the so-called hard sciences to the softer sciences<br />
of life-and perhaps even to social processes as well.<br />
(Such findings have at least analogical significance for social,<br />
economic or political realities. Words like "revolution,"<br />
"economic crash," "technological upheaval ," and "paradigm<br />
shift" all take on new shades of meaning when we begin thinking<br />
of them in terms of fluctuations, feedback amplification,<br />
dissipative structures, bifurcations, and the rest of the Prigoginian<br />
conceptual vocabulary.) It is these panoramic vistas that<br />
are opened to us by Order Out of Chaos.<br />
Beyond this, there is the even more puzzling, pervasive is·<br />
sue of time.<br />
Part of today's vast revolution in both science and culture is<br />
a reconsideration of time, and it is important enough to merit a<br />
brief digression here before returning to Prigogine's role in it.<br />
Take history, for example. One of the great contributions to<br />
historiography has been Braudel's division of time into three<br />
scales-" geographical time," in which events occur over the
ORDER OUT OF CHAOS<br />
xviii<br />
course of aeons; the much shorter "social time" scale by<br />
which economies, states, and civilizations are measured; and<br />
the even shorter scale of "individual time"-the history of<br />
human events.<br />
In social science, time remains a largely unmapped terrain.<br />
Anthropology has taught us that cultures differ sharply in the<br />
way they conceive of time. For some, time is cyclical-history<br />
endlessly recurrent. For other cultures, our own included,<br />
time is a highway stretched between past and future, and people<br />
or whole societies march along it. In still other cultures,<br />
human lives are seen as stationary in time; the future advances<br />
toward us, instead of us toward it.<br />
Each society, as I've written elsewhere, betrays its own<br />
characteristic "time bias"-the degree to which it places emphasis<br />
on past, present, or future. One lives in the past. Another<br />
may be obsessed with the future.<br />
Moreover, each culture and each person tends to think in<br />
terms of "time horizons." Some of us think only of the immediate-the<br />
now. Politicians, for example, are often criticized<br />
for seeking only immediate, short-term results. Their time<br />
horizon is said to be influenced by the date of the next election.<br />
Others among us plan for the long term. These differing<br />
time horizons are an overlooked source of social and political<br />
friction-perhaps among the most important.<br />
But despite the growing recognition that cultural conceptions<br />
of time differ, the social sciences have developed little<br />
in the way of a coherent theory of time. Such a theory might<br />
reach across many disciplines, from politics to group dynamics<br />
and interpersonal psychology. It might, for example, take<br />
account of what, in Future Shock, I called "durational expectancies"-our<br />
culturally induced assumptions about how long<br />
certain processes are supposed to take.<br />
We learn very early, for example, that brushing one's teeth<br />
should last only a few minutes, not an entire morning, or that<br />
when Daddy leaves for work, he is likely to be gone approximately<br />
eight hours, or that a "mealtime" may last a few minutes<br />
or hours, but never a year. (Television, with its division of<br />
the day into fixed thirty- or sixty-minute intervals, subtly<br />
shapes our notions of duration. Thus we normally expect the<br />
hero in a melodrama to get the girl or find the money or win<br />
the war in the last five minutes. In the United States we expect
xix<br />
FOREWORD: SCIENCE AND CHANGE<br />
commercials to break in at certain intervals.) Our minds are<br />
filled with such durational assumptions. Those of children are<br />
much different from those of fully socialized adults, and here<br />
again the diffe rences are a source of conflict.<br />
Moreover, children in an industrial society are "time<br />
trained"-they learn to read the clock, and they learn to distinguish<br />
even quite small slices of time, as when their parents<br />
tell them, "You've only got three more minutes till bedtime!"<br />
These sharply honed temporal skills are often absent in<br />
slower-moving agrarian societies that require less precision in<br />
daily scheduling than our time-obsessed society.<br />
Such concepts, which fit within the social and individual<br />
time scales of Braude!, have never been systematically developed<br />
in the social sciences. Nor have they, in any significant<br />
way, been articulated with our scientific theories of time,<br />
even though they are necessarily connected with our assumptions<br />
about physical reality. And this brings us back to Prigogine,<br />
who has been fascinated by the concept of time since<br />
boyhood. He once said to me that, as a young student, he was<br />
struck by a grand contradiction in the way science viewed<br />
time, and this contradiction has been the source of his life's<br />
work ever since.<br />
In the world model constructed by Newton and his followers,<br />
time was an afterthought. A moment, whether in the<br />
present, past, or future, was assumed to be exactly like any<br />
other moment. The endless cycling of the planets-indeed,<br />
the operations of a clock or a simple machine-can, in principle,<br />
go either backward or forward in time without altering the<br />
basics of the system. For this reason, scientists refer to time in<br />
Newtonian systems as "reversible."<br />
In the nineteenth century, however, as the main focus of<br />
physics shifted from dynamics to thermodynamics and the<br />
Second Law of thermodynamics was proclaimed, time suddenly<br />
became a central concern. For, according to the Second<br />
Law, there is an inescapable loss of energy in the universe.<br />
And, if the world machine is really running down and approaching<br />
the heat death, then it follows that one moment is no<br />
longer exactly like the last. Yo u cannot run the universe backward<br />
to make up for entropy. Events over the long term cannot<br />
replay themselves. And this means that there is a directionality<br />
or, as Eddington later called it, an "arrow" in time.
ORDER OUT OF CHAOS<br />
xx<br />
The whole universe is, in fact, aging. And, in turn, if this is<br />
true, time is a one-way street. It is no longer reversible, but<br />
irreversible.<br />
In short, with the rise of thermodynamics, science split<br />
down the middle with respect to time. Worse yet, even those<br />
who saw time as irreversible soon also split into two camps.<br />
After all, as energy leaked out of the system, its ability to<br />
sustain <strong>org</strong>anized structures weakened, and these, in turn,<br />
broke down into less <strong>org</strong>anized, hence more random elements.<br />
But it is precisely <strong>org</strong>anization that gives any system<br />
internal diversity. Hence, as entropy drained the system of energy,<br />
it also reduced the differences in it. Thus the Second<br />
Law pointed toward an increasingly homogeneous-and, from<br />
the human point of view, pessimistic-future.<br />
Imagine the problems introduced by Darwin and his followers!<br />
For evolution, far from pointing toward reduced <strong>org</strong>anization<br />
and diversity, points in the opposite direction.<br />
Evolution proceeds from simple to complex, from "lower" to<br />
"higher" forms of life, from undifferentiated to differentiated<br />
structures. And, from a human point of view, all this is quite<br />
optimistic. The universe gets "better" <strong>org</strong>anized as it ages,<br />
continually advancing to a higher level as time sweeps by.<br />
In this sense, scientific views of time may be summed up as<br />
a contradiction within a contradiction.<br />
It is these paradoxes that Prigogine and Stengers set out to<br />
illuminate, asking, "What is the specific structure of dynamic<br />
systems which permits them to 'distinguish' between past and<br />
future? What is the minimum complexity involved?"<br />
The answer, for them, is that time makes its appearance<br />
with randomness: "Only when a system behaves in a sufficiently<br />
random way may the difference between past and future,<br />
and therefore irreversibility, enter its description."<br />
In classical or mechanistic science, events begin with "initial<br />
conditions," and their atoms or particles follow "world<br />
lines" or trajectories. These can be traced either backward<br />
into the past or forward into the future. This is just the opposite<br />
of certain chemical reactions, for example, in which two<br />
liquids poured into the same pot diffuse until the mixture is<br />
uniform or homogeneous. These liquids do not de-diffuse<br />
themselves. At each moment of time the mixture is different,<br />
the entire process is "time-oriented."<br />
For classical science, at least in its early stages, such pro-
xxi<br />
FOREWORD: SCIENCE AND CHANGE<br />
cesses were regarded as anomalies, peculiarities that arose<br />
from highly unlikely initial conditions.<br />
It is Prigogine and Stengers' thesis that such time-dependent,<br />
one-way processes are not merely aberrations or deviations<br />
from a world in which time is irreversible. If anything,<br />
the opposite might be true, and it is reversible time, associated<br />
with "closed systems" (if such, indeed, exist in reality), that<br />
may well be the rare or aberrant phenomenon.<br />
What is more, irreversible processes are the source of<br />
order-hence the title Order Out of Chaos. It is the processes<br />
associated with randomness, openness, that lead to higher levels<br />
of <strong>org</strong>anization, such as dissipative structures.<br />
Indeed, one of the key themes of this book is its striking<br />
reinterpretation of the Second Law of thermodynamics. For<br />
according to the authors, entropy is not merely a downward<br />
slide toward dis<strong>org</strong>anization. Under certain conditions, entropy<br />
itself becomes the progenitor of order.<br />
What the authors are proposing, therefore, is a vast synthesis<br />
that embraces both reversible and irreversible time, and<br />
shows how they relate to one another, not merely at the level of<br />
macroscopic phenomena, but at the most minute level as well.<br />
It is a breathtaking attempt at "putting the pieces back together<br />
again." The argument is complex, and at times beyond<br />
easy reach of the lay reader. But it flashes with fresh insight<br />
and suggests a coherent way to relate seemingly unconnected-even<br />
contradictory-philosophical concepts.<br />
Here we begin to glimpse, in full richness, the monumental<br />
synthesis proposed in these pages. By insisting that irreversible<br />
time is not a mere aberration, but a characteristic of much<br />
of the universe, they subvert classical dynamics. For Prigogine<br />
and Stengers, it is not a case of either/or. Of course,<br />
reversibility still applies (at least for sufficiently long times)<br />
but in closed systems only. Irreversibility applies to the rest of<br />
the universe.<br />
Prigogine and Stengers also undermine conventional views<br />
of thermodynamics by showing that, under nonequilibrium<br />
conditions, at least, entropy may produce, rather than degrade,<br />
order, <strong>org</strong>anization-and therefore life.<br />
If this is so, then entropy, too, loses its either/or character.<br />
While certain systems run down, other systems simultaneously<br />
evolve and grow more coherent. This mutualistic,
ORDER OUT OF CHAOS<br />
xxii<br />
nonexclusive view makes it possible for biology and physics to<br />
coexist rather than merely contradict one another.<br />
Finally, yet another profound synthesis is implied-a new<br />
relationship between chance and necessity.<br />
The role of happenstance in the affairs of the universe has<br />
been debated, no doubt, since the first Paleolithic warrior accidently<br />
tripped over a rock. In the Old Te stament, God's will<br />
is sovereign, and He not only controls the orbiting planets but<br />
manipulates the will of each and every individual as He sees<br />
fit. As Prime Mover, all causality flows from Him, and all<br />
events in the universe are foreordained. Sanguinary conflicts<br />
raged over the precise meaning of predestination or free will,<br />
from the time of Augustine through the Carolingian quarrels.<br />
Wycliffe, Huss, Luther, C<strong>alvin</strong>-all contributed to the debate.<br />
No end of interpreters attempted to reconcile determinism<br />
with freedom of will. One ingenious view held that God did<br />
indeed determine the affairs of the universe, but that with respect<br />
to the free will of the individual, He never demanded a<br />
specific action. He merely preset the range of options available<br />
to the human decision-maker. Free will downstairs operated<br />
only within the limits of a menu determined upstairs.<br />
In the secular culture of the Machine Age, hard-line determinism<br />
has more or less held sway even after the challenges of<br />
Heisenberg and the "uncertaintists." Even today, thinkers<br />
such as Rene Thorn reject the idea of chance as illusory and<br />
inherently unscientific.<br />
Faced with such philosophical stonewalling, some defenders<br />
of free will, spontaneity, and ultimate uncertainty, especially<br />
the existentialists, have taken equally uncompromising stands.<br />
(For Sartre, the human being was "completely and always<br />
free," though even Sartre, in certain writings, recognized<br />
practical limitations on this freedom.)<br />
Two things seem to be happening to contemporary concepts<br />
of chance and determinism. To begin with, they are becoming<br />
more complex. As Edgar Morin, a leading French sociologistturned-epistemologist,<br />
has written:<br />
"Let us not f<strong>org</strong>et that the problem of determinism has<br />
changed over the course of a century .... In place of the idea<br />
of sovereign, anonymous, permanent laws directing all things<br />
in nature there has been substituted the idea of laws of interaction<br />
. ... There is more: the problem of determinism has be-
xxiii<br />
FOREWORD: SCIENCE AND CHANGE<br />
come that of the order of the universe. Order means that there<br />
are other things besides 'Jaws': that there are constraints, invariances,<br />
constancies, regularities in our universe . ... In<br />
place of the homogenizing and anonymous view of the old determinism,<br />
there has been substituted a diversifying and evolutive<br />
view of determinations."<br />
And as the concept of determinism has grown richer, new<br />
efforts have been made to recognize the co-presence of both<br />
chance and necessity, not with one subordinate to the· other,<br />
but as full partners in a universe that is simultaneously<br />
<strong>org</strong>anizing and de-<strong>org</strong>anizing itself.<br />
It is here that Prigogine and Stengers enter the arena. For they<br />
have taken the argument a step farther. They not only demonstrate<br />
(persuasively to me, though not to critics like the mathematician,<br />
Rene Thorn) that both determinism and chance operate, they also<br />
attempt to show how the two fit together.<br />
Thus, according to the theory of change implied in the idea<br />
of dissipative structures, when fluctuations force an existing<br />
system into a far-from-equilibrium condition and threaten its<br />
structure, it approaches a critical moment or bifurcation point.<br />
At this point, according to the authors, it is inherently impossible<br />
to determine in advance the next state of the system.<br />
Chance nudges what remains of the system down a new path<br />
of development. And once that path is chosen (from among<br />
many), determinism takes over again until the next bifurcation<br />
point is reached.<br />
Here, in short, we see chance and necessity not as irreconcilable<br />
opposites, but each playing its role as a partner in destiny.<br />
Yet another synthesis is achieved.<br />
When we bring reversible time and irreversible time, disorder<br />
and order, physics and biology, chance and necessity all<br />
into the same novel frame, and stipulate their interrelationships,<br />
we have made a grand statement-arguable, no doubt,<br />
but in this case both powerful and majestic.<br />
Yet this accounts only in part for the excitement occasioned<br />
by Order Out of Chaos. For this sweeping synthesis, as I have<br />
suggested, has strong social and even political overtones. Just<br />
as the Newtonian model gave rise to analogies in politics, diplomacy,<br />
and other spheres seemingly remote from science,<br />
so, too, does the Prigoginian model lend itself to analogical<br />
extension.
ORDER OUT OF CHAOS·<br />
xxfv<br />
By offering rigorous ways of modeling qualitative change,<br />
for example, they shed light on the concept of revolution. By<br />
explaining how successive instabilities give rise to transformatory<br />
change, they illuminate <strong>org</strong>anization theory. They throw a<br />
fresh light, as well, on certain psychological processes-innovation,<br />
for example, which the authors see as associated with<br />
"nonaverage" behavior of the kind that arises under nonequilibrium<br />
conditions.<br />
Even more significant, perhaps, are the implications for the<br />
study of collective behavior. Prigogine and Stengers caution<br />
against leaping to genetic or sociobiological explanations for<br />
puzzling social behavior. Many things that are attributed to<br />
biological pre-wiring are not produced by selfish, determinist<br />
genes, but rather by social interactions under nonequilibrium<br />
conditions.<br />
(In one recent study, for instance, ants were divided into<br />
two categories: One consisted of hard workers, the other of<br />
inactive or "lazy" ants. One might overhastily trace such<br />
traits to genetic predisposition. Yet the study found that if the<br />
system were shattered by separating the two groups from one<br />
another, each in turn developed its own subgroups of hard<br />
workers and idlers. A significant percentage of the "lazy" ants<br />
suddenly turned into hardworking Stakhanovites.)<br />
Not surprisingly, therefore, the ideas behind this remarkable<br />
book are beginning to be researched in economics, urban<br />
studies, human geography, ecology, and many other disciplines.<br />
No one-not even its authors-can appreciate the full implications<br />
of a work as crowded with ideas as Order Out of<br />
Chaos. Each reader will no doubt come away puzzled by some<br />
passages (a few are simply too technical for the reader without<br />
scientific training); startled or stimulated by others (as their<br />
implications strike home); occasionally skeptical; yet intellectually<br />
enriched by the whole. And if one measure of a book is<br />
the degree to which it generates good questions, this one is<br />
surely successful.<br />
Here are just a couple that have haunted me.<br />
How, outside a laboratory, might one define a .. fluctuation"?<br />
What, in Prigoginian terms, does one mean by ••cause"<br />
or "effect"? And when the authors speak of molecules communicating<br />
with one another to achieve coherent, synchro-
xxv<br />
FOREWORD: SCIENCE AND CHANGE<br />
nized change, one may assume they are not anthropomorphiz·<br />
ing. But they raise for me a host of intriguing issues about<br />
whether all parts of the environment are signaling all the time,<br />
or only intermittently; about the indirect, second, and nth<br />
order communication that takes place, permitting a molecule<br />
or an <strong>org</strong>anism to respond to signals which it cannot sense for<br />
lack of the necessary receptors. (A signal sent by the environment<br />
that is undetectable by A may be received by B and converted<br />
into a different kind of signal that A is properly<br />
equipped to receive-so that B serves as a relay/converter,<br />
and A responds to an environmental change that has been signaled<br />
to it via second-order communication.)<br />
In connection with time, what do the authors make of the<br />
idea put forward by Harvard astronomer David Layzer, that<br />
we might conceive of three distinct "arrows of time"-one<br />
based on the continued expansion of the universe since the Big<br />
Bang; one based on entropy; and one based on biological and<br />
historical evolution?<br />
Another question: How revolutionary was the Newtonian<br />
revolution? Taking issue with some historians, Prigogine and<br />
Stengers point out the continuity of Newton's ideas with alchemy<br />
and religious notions of even earlier vintage . Some<br />
readers might conclude from this that the rise of Newtonianism<br />
was neither abrupt nor revolutionary. Yet, to my mind,<br />
the Newtonian breakthrough should not be seen as a linear<br />
outgrowth of these earlier ideas. Indeed, it seems to me that<br />
the theory of change developed in Order Out of Chaos argues<br />
against just such a "continuist" view.<br />
Even if Newtonianism was derivative, this doesn't mean<br />
that the intc;:rnal structure of the Newtonian world-model was<br />
actually the same or that it stood in the same relationship to its<br />
external environment.<br />
The Newtonian system arose at a time when feudalism in<br />
Western Europe was crumbling-when the social system was,<br />
so to speak, far from equilibrium. The model of the universe<br />
proposed by the classical scientists (even if partially derivative)<br />
was applied analogously to new fields and disseminated<br />
successfully, not just because of its scientific power or "rightness,"<br />
but also because an emergent industrial society based<br />
on revolutionary principles provided a particularly receptive<br />
environment for it.<br />
As suggested earlier, machine civilization, in searching for
ORDER OUT OF CHAOS<br />
.xxvi<br />
an explanation of itself in the cosmic order of things, seized<br />
upon the Newtonian model and rewarded those who further<br />
developed it. It is not only in chemical beakers that we find<br />
auto-catalysis, as the authors would be the first to contend.<br />
For these reasons, it still makes sense to me to regard the<br />
Newtonian knowledge system as, itself, a "cultural dissipative<br />
structure" born of social fluctuation.<br />
Ironically, as I've said, I believe their own ideas are central<br />
to the latest revolution in science, and I cannot help but see<br />
these ideas in relationship to the demise of the Machine Age<br />
and the rise of what I have called a "Third Wave" civilization.<br />
Applying their own terminology, we might characterize today's<br />
breakdown of industrial or "Second Wave" society as a<br />
civilizational "bifurcation," and the rise of a more differentiated,<br />
"Third Wave" society as a leap to a new "dissipative<br />
structure" on a world scale. And if we accept this analogy,<br />
might we not look upon the leap from Newtonianism to Prigoginianism<br />
in the same way? Mere analogy, no doubt. But<br />
illuminating, nevertheless.<br />
Finally, we come once more to the ever-challenging issue of<br />
chance and necessity. For if Prigogine and Stengers are right<br />
and chance plays its role at or near the point of bifurcation,<br />
after which deterministic processes take over once more until<br />
the next bifurcation, are they not embedding chance, itself,<br />
within a deterministic framework? By assigning a particular<br />
role to chance, don't they de-chance it?<br />
This question, however, I had the pleasure of discussing<br />
with Prigogine, who smiled over dinner and replied, "Yes.<br />
That would be true. But, of course, we can never determine<br />
when the next bifurcation will arise." Chance rises phoenixlike<br />
once more.<br />
Order out of Chaos is a brilliant, demanding, dazzling bookchallenging<br />
for all and richly rewarding for the attentive reader. It<br />
is a book to study, to savor, to reread-and to question yet again. It<br />
places science and humanity back in a world in which ceteris<br />
paribus is a myth-a world in which other things are seldom held<br />
steady, equal, or unchanging. In short, it projects science into<br />
today's revolutionary world of instability, disequilibrium, and turbulence.<br />
In so doing, it serves the highest creative function-it<br />
helps us create fresh order.
PREFACE<br />
MAN'S NEW DIALOGUE<br />
WITH NATURE<br />
Our vision of nature is undergoing a radical change toward the<br />
multiple, the temporal, and the complex. For a long time a<br />
mechanistic world view dominated Western science. In this<br />
view the world appeared as a vast automaton. We now understand<br />
that we live in a pluralistic world. It is true that there are<br />
phenomena that appear to us as deterministic and reversible,<br />
such as the motion of a frictionless pendulum or the motion of<br />
the earth around the sun. Reversible processes do not know<br />
any privileged direction of time. But there are also irreversible<br />
processes that involve an arrow of time. If you bring together<br />
two liquids such as water and alcohol, they tend to mix in the<br />
forward direction of time as we experience it. We never observe<br />
the reverse process, the spontaneous separation of the<br />
mixture into pure water and pure alcohol. This is therefore an<br />
irreversible process. All of chemistry involves such irreversible<br />
processes.<br />
Obviously, in addition to deterministic processes, there<br />
must be an element of probability involved in some basic processes,<br />
such as, for example, biological evolution or the evolution<br />
of human cultures. Even the scientist who is convinced of<br />
the validity of deterministic descriptions would probably hesitate<br />
to imply that at the very moment of the Big Bang, the<br />
moment of the creation of the universe as we know it, the date<br />
of the publication of this book was already inscribed in the<br />
laws of nature. In the classical view the basic processes of<br />
nature were considered to be deterministic and reversible.<br />
Processes involving randomness or irreversibility were considered<br />
only exceptions. Today we see everywhere the role of<br />
irreversible processes, of fluctuations_<br />
Although Western science has stimulated an extremely fruitxxvii
ORDER OUT OF CHAOS<br />
xxviii<br />
ful dialogue between man and nature, some of its cultural consequences<br />
have been disastrous. The dichotomy between the<br />
"two cultures" is to a large extent due to the conflict between<br />
the atemporal view of classical science and the time-oriented<br />
view that prevails in a large part of the social sciences and<br />
humanities. But in the past few decades, something very dramatic<br />
has been happening in science, something as unexpected<br />
as the birth of geometry or the grand vision of the<br />
cosmos as expressed in Newton's work. We are becoming<br />
more and more conscious of the fact that on all levels, from<br />
elementary particles to cosmology, randomness and irreversibility<br />
play an ever-increasing role. Science is rediscovering<br />
time. It is this conceptual revolution that this book sets out to<br />
describe.<br />
This revolution is proceeding on all levels, on the level of<br />
elementary particles, in cosmology, and on the level of socalled<br />
macroscopic physics, which comprises the physics and<br />
chemistry of atoms and molecules either taken individually or<br />
considered globally as, for example, in the study of liquids or<br />
gases. It is perhaps particularly on this macroscopic level that<br />
the reconceptualization of science is most easy to follow. Classical<br />
dynamics and modern chemistry are going through a period<br />
of drastic change. If one asked a physicist a few years ago<br />
what physics permits us to explain and which problems remain<br />
open, he would have answered that we obviously do not<br />
have an adequate understanding of elementary particles or of<br />
cosmological evolution but that our knowledge of things in between<br />
was pretty satisfactory. Today a growing minority, to<br />
which we belong, would not share this optimism: we have only<br />
begun to understand the level of nature on which we live, and<br />
this is the level on which we have concentrated in this book.<br />
To appreciate the reconceptualization of physics taking<br />
place today, we must put it in proper historical perspective.<br />
The history of science is far from being a linear unfolding that<br />
corresponds to a series of successive approximations toward<br />
some intrinsic truth. It is full of contradictions, of unexpected<br />
turning points. We have devoted a large portion of this book to<br />
the historical pattern followed by Western science, starting<br />
with Newton three centuries ago. We have tried to place the<br />
history of science in the frame of the history of ideas to integrate<br />
it in the evolution of Western culture during the past
xxfx<br />
MAN'S NEW DIALOGUE WITH NATURE<br />
three centuries. Only in this way can we appreciate the unique<br />
moment in which we are presently living.<br />
Our scientific heritage includes two basic questions to<br />
which till now no answer was provided. One is the relation<br />
between disorder and order. The famous law of increase of<br />
entropy -describes the world as evolving from order to disorder<br />
; still, biological or social evolution shows us the complx<br />
emerging fr om the simple. How is this possible? How can<br />
structure arise from disorder? Great progress has been realized<br />
in this question. We know now that nonequilibrium, the<br />
flow of matter and energy, may be a source of order.<br />
But there is the second question, even more basic: classical<br />
or quantum physics describes the world as reversible, as<br />
static. In this description there is no evolution, neither to<br />
order nor to disorder ; the "information," as may be defined<br />
from dynamics, remains constant in time. Therefore there is<br />
an obvious contradiction between the static view of dynamics<br />
and the evolutionary paradigm of thermodynamics. What is<br />
irreversibility? What is entropy? Few questions have been discussed<br />
more often in the course of the history of science. We<br />
begin to be able to give some answers. Order and disorder are<br />
complicated notions: the units involved in the static description<br />
of dynamics are not the same as those that have to be<br />
introduced to achieve the evolutionary paradigm as expressed<br />
by the growth of entropy. This transition leads to a new concept<br />
of matter, matter that is "active," as matter leads to irreversible<br />
processes and as irreversible processes <strong>org</strong>anize<br />
matter.<br />
The evolutionary paradigm, including the concept of entropy,<br />
has exerted a considerable fascination that goes far<br />
beyond science proper. We hope that our unification of dynamics<br />
and thermodynamics will bring out clearly the radical novelty<br />
of the entropy concept in respect to the mechanistic world<br />
view. Time and reality are closely related. For humans, reality<br />
is embedded in the flow of time. As we shall see, the irreversibility<br />
of time is itself closely connected to entropy. To make<br />
time flow backward we would have to overcome an infinite<br />
entropy barrier.<br />
Tr aditionally science has dealt with universals, humanities<br />
with particulars. The convergence of science and humanities<br />
was emphasized in the French title of our book, La Nouvelle
ORDER OUT OF CHAOS<br />
xxx<br />
Alliance, published by Gallimard, Paris, in 1979. However, we<br />
have not succeeded in finding a proper English equivalent of<br />
this title. Furthermore, the text we present here differs from<br />
the French edition, especially in Chapters VII through IX. Although<br />
the origin of structures as the result of nonequilibrium<br />
processes was already adequately treated in the French edition<br />
(as well as in the translations that followed), we had to<br />
entirely rewrite the third part, which deals with our new results<br />
concerning the roots of time as well as with the formulation<br />
of the evolutionary paradigm in the frame of the physical<br />
sciences.<br />
This is all quite recent. The reconceptualization of physics<br />
is far from being achieved. We have decided, however, to present<br />
the situation as it seems to us today. We have a feeling of<br />
great intellectual excitement: we begin to have a glimpse of the<br />
road that leads from being to becoming. As one of us has devoted<br />
most of his scientific life to this problem, he may perhaps<br />
be excused for expressing his feeling of satisfaction, of<br />
aesthetic achievement, which he hopes the reader will share.<br />
For too long there appeared a conflict between what seemed<br />
to be eternal, to be out of time, and what was in time. We see<br />
now that there is a more subtle form of reality involving both<br />
time and eternity.<br />
This book is the outcome of a collective effort in which<br />
many colleagues and friends have been involved. We cannot<br />
thank them all individually. We would like, however, to single<br />
out what we owe to Erich Jantsch, Aharon Katchalsky, Pierre<br />
Resibois, and Leon Rosenfeld, who unfo rtunately are no<br />
longer with us. We have chosen to dedicate this book to their<br />
memory.<br />
We want also to acknowledge the continuous support we<br />
have received from the Instituts Internationaux de Physique et<br />
de Chimie, founded by E. Solvay, and from the Robert A.<br />
Welch Foundation.<br />
The human race is in a period of transition. Science is likely<br />
to play an important role at this moment of demographic explosion.<br />
It is therefore more important than ever to keep open<br />
the channels of communication between science and society.<br />
The present development of We stern science has taken it outside<br />
the cultural environment of the seventeenth centu.ry, in<br />
which it was born. We believe that science today carries a uni-
xxxi<br />
MAN'S NEW DIALOGUE WITH NATURE<br />
versal message that is more acceptable to different cultural<br />
traditions.<br />
During the past decades Alvin Toffier's books have been important<br />
in bringing to the attention of the public some features<br />
of the "Third Wave" that characterizes our time. We are therefore<br />
grateful to him for having written the Foreword to the<br />
English-language version of our book. English is not our native<br />
language. We believe that to some extent ever y language<br />
provides a different way of describing the common reality in<br />
which we are embedded. Some of these characteristics will<br />
survive even the most careful translation. In any case, we are<br />
most grateful to Joseph Early, Ian MacGilvray, Carol<br />
Thurston, and especially to Carl Rubino for their help in the<br />
preparation of this English-language version. We would also<br />
like to express our deep thanks to Pamela Pape for the careful<br />
typing of the successive versions of the manuscript.
INTRODUCTION<br />
THE CHALLENGE TO<br />
SCIENCE<br />
1<br />
It is hardly an exaggeration to state that one of the greatest<br />
dates in the history of mankind was April 28, 1686, when Newton<br />
presented his Principia to the Royal Society of London. It<br />
contained the basic laws of motion together with a clear formulation<br />
of some of the fundamental concepts we still use today,<br />
such as mass, acceleration, and inertia. The greatest<br />
impact was probably made by Book III of the Principia, titled<br />
The System of the World, which included the universal law of<br />
gravitation. Newton's contemporaries immediately grasped<br />
the unique importance of his work. Gravitation became a topic<br />
of conversation both in London and Paris.<br />
Three centuries have now elapsed since Newton's Principia.<br />
Science has grown at an incredible speed, permeating the life<br />
of all of us. Our scientific horizon has expanded to truly fantastic<br />
proportions. On the microscopic scale, elementary partide<br />
physics studies processes involving physical dimensions<br />
of the order of w- ts em and times of the order of I0-22 second.<br />
On the other hand, cosmology leads us to times of the<br />
order of 1010 years, the "age of the universe." Science and<br />
technology are closer than ever. Among other factors, new<br />
biotechnologies and the progress in information techniques<br />
promise to change our lives in a radical way.<br />
Running parallel to this quantitative growth are deep qualitative<br />
changes whose repercussions reach far beyond science<br />
proper and affect the very image of nature. The great founders<br />
of Western science stressed the universality and the eternal<br />
character of natural laws. They set out to formulate general<br />
schemes that would coincide with the very ideal of rationality.
ORDER OUT OF CHAOS 2<br />
As Roger Hausheer says in his fine introduction to Isaiah<br />
Berlin's Against the Current, "They sought all-embracing<br />
schemas, universal unifying frameworks, within which everything<br />
that exists could be shown to be systematically-i.e.,<br />
logically or causally-interconnected, vast structures in which<br />
there should be no gaps left open for spontaneous, unattended<br />
developments, where everything that occurs should be, at<br />
least in principle, wholly explicable in terms of immutable<br />
general laws." t<br />
The story of this quest is indeed a dramatic one. There were<br />
moments when this ambitious program seemed near completion.<br />
A fundamental level from which all other properties of<br />
matter could be deduced seemed to be in sight. Such moments<br />
can be associated with the formulation of Bohr's celebrated<br />
atomic model, which reduced matter to simple planetary systems<br />
formed by electrons and protons. Another moment of<br />
great suspense came when Einstein hoped to condense all the<br />
laws of physics into a single "unified field theory. " Great progress<br />
has indeed been realized in the unification of some of the<br />
basic forces found in nature. Still, the fundamental level remains<br />
elusive. Wherever we look we fi nd evolution, diversification,<br />
and instabilities. Curiously, this is true on all levels,<br />
in the field of elementary particles, in biology, and in astrophysics,<br />
with the expanding universe and the formation of<br />
black holes.<br />
As we said in the Preface, our vision of nature is undergoing<br />
a radical change toward the multiple, the temporal, and the<br />
complex. Curiously, the unexpected complexity that has been<br />
discovered in nature has not led to a slowdown in the progress<br />
of science, but on the contrary to the emergence of new conceptual<br />
structures that now appear as essential to our understanding<br />
of the physical world-the world that includes us. It<br />
is this new situation, which has no precedent in the history of<br />
science, that we wish to analyze in this book.<br />
The story of the transformation of our conceptions about<br />
science and nature can hardly be separated from another<br />
story, that of the feelings aroused by science. With every new<br />
intellectual program always come new hopes, fears, and expectations.<br />
In classical science the emphasis was on time-independent<br />
laws. As we shall see, once the particular state of a<br />
system has been measured, the reversible laws of classical sci-
3 THE CHALLENGE TO SCIENCE<br />
ence are supposed to determine its future, just as they had<br />
determined its past. It is natural that this quest for an eternal<br />
truth behind changing phenomena aroused enthusiasm. But it<br />
also came as a shock that nature described in this way was in<br />
fact debased: by the very success of science, nature was<br />
shown to be an automaton, a robot.<br />
The urge to reduce the diversity of nature to a web of illusions<br />
has been present in Western thought since the time of<br />
Greek atomists. Lucretius, following his masters Democritus<br />
and Epicurus, writes that the world is "just" atoms and void<br />
and urges us to look for the hidden behind the obvious: "Still,<br />
lest you happen to mistrust my words, because the eye cannot<br />
perceive prime bodies, hear now of particles you must admit<br />
exist in the world and yet cannot be seen. "2<br />
Yet it is well known that the driving force behind the work of<br />
the Greek atomists was not to debase nature but to free men<br />
from fear, the fear of any supernatural being, of any order that<br />
would transcend that of men and nature. Again and again Lucretius<br />
repeats that we have nothing to fear, that the essence of<br />
the world is the ever-changing associations of atoms in the<br />
void.<br />
Modern science transmuted this fundamentally ethical<br />
stance into what seemed to be an established truth; and this<br />
truth, the reduction of nature to atoms and void, in turn gave<br />
rise to what Lenoble3 has called the "anxiety of modern<br />
men." How can we recognize ourselves in the random world<br />
of the atoms? Must science be defined in terms of rupture between<br />
man and nature? 'JI bodies, the firmament, the stars,<br />
the earth and its kingdoms are not equal to the lowest mind;<br />
for mind knows all this in itself and these bodies nothing. "4<br />
This "Pensee" by Pascal expresses the same feeling of alienation<br />
we find among contemporary scientists such as Jacques<br />
Monod:<br />
Man must at last finally awake from his millenary<br />
dream; and in doing so, awake to his total solitude, his<br />
fundamental isolation. Now does he at last realize that,<br />
like a gypsy, he lives on the boundary of an alien world. A<br />
world that is deaf to his music. just as indifferent to his<br />
hopes as it is to his suffering or his crimes.s
ORDER OUT OF CHAOS 4<br />
This is a paradox. A brilliant breakthrough in molecular biology,<br />
the deciphering of the genetic code, in which Monod<br />
actively participated, ends upon a tragic note. This very progress,<br />
we are told, makes us the gypsies ,0f the universe. How<br />
can we explain this situation? Is not science a way of communication,<br />
a dialogue with nature?<br />
In the past, strong distinctions were frequently made between<br />
man's world and the supposedly alien natural world. A<br />
famous passage by Vico in The New Science describes this<br />
most vividly:<br />
... in the night of thick darkness enveloping the earliest<br />
antiquity, so remote from ourselves, there shines the eternal<br />
and never failing light of a truth beyond all question:<br />
that the world of civil society has certainly been made by<br />
men, and that its principles are therefore to be found<br />
within the modifications of our own human mind.<br />
Whoever reflects on this cannot but marvel that the philosophers<br />
should have bent all their energies to the study<br />
of the world of nature, which, since God made it, He<br />
alone knows; and that they should have neglected the<br />
study of the world of nations, or civil world, which, since<br />
men had made it, men could come to know. 6<br />
Present -day research leads us farther and farther away from<br />
the opposition between man and the natural world. It will be<br />
one of the main purposes of this book to show, instead of rupture<br />
and opposition, the growing coherence of our knowledge<br />
of man and nature.<br />
In the past, the questioning of nature has taken the most diverse<br />
forms. Sumer discovered writing; the Sumerian priests<br />
speculated that the future might be written in some hidden<br />
way in the events taking place around us in the present. They<br />
even systematized this belief, mixing magical and rational elements.<br />
7 In this sense we may say that Western science, which<br />
originated in the seventeenth century, only opened a new<br />
chapter in the everlasting dialogue between man and nature.
5 THE CHALLENGE TO SCIENCI:<br />
Alexandre Koyres has defined the innovation brought about<br />
by modern science in terms of "experimentation." Modern<br />
science is based on the discovery of a new and specific form of<br />
communication with nature-that is, on the conviction that<br />
nature responds to experimental interrogation. How can we<br />
define more precisely the experimental dialogue? Experimentation<br />
does not mean merely the faithful observation of facts<br />
as they occur, nor the mere search for empirical connections<br />
between phenomena, but presupposes a systematic interaction<br />
between theoretical concepts and observation.<br />
In hundreds of different ways scientists have expressed ttieir<br />
amazement when, on determining the right question, they discover<br />
that they can see how the puzzle fits together. In this<br />
sense, science is like a two-partner game ir;t which we have to<br />
guess the behavior of a reality unrelated to our beliefs, our<br />
ambitions, or our hopes. Nature cannot be forced to say anything<br />
we want it to. Scientific investigation is not a monologue.<br />
It is precisely the risk involved that makes this game<br />
exciting.<br />
But the uniqueness of Western science is far from being exhausted<br />
by such methodological considerations. When Karl<br />
Popper discussed the normative description of scientific rationality,<br />
he was forced to admit that in the final analysis rational<br />
science owes its existence to its success; the scientific<br />
method is applicable only by virtue of the astonishing points<br />
of agreement between preconceived models and experimental<br />
results.9 Science is a risky game, but it seems to have discovered<br />
questions to which nature provides consistent answers.<br />
The success of Western science is an historical fact, unpredictable<br />
a priori, but which cannot be ignored. The surprising<br />
success of modern science has led to an irreversible transformation<br />
of our relations with nature. In this sense, the term<br />
"scientific revolution" can legitimately be used. The history of<br />
mankind has been marked by other turning points, by other<br />
singular conjunctions of circumstances leading to irreversible<br />
changes. One such crucial event is known as the "Neolithic<br />
revolution." But there, just as in the case of the "choices"<br />
marking biological evolution, we can at present only proceed<br />
by guesswork, while there is a wealth of information concerning<br />
decisive episodes in the evolution of science. The so-called
ORDER OUT OF CHAOS 6<br />
"Neolithic revolution" took thousands of years. Simplifying<br />
somewhat, we may say the scientific revolution started only<br />
three ·Centuries ago. We have what is perhaps a unique opportunity<br />
to apprehend the specific and intelligible mixture of<br />
"chance" and "necessity" marking this revolution.<br />
Science initiated a successful dialogue with nature. On the<br />
other hand, the first outcome of this dialogue was the discovery<br />
of a silent world. This is the paradox of classical science.<br />
It revealed to men a dead, passive nature, a nature that behaves<br />
as an automaton which, once programmed, continues to<br />
follow the rules inscribed in the program. In this sense the<br />
dialogue with nature isolated man from nature instead of<br />
bringing him closer to it. A triumph of human reason turned<br />
into a sad truth. It seemed that science debased everything it<br />
touched.<br />
Modern science horrified both its opponents, for whom it<br />
appeared as a deadly danger, and some of its supporters, who<br />
saw in man's solitude as "discovered" by science the price we<br />
had to pay for this new rationality.<br />
The cultural tension associated with classical science can be<br />
held at least partly responsible for the unstable position of science<br />
within society; it led to an heroic assumption of the harsh<br />
implications of rationality, but it led also to violent rejection.<br />
We shall return later to present-day antiscience movements.<br />
Let us take an earlier example-the irrationalist movement in<br />
Germany in the 1920s that formed the cultural background to<br />
quantum mechanics. to In opposition to science, which was<br />
identified with a set of concepts such as causality, determinism,<br />
reductionism, and rationality, there was a violent upsurge<br />
of ideas denied by science but seen as the embodiment<br />
of the fundamental irrationality of nature. Life, destiny, freedom,<br />
and spontaneity thus became manifestations of a shadowy<br />
underworld impenetrable to reason. Without going into<br />
the peculiar sociopolitical context to which it owed its vehement<br />
nature, we can state that this rejection illustrates the<br />
risks associated with classical science. By admitting only a<br />
subjective meaning for a set of experiences men believe to be<br />
significant, science runs the risk of transferring these into the<br />
realm of the irrational, bestowing upon them a formidable<br />
power.<br />
As Joseph Needham has emphasized, Western thought has
7 THE CHALLENGE TO SCIENCE<br />
always oscillated between the world as an automaton and a<br />
theology in which God governs the universe. This is what<br />
Needham calls the "characteristic European schizophrenia.<br />
" II In fact, these visions are connected. An automaton<br />
needs an external god.<br />
Do we really have to make this tragic choice? Must we<br />
choose between a science that leads to alienation and an antiscientific<br />
metaphysical view of nature? We think such a choice<br />
is no longer necessary, since the changes that science is undergoing<br />
today lead to a radically new situation. This recent evolution<br />
of science gives us a unique opportunity to reconsider<br />
its position in culture in general. Modern science originated in<br />
the specific context of the European seventeenth century. We<br />
are now approaching the end of the twentieth century, and it<br />
seems that some more universal message is carried by science,<br />
a message that concerns the interaction of man and nature<br />
as well as of man with man.<br />
What are the assumptions of classical science from which we<br />
believe science has freed itself today? Generally those centering<br />
around the basic conviction that at some level the world is<br />
simple and is governed by time-reversible fundamental laws.<br />
Today this appears as an excessive simplification. We may<br />
compare it to reducing buildings to piles of bricks. Ye t out of<br />
the same bricks we may construct a factory, a palace, or a<br />
cathedral. It is on the level of the building as a whole that we<br />
apprehend it as a creature of time, as a product of a culture, a<br />
society, a style. But there is the additional and obvious problem<br />
that, since there is no one to build nature, we must give to<br />
its very "bricks"-that is, to its microscopic activity-a description<br />
that accounts for this building process.<br />
The quest of classical science is in itself an illustration of a<br />
dichotomy that runs throughout the history of We stern<br />
thought. Only the immutable world of ideas was traditionally<br />
recognized as "illuminated by the sun of the intelligible," to<br />
use Plato's expression. In the same sense, only eternal laws<br />
were seen to express scientific rationality. Te mporality was<br />
looked down upon as an illusion. This is no longer true today.
ORDER OUT OF CHAOS 8<br />
We have discovered that far from being an illusion, irrevers·<br />
ibility plays an essential role in nature and lies at the origin of<br />
most processes of self-<strong>org</strong>anization. We find ourselves in a<br />
world in which reversibility and determinism apply only to<br />
limiting, simple cases, while irreversibility and randomness<br />
are the rules. .<br />
The denial of time and complexity was central to the cultural<br />
issues raised by the scientific enterprise in its classical definition.<br />
The challenge of these concepts was also decisive for the<br />
metamorphosis of science we wish to describe. In his great<br />
book The Nature of the Physical World, Arthur Eddingtonl2<br />
introduced a distinction between primary and secondary laws.<br />
"Primary laws" control the behavior of single particles, while<br />
"secondary laws" are applicable to collections of atoms or<br />
molecules. To insist on secondary laws is to emphasize that<br />
the description of elementary behaviors is not sufficient for<br />
understanding a system as a whole. An outstanding case of a<br />
secondary law is, in Eddington's view, the second law of thermodynamics,<br />
the law that introduces the "arrow of time" in<br />
physics. Eddington writes: "From the point of view of philosophy<br />
of science the conception associated with entropy<br />
must, I think, be ranked as the great contribution of the nineteenth<br />
century to scientific thought. It marked a reaction from<br />
the view that everything to which science need pay attention is<br />
discovered by a microscopic dissection of objects." 13 This<br />
trend has been dramatically amplified today.<br />
It is true that some of the greatest successes of modern science<br />
are discoveries at the microscopic level, that of molecules,<br />
atoms, or elementary particles. For example, molecular<br />
biology has been immensely successful in isolating specific<br />
molecules that play a central role in the mechanism of life. In<br />
fact, this success has been so overwhelming that for many scientists<br />
the aim of research is identified with this "microscopic<br />
dissection of objects," to use Eddington's expression. However,<br />
the second law of thermodynamics presented the first<br />
challenge to a concept of nature that would explain away the<br />
complex and reduce it to the simplicity of some hidden world.<br />
Today interest is shifting from substance to relation, to communication,<br />
to time.<br />
This change of perspective is not the result of some arbitrary<br />
decision. In physics it was forced upon us by new dis-
9 THE CHALLENGE TO SCIENCE<br />
coveries no one could have foreseen. Who would have<br />
expected that most (and perhaps all) elementary particles<br />
would prove to be unstable? Who would have expected that<br />
with the experimental confirmation of an expanding universe<br />
we could conceive of the history of the world as a whole?<br />
At the end of the twentieth century we have learned to understand<br />
better the meaning of the two great revolutions that<br />
gave shape to the physics of our time, quantum mechanics and<br />
relativity. They started as attempts to correct classical mechanics<br />
and to incorporate into it the newly found universal<br />
constants. Today the situation has changed. Quantum mechanics<br />
has given us the theoretical frame to describe the incessant<br />
transformations of particles into each other. Similarly,<br />
general relativity has become the basic theory in terms of<br />
which we can describe the thermal history of our universe in<br />
its early stages.<br />
Our universe has a pluralistic, complex character. Structures<br />
may disappear, but also they may appear. Some processes<br />
are , as far as we know, well described by deterministic<br />
equations, but others involve probabilistic processes.<br />
How then can we overcome the apparent contradiction between<br />
these concepts? We are living in a single universe. As<br />
we shall see, we are beginning to appreciate the meaning of<br />
these problems. Moreover, the importance we now give to the<br />
various phenomena we observe and describe is quite different<br />
from, even opposite to, what was suggested by classical physics.<br />
There the basic processes, as we mentioned, are considered as<br />
deterministic and reversible. Processes involving randomness<br />
or irreversibility are considered to be exceptions. Today we<br />
see everywhere the role of irreversible processes, of fluctuations.<br />
The models considered by classical physics seem to us<br />
to occur only in limiting situations such as we can create artificially<br />
by putting matter into a box and then waiting till it<br />
reaches equilibrium.<br />
The artificial may be deterministic and reversible. The natural<br />
contains essential elements of randomness and irreversibility.<br />
This leads to a new view of matter in which matter is no<br />
longer the passive substance described in the mechanistic<br />
world view but is associated with spontaneous activity. This<br />
change is so profound that, as we stated in our Preface, we can<br />
really speak about a new dialogue of man with nature.
ORDER OUT OF CHAOS 10<br />
This book deals with the conceptual transformation of science<br />
from the Golden Age of classical science to the present.<br />
To describe this transformation we could have chosen many<br />
roads. We could have studied the problems of elementary particles.<br />
We could have followed recent fascinating developments<br />
in astrophysics. These are the subjects that seem to<br />
delimit the frontiers of science. However, as we stated in our<br />
Preface, over the past years so many new feaures of nature at<br />
our level have been discovered that we decided to concentrate<br />
on this intermediate level, on problems that belong mainly to<br />
our macroscopic world, which includes atoms, molecules, and<br />
especially biomolecules. Still it is important to emphasize that<br />
the evolution of science proceeds on somewhat parallel lines at<br />
every level, be it that of elementary particles, chemistry, biology,<br />
or cosmology. On every scale self-<strong>org</strong>anization, complexity,<br />
and time play a new and unexpected role.<br />
Therefore, our aim is to examine the significance of three<br />
centuries of scientific progress from a definite viewpoint.<br />
There is certainly a subjective element in the way we have<br />
chosen our material. The problem of time is really the center<br />
of the research that one of us has been pursuing all his life.<br />
When as a young student at the University of Brussels he<br />
came into contact with physics and chemistry for the first<br />
time, he was astonished that science had so little to say about<br />
time, especially since his earlier education had centered<br />
mainly around history and archaeology. This surprise could<br />
have led him to two attitudes, both of which we find exemplified<br />
in the past: one would have been to discard the prob<br />
lem, since classical science seemed to have no place for time;<br />
and the other would have been to look for some other way of<br />
apprehending nature, in which time would play a different,<br />
more basic role. This is the path Bergson and Whitehead, to<br />
mention only two philosophers. of our century, chose. The first<br />
position would be a "positivistic" one, the second a "metaphysical"<br />
one.<br />
There was, however, a third path, which was to ask whether<br />
the simplicity of the temporal evolution traditionally consid-
11 THE CHALLENGE TO SCIENCE<br />
ered in physics and chemistry was due to the fact that attention<br />
was paid mainly to some very simplified situations, to<br />
heaps of bricks in contrast with the cathedral to which we have<br />
alluded.<br />
This book is divided into three parts. The first part deals<br />
with the triumph of classical science and the cultural consequences<br />
of this triumph. Initially, science was greeted with<br />
enthusiasm. We shall then describe the cultural polarization<br />
that occurred as a result of the existence of classical science<br />
and its astonishing success. Is this success to be accepted as<br />
such, perhaps limiting its implications, or must the scientific<br />
method itself be rejected as partial or illusory? Both choices<br />
lead to the same result-the collision between what has often<br />
been called the "two cultures," science and the humanities.<br />
These questions have played a basic role in Western thought<br />
since the formulation of classical science. Again and again we<br />
come to the problem, "How to choose?" Isaiah Berlin has<br />
rightly seen in this question the beginning of the schism between<br />
the sciences and the humanities:<br />
The specific and unique versus the repetitive and the universal,<br />
the concrete versus the abstract, perpetual movement<br />
versus rest, the inner versus the outer, quality<br />
versus quantity, culture-bound versus timeless principles,<br />
mental strife and self-transformation as a permanent condition<br />
of man versus the possibility (and desirability) of<br />
peace, order, final harmony and the satisfaction of all ra-<br />
- tional human wishes-these are some of the aspects of<br />
the contrast.14<br />
We have devoted much space to classical mechanics. Indeed,<br />
in our view this is the best vantage point from which we<br />
may contemplate the present-day transformation of science.<br />
Classical dynamics seems to express in an especially clear and<br />
striking way the static view of nature. Here time apparently is<br />
reduced to a parameter, and future and past become equivalent.<br />
It is true that quantum theory has raised many new<br />
problems not covered by classical dynamics but it has nevertheless<br />
retained a number of the conceptual positions of classical<br />
dynamics, particularly as far as time and process are<br />
concerned.
ORDER OUT OF CHAOS 12<br />
As early as at the beginning of the nineteenth century,<br />
precisely when classical science was triumphant, when the<br />
Newtonian program dominated French science and the latter<br />
dominated Europe, the first threat to the Newtonian construction<br />
loomed into sight. In the second part of our study we<br />
shall follow the development of the science of heat, this rival to<br />
Newton's science of gravity, starting from the first gauntlet<br />
thrown down when Fourier formulated the law governing the<br />
propagation of heat. It was, in fact, the first quantitative description<br />
of something inconceivable in classical dynamicsan<br />
irreversible process.<br />
The two descendants of the science of heat, the science of<br />
energy conversion and the science of heat engines, gave birth<br />
to the first "nonclassical" science-thermodynamics. The<br />
most original contribution of thermodynamics is the celebrated<br />
second law, which introduced into physics the arrow of<br />
time. This introduction was part of a more global intellectual<br />
move. The nineteenth century was really the century of evolution;<br />
biology, geology, and sociology emphasized processes of<br />
becoming, of increasing complexity. As for thermodynamics,<br />
it is based on the distinction of two types of processes: reversible<br />
processes, which are independent of the direction of time,<br />
and irreversible processes, which depend on the direction of<br />
time. We shall see examples later. It was in order to distinguish<br />
the two types of processes that the concept of entropy was<br />
introduced, since entropy increases only because of the irreversible<br />
processes.<br />
During the nineteenth century the final state of thermodynamic<br />
evolution was at the center of scientific research. This<br />
was equilibrium thermodynamics. Irreversible processes were<br />
looked down on as nuisances, as disturbances, as subjects not<br />
worthy of study. Today this situation has completely changed.<br />
We now know that far from equilibrium, new types of structures<br />
may originate spontaneously. In far-from-equilibrium<br />
conditions we may have transformation from disorder, from<br />
thermal chaos, into order. New dynamic states of matter may<br />
originate, states that reflect the interaction of a given system<br />
with its surroundings. We have called these new structures dissipative<br />
structures to emphasize the constructive role of dissipative<br />
processes in their fo rmation.<br />
This book describes some of the methods that have been
13 THE CHALLENGE TO SCIENCE<br />
developed in recent years to deal with the appearance and evolution<br />
of dissipative structures. Here we find the key words<br />
that run throughout this book like leitmotivs: nonlinearity, instability,<br />
fluctuations. They have begun to permeate our view<br />
of nature even beyond the fields of physics and chemistry<br />
proper.<br />
We cited Isaiah Berlin when we discussed the opposition between<br />
the sciences and the humanities. He opposed the specific<br />
and unique to the repetitive and the universal. The remarkable<br />
feature is that when we move from equilibrium to far-fromequilibrium<br />
conditions, we move away from the repetitive and<br />
the universal to the specific and the unique. Indeed, the laws<br />
of equilibrium are universal. Matter near equilibrium behaves<br />
in a "repetitive" way. On the other hand, far from equilibrium<br />
there appears a variety of mechanisms corresponding to the<br />
possibility of occurrence of various types of dissipative structures.<br />
For example, far from equilibrium we may witness the<br />
appearance of chemical clocks, chemical reactions which behave<br />
in a coherent, rhythmical fashion. We may also have processes<br />
of self-<strong>org</strong>anization leading to nonhomogeneous<br />
structures to nonequilibrium crystals.<br />
We would like to emphasize the unexpected character of this<br />
behavior. Every one of us has an intuitive view of how a chemical<br />
reaction takes place; we imagine molecules floating<br />
through space, colliding, and reappearing in new forms. We<br />
see chaotic behavior similar to what the atomists described<br />
when they spoke about dust dancing in the air. But in a chemical<br />
clock the behavior is quite different. Oversimplifying somewhat,<br />
we can say that in a chemical clock all molecules change<br />
their chemical identity simultaneously, at regular time intervals.<br />
If the molecules can be imagined as blue or red, we<br />
would see their change of color following the rhythm of the<br />
chemical clock reaction.<br />
Obviously such a situation can no longer be described in<br />
terms of chaotic behavior. A new type of order has appeared.<br />
We can speak of a new coherence, of a mechanism of "communication"<br />
among molecules. But this type of communication<br />
can arise only in far-from-equilibrium conditions. It is<br />
quite interesting that such communication seems to be the rule<br />
in the world of biology. It may in fact be taken as the very basis<br />
of the definition of a biological system.
ORDER OUT OF CHAOS 14<br />
In addition, the type of dissipative structure depends critically<br />
on the conditions in which the structure is formed. External<br />
fields such as the gravitational field of earth, as well as<br />
the magnetic field, may play an essential role in the selection<br />
mechanism of self-<strong>org</strong>anization.<br />
We begin to see how, starting from chemistry, we may build<br />
complex structures, complex forms, some of which may have<br />
been the precursors of life. What seems certain is that these<br />
far-from-equilibrium phenomena illustrate an essential and unexpected<br />
property of matter: physics may henceforth describe<br />
structures as adapted to outside conditions. We meet in rather<br />
simple chemical systems a kind of prebiological adaptation<br />
mechanism. To use somewhat anthropomorphic language: in<br />
equilibrium matter is "blind," but in far-from-equilibrium conditions<br />
it begins to be able to perceive, to "take into account,"<br />
in its way of functioning, differences in the external world<br />
(such as weak gravitational or electrical fields).<br />
Of course, the problem of the origin of life remains a difficult<br />
one, and we do not think a simple solution is imminent.<br />
Still, from this perspective life no longer appears to oppose the<br />
"normal" laws of physics, struggling against them to avoid its<br />
normal fate-its destruction. On the contrary, life seems to<br />
express in a specific way the very conditions in which our biosphere<br />
is embedded, incorporating the nonlinearities of chemical<br />
reactions and the far-from-equilibrium conditions imposed<br />
on the biosphere by solar radiation.<br />
We have discussed the concepts that allow us to describe the<br />
formation of dissipative structures, such as the theory of bifurcations.<br />
It is remarkable that near-bifurcations systems present<br />
large fluctuations. Such systems seem to "hesitate"<br />
among various possible directions of evolution, and the famous<br />
law of large numbers in its usual sense breaks down. A<br />
small fluctuation may start an entirely new evolution that will<br />
drastically change the whole behavior of the macroscopic system.<br />
The analogy with social phenomena, even with history, is<br />
inescapable. Far from opposing "chance" and "necessity," we<br />
now see both aspects as essential in the description of nonlinear<br />
systems far from equilibrium.<br />
The first two parts of this book thus deal with two conflicting<br />
views of the physical universe: the static view of classical<br />
dynamics, and the evolutionary view associated with entropy.
15 THE CHALLENGE TO SCIENCE<br />
A confrontation between these views has become unavoidable.<br />
For a long time this confrontation was postponed by considering<br />
irreversibility as an illusion, as an approximation; it<br />
was man who introduced time into a timeless universe. However,<br />
this solution in which irreversibility is reduced to an illusion<br />
or to approximations can no longer be accepted, since we<br />
know that irreversibility may be a source of order, of coherence,<br />
of <strong>org</strong>anization.<br />
We can no longer avoid this confrontation. It is the subject<br />
of the third part of this book. We describe traditional attempts<br />
to approach the problem of irreversibility first in classical and<br />
then in quantum mechanics. Pioneering work was done here,<br />
especially by Boltzmann and Gibbs. However, we can state<br />
that the problem was left largely unsolved. As Karl Popper<br />
relates it, it is a dramatic story: first, Boltzmann thought he<br />
had given an objective formulation to the new concept of time<br />
implied in the second law. But as a result of his controversy<br />
with Zermelo and others, he had to retreat.<br />
In the light of history-or in the darkness of history<br />
Boltzmann was defeated, according to all accepted standards,<br />
though everybody accepts his eminence as a physicist.<br />
For he never succeeded in clearing up the status of<br />
his Ji-theorem; nor did he explain entropy increase ... .<br />
Such was the pressure that he lost faith in himself .. . . 15<br />
The problem of irreversibility still remains a subject of lively<br />
controversy. How is this possible one hundred fifty years after<br />
the discovery of the second law of thermodynamics? There are<br />
many aspects to this question, some cultural and some technical.<br />
There is a cultural component in the mistrust of time. We<br />
shall on several occasions cite the opinion of Einstein. His<br />
judgment sounds final: time (as irreversibility) is an illusion. In<br />
fact, Einstein was reiterating what Giordano Bruno had written<br />
in the sixteenth century and what had become for centuries<br />
the credo of science: "The universe is, therefore, one,<br />
infinite, immobile .... It does not move itself locally .... It<br />
does not generate itself .... It is not corruptible .... It is not<br />
alterable ... . "16 For a long time Bruno's vision dominated<br />
the scientific view of the Western world. It is therefore not<br />
surprising that the intrusion of irreversibility, coming mainly
ORDER OUT OF CHAOS 16<br />
from the engineering sciences and physical chemistry, was re·<br />
ceived with mistrust. But there are technical reasons in addi·<br />
tion to cultural ones. Every attempt to "derive" irreversibility<br />
from dynamics necessarily had to fail, because irreversibility<br />
is not a universal phenomenon. We can imagine situations that<br />
are strictly reversible, such as a pendulum in the absence of<br />
friction, or planetary motion. This failure has led to dis·<br />
couragement and to the feeling that, in the end, the whole concept<br />
of irreversibility has a subjective origin. We shall discuss<br />
all these problems at some length. Let us say here that today<br />
we can see this problem from a different point of view, since<br />
we now know that there are different classes of dynamic systems.<br />
The world is far from being homogeneous. Therefore the<br />
question can be put in different terms: What is the specific<br />
structure of dynamic systems that permits them to "distinguish"<br />
past and future? What is the minimum complexity<br />
involved?<br />
Progress has been realized along these lines. We can now be<br />
more precise about the roots of time in nature. This has farreaching<br />
consequences. The second law of thermodynamics,<br />
the law of entropy, introduced irreversibility into the macroscopic<br />
world. We now can understand its meaning on the<br />
microscopic level as well. As we shall see, the second law corresponds<br />
to a selection rule, to a restriction on initial conditions<br />
that is then propagated by the laws of dynamics.<br />
Therefore the second law introduces a new irreducible element<br />
into our description of nature. While it is consistent with<br />
dynamics, it cannot be derived from dynamics.<br />
Boltzmann already understood that probability and irreversibility<br />
had to be closely related. Only when a system behaves<br />
in a sufficiently random way may the difference between past<br />
and future, and therefore irreversibility, enter into its descrip:<br />
tion. Our analysis confirms this point of view. Indeed, what is<br />
the meaning of an arrow of time in a deterministic description<br />
of nature? If the future is already in some way contained in the<br />
present, which also contains the past, what is the meaning of<br />
an arrow of time? The arrow of time a manifestation of the<br />
fact that the future is not given, that, as the French poet Paul<br />
Valery emphasized, "time is construction." 17<br />
The experience of our everyday life manifests a radical difference<br />
between time and space. We can move from one point
17 THE CHALLENGE TO SCIENCE<br />
of space to another. However, we cannot turn time around. We<br />
cannot exchange past and future. As we shall see, this feeling<br />
of impossibility is now acquiring a precise scientific meaning.<br />
Permitted states are separated from states that are prohibited<br />
by the second law of thermodynamics by means of an infinite<br />
entropy barrier. There are other barriers in physics. One is the<br />
velocity of light, which in our present view limits the speed at<br />
which signals may be transmitted. It is essential that this barrier<br />
exist; if not, causality would fall to pieces. Similarly, the<br />
entropy barrier is the prerequisite for giving a meaning tQ communication.<br />
Imagine what would happen if our future would<br />
become the past for other people! We shall return to this later.<br />
The recent evolution of physics has emphasized the reality<br />
of time. In the process new aspects of time have been uncovered.<br />
A preoccupation with time runs all through our century.<br />
Think of Einstein, Proust, Freud, Te ilhard, Peirce, or<br />
Whitehead.<br />
One of the most surprising results of Einstein's special theory<br />
of relativity, published in 1905, was the introduction of a<br />
local time associated with each observer. However, this local<br />
time remained a reversible time. Einstein's problem both in<br />
the special and the general theories of relativity was mainly<br />
that of the "communication" between observers, the way they<br />
could compare time intervals. But we can now investigate time<br />
in other conceptual contexts.<br />
In classical mechanics time was a number characterizing the<br />
position of a point on its trajectory. But time may have a different<br />
meaning on a global level. When we look at a child and<br />
guess his or her age, this age is not located in any special part<br />
of the child's body. It is a global judgment. It has often been<br />
stated that science spatializes time. But we now discover that<br />
another point of view is possible. Consider a landscape and its<br />
evolution: villages grow, bridges and roads connect different<br />
regions and transform them. Space thus acquires a temporal<br />
dimension; following the words of geographer B. Berry, we<br />
have been led to study the "timing of space."<br />
But perhaps the most important progress is that we now<br />
may see the problem of structure, of order, from a different<br />
perspective. As we shall show in Chapter VIII, from the point<br />
of view of dynamics, be it classical or quantum, there can be<br />
no one time-directed evolution. The "information" as it can be
ORDER OUT OF CHAOS 18<br />
defined in terms of dynamics remains constant in time. This<br />
sounds paradoxical. When we mix two liquids, there would<br />
occur no "evolution" in spite of the fact that we cannot, without<br />
using some external device, undo the effect of the mixing.<br />
On the contrary, the entropy law describes the mixing as the<br />
evolution toward a "disorder," toward the most probable<br />
state. We can show now that there is no contradiction between<br />
the two descriptions, but to speak about information, or order,<br />
we have to redefine the units we are considering. The important<br />
new fact is that we now may establish precise rules to go<br />
from one type of unit to the other. In other words, we have<br />
achieved a microscopic formulation of the evolutionary paradigm<br />
expressed by the second law. As the evolutionary paradigm<br />
encompasses all of chemistry as well as essential parts of<br />
biology and the social sciences, this seems to us an important<br />
conclusion. This insight is quite recent. The process of reconceptualization<br />
occurring in physics is far from being complete.<br />
However, our intention is not to shed light on the definitive<br />
acquisitions of science, on its stable and well-established results.<br />
What we wish to do is emphasize the conceptual creativeness<br />
of scientific activity and the future prospects and<br />
new problems it raises. In any case, we now know that we are<br />
only at the beginning of this exploration. We shall not see the<br />
end of uncertainty or risk. Thus we have chosen to present<br />
things as we perceive them now, fully aware of how incomplete<br />
our answers are.<br />
Erwin Schrodinger once wrote, to the indignation of many philosophers<br />
of science:<br />
. . . there is a tendency to f<strong>org</strong>et that all science is boundl<br />
up with human culture in general, and that scientific findings,<br />
even those which at the moment appear the mostj<br />
advanced and esoteric and difficult to grasp, are meaning-!<br />
less outside their cultural context. A theoretical science!<br />
unaware that those of its constructs considered relevanti<br />
and momentou:s arc de:stined eventually to be framed inl<br />
concepts and words that have a grip on the educated com-I<br />
I<br />
I<br />
I
19 THE CHALLENGE TO SCIENCE<br />
munity and become part and parcel of the general world<br />
picture-a theoretical science, I say, where this is f<strong>org</strong>otten,<br />
and where the initiated continue musing to each<br />
other in terms that are, at best, understood by a small<br />
group of close fellow travellers, will necessarily be cut off<br />
from the rest of cultural mankind; in the long run it is<br />
bound to atrophy and ossify however virulently esoteric<br />
chat may continue within its joyfully isolated groups of<br />
experts.t8<br />
One of the main themes of this book is that of a strong interaction<br />
of the issues proper to culture as a whole and the internal<br />
conceptual problems of science in particular. We find<br />
questions about time at the very heart of science. Becoming,<br />
irreversibility-these are questions to which generations of<br />
philosophers have also devoted their lives. Today, when history-be<br />
it economic, demographic, or political-is moving at<br />
an unprecedented pace, new questions and new interests require<br />
·us to enter into new dialogues, to look for a new coherence.<br />
However, we know the progress of science has often been<br />
described in terms of rupture, as a shift away from concrete<br />
experience toward a level of abstraction that is increasingly<br />
difficult tc, grasp. We believe that this kind of interpretation is<br />
only a reflection, at the epistemological level, of the historical<br />
situation in which classical science found itself, a consequence<br />
of its inability to include in its theoretical frame vast<br />
areas of the relationship between man and his environment.<br />
There doubtless exists an abstract development of scientific<br />
theories. However, the conceptual innovations that have been<br />
decisive for the development of science are not necessarily of<br />
this type. The rediscovery of time has roots both in the internal<br />
history of science and in the social context in which science<br />
finds itself today. Discoveries such as those of unstable<br />
elementary particles or of the expanding universe clearly belong<br />
to the internal history of science, but the general interest<br />
in nonequilibrium situations, in evolving systems, may reflect<br />
our feeling that humanity as a whole is today in a transition<br />
period. Many results we shall report in Chapters V and VI, for<br />
example those on oscillating chemical reactions, could have<br />
been discovered many years ago, but the study of these non-
ORDER OUT OF CHAOS 20<br />
equilibrium problems was repressed in the cultural and ideological<br />
context of those times.<br />
We are aware that asserting this receptiveness to cultural<br />
content runs counter to the traditional conception of science.<br />
In this view science develops by freeing itself from outmoded<br />
forms of understanding nature; it purifies itself in a process<br />
that can be compared to an "ascesis" of reason. But this in<br />
turn leads to the conclusion that science should be practiced<br />
only by communities living apart, uninvolved in mundane<br />
matters. In this view, the ideal scientific community should be<br />
protected from the pressures, needs, and requirements of society.<br />
Scientific progress ought to be an essentially autonomous<br />
process that any "outside" influence, such as the<br />
scientists's participation in other cultural, social, or economic<br />
activities, would merely disturb or delay.<br />
This ideal of abstraction, of the scientist's withdrawal, finds<br />
an ally in still another ideal, this one concerning the vocation<br />
of a "true" researcher, namely, his desire to escape from<br />
worldly vicissitudes. Einstein describes the type of scientist<br />
who would find favor with the ·ngel of the Lord" should the<br />
latter be given the task of driving from the "Temple of Science"<br />
all those who are "unworthy' ' -it is not stated in what<br />
respects. They are generally<br />
... rather odd, uncommunicative, solitary fellows, who<br />
despite these common characteristics resemble one another<br />
really less than the host of the banished.<br />
What led them into the Te mple? . .. one of the strongest<br />
motives that lead men to art and science is flight<br />
from everyday life with its painful harshness and<br />
wretched dreariness, and from the fetters of one's own<br />
shifting desires. A person with a finer sensibility is driven<br />
to escape from personal existence and to the world of<br />
objective observing (Schauen) and understanding. This<br />
motive can be compared with the longing that irresistibly<br />
pulls the town-dweller away from his noisy, cramped<br />
quarters and toward the silent, high mountains, where<br />
the eye ranges freely through the still, pure air and traces<br />
the calm contours that seem to be made for eternity.<br />
With this negative motive there goes a positive one. Man<br />
seeks to form for himself, in whatever manner is suitable
21 tHE CHALLENGE TO SCIENCE<br />
for him, a simplified and. lucid image of the world (Bild<br />
der Welt), and so to overcome the world of experience by<br />
striving to replace it to some extent by this image.I9<br />
The incompatibility between the ascetic beauty sought after<br />
by science, on the one hand, and the petty swirl of worldly<br />
experience so keenly felt by Einstein, on the other, is likely to<br />
be reinforced by another incompatibility, this one openly Manichean,<br />
between science and society, or, more precisely, be·<br />
tween free human creativity and political power. In this case, it<br />
is not in an isolated community or in a temple that research<br />
would have to be carried out, but in a fortress, or else·in a<br />
madhouse, as Duerrenmatt imagined in his play The Physicists.20<br />
There, three physicists discuss the ways and means of<br />
advancing physics while at the same time safeguarding mankind<br />
from the dire consequences that result when political<br />
powers appropriate the results of its progress. The conclusion<br />
they reach is that the only possible way is that which has already<br />
been chosen by one of them; they all decide to pretend<br />
to be mad, to hide in a lunatic asylum. At the end of the play,<br />
as Fate would have it, this last refuge is discovered to be an<br />
illusion. The director of the asylum, who has been spying on<br />
her patient, steals his results and seizes world power.<br />
Duerrenmatt's play leads to a third conception of scientific<br />
activity: science progresses by reducing the complexity of reality<br />
to a hidden simplicity. What the physicist Moebius is trying<br />
to conceal in the madhouse is the fact that he has<br />
successfully solved the problem of gravitation, the unified theory<br />
of elementary particles, and, ultimately, the Principle of<br />
Universal Discovery, the source of absolute power. Of course,<br />
Duerrenmatt simplifies to make his point, yet it is commonly<br />
held that what is being sought in the "Temple of Science" is<br />
nothing less than the "formula" of the universe. The man of<br />
science, already portrayed as an ascetic, now becomes a kind<br />
of magician, a man apart, the potential holder of a universal<br />
key to all physical phenomena,. thus endowed with a potentially<br />
omnipotent knowledge. This brings us back to an issue<br />
we have already raised: it is only in a simple world (especially<br />
in the world of classical science, where complexity merely<br />
veils a fundamental simplicity) that a form of knowledge that<br />
provides a universal key can exist.
ORDER OUT OF CHAOS 22<br />
One of the problems of our time is to overcome attitudes<br />
that tend to justify and reinforce the isolation of the scientific<br />
community. We must open new channels of communication<br />
between science and society. It is in this spirit that this book<br />
has been written. We all know that man is altering his natural<br />
environment on an unprecedented scale. As Serge Moscovici<br />
puts it, he is creating a "new nature. "21 But to understand this<br />
man-made world, we need a science that is not merely a tool<br />
submissive to external interests, nor a cancerous tumor irresponsibly<br />
growing on a substrate society.<br />
1\.vo thousand years ago Chuang Tsu wrote:<br />
How [ceaselessly] Heaven revolves ! How [constantly]<br />
Earth abides at rest! Do the Sun and the Moon contend<br />
about their respective places? Is there someone presiding<br />
over and directing those things? Who binds and connects<br />
them together? Who causes and maintains them without<br />
trouble or exertion? Or is there perhaps some secret<br />
mechanism in consequence of which they cannot but be<br />
as they are ?22<br />
We believe that we are heading toward a new synthesis, a<br />
new naturalism. Perhaps we will eventually be able to combine<br />
the Western tradition, with its emphasis on experimentation<br />
and quantitative formulations, with a tradition such as the Chinese<br />
one, with its view of a spontaneous, self-<strong>org</strong>anizing<br />
world. Toward the beginning of this Introduction, we cited<br />
Jacques Monod. His conclusion was: "The ancient alliance<br />
has been destroyed; man knows at last that he is alone in the<br />
universe's indifferent immensity out of which he emerged only<br />
by chance. "23 Perhaps Monod was right. The ancient alliance<br />
has been shattered. Our role is not to lament the past. It is to<br />
try to discover in the midst of the extraordinary diversity of<br />
the sciences some unifying thread. Each great period of science<br />
has led to some model of nature. For classical science it<br />
was the clock; for nineteenth-century science, the period of<br />
the Industrial Revolution, it was an engine running down.<br />
What will be the symbol for us? What we have in mind may<br />
perhaps be expressed best by a reference to sculpture, from<br />
Indian or pre-Columbian art to our time. In some of the most
23 THE CHALLENGE TO SCIENCE<br />
beautiful manifestations of sculpture, be it in the dancing<br />
Shiva or in the miniature temples of Guerrero, there appears<br />
very clearly the search for a junction between stillness and<br />
motion, time arrested and time passing. We believe that this<br />
confrontation will give our period its uniqueness.
I<br />
I
BOOK ONE<br />
THE DEWSION OF<br />
THE UNIVERSAL
CHAPTER I<br />
THE TRIUMPH OF<br />
RE ASON<br />
The New Moses<br />
Nature and Natures laws lay hid in night:<br />
God said, let Newton be! and all was light.<br />
-Alexander Pope,<br />
Proposed Epitaph for Isaac 'Newton,<br />
who died in 1727<br />
There is nothing odd in the dramatic tone employedc by Pope.<br />
In the eyes of eighteenth-century England, Newton was the<br />
"new Moses" who had been shown the "tables of the law. "<br />
Poets, architects, and sculptors joined to propose monuments;<br />
a whole nation assembled to celebrate this unique event: a<br />
man had discovered the language that nature speaks-and<br />
obeys.<br />
Nature compelled, his piercing Mind obeys,<br />
And gladly shows him all her secret Ways;<br />
'Gainst Mathematicks she has no Defence,<br />
And yields t'experimental Consequence.1<br />
Ethics and politics drew upon the Newtonian episode for material<br />
on which to "ground" their arguments. Thus Desaguliers<br />
transposed the meaning of the new natural order into<br />
a political lesson: a constitutional monarchy is the best possible<br />
system of government, since the King, like the Sun, has<br />
his power limited by it.<br />
Like Ministers attending ev'ry Glance<br />
Six Worlds sweep round his Throne in Mystick Dance.<br />
27
ORDER OUT OF CHAOS 28<br />
He turns their Motion from his Devious Course,<br />
And bends their Orbits by Attractive Force;<br />
His Pow'r coerc'd by Laws, still leave them free,<br />
Directs, but not Destroys, their Liberty;2<br />
Although he himself did not encroach upon the domain of the<br />
moral sciences, Newton had no hesitation regarding the universal<br />
nature of the laws set out in his Principia. Nature is<br />
"very consonant and conformable to herself," he asserts in<br />
the celebrated Question 31 of his Opticks-and this strong and<br />
elliptical statement conceals a vast claim: combustion, fermentation,<br />
heat, cohesion, magnetism . .. there is no natural<br />
process which would not be produced by these active forcesattractions<br />
and repulsions-that govern both the motion of the<br />
stars and that of freely falling bodies.<br />
Already a national hero before his death, nearly a century later<br />
Newton was to become, mainly through the powerful influence<br />
exerted by Laplace, the symbol of the scientific revolution in<br />
Europe. Astronomers scanned a sky ruled by mathematics.<br />
The Newtonian system succeeded in overcoming all obstacles.<br />
Furthermore, it opened the way to mathematical methods<br />
by which apparent deviations could be accounted for and<br />
even be used to infer the existence of a hitherto unknown<br />
planet. The prediction of the existence of the planet Neptune<br />
was the consecration of the prophetic power inherent in the<br />
Newtonian vision.<br />
At the dawn of the nineteenth century, Newton's name<br />
tended to signify anything that claimed exemplarity. However,<br />
conflicting interpretations of his method are given. Some saw<br />
it as providing a blueprint for quantitative experimentation expressible<br />
in mathematics. For them, chemistry found its Newton<br />
in Lavoisier, who pioneered the systematic use of the<br />
balance. This was indeed a decisive step in the definition of a<br />
quantitative chemistry that took mass conservation as its<br />
Ariadne's thread. According to others, the Newtonian strategy<br />
consisted in isolating some central, specific fact and then<br />
using it as the basis for all further deductions concerning a<br />
given set of phenomena. In this perspective Newton's genius<br />
was located in his pragmatism. He did not try to explain gravitation;<br />
he took it as a fact. Similarly, each discipline should
29<br />
THE TRIUMPH OF REASON<br />
then take some central unexplained fact as its startm!; point.<br />
Physicians thus felt that they were authorized by Newton to<br />
refashion the vitalist conception and to speak of a "vital<br />
force" sui generis, the use of which would give the description<br />
of living phenomena a hoped-for systematic consistency. This<br />
is the same role that affinity, taken as the specificalJy chemical<br />
force of interaction, was calJed upon to play.<br />
Some "true Newtonians" took exception to this proliferation<br />
of forces and reasserted the universality of the explanatory<br />
power of gravitation. But it was too late. The term<br />
Newtonian was now applied to everything that dealt with a<br />
system of Jaws, with equilibrium, or even to all situations in<br />
which natural order on one side and moral, social, and political<br />
order on the other could be expressed in terms of an allembracing<br />
harmony. Romantic philosophers even discovered<br />
in the Newtonian universe an enchanted world animated by<br />
natural forces. More "orthodox" physicists saw in it a mechanical<br />
world governed by mathematics. For the positivists it<br />
meant the success of a procedure, a recipe to be identified<br />
with the very definition of science.3<br />
The rest is literature-often Newtonian literature: the harmony<br />
that reigns in the society of stars, the elective affinities<br />
and hostilities giving rise to the "social life" of chemical compounds<br />
appear as processes that can be transposed into the<br />
world of human society. No wonder that this period appears as<br />
the Golden Age of Classical Science.<br />
Thday Newtonian science still occupies a unique position.<br />
Some of the basic concepts it introduced represent a definitive<br />
acquisition that has survived all the mutations science has<br />
since undergone. However, today we know that the Golden<br />
Age of Classical Science is gone, and with it also the conviction<br />
that Newtonian rationality, even with its various conflicting<br />
interpretations, forms a suitable basis for our dialogue with<br />
nature.<br />
A central subject of this book is that of the Newtonian triumph,<br />
the continual opening up of new fields of investigation<br />
that have extended Newtonian thought right down to the pres<br />
. ent day. It also deals with doubts and struggles that arose from<br />
this triumph. Today we are beginning to see more clearly the<br />
limits of Newtonian rationality. A more consistent conception
ORDER OUT OF CHAOS 30<br />
of science and of nature seems to be emerging. This new conception<br />
paves the way for a new unity of knowledge and culture.<br />
A Dehumanized World<br />
... May God us keep<br />
From single Vision and Newtons sleep!<br />
-William Blake,<br />
in a letter to Thomas Butts<br />
dated November 22, 1802<br />
There is no better illustration of the instability of the cultural<br />
position of Newtonian science than the introduction to a<br />
UNESCO colloquium on the relationship between science and<br />
culture:<br />
For more than a century the sector of scientific activity<br />
has been growing to such an extent within the surrounding<br />
cultural space that it seems to be replacing the totality<br />
of the culture itself. Some believe that this is merely an<br />
illusion due to its high growth rate and that the lines of<br />
force of this culture will soon reassert themselves and<br />
bring science back into the service of man. Others consider<br />
that the recent triumph of science entitles it at last<br />
to rule over the whole of culture which, moreover, would<br />
deserve to go on being known as such only because it was<br />
transmitted through the scientific apparatus. Others<br />
again, appalled by the danger of man and society being<br />
manipulated if they come under the sway of science, perceive<br />
the spectre of cultural disaster looming in the distance.4<br />
In this statement science appears as a cancer in the body of<br />
culture, a cancer whose proliferation threatens to destroy the<br />
whole of cultural life. The question is whether we can dominate<br />
science and control its development, or whether we shall<br />
be enslaved. In only one hundred fifty years, science has been
31 THE TRIUMPH OF REASON<br />
downgraded from a source of inspiration for Western culture<br />
to a threat. Not only does it threaten man's material existence,<br />
but also, more subtly, it threatens to destroy the traditions and<br />
experiences that are most deeply rooted in our cultural life. It<br />
is not just the technological fallout of one or another scientific<br />
breakthrough that is being accused, but "the spirit of science"<br />
itself.<br />
Whether the accusation refers to a global skepticism exuded<br />
by scientific culture or to specific conclusions reached<br />
through scientific theories, it is often asserted today that science<br />
is debasing our world. What for generations had been a<br />
source of joy and amazement withers at its touch. Everything<br />
it touches is dehumanized.<br />
Oddly enough, the idea of a fatal disenchantment brought<br />
about by scientific progress is an idea held not only by the<br />
critics of science but often also by those who defend or glorify<br />
it. Thus, in his book The Edge of Objectivity, historian C. C.<br />
Gillispie expresses sympathy for those who criticize science<br />
and constantly endeavor to blunt the "cutting edge of objectivity":<br />
Indeed, the renewals of the subjective approach to nature<br />
make a pathetic theme. Its ruins lie strewn like good intentions<br />
aU along the ground traversed by science, until it<br />
survives only in strange corners like Lysenkoism and anthroposophy,<br />
where nature is socialized or moralized.<br />
Such survivals are relics of the perpetual attempt to escape<br />
the consequences of western man's most characteristic<br />
and successful campaign, which must doom to<br />
conquer. So like any thrust in the face of the inevitable,<br />
romantic natural philosophy has induced every nuance of<br />
mood from desperation to heroism. At the ugliest, it is<br />
sentimental or vulgar hostility to intellect. At the noblest,<br />
it inspired Diderot's naturalistic and moralizing science,<br />
Goethe's personification of nature, the poetry of Wordsworth,<br />
and the philosophy of Alfred North Whitehead, or<br />
of any other who would find a place in science for our<br />
qualitative and aesthetic appreciation of nature. It is the<br />
science of those who would make botany of blossoms and<br />
meteorology of sunsets. 5
ORDER OUT OF CHAOS 32<br />
Thus science leads to a tragic, metaphysical choice. Man has<br />
to choose between the reassuring but irrational temptation to<br />
seek in nature a guarantee of human values, or a sign pointing<br />
to a fundamental corelatedness, and fidelity to a rationality<br />
that isolates him in a silent world.<br />
The echoes of another leitmotiv-domination-mingle with<br />
that of disenchantment. A disenchanted world is, at the same<br />
time, a world liable to control and manipulation. Any science<br />
that conceives of the world as being governed according to a<br />
universal theoretical plan that reduces its various riches to the<br />
drab applications of general laws thereby becomes an instrument<br />
of domination. And man, a stranger to the world, sets<br />
himself up as its master.<br />
This disenchantment has taken various forms in recent decades.<br />
It is outside the aim of this book to study systematically<br />
the various forms of antiscience. In Chapter III we shall present<br />
a fuller reaction of Western thought to the surprising triumph<br />
of Newtonian rationality. Here let us only note that at<br />
present there is a shift of popular attitudes to nature associated<br />
with a widespread but in our opinion erroneous belief that<br />
there exists a fundamental antagonism between science and<br />
"naturalism." To illustrate at least some of the forms antiscientific<br />
criticism has taken in recent years, we have chosen<br />
three examples. First, Heidegger, whose philosophy holds a<br />
deep fascination for contemporary thought. We shall also refer<br />
to the criticisms stated by Arthur Koestler and by the great<br />
historian of science, Alexander Koyre.<br />
Martin Heidegger directs his criticism against the very core<br />
of the scientific endeavor, which he sees as fundamentally related<br />
to a permanent aim, the domination of nature. Therefore<br />
Heidegger claims that scientific rationality is the final accomplishment<br />
of something that has been implicitly present since<br />
ancient Greece, namely, the will to dominate, which is at work<br />
in any rational discussion or enterprise, the violence lurking in<br />
all positive and communicable knowledge. Heidegger emphasizes<br />
what he·calls the technological and scientific "framing"<br />
(Geste/1), 6 which leads to the general setting to work of the<br />
world and of men.<br />
Thus Heidegger does not present a detailed analysis of any<br />
particular technological or scientific product or process. What<br />
he challenges is the essence of technology, the way each thing
33 THE TRIUMPH OF REASON<br />
is taken into account. Each theory is part of the implementation<br />
of the master plan that makes up Western history. What<br />
we call a scientific "theory" implies, following Heidegger, a<br />
way of questioning things by which they are reduced to enslavement.<br />
The scientist, like the technologist, is a toy in the<br />
hands of the will to power disguised as thirst for knowledge;<br />
his very approach to things subjects them to systematic violence.<br />
Modern physics is not experimental physics because it<br />
uses experimental devices in its questioning of nature.<br />
Rather the reverse is true. Because physics, already as<br />
pure theory, requests nature to manifest itself in terms of<br />
predictable forces, it sets up the experiments precisely<br />
for the sole purpose of asking whether and how nature<br />
follows the scheme preconceived by science. 7<br />
Similarly, Heidegger is not concerned about the fact that industrial<br />
pollution, for example, has destroyed all animal life in<br />
the Rhine. What does concern him is that the river has been<br />
put to man's service.<br />
The hydroelectric plant is set into the current of the<br />
Rhine. It sets the Rhine to supplying its hydraulic pressure,<br />
which then sets the turbines turning . ... The hydroelectric<br />
plant is not built into the Rhine river as was<br />
the old bridge that joined bank wi'th bank for hundreds of<br />
years. Rather the river is dammed up into the power<br />
plant. What the river is now, namely, a water supplier,<br />
derives from out of the essence of the power station.s<br />
The old bridge over the Rhine is valued not as a proof of<br />
soundly tested ability, of painstaking and accurate observation,<br />
but because it does not "use" the river.<br />
Heidegger's criticisms, taking the very ideal of a positive,<br />
communicable knowledge as a threat, echo some themes of<br />
the antiscience movement to which we referred in the Introduction.<br />
But the idea of an indissociable link between science<br />
and the will to dominate also permeates some apparently very<br />
different assessments of our present-day situation. For instance,<br />
under the very suggestive title "The Coming of the
ORDER OUT OF CHAOS<br />
34<br />
Golden Age, "9 Gunther Stent states that science is now reaching<br />
its limits. We are close to a point of diminishing returns,<br />
where the questions we direct to things in order to master<br />
them become more and more complicated and devoid of interest.<br />
This marks the end of progress, but it is the opportunity<br />
for humanity to stop its frantic efforts, to end the age-old<br />
struggle against nature, and to accept a static and comfortable<br />
peace. We wish to show that the relative dissociation between<br />
the scientific knowledge of an object and the possibility of<br />
mastering it, far from marking the end of science, signals a<br />
host of new perspectives and problems. Scientific understanding<br />
of the world around us is just beginning. There is yet<br />
another idea of science that we feel is potentially just as detrimental,<br />
namely, the fascination with a mysterious science<br />
that, by paths of reasoning inaccessible to common mortals,<br />
will lead to results that can, in one fell swoop, challenge the<br />
meaning of basic concepts such as time, space, causality,<br />
mind, or matter. This kind of "mystery science," the results of<br />
which are imagined to be capable of shattering the framework<br />
of any traditional conception, has actually been encouraged<br />
by the successive "revelations" of relativity and quantum mechanics.<br />
It is certainly true that some of the most imaginative<br />
steps in the past, Einstein's interpretation of gravitation as a<br />
space curvature or Dirac's antiparticles, for example, have<br />
shaken some seemingly well-established conceptions. Thus<br />
there is a very delicate balance between the readiness to imagine<br />
that science can produce anything and a kind of down-toearth<br />
realism. Today the balance is strongly shifting toward a<br />
revival of mysticism, be it in the press media or even in science<br />
itself, especially among cosmologists.IO It has even been suggested<br />
by certain physicists and popularizers of science that<br />
mysterious relationships exist between parapsychology and<br />
quantum physics. Let us cite Koestler:<br />
We have heard a whole chorus of Nobel Laureates in<br />
physics informing us that matter is dead, causality is<br />
dead, determinism is dead. If that is so, let us give them a<br />
decent burial, with a requiem of electronic music. It is<br />
time for us to draw the lesson from twentieth-century<br />
post-mechanistic science, and to get out of the strait-
36<br />
THE TRIUMPH OF REASON<br />
jacket which nineteenth-century materialism imposed on<br />
our philosophical outlook. Paradoxically, had that outlook<br />
kept abreast with modern science itself, instead of<br />
lagging a century behind it, we would have been liberated<br />
from that strait-jacket long ago . ... But once this is recognized,<br />
we might become more receptive to phenomena<br />
around us which one-sided emphasis on physical science<br />
has made us ignore; might feel the draught that is blowing<br />
through the chinks of the causal edifice; pay more attention<br />
to confluential events; include the paranormal phenomena<br />
in our concept of normality; and realise that we<br />
have been living in the "Country of the Blind." I I<br />
We do not wish to judge or condemn a priori. There may be in<br />
some of the apparently fantastic propositions we hear today<br />
some seed of new knowledge. Nevertheless, we believe that<br />
leaps into the unimaginable are far too simple escapes from<br />
the concrete complexity of our world. We do not believe we<br />
shall leave the "Country of the Blind" in a day, since conceptual<br />
blindness is not the main reason for the problems and<br />
contradictions our society has failed to solve.<br />
Our disagreement with certain criticisms or distortions of<br />
science does not mean, however, that we wish to reject aJJ criticisms.<br />
Let us take, for instance, the position of Alexander<br />
Koyre, who has made outstanding contributions to the understanding<br />
of the development of modern science. In his study of<br />
the significance and implications of the Newtonian synthesis,<br />
Koyre wrote:<br />
Ye t there is something for which Newton-or better to<br />
say not Newton alone, but modern science in generalcan<br />
still be made responsible: it is the splitting of our<br />
world in two. I have been saying that modern science<br />
broke down the barriers that separated the heavens and<br />
the earth, and that it united and unified the universe. And<br />
that is true. But, as I have said, too, it did this by substituting<br />
for our world of quality and sense perception,<br />
the world in which we live, and love, and die, another<br />
world-the world of quantity, of reified geometry, a world<br />
in which, though there is a place for everything, there is
ORDER OUT OF CHAOS 36<br />
no place for man. Thus the world of science-the real<br />
world-became estranged and utterly divorced from the<br />
world of life, which science has been unable to explainnot<br />
even to explain away by calling it "subjective."<br />
True, these worlds are everyday-and even more and<br />
more-connected by the praxis. Yet for theory they are<br />
divided by an abyss.<br />
1\vo worlds: this means two truths. Or no truth at all.<br />
This is the tragedy of the modern mind which "solved<br />
the riddle of the universe," but only to replace it by another<br />
riddle: the riddle of itself. I2<br />
However, we hear in the conclusions of Koyre the same<br />
theme expressed by Pascal and Monod-this tragic feeling of<br />
estrangement. Koyre 's criticism does not challenge scientific<br />
thinking but rather classical science based on the Newtonian<br />
perspective. We no longer have to settle for the previous dilemma<br />
of choosing between a science that reduces man to<br />
being a stranger in a disenchanted world and antiscientific,<br />
irrational protests. Koyre's criticism does not invoke the limits<br />
of a "strait-jacket" rationality but only the incapacity of classical<br />
science to deal with some fundamental aspects of the<br />
world in which we live.<br />
Our position in this book is that the science described by<br />
Koyre is no longer our science. Not because we are concerned<br />
today with new, unimaginable objects, closer to magic than to<br />
logic, but because as scientists we are now beginning to find<br />
our way toward the complex processes forming the world with<br />
which we are most familiar, the natural world in which living<br />
creatures and their societies develop. Indeed, today we are beginning<br />
to go beyond what Koyre called "the world of quantity"<br />
into the world of "qualities" and thus of "becoming."<br />
This will be the main subject of Books One and 1\vo. We believe<br />
it is precisely this transition to a new description that<br />
makes this moment in the history of science so exciting. Perhaps<br />
it is not an exaggeration to say that it is a period like the<br />
time of the Greek atomists or the Renaissance, periods in<br />
which a new view of nature was being born. But let us first<br />
return to Newtonian science, certainly one of the great moments<br />
of human history.
37 THE TRIUMPH OF REASON<br />
The Newtonian Synthesis<br />
What lay behind the enthusiasm of Newton's contemporaries,<br />
their conviction that the secret of the universe, the truth about<br />
nature, had finally been revealed? Several lines of thought,<br />
probably present from the very beginning of humanity, converge<br />
in Newton's synthesis: first of all, science as a way of<br />
acting on our environment. Newtonian science is indeed an<br />
active science; one of its sources is the knowledge of the medieval<br />
craftsmen, the knowledge of the builders of machines.<br />
This science provides the means for systematically acting on<br />
the world, for predicting and modifying the course of natural<br />
processes, for conceiving devices that can harness and exploit<br />
the forces and material resources of nature.<br />
In this sense, modern science is a continuation of the ageless.efforts<br />
of man to <strong>org</strong>anize and exploit the world in which<br />
he lives. We have very scanty knowledge about the early<br />
stages of this endeavor. However, it is possible, in retrospect,<br />
to assess the knowledge and skills required for the "Neolithic<br />
Revolution" to take place, when man gradually began to <strong>org</strong>anize<br />
his natural and social environment, using new techniques<br />
to exploit nature and to <strong>org</strong>anize his society. We still use, or<br />
have used until quite recently, Neolithic techniques-for example,<br />
animal and plant species either bred or selected, weaving,<br />
pottery, metalworking. Our social <strong>org</strong>anization was for a<br />
long time based on the same techniques of writing, geometry,<br />
and arithmetic as those required to <strong>org</strong>anize the hierarchically<br />
differentiated and structured social groups of the Neolithic<br />
city-states. Thus we cannot help acknowledging the continuity<br />
that exists between Neolithic techniques and the scientific and<br />
industrial revolutions.t3<br />
Modern science has thus extended this ancient endeavor,<br />
amplifying it and constantly speeding up its rhythm. Nevertheless,<br />
this does not exhaust the significance of science in the<br />
sense given to it by the Newtonian synthesis.<br />
In addition to the various techniques used in a given society,<br />
we find a number of beliefs and myths that seek to understand<br />
man's place in nature. Like myths and cosmologies, science's
ORDER OUT OF CHAOS<br />
38<br />
endeavor is to understand the nature of the world, the way it is<br />
<strong>org</strong>anized, and man's place in it.<br />
From our standpoint it is quite irrelevant that the early speculations<br />
of the pre-Socratics appear to be adapted from the<br />
Hesiodic myth of creation-that is, the initial polarization of<br />
Heaven and Earth, the desire aroused by Eros, the birth of the<br />
first generations of gods to fo rm the differentiated cosmic<br />
powers, discord and strife, alternating atrocities and vendettas,<br />
until stability is finally reached under the rule of Justice<br />
(dike). What does matter is that, in the space of a few generations,<br />
the pre-Socratics collected, discussed, and criticized<br />
some of the concepts we are still trying to <strong>org</strong>anize in order to<br />
understand the relation between being and becoming, or the<br />
appearance of order out of a hypothetically undifferentiated<br />
initial environment.<br />
Where does the instability of the homogeneous come from?<br />
Why does it differentiate spontaneously? Why do things exist<br />
at all? Are they the fragile and mortal result of an injustice, a<br />
disequilibrium in the static equilibrium of forces between conflicting<br />
natural powers? Or do the forces that create and drive<br />
things exist autonomously-rival powers of love and hate leading<br />
to birth, growth, decline, and dispersion? Is change an illusion<br />
or is it, on the contrary, the unceasing struggle between<br />
opposites that constitutes things? Can qualitative change be<br />
reduced to the motion in a vacuum, of atoms differing only in<br />
their forms, or do atoms themselves consist of a multitude of<br />
qualitatively different germs, each unlike the others? And last,<br />
is the harmony of the world mathematical? Are numbers the<br />
key to nature?<br />
The numerical regularities among sounds that were discovered<br />
by the Pythagoreans are still part of our present theories.<br />
The mathematical schemes worked out by the Greeks<br />
form the first body of abstract thought in European historythat<br />
is, a thought whose results are communicable and reproducible<br />
for all reasoning human beings. The Greeks<br />
achieved for the first time a form of deductive knowledge that<br />
contained a degree of certainty unaffected by convictions, expectations,<br />
or passions.<br />
The most important aspect common to Greek thought and<br />
to modern science , which contrasts with the religious and
39<br />
THE TRIUMPH OF REASON<br />
mythicaI form of inquiry, is thus the emphasis on criticaI discussion<br />
and verification.14<br />
Little is known about this pre-Socratic philosophy that grew<br />
up in the lonian cities and the colonies of Magna Graecia.<br />
Thus we can only speculate about the relationships that might<br />
have existed between the development of theoretical and cosmological<br />
hypotheses and the crafts and technological activities<br />
that tlourished in those cities. Tr adition teUs that as a<br />
result of a hostile religious and social reaction, philosophers<br />
were accused of atheism and were either exiled or put to death.<br />
This early "recall to order" may serve as a symbol of the importance<br />
of social factors in the origin, and above alI the<br />
growth, of conceptual innovations. To understand the success<br />
of modem science we also have to explain why its fóunders<br />
were as a rule not unduly persecuted and their theoretical approach<br />
repressed in favor of a form of knowledge more consistent<br />
with social anticipations and convictions.<br />
Be that as it may, from Plato and Aristotle onward, the limits<br />
were set, and thought was channeled in socially acceptable<br />
directions. ln particular, the distinction between theoretical<br />
thinking and technological activity was established. The<br />
words we still use today-machine, mechanical, engineerhave<br />
a similar meaning. They do not refer to rational knowledge<br />
but to cunning and expediency. The idea was not to leam<br />
about natural processes in order to utilize them more effectively,<br />
but to deceive nature, to "machinate" against it-that<br />
is, to work wonders and create effects extraneous to the "natural<br />
order" of things. The fields of practical manipulation and<br />
that of the rational understanding of nature were thus rigidly<br />
separated. Archimedes' status is merely that of an engineer;<br />
his mathematical analysis of the equilibrium of machines is not<br />
considered to be applicable to the world of nature, at least<br />
within the framework of traditional physics. ln contrast, the<br />
Newtonian synthesis expresses a systematic alliance between<br />
manipulation and theoretical understanding.<br />
There is a third important element that found its expression<br />
in the Newtonian revolution. There is a striking contrast,<br />
which each of us has probably experienced, between the quiet<br />
world of the stars and planets and thé ephemeral, turbulent<br />
world around uso As Mircea Eliade has emphasized, in many
ORDER OUT OF CHAOS 40<br />
ancient civilizations there is a separation between profane<br />
space and sacred space, a division of the world into an ordi·<br />
nary space that is subject to chance and degradation and a<br />
sacred one that is meaningful, independent of contingency and<br />
history. This was the very contrast Aristotle established between<br />
the world of the stars and our sub lunar world. This contrast<br />
is crucial to the way in which Aristotle evaluated the<br />
possibility of a quantitative description of nature. Since the<br />
motion of the celestial bodies is not change but a "divine"<br />
state that is eternally the same, it ntay be described by means<br />
of mathematical idealizations. Mathematical precision and<br />
rigor are not relevant to the sub lunar world. Imprecise natural<br />
processes can only be subjected to an approximate description.<br />
In any case, for an Aristotelian it is more interesting to<br />
know why a process occurs than to describe how it occurs, or<br />
rather, these two aspects are indivisible. One of the main<br />
sources of Aristotle's thinking was the observation of embryonic<br />
growth, a highly <strong>org</strong>anized process in which interlocking,<br />
although apparently independent, events participate in a<br />
process that seems to be part of some global plan. Like the developing<br />
embryo, the whole of Aristotelian nature is <strong>org</strong>anized<br />
according to final causes. The purpose of all change, if it is in<br />
keeping with the nature of things, is to realize in each being the<br />
perfection of its intelligible essence. Thus this essence, which,<br />
in the case of living creatures, is at one and the same time their<br />
final, formal, and effective cause, is the key to the understanding<br />
of nature. In this sense the "birth of modern science," the<br />
clash between the Aristotelians and Galileo, is a clash between<br />
two forms of rationality. IS<br />
In Galileo 's view the question of "why," so dear to the Aristotelians,<br />
was a very dangerous way of addressing nature, at<br />
least for a scientist. The Aristotelians, on the other hand, considered<br />
Galileo's attitude as a form of irrational fanaticism.<br />
Thus, with the coming of the Newtonian system it was a<br />
new universality that triumphed, and its emergence unified<br />
what till then had appeared as divided.
41 THE TRIUMPH OF REASON<br />
The Experimental Dialogue<br />
We have already emphasized one of the essential elements of<br />
modern science: the marriage between theory and practice,<br />
the blending of the desire to shape the world and the desire to<br />
understand it. For this to be possible, it was not enough, despite<br />
the empiricists' beliefs, merely to respect observed facts.<br />
On certain points, including even the description of mechanical<br />
motion, it was in fact Aristotelian physics that was more<br />
easily brought into contact with empirical facts. The experimental<br />
dialogue with nature discovered by modern science involves<br />
activity rather than passive observation. What must be<br />
done is to manipulate physical reality, to "stage" it in such a<br />
way that it conforms as closely as possible to a theoretical<br />
description. The phenomenon studied must be prepared and<br />
isolated until it approximates some ideal situation that may be<br />
physically unattainable but that conforms to the conceptual<br />
scheme adopted.<br />
By way of example, let us take the description of a system of<br />
pulleys, a classic since the time of Archimedes, whose reasoning<br />
has been extended by modern scientists to cover all simple<br />
machines. It is astonishing to find that the modern explanation<br />
has eliminated, on the grounds that it is irrelevant, the very<br />
thing that Aristotelian physics set out to explain, namely, the<br />
fact that, using a typical image, a stone "resists" a horse's<br />
efforts to pull it and that this resistance can be "overcome" by<br />
applying traction through a system of pulleys. Nature, according<br />
to Galileo, never gives anything away, never does something<br />
for nothing, and can never be tricked; it is absurd to<br />
think that by cunning or by using some stratagem we can make<br />
it perform extra work.I6 Since the work the horse is able to<br />
perform is the same with or without the pulleys, the effect<br />
produced must be the same. This then becomes the starting<br />
point for a mechanical explanation, which thus refers to an<br />
idealized world. In this world the "new" effect-the stone finally<br />
set in motion-is of secondary importance; and the<br />
stone's resistance is described only qualitatively, in terms of<br />
friction and heating. Instead, what is described accurately is<br />
the ideal situation, in which a relationship of equivalence links
ORDER OUT OF CHAOS 42<br />
the cause, the work done by the horse, to the effect, the mo<br />
tion of the stone. In this ideal world, the horse can, in any<br />
case, shift the stone, and the system of pulleys has the sole<br />
effect of modifying the way the pulling efforts are transmitted;<br />
instead of moving the stone over a distance L, equal to the<br />
distance it travels while pulling the rope, the horse only moves<br />
it over a distance Lin, where n depends on the number of<br />
pulleys. Like all simple machines, the pulleys form a passive<br />
device that can only transmit motion without producing it.<br />
The experimental dialogue thus corresponds to a highly specific<br />
procedure. Nature is cross-examined through experimentation,<br />
as if in a court of law, in the name of a priori principles.<br />
Nature's answers are recorded with the utmost accuracy, but<br />
relevance of those answers is assessed in terms of the very<br />
idealizations that guided the experiment. All the rest does not<br />
count as information, but is idle chatter, negligible secondary<br />
effects. It may well be that nature rejects the theoretical hypothesis<br />
in question. Nevertheless, the latter is still used as a<br />
standard against which to measure the implications and the<br />
significance of the response, whatever it may be. It is precisely<br />
this imperative way of questioning nature that Heidegger refers<br />
to in his argument against scientific rationality.<br />
For us the experimental method is truly an art-that is, it is<br />
based on special skills and not on general rules. As such there<br />
are never any guarantees of success and one always remains at<br />
the mercy of triviality or poor judgment. No methodological<br />
principle can eliminate the risk, for instance, of persisting in a<br />
blind alley of inquiry. The experimental method is the art of<br />
choosing an interesting question and of scanning all the consequences<br />
of the theoretical framework thereby implied, all<br />
the ways nature could answer in the theoretical language<br />
chosen. Amid the concrete complexity of natural phenomena,<br />
one phenomenon has to be selected as the most likely to embody<br />
the theory's implications in an unambiguous way. This<br />
phenomenon will then be abstracted from its environment and<br />
"staged" to allow the theory to be tested in a reproducible and<br />
communicable way.<br />
Although this experimental procedure was criticized right<br />
from the outset, ignored by the empiricists, and attacked by<br />
others on the grounds that it was a kind of torture, a way of<br />
putting nature on the rack, it survived all the modifications of
43<br />
THE TRIUMPH OF REASON<br />
the theoretical content of scientific descriptions and ultimately<br />
defined the new method of investigation introduced by modern<br />
science.<br />
Experimental procedure can even become a tool for purely<br />
theoretical analysis. It is then a "thought experiment," the<br />
imagining of experimental situations governed entirely by theoretical<br />
principles, which permits the exploration of the consequences<br />
of these principles in a given situation. Such<br />
thought experiments played a crucial role in Galileo's work,<br />
and today they are at the center of investigations about the<br />
consequences of the conceptual upheavals in contemporary<br />
physics, namely, relativity and quantum mechanics. One of<br />
the most famous of such thought experiments is Einstein's famous<br />
train, from which an observer can measure the velocity<br />
of propagation of a ray of light emitted along an embankment,<br />
that is, moving at a velocity c in a reference system with respect<br />
to which the train is moving at a velocity v. According to<br />
classical reasoning, the observer on the train should attribute<br />
to the light, which is traveling in the same direction as he is, a<br />
velocity of c- v. However, this classical conclusion represents<br />
precisely the absurdity that the thought experiment was designed<br />
to expose. In relativity theory, the velocity of light appears<br />
as a universal constant of nature. Whatever inertial<br />
referer.ce system is used, the velocity of light is always the<br />
same. And since then Einstein's train has gone on exploring<br />
the physical consequences of this fundamental change.<br />
The experimental method is central to the dialogue with nature<br />
established by modern science. Nature questioned in this<br />
way is, of course, simplified and occasionally mutilated. This<br />
does not deprive it of its capacity to refute most of the hypotheses<br />
we can imagine. Einstein used to say that nature says<br />
"no" to most of the questions it is asked, and occasionally<br />
"perhaps." The scientist does not do as he pleases, and he<br />
cannot force nature to say only what he wants to hear. He<br />
cannot, at least in the long run, project upon it his most cherished<br />
desires and expectations. He actually runs a greater risk<br />
and plays a more dangerous game the better his tactics succeed<br />
in encircling nature, in setting it more squarely with its<br />
back to the wall.l7 Moreover, it is true that, whether the answer<br />
is "yes" or "no," it will be expressed in the same theoretical<br />
language as the question. However, this language, too,
ORDER OUT OF CHAOS 44<br />
develops according to a complex historical process involving<br />
nature's replies in the past and its relations with other theoretical<br />
languages. In addition, new questions arise corresponding<br />
to the changing interests of each period. This sets up a complex<br />
relationship between the specific rules of the scientific<br />
game-particularly the experimental method of reasoning<br />
with nature, which places the greatest constraint on the<br />
game-and a cultural network to which, sometimes unwittingly,<br />
the scientist belongs.<br />
We believe that the experimental dialogue is an irreversible<br />
acquisition of human culture. It actually provides a guarantee<br />
that when nature is explored by man it is treated as an independent<br />
being. It forms the basis of the communicable and<br />
reproducible nature of scientific results. However partially nature<br />
is allowed to speak, once it has expressed itself, there is<br />
no further dissent: nature never lies.<br />
The Myth at the Origin of Science<br />
The dialogue between man and nature was accurately perceived<br />
by the founders of modern science as a basic step toward<br />
the intelligibility of nature. But their ambitions went even<br />
farther. Galileo, and those who came after him, conceived of<br />
science as being capable of discovering global truths about<br />
nature. Nature not only would be written in a mathematical<br />
language that can be deciphered by experimentation, but there<br />
would actually exist only one such language. Following this<br />
basic conviction, the world is seen as homogeneous, and local<br />
experimentation can reveal global truth. The simplest phenomena<br />
studied by science can thus be interpreted as the key<br />
to understanding nature as a whole; the complexity of the latter<br />
is only apparent, and its diversity can be explained in terms<br />
of the universal truth embodied, in Galileo's case, in the mathematical<br />
laws of motion.<br />
This conviction has survived centuries. In an excellent set<br />
of lectures presented on the BBC several years ago, Richard<br />
Feynman ts compared nature to a huge chess game. The complexity<br />
is only apparent; each move follows simple rules. In its<br />
early days, modern science quite possibly needed this convic-
45 THE TRIUMPH OF REASON<br />
tion of being able to reach global truth. Such a conviction<br />
added an immense value to the experimental method and, to a<br />
certain extent, inspired it. Perhaps a revolutionary conception<br />
of the world, one as all-embracing as the "biological" conception<br />
of the Aristotelian world, was necessary to throw off<br />
the yoke of tradition, to give the champions of experimentation<br />
a strength of conviction and a power of argument that enabled<br />
them to hold their own against the previous forms of rationalism.<br />
Perhaps a metaphysical conviction was needed to transmute<br />
the craftsman's and machine builder's knowledge into a<br />
new method for the rational exploration of nature. We may<br />
also wonder what the implications of the existence of this kind<br />
of "mythical" conviction are for explaining the way modern<br />
science's first developments were accepted in the social context.<br />
On this highly controversial issue, we shall restrict ourselves<br />
to a few remarks of a quite general nature for the sole<br />
purpose of pinpointing the problem-that is, the problem of a<br />
science whose advance has been felt by some as the triumph<br />
of reason, but by others as a disillusionment, as the painful<br />
discovery of the robotlike stupidity of nature.<br />
It seems hard to deny the fundamental importance of social<br />
and economic factors-particularly the development of craftsmen's<br />
techniques in the monasteries, where the residual knowledge<br />
of a destroyed world was preserved, and later in the<br />
bustling merchant cities-in the birth of experimental science,<br />
which is a systematized form of part of the craftsmen's knowledge.<br />
Moreover, a comparative analysis such as Needham'sl9 exposes<br />
the decisive importance of social structures at the close<br />
of the Middle Ages. Not only was the class of craftsmen and<br />
potential technical innovators not held in contempt, as it was<br />
in ancient Greece, but, like the craftsmen, the intellectuals<br />
were, in the main, independent of the authorities. They were<br />
free entrepreneurs, craftsmen-inventors in search of patronage,<br />
who tended to look for novelty and to exploit all the opportunities<br />
it afforded, however dangerous they may have been<br />
for the social order. On the other hand, as Needham points<br />
out, Chinese men of science were officials, bound to observe<br />
the rules of the bureaucracy. They formed an integral part of<br />
the state, whose primary objective was to keep law and order.<br />
The compass, the printing press, and gunpowder, all of which
ORDER OUT OF CHAOS 46<br />
were to contribute to undermining the foundations of medieval<br />
society and to project Europe into the modern era, were discovered<br />
much earlier in China but had a much less destabilizing<br />
effect on its society. The enterprising European merchant<br />
society appears in contrast as particularly well suited to stimulate<br />
and sustain the dynamic and innovative growth of modern<br />
science in its early stages.<br />
However, the question remains. We know that the builders<br />
of machines used mathematical concepts-gear ratios, the displacements<br />
of the various working parts, and the geometry of<br />
their relative motions. But why was mathematization not restricted<br />
to machines? Why was natural motion conceived of in<br />
the image of a rationalized machine? This question may also<br />
be asked in connection with the clock, one of the triumphs of<br />
medieval craftsmanship that was soon to set the rhythm of life<br />
in the larger medieval towns. Why did the clock almost immediately<br />
become the very symbol of world order? In this last<br />
question lies perhaps some elements of an answer. A watch is<br />
a contrivance governed by a rationality that lies outside itself,<br />
by a plan that is blindly executed by its inner workings. The<br />
clock world is a metaphor suggestive of God the Watchmaker,<br />
the rational master of a robotlike nature. At the origin of modern<br />
science, a "resonance" appears to have been set up between<br />
theological discourse and theoretical and experimental<br />
activity-a resonance that was no doubt likely to amplify and<br />
consolidate the claim that scientists were in the process of discovering<br />
the secret of the "great machine of the universe."<br />
Of course, the term resonance covers an extremely complex<br />
problem. It is not our intention to state, nor are we in any<br />
position to affirm, that religious discourse in any way determined<br />
the birth of theoretical science, or of the "world view"<br />
that happened to develop in conjunction with experimental activity.<br />
By using the term resonance-that is, mutual amplification<br />
of two discourses-we have deliberately chosen an<br />
expression that does not assume whether it was theological<br />
discourse or the "scientific myth" that came first and triggered<br />
the other.<br />
Let us note that to some philosophers the question of the<br />
"Christian origin" of Western science is not only the question<br />
of the sta bilization of the concept of nature as an automaton,<br />
but also the question of some "essential" link between experi-
47 THE TRIUMPH OF REASON<br />
mental science as such and Western civilization in its Hebraic<br />
and Greek components. For Alfred North Whitehead this link<br />
is situated at the level of instinctive conviction. Such a conviction<br />
was "needed" to inspire the "scientific faith" of the<br />
founders of modern science:<br />
I mean the inexpugnable belief that every detailed occurrence<br />
can be correlated with its antecedents in a perfectly<br />
definite manner, exemplifying general principles. Without<br />
this belief the incredible labours of scientists would be<br />
without hope. It is this instinctive conviction, vividly<br />
poised before the imagination, which is the motive power<br />
of research: that there is a secret, a secret which can be<br />
unveiled. How has this conviction been so vividly implanted<br />
in the European mind?<br />
When we compare this tone of thought in Europe with<br />
the attitude of other civilizations when left to themselves,<br />
there seems but one source for its origin. It must come<br />
from the medieval insistence on the rationality of God,<br />
conceived as with the personal energy of Jehovah and<br />
with the rationality of a Greek philosopher. Every detail<br />
was supervised and ordered: the search into nature could<br />
only result in the vindication of the faith in rationality.<br />
. Remember that I am not talking of the explicit beliefs of a<br />
few individuals. What I mean is the impress on the European<br />
mind arising from the unquestioned faith of centuries.<br />
By this I mean the instinctive tone of thought and<br />
not a mere creed of words. 2 o<br />
We will not consider this matter further. It would be out of<br />
the question to .. prove" that modern science could have originated<br />
only in Christian Europe. It is not even necessary to ask<br />
if the founders of modern science drew any real inspiration<br />
from theological arguments. Whether or not they were sincere,<br />
the important point is that those arguments made the<br />
speculations of modern science socially credible and acceptable,<br />
over a period of time varying from country to country.<br />
Religious references were still frequent in English scientific<br />
texts of the nineteenth century. Remarkably enough, in the<br />
present-day revival of interest in mysticism, the direction of<br />
the argument appears reversed. It is now science that appears<br />
to lend credibility to mystical affirmation.
ORDER OUT OF CHAOS<br />
48<br />
The question we have confronted here obviously leads to·<br />
ward a multitude of problems in which theological and scien·<br />
tific issues are inextricably bound up with the "external"<br />
history of science, that is, the description of the relationship<br />
between the form and content of scientific knowledge on the<br />
one hand, and on the other, the use to which it is put in its<br />
social, economic, and institutional context. As we have already<br />
said, the only point we are presently interested in is the<br />
very particular character and implications of scientific dis·<br />
course that was amplified by resonance with theological discourses.<br />
Needham 2t tells of the irony with which Chinese men of let·<br />
ters of the eighteenth century greeted the Jesuits' announcement<br />
of the triumphs of modern science. The idea that nature<br />
was governed by simple, knowable laws appeared to them as a<br />
perfect example of anthropocentric foolishness. Needham believes<br />
that this "foolishness" has deep cultural roots. In order<br />
to illustrate the great differences between the Western and<br />
Chinese conceptions, he cites the animal trials held in the<br />
Middle Ages. On several occasions such freaks as a cock who<br />
supposedly laid eggs were solemnly condemned to death and<br />
burned for having infringed the laws of nature, which were<br />
equated with the laws of God. Needham explains how, in<br />
China, the same cock would, in all likelihood, merely have<br />
disappeared discreetly. It was not guilty of any crime, but its<br />
freakish behavior clashed with natural and social harmony.<br />
The governor of the province or even the emperor might find<br />
himself in a delicate situation if the misbehavior of the cock<br />
became known. Needham comments that, according to a<br />
philosophic conception dominant in China, the cosmos is in<br />
spontaneous harmony and the regularity of phenomena is not<br />
due to any external authority. On the contrary, this harmony in<br />
nature, society, and the heavens originates from the equilibrium<br />
among these processes. Stable and interdependent,<br />
they resonate with each other in a kind of nonconcerted harmony.<br />
If any law were involved, it would be a law that no one,<br />
neither God nor man, had ever conceived of. Such a law would<br />
also have to be expressed in a language undecipherable by<br />
man and not be a law established by a creator conceived in our<br />
own image.
49 THE TRIUMPH OF REASON<br />
Needham concludes by asking the following question:<br />
In the outlook of modern science there is, of course, no<br />
residue of the notions of command and duty in the<br />
"Laws" of Nature. They are now thought of as statistical<br />
regularities, valid only in given times and places, descriptions<br />
not prescriptions, as Karl Pearson put it in a famous<br />
chapter. The exact degree of subjectivity in the formulations<br />
of scientific law has been hotly debated during the<br />
whole period from Mach to Eddington, and such questions<br />
cannot be followed further here. The problem is<br />
whether the recognition of such statistical regularities<br />
and their mathematical expression could have been<br />
reached by any other road than that which Western science<br />
actually travelled. Was perhaps the state of mind in<br />
which an egg-laying cock could be prosecuted at law necessary<br />
in a culture which should later have the property<br />
of producing a Kepler? 22<br />
It must now be stressed that scientific discourse is in no way<br />
a mere transposition of traditional religious views. Obviously<br />
the world described by classical physics is not the world of<br />
Genesis, in which God created light, heaven, earth, and the<br />
living species, the world ·where Providence has never ceased<br />
to act, spurring man on toward a history where his salvation is<br />
at stake. The world of classical physics is an atemporal world<br />
which, if created, must have been created in one fell swoop,<br />
somewhat as an engineer creates a robot before letting it function<br />
alone. In this sense, physics has indeed developed in opposition<br />
to both religion and the traditional philosophies. And<br />
yet we know that the Christian God was actually called upon<br />
to provide a basis for the world's intelligibility. In fact, one can<br />
speak here of a kind of "convergence" between the interests<br />
of theologians, who held that the world had to acknowledge<br />
God's omnipotence by its total submission to Him, and of<br />
physicists seeking a world of mathematizable processes.<br />
In any case, the Aristotelian world destroyed by modern science<br />
was unacceptable to both these theologians and physicists.<br />
This ordered, harmonious, hierarchical, and rational world<br />
was too independent, the beings inhabiting it too powerful and
ORDER OUT OF CHAOS 50<br />
active, and their subservience to the absolute sovereign too<br />
suspect and limited for the needs of many theologians. 23 On<br />
the other hand, it was too complex and qualitatively differentiated<br />
to be mathematized.<br />
The "mechanized" nature of modern science, created and<br />
ruled according to a plan that totally dominates it, but of which<br />
it is unaware , glorifies its creator, and was thus admirably<br />
suited to the needs of both theologians and the physicists. Although<br />
Leibniz had endeavored to demonstrate that mathematization<br />
is compatible with a world that can display active and<br />
qualitatively differentiated behavior, scientists and theologians<br />
joined forces to describe nature as a mindless, passive mechanics<br />
that was basically alien to freedom and the purposes<br />
of the human mind. 'i\ dull affair, soundless, scentless, colourless,<br />
merely the hurrying of matter, endless, meaningless, "24<br />
as Whitehead observes. This Christian nature, stripped of any<br />
property that permits man to identify himself with the ancient<br />
harmony of natural "becoming," leaving man alone, face to<br />
face with God, thus converged with the nature that a single<br />
language, and not the thousand mathematical voices heard by<br />
Leibniz, was sufficient to describe.<br />
Theology may also help comment on man's odd position<br />
when he laboriously deciphers the laws governing the world.<br />
Man is emphatically not part of the nature he objectively describes;<br />
he dominates it from the outside. Indeed, for Galileo,<br />
the human soul, created in God's image, is capable of grasping<br />
the intelligible truths underlying the plan of creation. It can<br />
thus gradually approach a knowledge of the world that God<br />
himself possessed intuitively, fully, and instantaneously. 25<br />
Unlike the ancient atomists, who were persecuted on the<br />
grounds of atheism, and unlike Leibniz, who was sometimes<br />
suspected of denying the existence of grace or of human freedom,<br />
modern scientists have managed to come up with a<br />
culturally acceptable definition of their enterprise. The human<br />
mind, incorporated in a body subject to the laws of nature,<br />
can, by means of experimental devices, obtain access to the<br />
vantage point from which God himself surveys the world, to<br />
the divine plan of which this world is a tangible expression.<br />
Nevertheless, the mind itself remains outside the results of its<br />
achievement_ The scientist may descr ibe as secondary<br />
qualities, not part of nature but projected onto it by the mind,
51 THE TRIUMPH OF REASON<br />
everything that goes to make up the texture of nature, such as<br />
its perfumes and its colors. The debasement of nature is parallel<br />
to the glorification of all that eludes it, God and man.<br />
The Limits of Classical Science<br />
We have tried to describe the unique historical situation in<br />
which scientific practice and metaphysical conviction were<br />
closely coupled. Galileo and those who came after him raised<br />
the same problems as the medieval builders but broke away<br />
from their empirical knowledge to assert, with the help of<br />
God, the simplicity of the world and the universality of the<br />
language the experimental method postulated and deciphered.<br />
In this way, the basic myth underlying modern science can be<br />
seen as a product of the peculiar complex which, at the close<br />
of the Middle Ages, set up conditions of resonance and reciprocal<br />
amplification among economic, political, social, religious,<br />
philosophic, and technical factors. However, the rapid<br />
decomposition of this complex left classical science stranded<br />
and isolated in a transformed culture.<br />
Classical science was born in a culture dominated by the<br />
alliance between man, situated midway between the divine<br />
order and the natural order, and God, the rational and intelligible<br />
legislator, the sovereign architect we have conceived in our<br />
own image. It has outlived this moment of cultural consonance<br />
that entitled philosophers and theologians to engage in science<br />
and that entitled scientists to decipher and express opinions<br />
on the divine wisdom and power at work in creation. With the<br />
support of religion and philosophy, scientists had come to believe<br />
their enterprise was self-sufficient, that it exhausted the<br />
possibilities of a rational approach to natural phenomena. The<br />
relationship between scientific description and natural philosophy<br />
did not, in this sense, have to be justified. It could be<br />
seen as self-evident that science and philosophy were convergent<br />
and that science was discovering the principles of an<br />
authentic natural philosophy. But, oddly enough, the selfsufficiency<br />
experienced by scientists was to outlive the departure<br />
of the medieval God and the withdrawal of the epistemological<br />
guarantee offered by theology. The originally bold bet had become<br />
the triumphant science of the eighteenth century, 26 the
ORDER OUT OF CHAOS 52<br />
science that discovered the laws governing the motion of celestial<br />
and earthly bodies, a science that df\lembert and Euler<br />
incorporated into a complete and consistent system and<br />
whose history was defined by Lagrange as a logical achievement<br />
tending toward perfection. It was the science honored by<br />
the Academies founded by absolute monarchs such as Louis<br />
XIV, Frederick II, and Catherine the Great,27 the science that<br />
made Newton a national hero. In other words, it was a successful<br />
science, convinced that it had proved that nature is<br />
transparent. "Je n'ai pas besoin de cette hypothese" was<br />
Laplace's reply to Napoleon, who had asked him God's place<br />
in his world system.<br />
The dualist implications of modern science were to survive<br />
as well as its claims. For the science of Laplace which, in<br />
many respects, is still the classical conception of science today,<br />
a description is objective to the extent to which the observer<br />
is excluded and the description itself is made from a<br />
point lying de jure outside the world, that is, from the divine<br />
viewpoint to which the human soul, created as it was in God's<br />
image, had access at the beginning. Thus classical science still<br />
aims at discovering the unique truth about the world, the one<br />
language that will decipher the whole of nature-today we<br />
would speak of the fundamental level of description from ·<br />
which everything in existence can be deduced.<br />
On this essential point let us cite Einstein, who has translated<br />
into modern terms precisely what we may call the basic<br />
myth underlying modern science:<br />
What place does the theoretical physicist's picture of the<br />
world occupy among all these possible pictures? It demands<br />
the highest possible standard of rigorous precision<br />
in the description of relations, such as only the use of<br />
mathematical language can give. In regard to his subject<br />
matter, on the other hand, the physicist has to limit himself<br />
very severely: he must content himself with describing<br />
the most simple events which can be brought within<br />
the domain of our experience; all events of a more complex<br />
order are beyond the power of the human intellect to<br />
reconstruct with the subtle accuracy and logical perfection<br />
which the theoretical physicist demands. Supreme<br />
purity, clarity, and certainty at the cost of completeness.
53 THE TRIUMPH OF REASON<br />
But what can be the attraction of getting to know such a<br />
tiny section of nature thoroughly, while one leaves everything<br />
subtler and more complex shyly and timidly alone?<br />
Does the product of such a modest effort deserve to be<br />
called by the proud name of a theory of the universe?<br />
In my belief the name is justified; for the general laws<br />
on which the structure of theoretical physics is based<br />
claim to be valid for any natural phenomenon whatsoever.<br />
With them, it ought to be possible to arrive at the<br />
description, that is to say, the theory, of every natural<br />
process, including life, by means of pure deduction, if<br />
that process of deduction were not far beyond the capacity<br />
of the human intellect. The physicist's renunciation of<br />
completeness for his cosmos is therefore not a matter of<br />
fundamental principle. 2s<br />
For some time there were those who persisted in the illusion<br />
that attraction in the form in which it is expressed in the law of<br />
gravitation would justify attributing an intrinsic animation to<br />
nature and that if it were generalized it would explain the origins<br />
of increasingly specific forms of activity, including even<br />
the interactions that compose human society. But this hope<br />
was rapidly crushed, at least partly as a consequence of the<br />
demands created by the political, economic, and institutional<br />
setting where science developed. We shall not examine this<br />
aspect of the problem, important though it is. Our point here is<br />
to emphasize that this very failure seemed to establish the<br />
consistency of the classical view and to prove that what had<br />
once been an inspiring conviction was a sad truth. In fact, the<br />
only interpretation apparently capable of rivaling this interpretation<br />
of science was henceforth the positivistic refusal of<br />
the very project of understanding the world. For example,<br />
Ernst Mach, the influential philosopher-scientist whose ideas<br />
had a great impact on the young Einstein, defined the task of<br />
scientific knowledge as arranging experience in as economical<br />
an order as possible. Science has no other meaningful goal<br />
than the simplest and most economical abstract expression of<br />
facts:<br />
Here we have a clue which strips science of all its mystery,<br />
and shows us what its power really is. With respect
ORDER OUT OF CHAOS<br />
54<br />
to specific results it yields us nothing that we could not<br />
reach in a sufficiently long time without methods . ...<br />
Just as a single human being, restricted wholly to the<br />
fruits of his own labor, could never amass a fortune, but<br />
on the contrary the accumulation of the labor of many<br />
men in the hands of one is the foundation of wealth and<br />
power, so, also, no knowledge worthy of the name can be<br />
gathered up in a single human mind limited to the span of<br />
a human life and gifted only with finite powers, except by<br />
the most exquisite economy of thought and by the careful<br />
amassment of the economically ordered experience of<br />
thousands of co-workers. 29<br />
Thus science is useful because it leads to economy of<br />
thought. There may be some element of truth in such a statement,<br />
but does it tell the whole story? How far we have come<br />
from Newton, Leibniz, and the other founders of Western science,<br />
whose ambition was to provide an intelligible frame to<br />
the physical universe! Here science leads to interesting rules<br />
of action, but no more.<br />
This brings us back to our starting point, to the idea that it is<br />
classical science, considered for a certain period of time as<br />
the very symbol of cultural unity, and not science as such that<br />
led to the cultural crisis we have described. Scientists found<br />
themselves reduced to a blind oscillation between the thunderings<br />
of .. scientific myth" and the silence of "scientific seriousness,"<br />
between affirming the absolute and global nature of<br />
scientific truth and retreating into a conception of scientific<br />
theory as a pragmatic recipe for effective intervention in natural<br />
processes.<br />
As we have already stated, we subscribe to the view that<br />
classical science has now reached its limit. One aspect of this<br />
transformation is the discovery of the limitations of classical<br />
concepts that imply that a knowledge of the world "as it is"<br />
was possible. The omniscient beings, Laplace's or Maxwell's<br />
demon, or Einstein's God, beings that play such an important<br />
role in scientific reasoning, embody the kinds of extrapolation<br />
physicists thought they were allowed to make. As randomness,<br />
complexity, and irreversibility enter into physics as objects<br />
of positive knowledge, we are moving away from this<br />
rather naive assumption of a direct connection between our
55<br />
THE TRIUMPH OF REASON<br />
description of the world and the world itself. Objectivity in<br />
theoretical physics takes on a more subtle meaning.<br />
This evolution was forced upon us by unexpected supplemental<br />
discoveries that have shown that the existence of universal<br />
constants, such as the velocity of light, limit our power<br />
to manipulate nature. (We shall discuss this unexpected situation<br />
in Chapter VII.) As a result, physicists had to introduce<br />
new mathematical tools that make the relation between perception<br />
and interpretation more complex. Whatever reality may<br />
mean, it always corresponds to an active intellectual construction.<br />
The descriptions presented by science can no longer be<br />
disentangled from our questioning activity and therefore can<br />
no longer be attributed to some omniscient being.<br />
On the eve of the Newtonian synthesis, John Donne lamented<br />
the passing of the Aristotelian cosmos destroyed by<br />
Copernicus:<br />
And new Philosophy calls all in doubt,<br />
The Element of fire is quite put out,<br />
The Sun is lost, and th'earth, and no man's wit<br />
Can well direct him where to look for it.<br />
And freely men confess that this world's spent,<br />
When in the Planets and the Firmament,<br />
They seek so many new, then they see that this<br />
Is crumbled out again to his Atomies<br />
'Tis all in Pieces, all coherence gone.JO<br />
The scattered bricks and stones of our present culture seem,<br />
as in Donne's time, capable of being rebuilt into a new "coherence."<br />
Classical science, the mythical science of a simple,<br />
passive world, belongs to the past, killed not by philosophical<br />
criticism or empiricist resignation but by the internal development<br />
of science itself.
I<br />
I
CHAPTER II<br />
THE IDENTIFICATION<br />
OFTHEREAL<br />
Newtons Laws<br />
We shall now take a closer look at the mechanistic world view<br />
as it emerged from the work of Galileo, Newton, and their<br />
successors. We wish to describe its strong points, the aspects<br />
of nature it has succeeded in clarifying, but we also want to<br />
expose its limitations.<br />
·Ever since Galileo, one of the central problems of physics<br />
has been the description of acceleration. The surprising feature<br />
was that the change undergone by the state of motion of a<br />
body could be formulated in simple mathematical terms. This<br />
seems almost trivial to us today. Still, we should remember<br />
that Chinese science, so successful in many areas, did not produce<br />
a quantitative formulation of the laws of motion. Galileo<br />
discovered that we do not need to ask for the cause of a state<br />
of motion if the motion is uniform, any more than it is necessary<br />
to ask the reason for a state of rest. Both motion and rest<br />
remain indefinitely stable unless something happens to upset<br />
them. The central problem is the change from rest to motion,<br />
and from motion to rest, as well as, more generally, all changes<br />
of velocity. How do these changes occur? The formulation of<br />
the Newtonian laws of motion made use of two converging<br />
developments: one in physics, Kepler's laws for planetary motion<br />
and Galileo's laws for falling bodies, and the other in<br />
mathematics, the formulation of differential or "infinitesimal"<br />
calculus.<br />
How can a continuously varying speed be defined? How can<br />
we describe the instantaneous changes in the various quantities,<br />
such as position, velocity, and acceleration? How can<br />
we describe the state of a body at any given instant? To answer<br />
57
ORDER OUT OF CHAOS 58<br />
these questions, mathematicians have introduced the concept<br />
of infinitesimal quantities. An infinitesimal quantity is the result<br />
of a limiting process; it is typically the variation in a quantity<br />
occurring between two successive instants when the time<br />
elapsing between these instants tends toward zero. In this way<br />
the change is broken up into an infinite series of infinitely<br />
small changes.<br />
At each instant the state of a moving body can be defined by<br />
its position r, by its velocity v, which expresses its "instantaneous<br />
tendency" to modify this position, and by its acceleration<br />
a, again its "instantaneous tendency," but now to modify<br />
its velocity. Instantaneous velocities and accelerations are limiting<br />
quantities that measure the ratio between two infinitesimal<br />
quantities: the variation of r (or v) during a temporal<br />
interval 6.t, and this interval 6.t when 6.t tends to zero. Such<br />
quantities are "derivatives with respect to time," and since<br />
Leibniz they have been written as v=drldt and a=dv/dt.<br />
Therefore, acceleration, the derivative of a derivative, a= d2r!<br />
dt2, becomes a "second derivative." The problem on which<br />
Newtonian physics concentrates is the calculation of this second<br />
derivative, that is, of the acceleration undergone at each<br />
instant by the points that form a system. The motion of each of<br />
these points over a finite interval of time can then be calculated<br />
by integration, by adding up the infinitesimal velocity<br />
changes occurring during this interval. The simplest case is<br />
when a is constant (for example, for a freely falling body a is<br />
the gravitational constant g). Generally speaking, acceleration<br />
itself varies in time, and the physicist's task is to determine<br />
precisely the nature of this variation.<br />
In Newtonian language, to study acceleration means to determine<br />
the various "forces" acting on the points in the system<br />
under examination. Newton's second law, F= ma, states that<br />
the force applied at any point is proportional to the acceleration<br />
it produces. In the case of a system of material points, the<br />
problem is more complicated, since the forces acting on a<br />
given body are determined at each instant by the relative distances<br />
between the bodies of the system, and thus vary at each<br />
instant as a result of the motion they themselves produce.<br />
A problem in dynamics is expressed in the form of a set of<br />
"differential .. equations. The instantaneous state of each of<br />
the bodies in a system is described as a point and defined by
59 THE IDENTIFICATION OF THE REAL<br />
means of its position as well as by its velocity and acceleration,<br />
that is, by the first and second derivatives of the position.<br />
At each instant, a set of forces, which is a function of the<br />
distance between the points in the system (a function of r),<br />
gives a precise acceleration to each point; the accelerations<br />
then bring about changes in the distances separating these<br />
points and therefore in the set of forces acting at the following<br />
instant.<br />
While the differential equations set up the dynamics problem,<br />
their "integration" represents the solution of this problem.<br />
It leads to the calculation of the trajectories, r(t). These<br />
trajectories contain all the information acknowledged as relevant<br />
by dynamics; it provides a complete description of the<br />
dynamic system.<br />
The description therefore implies two elements: the positions<br />
and velocities of each of the points at one instant, often<br />
called the "initial instant," and the equations of motion that<br />
relate the dynamic forces to the accelerations. The integration<br />
of the dynamic equations starting from the "initial state" unfold<br />
the succession of states, that is, the set of trajectories of<br />
its constitutive bodies.<br />
The triumph of Newtonian science is the discovery that a<br />
single force, gravity, determines both the motion of planets<br />
and comets in the sky and the motions of bodies falling toward<br />
the earth. Whatever pair of material bodies is considered, the<br />
Newtonian system implies that they are linked by the same<br />
force of attraction. Newtonian dynamics thus appears to be<br />
doubly universal. The definition of the law of gravity that describes<br />
how masses tend to approach one another contains no<br />
reference to any scale of phenomena. It can be applied equally<br />
well to the motion of atoms, of planets, or of the stars in a<br />
galaxy. Every body, whatever its size, has a mass and acts as a<br />
source of the Newtonian forces of interaction.<br />
Since gravitational forces connect any two bodies (for two<br />
bodies of mass m and m ' and separated by a distance r, the<br />
gravitational force is kmm'fr2, where k is the Newtonian force<br />
of attraction equal to 6. 67cm3g-1sec-2), the only true dynamic<br />
system is the universe as a whole. Any local dynamic<br />
system, such as our planetary system, can only be defined<br />
approximately, by neglecting forces that are small in comparison<br />
to those whose effect is being considered.
ORDER OUT OF CHAOS<br />
60<br />
It must be emphasized that whatever the dynamic system<br />
chosen, the laws of motion can always be expressed in the<br />
form F= ma. Other types of forces apart from those due to<br />
gravity may be discovered (and actually have been discovered-for<br />
instance, electric forces of attraction and repulsion)<br />
and would thereby modify the empirical content of the laws of<br />
motion. They would not, however, modify the form of those<br />
laws. In the world of dynamics, change is identified with acceleration<br />
or deceleration. The integration of the laws of motion<br />
leads to the trajectories that the particles follow. Therefore the<br />
laws of change, of time's impact on nature, are expressed in<br />
terms of the characteristics of trajectories.<br />
The basic characteristics of trajectories are lawfulness, determinism,<br />
and reversibility. We have seen that in order to calculate<br />
a trajectory we need, in addition to our knowledge of<br />
the laws of motion, an empirical definition of a single instantaneous<br />
state of the system. The general law then deduces<br />
from this "initial state" the series of states the system passes<br />
through as time progresses, just as logic deduces a conclusion<br />
from basic premises. The remarkable feature is that once the<br />
forces are known, any single state is sufficient to define the<br />
system completely, not only its future but also its past. At<br />
each instant, therefore, everything is given. Dynamics defines<br />
all states as equivalent: each of them allows all the others to be<br />
calculated along with the trajectory which connects all states,<br />
be they in the past or the future.<br />
"Everything is given." This conclusion of classical dynamics,<br />
which Bergson repeatedly emphasized, characterizes the<br />
reality that dynamics describes. Everything is given, but everything<br />
is also possible. A being who has the power to control<br />
a dynamic system may calculate the right initial state in such a<br />
way that the system "spontaneously" reaches any chosen<br />
state at some chosen time. The generality of dynamic laws is .<br />
matched by the arbitrariness of the initial conditions.<br />
The reversibility of a dynamic trajectory was explicitly<br />
stated by all the founders of dynamics. For instance, when<br />
Galilee or Huyghens described the implications of the equivalence<br />
between cause and effect postulated as the basis of<br />
their mathematization of motion, they staged thought experiments<br />
such as an elastic ball bouncing on the ground. As the
61 THE IDENTIFICATION OF THE REAL<br />
result of its instantaneous velocity inversion, such a body<br />
would return to its initial position. Dynamics assigns this<br />
property of reversibility to all dynamic changes. This early<br />
"thought experiment" illustrates a general mathematical property<br />
of dynamic equations. The structure of these equations<br />
implies that if the velocities of all the points of a system are<br />
reversed, the system will go "backward in time. " The system<br />
would retrace all the states it went through during the previous<br />
change. Dynamics defines as mathematically equivalent<br />
changes such as t-+- t, time inversion, and v-+- v, velocity<br />
reversal. What one dynamic change has achieved, another<br />
change, defined by velocity inversion, can undo, and in this<br />
way exactly restore the original conditions.<br />
This property of reversibility in dynamics leads, however, to<br />
a difficulty whose full significance was realized only with the<br />
introduction of quantum mechanics. Manipulation and measurement<br />
are essentially irreversible. Active science is thus,<br />
by definition, extraneous to the idealized, reversible world it is<br />
describing. From a more general point of view, reversibility<br />
may be taken as the very symbol of the "strangeness" of the<br />
world described by dynamics. Everyone is familiar with the<br />
absurd effects produced by projecting a film backward-the<br />
sight of a match being regenerated by its flame, broken ink<br />
pots that reassemble and return to a tabletop after the ink has<br />
poured back into them, branches that grow young again and<br />
turn into fresh shoots. In the world of classical dynamics, such<br />
events are considered to be just as likely as the normal ones.<br />
We are so accustomed to the laws of classical dynamics that<br />
are taught to us early in school that we often fail to sense the<br />
boldness of the assumptions on which they are based. A world<br />
in which all trajectories are reversible is a strange world indeed.<br />
Another astonishing assumption is that of the complete<br />
independence of initial conditions from the laws of motion. It<br />
is possible to take a stone and throw it with some initial velocity<br />
limited only by one's physical strength, but what about<br />
a complex system SlJCh as a gas formed by many particles? It<br />
is obvious that we can no longer impose arbitrary initial conditions.<br />
The initial conditions must be the outcome of the dynamic<br />
evolution itself. This is an important point to which we<br />
shall come back in the third part of this book. But whatever its
ORDER OUT OF CHAOS 62<br />
limitations, today, three centuries later, we can only admire<br />
the logical coherence and the power of the methods discovered<br />
by the founders of classical dynamics.<br />
Motion and Change<br />
Aristotle made time the measure of change. But he was fully<br />
aware of the qualitative multiplicity of change in nature. Still<br />
there is only one type of change surviving in dynamics, one<br />
"process," and that is motion. The qualitative diversity of<br />
changes in nature is reduced to the study of the relative displacement<br />
of material bodies. Time is a parameter in terms of<br />
which these displacements may be described. In this way<br />
space and time are inextricably tied together in the world of<br />
classical dynamics. (Also see Chapter IX.)<br />
It is interesting to compare dynamic change with the atomists'<br />
conception of change, which enjoyed considerable favor<br />
at the time Newton formulated his laws. Actually, it seems that<br />
not only Descartes, Gassendi, and d'Alembert, but even Newton<br />
himself believed that collisions between hard atoms were<br />
the ultimate, and perhaps the only, sources of changes of motion.t<br />
Nevertheless, the dynamic and the atomic descriptions<br />
differ radically. Indeed, the continuous nature of the acceleration<br />
described by the dynamic equations is in sharp contrast<br />
with the discontinuous, instantaneous collisions between hard<br />
particles. Newton had already noticed that, in contradiction to<br />
dynamics, an irreversible loss of motion is involved in each<br />
hard collision. The only reversible collision-that is, the only<br />
one in agreement with the laws of dynamics-is the "elastic,"<br />
momentum-conserving collision. But how can the complex<br />
property of "elasticity" be applied to atoms that are supposed<br />
to be the fundamental elements of nature?<br />
On the other hand, at a less technical level, the laws of dynamic<br />
motion seem to contradict the randomness generally<br />
attributed to collisions between atoms. The ancient philosophers<br />
had already pointed out that any natural process can be<br />
interpreted in many different ways in terms of the motion of<br />
and collisions between atoms. This was not a problem for the<br />
atomists, since their main aim was to describe a godless, law-
63<br />
THE IDENTIFICATION OF THE REAL<br />
less world in which man is free and can expect to receive neither<br />
punishment nor reward from any divine or natural order.<br />
But classical science was a science of engineers and astronomers,<br />
a science of action and prediction. Speculations based<br />
on hypothetical atoms could not satisfy its needs. In contrast,<br />
Newton's law provided a means of predicting and manipulating.<br />
Nature thus becomes law-abiding, docile, and predictable,<br />
instead of being chaotic, unruly, and stochastic. But<br />
what is the connection between the mortal, unstable world in<br />
which atoms unceasingly combine and separate, and the immutable<br />
world of dynamics governed by Newton's law, a siagle<br />
mathematical formula corresponding to an eternal truth unfolding<br />
toward a tautological future? In the twentieth century<br />
we are again witnessing the clash between lawfulness and random<br />
events, which, as Koyre has shown, had already tormented<br />
Descartes. 2 Ever since the end of the nineteenth<br />
century, with the kinetic theory of gases, the atomic chaos has<br />
reintegrated physics, and the problem of the relationship between<br />
dynamic law and statistical description has penetrated<br />
to the very core of physics. It is one of the key elements in the<br />
present renewal of dynamics (see Book III).<br />
In the eighteenth century, however, this contradiction<br />
seemed to produce a deadlock. This may partly explain the<br />
skepticism of some eighteenth-century physicists regarding<br />
the significance of Newton's dynamic description. We have already<br />
noted that collisions may lead to a loss of motion. They<br />
thereby concluded that in such nonideal cases, "energy" is<br />
not conserved but is irreversibly dissipated (see Chapter IV,<br />
section 3). Therefore, the atomists could not help considering<br />
dynamics as an idealization of limited value. Continental<br />
physicists and mathematicians such as dl\lembert, Clairaut,<br />
and Lagrange resisted the seductive charms of Newtonianism<br />
for a long time.<br />
Where do the roots of the Newtonian concept of change lie?<br />
It appears to be a synthesis3 of the science of ideal machines,<br />
where motion is transmitted without collision or friction between<br />
parts already in contact, and the science of celestial<br />
bodies interacting at a distance. We have seen that it appears<br />
as the very antithesis of atomism, which is based on the concept<br />
of random collisions. Does this, however, vindicate the<br />
view of those who believe that Newtonian dynamics repre-
ORDER OUT OF CHAOS<br />
64<br />
sents a rupture in the history of thinking, a revolutionary novelty?<br />
This is what positivist historians have claimed when they<br />
described how Newton escaped the spell of preconceived notions<br />
and had the courage to infer from the mathematical study of<br />
planetary motions and the laws of falling bodies the action of a<br />
"universal" force. We know that on the contrary the eighteenthcentury<br />
rationalists emphasized the apparent similarity between<br />
his "mathematical" forces and traditional occult<br />
qualities. Fortunately, these critics did not know the strange<br />
story behind the Newtonian forces! For behind Newton's cautious<br />
declaration-"! frame no hypotheses"-concerning the<br />
nature of the forces lurked the passion of an alchemist.4 We<br />
now know that, side by side with his mathematical studies,<br />
Newton had studied the ancient alchemists for thirty years<br />
and had carried out painstaking laboratory experiments on<br />
ways of achieving the master work, the synthesis of gold.<br />
Recently some historians have gone so far as to propose that<br />
the Newtonian synthesis of heaven and earth was the achievement<br />
of a chemist, not an astronomer. The Newtonian force<br />
"animating" matter and, in the stronger sense of the term,<br />
making up the very activity of nature would then be the inheritor<br />
of the forces Newton the chemist observed and manipulated,<br />
the chemical "affinities" forming and disrupting ever<br />
new combinations of matter.s The decisive role played by celestial<br />
orbits of course remains. Still, at the start of his intense<br />
astronomical studies-about 1679-Newton apparently expected<br />
to find new forces of attraction only in the heavens,<br />
forces similar to chemical forces and perhaps easier to study<br />
mathematically. Six years later this mathematical study produced<br />
an unexpected conclusion: the forces between the planets<br />
and those accelerating freely falling bodies are not merely<br />
similar but are the same. Attraction is not specific to each<br />
planet; it is the same-for the moon circling the earth, for the<br />
planets, and even for comets passing through the solar system.<br />
Newton set out to discover in the sky forces similar to the<br />
chemical forces: the specific affinities, different for each<br />
chemical compound and giving each compound qualitatively<br />
differentiated activities. What he actually found was a universal<br />
law, which, as he emphasized, could be applied to all phenomena-whether<br />
chemical, mechanical, or celestial in<br />
nature.
65 THE IDENTIFICATION OF THE REAL<br />
The Newtonian synthesis is thus a surprise. It is an unexpected,<br />
staggering discovery that the scientific world has commemorated<br />
by making Newton the symbol of modern science.<br />
What is particularly astonishing is that the basic code of nature<br />
appeared to have been cracked in a single creative act.<br />
For a long time this sudden loquaciousness of nature, this<br />
triumph of the English Moses, was a source of intellectual<br />
scandal for continental rationalists. Newton's work was<br />
viewed as a purely empirical discovery that could thus equally<br />
well be empirically disproved. In 1747 Euler, Clairaut, and<br />
dlembert, without doubt some of the greatest scientists of<br />
the time, came to the same conclusion: Newton was wrong. In<br />
order to describe the moon's motion, a more complex mathematical<br />
form must be given to the force of attraction, making it<br />
the sum of two terms. For the following two years, each of<br />
them believed that nature had proved Newton wrong, and this<br />
belief was a source of excitement, not of dismay. Far from considering<br />
Newton's discovery synonymous with physical science<br />
itself, physicists were blithely contemplating dropping it<br />
altogether. Dlembert went so far as to express scruples<br />
about seeking fresh evidence against Newton and giving him<br />
"le coup de pied de l'iine."6<br />
Only one courageous voice against this verdict was raised in<br />
France. In 1748, Buffon wrote:<br />
A physical law is a law only by virtue of the fact that it is<br />
easy to measure, and that the sale it represents is not<br />
only always the same, but is actually unique . ... M.<br />
Clairaut has raised an objection against Newton's system,<br />
but it is at best an objection and must not and cannot<br />
become a principle; an attempt should be ,made to overcome<br />
it and not to turn it into a theory the entire consequences<br />
of which merely rest on a calculation; for, as I<br />
have said, one may represent anything by means of calculation<br />
and achieve nothing; and if it is allowed to add<br />
one or more terms to a physical law such as that of attraction,<br />
we are only adding to arbitrariness instead of representing<br />
reality. 7<br />
Later Buffon was to announce what was to become, although<br />
for only a short time, the research program for chemistry:
ORDER OUT OF CHAOS<br />
66<br />
The laws of affinity by means of which the constituent<br />
parts of different substances separate from others to<br />
combine together to form homogeneous substances are<br />
the same as the general law governing the reciprocal action<br />
of all the celestial bodies on one another: they act in<br />
(he same way and with the same ratios of mass and distance;<br />
a globule of water, of sand or metal acts upon another<br />
globule just as the terrestrial globe acts on the<br />
moon, and if the laws of affinity have hitherto been regarded<br />
as different from those of gravity, it is because<br />
they have not been fully understood, not grasped completely;<br />
it is because the whole extent of the problem has<br />
not been taken in. The figure which, in the case of celestial<br />
bodies has little or no effect upon the law of interaction<br />
between bodies because of the great distance<br />
involved, is, on the contrary, all important when the distance<br />
is very small or zero . ... Our nephews will be<br />
able, by calculation, to gain access to this new field of<br />
knowledge [that is, to deduce the law of interaction between<br />
elementary bodies from their figures].&<br />
History was to vindicate the naturalist, for whom force was<br />
not mer;ely a mathematical artifice but the very essence of the<br />
new science of nature. The physicists were later compelled to<br />
admit their mistake. Fifty years afterward, Laplace could<br />
write his Systeme du Monde. The law of universal gravity had<br />
stood all tests successfully: the numerous cases apparently<br />
disproving it had been transformed into a brilliant demonstration<br />
of its validity. At the same time, under Buffon's influence,<br />
the French chemists rediscovered the odd analogy between<br />
physical attraction and chemical affinity.9 Despite the sarcasms<br />
of d/\lembert, Condillac, and Condorcet, whose unbending<br />
rationalism was quite incompatible with these obscure and<br />
barren "analogies," they trod Newton's path in the opposite<br />
direction-from the stars to matter.<br />
By the early nineteenth century, the Newtonian programthe<br />
reduction of all physicochemical phenomena to the action<br />
of forces (in addition to gravitational attraction, this included<br />
the repelling force of heat, which makes bodies expand and<br />
favors dissolution, as well as electric and magnetic forces)<br />
had become the official program of Laplace's school, which
67 THE IDENTIFICATION OF THE REAL<br />
dominated the scientific world at the time when Napoleon<br />
dominated Europe. JO<br />
The early nineteenth century saw the rise of the great<br />
French ecoles and the re<strong>org</strong>anization of the universities. This<br />
is the time when scientists became teachers and professional<br />
researchers and took up the tak of training their successors.••<br />
It is also the time of the first attempts to present a synthesis of<br />
knowledge, to gather it together in textbooks and works of<br />
popularization. Science was no longer discussed in the salons;<br />
it was taught or popularized.1 2 It had become a matter of professional<br />
consensus and magistral authority. The first consensus<br />
centered around the Newtonian system: in France<br />
Buffon's confidence finally triumphed over the rational skepticism<br />
of the Enlightenment.<br />
One century after Newton's apotheosis in England, the<br />
grandiloquence of these lines written by Ampere's son echoes<br />
that of Pope's epitaph: 13<br />
Announcing the coming of science's Messiah<br />
Kepler had dispelled the clouds around the Arch.<br />
Then the Word was made man, the Word of the seeing<br />
God<br />
Whom Plato revered, and He was called Newton.<br />
He came, he revealed the principle supreme,<br />
Eternal, universal, One and unique as God Himself.<br />
The worlds were hushed, he spoke: ATTRACTION.<br />
This word was the very word of creation.*<br />
For a short time, which nevertheless left an indelible mark,<br />
science was triumphant, acknowledged and honored by<br />
powerful states and acclaimed as the possessor of a consistent<br />
conception of the world. Worshiped by Laplace, Newton became<br />
the universal symbol of this golden age. It was a happy<br />
moment, indeed, a moment in which scientists were regarded<br />
both by themselves and others as the pioneers of progress,<br />
achieving an enterprise sustained and fostered by society as a<br />
whole.<br />
What is the significance of the Newtonian synthesis today,<br />
after the advent of field theory, relativity, and quantum me-<br />
*Our translation-authors.
ORDER OUT OF CHAOS<br />
68<br />
chanics? This is a complex problem, to which we shall return.<br />
We now know that nature is not always "comfortable and consonant<br />
with herself. " At the microscopic level, the laws of<br />
classical mechanics have been replaced by those of quantum<br />
mechanics. Likewise, at the level of the universe, relativistic<br />
physics has displaced Newtonian physics. Classical physics<br />
nevertheless remains the natural reference point. Moreover, in<br />
the sense that we have defined it-that is, as the description of<br />
deterministic, reversible, static trajectories-Newtonian dynamics<br />
still may be said to form the core of physics.<br />
Of course, since Newton the formulation of classical dynamics<br />
has undergone great changes. This was a result of the<br />
work of some of the greatest mathematicians and physicists,<br />
such as Hamilton and Poincare. In brief, we may distinguish<br />
two periods. First there was a period of clarification and of<br />
generalization. During the second period, the very concepts<br />
upon which classical dynamics rests, such as initial conditions<br />
and the meaning of trajectories, have undergone a critical revision<br />
even in the fields in which (in contrast to quantum mechanics<br />
and relativity) classical dynamics remains valid. At<br />
the moment this book is being written, at the end of the twentieth<br />
century, we are still in this second period. Let us turn<br />
now to the general language of dynamics that was discovered<br />
by nineteenth-century scientists. (In Chapter IX we shall describe<br />
briefly the revival of classical dynamics in our time.)<br />
The Language of Dynarnics<br />
Today classical dynamics can be formulated in a compact and<br />
elegant way. As we shall see, all the properties of a dynamic<br />
system can be summarized in terms of a single function, the<br />
Hamiltonian. The language of dynamics presents a remarkable<br />
consistency and completeness. An unambiguous formulation<br />
can be given to each "legitimate" problem. No wonder the<br />
structure of dynamics has both fascinated and terrified the<br />
imagination since the eighteenth century.<br />
In dynamics,the same system can be studied from different<br />
points of view. In classical dynamics all these points of view<br />
are equivalent in the sense that we can go from one to another<br />
by a transformation, a change of variables. We may speak of
69 THE IDENTIFICATION OF THE REAL<br />
various equivalent representations in which the laws of dynamics<br />
are valid. These various equivalent representations<br />
form the general language of dynamics. This language can be<br />
used to make explicit the static character classical dynamics<br />
attributes to the systems it describes: for many classes of dynamic<br />
systems, time appears merely as an accident, since<br />
their description can be reduced to that of noninteracting mechanical<br />
systems. To introduce these concepts in a simple way,<br />
let us start with the principle of conservation of energy.<br />
In the ideal world of dynamics, devoid of frictions and collisions,<br />
machines have an efficiency of one-the dynamic system<br />
comprising the machine merely transmits the whole of the<br />
motion it receives. A machine receiving a certain quantity of<br />
potential energy (for example, a compressed spring, a raised<br />
weight, compressed air) can produce a motion corresponding<br />
to an "equal" quantity of kinetic energy, exactly the quantity<br />
that would be needed to restore the potential energy the machine<br />
has used in producing the motion. The simplest case is<br />
that in which the only force considered is gravity (which applies<br />
to simple machines, pulleys, levers, capstans, etc.). In<br />
this case it is easy to establish an overall relationship of equivalence<br />
between cause and effect. The height (h) through which<br />
a body falls entirely determines the velocity acquired during<br />
its fall. Whether a body of mass m falls vertically, runs down<br />
an inclined plane, or follows a roller-coaster path, the acquired<br />
velocity (v) and the kinetic energy (mv2/2) depend only on the<br />
drop in level h (v = Vfiii) and enable the body to return to its<br />
original height. The work done against the force of gravity implied<br />
in this upward motion restores the potential energy, mgh,<br />
that the system lost during the fall. Another example is the<br />
pendulum, in which kinetic energy and potential energy are<br />
continuously transformed into one another.<br />
Of course, if instead of a body falling toward the earth, we<br />
are dealing with a system of interacting bodies, the situation is<br />
less easily visualized. Still, at each instant the global variation<br />
in kinetic energy compensates for the variation in potential<br />
energy (bound to the variation in the distances between the<br />
points in the system). Here also energy is conserved in an isolated<br />
system.<br />
Potential energy (or ''potential," conventionally denoted as<br />
V), which depends on the relative positions of the particles, is
ORDER OUT OF CHAOS 70<br />
thus a generalization of the quantity that enabled builders of<br />
machines to measure the motion a machine could produce as<br />
the result of a change in its spatial configuration (for example,<br />
the change in the height of a mass m, which is part of the<br />
machine, gives it a potential energy mgh). Moreover, potential<br />
energy allows us to calculate the set of forces applied at each<br />
instant to the different points of the system to be described. At<br />
each point the derivative of the potential with respect to the<br />
space coordinate q measures the force applied at this point in<br />
the direction of that coordinate. Newton's laws of motion thus<br />
can be formulated using the potential function instead of force<br />
as the main quantity: the variation in the velocity of a point<br />
mass at each instant (or the momentum p, the product of the<br />
mass and the velocity) is measured by the derivative of the<br />
potential with respect to the coordinate q of the mass.<br />
In the nineteenth century this formulation was generalized<br />
through the introduction of a new function, the Hamiltonian<br />
(H). This function is simply the total energy, the sum of the<br />
system's potential and kinetic energy. However, this energy is<br />
no longer expressed in terms of positions and velocities, conventionally<br />
denoted by q and dq/dt, but in terms of so-called<br />
canonical variables-coordinates and momenta-for which<br />
the standard notation is q and p. In simple cases, such as with<br />
a free particle, there is a straightforward relation between velocity<br />
and momentum (p = m dqldt), but in general the relation<br />
is more complicated.<br />
A single function, the Hamiltonian, H(p, q), describes the<br />
dynamics of a system completely. All our empirical knowledge<br />
is put into the form of H. Once this function is known, we may<br />
solve, at least in principle, all possible problems. For example,<br />
the time variation of the coordinate and of the momenta is<br />
simply given by the derivatives of H in respect to p or q. This<br />
Hamiltonian formulation of dynamics is one of the greatest<br />
achievements in the history of science. It has been progressively<br />
extended to cover the theory of electricity and magnetism.<br />
It has also been used in quantum mechanics. It is true<br />
that in quantum mechanics, as we shall see later, the meaning<br />
of the Hamiltonian H had to be generalized: here it is no<br />
longer a simple function of the coordinates and momenta, but<br />
it becomes a new kind of entity, an operator. (We shall return<br />
to this question in Chapter VII.) In any case, the Hamiltonian
71 THE IDENTIFICATION OF THE REAL<br />
description is still of the greatest importance today. The equations<br />
which, through the derivatives of the Hamiltonian, give<br />
the time variation of the coordinates and momenta are the socalled<br />
canonical equations. They contain the general properties<br />
of all dynamic changes. Here we have the triumph of the<br />
mathematization of nature. All dynamic change to which classical<br />
dynamics applies can be reduced to these simple mathematical<br />
equations.<br />
Using these equations, we can verify the above-mentioned<br />
general properties implied by classical dynamics. The canonical<br />
equations are reversible: time inversion is mathematically<br />
the equivalent of velocity inversion. They are also conservative:<br />
the Hamiltonian, which expressed the system's energy in<br />
the canonical variables-coordinates and momenta-is itself<br />
conserved by the changes it brings about in the course of time.<br />
We have already noticed that there exist many points of view<br />
or "representations" in which the Hamiltonian form of the<br />
equations of motion is maintained. They correspond to various<br />
choices of coordinates and momenta. One of the basic problems<br />
of dynamics is to examine precisely how we can select<br />
the pair of canonical variables q and p to obtain as simple a<br />
description of dynamics as possible. For example, we could<br />
look for canonical variables by which the Hamiltonian is reduced<br />
to kinetic energy and depends only on the momenta<br />
(and not on the coordinates). What is remarkable is that in this<br />
case momenta become constants of motion. Indeed, as we<br />
have seen, the time variation of the momenta depends, according<br />
to the canonical equation, on the derivative of the Hamiltonian<br />
in respect to the coordinates. When this derivative<br />
vanishes, the momenta indeed become constants of motion.<br />
This is similar to what happens in a "free particle" system.<br />
What we have done when we go to a free particle system is<br />
"eliminate" the interaction through a change of representation.<br />
We will define systems for which this is possible as "integrable<br />
systems." Any integrable system may thus be<br />
represented as a set of units, each changing in isolation, quite<br />
independently of all the others, in that eternal and immutable<br />
motion Aristotle attributed to the heavenly bodies (Figure 1).<br />
We have already noted that in dynamics "everything is<br />
given." Here this means that, from the very first instant, the<br />
value of the various invariants of the motion is fixed; nothing
ORDER OUT OF CHAOS 72<br />
•<br />
><br />
•<br />
•<br />
•<br />
•<br />
•<br />
•<br />
(a )<br />
(b)<br />
Figure 1. Two representations of the same dynamic system: (a) as a set of<br />
interacting points; the interaction between the points is represented by wavy<br />
lines; (b) as a set where each point behaves independently from the others.<br />
The potential energy being eliminated, their respective · motions are not explicitly<br />
dependent on their relative positions.<br />
may "happen" or "take place." Here we reach one of those<br />
dramatic moments in the history of science when the description<br />
of nature was nearly reduced to a static picture. Indeed,<br />
through a clever change of variables, all interaction could be<br />
made to disappear. It was believed that integrable systems,<br />
reducible to free particles, were the prototype of dynamic systems.<br />
Generations of physicists and mathematicians tried hard<br />
to find for each kind of systems the "right" variables that<br />
would eliminate the interactions. One widely studied example<br />
was the three-body problem, perhaps the most important<br />
problem in the history of dynamics. The moon's motion, influenced<br />
by both the earth and the sun, is one instance of this<br />
problem. Countless attempts were made to express it in the<br />
form of an integrable system until, at the end of the nineteenth<br />
century, Bruns and Poincare showed that this was impossible.<br />
This came as a surprise and, in fact, announced the end of all<br />
simple extrapolations of dynamics based on integrable systems.<br />
The discovery of Bruns and Poincare shows that dynamic<br />
systems are not isomorphic. Simple, integrable systems<br />
can indeed be reduced to noninteracting units, but in general,<br />
interactions cannot be eliminated. Although this discovery<br />
was not clearly understood at the time, it implied the demise of<br />
the conviction that the dynamic world is homogeneous, reducible<br />
to the concept of integrable systems. Nature as an
73 THE IDENTIFICATION OF THE REAL<br />
evolving, interactive multiplicity thus resisted its reduction to<br />
a timeless and universal scheme.<br />
There were other indications pointing in the same direction.<br />
We have mentioned that trajectories correspond to deterministic<br />
laws; once an initial state is given, the dynamic laws of motion<br />
permit the calculation of trajectories at each point in the<br />
future or the past. However, a trajectory may become intrinsically<br />
indeterminate at certain singular points. For instance, a<br />
rigid pendulum may display two qualitatively different types of<br />
behavior-it may either oscillate or swing around its points of<br />
suspension. If the initial push is just enough to bring it into a<br />
vertical position with zero velocity, the direction in which it<br />
will fall, and therefore the nature of its motion, are indeterminate.<br />
An infinitesimal perturbation would be enough to set it<br />
rotating or oscillating. (This problem of the "instability" of<br />
motion will be discussed fully in Chapter IX.)<br />
It is significant that Maxwell had already stressed the importance<br />
of these singular points. After describing the explosion<br />
of gun cotton, he goes on to say:<br />
In all such cases there is one common circumstancethe<br />
system has a quantity of potential energy, which is<br />
capable of being transformed into motion, but which cannot<br />
begin to be so transformed till the system has reached<br />
a certain configuration, to attain which requires an expenditure<br />
of work, which in certain cases may be infinitesimally<br />
small, and in general bears no definite proportion to the<br />
energy developed in consequence thereof. For example,<br />
the rock loosed by frost and balanced on a singular point<br />
of the mountain-side, the little spark which kindles the<br />
great forest, the little word which sets the world a fighting,<br />
the little scruple which prevents a man from doing his<br />
will, the little spore which blights all the potatoes, the<br />
little gemmule which makes us philosophers or idiots.<br />
Every existence above a certain rank has its singular<br />
points: the higher the rank, the more of them. At these<br />
points, influences whose physical magnitude is too small<br />
to be taken account of by a finite being, may produce<br />
results of the greatest importance. All great results produced<br />
by human endeavour depend on taking advantage<br />
of these singular states when they occur. I4
ORDER OUT OF CHAOS 74<br />
This conception received no further elaboration owing to the<br />
absence of suitable mathematical techniques for identifying<br />
systems containing such singular points and the absence of the<br />
chemical and biological knowledge that today affords, as we<br />
shall see later, a deeper insight into the truly essential role<br />
played by such singular points.<br />
Be that as it may, from the time of Leibniz' monads (see the<br />
conclusion to section 4) down to the present day (for example,<br />
the stationary states of the electrons in the Bohr model-see<br />
Chapter VII), integrable systems have been the model par excellence<br />
of dynamic systems, and physicists have attempted to<br />
extend the properties of what is actually a very special class of<br />
Hamiltonian equations to cover all natural processes. This is<br />
quite understandable. The class of integrable systems is the<br />
only one that, until recently, had been thoroughly explored.<br />
Moreover, there is the fascination always associated with a<br />
closed system capable of posing all problems, provided it does<br />
not define them as meaningless. Dynamics is such a language;<br />
being complete, it is by definition coextensive with the world<br />
it is describing. It assumes that all problems, whether simple<br />
or complex, resemble one another since it can always pose<br />
them in the same general form. Thus the temptation to conclude<br />
that all problems resemble one another from the point of<br />
view of their solutions as well, and that nothing new can appear<br />
as a result of the greater or lesser complexity of the integration<br />
procedure. It is this intrinsic homogeneity that we now<br />
know to be false. Moreover, the mechanical world view was<br />
acceptable as long as all observables referred in one way or<br />
another to motion. This is no longer the case. For example ,<br />
unstable particles have an energy that can be related to motion<br />
but that also has a lifetime that is a quite different type of observable,<br />
more closely related to irreversible processes, as we<br />
shall describe them in Chapters IV and V. The necessity of<br />
introducing new observables into the theoretical sciences was,<br />
and still is today, one of the driving forces that move us beyond<br />
the mechanical world view.
75 THE IDENTIFICATION OF THE REAL<br />
Laplaces Demon<br />
Extrapolations from the dynamic description discussed above<br />
have a symbol-the demon imagined by Laplace, capable at<br />
any given instant of observing the position and velocity of<br />
each mass that forms part of the universe and of inferring its<br />
evolution, both toward the past and toward the future. Of<br />
course, no one has ever dreamed that a physicist might one<br />
day benefit from the knowledge possessed by Laplace's demon.<br />
Laplace himself only used this fiction to demonstrate the<br />
extent of our ignorance and the need for a statistical description<br />
of certain processes. The problematics of Laplace's demon<br />
are not related to the question of whether a deterministic<br />
prediction of the course of events is actually possible, but<br />
whether it is possible in principle, de jure. This possibility<br />
seems to be implied in mechanistic description, with its<br />
characteristic duality based on dynamic law and initial conditions.<br />
Indeed, the fact that a dynamic system is governed by a<br />
deterministic law, even though in practice our ignorance of the<br />
initial state precludes any possibility of deterministic predictions,<br />
allows the "objective truth" of the system as it would be<br />
seen by Laplace's demon to be distinguished from empirical<br />
limitations due to our ignorance. In the context of classical<br />
dynamics, a deterministic description may be unattainable in<br />
practice; nevertheless, it stands as a limit that defines a series<br />
of increasingly accurate descriptions.<br />
It is precisely the consistency of this duality formed by dynamic<br />
law and initial conditions that is challenged in the revival<br />
of classical mechanics, which we will describe in Chapter<br />
IX. We shall see that the motion may become so complex, the<br />
trajectories so varied, that no observation, whatever its precision,<br />
can lead us to the determination of the exact initial conditions.<br />
But at that point the duality on which classical mechanics<br />
was constructed breaks down. We can predict only the average<br />
behavior of bundles of trajectories.<br />
Modern science was born out of the breakdown of the animistic<br />
alliance with nature. Man seemed to possess a place in<br />
the Aristotelian world as both a living and a knowing creature.
ORDER OUT OF CHAOS 76<br />
The world was made to his measure. The first experimental<br />
dialogue received part of its social and philosophic justification<br />
from another alliance, this time with the rational God of<br />
Christianity. To the extent to which dynamics has become and<br />
still is the model of science, certain implications of this historical<br />
situation have persisted to our day.<br />
Science is still the prophetic announcement of a description<br />
of the world seen from a divine or demonic point of view. It is<br />
the science of Newton, the new Moses to whom the truth of<br />
the world was unveiled; it is a revealed science that seems<br />
alien to any social and historical context identifying it as the<br />
result of the activity of human society. This type of inspired<br />
discourse is found throughout the history of physics. It has accompanied<br />
each conceptual innovation, each occasion at<br />
which physics seemed at the point of unification and the prudent<br />
mask of positivism was dropped. Each time physicists<br />
repeated what Ampere's son stated so explicitly: this worduniversal<br />
attraction, energy, field theory, or elementary particles-is<br />
the word of creation. Each time-in Laplace's time,<br />
at the end of the nineteenth century, or even today-physicists<br />
announced that physics was a closed book or about to become<br />
so. There was only one final stronghold where nature continued<br />
to resist, the fall of which would leave it defenseless,<br />
conquered, and subdued by our knowledge. They were thus<br />
unwittingly repeating the ritual of the ancient faith. They were<br />
announcing the coming of the new Moses, and with him a new<br />
Messianic period in science.<br />
Some might wish to disregard this prophetic claim, this<br />
somewhat naive enthusiasm, and it is certainly true that dialogue<br />
with nature has gone on all the same, together with a<br />
search for new theoretical languages, new questions, and new<br />
answers. But we do not accept a rigid separation between the<br />
scientist's "actual" work and the way he judges, interprets,<br />
and orientates this work. To accept it would be to reduce science<br />
to an ahistorical accumulation of results and to pay no<br />
attention to what scientists are looking for, the ideal knowledge<br />
they try to attain, the reasons why they occasionally<br />
quarrel or remain unable to communicate with each other. ts<br />
Once again, it was Einstein who formulated the enigma produced<br />
by the myth of modern science. He has stated that the<br />
miracle, the only truly astonishing feature, is that science ex-
77 THE IDENTIFICATION OF THE REAL<br />
ists at all, that we find a convergence between nature and the<br />
human mind. Similarly, when, at the end of the nineteenth<br />
century, du Bois Reymond made Laplace's demon the very<br />
incarnation of the logic of modern science, he added, "Ignoramus,<br />
ignorabimus": we shall always be totally ignorant of<br />
the relationship between the world of science and the mind<br />
which knows, perceives, and creates this science.I6<br />
Nature speaks with a thousand voices, and we have only begun<br />
to listen. Nevertheless, for nearly two centuries Laplace's<br />
demon has plagued our imagination, bringing a nightmare in<br />
which all things are insignificant. If it were really true that the<br />
world is such that a demon-a being that is, after all, like us,<br />
possessing the same science, but endowed with sharper<br />
senses and greater powers of calculation-could, starting from<br />
the observation of an instantaneous state, calculate its future<br />
and past, if nothing qualitatively differentiates the simple systems<br />
we can describe from the more complex ones for which a<br />
demon is needed, then the world is nothing but an immense<br />
tautology. This is the challenge of the science we have inherited<br />
from our predecessors, the spell we have to exorcise today.
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I
CHAPTER Ill<br />
THE TWO CULTURES<br />
Diderot and the Discourse of the Living<br />
In his interesting book on the history of the idea of progress,<br />
Nisbet writes:<br />
No single idea has been more important than, perhaps as<br />
important as, the idea of progress in Western civilization<br />
for nearly three thousand years.•<br />
There has been no stronger support for the idea of progress<br />
than the accumulation of knowledge. The grandiose spectacle<br />
of this gradual increase of knowledge is indeed a magnificent<br />
example of a successful collective human endeavor.<br />
Let us recall the remarkable discoveries achieved at the end<br />
of the eighteenth century and the beginning of the nineteenth<br />
century: the theories of heat, electricity, magnetism, and optics.<br />
It is not surprising that the idea of scientific progress,<br />
already clearly formulated in the eighteenth century, dominated<br />
the nineteenth. Still, as we have pointed out, the position<br />
of science in We stern culture remained unstable. This<br />
lends a dramatic aspect to the history of ideas from the high<br />
point of the Enlightenment.<br />
We have already stated the alternative: to accept science<br />
with what appears to be its alienating conclusions or to turn to<br />
an antiscientific metaphysics. We have also emphasized the<br />
solitude felt by modern men, the loneliness described by Pascal,<br />
Kierkegaard, or Monod. We have mentioned the antiscientific<br />
implications of Heidegger's metaphysics. Now we<br />
wish to discuss more fully some aspects of the intellectual history<br />
of the West, from Diderot, Kant, and Hegel to Whitehead<br />
and Bergson; all of them attempted to analyze and limit the<br />
scope of modern science as well as to open new perspectives<br />
79
ORDER OUT OF CHAOS<br />
80<br />
seen as radically alien to that science. Today it is usually<br />
agreed that those attempts have for the most part failed. Few<br />
would accept, for example, Kant's division of the world into<br />
phenomenal and noumenal spheres, or Bergson's "intuition"<br />
as an alternative path to a knowledge whose significance<br />
would parallel that of science. Still these attempts are part of<br />
our heritage. The history of ideas cannot be understood without<br />
reference to them.<br />
We shall also briefly discuss scientific positivism, which is<br />
based on the separation of what is true from what is scientifically<br />
useful. At the outset this positivistic view may seem to op<br />
pose clearly the metaphysical views we have mentioned, views<br />
that I. Berlin described as the "Counter-Enlightenment."<br />
However, their fundamental conclusion is the same: we must<br />
reject science as a basis for true knowledge even if at the same<br />
time we recognize its practical importance or we deny, as positivists<br />
do, the possibility of any other cognitive enterprise.<br />
We must remember all these developments to understand<br />
what is at stake To what extent is science a basis for the intelligibility<br />
of nature, including man? What is the meaning of the<br />
idea of progress today?<br />
Diderot, one of the towering figures of the Enlightenment, is<br />
certainly no representative of antiscientific thought. On the<br />
contrary, his confidence in science, in the possibilities of<br />
knowledge, was total. Yet this is the very reason why science<br />
had, following Diderot, to understand life before it could hope<br />
to achieve any coherent vision of nature.<br />
We have already mentioned that the birth of modern science<br />
was marked by the abandonment of vitalist inspiration and, in<br />
particular, of Aristotelian final causes. However, the issue of<br />
the <strong>org</strong>anization of living matter remained and became a challenge<br />
for classical science. Diderot, at the height of the Newtonian<br />
triumph, emphasizes that this problem was repressed by<br />
physics. He imagines it as haunting the dreams of physicists<br />
who cannot conceive of it while they are awake. The physicist<br />
d /\lembert is dreaming:<br />
· living point . .. No, that's wrong. Nothing at all to<br />
begin with, and then a living point. This living point is<br />
joined by another, and then another, and from these successive<br />
joinings there results a unified being, for I am a
81 THE TWO CULTURES<br />
unity, of that I am certain. . . . (As he said this he felt<br />
himself all over.) But how did this unity come about?"<br />
"Now listen, Mr. Philosopher, I can understand an aggregate,<br />
a tissue of tiny sensitive bodies, but an animal! ...<br />
A whole, a system that is a unit, an individual conscious<br />
of its own unity! I can't see it, no, I can't see it. "2<br />
In an imaginary conversation with dj\lembert, Oiderot speaks<br />
in the first person, demonstrating the inadequacy of mechanistic<br />
explanation:<br />
Look at this egg: with it you can overthrow all the schools<br />
of theology and all the churches in the world. What is this<br />
egg? An insensitive mass before the germ is put into<br />
it ... How does this mass evolve into a new <strong>org</strong>anization,<br />
into sensitivity, into life? Through heat. What will<br />
generate heat in it? Motion. What will the successive<br />
effects of motion be? Instead of answering me, sit down<br />
and let us follow out these effects with our eyes from one<br />
moment to the next. First there is a speck which moves<br />
about, a thread growing and taking colour, flesh being<br />
formed, a beak, wing-tips, eyes, feet coming into view, a<br />
yellowish substance which unwinds and turns into intestines-and<br />
you have a living creature .... Now the wall<br />
is breached and the bird emerges, walks, flies, feels pain,<br />
runs away, comes back again, complains, suffers, loves,<br />
desires, enjoys, it experiences all your affections and<br />
does all the things you do. And will you maintain, with<br />
Descartes, that it is an imitating machine pure and simple?<br />
Why, even little children will laugh at you, and philosophers<br />
will answer that if it is a machine you are one<br />
too! If, however, you admit that the only difference between<br />
you and an animal is one of <strong>org</strong>anization, you will<br />
be showing sense and reason and be acting in good faith;<br />
but then it will be concluded, contrary to what you had<br />
said, that from an inert substance arranged in a certain<br />
way and impregnated by another inert substance, subjected<br />
to heat and motion, you will get sensitivity, life,<br />
memory, consciousness, passions, thought ... Just listen<br />
to your own arguments and you will feel how pitiful
ORDER OUT OF CHAOS 82<br />
they are. You will come to feel that by refusing to entertain<br />
a simple hypothesis that explains everything-sensitivity<br />
as a property common to all matter or as a result<br />
of the <strong>org</strong>anization of matter-you are flying in the face of<br />
common sense and plunging into a chasm of mysteries,<br />
contradictions and absurdities.3<br />
In opposition to rational mechanics, to the claim that material<br />
nature is nothing but inert mass and motion, Diderot appeals<br />
to one of physics' most ancient sources of inspiration,<br />
namely, the growth, differentiation, and <strong>org</strong>anization of the<br />
embryo. Flesh forms, and so does the beak, the eyes, and the<br />
intestines; a gradual <strong>org</strong>anization occurs in biological "space,"<br />
out of an apparently homogeneous environment differentiated<br />
forms appear at exactly the right time and place through the<br />
effects of complex and coordinated processes.<br />
How can an inert mass, even a Newtonian mass animated<br />
by the forces of gravitational interaction, be the starting point<br />
for <strong>org</strong>anized active local structures? We have seen that the<br />
Newtonian system is a world system: no local configuration of<br />
·bodies can claim a particular identity; none is more than a<br />
contingent proximity between bodies connected by general relations.<br />
But Diderot does not despair. Science is only beginning; rational<br />
mechanics is merely a first, overly abstract attempt. The<br />
spectacle of the embryo is enough to refute its claims to universality.<br />
This is why Diderot compares the work of great<br />
"mathematicians" such as Euler, Bernoulli, and di\lembert to<br />
the pyramids of the Egyptians, awe-inspiring witnesses to the<br />
genius of their builders, now lifeless ruins, alone and forlorn.<br />
True science, alive and fruitful, will be carried on elsewhere."<br />
Moreover, it seems to him that this new science of <strong>org</strong>anized<br />
living matter has already begun. His friend d'Holbach is busy<br />
studying chemistry, Diderot himself has chosen medicine. The<br />
problem in chemistry as well as in medicine is to replace inert<br />
matter with active matter capable of <strong>org</strong>anizing itself and producing<br />
living beings. Diderot claims that matter has to be sensitive.<br />
Even a stone has sensation in the sense that the<br />
molecules of which it is composed actively seek certain combinations<br />
rather than others and thus are governed their likes<br />
and dislikes. The sensitivity of the whole <strong>org</strong>anism is then
83<br />
THE TWO CULTURES<br />
simply the sum of that of its parts, just as a swarm of bees with<br />
its globally coherent behavior is the result of interactions between<br />
one bee and another; and, Diderot thereby concludes,<br />
the human soul does not exist any more than the soul of the<br />
beehive does. s<br />
Diderot's vitalist protest against physics and the universal<br />
laws of motion thus stems from his rejection of any form of<br />
spiritualist dualism. Nature must be described in such a way<br />
that man's very existence becomes understandable. Otherwise,<br />
and this is what happens in the mechanistic world view,<br />
the scientific description of nature will have its counterpart in<br />
man as an automaton endowed with a soul and thereby alien to<br />
nature.<br />
The twofold basis of materialistic naturalism, at once chemical<br />
and medical, that Diderot employed to counter the physics<br />
of his time is recurrent in the eighteenth century. While biologists<br />
speculated about the animal-machine, the preexistence of<br />
germs, and the chain of living creatures-all problems close to<br />
theology6-chemists and physicians had to face directly the<br />
complexity of real processes in both chemistry and life. Chemistry<br />
and medicine were, in the late eighteenth century, privileged<br />
sciences for those who fought against the physicists'<br />
esprit de systeme in favor of a science that would take into<br />
account the diversity of natural processes. A physicist could<br />
be pure esprit, a precocious child, but a physician or a chemist<br />
must be a man of experience: he must be able to decipher the<br />
signs, to spot the clues. In this sense, chemistry and medicine<br />
are arts. They demand judgment, application, and tenacious<br />
observation. Chemistry is a madman's passion, Venel concluded<br />
in the article he wrote for Diderot's Encyclopedie, an<br />
eloquent defense of chemistry against the abstract imperialism<br />
of the Newtonians.7 To emphasize the fact that protests raised<br />
by chemists and physicians against the way physicists reduced<br />
living processes to peaceful mechanisms and the quiet unfolding<br />
of universal laws were common in Diderot's day, we invoke<br />
the eminent figure of Stahl, the father of vitalism and inventor<br />
of the first consistent chemical systematics.<br />
According to Stahl, universal laws apply to the living only in<br />
the sense that these laws condemn them to death and corruption;<br />
the matter of which living beings are composed is so frail,<br />
so easily decomposed, that if it were governed solely by the
ORDER OUT OF CHAOS<br />
84<br />
common laws of matter, it would not withstand decay or dissolution<br />
for a moment. If a living creature is to survive in spite<br />
of the general laws of physics, however short its life when it is<br />
compared to that of a stone or another inanimate object, it has<br />
to possess in itself a "principle of conservation" that maintains<br />
the harmonious equilibrium of the texture and structure<br />
of its body. The astonishing longevity of a living body in view<br />
of the extreme corruptibility of its constitutive matter is thus<br />
indicative of the action of a "natural, permanent, immanent<br />
principle," of a particular cause that is alien to the laws of<br />
inanimate matter and that constantly struggles against the constantly<br />
active corruption whose inevitability these laws imply. s<br />
To us this analysis of life sounds both near and remote. It is<br />
close to us in its acute awareness of the singularity and the precariousness<br />
of life. It is remote because, like Aristotle, Stahl<br />
defined life in static terms, in terms of conservation, not of<br />
becoming or evolution. Still, the terminology used by Stahl<br />
can be found in recent biological literature, for example,<br />
where we read that enzymes "combat" decay and allow the<br />
body to ward off the death to which it is inexorably doomed by<br />
physics. Here also, biological <strong>org</strong>anization defies the laws of<br />
nature, and the only "normal" trend is that which leads to<br />
death (see Chapter V).<br />
Indeed, Stahl's vitalism is relevant as long as the laws of<br />
physics are identified with evolution toward decay and dis<strong>org</strong>anization.<br />
Today the "vitalist principle" has been superseded<br />
by the succession of improbable mutations preserved in the<br />
genetic message "governing" the living structure. Nonetheless,<br />
some extrapolations starting from molecular biology relegate<br />
life to the confines of nature-that is, conclude life is<br />
compatible with the basic laws of physics but purely contingent.<br />
This was explicitly stated by Monod: life does not "follow<br />
from the laws of physics, it is compatible with them. Life<br />
is an event whose singularity we have to recognize."<br />
But the transition from matter to life can also be viewed in a<br />
different way. As we shall see, far from equilibrium, new self<strong>org</strong>anizational<br />
processes arise. (These questions will be studied<br />
in detail in Chapters V and VI.) In this way biological<br />
<strong>org</strong>anization begins to appear as a natural process.<br />
However, long before these recent developments, the problematics<br />
of life had been transformed. In a politically trans-
85 THE TWO CULTURES<br />
formed Europe the intellectual landscape was remodeled as<br />
the Romantic movement, closely linked with the Counter<br />
Enlightenment, shows.<br />
Stahl criticized the metaphor of the automaton because, unlike<br />
a living being, the purpose of an automaton does not lie<br />
within itself; its <strong>org</strong>anization is imposed upon it by its maker.<br />
Diderot, far from situating the study of life outside the reach of<br />
science, saw it as representing the future of a science he considered<br />
to be still in its infancy. A few years later, such points<br />
of view were to be challenged.9 Mechanical change, activity as<br />
described by the laws of motion, had now become synonymous<br />
with the artificial and with death. Opposed to it,<br />
united in a complex with which we are now quite familiar, were<br />
the concepts of life, spontaneity, freedom, and spirit. This opposition<br />
was paralleled by the opposition between calculation<br />
and manipulation on the one hand, and the free speculative<br />
activity of the mind on the other. Through speculation the philosopher<br />
would reach the spiritual activity at the core of nature.<br />
As for the scientist, his concern with nature would be<br />
reduced to taking it as a set of manipulable and measurable<br />
objects; he would thus be able to take possession of nature, to<br />
dominate and control it but not understand it. Thus the intelligibility<br />
of nature would lie beyond the grasp of science.<br />
We are not concerned here with the history of philosophy<br />
but merely with emphasizing the extent to which the philosophical<br />
criticism of science had at this time become harsher,<br />
resembling certain modern forms of antiscience. It was no<br />
longer a question of refuting rather naive and shortsighted<br />
generalizations that only have to be repeated aloud-to use<br />
Diderot's language-to make even children laugh, but of refuting<br />
the type of approach that produced experimental and<br />
mathematical knowledge of nature. Scientific knowledge is not<br />
being criticized for its limitations but for its nature, and a rival<br />
knowledge, based on another approach, is being announced.<br />
Knowledge is fragmented into two opposed modes of inquiry.<br />
From a philosophical point of view, the transition from Diderot<br />
to the Romantics and, more precisely, from one of these<br />
two types of critical attitudes toward science to the other, can<br />
be found in Kant's transcendental philosophy, the essential<br />
point being that the Kantian critique identified science in general<br />
with its Newtonian realization. It thereby branded as im-
ORDER OUT OF CHAOS 86<br />
possible any opposition to classical science that was not an<br />
opposition to science itself. Any criticism against Newtonian<br />
physics must then be seen as aimed at downgrading the rational<br />
understanding of nature in favor of a different form of<br />
knowledge. Kant's approach had immense repercussions,<br />
which continue down to our day. Let us therefore summarize<br />
his point of view as presented in Critique of Pure Reason,<br />
which, in opposition to the progressist views of the Enlightenment,<br />
presents the closed and limiting conception of science<br />
we have just defined.<br />
Kants Critical Ratification<br />
How to restore order in the intellectual landscape left in disarray<br />
with the disappearance of God conceived as the rational<br />
principle that links science and nature? How could scientists<br />
ever have access to global truth when it could no longer be<br />
asserted, except metaphorically, that science deciphers the<br />
word of creation? God was now silent or at least no longer<br />
spoke the same language as human reason. Moreover, in a nature<br />
from which time was eliminated, what remained of our<br />
subjective experience? What was the meaning of freedom,<br />
destiny, or ethical values?<br />
Kant argued that there were two levels of reality: a phenomenal<br />
level that corresponds to science, and a noumenal level<br />
corresponding to ethics. The phenomenal order is created by<br />
the human mind. The noumenal level transcends man's intellect;<br />
it corresponds to a spiritual reality that supports his ethical<br />
and religious life. In a way, Kant's solution is the only one<br />
possible for those who assert both the reality of ethics and the<br />
reality of the objective world as it is expressed by classical<br />
science. Instead of God, it is now man himself who is the<br />
source of the order he perceives in nature. Kant justifies both<br />
scientific knowledge and man's alienation from the phenomenal<br />
world described by science. From this perspective we can<br />
see that Kantian philosophy explicitly spells out the philosophical<br />
content of classical science.<br />
Kant defines the subject of critical philosophy as transcendental.<br />
It is not concerned with the objects of experience but<br />
is based on the a priori fact that a systematic knowledge of
87 THE TWO CULTURES<br />
these objects is possible (this is for him proved by the existence<br />
of physics), going on to state the a priori conditions of<br />
possibility for this mode of knowledge.<br />
To do so a distinction must be made between the direct sensations<br />
we receive from the outside world and the objective,<br />
"rational" mode of knowledge. Objective knowledge is not<br />
passive; it forms its objects. When we take a phenomenon as<br />
the object of experience, we assume a priori before we actually<br />
experience it that it obeys a given set of principles. Insofar as it<br />
is perceived as a possible object of knowledge, it is the product<br />
of our mind's synthetic activity. We find ourselves in the<br />
objects of our knowledge, and the scientist himself is thus the<br />
source of the universal laws he discovers in nature.<br />
The a priori conditions of experience are also the conditions<br />
for the existence of the objects of experience. This celebrated<br />
statement sums up the "Copernican revolution" achieved by<br />
Kant's "transcendental" inquiry. The subject no longer "revolves"<br />
around its object, seeking to discover the laws by<br />
which it is governed or the language by which it may be deciphered.<br />
Now the subject itself is at the center, imposing its<br />
laws, and the world perceived speaks the language of that subject.<br />
No wonder, then, that Newtonian science is able to describe<br />
the world from an external, almost divine point of view!<br />
That all perceived phenomena are governed by the laws of<br />
our mind does not mean that a concrete knowledge of these<br />
objects is useless. According to Kant, science does not engage<br />
in a dialogue with nature but imposes its own language upon it.<br />
Still it must discover, in each case, the specific message expressed<br />
in this general language. A knowledge of the a priori<br />
concepts alone is vain and empty.<br />
From the Kantian point of view Laplace's demon, the symbol<br />
of the scientific myth, is an illusion, but it is a rational<br />
illusion. Although it is the result of a limiting process and, as<br />
such, illegitimate, it is still the expression of a legitimate conviction<br />
that is the driving force of science-the conviction<br />
that, in its entirety, nature is rightfully subjected to the laws<br />
that scientists succeed in deciphering. Wherever it goes, whatever<br />
it questions, science will always obtain, if not the same<br />
answer, at least the same kind of answer. There exists a single<br />
universal syntax that includes all possible answers.<br />
Transcendental philosophy thus ratified the physicist's
ORDER OUT OF CHAOS 88<br />
claim to have found the definitive form of all positive knowledge.<br />
At the same time, however, it secured for philosophy a<br />
dominant position in. respect to science. It was no longer necessary<br />
to look for the philosophic significance of the results of<br />
scientific activity. From the transcendental standpoint, those<br />
results cannot lead to anything really new. It is science, not its<br />
results, that is the subject of philosophy; science taken as a<br />
repetitive and closed enterprise provides a stable foundation<br />
for transcendental reflection.<br />
Therefore, while it ratifies all the claims of science, Kant's<br />
critical philosophy actually limits scientific activity to problems<br />
that can be considered both easy and futile. It condemns<br />
science to the tedious task of deciphering the monotonous language<br />
of phenomena while keeping for itself questions of human<br />
"destiny": what man may know, what he must do, what<br />
he may hope for. The world studied by science, the world accessible<br />
to positive knowledge is "only" the world of phenomena.<br />
Not only is the scientist unable to know things in themselves,<br />
but even the questions he asks are irrelevant to the real problems<br />
of mankind. Beauty, freedom, and ethics cannot be objects<br />
of positive knowledge. They belong to the noumenal<br />
world, which is the domain of philosophy, and they are quite<br />
unrelated to the phenomenal world.<br />
We can accept Kant's starting point, his emphasis on the<br />
active role man plays in scientific description. Much has already<br />
been said about experimentation as the art of choosing<br />
situations that are hypothetically governed by the law under<br />
investigation and staging them to give clear, experimental answers.<br />
For each experiment certain principles are presupposed<br />
and thus cannot be established by that experiment.<br />
However, as we have seen, Kant goes much further. He denies<br />
the diversity of possible scientific points of view, the diversity<br />
of presupposed principles. In agreement with the myth of classical<br />
science, Kant is after the unique language that science<br />
deciphers in nature, the unique set of a priori principles on<br />
which physics is based and that are thus to be identified with<br />
the categories of human understanding. Thus Kant denies the<br />
need for the scientist's active choice, the need for a selection<br />
of a problematic situation corresponding to a particular theoretical<br />
language in which definite questions may be asked and<br />
experimental answers sought.
89 THE TWO CULTURES<br />
Kant's critical ratification defines scientific endeavor as<br />
silent and systematic, closed within itself. By so doing, philosophy<br />
endorses and perpetuates the rift, debasing and surrendering<br />
the whole field of positive knowledge to science<br />
while retaining for itself the field of freedom and ethics, conceived<br />
as alien to nature.<br />
A Prlilosophy of Nature? Hegel and Bergson<br />
The Kantian truce between science and philosophy was a fragile<br />
one. Post-Kantian philosophers disrupted this truce in favor of<br />
a new philosophy of science, presupposing a new path to knowledge<br />
that was distinct from science and actually hostile to it.<br />
Speculation released from the constraints of any experimental<br />
dialogue reigned supreme, with disastrous consequences for<br />
the dialogue between scientists and philosophers. For most<br />
scientists, the philosophy of nature became synonymous with<br />
arrogant, absurd speculation riding roughshod over facts, and<br />
indeed regularly proven wrong by the facts. On the other side,<br />
for most philosophers it has become a symbol of the dangers<br />
involved in dealing with nature and in competing with science.<br />
The rift among science, philosophy, and humanistic studies<br />
was thus made greater by mutual disdain and fear.<br />
As an example of this speculative approach to nature, let us<br />
first consider Hegel. Hegel's philosophy has cosmic dimensions.<br />
In his system increasing levels of complexity are specified,<br />
and nature's purpose is the eventual self-realization of its<br />
spiritual element. Nature's history is fulfilled with the appearance<br />
of man-that is, with the coming of Spirit apprehending<br />
itself.<br />
The Hegelian philosophy of nature systematically incorporates<br />
all that is denied by Newtonian science. In particular, it<br />
rests on the qualitative difference between the simple behavior<br />
described by mechanics and the behavior of more complex entities<br />
such as living beings. It denies the possibility of reducing<br />
those levels, rejecting the idea that differences are merely apparent<br />
and that nature is basically homogeneous and simple. It<br />
affirms the existence of a hierarchy, each level of which presupposes<br />
the preceding ones.
ORDER OUT OF CHAOS<br />
90<br />
Unlike the Newtonian authors of romans de Ia matiere, of<br />
world-embracing panoramas ranging from gravitational interactions<br />
to human passions, Hegel knew perfectly well that his<br />
distinctions among levels (which, quite apart from his own interpretation,<br />
we may acknowledge as corresponding to the<br />
idea of an increasing complexity in nature and to a concept of<br />
time whose significance would be richer on each new level)<br />
ran counter to his day's mathematical science of nature. He<br />
therefore set out to limit the significance of this science, to<br />
show that mathematical description is restricted to the most<br />
trivial situations. Mechanics can be mathematized because it<br />
attributes only space-time properties to matter. · brick does<br />
not kill a man merely because it is a brick, but solely because<br />
of its acquired velocity; this means that the man is killed tzy<br />
space and time." IO The man is killed by what we call kinetic<br />
energy (mv2/2)-by an abstract quantity defining mass and velocity<br />
as interchangeable; the same murderous effect can be<br />
achieved by reducing one and increasing the other.<br />
It is precisely this interchangeability that Hegel sets as a<br />
condition for mathematization that is no longer satisfied when<br />
the mechanical level of description is abandoned for a "higher"<br />
one involving a larger spectrum of physical properties.<br />
In a sense Hegel's system provides a consistent philosophic<br />
response to the crucial problems of time and complexity.<br />
However, for generations of scientists it represented the epitome<br />
of abhorrence and contempt. In a few years, the intrinsic<br />
difficulties of Hegel's philosophy of nature were aggravated by<br />
the obsolescence of the scientific background on which his<br />
system was based, for Hegel, of course, based his rejection of<br />
the Newtonian system on the scientific conceptions of his<br />
time.11 And it was precisely those conceptions that were to fall<br />
into oblivion with astonishing speed. It is difficult to imagine a<br />
less opportune time than the beginning of the nineteenth century<br />
for seeking experimental and theoretical support for an<br />
alternative to classical science. Although this time was characterized<br />
by a remarkable extension of the experimental scope of<br />
science (see Chapter IV) and by a proliferation of theories that<br />
seemed to contradict Newtonian science, most of those theories<br />
had to be given up only a few years after their appearance.<br />
At the end of the nineteenth c:entury, when Bergson under-
91 THE TWO CULTURES<br />
took his search for an acceptable alternative to the science of<br />
his time, he turned to intuition as a form of speculative knowledge,<br />
but he presented it as quite different from that of the<br />
Romantics. He explicitly stated that intuition is unable to produce<br />
a system but produces only results that are always partial<br />
and nongeneralizable, results to be formulated with great caution.<br />
In contrast, generalization is an attribute of "intelligence,"<br />
the greatest achievement of which is classical<br />
science. Bergsonian intuition is a concentrated attention, an<br />
increasingly difficult attempt to penetrate deeper into the singularity<br />
of things. Of course, to communicate, intuition must<br />
have recourse to language-"in order to be transmitted, it will<br />
have to use ideas as a conveyance." 12 This it does with infinite<br />
patience and circumspection, at the same time accumulating<br />
images and comparisons in order to "embrace reality," 13 thus<br />
suggesting in an increasingly precise way what cannot be communicated<br />
by means of general terms and abstract ideas.<br />
Science and intuitive metaphysics "are or can become<br />
equally precise and definite. They both bear upon reality itself.<br />
But each one of them retains only half of it so that one<br />
could see in them, if one wished, two subdivisions of science<br />
or two departments of metaphysics, if they did not mark divergent<br />
directions of the activity of thought." 14<br />
The definition of these two divergent directions may also be<br />
considered as the historical consequence of scientific evolution.<br />
For Bergson, it is no longer a question of finding scientific<br />
alternatives to the physics of his time. In his view,<br />
chemistry and biology had definitely chosen mechanics as<br />
their model. The hopes that Diderot had cherished for the future<br />
of chemistry and medicine had thus been dashed. In<br />
Bergson's view, science is a whole and must therefore be<br />
judged as a whole. And this is what he does when he presents<br />
science as the product of a practical intelligence whose aim is<br />
to dominate matter and that develops by abstraction and generalization<br />
the intellectual categories needed to achieve this<br />
domination. Science is the product of our vital need to exploit<br />
the world, and its concepts are determined by the necessity of<br />
manipulating objects, of making predictions, and of achieving<br />
reproducible actions. This is why rational mechanics represents<br />
the very essence of science, its actual embodiment. The
ORDER OUT OF CHAOS 92<br />
other sciences are more vague, awkward manifestations of an<br />
approach that is all the more successful the more inert and<br />
dis<strong>org</strong>anized the terrain it explores.<br />
For Bergson all the limitations of scientific rationality can<br />
be reduced to a single and decisive one: it is incapable of understanding<br />
duration since it reduces time to a sequence of<br />
instantaneous states linked by a deterministic law.<br />
"Time is invention, or it is nothing at all." 15 Nature is<br />
change, the continual elaboration of the new, a totality being<br />
created in an essentially open process of development without<br />
any preestablished model. "Life progresses and endures in<br />
time." 16 The only part of this progression that intelligence can<br />
grasp is what it succeeds in fixing in the form of manipulable<br />
and calculable elements and in referring to a time seen as<br />
sheer juxtaposition of instants.<br />
Therefore, physics "is limited to coupling simultaneities<br />
between the events that make up this time and the positions<br />
of the mobile T on its trajectory. It detaches these<br />
events from the whole, which at every moment puts on a<br />
new form and which communicates to them something of<br />
its novelty. It considers them in the abstract, such as they<br />
would be outside of the living whole, that is to say, in a<br />
time unrolled in space. It retains only the events or systems<br />
of events that can be thus isolated without being<br />
made to undergo too profound a deformation, because<br />
only these lend themselves to the application of its<br />
method. Our physics dates from the day when it was<br />
known how to isolate such systems. " 1 7<br />
When it comes to understanding duration itself, science is<br />
powerless. What is needed is intuition, a "direct vision of the<br />
mind by the mind. "18 "Pure change, real duration, is something<br />
spiritual. Intuition is what attains the spirit, duration,<br />
pure change.I9<br />
Can we say Bergson has failed in the same way that the<br />
post-Kantian philosophy of nature failed? He has failed insofar<br />
as the metaphysics based on intuition he wished to create<br />
has not materialized. He has not failed in that, unlike Hegel,<br />
he had the good fortune to pass judgment upon science that<br />
was, on the whole, firmly established-that is, classical sci-
93 THE TWO CULTURES<br />
ence at its apotheosis, and thus identified problems which are<br />
indeed still our problems. But, like the post-Kantian critics,<br />
he identified the science of his time with science in general.<br />
He thus attributed to science de jure limitations that were only<br />
de facto. As a consequence he tried to define once and for all a<br />
statu quo for the respective domains of science and other intellectual<br />
activities. Thus the only perspective remaining open<br />
for him was to introduce a way in which antagonistic approaches<br />
could at best merely coexist.<br />
In conclusion, even if the way in which Bergson sums up the<br />
achievement of classical science is still to some extent acceptable,<br />
we can no longer accept it as a statement of the eternal<br />
limits of the scientific enterprise. We conceive of it more as a<br />
program that is beginning to be implemented by the metamorphosis<br />
science is now undergoing. In particular, we know<br />
that time linked with motion does not exhaust the meaning of<br />
time in physics. Thus the limitations Bergson criticized are<br />
beginning to be overcome, not by abandoning the scientific<br />
approach or abstract thinking but by perceiving the limitations<br />
of the concepts of classical dynamics and by discovering new<br />
formulations valid in more general situations.<br />
Process and Reality: Whitehead<br />
As we have emphasized, the element common to Kant, Hegel,<br />
and Bergson is the search for an approach to reality that is<br />
different from the approach of classical science. This is also<br />
the fundamental aim of Whitehead's philosophy, which is resolutely<br />
pre-Kantian. In his most important book, Process and<br />
Reality, he puts us back in touch with the great philosophies of<br />
the Classical Age and their quest for rigorous conceptual experimentation.<br />
Whitehead sought to understand human experience as a pro<br />
cess belonging to nature, as physical existence. This challenge<br />
led him, on the one hand, to reject the philosophic tradition<br />
that defined subjective experience in terms of consciousness,<br />
thought, and sense perception, and, on the other, to conceive<br />
of all physical existence in terms of enjoyment, feeling, urge,<br />
appetite, and yearning-that is, to cross swords with what he
ORDER OUT OF CHAOS 94<br />
calls "scientific materialism," born in the seventeenth century.<br />
Like Bergson, Whitehead was thus led to point out the<br />
basic inadequacies of the theoretical scheme developed by<br />
seventeenth-century science:<br />
The seventeenth century had finally produced a scheme<br />
of scientific thought framed by mathematicians, for the<br />
use of mathematicians. The great characteristic of the<br />
mathematical mind is its capacity for dealing with abstractions;<br />
and for eliciting from them clear-cut demonstrative<br />
trains of reasoning, entirely satisfactory so long<br />
as it is those abstractions which you want to think about.<br />
The enormous success of the scientific abstractions,<br />
yielding on the one hand matter with its simple location<br />
in space and time, on the other hand mind, perceiving,<br />
suffering, reasoning, but not interfering, has foisted on to<br />
philosophy the task of accepting them as the most concrete<br />
rendering of fact.<br />
Thereby, modern philosophy has been ruined. It has<br />
oscillated in a complex manner between three extremes.<br />
There are the dualists, who accept matter and mind as on<br />
equal basis, and the two varieties of monists, those who<br />
put mind inside matter, and those who put matter inside<br />
mind. But this juggling with abstractions can never overcome<br />
the inherent confusion introduced by the ascription<br />
of misplaced concreteness to the scientific scheme of the<br />
seventeenth century. 20<br />
However, Whitehead considered this to be only a temporary<br />
situation. Science is not doomed to remain a prisoner of confusion.<br />
We have already raised the question of whether it is possible<br />
to formulate a philosophy of nature that is not directed against<br />
science. Whitehead's cosmology is the most ambitious attempt<br />
to do so. Whitehead saw no basic contradiction between<br />
science and philosophy. His purpose was to define the<br />
conceptual field within which the problem of human experience<br />
and physical processes could be dealt with consistently<br />
and to determine the conditions under which the problem<br />
could be solved. What had to be done was to formulate: the:<br />
principles necessary to characterize all forms of existence,
95<br />
THE TWO CULTURES<br />
from that of stones to that of man. It is precisely this universality<br />
that, in Whitehead's opinion, defines his enterprise as<br />
"philosophy." While each scientific theory selects and abstracts<br />
from the world's complexity a peculiar set of relations,<br />
philosophy cannot favor any particular region of human experience.<br />
Through conceptual experimentation it must construct<br />
a consistency that can accommodate all dimensions of experience,<br />
whether they belong to physics, physiology, psychology,<br />
biology, ethics, etc.<br />
Whitehead understood perhaps more sharply than anyone<br />
else that the creative evolution of nature could never be conceived<br />
if the elements composing it were defined as permanent,<br />
individual entities that maintained their identity throughout all<br />
changes and interactions. But he also understood that to make<br />
all permanence illusory, to deny being in the name of becoming,<br />
to reject entities in favor of a continuous and ever-changing<br />
flux meant falling once again into the trap always lying in wait<br />
for philosophy-to "indulge in brilliant feats of explaining<br />
away." 21<br />
Thus for Whitehead the task of philosophy was to reconcile<br />
permanence and change, to conceive of things as processes, to<br />
demonstrate that becoming forms entities, individual identities<br />
that are born and die. It is beyond the scope of this book to<br />
give a detailed presentation of Whitehead's system. Let us<br />
only emphasize that he demonstrated the connection between<br />
a philosophy of relation-no element of nature is a permanent<br />
support for changing relations; each receives its identity from<br />
its relations with others-and a philosophy of innovating becoming.<br />
In the process of its genesis, each existent unifies the<br />
multiplicity of the world, since it adds to this multiplicity an<br />
extra set of relations. At the creation of each new entity "the<br />
many become one and are increased by one. "22<br />
In the conclusion of this book, we shall again encounter<br />
Whitehead's question of permanence and change, this time as<br />
it is raised in physics; we shall speak of entities formed by<br />
their irreversible interaction with the world. Today physics has<br />
discovered the need to assert both the distinction and interdependence<br />
between units and relations. It now recognizes that,<br />
for an interaction to be real, the "nature" of the related things<br />
must derive from these relations, while at the same time the relations<br />
must derive from the "nature" of the things (see Chap-
ORDER OUI OF CHAOS<br />
96<br />
ter X). This is the forerunner of "self-consistent" descriptions<br />
as expressed, for instance, by the "bootstrap" philosophy in<br />
elementary-particle physics, which asserts the universal connectedness<br />
of all particles. However, when Whitehead wrote<br />
Process and Reality, the situation of physics was quite different,<br />
and Whitehead's philosophy found an echo only in biology.2<br />
3<br />
Whitehead's case as well as Bergson's convince us that only<br />
an opening, a widening of science can end the dichotomy between<br />
science and philosophy. This widening of science is possible<br />
only if we revise our conception of time. To deny timethat<br />
is, to reduce it to a mere deployment of a reversible lawis<br />
to abandon the possibility of defining a conception of nature<br />
coherent with the hypothesis that nature produced living<br />
beings, particularly man. It dooms us to choosing between an<br />
antiscientific philosophy and an alienating science.<br />
"Ignoramus, lgnoramibus": The Positivists Strain<br />
Another method of overcoming the difficulties of classical rationality<br />
implied in classical science was to separate what was<br />
scientifically most fruitful from what is "true." This is another<br />
form of the Kantian cleavage. In his 1865 address "On the Goal<br />
of the Natural Sciences," Kirchoff stated that the ultimate<br />
goal of science is to reduce every phenomenon to motion, mo<br />
tion that in turn is described by theoretical mechanics. A similar<br />
statement was made by Helmholtz, a chemist, physician,<br />
physicist, and physiologist who dominated the German universities<br />
at the time when they were becoming the hub of European<br />
science. He stated: "the phenomena of nature are to be<br />
referred back to motions of material particles possessing unchangeable<br />
moving forces, which are dependent upon conditions<br />
of space alone. "2 4<br />
The aim of the natural sciences, therefore, was to reduce all<br />
observations to the laws formulated by Newton and extended<br />
by such illustrious physicists and mathematicians as Lagrange,<br />
Hamilton, and others. We were not to ask why these forces<br />
exist and enter Newton's equation. In any case, we could not<br />
"understand" matter or forces even if we used these concepts
97 THE TWO CULTURES<br />
to formulate the laws of dynamics. The why, the basic nature<br />
of forces and masses, remains hidden from us. Du Bois Reymond,<br />
as we already mentioned, expressed concisely the<br />
limitations of our knowledge: "Ignoramus, ignoramibus." Science<br />
provides no access to the mysteries of the universe. What<br />
then is science?<br />
We have already referred to Mach's influential view: Science<br />
is part of the Darwinian struggle for life. It helps us to <strong>org</strong>anize<br />
our experience. It leads to an economy of thought. Mathematical<br />
laws are nothing more than conventions useful for summarizing<br />
the results of possible experiments. At the end of the<br />
nineteenth century, scientific positivism exercised a great intellectual<br />
appeal. In France it influenced the work of eminent<br />
thinkers such as Duhem and Poincare.<br />
One more step in the elimination of "contemptible metaphysics"<br />
and we come to the Vienna school. Here science is<br />
granted jurisdiction over all positive knowledge and philosophy<br />
needed to keep this positive knowledge in .order. This<br />
meant a radical submission of all rational knowledge and questions<br />
to science. When Reichenbach, a distinguished neopositivist<br />
philosopher, wrote a book on the "direction of<br />
time," he stated:<br />
There is no other way to solve the problem of time than<br />
the way through physics. More than any other science,<br />
physics has been concerned with the nature of time. If<br />
time is objective the physicist must have discovered the<br />
fact. If there is Becoming, the physicist must know it; but<br />
if time is merely subjective and Being is timeless, the<br />
physicist must have been able to ignore time in his construction<br />
of reality and describe the world without the<br />
help of time .... It is a hopeless enterprise to search for<br />
the nature of time without studying physics. If there is a<br />
solution to the philosophical problem of time, it is written<br />
down in the equations of mathematical physics.25<br />
Reichenbach's work is of great interest to anyone wishing to<br />
see what physics has to say on the subject of time, but it is not<br />
so much a book on the philosophy of nature as an account of<br />
the way in which the problem of time challenges scientists, not<br />
philosophers.
ORDER OUT OF CHAOS 98<br />
What then is the role of philosophy? It has often been said<br />
that philosophy should become the science of science. Philosophy's<br />
objective would then be to analyze the methods of<br />
science, to axiomatize and to clarify the concepts used. Such<br />
a role would make of the former "queen of sciences" something<br />
like their housemaid. Of course, there is the possibility<br />
that this clarification of concepts would permit further progress,<br />
that philosophy understood in this way would, through<br />
the use of other methods-logic, semantics-produce new<br />
knowledge comparable to that of science proper. It is this hope<br />
that sustains the "analytic philosophy" so prevalent in Anglo<br />
American circles. We do not want to minimize the interest of<br />
such an inquiry. However, the problems that concern us here<br />
are quite different. We do not aim to clarify or axiomatize existing<br />
knowledge but rather to close some fundamental gaps in<br />
this knowledge.<br />
A New Start<br />
In the first part of this book we described, on the one hand,<br />
dialogue with nature that classical science made possible and,<br />
on the other, the precarious cultural position of science. Is<br />
there a way out? In this chapter we have discussed some attempts<br />
to reach alternative ways of knowledge. We have also<br />
considered the positivist view, which separates science from<br />
reality.<br />
The moments of greatest excitement at scientific meetings<br />
very often occur when scientists discuss questions that are<br />
likely to have no practical utility whatsoever, no survival<br />
value-topics such as possible interpretations of quantum mechanics,<br />
or the role of the expanding universe in our concept<br />
of time. If the positivistic view, which reduces science to a<br />
symbolic calculus, was accepted , science would lose much of<br />
its appeal. Newton's synthesis between theoretical concepts<br />
and active knowledge would be shattered. We would be back<br />
to the situation familiar from the time of Greece and Rome,<br />
with an unbridgeable gap between technical, practical knowledge<br />
on one side and theoretical knowledge on the other.<br />
For the ancients. nature was a source of wisdom. Medieval<br />
nature spoke of God. In modern times nature has become so
99<br />
THE TWO CULTURES<br />
silent that Kant considered that science and wisdom, science<br />
and truth, ought to be completely separated. We have been<br />
living with this dichotomy for the past two centuries. It is time<br />
for it to come to an end. As far as science is concerned, the<br />
time is ripe for this to happen. From our present perspective,<br />
the first step toward a possible reunification of knowledge was<br />
the discovery in the nineteenth century of the theory of heat,<br />
of the laws of thermodynamics. Thermodynamics appears as<br />
the first form of a "science of complexity." This is the science<br />
we now wish to describe, from its formulation to recent developments.
I<br />
I
BOOK TWO<br />
THE SCIENCE OF<br />
COMPLEXITY
I<br />
I<br />
I<br />
I<br />
I
CHAPTER IV<br />
ENERGY AND THE<br />
INDUSTRIAL AGE<br />
Heat, the Rival of Gravitation<br />
Ignis mutat res. Ageless wisdom has always linked chemistry<br />
to the "science of fire." Fire became part of experimental science<br />
during the eighteenth century, starting a conceptual<br />
transformation that forced science to reconsider what it had<br />
previously .. rejected in the name of a mechanistic world view,<br />
topics such as irreversibility and complexity.<br />
Fire transforms matter; fire leads to chemical reactions, to<br />
processes such as melting and evaporation. Fire makes fuel<br />
burn and release heat. Out of all this common knowledge,<br />
nineteenth-century science concentrated on the single fact<br />
that combustion produces heat and that heat may lead to an<br />
increase in volume; as a result, combustion produces work.<br />
Fire leads, therefore, to a new kind of machine, the heat engine,<br />
the technological innovation on which industrial society<br />
has been founded. I<br />
It is interesting to note that Adam Smith was working on his<br />
Wealth of Nations and collecting data on the prospects and<br />
determinants of industrial growth at the same university at<br />
which James Watt was putting the finishing touches on his<br />
steam engine. Yet the only use for coal that Adam Smith could<br />
find was to provide heat for workers. In the eighteenth century,<br />
wind, water, and animals, and the simple machines<br />
driven by them, were still the only conceivable sources of<br />
power.<br />
The rapid spread of the British steam engine brought about<br />
a new interest in the mechanical effect of heat, and thermodynamics.<br />
born out of this interest, was thus not so much<br />
103
ORDER OUT OF CHAOS<br />
104<br />
concerned with the nature of heat as with heat's possibilities<br />
for producing "mechanical energy."<br />
As for the birth of the "science of complexity," we propose<br />
to date it in 1811, the year Baron Jean-Joseph Fourier, the prefect<br />
oflsere, won the prize of the French Academy of Sciences<br />
for his mathematical description of the propagation of heat in<br />
solids.<br />
The result stated by Fourier was surprisingly simple and elegant:<br />
heat flow is proportional to the gradient of temperature.<br />
It is remarkable that this simple law applies to matter, whether<br />
its state is solid, liquid, or gaseous. Moreover, it remains valid<br />
whatever the chemical composition of the body is, whether it<br />
is iron or gold. It is only the coefficient of proportionality between<br />
the heat flow and the gradient of temperature that is<br />
specific to each substance.<br />
Obviously, the universal character of Fourier's law is not<br />
directly related to dynamic interactions as expressed by Newton's<br />
law, and its formulation may thus be considered the starting<br />
point of a new type of science. Indeed, the simplicity of<br />
Fourier's mathematical description of heat propagation stands<br />
in sharp contrast to the complexity of matter considered from<br />
the molecular point of view. A solid, a gas, or a liquid are<br />
macroscopic systems formed by an immense number of molecules,<br />
and yet heat conductivity is described by a single law.<br />
Fourier formulated his result at the time when Laplace's<br />
school dominated European science. Laplace, Lagrange, and<br />
their disciples vainly joined forces to criticize Fourier's theory,<br />
but they were forced to retreat. 2 At the peak of its glory,<br />
the Laplacian dream met with its first setback. A physical theory<br />
had been created that was every bit as mathematically<br />
rigorous as the mechanical laws of motion but that remained<br />
completely alien to the Newtonian world. From this time on,<br />
mathematics, physics, and Newtonian science ceased to be<br />
synonymous.<br />
The formulation of the law of heat conduction had a lasting<br />
influence. Curiously, in France and Britain it was the starting<br />
point of different historical paths leading to our time.<br />
In France, the failure of Laplace's dream led to the positivist<br />
classification of science into the well-defined compartments<br />
introduced by Auguste Comte. The Comtean division of science<br />
has been well analyzed by Michel Serres3-heat and
105 ENERGY AND THE INDUSTRIAL AGE<br />
gravity, two universals, coexist in physics. Worse, as Comte<br />
was to state later, they are antagonistic. Gravitation acts on an<br />
inert mass that submits to it without being affected by it in any<br />
other way than by the motion it acquires or transmits. Heat<br />
transforms matter, determines changes of state, and leads to a<br />
modification of intrinsic properties. This was, in a sense, a<br />
confirmation of the protest made by the anti-Newtonian chemists<br />
of the eighteenth century and by all those who emphasized<br />
the difference between the purely spatiotemporal behavior at-<br />
. tributed to mass and the specific activity of matter. This distinction<br />
was used as a foundation for the classification of the<br />
sciences, all placed by Comte under the common sign of<br />
order-that is, of equilibrium. To the mechanical equilibrium<br />
between forces the positivist classification simply adds the<br />
concept of thermal equilibrium.<br />
In Britain, on the other hand, the theory of heat propagation<br />
did not mean giving up the attempt to unite the fields of knowledge<br />
but opened a new line of inquiry, the progressive formulation<br />
of a theory of irreversible processes.<br />
Fourier's law, when applied to an isolated body with an unhomogeneous<br />
temperature distribution, describes the gradual<br />
onset of thermal equilibrium. The effect of heat propagation is<br />
to equalize progressively the distribution of temperature until<br />
homogeneity is reached. Everyone knew that this was an irreversible<br />
process; a century before, Boerhave had stressed that<br />
heat always spread and leveled out. The science of complex<br />
phenomena-involving interaction among a large number of<br />
particles-and the occurrence of temporal asymmetry thus<br />
were linked from the outset. But heat conduction did not become<br />
the starting point of an investigation into the nature of<br />
irreversibility before it was first linked with the notion of dissipation<br />
as seen from an engineering point of view. 4<br />
Let us go into some detail about the structure of the new<br />
"science of heat" as it took shape in the early nineteenth century.<br />
Like mechanics, the science of heat implied both an original<br />
conception of the physical object and a definition of<br />
machines or engines-that is, an identification of cause and<br />
effect in a specific mode of production of mechanical work.<br />
The study of the physical processes involving heat entails<br />
defining a system, not, as in the case of dynamics, by the position<br />
and velocity of its constituents (there are some IQ23 mole-
ORDER OUT OF CHAOS<br />
106<br />
cutes in a volume of gas or a solid fragment of the order of a<br />
cm3), but by a set of macroscopic parameters such as temperature,<br />
pressure, volume, and so on. In addition, we have to<br />
take into account the boundary conditions that describe the<br />
relation of the system to its environment.<br />
Let us consider specific heat, one of the characteristic properties<br />
of a macroscopic system, as an example. The specific<br />
heat is a measure of the amount of heat required to raise the<br />
temperature of a system by one degree while its volume or<br />
pressure is kept constant. To study the specific heat-for instance,<br />
at constant volume-the system must be brought into<br />
interaction with its environment; it mu s t receive a certain<br />
amount of heat while at the same time its volume is kept constant<br />
and its pressure is allowed to vary.<br />
More generally, a system may be subjected to mechanical<br />
action (for example, either the pressure or the volume may be<br />
fixed by using a piston device), thermal action (a certain<br />
amount of heat may be given to or removed from the system,<br />
or the system itself may be brought to a given temperature<br />
through heat exchange), or chemical action (a flux of reactants<br />
and reaction products between the system and the environment).<br />
As we have already mentioned, pressure, volume,<br />
chemical composition, and temperature are the classical physicochemical<br />
parameters in terms of which the properties of<br />
macroscopic systems are defined. Thermodynamics is the science<br />
of the correlation among the variations in these properties.<br />
In comparison with dynamic objects, thermodynamic<br />
objects therefore lead to a new point of view. The aim of the<br />
theory is not to predict the changes in the system in terms of<br />
the interactions among particles; it aims instead to predict<br />
how the system will react to modifications we may impose on<br />
it from the outside.<br />
A mechanical engine gives back in the form of work the<br />
potential energy it has received from the outside world. Both<br />
cause and effect are of the same nature and, at least ideally,<br />
equivalent. In contrast, the heat engine implies material<br />
changes of states, including the transformation of the system's<br />
mechanical properties, dilatation, and expansion. The mechanical<br />
work produced must be seen as the result of a true<br />
process of transformation and not only as a transmission of<br />
movement. Thus the heat engine is not merely a passive de-
107 ENERGY AND THE INDUSTRIAL AGE<br />
vice; strictly speaking, it produces motion. This is the origin of<br />
a new problem: in order to restore the system's capacity to<br />
produce motion, the system must be brought back to its initial<br />
state. Thus a second process is needed, a second change of<br />
state that compensates for the change producing the motion.<br />
In a heat engine, this second process, which is opposite to the<br />
first, involves cooling the system until it regains its initial temperature,<br />
pressure, and volume.<br />
The problem of the efficiency of heat engines, of the ratio<br />
between the work done and the heat that must be supplied to<br />
the system to produce the two mutually compensating processes,<br />
is the very point at which the concept of irreversible<br />
process was introduced into physics. We shall return to the<br />
importance of Fourier's law in this context. Let us first describe<br />
the essential role played by the principle of energy conservation.<br />
The Principle of the Conservation of Energy<br />
We have already emphasized the central place of energy in<br />
classical dynamics. The Hamiltonian (the sum of the kinetic<br />
and potential energies) is expressed in terms of canonical variables-coordinates<br />
and momenta-and leads to changes in<br />
these variables while itself remaining constant throughout the<br />
motion. Dynamic change merely modifies the respective importance<br />
of potential and kinetic energy, conserving their totality.<br />
The early nineteenth century was characterized by unprecedented<br />
experimental ferment. 5 Physicists realized that motion<br />
does more than bring about changes in the relative position of<br />
bodies in space. New processes identified in the laboratories<br />
gradually formed a network that ultimately linked all the new<br />
fields of physics with other, more traditional branches, such as<br />
mechanics. One of these connections was accidentally discovered<br />
by Galvani. Before him, only static electric charges<br />
were known. Galvani, using a frog's body, set up the first experimental<br />
electric current. Volta soon recognized that the<br />
··galvanic" contractions in the frog were actually the effect of<br />
an electric current passing through it. In 1800, Volta con-
ORDER OUT OF CHAOS<br />
108<br />
structed a chemical battery; electricity could thus be produced<br />
by chemical reactions. Then came electrolysis: electric<br />
current can modify chemical affinities and produce chemical<br />
reactions. But this current can also produce light and heat;<br />
and, in 1820, Oersted discovered the magnetic effects produced<br />
by electrical currents. In 1822, Seebeck showed that,<br />
inversely, heat could produce electricity and, in 1834, how<br />
matter could be cooled by electricity. Then, in 183 1, Faraday<br />
induced an electric current by means of magnetic effects. A<br />
whole network of new effects was gradually uncovered. The<br />
scientific horizon was expanding at an unprecedented rate.<br />
In 1847 a decisive step was taken by Joule: the links among<br />
chemistry, the science of heat, electricity, magnetism, and biology<br />
were recognized as a "conversion." The idea of conversion,<br />
which postulates that "something" is quantitatively<br />
conserved while it is qualitatively transformed, generalizes<br />
what occurs during mechanical motion. As we have seen, total<br />
energy is conserved while potential energy is converted into<br />
kinetic energy, or vice versa. Joule defined a general equivalent<br />
for physicochemical transformations, thus making it<br />
possible to measure the quantity conserved. This quantity was<br />
later6 to become known as "energy." He established the first<br />
equivalence by measuring the mechanical work required to<br />
raise the temperature of a given quantity of water by one degree.<br />
A unifying element had been discovered in the middle of<br />
a bewildering variety of new discoveries. The conservation of<br />
energy, throughout the various transformations undergone by<br />
physical, chemical, and biological systems, was to provide a<br />
guiding principle in the exploration of the new processes.<br />
No wonder that the principle of the conservation of energy<br />
was so important to nineteenth-century physicists. For many<br />
of them it meant the unification of the whole of nature. Joule<br />
expressed this conviction in an English context:<br />
Indeed the phenomena of nature, whether mechanical,<br />
chemical, or vital, consist almost entirely in a continual<br />
conversion of attraction through space, living force<br />
(N.B., kinetic energy) and heat into one another. Thus it<br />
is that order is maintained in the universe-nothing is deranged,<br />
nothing ever lost, but the entire machinery, com-
109<br />
ENERGY AND THE INDUSTRIAL AGE<br />
plicated as it is, works smoothly and harmoniously. And<br />
though, as in the awful vision of Ezekiel, "wheel may be<br />
in the middle of wheel," and everything may appear complicated<br />
and involved in the apparent confusion and intricacy<br />
of an almost ·endless variety of causes, effects,<br />
conversions, and arrangements, yet is the most perfect<br />
regularity preserved-the whole being governed by the<br />
sovereign will of God. 7<br />
The case of the Germans Helmholtz, Mayer, and Lie big-all<br />
three belonging to a culture that would have rejected Joule's<br />
conviction on the grounds of strictly positivist practice-is<br />
even more striking. At the time of their discoveries, none of<br />
the three was, strictly speaking, a physicist. On the other<br />
hand, all of them were interested in the physiology of respiration.<br />
This had become, since Lavoisier, a model problem in<br />
which the functioning of a living being could be described in<br />
precise physical and chemical terms, such as the combustion<br />
of oxygen, the release of heat, and muscular work. It was thus<br />
a question that would attract physiologists and chemists hostile<br />
to Romantic speculation and eager to contribute to experimental<br />
science. However, judging from the account of how<br />
these three scientists came to the conclusion that respiration,<br />
and then the whole of nature, was governed by some fundamental<br />
"equivalence," we may state that the German philosophic<br />
tradition had imbued them with a conception that was<br />
quite alien to a positivist position: without hesitation they all<br />
concluded that the whole of nature, in each of its details, is<br />
ruled by this single principle of conservation.<br />
The case of Mayer is the most remarkable.s As a young doctor<br />
working in the Dutch colonie s in Java, he noticed the bright<br />
red color of the venous blood of one of his patients. This led<br />
him to conclude that, in a warm, tropical climate, the inhabitants<br />
need to burn less oxygen to maintain body temperature;<br />
this results in the bright color of their blood. Mayer went on to<br />
establish the balance between oxygen consumption, which is<br />
the source of energy, and the energy consumption involved in<br />
maintaining body temperature despite heat losses and manual<br />
work. This was quite a leap, since the color of the blood could<br />
as well be due to the patient's ··taziness. ,, But Mayer went
ORDER OUT OF CHAOS 110<br />
further and concluded that the balance between oxygen consumption<br />
and heat loss was merely the particular manifestation<br />
of the existence of an indestructible "force" underlying all<br />
phenomena.<br />
This tendency to see natural phenomena as the products of<br />
an underlying reality that remains constant throughout its<br />
transformations is strikingly reminiscent of Kant. Kant's influence<br />
can also be recognized in another idea held by some<br />
physiologists, the distinction between vitalism as philosophical<br />
speculation and the problem of scientific methodology. For<br />
those physiologists, even if there was a "vital" force underlying<br />
the function of living beings, the object of physiology<br />
would nonetheless be purely physicochemical in nature. From<br />
the two points of view mentioned, Kantianism, which ratified<br />
the systematic form taken by mathematical physics during the<br />
eighteenth century, can also be identified as one of the roots of<br />
the renewal of physics in the nineteenth century.9<br />
Helmholtz quite openly acknowledged Kant's influence.<br />
For Helmholtz, the principle of the conservation of energy was<br />
merely the embodiment in physics of the general a priori requirement<br />
on which all science is based-the postulate that<br />
there is a basic invariance underlying natural transformations:<br />
The problem of the sciences is, in the first place, to seek<br />
the laws by which the particular processes of nature may<br />
be referred to, and deduced from, general rules.<br />
We are justified, and indeed impelled in this proceeding,<br />
by the conviction that every change in nature must<br />
have a sufficient cause. The proximate causes to which<br />
we refer phenomena may, in themselves, be either variable<br />
or invariable; in the former case the above conviction<br />
impels us to seek for causes to account for the<br />
change, and thus we proceed until we at length arrive at<br />
final causes which are unchangeable, and which therefore<br />
must, in all cases where the exterior conditions are<br />
the same, produce the same invariable effects. The final<br />
aim of the theoretic natural sciences is therefore to discover<br />
the ultimate and unchangeable causes of natural<br />
phenomena. tO<br />
With the principle of the conservation of energy, the idea of
111 ENERGY AND THE INDUSTRIAL AGE<br />
a new golden age of physics began to take shape, an age that<br />
would lead to the ultimate generalization of mechanics.<br />
The cultural implications were far-reaching, and they included<br />
a conception of society and men as energy-transforming<br />
engines. But energy conversion cannot be the whole story. It<br />
represents the aspects of nature that are peaceful and controllable,<br />
but below there must be another, more "active" level.<br />
Nietzsche was one of those who detected the echo of creations<br />
and destructions that go far beyond mere conservation or conversion.<br />
Indeed, only difference, such as a difference of temperature<br />
or of potential energy, can produce results that are<br />
also differences.11 Energy conversion is merely the destruction<br />
of a difference, together with the creation of another difference.<br />
The power of nature is thus concealed by the use of<br />
equivalences. However, there is another aspect of nature that<br />
involves the boilers of steam engines, chemical transformations,<br />
life and death, and that goes beyond equivalences and<br />
conservation of energy.12 Here we reach the most original contribution<br />
of thermodynamics, the concept of irreversibility.<br />
Heat Engines and the Arrow of Time<br />
When we compare mechanical devices to thermal engines, for<br />
example, to the red-hot boilers of locomotives, we can see at a<br />
glance the gap between the classical age and nineteenthcentury<br />
technology. Still, physicists first thought that this gap<br />
could be ignored, that thermal engines could be described like<br />
mechanical ones, neglecting the crucial fact that fuel used by<br />
the steam engine disappears forever. But such complacency<br />
soon became impossible. For classical mechanics the symbol<br />
of nature was the clock; for the Industrial Age, it became a<br />
reservoir of energy that is always threatened with exhaustion.<br />
The world is burning like a furnace; energy, although being<br />
conserved, also is being dissipated.<br />
The original formulation of the second law of thermodynamics,<br />
which would lead to the first quantitative expression<br />
of irreversibility, was made by Sadi Carnot in 1824, before<br />
the general formulation of the principle of ;onservation of energy<br />
by Mayer (1842) and Helmholtz (1847). Carnot analyzed
ORDER OUT OF CHAOS 112<br />
the heat engine, closely following the work of his father, Lazare<br />
Carnot, who had produced an influential description of me ..<br />
chanical engines.<br />
The description of mechanical engines assumes motion as a<br />
given. In modern language this corresponds to conservation of<br />
energy and momentum. Motion is merely converted and<br />
transferred to other bodies. But the analogy between mechani ..<br />
cal and thermal engines was a natural one for Sadi Carnot,<br />
since he assumed, with most of the scientists of his time, that<br />
heat as well as mechanical energy are conserved.<br />
Water falling from one level to another can drive a mill. Sim ..<br />
ilarly, Sadi Carnot assumed two sources, one of which gives<br />
heat to the engine system, and the other, at a different tern ..<br />
perature, which absorbs the heat given by the former. It is the<br />
motion of the heat through the engine, between the two<br />
sources at different temperatures-that is, the driving force of<br />
fire-that will make the engine work.<br />
Carnot repeated his father's questions. l3 Which machine<br />
will have the highest efficiency? What are the sources of loss?<br />
What are the processes whereby heat propagates without producing<br />
work? Lazare Carnot had concluded that in order to<br />
obtain maximum efficiency from a mechanical machine it<br />
must be built and made to function to reduce to a minimum<br />
shocks, friction, or discontinuous changes of speed-in short,<br />
all that is caused by the sudden contact of bodies moving at<br />
different speeds. In doing so he had merely applied the physics<br />
of his time: only continuous phenomena are conservative;<br />
all abrupt changes in motion cause an irreversible loss of the<br />
"living force. " Similarly, the ideal heat engine, instead of having<br />
to avoid all contacts between bodies moving at different<br />
speeds, will have to avoid all contact between bodies having<br />
different temperatures.<br />
The cycle therefore has to be designed so that no temperature<br />
change results from direct heat flow between two bodies<br />
at different temperatures. Since such flows have no mechanical<br />
effect, they would merely lead to a loss of efficiency.<br />
The ideal Carnot cycle is thus a rather tricky device that<br />
achieves the paradoxical result of a heat transfer between two<br />
sources at different temperatures without any contact between<br />
bodies of different temperatures. It is divided into four phases.<br />
During each of the two isothermal phases, the system is in
..<br />
...<br />
113 ENERGY AND THE INDUSTRIAL AGE<br />
contact with one of the two heat sources and is kept at the<br />
temperature of this source. When in contact with the hot<br />
source, it absorbs heat and expands; when in contact with the<br />
cold source, it loses heat and contracts. The two isothermal<br />
phases are linked up by two phases in which the system is<br />
isolated from the sources-that is, heat no longer enters or<br />
leaves the system, but the temperature of the latter changes as<br />
a result, respectively, of expansion and compression. The volume<br />
continues to change until the system has passed from the<br />
temperature of one source to that of the other.<br />
p<br />
'<br />
'<br />
'<br />
...<br />
..<br />
..<br />
..<br />
...<br />
',Q<br />
...<br />
..<br />
...<br />
..<br />
... .........<br />
.. _<br />
_ _<br />
T<br />
H<br />
---- T<br />
c L<br />
v<br />
Figure 2. Pressure-volume diagram of the Carnot cycle: a thermodynamic<br />
engine, functioning between two sources, one "hot" at temperature TH, the<br />
other "cold" at temperature TL. Between state a and state b, there is an<br />
isothermal change: The system, kept at temperature TH, absorbs heat and<br />
expands. Between b and c, the system is kept expanding while in thermal<br />
isolation; its temperature goes down from TH to TL. Those two steps produce<br />
mechanical energy. Between c and d, there is a second isothermal change:<br />
the system is compressed and releases heat while being kept at temperature<br />
TL. Between d and a, the system, again isolated, is compressed while its<br />
temperature increases to temperature TH.
ORDER OUT OF CHAOS 114<br />
It is quite remarkable that this description of an ideal thermal<br />
engine does not mention the irreversible processes that<br />
are at the basis of its realization. No mention is made of the<br />
furnace in which the coal is burning. The model is only concerned<br />
with the effect of the combustion, which permits the<br />
maintenance of the temperature difference between the two<br />
sources.<br />
In 1850, Clausius described the Carnot cycle from the new<br />
perspective provided by the conservation of energy. He discovered<br />
that the need for two sources and the formula for theoretical<br />
efficiency stated by Carnot express a specific problem<br />
with heat engines: the need for a process compensating for<br />
conversion (in the present instance, cooling by contact with a<br />
cold source) to restore the engine to its initial mechanical and<br />
thermal conditions. Balance relations expressing energy conversion<br />
are now joined by new equivalence relations between<br />
the effects of two processes on the state of the system, heat<br />
flux between the sources and conversion of heat into work. A<br />
new science, thermodynamics, which linked mechanical and<br />
thermal effects, came into being.<br />
The work of Clausius explicitly demonstrated that we cannot<br />
use without restriction the seemingly inexhaustible energy<br />
reservoir that nature provides. Not all energy-conserving processes<br />
are possible. An energy difference, for instance, cannot<br />
be created without the destruction of an at least equivalent<br />
energy difference. Thus in the ideal Carnot cycle, the price for<br />
the work produced is paid by the heat, which is transferred<br />
from one source to the other. The outcome, as expressed by<br />
the mechanical work produced on one side, and the transfer of<br />
heat on the other, is linked by an equivalence. This equivalence<br />
is valid in both directions. By working in reverse, the<br />
same machine can restore the initial temperature difference<br />
while consuming the work produced. No heat engine can be<br />
constructed using a single source of heat.<br />
Clausius was no more concerned than Carnot with the<br />
losses whereby all real engines have an efficiency lower than<br />
the ideal value predicted by the theory. His description, like<br />
that of Carnot, corresponds to an idealization. It leads to the'<br />
definition of the limit nature imposes on the yield of thermal<br />
engines.
115 ENERGY AND THE INDUSTRIAL AGE<br />
However, since the eighteenth century, the status of idealizations<br />
had changed. Based as it was on the principle of the conservation<br />
of energy, the new science claimed to describe not<br />
only idealizations, but also nature itself, including "losses."<br />
This raised a new problem, whereby irreversibility entered<br />
physics. How does one describe what happens in a real engine?<br />
How does one include losses in the energy balance?<br />
How do they reduce efficiency? These questions paved the<br />
way to the second law of thermodynamics.<br />
From Technology to Cosmology<br />
As we have seen, the question raised by Carnot and Clausius<br />
led to a description of ideal engines that was based on conservation<br />
and compensation. In addition, it provided an opportunity<br />
for presenting new problems, such as the dissipation of<br />
energy. William Thomson, who had great respect for Fourier's<br />
work, was quick to grasp the importance of the problem, and<br />
in 1852 he was the first to formulate the second law of thermodynamics.<br />
It was Fourier's heat propagation that Carnot had identified<br />
as a possible cause for the power losses in a heat engine. Carnot's<br />
cycle, no longer the ideal cycle but the "real" cycle, thus<br />
became the point of convergence of the two universalities discovered<br />
in the nineteenth century-energy conversion and<br />
heat propagation. The combination of these two discoveries<br />
led Thomson to formulate his new principle: the existence in<br />
nature of a universal tendency toward the degradation of mechanical<br />
energy. Note the word "universal," which has obvious<br />
cosmological connotations.<br />
The world of Laplace was eternal, an ideal perpetual-motion<br />
machine. Since Thomson's cosmology is not merely a reflection<br />
of the new ideal heat engine but also incorporates the consequences<br />
of the irreversible propagation of heat in a world in<br />
which energy is conserved. This world is described as an engine<br />
in which heat is converted into motion only at the price of some<br />
irreversible waste and useless dissipation. Effect-producing<br />
differences in nature progressively diminish. The world uses
ORDER OUT OF CHAOS 116<br />
up its differences as it goes from one conversion to another<br />
and tends toward a final state of thermal equilibrium, "heat<br />
death." In accordance with Fourier's law, in the end there will<br />
no longer be any differences of temperature to produce a mechanical<br />
effect.<br />
Thomson thus made a dizzy leap from engine technology to<br />
cosmology. Hs formulation of the second law was couched in<br />
the scientific terminology of his time: the conservation of energy,<br />
engines, and Fourier's law. It is clear, moreover, that the<br />
part played by the cultural context was important. It is generally<br />
accepted that the problem of time took on a new importance<br />
during the nineteenth century. Indeed, the essential role<br />
of time began to be noticed in all fields-in geology, in biology,<br />
in language, as well as in the study of human social evolution<br />
and ethics. But it is interesting that the specific form in which<br />
time was introduced in physics, as a tendency toward homogeneity<br />
and death, reminds us more of ancient mythological and<br />
religious archetypes than of the progressive complexification<br />
and diversification described by biology and the social sciences.<br />
The return of these ancient themes can be seen as a<br />
cultural repercussion of the social and economic upheavals of<br />
the time. The rapid transformation of the technological mode<br />
of interaction with nature, the constantly accelerating pace of<br />
change experienced by the nineteenth century, produced a<br />
deep anxiety. This anxiety is still with us and takes various<br />
forms, from the repeated proposals for a "zero growth" society<br />
or for a moratorium on scientific research to the<br />
announcement of "scientific truths" concerning our disintegrating<br />
universe. Present knowledge in astrophysics is still<br />
scanty and very problematic, since in this field gravitational<br />
effects play an essential role and problems imply the simultaneous<br />
use of thermodynamics and relativity. Yet most texts<br />
in this field are unanimous in predicting final doom. The conclusion<br />
of a recent book reads:<br />
The unpalatable truth appears to be that the inexorable<br />
disintegration of the universe as we know it seems assured,<br />
the <strong>org</strong>anization which sustains all ordered activity,<br />
frem men to galaxies, is slowly but inevitably<br />
running down, and may even be overtaken by total gravitational<br />
collapse into oblivion.t4
117 ENERGY AND THE INDUSTRIAL AGE<br />
Others are more optimistic. In an excellent article on the<br />
energy of the universe, Freeman Dyson has written:<br />
It is conceivable however that life may have a larger role<br />
to play than we have yet imagined. Life may succeed<br />
against all of the odds in molding the universe to its own<br />
purpose. And the design of the inanimate universe may<br />
not be as detached from the potentialities of life and intelligence<br />
as scientists of the twentieth century have tended<br />
to suppose.15<br />
In spite of the important progress made by Hawking and others,<br />
our knowledge of large-scale transformations in our universe<br />
remains inadequate.<br />
The Birth of Entropy<br />
In 1865, it was Clausius' turn to make the leap from technology<br />
to cosmology. At the outset he merely reformulated his<br />
earlier conclusions, but in doing so he introduced a new concept,<br />
entropy. His first goal was to distinguish clearly between<br />
the concepts of conservation and of reversibility. Unlike mechanical<br />
transformations, where reversibility and conservation<br />
coincide, a physicochemical transformation may conserve energy<br />
even though it cannot be reversed. This is true, for instance,<br />
in the case of friction, in which motion is converted<br />
into heat, or in the case of heat conduction as it was described<br />
by Fourier.<br />
We are already familiar with energy, which is a function of<br />
the state of a system-that is, a function dependent only on<br />
the value of the parameters (pressure, volume, temperature)<br />
by which that state may be defined.t6 But we must go beyond<br />
the principle of energy conservation and find a way to express<br />
the distinction between "useful" exchanges of energy in the<br />
Carnot cycle and "dissipated" energy that is irreversibly<br />
wasted.<br />
This is precisely the role of Clausius' new function, entropy,<br />
generally denoted by S.<br />
Apparently Clausius merely wished to express in a new form
ORDER OUT OF CHAOS 118<br />
the obvious requirement that an engine return to its initial<br />
state at the end of its cycle. The first definition of entropy is<br />
centered on conservation: at the end of each cycle, whether<br />
ideal or not, the function of the system's state, entropy, returns<br />
to its initial value. But the parallel between entropy and energy<br />
ends as soon as we abandon idealizations.t7<br />
Let us consider the variation of the entropy dS over a short<br />
time interval dt. The situation is quite different for ideal and<br />
real engines. In the first case, dS may be expressed completely<br />
in terms of the exchanges between the engine and its environment.<br />
We can set up experiments in which heat is given up by<br />
the system instead of flowing into the system. The corresponding<br />
change in entropy would simply have its sign<br />
changed. This kind of contribution to entropy, which we shall<br />
call deS, is therefore reversible in the sense that it can have<br />
either a positive or a negative sign. The situation is drastically<br />
different in a real engine. Here, in addition to reversible exchanges,<br />
we have irreversible processes inside the system,<br />
such as heat losses, friction, and so on. These produce an entropy<br />
increase or "entropy production" inside the system.<br />
This increase of entropy, which we shall call diS, cannot<br />
change its sign through a reversal of the heat exchange with<br />
the outside world. Like all irreversible processes (such as heat<br />
conduction), entropy production always proceeds in the same<br />
direction. In other words, diS can only be positive or vanish in<br />
the absence of irreversible processes. Note that the positive<br />
sign of diS is chosen merely by convention; it could just as<br />
well have been negative. The point is that the variation is monotonous,<br />
that entropy production cannot change its sign as<br />
time goes on.<br />
The notations deS and diS have been chosen to remind the<br />
reader that the first term refers to exchanges (e) with the outside<br />
world, while the second refers to the irreversible processes<br />
inside (i) the system. The entropy variation dS is<br />
therefore the sum of the two terms deS and diS, which have<br />
quite different physical meanings.18<br />
To grasp the peculiar feature of this decomposition of entropy<br />
variation into two parts, it is useful to apply our formulation<br />
to energy. Let us denote energy by E and variation over a<br />
short time dt by dE. Of course, we would still write that dE is<br />
equal to the sum of a term deE due to the exchanges of energy
119 ENERGY AND THE INDUSTRIAL AGE<br />
and a term diE linked to the "internal production" of energy.<br />
However, the principle of the conservation of energy states<br />
that energy is never "produced" but only transferred from one<br />
place to another. The variation in energy dE is then reduced to<br />
deE. On the other hand, if we take a nonconserved quantity,<br />
such as the quantity of hydrogen molecules contained in a vessel,<br />
this quantity may vary both as the result of adding hydrogen<br />
to the vessel or through chemical reactions occurring<br />
inside the vessel. But in this case, the sign of the "production"<br />
is not determined. Depending on the circumstances, we can<br />
produce or destroy hydrogen molecules by transferring hydrogen<br />
atoms to other chemical components. The peculiar feature<br />
of the second law is the fact that the production term diS is<br />
always positive. The production of entropy expresses the occurrence<br />
of irreversible changes inside the system.<br />
Clausius was able to express quantitatively the entropy flow<br />
deS in terms of the heat received (or given up) by the system.<br />
In a world dominated by the concepts of reversibility and conservation,<br />
this was his main concern. Regarding the irreversible<br />
processes involved in entropy production, he merely stated<br />
the existence of the inequality diS/dt>O. Even so, important<br />
progress had been made, for, if we leave the Carnot cycle and<br />
consider other thermodynamic systems, the distinction between<br />
entropy flow and entropy production can still be made.<br />
For an isolated system .that has no exchanges with its environment,<br />
the entropy flow is, by definition, zero. Only the production<br />
term remains, and the system's entropy can only<br />
increase or remain constant. Here, then, it is no longer a question<br />
of irreversible transformations considered as approximations<br />
of reversible transformations; increasing entropy corresponds<br />
to the spontaneous evolution of the system. Entropy thus becomes<br />
an "indicator of evolution," or an "arrow of time," as<br />
Eddington aptly called it. For all isolated systems, the future<br />
is the direction of increasing entropy.<br />
What system would be better "isolated" than the universe<br />
as a whole? This concept is the basis of the cosmological formulation<br />
of the two laws of thermodynamics given by Clausius<br />
in 1865:<br />
Die Energie der Welt ist konstant.<br />
Die Entropie der Welt strebt einem Maximum zu.19<br />
The statement that the entropy of an isolated system in-
ORDER OUT OF CHAOS<br />
120<br />
creases to a maximum goes far beyond the technological problem<br />
that gave rise to thermodynamics. Increasing entropy is<br />
no longer synonymous with loss but now refers to the natural<br />
processes within the system. These are the processes that ultimately<br />
lead the system to thermodynamic "equilibrium"<br />
corresponding to the state of maximum entropy.<br />
In Chapter I we emphasized the element of surprise involved<br />
in the discovery of Newton's universal laws of dynamics.<br />
Here also the element of surprise is apparent. When Sadi<br />
Carnot formulated the laws of ideal thermal engines, he was<br />
far from imagining that his work would lead to a conceptual<br />
revolution in physics.<br />
Reversible transformations belong to classical science in the<br />
sense that they define the possibility of acting on a system, of<br />
controlling it. The dynamic object could be controlled through<br />
its initial conditions. Similarly, when defined in terms of its<br />
reversible transformations, the thermodynamic object may be<br />
controlled through its boundary conditions: any system in<br />
thermodynamic equilibrium whose temperature, volume, or<br />
pressure are gradually changed passes through a series of<br />
equilibrium states, and any reversal of the manipulation leads<br />
to a return to its initial state. The reversible nature of such<br />
change and controlling the object through its boundary conditions<br />
are interdependent processes. In this context irreversibility<br />
is "negative"; it appears in the form of "uncontrolled"<br />
changes that occur as soon as the system eludes control. But<br />
inversely, irreversible processes may be considered as the last<br />
remnants of the spontaneous and intrinsic activity displayed<br />
by nature when experimental devices are employed to harness<br />
it.<br />
Thus the "negative" property of dissipation shows that, unlike<br />
dynamic objects, thermodynamic objects can only be partially<br />
controlled. Occasionally they "break loose" into<br />
spontaneous change.<br />
All changes are not equivalent for a thermodynamic system.<br />
This is the meaning of the expression dS =deS+ diS.<br />
Spontaneous change toward equilibrium diS is different from<br />
the change deS, which is determined and controlled by a modification<br />
of the boundary conditions (for example, ambient<br />
temperature). For an isolated system, equilibrium appears as
121 ENERGY AND THE INDUSTRIAL AGE<br />
an ••attractor" of nonequilibrium states. Our initial assertion<br />
may thus be generalized by saying that evolution toward an<br />
attractor state differs from all other changes, especially from<br />
changes determined by boundary conditions.<br />
Max Planck often emphasized the difference between the<br />
two types of change found in nature. Nature, wrote Planck,<br />
seems to "favor" certain states. The irreversible increase in<br />
entropy diS/dt describes a system's approach to a state which<br />
"attracts" it, which the system prefers and from which it will<br />
not move of its own "free will." "From this point of view, Nature<br />
does not permit processes whose final states she finds<br />
less attractive than their initial states. Reversible processes are<br />
limiting cases. In them, Nature has an equal propensity for<br />
initial and final states; this is why the passage between them<br />
can be made in both directions. "20<br />
How foreign such language sounds when compared with the<br />
language of dynamics! In dynamics, a system changes according<br />
to a trajectory that is given once and for all, whose starting<br />
point is never f<strong>org</strong>otten (since initial conditions determine the<br />
trajectory for all time). However, in an isolated system all nonequilibrium<br />
situations produce evolution toward the same kind<br />
of equilibrium state. By the time equilibrium has been<br />
reached, the system has f<strong>org</strong>otten its initial conditions-that<br />
is, the way it had been prepared.<br />
Thus specific heat or the compressibility of a system in<br />
equilibrium are properties independent of the way the system<br />
has been set up. This fortunate circumstance greatly simplifies<br />
the study of the physical states of matter. Indeed, complex<br />
systems consist of an immense number of particles.* From the<br />
dynamic standpoint it is practically impossible to reproduce<br />
any state of such systems in view of the infinite variety of dynamic<br />
states that may occur.<br />
We are now confronted with two baically different descriptions:<br />
dynamics, which applies to the world of motion, and<br />
*Physical chemistry often employs Avogadro's number-that is, the number<br />
of molecules in a "mole" of matter (a mole always contains the same<br />
number of particles, the number of atoms contained in one gram of hydro<br />
gen). This number is of the order of 6.1023, and it is the characteristic order<br />
of magnitude of the number of particles forming systems governed by the<br />
laws of classical thermodynamics.
ORDER OUT OF CHAOS 122<br />
thermodynamics, the science of complex systems with its intrinsic<br />
direction of evolution toward increasing entropy. This<br />
dichotomy immediately raises the question of how these descriptions<br />
are related, a problem that has been debated since<br />
the laws of thermodynamics were formulated.<br />
Boltzmanns Order Principle<br />
The second law of thermodynamics contains two fundamental<br />
elements: ( 1) a "negative " one that expresses the impossibility<br />
of certain processes (heat flows from the hot source to the cold<br />
and not vice versa) and (2) a "positive," constructive one. The<br />
second is a consequence of the first; it is the impossibility of<br />
certain processes that permits us to introduce a function, entropy,<br />
which increases uniformly for isolated systems. Entropy<br />
behaves as an attractor for isolated systems.<br />
How could the formulations of thermodynamics be reconciled<br />
with dynamics? At the end of the nineteenth century,<br />
most scientists seemed to think this was impossible. The principles<br />
of thermodynamics were new laws forming the basis of a<br />
new science that could not be reduced to traditional physics.<br />
Both the qualitative diversity of energy and its tendency toward<br />
dissipation had to be accepted as new axioms. This was<br />
the argument of the "energeticists" as opposed to the "atomists,"<br />
who refused to abandon what they considered to be the<br />
essential mission of physics-to reduce the complexity of natural<br />
phenomena to the simplicity of elementary behavior expressed<br />
by the laws of motion.<br />
Actually, the problems of the transition from the microscopic<br />
to the macroscopic level were to prove exceptionally<br />
fruitful for the development of physics as a whole. Boltzmann<br />
was the first to take up the challenge. He felt that new concepts<br />
had to be developed to extend the physics of trajectories<br />
to cover the situation described by thermodynamics. Following<br />
in Maxwell's footsteps, Boltzmann sought this conceptual<br />
innovation in the theory of probability.<br />
That probability could play a role in the description of complex<br />
phenomena was not surprising: Maxwell himself appears
123 ENERGY AND THE INDUSTRIAL AGE<br />
to have been influenced by the work of Quetelet, the inventor<br />
of the "average" man in sociology. The innovation was to introduce<br />
probability in physics not as a means of approximation<br />
but rather as an explanatory principle, to use it to show<br />
that a system could display a new type of behavior by virtue of<br />
its being composed of a large population to which the laws of<br />
probability could be applied.<br />
Let us consider a simple example of the application of the<br />
concept of probability in physics. An ensemble composed of<br />
N particles is contained in a box divided into two equal compartments.<br />
The problem is to find the probability of the<br />
various possible distributions of particles between the compartments-that<br />
is, the probability of finding N1 particles in<br />
the first compartment (and N2 = N-N1 in the second).<br />
Using combinatorial analysis, it is easy to calculate the<br />
number of ways in which each different distribution of N particles<br />
can be achieved. Thus if N = 8, there is only one way of<br />
placing the eight particles in a single half. There are, however,<br />
eight different ways of putting one particle in one half and<br />
seven in the other half, if we suppose the particles to be distinguishable,<br />
as is assumed in classical physics. Furthermore,<br />
equal distribution of the eight particles between the two halves<br />
can be carried out in 8!14!4! = 70 different ways (where<br />
n! = 1·2·3 ... (n-l)·n). Likewise, whatever the value of N, a<br />
number P of situations called complexions in physics may be<br />
defined, giving the number of ways of achieving any given distribution<br />
N1,N2• Its expression is P=N!IN1!N2!.<br />
For any given population, the larger the number of complexions<br />
the smaller the difference between N1 and N2• It is maximum<br />
when the population is equally distributed over the two<br />
halves. Moreover, the larger the value of N, the greater the<br />
difference between the number of complexions corresponding<br />
to the different ways of distribution. For values of N of the<br />
order of 1Q 2 3 values found in macroscopic systems, the overwhelming<br />
majority of possible distributions corresponds to<br />
the distribution N1 =N2 =NI2. For systems composed of a<br />
large number of particles, all states that differ from the state<br />
corresponding to an equal distribution are thus highly improbable.<br />
Boltzmann was the .first to realize that irreversible increase
ORDER OUT OF CHAOS 124<br />
in entropy could be considered as the expression of a growing<br />
molecular disorder, of the gradual f<strong>org</strong>etting of any initial dissymmetry,<br />
since dissymmetry decreases the number of complexions<br />
when compared to the state corresponding to the<br />
maximum of P. Boltzmann thus aimed to identify entropy S<br />
with the number of complexions: entropy characterizes each<br />
macroscopic state in terms of the number of ways of achieving<br />
this state. Boltzmann's famous equation S = k lg pt expresses<br />
this idea in quantitative form. The proportionality factor k in<br />
this formula is a universal constant, known as Boltzmann's<br />
constant.<br />
Boltzmann's results signify that irreversible thermodynamic<br />
change is a change toward states of increasing probability and<br />
that the attractor state is a macroscopic state corresponding to<br />
maximum probability. This takes us far beyond Newton. For<br />
the first time a physical concept has been explained in terms of<br />
probability. Its utility is immediately apparent. Probability can<br />
adequately explain a system's f<strong>org</strong>etting of all initial dissymmetry,<br />
of all special distributions (for example, the set of particles<br />
concentrated in a subregion of the system, or the<br />
distribution of velocities that is created when two gases of different<br />
temperatures are mixed). This f<strong>org</strong>etting is possible because,<br />
whatever the evolution peculiar to the system, it will<br />
ultimately lead to one of the microscopic states corresponding<br />
to the macroscopic state of disorder and maximum symmetry,<br />
since these macroscopic states correspond to the overwhelming<br />
majority of possible microscopic states. Once this state<br />
has been reached, the system will move only short distances<br />
from the state, and for short periods of time. In other words,<br />
the system will merely fluctuate around the attractor state.<br />
Boltzmann's order principle implies that the most probable<br />
state available to a system is the one in which the multitude of<br />
events taking place simultaneously in the system compensates<br />
for one another statistically. In the case of our first example,<br />
whatever the initial distribution, the system's evolution will<br />
ultimately lead it to the equal distribution N1 = N 2 • This state<br />
will put an end to the system's irreversible macroscopic evolutThe<br />
logarithmic expression indicates that entropy is an additive quantity<br />
(S 1 +2 = S 1 + S2), while the number of complexions is multiplicative<br />
(PI +2 =PI· P2).
125 ENERGY AND THE INDUSTRIAL AGE<br />
tion. Of course, the particles will go on moving from one half<br />
to the other, but on the average, at any given instant, as many<br />
will be going in one direction as in the other. As a result, their<br />
motion will cause only small, short-lived fluctuations around<br />
the equilibrium state N1 =N2• Boltzmann's probabilistic interpretation<br />
thus makes it possible to understand the specificity<br />
of t h e attractor studied by equilibrium thermodynamics.<br />
This is not the whole story, and we shall devote the third<br />
part of this book to a more detailed discussion. A few remarks<br />
suffice here. In classical mechanics (and, as we shall see, in<br />
quantum mechanics as well), everything is determined in<br />
terms of initial states and the laws of motion. How then does<br />
probability enter the description of nature? Here it is common<br />
to invoke our ignorance of the exact dynamic state of the system.<br />
This is the subjectivistic interpretation of entropy. Such<br />
an interpretation was acceptable when irreversible processes<br />
were considered to be mere nuisances corresponding to friction<br />
or, more generally, to losses in the functioning of thermal<br />
engines. But today the situation has changed. As we shall see,<br />
irreversible processes have an immense constructive importance:<br />
life would not be possible without them. The subjectivistic<br />
interpretation is therefore highly questionable. Are we<br />
ourselves merely the result of our ignorance, of the fact that<br />
we only observe macroscopic states.<br />
Moreover, both in thermodynamics as well as in its probabilistic<br />
interpretation, there appears a dissymmetry in time:<br />
entropy increases in the direction of the future, not of the past.<br />
This seems impossible when we consider dynamic equations that<br />
are invariant in respect to time inversion. As we shall see, the<br />
second law is a selection principle compatible with dynamics<br />
but not deducible from it. It limits the possible initial conditions<br />
available to a dynamic system. The second law therefore<br />
marks a radical departure from the mechanistic world of classical<br />
or quantum dynamics. Let us now return to Boltzmann's<br />
work.<br />
So far we have discussed isolated systems in which the number<br />
of particles as well as the total energy are fixed by the<br />
boundary conditions. However, it is possible to extend<br />
Boltzmann's explanation to open systems that interact with<br />
their environment. In a closed system, defined by boundary<br />
conditions such that its temperature Tis kept constant by heat
ORDER OUT OF CHAOS<br />
126<br />
exchange with the environment, equilibrium is not defined in<br />
terms of maximum entropy but in terms of the minimum of a<br />
similar function, free energy: F=E-TS , where E is the energy<br />
of the system and Tis the temperature (measured on the<br />
so-called Kelvin scale, where the freezing point of water is<br />
273°C and its boiling point is 373°C).<br />
This formula signifies that equilibrium is the result of competition<br />
between energy and entropy. Temperature is what<br />
determines the relative weight of the two factors. At low<br />
temperatures, energy prevails, and we have the formation of<br />
ordered (weak-entropy) and low-energy structures such as<br />
crystals. Inside these structures each molecule interacts with<br />
its neighbors, and the kinetic energy involved is small compared<br />
with the potential energy that results from the interactions<br />
of each molecule with its neighbors. We can imagine<br />
each particle as imprisoned by its interactions with its neighbors.<br />
At high temperatures, however, entropy is dominant and<br />
so is molecular disorder. The importance of relative motion<br />
increases, and the regularity of the crystal is disrupted; as the<br />
temperature increases, we first have the liquid state, then the<br />
gaseous state.<br />
The entropy S of an isolated system and the free energy F of<br />
a system at fixed temperature are examples of "thermodynamic<br />
potentials." The extremes of thermodynamic potentials<br />
such as S or F define the attractor states toward which systems<br />
whose boundary conditions correspond to the definition<br />
of these potentials tend spontaneously.<br />
Boltzmann's principle can also be used to study the coexistence<br />
of structures (such as the liquid phase and the solid<br />
phase) or the equilibrium between a crystallized product and<br />
the same product in solution. It is important to remember,<br />
however, that equilibrium structures are defined on the molecular<br />
level. It is the interaction between molecules acting<br />
over a range of the order of some to-s em, the same order of<br />
magnitude as the diameter of atoms in molecules, that makes a<br />
crystal structure stable and endows it with its macroscopic<br />
properties. Crystal size, on the other hand, is not an intrinsic<br />
property of structure. It depends on the quantity of matter in<br />
the crystalline phase at equilibrium.
127 ENERGY AND THE INDUSTRIAL AGE<br />
Carnot and Darwin<br />
Equilibrium thermodynamics provides a satisfactory explanation<br />
for a vast number of physicochemical phenomena. Yet it<br />
may be asked whether the concept of equilibrium structures<br />
encompasses the different structures we encounter in nature.<br />
Obviously the answer is no.<br />
Equilibrium structures can be seen as the results of statistical<br />
compensation for the activity of microscopic elements<br />
(molecules, atoms). By definition they are inert at the global<br />
level. For this reason they are also "immortal." Once they<br />
have been formed, they may be isolated and maintained indefinitely<br />
without further interaction with their environment.<br />
When we examine a biological cell or a city, however, the situation<br />
is quite different: not only are these systems open, but<br />
also they exist only because they are open. They feed on the<br />
flux of matter and energy coming to them from the outside<br />
world. We can isolate a crystal, but cities and cells die when<br />
cut off from their environment. They form an integral part of<br />
the world from which they draw sustenance, and they cannot<br />
be separated from the fluxes that they incessantly transform.<br />
However, it is not only living nature that is profoundly alien<br />
to the models of thermodynamic equilibrium. Hydrodynamics<br />
and chemical reactions usually involve exchanges of matter<br />
and energy with the outside world.<br />
It is difficult to see how Boltzmann's order principle can be<br />
applied to such situations. The fact that a system becomes<br />
more uniform in the course of time can be understood in terms<br />
of complexions; in a state of uniformity, when the "differences"<br />
created by the initial conditions have been f<strong>org</strong>otten,<br />
the number of complexions will be maximum. But it is impossible<br />
to understand spontaneous convection from this point of<br />
view. The convection current calls for coherence, for the cooperation<br />
of a vast number of molecules. It is the opposite of<br />
disorder, a privileged state to which only a comparatively<br />
small number of complexions may correspond. In Boltzmann's<br />
terms, it is an "improbable" state. If convection must<br />
be considered a "miracle,·· what then is there to say about life,
ORDER OUT OF CHAOS 128<br />
with its highly specific features present in the simplest <strong>org</strong>anisms?<br />
The question of the relevance of equilibrium models can be reversed.<br />
In order to produce equilibrium, a system must be<br />
"protected" from the fluxes that compose nature. It must be<br />
"canned," so to speak, or put in a bottle, like the homunculus<br />
in Goethe's Fa ust, who addresses to the alchemist who created<br />
him: "Come, press me tenderly to your breast, but not<br />
too hard, for fear the glass might break. This is the way things<br />
are: something natural, the whole world hardly suffices what<br />
is, but what is artificial demands a closed space." In the world<br />
that we are familiar with, equilibrium is a rare and precarious<br />
state. Even evolution toward equilibrium implies a world like<br />
ours, far enough away from the sun for the partial isolation of a<br />
system to be conceivable (no "canning" is possible at the temperature<br />
of the sun), but a world in which nonequilibrium remains<br />
the rule, a "lukewarm" world where equilibrium and<br />
nonequilibrium coexist.<br />
For a long time, however, physicists thought they could define<br />
the inert structure of crystals as the only physical order<br />
that is predictable and reproducible and approach equilibrium<br />
as the only evolution that could be deduced from the fundamental<br />
laws of physics. Thus any attempt at extrapolation<br />
from thermodynamic descriptions was to define as rare and<br />
unpredictable the kind of evolution described by biology and<br />
the social sciences. How, for example, could Darwinian evolution-the<br />
statistical selection of rare events-be reconciled<br />
with the statistical disappearance of all peculiarities, of all rare<br />
configurations, described by Boltzmann? As Roger Caillois21<br />
asks: "Can Carnot and Darwin both be right?"<br />
It is interesting to note how similar in essence the Darwinian<br />
approach is to the path explored by Boltzmann. This may be<br />
more than a coincidence. We know that Boltzmann had immense<br />
admiration for Darwin. Darwin's theory begins with an<br />
assumption of the spontaneous fluctuations of species; then<br />
selection leads to irreversible biological evolution. Therefore,<br />
as with Boltzmann, a randomness leads to irreversibility. Yet<br />
the result is very different. Boltzmann's interpretation implies<br />
the f<strong>org</strong>etting of initial conditions, the "destruction" of initial<br />
structures, while Darwinian evolution is associated with self<strong>org</strong>anization,<br />
ever-increasing complexity.
129 ENERGY AND THE INDUSTRIAL AGE<br />
To sum up our argument so far, equilibrium thermodynamics<br />
was the first response of physics to the problem of nature's<br />
complexity. This response was expressed in terms of the dissipation<br />
of energy, the f<strong>org</strong>etting of initial conditions, and evolution<br />
toward disorder. Classical dynamics, the science of eternal,<br />
reversible trajectories, was alien to the problems facing the<br />
nineteenth century, which was dominated by the concept of<br />
evolution. Equilibrium thermodynamics was in a position to<br />
oppose its view of time to that of other sciences: for thermodynamics,<br />
time implies degradation and death. As we have seen,<br />
Diderot had already asked the question: Where do we, <strong>org</strong>anized<br />
beings endowed with sensations, fit in an inert world<br />
subject to dynamics? There is another question, which has<br />
plagued us for more than a century: What significance does<br />
the evolution of a living being have in the world described by<br />
thermodynamics, a world of ever-increasing disorder? What is<br />
the relationship between thermodynamic time, a time headed<br />
toward equilibrium, and the time in which evolution toward<br />
increasing complexity is occurring?<br />
Was Bergson right? Is time the very medium of innovation,<br />
or is it nothing at all?
CHAPTERV<br />
THE THREE STAGES<br />
OF THERMODYNAMICS<br />
Flux and Force<br />
Let us return I to the description of the second law given in the<br />
previous chapter. The concept of entropy plays a central role<br />
in the description of evolution. As we have seen, its variation<br />
can be written as the sum of two terms-the term deS, linked<br />
to the exchanges between the system and the rest of the world,<br />
and a production term, diS, resulting from irreversible phenomena<br />
inside the system. This term is always positive except<br />
at thermodynamic equilibrium, when it becomes zero. For isolated<br />
systems (deS= 0), the equilibrium state corresponds to a<br />
state of maximum entropy.<br />
In order to appreciate the significance of the second law for<br />
physics, we need a more detailed description of the various<br />
irreversible phenomena involved in the entropy production diS<br />
or in the entropy production per unit time P= diS/dt.<br />
For us chemical reactions are of particular significance. Together<br />
with heat conduction, they form the prototype of irreversible<br />
processes. In addition to their intrinsic importance,<br />
chemical processes play a fundamental role in biology. The<br />
living cell presents an incessant metabolic activity. There<br />
thousands of chemical reactions take place simultaneously to<br />
transform the matter the cell feeds on, to synthesize the fundamental<br />
biomolecules, and to eliminate waste products. As<br />
regards both the different reaction rates and the reaction sites<br />
within the cell, this chemical activity is highly coordinated.<br />
The biological structure thus combines order and activity. In<br />
contrast, an equilibrium state remains inert even though it may<br />
be structured, as, for example, with a crystal. Can chemical<br />
131
ORDER OUT OF CHAOS 132<br />
processes provide us with the key to the difference between<br />
the behavior of a cry stal and that of a cell?<br />
We will have to consider chemical reactions from a dual<br />
point of view, both kinetic and thermodynamic.<br />
From the kinetic point of view, the fundamental quantity is the<br />
reaction rate. The classical theory of chemical kinetics is<br />
based on the assumption that the rate of a chemical reaction is<br />
proportional to the concentrations of the products taking part<br />
in it. Indeed, it is through collisions between molecules that a<br />
reaction takes place, and it is quite natural to assume that the<br />
number of collisions is proportional to the product of the concentrations<br />
of the reacting molecules.<br />
For the sake of example, let us take a simple reaction such<br />
as A + X B + Y. This "reaction equation" means that whenever<br />
a molecule of component A encounters a molecule of X,<br />
there is a certain probability that a reaction will take place and<br />
a molecule of B and a molecule of Y will be produced. A collision<br />
producing such a change in the molecules involved is a<br />
"reactive collision." Only a usually very small fraction (for<br />
example, 111 (6) of all collisions are of this kind. In most cases,<br />
the molecules retain their original nature and merely exchange<br />
energy.<br />
Chemical kinetics deals with changes in the concentration<br />
of the different products involved.in a reaction. This kinetics is<br />
described by differential equations, just as motion is described<br />
by the Newtonian equations. However, in this case, we are not<br />
calculating acceleration but the rates of change of concentration,<br />
and these rates are expressed as a function of the<br />
concentrations of the reactants. The rate of change of concentration<br />
of X, dXldt, is thus proportional to the product of<br />
the concentrations of A and X in the solution-that is,<br />
dXldt= -kA'X, where k is a proportionality factor that is<br />
linked to quantities such as temperature and pressure and that<br />
provides a measure for the fraction of reactive collisions taking<br />
place and leading to the reaction A + X Y + B. Since, in<br />
the example taken, whenever a molecule of X disappears, a<br />
molecule of A disappears too, and a molecule of Yand one of<br />
B are formed, the rates of change of concentration are related:<br />
dXldt=dAldt= -dYldt= -dBldt.<br />
But if the collision between a molecule of X and a molecule
133 THE THREE STAGES OF THERMODYNAMICS<br />
of A can set off a chemical reaction, the collision between molecules<br />
of Y and B can set off the opposite reaction. A second<br />
reaction Y + B-.X +A thus occurs within the system described,<br />
bringing about a supplementary variation in the concentration<br />
of X, dX/dt = k' YB. The total variation in<br />
concentration of a chemical compound is given by the balance<br />
between the forward and the reverse reaction. In our example,<br />
dX/dt (= -dY/dt= . . .)= -kAX+k'YB.<br />
If left to itself, a system in which chemical reactions occur<br />
tends toward a state of chemical equilibrium. Chemical equilibrium<br />
is therefore a typical example of an "attractor" state.<br />
Whatever its initial chemical composition, the system spontaneously<br />
reaches this final stage, where the forward and reverse<br />
reactions compensate one another statistically so that<br />
there is no longer any overall variation in the concentrations<br />
(dX/dt=O). This compensation implies that the ratio between<br />
equilibrium concentrations is given by AXIYB= k'lk= K. This<br />
result is known as the "law of mass action," or Guldberg and<br />
Waage's law, and K is the equilibrium constant. The ratio between<br />
concentrations determined by the law of mass action<br />
corresponds to chemical equilibrium in the same way that uniformity<br />
of temperature (in the case of an isolated system) corresponds<br />
to thermal equilibrium. The corresponding entropy<br />
production vanishes.<br />
Before we deal with the thermodynamic description of<br />
chemical reactions, let us briefly consider an additional aspect<br />
of the kinetic description. The rate of chemical reactions is<br />
affected not only by the concentrations of the reacting molecules<br />
and thermodynamic parameters (for example, pressure<br />
and temperature) but also may be affected by the presence in<br />
the system of chemical substances that modify the reaction<br />
rate without themselves being changed in the process. Substances<br />
of this kind are known as "catalysts." Catalysts can,<br />
for instance, modify the value of the kinetic constants k or k'<br />
or even allow the system to follow a new "reaction path. " In<br />
biology, this role is played by specific proteins, the "enzymes."<br />
These macromolecules have a spatial configuration<br />
that allows them to modify the rate of a given reaction. Often<br />
they are highly specific and affect only one reaction. A possible<br />
mechanism for the catalytic effect of enzymes is to present
ORDER OUT OF CHAOS<br />
134<br />
different "reaction sites" to which the different molecules involwd<br />
in the reaction tend to attach themselves, thus increasing<br />
the likelihood of their coming into contact and reacting.<br />
One very important type of catalysis, particularly in biology,<br />
is the one in which the presence of a product is required<br />
for its own synthesis. In other words, in order to produce the<br />
molecule X we must begin with a system already containing X.<br />
Very frequently, for instance, the molecule X activates an enzyme.<br />
By attaching itself to the enzyme it stabilizes that particular<br />
configuration in which the reaction site is available. To<br />
such an autocatalysis process correspond reaction schemes<br />
such as A+ 2X 3X; in the presence of molecules X, a molecule<br />
A is converted into a molecule X. Therefore we need X to<br />
produce more X. This reaction may be symbolized by the reaction<br />
"loop":<br />
A<br />
One important feature of systems involving such "reaction<br />
loops" is that the kinetic equations describing the changes occurring<br />
in them are nonlinear differential equations .<br />
. If we apply the same method as above, the kinetic equation<br />
obtained for the reaction A+ 2X 3X is dX/dt = kA)(2, where<br />
the rate of variation of the concentration of X is proportional<br />
to the square of its concentration.<br />
Another very important class of catalytic reactions in biology<br />
is that of crosscatalysis-for example, 2X + Y3X, B +X<br />
Y + D, which may be represented by the loop of Figure 3.<br />
This is a case of crosscatalysis, since X is produced from Y,<br />
and simultaneously Y from X. Catalysis does not necessarily<br />
increase the reaction rate; it may, on the contrary, lead to inhibition,<br />
which can also be represented by suitable feedback<br />
loops.<br />
The peculiar mathematical properties of the nonlinear differential<br />
equations describing chemical processes with catalytic<br />
steps are vitally important, as we shall see later, for the<br />
thermodynamics of far-from-equilibrium chemical processes.<br />
In addition, as we have already mentioned, molecular biology
135 THE THREE STAGES OF THERMODYNAMICS<br />
,.... -----£ a<br />
D<br />
or<br />
A -.x<br />
e.x -.v.o<br />
2.X•V -..Jx<br />
x-.E<br />
Figure 3. This graph represents the reaction paths for the "Brusselator"<br />
reactions, which are further described in the text.<br />
has established that these loops play an essential role in metabolic<br />
functions. For example, the relation between nucleic<br />
acids and proteins can be described in terms of a crosscatalytic<br />
effect: nucleic acids contain the information to produce<br />
proteins, which in turn produce nucleic acids.<br />
In addition to the rates of chemical reactions, we must also<br />
consider the rates of other irreversible processes, such as heat<br />
transfer and the diffusion of matter. The rates of irreversible<br />
processes are also called fluxes and are denoted by the symbol<br />
J. There is no general theory from which we can derive the<br />
form of the rates or fluxes. In chemical reactions the rate depends<br />
on the molecular mechanism, as can be verified by the<br />
examples already indicated. The thermodynamics of irreversible<br />
processes introduces a second type of quantity: in addition<br />
to the rates, or fluxes, J, it uses "generalized forces," X, that<br />
"cause" the fluxes. The simplest example is that of heat conduction.<br />
Fourier's law tells us that the heat flux J is proportional<br />
to the temperature gradient. This temperature gradient<br />
is the "force" causing the heat flux. By definition, flux and<br />
forces both vanish at thermal equilibrium. As we shall see, the<br />
production of entropy P= diS/dt can be calculated from the<br />
flux and the forces.<br />
Let us consider the definition of the generalized force corresponding<br />
to a chemical reaction. Recall the reaction A+ X
ORDER OUT OF CHAOS 136<br />
-+ Y + B. We have seen how, at equilibrium, the ratio between<br />
concentrations is given by the law of mass action. As Theophile<br />
De Donder has shown, a "chemical force" can be introduced,<br />
the "affinity" a that determines the direction of the<br />
chemical reaction rate just as the temperature gradient determines<br />
the direction in which heat will flow. In the case of the<br />
reaction we are considering, the affinity is proportional to log<br />
KB YIAX, where K is the equilibrium constant. It is immediately<br />
apparent that the affinity a vanishes at equilibrium<br />
where, following the law of mass action, we have AXIBY=K.<br />
The affinity increases (in absolute value) when we drive the<br />
system away from equilibrium. We can see this if we eliminate<br />
from the system a fraction of the molecules B once they are<br />
formed through the reaction A + X-+ Y + B. Affinity can be said<br />
to measure the distance between the actual state of the system<br />
and its equilibrium state. Moreover, as we have mentioned, its<br />
sign determines the direction of the chemical reaction. If a is<br />
positive, then there are "too many" molecules B and Y, and<br />
the net reaction proceeds in the direction B + Y -+A+ X. On the<br />
contrary, if a is negative there are "too few" B and Y, and the<br />
net reaction proceeds in the opposite direction.<br />
Affinity as we have defined it is a way of rendering more<br />
precise the ancient affinity described by the alchemists, who<br />
deciphered the elective· relationships between chemical<br />
bodies-that is, the "likes" and "dislikes" of molecules. The<br />
idea that chemical activity cannot be reduced to mechanical<br />
trajectories, to the calm domination of dynamic laws, has been<br />
emphasized from the beginning. We could cite Diderot at<br />
length. Later, Nietzsche, in a different context, asserted that it<br />
was ridiculous to speak of "chemical laws," as though chemical<br />
bodies were governed by laws similar to moral laws. In<br />
chemistry, he protested, there is no constraint, and each body<br />
does as it pleases. It is not a matter of "respect" but of a power<br />
struggle, of the ruthless domination of the weaker by the<br />
stronger.2 Chemical equilibrium, with vanishing affinity, corresponds<br />
to the resolution of this conflict. Seen from this point<br />
of view, the specificity of thermodynamic affinity thus rephrases<br />
an age-old problem in modern language,3 the problem<br />
of the distinction between the legal and indifferent world of<br />
dynamic law, and the world of spontaneous and productive<br />
activity to which chemical reactions belong.
137 THE THREE STAGES OF THERMODYNAMICS<br />
Let us emphasize the basic conceptual distinction between<br />
physics and chemistry. In classical physics we can at least<br />
conceive of reversible processes such as the motion of a frictionless<br />
pendulum. To neglect irreversible processes in dynamics<br />
always corresponds to an idealization, but, at least in<br />
some cases, it is a meaningful one. The situation in chemistry<br />
is quite different. Here the processes that define chemistrychemical<br />
transformations characterized by reaction ratesare<br />
irreversible. For this reason chemistry cannot be reduced<br />
to the idealization that lies at the basis of classical or quantum<br />
mechanics, in which past and future play equivalent roles.<br />
As could be expected, all possible irreversible processes appear<br />
in entropy production. Each of them enters through the<br />
product of its rate or flux J multiplied by the corresponding<br />
force X. The total entropy production per unit time, P= diS/dt,<br />
is the sum of these contributions. Each of them appears<br />
through the product JX.<br />
We can divide thermodynamics into three large fields, the<br />
study of which corresponds to three successive stages in its<br />
development. Entropy production, the fluxes, and the forces<br />
are all zero at equilibrium. In the close-to-equilibrium region,<br />
where thermodynamic forces are "weak," the rates Jk are linear<br />
functions of the forces. The third field is called the "nonlinear"<br />
region, since in it the rates are in general more<br />
complicated functions of the forces. Let us first emphasize<br />
some general features of linear thermodynamics that apply to<br />
close-to-equilibrium situations.<br />
Linear Thermodynamics<br />
In 1931, Lars Onsager discovered the first general relations<br />
in nonequilibrium thermodynamics for the linear, near-toequilibrium<br />
region. These are the famous "reciprocity relations."<br />
In qualitative terms, they state that if a force-say,<br />
"one" (corresponding, for example, to a temperature gradient)-may<br />
influence a flux "two" (for example, a diffusion<br />
process), then force "two" (a concentration gradient) will also<br />
influence the flux .. one .. (the heat flow). This has indeed been<br />
verified. For example, in each case where a thermal gradient
ORDER OUT OF CHAOS<br />
138<br />
induces a process of diffusion of matter, we find that a concentration<br />
gradient can set up a heat flux through the system.<br />
The general nature of Onsager's relations has to be emphasized.<br />
It is immaterial, for instance, whether the irreversible<br />
processes take place in a gaseous, liquid, or solid medium.<br />
The reciprocity expressions are valid independently of any microscopic<br />
assumptions.<br />
Reciprocity relations have been the first results in the thermodynamics<br />
of irreversible processes to indicate that this was<br />
not some ill-defined no-man 's-tand but a worthwhile subject of<br />
study whose fertility could be compared with that of equilibrium<br />
thermodynamics. Equilibrium thermodynamics was<br />
an achievement of the nineteenth century, nonequilibrium<br />
thermodynamics was developed in the twentieth century, and<br />
Onsager's relations mark a crucial point in the shift of interest<br />
away from equilibrium toward nonequilibrium.<br />
A second general result in this field of linear, nonequilibrium<br />
thermodynamics bears mention here. We have already<br />
spoken of thermodynamic potentials whose extrema correspond<br />
to the states of equilibrium toward which thermodynamic evolution<br />
tends irreversibly. Such are the entropy S for is
139 THE THREE STAGES OF THERMODYNAMICS<br />
of time. Therefore its time variation dS = 0 vanishes. But we<br />
have seen that the time variation of entropy is made up of two<br />
terms-the entropy flow deS and the positive entropy production<br />
diS. Therefore, dS=O implies that deS= -diS
ORDER OUT OF CHAOS 140<br />
any specificity. Carnot or Darwin? The paradox mentioned in<br />
Chapter IV remains. There is still no connection between the<br />
appearance of natural <strong>org</strong>anized forms on one side, and on the<br />
other the tendency toward "f<strong>org</strong>etting" of initial conditions,<br />
along with the resulting dis<strong>org</strong>anization.<br />
Far "from Equilibrium<br />
At the root of nonlinear thermodynamics lies something quite<br />
surprising, something that first appeared to be a failure: in<br />
spite of much effort, the generalization of the theorem of minimum<br />
entropy production for systems in which the fluxes are<br />
no longer linear functions of the forces appeared impossible.<br />
Far from equilibrium, the system may still evolve to some<br />
steady state, but in general this state can no longer be characterized<br />
in terms of some suitably chosen potential (such as<br />
entropy production for near-equilibrium states).<br />
The absence of any potential function raises a new question:<br />
What can we say about the stability of the states toward which<br />
the system evolves? Indeed, as long as the attractor state is<br />
defined by the minimum of a potential such as the entropy<br />
production, its stability is guaranteed. It is true that a fluctuation<br />
may shift the system away from this minimum. The second<br />
law of thermodynamics, however, imposes the return<br />
toward the attractor. The system is thus "immune" with respect<br />
to fluctuations. Thus whenever we define a potential, we<br />
are describing a "stable world" in which systems follow an<br />
evolution that leads them to a static situation that is established<br />
once and for all.<br />
When the thermodynamic forces acting on a system become<br />
such that the linear region is exceeded, however, the stability<br />
of the stationary state, or its independence from fluctuations,<br />
can no longer be taken for granted. Stability is no longer the<br />
consequence of the general laws of physics. We must examine<br />
the way a stationary state reacts to the different types of fluctuation<br />
produced by the system or its environment. In some<br />
cases, the analysis leads to the conclusion that a state is "unstable"-in<br />
such a state, certain fluctuations, instead of re-
141 THE THREE STAGES OF THERMODYNAMICS<br />
gressing, may be amplified and invade the entire system,<br />
compelling it to evolve toward a new regime that may be<br />
qualitatively quite different from the stationary states corresponding<br />
to minimum entropy production.<br />
Thermodynamics leads to an initial general conclusion concerning<br />
systems that are liable to escape the type of order governing<br />
equilibrium. These systems have to be "far from<br />
equilibrium." In cases where instability is possible, we have to<br />
ascertain the threshold, the distance from equilibrium, at<br />
which fluctuations may lead to new behavior, different from<br />
the "normal" stable behavior characteristic of equilibrium or<br />
near-equilibrium systems.<br />
Why is this conclusion so interesting?<br />
Phenomena of this kind are well known in the field of hydrodynamics<br />
and fluid flow. For instance, it has long been known<br />
that once a certain flow rate of flux has been reached, turbulence<br />
may occur in a fluid. Michel Serres has recently recalled4<br />
that the early atomists were so concerned about<br />
turbulent flow that it seems legitimate to consider turbulence<br />
as a basic source of inspiration of Lucretian physics. Sometimes,<br />
wrote Lucretius, at uncertain times and places, the<br />
eternal, universal fall of the atoms is disturbed by a very slight<br />
deviati0n-the "clinamen." The resulting vortex gives rise to<br />
the world, to all natural things. The clinamen, this spontaneous,<br />
unpredictable deviation, has often been criticized as<br />
one of the main weaknesses of Lucretian physics, as being<br />
something introduced ad hoc. In fact, the contrary is truethe<br />
clinamen attempts to explain events such as laminar flow<br />
ceasing to be stable and spontaneously turning into turbulent<br />
flow. Today hydrodynamic experts test the stability of fluid<br />
flow by introducing a perturbation that expresses the effect of<br />
molecular disorder added to the average flow. We are not so far<br />
from the clinamen of Lucretius!<br />
For a long time turbulence was identified with disorder or<br />
noise. Today we know that this is not the case. Indeed, while<br />
turbulent motion appears as irregular or chaotic on the macroscopic<br />
scale, it is, on the contrary, highly <strong>org</strong>anized on the<br />
microscopic scale. The multiple space and time scales involved<br />
in turbulence correspond to the coherent behavior of<br />
millions and millions of molecules. Viewed in this way, the<br />
transition from laminar flow to turbulence is a process of self-
ORDER OUT OF CHAOS 142<br />
<strong>org</strong>anization. Part of the energy of the system, which in laminar<br />
flow was in the thermal motion of the molecules, is being<br />
transferred to macroscopic <strong>org</strong>anized motion.<br />
The "Benard instability" is another striking example of the<br />
instability of a stationary state giving rise to a phenomenon of<br />
spontaneous self-<strong>org</strong>anization. The instability is due to a vertical<br />
temperature gradient set up in a horizontal liquid layer. The<br />
lower surface of the latter is heated to a given temperature,<br />
which is higher than that of the upper surface. As a result of<br />
these boundary conditions, a permanent heat flux is set up,<br />
moving from the bottom to the top. When the imposed gradient<br />
reaches a threshold value, the fluid's state of rest-the<br />
stationary state in which heat is conveyed by conduction<br />
alone, without convection-becomes unstable. A convection<br />
corresponding to the coherent motion of ensembles of molecules<br />
is produced, increasing the rate of heat transfer. Therefore,<br />
for given values of the constraints (the gradient of<br />
temperature), the entropy production of the system is increased;<br />
this contrasts with the theorem of minimum entropy<br />
production. The Benard instability is a spectacular phenomenon.<br />
The convection motion produced actually consists<br />
of the complex spatial <strong>org</strong>anization of the system. Millions of<br />
molecules move coherently, forming hexagonal convection<br />
cells of a characteristic size.<br />
In Chapter IV we introduced Boltzmann's order principle,<br />
which relates entropy to probability as expressed by the number<br />
of complexions P. Can we apply this relation here? To each<br />
distribution of the velocities of the molecules corresponds a<br />
number of complexions. This number measures the number of<br />
ways in which we can realize the velocity distribution by attributing<br />
some velocity to each molecule. The argument runs<br />
parallel to that in Chapter IV, where we expressed the number<br />
of complexions in terms of the distributions of molecules between<br />
two boxes. Here also the number of complexions is<br />
large when there is disorder-that is, a wide dispersion of velocities.<br />
In contrast, coherent motion means that many molecules<br />
travel with nearly the same speed (small dispersion of<br />
velocities). To such a distribution corresponds a number of<br />
complexions P so low that there seems almost no chance for<br />
the phenomenon of self-<strong>org</strong>anization to occur. Yet it occurs!<br />
We see, therefore, that calculating the number of complexions,
1-43<br />
THE THREE STAGES OF THERMODYNAMICS<br />
which entails the hypothesis of an equal a priori probability for<br />
each molecular state, is misleading. Its irrelevance is particularly<br />
obvious as far as the genesis of the new behavior is<br />
concerned. In the case of the Benard instability it is a fluctuation,<br />
a microscopic convection current, which would have<br />
been doomed to regression by the application of Boltzmann's<br />
order principle, but which on the contrary is amplified until it<br />
invades the whole system. Beyond the critical value of the imposed<br />
gradient, a new molecular order has thus been produced<br />
spontaneously. It corresponds to a giant fluctuation stabilized<br />
through energy exchanges with the outside world.<br />
In far-from-equilibrium conditions, the concept of probability<br />
that underlies Boltzmann's order principle is no longer<br />
valid in that the structures we observe do not correspond to a<br />
maximum of complexions. Neither can they be related to a<br />
minimum of the free energy F = E- TS. The tendency toward<br />
leveling out and f<strong>org</strong>etting initial conditions is no longer a general<br />
property. In this context, the age-old problem of the origin<br />
cf life appears in a different perspective. It is certainly true that<br />
life is incompatible with Boltzmann's order principle but not with<br />
the kind of behavior that can occur in far-from-equilibrium<br />
conditions.<br />
Classical thermodynamics leads to the concept of "equilibrium<br />
structures" such as crystals. Benard cells are structures<br />
too, but of a quite different nature. That is why we have<br />
introduced the notion of "dissipative structures," to emphasize<br />
the close association, at first paradoxical, in such situations<br />
between structure and order on the one side, and<br />
dissipation or waste on the other. We have seen in Chapter IV<br />
that heat transfer was considered a source of waste in classical<br />
thermodynamics. In the Benard cell it becomes a source of<br />
order.<br />
The interaction of a system with the outside world, its embedding<br />
in nonequilibrium conditions, may become in this way<br />
the starting point for the formation of new dynamic states of<br />
matter-dissipative structures. Dissipative structures actually<br />
correspond to a form of supramolecular <strong>org</strong>anization. Although<br />
the parameters describing crystal structures may be<br />
derived from the properties of the molecules of which they are<br />
composed, and in particular from the range of their forces of<br />
attraction and repulsion, Benard cells, like all dissipative
ORDER OUT OF CHAOS<br />
144<br />
structures, are essentially a reflection of the global situation of<br />
nonequilibrium producing them. The parameters describing<br />
them are macroscopic; they are not of the order of 10-8 cm,<br />
like the distance between the molecules of a crystal, but of the<br />
order of centimeters. Similarly, the time scales are differentthey<br />
correspond not to molecular times (such as periods of<br />
vibration of individual molecules, which may correspond to<br />
about 10-15 sec) but to macroscopic times: seconds, minutes,<br />
or hours.<br />
Let us return to the case of chemical reactions. There are<br />
some fundamental differences from the Benard problem. In<br />
the Benard cell the instability has a simple mechanical origin.<br />
When we heat the liquid layer from below, the lower part of the<br />
fluid becomes less dense, and the center of gravity rises. It is<br />
therefore not surprising that beyond a critical point the system<br />
tilts and convection sets in.<br />
But in chemical systems there are no mechanical features of<br />
this type. Can we expect any self-<strong>org</strong>anization? Our mental<br />
image of chemical reactions corresponds to molecules speeding<br />
through space, colliding at random in a chaotic way. Such<br />
an image leaves no place for self-<strong>org</strong>anization, and this may be<br />
one of the reasons why chemical instabilities have only recently<br />
become a subject of interest. There is also another difference.<br />
All flows become turbulent at a "sufficiently" large<br />
distance from equilibrium (the threshold is measured by dimensionless<br />
numbers such as Reynolds' number). This is not<br />
true for chemical reactions. Being far from equilibrium is a<br />
necessary requirement but not a sufficient one. For many<br />
chemical systems, whatever the constraints imposed and the<br />
rate of the chemical changes produced, the stationary state<br />
remains stable and arbitrary fluctuations are damped, as is the<br />
case in the close-to-equilibrium range. This is true in particular<br />
of systems in which we have a chain of transformations of<br />
the type A-+B-+C-+D . .. and that may be described by linear<br />
differential equations.<br />
The fate of the fluctuations perturbing a chemical system,<br />
as well as the kinds of new situations to which it may evolve,<br />
thus depend on the detailed mechanism of the chemical reactions.<br />
In contrast with close-to-equilibrium situations, the<br />
behavior of a far-from-equilibrium system becomes highly specific.<br />
There is no longer any universally valid law from which
145 THE THREE STAGES OF THERMODYNAMICS<br />
the overall behavior of the system can be deduced. Each system<br />
is a separate case; each set of chemical reactions must be<br />
investigated and may well produce a qualitatively different behavior.<br />
Nevertheless, one general result has been obtained, namely<br />
a necessary condition for chemical instability: in a chain of<br />
chemical reactions occurring in the system, the only reaction<br />
stages that, under certain conditions and circumstances, may<br />
jeopardize the stability of the stationary state are precisely the<br />
"catalytic loops"-stages in which the product of a chemical<br />
reaction is involved in its own synthesis. This is an interesting<br />
conclusion, since it brings us closer to some of the fundamental<br />
achievements of modern molecular biology (see Figure 4).<br />
Figure 4. Catalytic loops correspond to nonlinear terms. In the case of a<br />
one-independent-variable problem, this means the occurrence of at least<br />
one term where the independent variable appears with a power higher than<br />
1; in this simple case, it is easy to see the relation between such nonlinear<br />
terms and the potential instability of stationary states.<br />
Let us take for the independent variable X the time evolution dXIdt= f(X). It<br />
is always possible to decompose f(X) in two functions representing a gain<br />
and a loss f + (X) and f _ (X) , each of which is positive or 0, such that<br />
f(X) = f +(X)- f _(X). In this way, stationary states (dX!dt= 0) correspond to<br />
values where f +(X)= f _(X).<br />
Those states are graphically given by the intersections of the two graphs<br />
plotting f + and f _. If f + and f _ are linear, there can only be one intersection.<br />
In other cases, the type of the intersection permits us to infer the stability of<br />
the stationary state.<br />
Four cases are possible:<br />
Sl: stable with respect to negative fluctuations, unstable with respect to<br />
positive ones: If the system deviates slightly to the left of Sl , the positive<br />
difference between f + and f _ will reduce this deviation back to Sl; deviations<br />
to the right will be amplified.<br />
SS: stable with respect to positive and negative fluctuations.<br />
IS: stable only with respect to positive fluctuations.<br />
II: unstable with respect to positive and negative fluctuations.<br />
X
ORDER OUT 0 CHAOS 146<br />
Beyond the Threshold of Chemical Instability<br />
Today the study of chemical instabilities is common. Both theoretical<br />
and experimental work are being pursued in a large<br />
number of institutions and laboratories. Indeed, as will become<br />
clear, these investigations are of interest to a wide range<br />
of scientists-not only to mathematicians, physicists, chemists,<br />
and biologists, but also to economists and sociologists.<br />
In far-from-equilibrium conditions various new phenomena<br />
appear beyond the threshold of chemical instability. To describe<br />
them in a concrete fashion, it is useful to start with a<br />
simplified theoretical model, one that has been developed at<br />
Brussels during the past decade. American scientists have<br />
called this model the "Brusselator," and this name is used in<br />
scientific literature (Geographical associations seem to have<br />
become the rule in this field; in addition to the Brusselator,<br />
there is an "Oregonator," and most recently a "Paloaltonator"<br />
!). Let us briefly describe the Brusselator. The steps<br />
responsible for instability have already been noted (see Figure<br />
3). The product X, synthetized from A and broken down into<br />
the form of E, is linked by a relationship of crosscatalysis to<br />
produce Y. X is produced from Y during a trimolecular step<br />
but, conversely, Y is synthetized by a reaction between X and<br />
a product B.<br />
In this model, the concentrations of the products A, B, D, and<br />
E are given parameters (the "control substances"). The behavior<br />
of the system is explored for increasing values of B, with A<br />
remaining constant. The stationary state toward which such a<br />
system is likely to evolve-the state for which dX/dt = dY/dt<br />
= 0-corresponds to concentrations X0 =A and Y0 = BIA. This<br />
can be easily verified by writing the kinetic equations and<br />
looking for the stationary state. However, this stationary state<br />
ceases to be stable as soon as the concentration of B exceeds a<br />
critical threshold (everything else being kept equal). After the<br />
critical threshold has been reached, the stationary state becomes<br />
an unstable "focus" and the system leaves this focus to<br />
reach a "limit cycle."
147 THE THREE STAGES OF THERMODYNAMICS<br />
0<br />
2 3 4 y<br />
Figure 5. This scheme represents concentration of component X vs. concentration<br />
of component Y. The cycle's focus (point S) is the stationary state,<br />
which is unstable for 8>(1 + A2). All the trajectories (of which five are plotted),<br />
whatever their intitial state, lead to the same cycle.<br />
Instead of remaining stationary, the concentrations of X and Y<br />
begin to oscillate with a well-defined periodicity. The oscillation<br />
period depends both on the kinetic constants characterizing<br />
the reaction rates and the boundary conditions imposed on<br />
the system as a whole (temperature, concentration of A., B,<br />
etc.).<br />
Beyond the critical threshold the system spontaneously<br />
leaves the stationary state X0=A, Y0=BIA as the result of<br />
fluctuations. Whatever the initial conditions, it approaches the<br />
limit cycle, the periodic behavior of which is stable. We therefore<br />
have a periodic chemical process-a chemical clock. Let<br />
us pause a moment to emphasize how unexpected such a phenomenon<br />
is. Suppose we have two kinds of molecules, "red"<br />
and "blue." Because of the chaotic motion of the molecules,<br />
we would expect that at a given moment we would have more<br />
red molecules, say, in the left part of a vessel. Then a bit later<br />
more blue molecules would appear, and so on. The vessel<br />
would appear to us as "violet," with occasional irregular
ORDER OUT OF CHAOS 148<br />
flashes of red or blue. However, this is not what happens with<br />
a chemical clock; here the system is all blue, then it abruptly<br />
changes its col or to red, then again to blue. Because all these<br />
changes occur at regular time intervals, we have a coherent<br />
process.<br />
Such a degree of order stemming from the activity of biIlions<br />
of molecules seems incredible, and indeed, if chemical clocks<br />
had not been observed, no one would believe that such a process<br />
is possible. To change color all at once, molecules must<br />
have a way to "communicate." The system has to act as a<br />
whole. We will return repeatedly to this key word, communicate,<br />
which is of obvious importance in so many fields, from<br />
chemistry to neurophysiology. Dissipative structures introduce<br />
probably one of the simplest physical mechanisms for<br />
communication.<br />
There is an interesting difference between the simplest kind<br />
of mechanical oscillator, the spring, and a chemical clock. The<br />
chemical clock has a well-defined periodicity corresponding<br />
to the limit cycle its trajectory is following. On the contrary, a<br />
spring has a frequency that is amplitude-dependent. From this<br />
point of view a chemical clock is more reliable as a timekeeper<br />
than a spring.<br />
But chemical clocks are not the only type of self-<strong>org</strong>anization.<br />
Until now diffusion has been neglected. All substances were<br />
assumed to be evenly distributed over the reaction space. This<br />
is an idealization; small fluctuations will always lead to differences<br />
in concentrations and thus to diffusion. We therefore<br />
have to add diffusion to the chemical reaction equations. The<br />
diffusion-reaction equations of the Brusselator display an astonishing<br />
range of behaviors available to this system. Indeed,<br />
whereas at equilibrium and near-equilibrium the system remains<br />
spatially homogeneous, the diffusion of the chemical<br />
throughout the system induces, in the far-from-equilibrium region,<br />
the possibility of new types of instability, including the<br />
amplification of fluctuations breaking the initial spatial symmetry.<br />
Oscillations in time, chemical clocks, thus cease to be<br />
the only kind of dissipative structure available to the system.<br />
Far from it; for example, oscillations may appear that are now<br />
both time- and space-dependent. They correspond to chemical<br />
waves of X and Y concentrations that periodically pass through<br />
the system.
149<br />
THE THREE STAGES OF THERMODYNAMICS<br />
X hO<br />
hO.S8<br />
3<br />
o<br />
X h 1.10<br />
o<br />
h 1.88<br />
I--_ror.-____ 1 _ _____ _...,<br />
o<br />
X h2.04<br />
o<br />
h 3.435<br />
3<br />
21....-- - - -<br />
-- - --'"<br />
1 0<br />
o<br />
Figure 6. Chemical waves simulated on computer: successive steps of<br />
evolution of spatial profile of concentration of constituent X in the "Brusselator"<br />
trimolecular model. At time t= 3.435 we recover the same distribution as<br />
at time t= O. Concentration of A and B: 2, 5.45 (B>[1 + A2]). Diffusion coefficients<br />
for X and Y: 810-3,410-3•<br />
In addition, especially when the values of the diffusion constants<br />
of X and Y are quite different from each other, the system<br />
may display a stationary, time-independent behavior, and<br />
stable spatial structures may appear.
ORDER OUT OF CHAOS<br />
150<br />
Here we must pause once again, this time to emphasize how<br />
much the spontaneous formation of spatial structures contradicts<br />
the laws of equilibrium physics and Boltzmann's order<br />
principle. Again, the number of complexions corresponding to<br />
such structures would be extremely small in comparison with<br />
the number in a uniform distribution. Still , nonequilibrium<br />
processes may lead to situations that would appear impossible<br />
from the classical point of view.<br />
The number of different dissipative structures compatible<br />
with a given set of boundary conditions may be increased still<br />
further when the problem is studied in two or three dimensions<br />
instead of one. In a circular, two-dimensional space, for<br />
instance, the spatially structured stationary state may be<br />
characterized by the occurrence of a privileged axis.<br />
X<br />
--· --·<br />
Figure 7. Stationary state with privileged axis obtained by computer simulation.<br />
Concentration X is a function of geometrical coordinates p,a in the<br />
horizontal plane. The location of the perturbation applied to the uniform unstable<br />
solution (X Q<br />
, Y0) is indicated by an arrow.<br />
This corresponds to a new, extremely interesting symmetrybreaking<br />
process, especially when we recall that one of the<br />
first stages in morphogenesis of the embryo is the formation of<br />
a gradient in the system. We will return to these problems later<br />
in this chapter and again in Chapter VI.
151 THE THREE STAGES OF THERMODYNAMICS<br />
Up to now it has been assumed that the "control substances"<br />
(A, B, D, and E) are uniformly distributed throughout<br />
the reaction system. If this simplification is abandoned, additional<br />
phenomena can occur. For example, the system takes<br />
on a "natural size ," which is a function of the parameters describing<br />
it. In this way the system determines its own intrinsic<br />
size-that is, it determines the region that is spatially structured<br />
or crossed by periodic concentration waves.<br />
These results still give a very incomplete picture of the variety<br />
of phenomena that may occur far from equilibrium. Let us<br />
first mention the possibility of multiple states far from equilibrium.<br />
For given boundary conditions there may appear<br />
more than one stationary state-'-for instance one rich in the<br />
chemical X, the other poor. The shift from one state to another<br />
plays an important role in control mechanisms as they have<br />
been described in biological systems.<br />
Since the classical work of Lyapounov and Poincare,<br />
characteristic points such as focus or lines such as limit cycles<br />
Iii<br />
y<br />
Figure 8. (a) Bromide-ion concentration in the Belousov-Zhabotinsky reaction<br />
at times t1 and t1 + T (cf. R. H. Simoyi, A. Wolf, and H. L. Swinney,<br />
Physics Review Letters, Vol. 49 (1982), p. 245; see J. Hirsch, "Condensed<br />
Matter Physics," and on computers, Physics Today (May 1983), pp. 44-52).<br />
(b) Attractor lines calculated by Hao Bai-lin for a Brusselator with external<br />
periodic supply of component X (personal communication).
ORDER OUT OF CHAOS 152<br />
were known to mathematicians as the "attractors" of stable<br />
systems. What is new is their application to chemical systems.<br />
It is worth noting that the first paper dealing with instabilities<br />
in reaction-diffusion systems was published by Thring in 1952.<br />
In recent years new types of attractors have been identified.<br />
They appear only when the number of independent variables<br />
increases (there are two independent variables in the Brusselator,<br />
the variables X and Y). In particular, we can get "strange<br />
attractors" that do not correspond to periodic behavior.<br />
Figure 8, which summarizes some calculations by Hao Bailin,<br />
gives an idea of such very complicated attractor lines calculated<br />
for a model generalizing the Brusselator through the<br />
addition of an external periodic supply of X. What is remarkable<br />
is that most of the possibilities we have described have<br />
been observed in in<strong>org</strong>anic chemistry as well as in a number of<br />
biological situations.<br />
In in<strong>org</strong>anic chemistry the best-known example is the<br />
Belousov-Zhabotinsky reaction discovered in the early 1960s.<br />
The corresponding reaction scheme, the Oregonator, introduced<br />
by Noyes and his colleagues, is in essence similar to the<br />
Brusselator though more complex. The Belousov-Zhabotinsky<br />
reaction consists of the oxidation of an <strong>org</strong>anic acid (malonic<br />
acid) by a potassium bromate in the presence of a suitable catalyst,<br />
cerium, manganese, or ferroin.<br />
INFLOW<br />
MALONIC . ::: ...rP U ::- M :: P :
153 THE THREE STAGES OF THERMODYNAMICS<br />
Various experimental conditions may be set up giving different<br />
forms of auto<strong>org</strong>anization within the same system-a chemical<br />
clock, a stable spatial differentiation, or the formation of<br />
waves of chemical activity over macroscopic distances.5<br />
Let us now turn to a matter of the greatest interest: the relevance<br />
of these results for the understanding of living systems.<br />
The Encounter with Molecular Biology<br />
Earlier in this chapter we showed that in far-from-equilibrium<br />
conditions various types of self-<strong>org</strong>anization processes may<br />
occur. They may lead to the appearance of chemical oscillations<br />
or to spatial structures . We have seen that the basic condition<br />
for the appearance of such phenomena is the existence<br />
of catalytic effects.<br />
Although the effects of "nonlinear" reactions (the presence<br />
of the reaction product) have a feedback action on their<br />
"cause" and are comparatively rare in the in<strong>org</strong>anic world,<br />
molecular biology has discovered that they are virtually the<br />
rule as far as living systems are concerned. Autocatalysis (the<br />
presence of X accelerates its own synthesis), autoinhibition<br />
(the presence of X blocks a catalysis needed to synthesize it),<br />
and crosscatalysis (two products belonging to two different reaction<br />
chains activate each other's synthesis) provide the classical<br />
regulation mechanism guaranteeing the coherence of the<br />
metabolic function.<br />
Let us emphasize an interesting difference. In the examples<br />
known in in<strong>org</strong>anic chemistry, the molecules involved are<br />
simple but the reaction mechanisms are complex-in the<br />
Belousov-Zhabotinsky reaction, about thirty compounds have<br />
been identified. On the contrary, in the many biological examples<br />
we have, the reaction scheme is simple but the molecules<br />
(proteins, nucleic acids, etc.) are highly complex and specific.<br />
This can hardly be an accident. Here we encounter an initial<br />
element marking the difference between physics and biology.<br />
Biological systems have a past. Their constitutive molecules<br />
are the result of an evolution; they have been selected to take<br />
part in the autocatalytic mechanisms to generate very specific<br />
forms of <strong>org</strong>anization processes.<br />
A description of the network of metabolic activations and
ORDER OUT OF CHAOS<br />
154<br />
inhibitions is an essential step in understanding the functional<br />
logic of biological systems. This includes the triggering of syntheses<br />
the moment they are needed and the blocking of those<br />
chemical reactions whose unused products would accumulate<br />
in the cell.<br />
The basic mechanism through which molecular biology explains<br />
the transmission and exploitation of genetic information<br />
is itself a feedback loop, a "nonlinear" mechanism. Deoxyribonucleic<br />
acid (DNA), which contains in sequential form all<br />
the information required for the synthesis of the various basic<br />
proteins needed in cell building and functioning, participates<br />
in a sequence of reactions during which this information is translated<br />
into the form of different protein sequences. Among the<br />
proteins synthesized, some enzymes exert a feedback action<br />
that activates or controls not only the different transformation<br />
stages but also the autocatalytic mechanism of DNA replication,<br />
by which genetic information is copied at the same rate<br />
as the cells multiply.<br />
Here we have a remarkable case of the convergence of two<br />
sciences. The understanding attained here required complementary<br />
developments in physics and biology, one toward the<br />
complex and the other toward the elementary.<br />
Indeed, from the point of view of physics, we now investigate<br />
"complex" situations far removed from the ideal situations<br />
that can be described in terms of equilibrium thermodynamics.<br />
On the other hand, molecular biology succeeded in relating<br />
living structures to a relatively small number of basic<br />
biomolecules. Investigating the diversity of chemical mechanisms,<br />
it discovered the intricacy of the metabolic reaction<br />
chains, the subtle, complex logic of the control, inhibition, and<br />
activation of the catalytic function of the enzymes associated<br />
with the critical step of each of the metabolic chains. In this<br />
way molecular biology provides the microscopic basis for the<br />
instabilities that may occur in far-from-equilibrium conditions.<br />
In a sense, living systems appear as a well-<strong>org</strong>anized factory:<br />
on the one hand, they are the site of multiple chemical<br />
transformations; on the other, they present a remarkable "spacetime"<br />
<strong>org</strong>anization with highly nonuniform distribution of biochemical<br />
material. We can now link function and structure.<br />
Let us briefly consider two examples that have been studied<br />
extensively in the past few years.
155 THE THREE STAGES OF THERMODYNAMICS<br />
First we shall consider glycolysis, the chain of metabolic<br />
reactions during which glucose is broken down and an energyrich<br />
substance ATP (adenosine triphosphate) is synthetized,<br />
providing an essential source of energy common to all living<br />
cells. For each glucose molecule that is broken down , two<br />
molecules of ADP (adenosine disphosphate) are transformed<br />
into two molecules of ATP. Glycolysis provides a fine example<br />
of how complemetary the analytical approach of biology and<br />
the investigation of stability in far-from-equilibrium conditions<br />
are. 6<br />
Biochemical experiments have discovered the existence of<br />
temporal oscillations in concentrations related to the glycolytic<br />
cycle. 7 It has been shown that these oscillations are determined<br />
by a key step in the reaction sequence, a step activated<br />
by ADP and inhibited by ATP. This is a typical nonlinear phenomenon<br />
well suited to regulate metabolic functioning. Indeed,<br />
each time the cell draws on its energy reserves, it is<br />
exploiting the phosphate bonds, and AT P is converted into<br />
ADP. ADP accumulation inside the cell thus signifies intensive<br />
energy consumption and the need to replenish stocks. ATP<br />
accumulation, on the other hand, means that glucose can be<br />
broken down at a slower rate.<br />
Theoretical investigation of this process has shown that this<br />
mechanism is indeed liable to produce an oscillation phenomenon,<br />
a chemical clock. The theoretically calculated values<br />
of the chemical concentrations necessary to produce<br />
oscillation and the period of the cycle agree with the experimental<br />
data. Glycolytic oscillation produces a modulation of<br />
all the cell's energy processes which are dependent on ATP<br />
concentration and therefore indirectly on numerous other metabolic<br />
chains.<br />
We may go farther and show that in the glycolytic pathway<br />
the reactions controlled by some of the key enzymes are in farfrom-equilibrium<br />
conditions. Such calculations have been reported<br />
by Benno Hesss and have since been extended to other<br />
systems. Under usual conditions the glycolytic cycle corresponds<br />
to a chemical clock, but changing these conditions can<br />
induce spatial pattern formations in complete agreement with<br />
the predictions of existing theoretical models.<br />
A living system appears very complex from the thermodynamic<br />
point of view. Certain reactions are close to equi-
ORDER OUT OF CHAOS 156<br />
librium, and others are not. Not everything in a living system<br />
is "alive." The energy flow that crosses it somewhat resembles<br />
the flow of a river that generally moves smoothly but that<br />
from time to time tumbles down a waterfall; which liberates<br />
part of the energy it contains.<br />
Let us consider another biological process that also has<br />
been studied from the point of view of stability: the aggregation<br />
of slime molds, the Acrasiales amoebas (Dictyostelium<br />
discoideum). This process9A is an interesting case on the borderline<br />
between unicellular and pluricellular biology. When<br />
The aggregation of cellular slime molds furnishes a particularly re<br />
markable example of a self-<strong>org</strong>anization phenomenon in a biological<br />
sYltem in which a chemical clock plays an essential role. See Figure A.<br />
fruiting ?<br />
body <br />
spores 0<br />
00<br />
I<br />
(Y:J growth<br />
Q)<br />
\<br />
((t'': kx<br />
\ , .. , I<br />
, -<br />
.. - - '<br />
;'; aggregation<br />
\<br />
157 THE THREE STAGES OF THERMODYNAMICS<br />
In Dictyostelium discoideum, the aggregation proceeds in a periodic manner.<br />
Movies of aggregation process show the existence of concentric waves<br />
of amoebae moving toward the center with a periodicity of several minutes.<br />
The nature of the chemotactic factor is known: it is cyclic AMP (cAMP), a<br />
substance involved in numerous biochemical processes such as hormonal<br />
regulations. The aggregation centers release the signals of cAMP in a periodic<br />
fashion. The other cells respond by moving toward the centers and by<br />
relaying the signals to the periphery of the aggregation territory. The existence<br />
of a mechanism of relay of the chemotactic signals allows each center<br />
to control the aggregation of some 105 amoebae.<br />
The analysis of a model of the process of aggregation reveals the existence<br />
of two types of bifurcations. First the aggregation itself represents a<br />
breaking of spatial symmetry. The second bifurcation breaks the temporal<br />
symmetry.<br />
Initially the amoebae are homogeneously distributed. When some of them<br />
begin to secrete the chemotactic signals, there appear local fluctuations in<br />
the concentration of cAMP. For a critical value of some parameter of the<br />
system (diffusion coefficient of cAMp, motility of the amoebae, etc.), fluctuations<br />
are amplified: the homogeneous distribution becomes unstable and the<br />
amoebae evolve toward an inhomogeneous. distribution in space. This new<br />
distribution corresponds to the accumulation of amoebae around aggregation<br />
centers.<br />
To understand the origin of the periodicity in the aggregation of D. discoideum,<br />
it is necessary to study the mechanism of synthesis of the chemotactic<br />
signal. On the basis of experimental observations one can describe<br />
this mechanism by the scheme of Figure B.<br />
_----+:.-- cAM P <br />
Figure B<br />
ATP<br />
Y<br />
cAMP<br />
On the surface of the cell, receptors (R) bind the molecules of cAMP. The<br />
receptor faces the extracellular medium and is functionally linked to an enzyme,<br />
adenylate cyclase (C), which transforms intracellular ATP into cAMP.<br />
The cAMP thus synthesized is transported across the membrane into the<br />
extracellular medium, where it is degraded by phosphodiesterase, an enzyme<br />
that is secreted by the amoebae. The experiments show that binding of<br />
extracellular cAMP to the membrane receptor activates adenylate cyclase<br />
(positive feedback indicated by +).<br />
On the basis of this autocatalytic regulation, the analysiS of a model for
ORDER OUT OF CH AOS<br />
158<br />
cAMP synthesis has permitted unification of different types of behavior observed<br />
during aggregation.se<br />
Two key parameters of the model are the concentrations of adenylate<br />
cyclase (s) and of phosphodiesterase (k). Figure C (redrawn from A. GoLD·<br />
BETER and L. SEGEL, Differentiation, Vol. 17 [1980], pp. 127-35), shows the<br />
behavior of the modelized system in the space formed by s and k.<br />
cu<br />
-<br />
0<br />
:>.<br />
c<br />
cu<br />
"'Q<br />
<<br />
A<br />
Phosphodiesterase ,<br />
k<br />
Figure C<br />
Three regions can be distinguished for different values of k and s. Region<br />
A corresponds to a stable, nonexcitable stationary state; region B to a stationary<br />
state stable but excitable: the system is capable of amplifying small<br />
perturbations in the concentration of cAMP in a pulsatory manner (and thus<br />
of relaying cAMP signals); region C corresponds to a regime of sustained<br />
oscillations around an unstable stationary state.<br />
The arrow indicates a possible "developmental path" corresponding to a<br />
rise in phosphodiesterase (k) and adenylate cyclase (s), a rise that is observed<br />
to occur after the beginning of starvation. The crossing of regions A,<br />
B and C corresponds to the observed change of behavior: cells are at first<br />
incapable of responding to extracellular cAMP signals; thereafter they relay<br />
these signals and, finally, they become capable of synthetizing them periodically<br />
in an autonomous way. The aggregation centers would thus be the<br />
cells for which the parameters s and k have reached the more rapidly a point<br />
located inside region 0 after starvation has begun.
159 THE THREE STAGES OF THERMODYNAMICS<br />
the environment in which these amoebas live and multiply becomes<br />
poor in nutrients, they undergo a spectacular transformation.<br />
(See Figure A.) Starting as a population of isolated<br />
cells, they join to form a mass composed of several tens of<br />
thousands of cells. This "pseudoplasmodium" then undergoes<br />
differentiation, all the while changing shape. A "foot" forms,<br />
consisting of about one third of the cells and containing abundant<br />
cellulose. This foot supports a round mass of spores,<br />
which will detach themselves and spread, multiplying as soon<br />
as they come in contact with a suitable nutrient medium and<br />
thus forming a new colony of amoebas. This is a spectacular<br />
example of adaptation to the environment. The population<br />
lives in one region until it has exhausted the available resources.<br />
It then goes through a metamorphosis by means of<br />
which it acquires the mobility to invade other environments.<br />
An investigation of the first stage of the aggregation process<br />
reveals that it begins with the onset of displacement waves in<br />
the amoeba population, with a pulsating motion of convergence<br />
of the amoebaes toward a "center of attraction,"<br />
which appears to be produced spontaneously. Experimental<br />
investigation and modelization have shown that this migration<br />
is a response by the cells to the existence in the environment<br />
of a concentration gradient in a key substance, cyclic AMP,<br />
which is periodically produced by an amoeba which is the attractor<br />
center and later by other cells through a relay mechanism.<br />
Here we again see the remarkable role of chemical<br />
clocks. They provide, as we have already stressed, new means<br />
of communication. In the present case, the self-<strong>org</strong>anization<br />
mechanism leads to communication between cells.<br />
There is another aspect we wish to emphasize. Slime mold<br />
aggregation is a typical example of what may be termed "order<br />
through fluctuations" : the setting up of the attractor center<br />
giving off the AMP indicates that the metabolic regime corresponding<br />
to a normal nutritive environment has become unstable-that<br />
is, the nutritive environment has become<br />
exhausted. The fact that under such conditions of food shortage<br />
any given amoeba can be the first to emit cyclic AMP and<br />
thus become an attractor center corresponds to the random<br />
behavior of fluctuations. This fluctuation is then amplified and<br />
<strong>org</strong>anizes the medium.
ORDER OUT OF CHAOS<br />
160<br />
Bifurcations and Symmetry-Breaking<br />
Let us take a closer look at the emergence of self-<strong>org</strong>anization<br />
and the processes that occur when we go beyond this threshold.<br />
At equilibrium or near-equilibrium, there is only one<br />
steady state that will depend on the values of some control<br />
parameters. We shall call A the control parameter, which, for<br />
example, may be the concentration of substance B in the<br />
Brusselator described in section 4. We now follow the change<br />
in the state of the system as the value of B increases. In this<br />
way the system is pushed farther and farther away from equilibrium.<br />
At some point we reach the threshold of the stability<br />
of the "thermodynamic branch." Then we reach what is generally<br />
called a "bifurcation point." (These are the points whose<br />
role Maxwell emphasized in his thoughts on the relation between<br />
determinism and free choice [see Chapter II, section<br />
3].)<br />
X<br />
,.<br />
,<br />
,<br />
I<br />
I<br />
\<br />
\<br />
\<br />
\<br />
A B\ E<br />
t-------=--=-...·--<br />
Figure 10. Bifurcation diagram. The diagram plots the steady-state values<br />
of X as function of a bifurcation parameter . Continuous lines are stable<br />
stationary states; broken lines are unstable stationary states. The only way<br />
to get to branch D is to start with some concentration X0 higher than the<br />
value of X corresponding to branch E.
161 THE THREE STAGES OF THERMODYNAMICS<br />
Let us consider some typical bifurcation diagrams. At bifurcation<br />
point B, the thermodynamic branch becomes unstable<br />
in respect to fluctuations. For the value Ac of the control parameter<br />
A, the system may be in three different steady states:<br />
C, E, D. 1\vo of these states are stable, one unstable. It is very<br />
important to emphasize that the behavior of such systems depends<br />
on their history. Suppose we slowly increase the value<br />
of the control parameter A; we are likely to follow the path A,<br />
B, C in Figure 10. On the contrary, if we start with a large<br />
value of the concentration X and maintain the value of the control<br />
parameter constant, we are likely to come to point D. The<br />
state we reach depends on the previous history of the system.<br />
Until now history has been commonly used in the interpretation<br />
of biological and social phenomena, but that it may play<br />
an important role in simple chemical processes is quite unexpected.<br />
Consider the bifurcation diagram represented in Figure 11.<br />
This differs from the previous diagram in that at the bifurcation<br />
point two new stable solutions emerge. Thus a new question:<br />
Where will the system go when we reach the bifurcation<br />
point? We have here a "choice" between two possibilities;<br />
X<br />
Figure 11. Symmetrical bifurcation diagram. X is plotted as a function of A.<br />
For AAc there<br />
are two stable stationary states for each value of A (the formerly stable state<br />
becomes unstable).
ORDER OUT OF CHAOS 162<br />
they may represent either of the two nonuniform distributions<br />
of chemical X in space, as represented in Figures 12 and 13.<br />
X<br />
----r<br />
X<br />
---- r<br />
Figures 12 and 13. Two possible spatial distributions of the chemical component<br />
X, corresponding to each of the two branches in Figure 11. Figure 12<br />
corresponds to a "left" structure as component X has a higher concentration<br />
in the left part; similarly, Figure 13 corresponds to a "right" structure.<br />
The two structures are mirror images of one another. In Figure<br />
12 the concentration of X is larger at the left; in Figure 13 it is<br />
larger at the right. How will the system choose between left<br />
and right? There is an irreducible random element; the macroscopic<br />
equation cannot predict the path the system will<br />
take. Turning to a microscopic description will not help. There<br />
is also no distinction between left and right. We are faced with<br />
chance events very similar to the fall of dice.
163<br />
THE THREE STAGES OF THERMODYNAMICS<br />
We would expect that if we repeat the experiment many<br />
times and lead the system beyond the bifurcation point, half of<br />
the system will go into the left configuration, half into the<br />
right. Here another interesting question arises: In the world<br />
around us, some basic simple symmetries seem to be broken.to<br />
Everybody has obser ved that shells often have a preferential<br />
chirality. Pasteur went so far as to see in dissymmetry, in<br />
the breaking of symmetry, the very characteristic of life. We<br />
know today that DNA, the most basic nucleic acid, takes the<br />
form of a left-handed helix. How did this dissymmetry arise?<br />
One common answer is that it comes from a unique event that<br />
has by chance favored one of the two possible outcomes; then<br />
an autocatalytic process sets in, and the left-handed structure<br />
produces other left-handed structures. Others imagine a<br />
"war" between left- and right-handed structures in which one<br />
of them has annihilated the other. These are problems for<br />
which we have not yet found a satisfactory answer. To speak of<br />
unique events is not satisfactory; we need a more "systematic"<br />
explanation.<br />
We have recently discovered a striking example of the fundamental<br />
new properties that matter acquires in far-fromequilibrium<br />
conditions: external fields, such as the gravitational<br />
field, can be "perceived" by the system, creating the possibility<br />
of pattern selection.<br />
How would an external field-a gravitational field-change<br />
an equilibrium situation? The answer is provided by Boltzmann's<br />
order principle: the basic quantity involved is the ratio<br />
of potential energy/thermal energy. This is a small quantity for<br />
the gravitational field of earth; we would have to climb a mountain<br />
to achieve an appreciable change in pressure or in the<br />
composition of the atmosphere. But recall the Benard cell;<br />
from a mechanical perspective, its instability is the raising of<br />
its center of gravity as the result of thermal dilatation. In other<br />
words, gravitation plays an essential role here and leads to a<br />
new structure in spite of the fact that the Benard cell may have<br />
a thickness of only a few millimeters. The effect of gravitation<br />
on such a thin layer would be negligible at equilibrium, but<br />
because of the nonequilibrium induced by the difference in<br />
temperature, macroscopic effects due to gravitation become<br />
visible even in this thin layer. Nonequilibrium magnifies the<br />
effect of gravitation.tl
. . .. .. . . .<br />
ORDER OUT OF CHAOS<br />
164<br />
Gravitation obviously will modify the diffusion flow in a reaction<br />
diffusion equation. Detailed calculations show that this<br />
can be quite dramatic near a bifurcation point of an unperturbed<br />
system. In particular, we can conclude that very small<br />
gravitational fields can lead to pattern selection.<br />
Let us again consider a system with a bifurcation diagram<br />
such as represented in Figure 11. Suppose that for no gravitation,<br />
g=O, we have, as in Figures 12 and 13, an asymmetric<br />
"up/down" pattern as well as its mirror image, "down/up."<br />
Both are equally probable, but when g is taken into account,<br />
the bifurcation equations are modified because the diffusion<br />
flow contains a term proportional to g. As a result, we now<br />
obtain the bifurcation diagram represented in Figure 14. The<br />
original bifurcation has disappeared-this is true whatever the<br />
value of the field. One structure (a) now emerges continuously<br />
as the bifurcation parameter grows, while the other (b) can be<br />
attained only through a finite perturbation.<br />
x<br />
. .. ... .<br />
.<br />
.<br />
.<br />
.<br />
.<br />
<br />
.•. .<br />
.<br />
.<br />
...<br />
'.<br />
" ,<br />
.------<br />
• •••••••<br />
b)<br />
Figure 14. Phenomenon of assisted bifurcation in the presence of an external<br />
field. X is plotted as a function of parameter ". The symmetrical bifurcation<br />
that would occur in the absence of the field is indicated by the dotted<br />
line. The bifurcation value is "0; the stable branch (b) is at finite di'stance from<br />
branch (a).
165 THE THREE STAGES OF THERMODYNAMICS<br />
Therefore, if we follow the path (a), we expect the system to<br />
follow the continuous path. This expectation is correct as long<br />
as the distance s between the two branches remains large in respect<br />
to thermal fluctuations in the concentration of X. There<br />
occurs what we would like to call an "assisted" bifurcation.<br />
As before, at about the value Ac a self-<strong>org</strong>anization process<br />
may occur. But now one of the two possible patterns is preferred<br />
and will be selected.<br />
The important point is that, depending on the chemical process<br />
responsible for the bifurcation, this mechanism expresses<br />
an extraordinary sensitivity. Matter, as we mentioned earlier<br />
in this chapter, perceives differences that would be insignificant<br />
at equilibrium. Such possibilities lead us to think of the<br />
simplest <strong>org</strong>anisms, such as bacteria, which we know are able<br />
to react to electric or magnetic fields. More generally they<br />
show that far-from-equilibrium chemistry leads to possible<br />
"adaptation" of chemical processes to outside conditions. This<br />
contrasts strongly with equilibrium situations, in which large<br />
perturbations or modifications of the boundary conditions are<br />
necessary to determine a shift for one structure to another.<br />
The sensitivity of far-from-equilibrium states to external fluctuations<br />
is another example of a system's spontaneous "adaptative<br />
<strong>org</strong>anization" to its environment. Let us give an example12<br />
of self-<strong>org</strong>anization as a function of fluctuating external conditions.<br />
The simplest conceivable chemical reaction is the isomerization<br />
reaction where AB. In our model the product A can<br />
also enter into another reaction: A+ light-+ A *-+A+ heat. A<br />
absorbs light and gives it back as heat while leaving its excited<br />
state A*. Consider these two processes as taking place in a<br />
closed system: only light and heat can be exchanged with the<br />
outside. Nonlinearity exists in the system because the transformation<br />
from B to A absorbs heat: the higher the temperature,<br />
the faster the formation of A. But also the higher the concentration<br />
of A, the higher the absorption of light by A and its transformation<br />
into heat, and the higher the temperature. A<br />
catalyzes its own formation.<br />
We expect to find that the concentration of A corresponding<br />
to the stationary state increases with the light intensity. This is<br />
indeed the case. But starting from a critical point, there appears<br />
one of the standard far-from-equilibrium phenomena:<br />
the coexistence of multiple stationary states. For the same val-
ORDER OUT OF CHAOS<br />
166<br />
ues of light intensity and temperature, the system can be<br />
found in two different stable stationary states with different<br />
concentrations of A. A third, unstable state marks the threshold<br />
between the first two. Such a coexistence of stationary<br />
states gives birth to the well-known phenomenon of hysteresis.<br />
But this is not the whole story. If the light intensity, instead<br />
of being constant, is taken as randomly fluctuating, the<br />
situation is altered profoundly. The zone of coexistence between<br />
the two stationary states increases, and for certain values<br />
of the parameters coexistence among three stationary<br />
stable states becomes possible.<br />
In such a case, a random fluctuation in the external flux,<br />
often termed "noise," far from being a nuisance, produces<br />
new types of behavior, which would imply, under deterministic<br />
fluxes, much more complex reaction schemes. It is important<br />
to remember that random noise in the fluxes may be consid-<br />
X<br />
p<br />
1\ ..<br />
•<br />
'<br />
•<br />
I<br />
<br />
I ' ' I<br />
J ' 'I<br />
v<br />
•<br />
p':<br />
Q<br />
b1 b2 b<br />
Figure 15. This figure shows how a "hysteresis" phenomenon occurs if we<br />
have the value of the bifurcation parameter b first growing and then diminishing.<br />
If the system is initially in a stationary state belonging to the lower<br />
branch, it will stay there while b grows. But at b = b2, there will be a discontinuity:<br />
The system jumps from Q to Q', on the higher branch. Inversely,<br />
starting from a state on the higher branch, the system will remain there till<br />
b=b1, when it will jump down toP'. Such types of bistable behavior are<br />
observed in many fields, such as lasers, chemical reactions or biological<br />
membranes.
167 THE THREE STAGES OF THERMODYNAMICS<br />
ered as unavoidable in any "natural system." For example, in<br />
biological or ecological systems the parameters defining interaction<br />
with the environment cannot generally be considered as<br />
constants. Both the cell and the ecological niche draw their<br />
sustenance from their environment ; and humidity, pH, salt<br />
concentration, light, and nutrients form a permanently fluctuating<br />
environment. The sensitivity of nonequilibrium states,<br />
not only to fluctuations produced by their internal activity but<br />
also to those coming from their environment, suggests new<br />
perspectives for biological inquiry.<br />
Cascading Bifurcations<br />
and the Transitions to Chaos<br />
The preceding paragraph dealt only with the first bifurcation<br />
or, as mathematicians put it, the primary bifurcation, which<br />
occurs when we push a system beyond the threshold of stability.<br />
Far from exhausting the new solutions that may appear,<br />
this primary bifurcation introduces only a single characteristic<br />
time (the period of the limit cycle) or a single characteristic<br />
length. To generate the complex spatial temporal activity observed<br />
in chemical or biological systems, we have to follow the<br />
bifurcation diagram farther.<br />
We have already alluded to phenomena arising from the<br />
complex interplay of a multitude of frequences in hydrodynamical<br />
or chemical systems. Let us consider Benard structures,<br />
which appear at a critical distance from equilibrium.<br />
Farther away from thermal equilibrium the convection flow<br />
begins to oscillate in time; as the distance from equilibrium is<br />
increased still farther, more and more oscillation frequencies<br />
appear, and eventually the transition to equilibrium is complete.13<br />
The interplays among the frequencies produce possibilities<br />
of large fluctuations; the "region" in the bifurcation<br />
diagram defined by such values of the parameters is often<br />
called "chaotic." In cases such as the Benard instability, order<br />
or coherence is sandwiched between thermal chaos and nonequilibrium<br />
turbulent chaos. Indeed, if we continue to in<br />
rease the gradient of temperature, the onvcction patterns<br />
become more complex; oscillations set in, and the ordered as-
ORDER OUT OF CHAOS ' 168<br />
TRACES OF Br- CONCENTRATION<br />
Homogeneous Steady State<br />
f\/l.J\NVVVV\ Sinusoidal Oscillations<br />
•<br />
•<br />
•<br />
Complex Periodic States<br />
(Subharmonic bifurcation)<br />
2:<br />
o<br />
0:<br />
u..<br />
&£J<br />
U<br />
Z<br />
<br />
!:!!<br />
c<br />
Chaos<br />
Mixed - Mode Oscillations<br />
Chaotic<br />
and<br />
Periodic<br />
Relaxation Oscillations<br />
TIME<br />
Figure 16. Temporal oscillations of the ion Br- in the Belousov<br />
Zhabotinski reaction. The figure represents a succession of regions corresponding<br />
to qualitative differences. This is a schematic representation. The<br />
experimental data indicate the existence of much more complicated sequences.<br />
pect of the convection is largely destroyed. However, we<br />
should not confuse "equilibrium thermal chaos" and "nonequilibrium<br />
turbulent chaos." In thermal chaos as realized in<br />
equilibrium, all characteristic space and time scales are of molecular<br />
range, while in turbulent chaos we have such an abundance<br />
of macroscopic time and length scales that the system<br />
appears chaotic. In chemistry the relation between order and<br />
chaos appears highly complex: successive regimes of ordered<br />
(oscillatory) situations follow regimes of chaotic .behavior.<br />
This has, for instance, been observed as a function of the flow<br />
rate in the Belousov-Zhabotinsky reaction.
169 THE. THREE STAGES OF THERMODYNAMICS<br />
In many cases it is difficult to disentangle the meaning of<br />
words such as "order" and "chaos." Is a tropical forest an<br />
ordered or a chaotic system? The history of any particular<br />
animal species will appear very contingent, dependent on<br />
other species and on environmental accidents. Nevertheless,<br />
the feeling persists that, as such, the overall pattern of a tropical<br />
forest; as represented, for instance, by the diversity of species,<br />
corresponds to the very archetype of order. Whatever the<br />
precise meaning we will eventually give to this terminology, it<br />
is clear that in some cases the succession of bifurcations forms<br />
an irreversible evolution where the determinism of characteristic<br />
frequencies produces an increasing randomness stemming<br />
from the multiplicity of those frequencies.<br />
A remarkably simple road to "chaos" that has already attracted<br />
a lot of attention is the "Feigenbaum sequence." It<br />
concerns any system whose behavior is characterized by a<br />
very general feature-that is, for a determined range of parameter<br />
values the system's behavior is periodic, with a period T;<br />
beyond this range, the period becomes 2T, and beyond yet another<br />
critical threshold, the system needs 4 Tin order to repeat<br />
itself. The system is thus characterized by a succession of bifurcations,<br />
with successive period doubling. This constitutes a<br />
typical route going from simple periodic behavior to the complex<br />
aperiodic behavior occurring when the period has doubled<br />
ad infinitum. This route, as Feigenbaum discovered, is<br />
characterized by universal numerical features independent of<br />
the mechanism involved as long as the system possesses the<br />
qualitative property of period doubling. "In fact, most measurable<br />
properties of any such system in this aperiodic limit<br />
now can be determined in a way that essentially bypasses the<br />
details of the equations governing each specific system . . . . "14<br />
In other cases, such as those represented in Figure 16, both<br />
deterministic and stochastic elements characterize the history<br />
of the system.<br />
If we consider Figure 17 and a value of the control parameter<br />
of the order of X 6 , we see that the system already has a<br />
wealth of possible stable and unstable behaviors. The "historical"<br />
path along which the system evolves as the control parameter<br />
grows is characterized by a succession of stable regions,<br />
where deterministic laws dominate, and of instable ones, near<br />
the bifurcation points, where the system can "choose" be-
ORDER OUT OF CHAOS 170<br />
tween or among more than one possible future. Both the deterministic<br />
character of the kinetic equations whereby the set of<br />
possible states and their respective stability can be calculated,<br />
and the random fluctuations "choosing" between or among<br />
the states around bifurcation points are inextricably connected.<br />
This mixture of necessity and chance constitutes the<br />
history of the system.<br />
Solutions<br />
\ (c'),,<br />
' ,,<br />
' ,<br />
'<br />
I<br />
I<br />
,<br />
(/ ..<br />
, -<br />
-<br />
-<br />
t---
171 THE THREE STAGES OF THERMODYNAMICS<br />
From Euclid to Aristotle<br />
One of the most interesting aspects of dissipative structures is<br />
their coherence. The system behaves as a whole, as if it were<br />
the site of long-range forces. In spite of the fact that interactions<br />
among molecules do not exceed a range of some I0-8<br />
em, the system is structured as though each molecule were<br />
"informed" about the overall state of the system.<br />
It has often been said-and we have already repeated itthat<br />
modern science was born when Aristotelian space, for<br />
which one source of inspiration was the <strong>org</strong>anization and solidarity<br />
of biological functions, was replaced by the homogeneous<br />
and isotropic space of Euclid. However, the theory of<br />
dissipative structures moves us closer to Aristotle's conception.<br />
Whether we are dealing with a chemical clock, concentration<br />
waves, or the inhomogeneous distribution of chemical<br />
products, instability has the effect of breaking symmetry, both<br />
temporal and spatial. In a limit cycle, no two instants are<br />
equivalent; the chemical reaction acquires a phase similar to<br />
that characterizing a light wave, for example. Again, when a<br />
favored direction results from an instability, space ceases to be<br />
isotropic. We move from Euclidian to Aristotelian space!<br />
It is tempting to speculate that the breaking of space and<br />
time symmetry plays an important part in the fascinating phenomena<br />
of morphogenesis. These phenomena have often led<br />
to the conviction that some internal purpose must be involved,<br />
a plan realized by the embryo when its growth is complete. At<br />
the beginning of this century, German embryologist Hans<br />
Driesch believed that some immaterial "entelechy" was responsible<br />
for the embryo's development. He had discovered<br />
that the embryo at an early stage was capable of withstanding<br />
the severest perturbations and, in spite of them, of developing<br />
into a normal, functional <strong>org</strong>anism. On the other hand, when<br />
we observe embryological development on film, we "see"<br />
jumps corresponding to radical re<strong>org</strong>anizations followed by<br />
periods of more "pacific" quantitative growth. There are, fortunately,<br />
few mistakes. The jumps are performed in a reproducible<br />
fashion. We might speculate that the basic<br />
mechanism of evolution is based on the play between bifurca-
ORDER OUT OF CHAOS 172<br />
tions as mechanisms of exploration and the selection of chemical<br />
interactions stabilizing a particular trajectory. Some forty<br />
years ago, the biologist Waddington introduced such an idea.<br />
The concept of "chreod" that he introduced to describe the<br />
stabilized paths of development would correspond to possible<br />
lines of development produced as a result of the double imperative<br />
of flexibility and security. 15 Obviously the problem is<br />
very complex and can be dealt with only briefly here.<br />
Many years ago embryologists introduced the concept of a<br />
morphogenetic field and put forward the hypothesis that the<br />
differentiation of a cell depends on its position in that field.<br />
But how does a cell "recognize" its position? One idea that is<br />
often debated is that of a "gradient" of a characteristic substance,<br />
of one or more "morphogens." Such gradients could<br />
actually be produced by symmetry-breaking instabilities in<br />
far-from-equilibrium conditions. Once it has been produced, a<br />
chemical gradient can provide each cell with a different chemical<br />
environment and thus induce each of them to synthesize a<br />
specific set of proteins. This model, which is now widely used,<br />
seems to be in agreement with experimental evidence. In particular,<br />
we may refer to Kauffman's work16 on drosophila. A<br />
reaction-diffusion system is taken as responsible for the commitment<br />
to alternative development programs that appear to<br />
occur in different groups of cells in the early embryo. Each<br />
3 1 3<br />
4 r--------r------------ 4<br />
Figure 18. Schematic representation of the structure of the drosophila embryo<br />
as it results from successive binary choices. See text for more detail.
173 THE THREE STAGES OF THERMODYNAMICS<br />
compartment would be specified by a unique combination of<br />
binary choices, each of these choices being the result of a spatial<br />
symmetry-breaking bifurcation. The model leads to successful<br />
predictions about the result of transplantations as a<br />
function of the "distance" between the original and final regions-that<br />
is, of the number of differences among the states<br />
of the binary choices or "switches" that specify each of them.<br />
Such ideas and models are especially important in biological<br />
systems where the embryo begins to develop in an apparently<br />
symmetrical state (for example, Fucus, Acetabularia).<br />
We may ask if the embryo is really homogeneous at the beginning.<br />
And even if small inhomogeneities are present in the initial<br />
environment, do they cause or channel evolution toward a<br />
given structure? Precise answers to such questions are not<br />
available at present. However, one thing seems established:<br />
the instability connected with chemical reactions and transport<br />
appears as the only general mechanism capable of breaking<br />
the symmetry of an initially homogeneous situation.<br />
The very possibility of such a solution takes us far beyond<br />
the age-old conflict between reductionists and antireductionists.<br />
Ever since Aristotle (and we have cited Stahl, Hegel,<br />
Bergson, and other antireductionists), the same conviction<br />
has been expressed: a concept of complex <strong>org</strong>anization is required<br />
to connect the various levels of description and account<br />
for the relationship between the whole and the behavior of the<br />
parts. In answer to the reductionists, for whom the sole<br />
"cause" or <strong>org</strong>anization can lie only in the part, Aristotle with<br />
his formal cause, Hegel with his emergence of Spirit in Nature,<br />
and Bergson with his simple, irrepressible, <strong>org</strong>anizationcreating<br />
act, assert that the whole is predominant. To cite<br />
Bergson,<br />
In general, when the same object appears in one aspect<br />
as simple and in another as infinitely complex, the two<br />
aspects have by no means the same importance, or rather<br />
the same degree of reality. In such cases, the simplicity<br />
belongs to the object itself, and the infinite complexity to<br />
the views we take in turning around it, to the symbols by<br />
which our senses or intellect represent it to us, or, more<br />
gc::nc::rally, to c::lc::mc::nts of a different order, with which we:<br />
try to imitate it artificially, but with which it remains in-
ORDER OUT OF CHAOS 174<br />
commensurable, being of a different nature. An artist of<br />
genius has painted a figure on his canvas. We can imitate<br />
his picture with many-coloured squares of mosaic. And<br />
we shall reproduce the curves and shades of the model so<br />
much the better as our squares are smaller, more numerous<br />
and more varied in tone. But an infinity of elements<br />
infinitely small, presenting an infinity of shades, would<br />
be necessary to obtain the exact equivalent of the figure<br />
that the artist has conceived as a simple thing, which he<br />
has wished to transport as a whole to the canvas, and<br />
which is the more complete the more it strikes us as the<br />
projection of an indivisible intuition. 17<br />
In biology, the conflict between reductionists and antireductionists<br />
has often appeared as a conflict between the assertion<br />
of an external and an internal purpose. The idea of an immanent<br />
<strong>org</strong>anizing intelligence is thus often opposed by an <strong>org</strong>anizational<br />
model borrowed from the technology of the time<br />
(mechanical, heat, cybernetic machines), which immediately<br />
elicits the retort: "Who" built the machine, the automaton<br />
that obeys external purpose?<br />
As Bergson emphasized at the beginning of this century,<br />
both the technological model and the vitalist idea of an internal<br />
<strong>org</strong>anizing power are expressions of an inability to conceive<br />
evolutive <strong>org</strong>anization without immediately referring it<br />
to some preexisting goal. Today, in spite of the spectacular success<br />
of molecular biology, the conceptual situation remains<br />
about the same: Bergson's argument could be applied to contemporary<br />
metaphors such as "<strong>org</strong>anizer," "regulator," and<br />
"genetic program." Unorthodox biologists such as Paul Weiss<br />
and Conrad Waddington 18 have rightly criticized the way this<br />
kind of qualification attributes to individual molecules the<br />
power to produce the global order biology aims to understand,<br />
and, by so doing, mistakes the formulation of the problem for<br />
its solution.<br />
It must be recognized that technological analogies in biology<br />
are not without interest. However, the general validity of<br />
such analogies would imply that, as in an electronic circuit, for<br />
example, there is a basic homogeneity between the description<br />
of molecular interaction and that of global behavior: The functioning<br />
of a circuit may be deduced from the nature and posi-
175 THE THREE STAGES OF THERMODYNAMICS<br />
tion of its relays; both refer to the same scale, since the relays<br />
were designed and installed by the same engineer who built<br />
the whole machine. This cannot be the rule in biology.<br />
It is true that when we come to a biological system such as<br />
the bacterial chemotaxis, it is hard not to speak of a molecular<br />
machine consisting of receptors, sensory and regulatory processing<br />
systems, and motor response. We know of approximately<br />
twenty or thirty receptors that can detect highly<br />
specific classes of compounds and make a bacterium swim up<br />
spatial gradients of attractants or down gradients of repellents.<br />
This "behavior" is determined by the output of the processing<br />
system-that is, the switching on or off of a tumble that generates<br />
a change in the bacterium's direction.l9<br />
But such cases, fascinating as they are, do not tell the whole<br />
story. In fact it is tempting to see them as limiting cases, as the<br />
end products of a specific kind of selective evolution, emphasizing<br />
stability and reproducible behavior against openness<br />
and adaptability. In such a perspective, the relevance of the<br />
technological metaphor is not a matter of principle but of opportunity.<br />
The problem of biological order involves the transition from<br />
the molecular activity to the supermolecular order of the cell.<br />
This problem is far from being solved.<br />
Often biological order is simply presented as an improbable<br />
physical state created and maintained by enzymes resembling<br />
Maxwell's demon, enzymes that maintain chemical differences<br />
in the system in the same way as the demon maintains<br />
temperature and pressure differences. If we accept this, biology<br />
would be in the position described by Stahl. The laws of<br />
nature allow only death. Stahl's notion of the <strong>org</strong>anizing action<br />
of the soul is replaced by the genetic information contained in<br />
the nucleic acids and expressed in the formation of enzymes<br />
that permit life to be perpetuated. Enzymes postpone death<br />
and the disappearance of life.<br />
In the context of the physics of irreversible processes, the<br />
results of biology obviously have a different meaning and different<br />
implications. We know today that both the biosphere as<br />
a whole as well as its components, living or dead, exist in farfrom-equilibrium<br />
conditions. In this context life, far from<br />
being outside the natural order, appears as the supreme expression<br />
of the self-<strong>org</strong>anizing processes that occur.
ORDER OUT OF CHAOS<br />
176<br />
We are tempted to go so far as to say that once the condi·<br />
tions for self-<strong>org</strong>anization are satisfied, life becomes as predictable<br />
as the Benard instability or a falling stone. It is a<br />
. remarkable fact that recently discovered fossil forms of life<br />
appear nearly simultaneously with the first rock formations<br />
(the oldest microfossils known today date back 3.8 . 109 years,<br />
while the age of the earth is supposed to be 4. 6,109 years; the<br />
formation of the first rocks is also dated back to 3.8 . 109 years).<br />
The early appearance of life is certainly an argument in favor of<br />
the idea that life is the result of spontaneous seif-<strong>org</strong>anization<br />
that occurs whenever conditions for it permit. However, we<br />
must admit that we remain far from any quantitative theory.<br />
To return to our understanding of life and evolution, we are<br />
now in a better position to avoid the risks implied by any de·<br />
nunciation of reductionism. A system far from equilibrium may<br />
be described as <strong>org</strong>anized not because it realizes a plan alien<br />
to elementary activities, or transcending them, but, on the<br />
contrary, because the amplification of a microscopic fluctuation<br />
occurring at the "right moment" resulted in favoring one reaction<br />
path over a number of other equally possible paths. Under cer·<br />
tain circumstances, therefore, the role played by individual behavior<br />
can be decisive. More generally, the "overall" behavior<br />
cannot in general be taken as dominating in any way the elementary<br />
processes constituting it. Self-<strong>org</strong>anization processes<br />
in far-from-equilibrium conditions correspond to a delicate interplay<br />
between chance and necessity, between fluctuations<br />
and deterministic laws. We expect that near a bifurcation, fluctuations<br />
or random elements would play an important role,<br />
while between bifurcations the deterministic aspects would<br />
become dominant. These are the questions we now need to<br />
investigate in more detail.
CHAPTER VI<br />
ORDER THROUGH<br />
FWCTUATIONS<br />
Fluctuations and Chemistry<br />
In our Introduction we noted that a reconceptualization of the<br />
physical sciences is occurring today. They are moving from<br />
deterministic, reversible processes to stochastic and irreversible<br />
ones. This change of perspective affects chemistry in a<br />
striking way. As we have seen in Chapter V, chemical processes,<br />
in contrast to the trajectories of classical dynamics,<br />
correspond to irreversible processes. Chemical reactions lead<br />
to entropy production. On the other hand, classical chemistry<br />
continues to rely on a deterministic description of chemical<br />
evolution. As we have seen in Chapter V, it is necessary to<br />
produce differential equations involving the concentration of<br />
the various chemical components. Once we know these concentrations<br />
at some initial time (as well as at appropriate<br />
boundary conditions when space-dependent phenomena such as<br />
diffusion are involved), we may calculate what the concenK-ation<br />
will be at a later time. It is interesting to note that the deterministic<br />
view of chemistry fails when far-from-equilibrium<br />
processes are involved.<br />
We have repeatedly emphasized the role of fluctuations. Let<br />
us summarize here some of the more striking features. Whenever<br />
we reach a bifurcation point, deterministic description<br />
breaks down. The type of fluctuation present in the system<br />
will lead to the choice of the branch it will follow. Crossing a<br />
bifurcation is a stochastic process, such as the tossing of a<br />
coin. Chemical chaos provides another example (see Chapter<br />
V). Here we can no longer follow an individual chemical trajectory.<br />
We cannot predict the details of temporal evolution.<br />
177
ORDER OUT OF CHAOS 178<br />
Once again, only a statistical description is possible. The existence<br />
of an instability may be viewed as the result of a fluctuation<br />
that is first localized in a small part of the system and then<br />
spreads and leads to a new macroscopic state.<br />
This situation alters the traditional view of the relation between<br />
the microscopic level as described by molecules or<br />
atoms and the macroscopic level described in ter ms of global<br />
variables such as concentration. In many situations fluctuations<br />
correspond only to small corrections. As an example, let<br />
us take a gas composed of N molecules enclosed in a vessel of<br />
volume V. Let us divide this volume into two equal parts.<br />
What is the number of particles X in one of these two parts?<br />
Here the variable X is a "random" variable, and we would<br />
expect it to have a value in the neighborhood of N/2.<br />
A basic theorem in probability theory, the law of large numbers,<br />
provides an estimate of the "error" due to fluctuations.<br />
In essence, it states that if we measure X we have to expect a<br />
value of the order N/2±v'Nii. If N is large, the difference<br />
introduced by fluctuations v'Nfi may also be large (if<br />
N= 1Q24, VN= 1012); however, the relative error introduced<br />
by fluctuations is of the order of (v'N!i)/(N/2) or llYN and<br />
thus tends toward zero for a sufficiently large value of N. As<br />
soon as the system becomes large enough, the law of large<br />
numbers enables us to make a clear distinction between mean<br />
values and fluctuations, and the latter may be neglected.<br />
Hewever, in nonequilibrium processes we may find just the<br />
opposite situation. Fluctuations determine the global outcome.<br />
We could say that instead of being corrections in the<br />
average values, fluctuations now modify those averages. This<br />
is a new situation. For this reason we would like to introduce a<br />
neologism and call situations resulting from fluctuation "order<br />
through fluctuation." Before giving examples, let us make<br />
some general remarks to illustl·ate the conceptual novelty of<br />
this situation.<br />
Readers may be familiar with the Heisenberg uncertainty<br />
relations, which express in a striking way the probabilistic aspects<br />
of quantum theory. Since we can no longer simultaneously<br />
measure position and coordinates in quantum theory,<br />
classical determinism is breaking down. This was believed to<br />
be of no importance for the description of macroscopic objects
179 ORDER THROUGH FLUCTUATIONS<br />
such as living systems. But the role of fluctuations in nonequilibrium<br />
systems shows that this is not the case. Randomness<br />
remains essential on the macroscopic level as well. It is interesting<br />
to note another analogy with quantum theory, which<br />
assigns a wave behavior to all elementary particles. As we<br />
have seen, chemical systems far from equilibrium may also<br />
lead to coherent wave behavior: these are the chemical clocks<br />
discussed in Chapter V. Once again, some of the properties<br />
quantum mechanics discovered on the microscopic level now<br />
appear on the macroscopic level.<br />
Chemistry is actively involved in the reconceptualization of<br />
science. I We are probably only at the beginning of new directions<br />
of research. It may well be, as some recent calculations<br />
suggest, that the idea of reaction rate has to be replaced in<br />
some cases by a statistical theory involving a distribution of<br />
reaction probabilities.2<br />
Fluctuations and Correlations<br />
Let us go back to the types of chemical reaction discussed in<br />
Chapter V. To take a specific example, consider a chain of<br />
reactions such as APXPF. The kinetic equations in Chapter V<br />
refer to the average concentrations. To emphasize this we shall<br />
now write instead of X. We can then ask what is the<br />
probability at a given time of finding a number X for the concentration<br />
of this component. Obviously this probability will<br />
fluctuate, as do the number of collisions among the various<br />
molecules involved. It is easy to write an equation that describes<br />
the change in this probability distribution P(X,t) as a<br />
result of processes that produce molecule X and of processes<br />
that destroy that molecule. We may perform the calculation for<br />
equilibrium systems or for steady-state systems. Let us first<br />
mention the results obtained for equilibrium systems.<br />
At equilibrium we virtually recover a classical probabilistic<br />
distribution, the Poisson distribution, which is described in<br />
every textbook on probabilities, since it is valid in a variety of<br />
situations, such as the distribution of telephone calls, waiting<br />
times in restaurants, or the fluctuation of the concentration of
ORDER OUT OF CHAOS 180<br />
particles in a gas or a liquid. The mathematical form of this<br />
distribution is of no importance here. We merely want to emphasize<br />
two of its aspects. First, it leads to the law of large<br />
numbers as formulated in the first section of this chapter. Thus<br />
fluctuations indeed become negligible in a large system. Moreover,<br />
this law enables us to calculate the correlation between<br />
the number of particles X at two different points in space separated<br />
by some distance r. The calculation demonstrates that at<br />
equilibrium there is no such correlation. The probability of<br />
finding two molecules X and X' at two different points rand r'<br />
is the product of finding X at rand X' at r' (we cons.ider distances<br />
that are large in respect to the range of intermolecular<br />
forces).<br />
One of the most unexpected results of recent research is that<br />
this situation changes drastically when we move to nonequilibrium<br />
situations. First, when we come close to bifurcation<br />
points the fluctuations become abnormally high and the law of<br />
large numbers is violated. This is to be expected, since the<br />
system may then "choose" among various regimes. Fluctuations<br />
can even reach the same order of magnitude as the mean<br />
macroscopic values. Then the distinction between fluctuations<br />
and mean values breaks down. Moreover, in the case of a nonlinear<br />
type of chemical reaction discussed in Chapter V, longrange<br />
correlations appear. Particles separated by macroscopic<br />
distances become linked. Local events have repercussions<br />
throughout the whole system. It is interesting to note3 that<br />
such long-range correlations appear at the precise point of<br />
transition from equilibrium to nonequilibrium. From this point<br />
of view the transition resembles a phase transition. However,<br />
the amplitudes of these long-range correlations are at first<br />
small but increase with distance from equilibrium and may become<br />
infinite at the bifurcation points.<br />
We believe that this type of behavior is quite interesting,<br />
since it gives a molecular basis to the problem of communication<br />
mentioned in our discussion of the chemical clock. Even<br />
before the macroscopic bifurcation, the system is <strong>org</strong>anized<br />
through these long-range correlations. We come back to one of<br />
the main ideas of this book: nonequilibrium as a source of order.<br />
Here the situation is especially clear. At equilibrium molecules<br />
behave as essentially independent entities; they· ignore<br />
one another. We would like to call them "hypnons," "sleep-
181 ORDER THROUGH FLUCTUATIONS<br />
walkers." Though each of them may be as complex as we like,<br />
they ignore one another. However, nonequilibrium wakes them<br />
up and introduces a coherence quite foreign to equilibrium.<br />
The microscopic theory of irreversible processes that we shall<br />
develop in Chapter IX will present a similar picture of matter.<br />
Matter's activity is related to the nonequilibrium conditions<br />
that it itself may generate. Just as in macroscopic behavior, the<br />
laws of fluctuations and correlations are universal at equilibrium<br />
(when we find the Poisson type of distribution); they<br />
become highly specific depending on the type of nonlinearity<br />
involved when we cross the boundary between equilibrium<br />
and nonequilibrium.<br />
The Amplification of Fluctuations<br />
Let us first take two examples wherein the growth of a fluctuation<br />
preceding the formation of a new structure can be followed<br />
in detail. The first is the aggregation of slime molds,<br />
which when threatened with starvation coalesce into a single<br />
supracellular mass. We have already mentioned this in Chapter<br />
V. Another illustration of the role of fluctuations is the first<br />
stage in the construction of a termites' nest. This was first<br />
described by Grasse, and Deneubourg has studied it from the<br />
standpoint that interests us here."'<br />
Self-Aggregation Process in an Insect Population<br />
Larvae of a coleoptera (Dendroctonus micans [Scot.]). are initially distributed<br />
at random between two horizontal sheets of glass, 2 mm apart. The<br />
borders are open and the surface is equal to 400 cm2.<br />
The aggregation process appears to result from the competition between<br />
two factors: the random moves of the larvae, and their reaction to a chemical<br />
product, a "pheromon" they synthetize from terpanes contained in the tree<br />
on which they feed and that each of them emits at a rate depending on its<br />
nutrition state. The pheromon diffuses in space, and the larvae move in the<br />
direction of its concentration gradient. Such a reaction provides an autocatalytic<br />
mechanism since, as they gather in a cluster, the larvae contribute<br />
to enhance the attractiveness of the corresponding region. The higher the<br />
local density of larvae in this region, the stronger the gradient and the more<br />
intense the tendency to move toward the crowded point.<br />
The experiment shows that the density of the larvae population determines<br />
not only the rate of the aggregation process but its effectiveness as<br />
well-that is, the number of larvae that will finally be part of the cluster. At
ORDER OUT OF CHAOS<br />
182<br />
high density (Figure A) a cluster appears and rapidly grows at the center of<br />
the experimental setup. At very low density (Figure B), no stable cluster appears.<br />
Moreover, other experiments have explored the possibility for a cluster to<br />
develop starting from a "nucleus" artificially created in a peripheral region of<br />
the system. Different solutions appear depending on the number of larvae in<br />
this initial nucleus.<br />
,.<br />
J<br />
(<br />
I'<br />
\<br />
t<br />
'<br />
I<br />
Figure A. Self-aggregation at high density. The times are 0 minutes and 21<br />
minutes.
..<br />
183 ORDER THROUGH FLUCTUATIONS<br />
If this number is small compared with the total number of larvae, the cluster<br />
fails to develop (Figure D). If it is large, the cluster grows (Figure E). For<br />
intermediate values of the initial nucleus, new types of structure may develop:<br />
Two, three or four other clusters appear and coexist, with a time of life<br />
at least greater than the time of observation (Figures F and G).<br />
No such multicluster structure was ever observed in experiments with homogeneous<br />
initial conditions. It would seem they correspond, in a bifurcation<br />
4)<br />
"'<br />
\1<br />
1<br />
"-<br />
.,..<br />
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I .,.<br />
,.<br />
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"<br />
\<br />
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A<br />
...<br />
-<br />
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l<br />
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Figure B. Self-aggregation at low density. The times are 0 minutes and 22<br />
minutes.
ORDER OUT OF CH AOS<br />
184<br />
diagram, to stable states compatible with the value of the parameters<br />
characterizing the system but that cannot be attained by this system starting<br />
from homogeneous conditions. The nucleus would play the part of a finite<br />
perturbation necessary to excite the system and deport it in a region of the<br />
bifurcation diagram corresponding to such families of multicluster solutions .<br />
•<br />
Q<br />
100<br />
•<br />
80<br />
60<br />
40<br />
20<br />
10<br />
--f--- -- -<br />
,--..--- -<br />
-<br />
.. .... ..<br />
. . .<br />
-<br />
...... .. ....<br />
. ..... ..<br />
f<br />
--- r<br />
····]<br />
20 30 40<br />
. -- ..<br />
,-<br />
,-f<br />
.. · ·······t<br />
Figure C. Percent of the total number of larvae in the central cluster in<br />
function of time at three different densities.<br />
N<br />
CRITICAl NUCLEUS dKnc of e10 lot- nt ... nud
185 ORDER T;-iROUGH FWCTUATIONS<br />
N<br />
CR.tTiCAL NUCLEUS !l'owlh of• :or.-- inilinlnutk .. 1•··•1<br />
growth of a 30 la
ORDER OUT OF CHAOS<br />
186<br />
• > <br />
, , ..<br />
"<br />
.,<br />
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., '<br />
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., .I I<br />
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,J<br />
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Figure G. Growth of a cluster (I) introduced peripherally, which induce the<br />
formation of a second little cluster (II).<br />
The construction of a termites' nest is one of those coherent<br />
activities that have led some scientists to speculate about a<br />
.. collective mind" in insect communities. But curiously, it appears<br />
that in fact the termites need very little information to<br />
participate in the construction of such a huge and complex<br />
edifice as the nest. The first stage in this activity, the construction<br />
of the base, has been shown by Grasse to be the<br />
result of what appears to be disordered behavior among termites.<br />
At this stage, they transport and drop lumps of earth in<br />
a random fashion, but in doing so they impregnate the lumps<br />
with a hormone that attracts other termites. The situation<br />
could thus be represented as follows: the initial "fluctuation"<br />
would be the slightly larger concentration of lumps of earth,<br />
which inevitably occurs at one time or another at some point<br />
in the area. The amplification of this event is produced by the
187 ORDER THROUGH FLUCTUATIONS<br />
increased density of termites in the region, attracted by the<br />
· slightly higher hormone concentration. As termites become<br />
more numerous in a region, the probability of their dropping<br />
lumps of earth there increases, leading in turn to a still higher<br />
concentration of the hormone. In this way "pillars" are<br />
formed, separated by a distance related to the range over<br />
which the hormone spreads. Similar examples have recently<br />
been described.<br />
Although Boltzmann's order principle enables us to describe<br />
chemical or biological processes in which differences<br />
are leveled out and initial conditions f<strong>org</strong>otten, it cannot explain<br />
situations such as these, where a few "decisions" in an<br />
unstable situation may channel a system formed by a large<br />
number of interactive entities toward a global structure.<br />
When a new structure results from a finite perturbation, the<br />
fluctuation that leads from one regime to the other cannot possibly<br />
overrun the initial state in a single move. It must first estab<br />
lish itself in a limited region and then invade the whole space:<br />
there is a nucleation mechanism. Depending on whether the<br />
size of the initial fluctuating region lies below or above some<br />
critical value (in the case of chemical dissipative structures,<br />
this threshold depends in particular on the kinetic constants<br />
and diffusion coefficients), the fluctuation either regresses or<br />
else spreads to the whole system. We are familiar with nucleation<br />
phenomena in the classical theory of phase change: in a<br />
gas, for example, condensation droplets incessantly form and<br />
evaporate. That temperature and pressure reach a point where<br />
the liquid state becomes stable means that a critical droplet<br />
size can be defined (which is smaller the lower the temperature<br />
and the higher the pressure). If the size of a droplet exceeds<br />
this "nucleation threshold," the gas almost instantaneously<br />
transforms into a liquid.<br />
Moreover, theoretical studies and numerical simulations<br />
show that the critical nucleus size increases with the efficacy<br />
of the diffusion mechanisms that link all the regions of systems.<br />
In other words, the faster communication takes place<br />
within a system, the greater the percentage of unsuccessful<br />
fluctuations and thus the more stable the system. This aspect<br />
of the critical-size problem means that in such situations the<br />
"outside world,·· the environment of the fluctuating region,<br />
always tends to damp fluctuations. These will be destroyed or
ORDER OUT OF CHAOS<br />
188<br />
(a)<br />
(b)<br />
Figure 19. Nucleation of a liquid droplet in a supersaturated vapor. (a)<br />
droplet smaller than the critical size; (b) droplet larger than the critical size.<br />
The existence of the threshold has been experimentally verified for dissipative<br />
structures.<br />
amplified according to the effectiveness of the communication<br />
between the fluctuating region and the outside world. The critical<br />
size is thus determined by the competition between the<br />
system's "integrative power" and the chemical mechanisms<br />
amplifying the fluctuation.<br />
This model applies to the results obtained recently in in<br />
vitro experimental studies of the onset of cancer tumors.s An<br />
individual tumor cell is seen as a "fluctuation," uncontrollably<br />
and permanently able to appear and to develop through replication.<br />
It is then confronted with the population of cytotoxic<br />
cells that either succeeds in destroying it or fails. Following<br />
the values of the different parameters characteristic of the replication<br />
and destruction processes, we can predict a regression<br />
or an amplification of the tumor. This kind of kinetic study has<br />
led to the recognition of unexpected features in the interaction<br />
between cytotoxic cells and the tumor. It seems that cytotoxic<br />
cells can confuse dead tumor cells with living ones. As a result,<br />
the destruction of the cancer cells becomes increasingly<br />
difficult.<br />
The question of the limits of complexity has often been<br />
raised. Indeed, the more complex a system is, the more numerous<br />
are the types of fluctuations that threaten its stability.<br />
How then, it has been asked, can systems as complex as eco<br />
logical or human <strong>org</strong>anizations possibly exist? How do they
189 ORDER THROUGH FLUCTUATIONS<br />
manage to avoid permanent chaos? The stabilizing effect of<br />
communication, of diffusion processes, could be a partial an·<br />
swer to these questions. In complex systems, where species<br />
and individuals interact in many different ways, diffusion and<br />
communication among various parts of the system are likely<br />
to be efficient. There is competition between stabilization<br />
through communication and instability through fluctuations.<br />
The outcome of that competition determines the threshold of<br />
stability.<br />
Structural Stability<br />
When can we begin to speak about "evolution" in its proper<br />
sense? As we have seen, dissipative structures require far-fromequilibrium<br />
conditions. Yet the reaction diffusion equations contain<br />
parameters that can be shifted back to near-equilibrium<br />
conditions. The system can explore the bifurcation diagram in<br />
both directions. Similarly, a liquid can shift from laminar flow<br />
to turbulence and back. There is no definite evolutionary pat·<br />
tern involved.<br />
The situation for models involving the size of the system as<br />
a bifurcation parameter is quite different. Here, growth occurring<br />
irreversibly in time produces an irreversible evolution.<br />
But this remains a special case, even if it can be relevant for<br />
morphogenetic development.<br />
Be it in biological, ecological, or social evolution, we cannot<br />
take as given either a definite set of interacting units, or a definite<br />
set of transformations of these units. The definition of the<br />
system is thus liable to be modified by its evolution. The simplest<br />
example of this kind of evolution is associated with the<br />
concept of structural stability. It concerns the reaction of a<br />
given system to the introduction of new units able to multiply<br />
by taking part in the system's processes.<br />
The problem of the stability of a system vis-a-vis this kind of<br />
change may be formulated as follows: the new constituents, introduced<br />
in small quantities, lead to a new set of reactions among<br />
the system's components. This new set of reactions then enters<br />
into competition with the system·s previous mode of functioning.<br />
If the system is "structurally stable" as far as this
ORDER OUT OF CHAOS<br />
190<br />
intrusion is concerned, the new mode of functioning will be<br />
unable to establish itself and the "innovators" will not survive.<br />
If, however, the structural fluctuation successfully imposes itself-if,<br />
for example, the kinetics whereby the "innovators"<br />
multiply is fast enough for the latter to invade the system instead<br />
of being destroyed-the whole system will adopt a new<br />
mode of functioning: its activity will be governed by a new<br />
"syntax. "6<br />
The simplest example of this situation is a population of<br />
macromolecules reproduced by polymerization inside a system<br />
being fed with the monomers A and B. Let us assume the<br />
polymerization process to be autocatalytic-that is, an already<br />
synthesized polymer is used as a model to form a chain having<br />
the same sequence. This kind of synthesis is much faster than<br />
a synthesis in which there is no model to copy. Each type of<br />
polymer, characterized by a particular sequence of A and B,<br />
can be described by a set of parameters measuring the speed<br />
of the synthesis of the copy it catalyzes, the accuracy of the<br />
copying process, and the mean life of the macromolecule itself.<br />
It may be shown that, under certain conditions, a single<br />
type of polymer having a sequence, shall we say, ABABABA . ..<br />
dominates the population, the other polymers being reduced<br />
to mere "fluctuations" with respect to the first. The . problem<br />
of structural stability arises each time that, as a result of a<br />
copying "error," a new type of polymer characterized by a<br />
hitherto unknown sequence and by a new set of parameters<br />
appears in the system and begins to multiply, competing with<br />
the dominant species for the available A and B monomers.<br />
Here we encounter an elementary case of the classic Darwinian<br />
idea of the "survival of the fittest."<br />
Such ideas form the basis for the model of prebiotic evolution<br />
developed by Eigen and his coworkers. The details of<br />
Eigen's argument are easily accessible elsewhere.? Let us<br />
briefly state that it seems to show that there is only one type of<br />
system that can resist the "errors" that autocatalytic populations<br />
continually make-a polymer system structurally stable<br />
for any possible "mutant polymer." This system is composed<br />
of two sets of polymer molecules. The molecules of the first<br />
set are of the "nucleic acid" type; each molecule is capable of<br />
reproducing itself and act s as a catalyst in the synthesis of
191 ORDER THROUGH FLUCTUATIONS<br />
a molecule of the second set, which is of the proteic type: each<br />
molecule of this second set catalyzes the self-reproduction of a<br />
molecule of the first set. This transcatalytic association between<br />
molecules of the two sets may turn into a cycle (each<br />
"nucleic acid" reproduces itself with the help of a "protein").<br />
It is then capable of stable survival, sheltered from the continual<br />
emergence of new polymers with higher reproductive<br />
efficiency: indeed, nothing can intrude into the self-replicating<br />
cycle formed by "proteins" and "nucleic acids." A new kind<br />
of evolution may thus begin to grow on this stable foundation,<br />
heralding the genetic code.<br />
Eigen 's approach is certainly of great interest. Darwinian<br />
selection for faithful self-reproduction is certainly important<br />
in an environment with a limited capacity. But we tend to believe<br />
that this is not the only aspect involved in pre biotic evolution.<br />
The "far-from-equilibrium" conditions related to critical<br />
amounts of flow of energy and matter are also important. It<br />
seems reasonable to assume that some of the first stages moving<br />
toward life were associated with the formation of mechanisms<br />
capable of absorbing and transforming chemical energy,<br />
so as to push the system into "far-from-equilibrium" conditions.<br />
At this stage life, or "prelife," probably was so diluted<br />
that Darwinian selection did not play the essential role it did in<br />
later stages.<br />
Much of this book has centered around the relation between<br />
the microscopic and the macroscopic. One of the most important<br />
problems in evolutionary theory is the eventual feedback<br />
between macroscopic structures and microscopic events: macroscopic<br />
structures emerging from microscopic events would<br />
in turn lead to a modification of the microscopic mechanisms.<br />
Curiously, at present, the better understood cases concern social<br />
situations. When we build a road or a bridge, we can predict<br />
how this will affect the behavior of the population, and<br />
this will in turn determine other modifications of the modes of<br />
communication in the region. Such interrelated processes generate<br />
very complex situations, the understanding of which is<br />
needed before any kind of modelization. This is why what we<br />
will now describe are only very simple cases.
ORDER OUT OF CHAOS 192<br />
Logistic Evolution<br />
In social cases, the problem of structural stability has a large<br />
number of applications. But it must be emphasized that such<br />
applications imply a drastic simplification of a situation defined<br />
simply in terms of competition between self-replicating<br />
processes in an environment where only a limited amount of<br />
the needed resources exists.<br />
In ecology the classic equation for such a problem is called<br />
the "logistic equation." This equation describes the evolution<br />
of a population containing N individuals, taking into account<br />
the birthrate, the death rate, and the amount of resources available<br />
to the population. The logistic equation can be written<br />
dN/dt = rN(K- N) - mN, where r and m are characteristic<br />
birth and death constants and K the "carrying capacity" of the<br />
environment. Whatever the initial value of N, as time goes on<br />
it will reach the steady-state value N = K-mlr determined by<br />
the differences of the carrying capacity and the ratio of death<br />
and birth constants. When this value is reached, the environ-<br />
K - ..!Il<br />
r<br />
N<br />
Figure 20. Evolution of a population N as a function of time t according to<br />
the logistic curve. The stationary state N=O is unstable while the stationary<br />
state N = K- mlr is stable with respect to fluctuations of N.<br />
t
193 ORDER THROUGH FLUCTUATIONS<br />
ment is saturated, and at each instant as many individuals die<br />
as are born.<br />
The apparent simplicity of the logistic equation conceals to<br />
some extent the complexity of the mechanisms involved. We<br />
have already mentioned the effect of external noise, for example.<br />
Here it has an especially simple meaning. Obviously, if<br />
only because of climatic fluctuations, the coefficients K, m,<br />
and r cannot be taken as constant. We know that such fluctuations<br />
can completely upset the ecological equilibrium and<br />
even drive the population to extinction. Of course, as a result,<br />
new processes, such as the storage of food and the formation<br />
of new colonies, will begin and eventually evolve so that some<br />
effects of external fluctuation may be avoided.<br />
But there is more. Instead of writing the logistic equation as<br />
continuous in time, let us compare the population at fixed time<br />
intervals (for example, separated by a year). This "discrete"<br />
logistic equation can be written in the form Nt + 1 = Nt(l + r<br />
[1-N/KJ), wliere Nt and Nt+ 1 are the populations separated<br />
by a one-year interval (we neglect here the death term). The<br />
remarkable feature, noted by R. May,8 is that such equations,<br />
in spite of their simplicity, admit a bewildering number of solu ·<br />
tions. For values of the parameter Or2, we have, as in the<br />
continuous case, a uniform approach to equilibrium. For values<br />
of r lower than 2.444, a limit cycle sets in: we now have a<br />
periodic behavior with a two-year period. This is followed by<br />
four-, eight-, etc., year cycles, until the behavior can only be<br />
described as chaotic (if r is larger than 2.57). Here we have a<br />
transition to chaos as described in Chapter V. Does this chaos<br />
arise in nature? Recent studies9 seem to indicate that the parameters<br />
characterizing natural populations keep them from<br />
the chaotic region. Why is this so? Here we have one of the<br />
very interesting problems created by the confluence of evolutionary<br />
problems with the mathematics produced by computer<br />
simulation.<br />
Up to now we have taken a static point of view. Let us now<br />
move to mechanisms, whereby the parameters K, r, and m<br />
may vary during biological or ecological evolution.<br />
We have to expect that during evolution the values of the<br />
ecological parameters K, r, and m will vary (as well as many<br />
other parameters and variables, whether they are quantifiable<br />
or not). Living societies continually introduce new ways of ex-
ORDER OUT OF CHAOS 194<br />
ploiting exi.;ting resources or of discovering new ones (that is,<br />
K increases) and continually discover new ways of extending<br />
their lives or of multiplying more quickly. Each ecological<br />
equilibrium defined by the logistic equation is thus only temporary,<br />
and a logistically defined niche will be occupied successively<br />
by a series of species, each capable of ousting the<br />
preceding one when its "aptitude" for exploiting the niche, as<br />
measured by the quantity K-mlr, becomes greater. (See Figure<br />
21.) Thus the logistic equation leads to the definition of a<br />
very simple situation where we can give a quantitative formulation<br />
of the Darwinian idea of the "survival of the fittest."<br />
The "fittest" is the species for which at a given time the quantity<br />
K-mlr is the largest.<br />
As restricted as the problem described by the logistic equation<br />
is, it nonetheless leads to some marvelous examples of<br />
nature's inventiveness .<br />
Take the example of caterpillars, who must remain undetected,<br />
since the slowness of their movement
195 ORDER THROUGH FLUCTUATIONS<br />
of these strategies is effective against all predators at all times,<br />
particularly if a predator is hungry enough. The ideal strategy<br />
is to remain totally undetected. Some caterpillars approach<br />
this ideal, and the variety and sophistication of the strategies<br />
used by the hundreds of lepidopteran species to remain undetected<br />
bring to mind the words of distinguished nineteenthcentury<br />
naturalist Louis Agassiz: "The possibilities of existence<br />
run so deeply into the extravagant that there is scarcely<br />
any conception too extraordinary for Nature to realize. " I O<br />
We cannot resist giving an example reported by Milton<br />
Love. '' The sheep liver trematode has to pass from an ant to a<br />
sheep, where it will finally reproduce itself. The chances of<br />
sheep swallowing an infected ant are very small, but the ant<br />
behaves in a remarkable way: it starts to maximize the probability<br />
of its encounter with a sheep. The trematode has truly<br />
"body snatched" its host. It has burrowed into the ant's brain,<br />
compelling its victim to behave in a suicidal way: the possessed<br />
ant, instead of staying on the ground, climbs to the tip<br />
of a blade of grass and there, immobile, waits for a sheep. This<br />
is indeed an incredibly "clever" solution to the parasites problem.<br />
How it was selected remains a puzzle.<br />
Other situations in biological evolution may be investigated<br />
using models similar to the logistic equation. For instance, it is<br />
possible to calculate the conditions of interspecies competition<br />
under which it may be advantageous for a fraction of the<br />
population to specialize in warlike and nonproductive activity<br />
(for example, the "soldiers" among the social insects). We can<br />
also determine the kind of environment in which a species that<br />
has become specialized, that has restricted the range of its<br />
food resources, will survive more easily than a nonspecialized<br />
species that consumes a wider range of resources. t2 But here<br />
we are approaching some very different problems, which concern<br />
the <strong>org</strong>anization of internally differentiated populations.<br />
Clear distinctions are absolutely necessary if we are to avoid<br />
confusion. In populations where individuals are not interchangeable<br />
and where each, with its own memory, character,<br />
and experience, is called upon to play a singular role, the relevance<br />
of the logistic equation and, more generally, of any simple<br />
Darwinian reasoning becomes quite relative. We shall<br />
return to this problem.<br />
It is interesting to note that the type of curve represented in
ORDER OUT OF CHAOS<br />
196<br />
Figure 21 showing the succession of growths and peaks defined<br />
by a given logistic equation's family with increasing<br />
K - mlr has also been used to describe the multiplication of<br />
certain technical procedures or products. Here too, the discovery<br />
or introduction of a new technique or product breaks<br />
some kind of social, technological, or economic equilibrium.<br />
This equilibrium would correspond to the maximum reached<br />
by the growth curve of the techniques or products with which<br />
the innovation is going to have to compete and that play a similar<br />
role in the situation described by the equation. 13 Thus, to<br />
choose but one example, not only did the spread of the steamship<br />
lead to the disappearance of most sailing ships, but, by<br />
reducing the cost of transportation and increasing its speed, it<br />
caused an increase in the demand for sea transport ("K") and<br />
consequently an increase in the population of ships. We are<br />
obviously representing here an extremely simple situation,<br />
supposedly governed by purely economic logic. Indeed, in<br />
this case innovation seems merely to satisfy, albeit in a different<br />
way, a preexisting need that remains unchanged. However,<br />
in ecology as in human societies, many innovations are<br />
successful without such a preexisting "niche." Such innovations<br />
transform the environment in which they appear, and as<br />
they spread, they create the conditions necessary for their<br />
own multiplication. their "niche." In social situations. in particular,<br />
the creation of a "demand," and even of a "need" for<br />
this demand to fulfill, often appears as correlated with the production<br />
of the goods or techniques that satisfy the demand.<br />
Evolutionary Feedback<br />
A first step toward accounting for this dimension of the evolutionary<br />
process can be achieved by making the "carrying capacity"<br />
of a system a function of the way it is exploited instead<br />
of taking it as given.<br />
In this way some supplementary dimensions of economic<br />
activities, and more particularly the "multiplying effects," can<br />
be represented. Thus we can describe the self-accelerating<br />
properties of systems and the spatial differentiation between<br />
different levels of activity.
197 ORDER THROUGH FLUCTUATIONS<br />
Geographers have already constructed a model correlating<br />
these processes, the Christaller model, defining the optimal<br />
spatial distribution of centers of economic activity. Important<br />
centers would be at the intersection of an hexagonal network,<br />
each being surrounded by a ring of towns of the next smallest<br />
size, each being, etc . . . . Obviously, in actual cases, such a<br />
regular hierarchical distribution is very infrequent: historical,<br />
political, and geographical factors abound, disrupting the spatial<br />
symmetry. But there is more. Even if all the important<br />
sources of asymmetrical development were excluded and we<br />
started from a homogeneous economic and geographical space,<br />
the modeling of the genesis of a distribution such as defined by<br />
Christaller establishes that the kind of static optimalization he<br />
describes constitutes a possible but quite unlikely result of the<br />
process.<br />
The model in question14 stages only the minimal set of variables<br />
implied by a calculation such as Christaller. A set of<br />
equations extending the logistic equations is constructed,<br />
starting from the basic supposition that populations tend to<br />
migrate as a function of local levels of economic activity,<br />
which thus define a kind of local "carrying capacity," here<br />
reduced to an "employment" capacity. But the local population<br />
is .also a potential consumer for locally produced goods.<br />
We have, in fact, a double positive feedback, called the "urban<br />
multiplier," for a local development: both the local population<br />
and the economic infrastructute produced by the already attained<br />
level of activity accelerate the increase of this activity.<br />
But each local level of activity is also determined by competition<br />
with similar centers of activity located elsewhere. The<br />
sale of produced goods or services depends on the cost of<br />
transporting them to consumers and on the size of the "enterprise."<br />
The expansion of each such enterprise depends on a<br />
demand that this expansion itself helps to create and for which<br />
it competes. Thus the respective growth of population and<br />
manufacturing or service activities is linked by strong feedback<br />
and nonlinearities.<br />
The model starts with a hypothetical initial condition, where<br />
"level 1" activity (rural) exists at the different points; it then<br />
permits us to follow successive launchings of activities corresponding<br />
to "superior" levels in Christaller's hierarchy-that<br />
is, implying exportation oo a greater range. Even if the initial
ORDER OUT OF CHAOS<br />
198<br />
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•<br />
65 66<br />
• •<br />
61<br />
•<br />
60<br />
•<br />
69<br />
•<br />
&9<br />
•<br />
68 68<br />
• •<br />
Figure 22. A possible history of "urbanization." • have only function 1;<br />
• have functions 1 and 2; A have functions 1, 2 and 3. are the largest<br />
centers, with functions 1, 2, 3, and 4. At t=O (not represented), all points
199<br />
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61<br />
•<br />
66<br />
•<br />
67<br />
•<br />
64<br />
•<br />
62<br />
•<br />
64<br />
•<br />
62<br />
•<br />
67<br />
•<br />
66<br />
•<br />
63<br />
•<br />
67<br />
•<br />
64 69<br />
• •<br />
have a "population" of 67 units. At C, the largest center is gOing through a<br />
maximum (152 population units); this is followed by an "urban sprawl," with<br />
creation of satellite cities; this also occurs around the second min center.
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200<br />
62<br />
•<br />
65<br />
•<br />
65<br />
•<br />
62<br />
•<br />
64<br />
•<br />
61<br />
•<br />
67<br />
•<br />
65<br />
•<br />
66<br />
•<br />
66<br />
•<br />
64<br />
•<br />
66 66<br />
• •
201 ORDER THROUGH FLUCTUATIONS
ORDER OUT OF CHAOS<br />
202<br />
i7<br />
•<br />
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•
203<br />
ORDER THROUGH FLUCTUATIONS<br />
state is quite homogeneous, the model shows that the mere<br />
play of chance factors-factors uncontrolled by the model,<br />
such as the place and time where the different enterprises<br />
start-is sufficient to produce symmetry breakings: the appearance<br />
of highly concentrated zones of activity while others<br />
suffer a reduction in economic activity and are depopulated.<br />
The different computer simulations show growth and decay,<br />
capture and domination, periods of opportunity for alternative<br />
developments followed by solidification of the existing domination<br />
structures.<br />
Whereas Christaller's symmetrical distribution ignores history,<br />
this scenario takes it into account, at least in a very minimal<br />
sense, as an interplay between .. laws," in this case of a<br />
purely economic nature, and the .. chance" governing the sequence<br />
of launchings.<br />
Modelizations of Corrplexity<br />
In spite of its simplicity, our model succeeds in showing some<br />
properties of the evolution of complex systems, and in particular,<br />
the difficulty of .. governing" a development determined by<br />
multiple interacting elements. Each individual action or each<br />
local intervention has a collective aspect that can result in<br />
quite unanticipated global changes. As Waddington emphasized,<br />
at present we have very little understanding of how a<br />
complex system is likely to respond to a given change. Often<br />
this response runs counter to our intuition. The term .. counterintuitive"<br />
was introduced at MIT to express our frustration:<br />
.. The damn thing just does not do what it should do!" To take<br />
the classic example cited by Waddington, a program of slum<br />
clearance results in a situation worse than before. New buildings<br />
attract a larger number of people into the area, but if there<br />
are not enough jobs for them, they remain poor, and their<br />
dwellings become even more overcrowded.t5 We are trained to<br />
think in terms of linear causality, but we need new "tools of<br />
thought": one of the greatest benefits of models is precisely to<br />
help us discover these tools and learn how to use them.<br />
As we have already emphasized, logistic equations are most<br />
relevant when the crucial dimension is the growth of a popula-
ORDER OUT OF CHAOS<br />
204<br />
tion, be it of animals, activities, or habits. What is presupposed<br />
is that each member of a given population can be taken<br />
as the equivalent of any of the others. But this general equivalence<br />
can itself be seen not as a simple general fact but as an<br />
approximation, the validity of which depends on the constraints<br />
and pressures to which this population was submitted<br />
and on the strategy it used to cope with them.<br />
Take, for example, the distinction ecologists have proposed<br />
between K and r strategies. K and r refer to the parameters in<br />
logistic equations. Though this distinction is only relative, it is<br />
especially clear when it characterizes the divergence resulting<br />
from a systematic interaction between two populations, particularly<br />
the prey-predator interaction. In this view, the typical<br />
evolution for a prey population will be the increase in the reproduction<br />
rate r. The predator will evolve toward more effective<br />
ways of capturing its prey-that is, toward an amelioration<br />
of K. But this amelioration, defined in a logistic frame, is liable<br />
to have consequences that go beyond the situations defined by<br />
logistic equations.<br />
As Stephen 1. Gould remarked, 16 a K strategy implies individuals<br />
becoming more and more able to learn from experience<br />
and to store memories-that is, individuals more<br />
complex with a longer period of maturation and apprenticeship.<br />
This in turn means individuals both more "valuable"<br />
representing a larger biological investment-and characterized<br />
by a longer period of vulnerability. The development of<br />
"social" and "family" ties thus appears as a logical counterpart<br />
of the K strategy. From that point on, other factors, besides<br />
the mere number of individuals in the population,<br />
become more and more relevant and the logistic equation measuring<br />
the success by the number of individuals becomes misleading.<br />
We have here a particular example of what makes<br />
modelization so risky. In complex systems, both the definition<br />
of entities and of the interactions among them can be modified<br />
by evolution. Not only each state of a system but also the very<br />
definition of the system as modelized is generally unstable, or<br />
at least metastable.<br />
We come to problems where methodology cannot be separated<br />
from the question of the nature of the object investigated.<br />
We cannot ask the same questions about a population of<br />
flies that reproduce and die by millions without apparently
205 ORDER THROUGH FLUCTUATIONS<br />
learning from or enlarging their experience and about a population<br />
of primates where each individual is an entanglement of<br />
its own experiences and the traditions of the populations in<br />
which he lives.<br />
We also find that, within anthropology itself, basic choices<br />
must be made between various approaches to collective phenomena.<br />
It is well known, for example, that structural anthropology<br />
privileges those aspects of society where the tools<br />
of logic and finite mathematics can be used, aspects such as<br />
the elementary structures of kinship or the analysis of myths,<br />
whose transformations are often compared to crystalline<br />
growth. Discrete elements are counted and combined. This<br />
contrasts with approaches that analyze evolution in terms of<br />
processes involving large, partially chaotic populations. We<br />
are dealing with two different outlooks and two types of models:<br />
Levi-Strauss defines them respectively as "mechanical"<br />
and "statistical." In the mechanical model "the elements are<br />
of the same scale as the phenomena" and individual behavior<br />
is based on prescriptions referring to the structural <strong>org</strong>anization<br />
of society. The anthropologist makes the logic of this behavior<br />
explicit. The sociologist, on the other hand, works with<br />
statistical models for large populations and defines averages<br />
and thresholds.I7<br />
A society defined entirely in terms of a functional model<br />
would correspond to the Aristotelian idea of natural hierarchy<br />
and order. Each official would perform the duties for which he<br />
has been appointed. These duties would translate at each level<br />
the different aspects of the <strong>org</strong>anization of the society as a<br />
whole. The king gives orders to the architect, the architect to<br />
the contractor, the contractor to the workers. Everywhere a<br />
mastermind is at work. On the contrary, termites and other<br />
social insects seem to approach the "statistical" model. As we<br />
have seen, there seems to be no mastermind behind the construction<br />
of the termites' nest, when interactions among individuals<br />
produce certain types of collective behavior in some<br />
circumstances, but none of these interactions refer to any<br />
global task, being all purely local. Such a description necessarily<br />
implies averages and reintroduces the question of stability<br />
and bifurcations.<br />
Which events will regress, and which arc likely to affect the<br />
whole system? What are the situations of choice, and what are
ORDER OUT OF CHAOS<br />
206<br />
the regimes of stability? Since size or the system's density<br />
may play the role of a bifurcation parameter, how may purely<br />
quantitative growth lead to qualitatively new choices? Questions<br />
such as these call for an ambitious program indeed. As<br />
with the rand K strategies, they lead us to connect the choice<br />
of a "good" model for social behavior and history. How does<br />
the evolution of a population lead it to become more "mechanical"?<br />
This question seems parallel to questions we have already<br />
met in biology. How, for example, does the selection of<br />
the genetic information governing the rates and regulations of<br />
metabolic reactions favor certain paths to such an extent that<br />
development seems to be purposive or appear as the translation<br />
of a "message"?<br />
We believe that models inspired by the concept of "order<br />
through fluctuations" will help us with these questions and<br />
even permit us in some circumstances to give a more precise<br />
formulation to the complex interplay between individual and<br />
collective aspects of behavior. From the physicist's point of<br />
view, this involves a distinction between states of the system<br />
in which all individual initiative is doomed to insignificance on<br />
the one hand, and on the other, bifurcation regions in which an<br />
individual, an idea, or a new behavior can upset the global<br />
state. Even in those regions, amplification obviously does not<br />
occur with just any individual, idea, or behavior, but only with<br />
those that are "dangerous"-that is, those that can exploit to<br />
their advantage the nonlinear relations guaranteeing the stability<br />
of the preceding regime. Thus we are led to conclude<br />
that the same nonlinearities may produce an order out of the<br />
chaos of elementary processes and still, under different circumstances,<br />
be responsible for the destruction of this same<br />
order, eventually producing a new coherence beyond another<br />
bifurcation.<br />
"Order through fluctuations" models introduce an unstable<br />
world where small causes can have large effects, but this world<br />
is not arbitrary. On the contrary, the reasons for the amplification<br />
of a small event are a legitimate matter for rational inquiry.<br />
Fluctuations do not cause the transformation of a systefll 's activity.<br />
Obviously, to use an image inspired by Maxwell, the<br />
match is responsible for the forest fire, but reference to a<br />
match does not suffice to understand the fire. Moreover, the<br />
fact that a fluctuation evades control does not mean that we
207 ORDER THROUGH FLUCTUATIONS<br />
cannot locate the reasons for the instability its amplification<br />
causes.<br />
An Open World<br />
In view of the complexity of the questions raised here, we can<br />
hardly avoid stating that the way in which biological and social<br />
evolution has traditionally been interpreted represents a particularly<br />
unfortunate use of the concepts and methods borrowed<br />
from physics18-unfortunate because the area of<br />
physics where these concepts and methods are valid was very<br />
restricted, and thus the analogies between them and social or<br />
economic phenomena are completely unjustified.<br />
The foremost example of this is the paradigm of optimization.<br />
It is obvious that the management of human society as<br />
well as the action of selective pressures tends to optimize<br />
some aspects of behaviors or modes of connection, but to consider<br />
optimization as the key to understanding how populations<br />
and individuals survive is to risk confusing causes with<br />
effects.<br />
Optimization models thus ignore both the possibility of radical<br />
transformations-that is, transformations that change the<br />
definition of a problem and thus the kind of solution soughtand<br />
the inertial constraints that may eventually force a system<br />
into a disastrous way of functioning. Like doctrines such as<br />
Adam Smith's invisible hand or other definitions of progress in<br />
terms of maximization or minimization criteria, this gives a<br />
reassuring representation of nature as an all-powerful and rational<br />
calculator, and of a coherent history characterized by<br />
global progress. To restore both inertia and the possibility of<br />
unanticipated events-that is, restore the open character of<br />
history-we must accept its fundamental uncertainty. Here we<br />
could use as a symbol the apparently accidental character of<br />
the great cretaceous extinction that cleared the path for the<br />
development of mammals, a small group of ratlike creatures.'<br />
This has been a general presentation, a kind of "bird's-eye<br />
view," and thus has omitted many topics of great interest:<br />
flames, plasmas, and lasers, for example, present nonequilibrium<br />
instabilities of great theoretical and practical interest.
ORDER OUT OF CHAOS<br />
208<br />
Everywhere we look, we find a nature that is rich in diversity<br />
and innovations. The conceptual evolution we have described<br />
is itself embedded in a wider history, that of the progressive<br />
rediscovery of time.<br />
We have seen new aspects of time being progressively incorporated<br />
into physics, while the ambitions to omniscience inherent<br />
in classical science were progressively rejected. In this<br />
chapter we have moved from physics through biology and<br />
ecology to human society, but we could have proceeded in the<br />
inverse order. Indeed, history began by concentrating mainly<br />
on human societies, after which attention was given to the<br />
temporal dimensions of life and of geology. The incorporation<br />
of time into physics thus appears as the last stage of a progressive<br />
reinsertion of history into the natural and social sciences.<br />
Curiously, at every stage of the process, a decisive feature of<br />
this "historicization" has been the discovery of some temporal<br />
heterogeneity. Since the Renaissance, We stern society has<br />
come into contact with different populations that were seen as<br />
corresponding to different stages of development; nineteenthcentury<br />
biology and geology learned to discover and classify<br />
fossils and to recognize in landscapes the memories of a past<br />
with which we coexist; finally, twentieth-century physics has<br />
also discovered a kind of fossil, residual black-body radiation,<br />
which tells us about the beginnings of the universe. Today we<br />
know that we live in a world where different interlocked times<br />
and the fossils of many pasts coexist.<br />
We must now proceed to another question. We have said<br />
that life is starting to seem as "natural as a falling body." What<br />
has the natural process of self-<strong>org</strong>anization to do with a falling<br />
body? What possible link can there be between dynamics, the<br />
science of force and trajectories, and the science of complexity<br />
and becoming, the science of living processes and of the<br />
natural evolution of which they are part? At the end of the<br />
nineteenth century, irreversibility was associated with the phenomena<br />
of friction, viscosity, and heating. Irreversibility lay at<br />
the origin of energy losses and waste. At that time it was still<br />
possible to subscribe to the fiction that irreversibility was only<br />
a result of our ineptitude, of our unsophisticated machines,<br />
and that nature remained fundamentally reversible. Now it is
209 ORDER THROUGH FLUCTUATIONS<br />
no longer possible: today even physics tells us that irreversible<br />
processes play a constructive and indispensable role.<br />
So we come to a question that can be avoided no longer.<br />
What is the relation between this new science of complexity<br />
and the science of simple, elementary behavior? What is the<br />
relation between these two opposing views of nature? Are<br />
there two sciences, two truths for a single world? How is that<br />
possible?<br />
In a certain sense, we have come back to the beginning of<br />
modern science. Now, as at Newton's time, two sciences<br />
come face to face-the science of gravitation, which describes<br />
an atemporal nature subject to laws, and the science of fire,<br />
chemistry. We now understand why it was impossible for the<br />
first synthesis produced by science, the Newtonian synthesis,<br />
to be complete; the forces of interaction described by dynamics<br />
cannot explain the complex and irreversible behavior of<br />
matter. Ignis mutat res. According to this ancient saying,<br />
chemical structures are the creatures of fire, the results of irreversible<br />
processes. How can we bridge the gap between being<br />
and becoming-two concepts in conflict, yet both necessary<br />
to reach a coherent description of this strange world in which<br />
we live?
BOOK THREE<br />
FROM BEING TO<br />
BECOMING
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I
CHAPTER VII<br />
REDISCOVERING TIME<br />
A Change of Emphasis<br />
Whitehead wrote that a "clash of doctrines is not a disaster, it<br />
is an opportunity. " t If this statement is true, few opportunities<br />
in the history of science have been so promising: two worlds<br />
have come face to face, the world of dynamics and the world of<br />
thermodynamics.<br />
Newtonian science was the outcome, the crowning synthesis<br />
of centuries of experimentation as well as of converging<br />
lines of theoretical research. The same is true for thermodynamics.<br />
The growth of science is quite different from the<br />
uniform unfolding of scientific disciplines, each in turn divided<br />
into an increasing number of watertight compartments. Quite<br />
the contrary, the convergence of different problems and points<br />
of view may break open the compartments and stir up scientific<br />
culture. These turning points have consequences that go<br />
beyond their scientific context and influence the intellectual<br />
scene as a whole. Inversely, global problems often have been<br />
sources of inspiration to science.<br />
The clash of doctrines, the conflict between being and becoming,<br />
indicates that a new turning point has been reached,<br />
that a new synthesis is needed. Such a synthesis is taking<br />
shape today, every bit as unexpected as the preceding ones.<br />
We again find a remarkable convergence of research, all of<br />
which contributes to identifying the difficulties inherent in the<br />
Newtonian concept of a scientific theory.<br />
The ambition of Newtonian science was to present a vision<br />
of nature that would be universal, deterministic, and objective<br />
inasmuch as it contains no reference to the observer, complete<br />
inasmuch as it attains a level of description that escapes the<br />
clutches of time.<br />
213
ORDER OUT OF CHAOS 214<br />
We have reached the core of the problem. "What is time?''<br />
Must we accept the opposition, traditional since Kant, between<br />
the static time of classical physics and the existential<br />
time we experience in our lives? According to Carnap:<br />
Once Einstein said that the problem of the Now worried<br />
him seriously. He explained that the experience of<br />
the Now means something special for man, something<br />
essentially different from the past and the future, but<br />
that this important difference does not and cannot occur<br />
within physics. That this experience cannot be grasped<br />
by science seemed to him a matter of painful but inevitable<br />
resignation. I remarked that all that occurs objectively<br />
can be described in science; on the one hand the<br />
temporal sequence of events is described in physics; and,<br />
on the other hand, the peculiarities of man's experiences<br />
with respect to time, including his different attitude towards<br />
past, present and future, can be described and (in<br />
principle) explained in psychology. But Einstein thought<br />
that these scientific descriptions cannot possibly satisfy<br />
our human needs; that there is something essential about<br />
the Now which is just outside of the realm of science.2<br />
It is interesting to note that Bergson, in a sense following an<br />
opposite road, also reached a dualistic conclusion (see Chapter<br />
III). Like Einstein, Bergson started with a subjective time<br />
and then moved to time in nature, time as objectified by physics.<br />
However, for him this objectivization led to a debasement<br />
of time. Internal existential time has qualitative features that<br />
are lost in the process. It is for this reason that Bergson introduced<br />
the distinction between physical time and duration, a<br />
concept referring to existential time.<br />
But we cannot stop here. As J. T. Fraser says, "The resulting<br />
dichotomy between time felt and time understood is a hallmark<br />
of scientific-industrial civilization, a sort of collective<br />
schizophrenia. "3 As we have already emphasized, where classical<br />
science used to emphasize permanence, we now find<br />
change and evolution; we no longer see in the skies the trajectories<br />
that filled Kant's heart with the same admiration as the<br />
moral law residing in him. We now see strange objects: quasars,·<br />
pulsars, galaxies exploding and being torn apart, stars
215 REDISCOVERING TIME<br />
that, we are told, collapse into "black holes" irreversibly devouring<br />
everything they manage to ensnare.<br />
Time has penetrated not only biology, geology, and the social<br />
sciences but also the two levels from which it has been<br />
traditionally excluded, the microscopic and the cosmic. Not<br />
only life, but also the universe as a whole has a history; this<br />
has profound implications.<br />
The first theoretical paper dealing with a cosmological<br />
model from the point of view of general relativity was published<br />
by Einstein in 1917. It presented a static, timeless view<br />
of the universe, Spinoza's vision translated into physics. But<br />
then comes the unexpected. It became immediately evident<br />
that there were other, time-dependent solutions to Einstein's cosmological<br />
equations. We owe this discovery to the Russian astrophysicist<br />
A. Friedmann and the Belgian G. Lemaitre. At the<br />
same time Hubble and his coworkers were studying the motions<br />
of galaxies, and they demonstrated that the velocity of<br />
distant galaxies is proportional to their distance from earth.<br />
The relation with the expanding universe discovered by Friedmann<br />
and Lemaitre was obvious. Yet for many years physicists<br />
remained reluctant to accept such an "historical"<br />
description of cosmic evolution. Einstein himself was wary of<br />
it. Lemaitre often said that when he tried to discuss with Einstein<br />
the possibility of making the initial state of the universe<br />
more precise and perhaps finding there the explanation of cosmic<br />
rays, Einstein showed no interest.<br />
Today there is new evidence, the famous residual blackbody<br />
radiation, the light that illuminated the explosion of the<br />
hyperdense fireball with which our universe began. The whole<br />
story appears as another irony of history. In a sense, Einstein<br />
has, against his will, become the Darwin of physics. Darwin<br />
taught us that man is embedded in biological evolution; Einstein<br />
has taught us that we are embedded in an evolving universe.<br />
Einstein's ideas led him to a new continent, as unexpected to<br />
him as was America to Columbus. Einstein, like many physicists<br />
of his generation, was guided by a deep conviction that<br />
there was a fundamental, simple level in nature. Yet today this<br />
level is becoming less and less accessible to experiment. The<br />
only objects whose behavior is truly "simple" exist in our own<br />
world, at the macroscopic level. Classical science carefully<br />
chose its objects from this intermediate range. The first ob-
ORDER OUT OF CHAOS 216<br />
jects singled out by Newton-falling bodies, the pendulum,<br />
planetary motion-were simple. We know now, however, that<br />
this simplicity is not the hallmark of the fundamental: it cannot<br />
be attributed to the rest of the world.<br />
Does this suffice? We now know that stability and simplicity<br />
are exceptions. Should we merely disregard the totalizing totalitarian<br />
claims of a conceptualization that, in fact, applies<br />
only to simple and stable objects? Why worry about the incompatibility<br />
between dynamics and thermodynamics?<br />
We must not f<strong>org</strong>et the words of Whitehead, words constantly<br />
confirmed by the history of science: a clash of doctrines<br />
is an opportunity, not a disaster. It has often been<br />
suggested that we simply ignore certain issues for practical<br />
reasons on the grounds that they are based on idealizations<br />
that are difficult to implement. At the beginning of this century,<br />
several physicists suggested abandoning determinism on<br />
the grounds that it was inaccessible in real experience. 4 Indeed,<br />
as we have already emphasized, we never know the exact<br />
positions and velocities of the molecules in a large system;<br />
thus an exact prediction of the system's future evolution is impossible.<br />
More recently, Brillouin hoped to destroy determinism<br />
by appealing to the commonsense truth that accurate<br />
prediction requires an accurate knowledge of the initial conditions<br />
and that this knowledge must be paid for; the exact prediction<br />
necessary to make determinism work requires that an<br />
"infinite" price be paid.<br />
These objections, while reasonable, do not affect the conceptual<br />
world of dynamics. They shed no new light on reality.<br />
Moreover, the improvements in technology could bring us<br />
closer and closer to the idealization implied by classical dynamics.<br />
In contrast, demonstrations of "impossibility" have a fundamental<br />
importance. They imply the discovery of an unexpected<br />
intrinsic structure of reality that dooms an intellectual<br />
enterprise to failure. Such discoveries will exclude the possibility<br />
of an operation that previously could have been imagined<br />
as feasible, at least in principle. "No engine can have an<br />
efficiency greater than one," "no heat engine can produce<br />
useful work unless it is in contact with two sources" are examples<br />
of statements of impossibility which have led to profound<br />
conceptual innovations.
217 .REDISCOVERING TIME<br />
Thermodynamics, relativity, and quantum mechanics are all<br />
rooted in the discovery of impossibilities, of limits to the ambitions<br />
of classical physics. Thus they marked the end of an exploration<br />
that had reached its limits. But we can now see these<br />
scientific innovations in a different light, not as an end but a<br />
beginning, as the opening up of new opportunities. We shall<br />
see in Chapter IX that the second law of thermodynamics expresses<br />
an "impossibility," even on the microscopic level, but<br />
even there the newly discovered impossibility becomes a start·<br />
ing point for the emergence of new concepts.<br />
The End of Universality<br />
Scientific description must be consistent with the resources<br />
available to an observer who belongs to the world he describes<br />
and cannot refer to some being who contemplates the physical<br />
world "from the outside." This is one of the fundamental requirements<br />
of relativity theory. In connection with the propagation<br />
of signals a limit appears that cannot be transgressed<br />
by any observer. Indeed, c, the velocity of light in vacuum<br />
(c=300,000 km/sec), is the limiting velocity for the propagation<br />
of all signals. Thus this limiting velocity plays a fundamental<br />
role. It limits the region in space that may influence the<br />
point where an observer is located.<br />
There is no universal constant in Newtonian physics. This is<br />
the reason for its claim to universality, why it can be applied in<br />
the same way whatever the scale of the objects: the motion of<br />
atoms, planets, and stars are governed by a single law.<br />
The discovery of universal constants signified a radical<br />
change. Using the velocity of light as the comparison standard,<br />
physics has established a distinction between low and<br />
high velocities, those approaching the speed of light.<br />
Likewise, Planck's constant, h; sets up a natural scale according<br />
to the object's mass. The atom can no longer be regarded<br />
as a tiny planetary system. Electrons belong to a<br />
different scale than planets and all other heavy, slow-moving,<br />
macroscopic objects, including ourselves.<br />
Universal constants not only destroy the homogeneity of the<br />
universe by introducing physical scales in terms of which vari-
ORDER OUT OF CHAOS 218<br />
ous behaviors become qualitatively different, they also lead to<br />
a new conception of objectivity. No observer can transmit signals<br />
at a velocity higher than that of light in a vacuum. Hence<br />
Einstein's remarkable conclusion: we can no longer define the<br />
absolute simultaneity of two distant events; simultaneity can<br />
be defined only in terms of a given reference frame. The scope<br />
of this book does not permit an extensive account of relativity<br />
theory. Let us merely point out that Newton's laws did not<br />
assume that the observer was a "physical being." Objective<br />
description was defined precisely as the absence of any reference<br />
to its author. For "nonphysical" intelligent beings capable<br />
of communicating at an infinite velocity, the laws of<br />
relativity would be irrelevant. The fact that relativity is based<br />
on a constraint that applies only to physically localized observers,<br />
to beings who can be in only one place at a time and<br />
not everywhere at once, gives this physics a "human" quality.<br />
This does not mean, however, that it is a "subjective" physics,<br />
the result of our preferences and convictions; it remains subject<br />
to intrinsic constraints that identify us as part of the physical<br />
world we are describing. It is a physics that presupposes an<br />
observer situated within the observed world . .Our dialogue<br />
with nature will be successful only if it is carried on from<br />
within nature.<br />
The Rise of Quantum Mechanics<br />
Relativity altered the classical concept of objectivity. However,<br />
it left unchanged another fundamental characteristic of<br />
classical physics, namely, the ambition to achieve a "complete"<br />
description of nature. After relativity, physicists could<br />
no longer appeal to a demon who observed the entire universe<br />
from outside, but they could still conceive of a supreme mathematician<br />
who, as Einstein claimed, neither cheats nor plays<br />
dice. This mathematician would possess the formula of the<br />
universe, which would include a complete description of nature.<br />
In this sense, relativity remains a continuation of classical<br />
physics.<br />
Quantum mechanics, on the other hand, is the first physical<br />
theory truly to have broken with the past. Quantum mechanics<br />
not only situates us in nature, it also labels us as "heavy"
219 REDISCOVERING TIME<br />
beings composed of a macroscopic number of atoms. In order<br />
to visualize more clearly the consequences of the velocity of<br />
light as a universal constant, Einstein imagined himself riding<br />
a photon. But quantum mechanics discovered that we are too<br />
heavy to ride photons or electrons. We cannot possibly replace<br />
such airy beings, identify ourselves with them, and describe<br />
what they would think, if they were able to think, and what<br />
they would experience, if they were able to feel anything.<br />
The history of quantum mechanics, like that of all conceptual<br />
innovations, is complex, full of unexpected events; it is<br />
the history of a logic whose implications were discovered long<br />
after it was conceived in the urgency of experiment and in a<br />
difficult political and cultural environment.5 This history cannot<br />
be related here; we only wish to emphasize its role in the<br />
construction of the bridge from being to becoming, which is<br />
our main subject.<br />
The birth of quantum mechanics was in itself part of the<br />
quest for this bridge. Planck was interested in the interaction<br />
between matter and radiation. Underlying his work was the<br />
ambition to accomplish for the matter-light interaction what<br />
Boltzmann had achieved for the matter-matter interaction,<br />
namely, to discover a kinetic model for irreversible processes<br />
leading to equilibrium. 6 To his surprise, he was forced, in order<br />
to reach experimental results valid at thermal equilibrium,<br />
to assume that an exchange of energy between matter and radiation<br />
occurred only in discrete steps involving a new universal<br />
constant. This universal constant "h" measures the "size"<br />
of each step.<br />
In this case, as in many others, the challenge of irreversibility<br />
led to decisive progress in physics.<br />
This discovery remained isolated until Einstein presented<br />
the first general interpretation of Planck's constant. He understood<br />
that it had far-reaching implications for the nature of<br />
light. He introduced a revolutionary concept: the waveparticle<br />
duality of light.<br />
Since the beginning of the nineteenth century, light had<br />
been associated with wave properties manifest in phenomena<br />
such as diffraction or interference. However, at the end of the<br />
nineteenth century, new phenomena were discovered, notably<br />
the photoelectric effect-that is, the expulsion of electrons as<br />
the result of the absorption of light. These new experimental
ORDER OUT OF CHAOS 220<br />
results were difficult to explain in terms of the traditional wave<br />
properties of light. Einstein solved the riddle by assuming that<br />
light may be both wave and particle and that these two aspects<br />
are related through Planck's constant. More precisely, a light<br />
wave is characterized by its frequency u and its wavelength X;<br />
h permits us to go from frequency and wavelength to mechanical<br />
quantities such as energy e and momentum p. The relations<br />
between u and A on the one side and e and p on the other are<br />
Very simple: B = hu, p = h/X, and both involve h. 1\.venty years<br />
later, Louis de Broglie extended this wave-particle duality<br />
from light to matter; thus the starting point for the modern<br />
formulation of quantum mechanics.<br />
In 1913 Niels Bohr had linked the new quantum physics to<br />
the structure of atoms (and later of molecules). As a result of<br />
the wave-particle duality, he showed that there exist discrete<br />
sequences of electron orbits. When an atom is excited, the<br />
electron jumps from one orbit to another. At this very instant<br />
the atom emits or absorbs a photon the frequency of which<br />
corresponds to the difference between the energies characterizing<br />
the electron's motion in each of the two orbits. This<br />
difference is calculated in terms of Einstein's formula relating<br />
energy to frequency.<br />
Thus we reach the decisive years 1925-27, a "golden age" of<br />
physics.7 During this short period, Heisenberg, Born, Jordan,<br />
Schrodinger, and Dirac made quantum physics into a consistent<br />
new theory. This theory incorporates Einstein's and de<br />
Broglie's wave-particle duality in the framework of a new generalized<br />
form of dynamics: quantum mechanics. For our purposes<br />
here, the conceptual novelty of quantum mechanics is<br />
essential.<br />
First and foremost, a new formulation, unknown in classical<br />
physics, had to be introduced to allow "quantitization" to be<br />
incorporated into the theoretical language. The essential fact<br />
is that an atom can be found only in discrete energy levels<br />
corresponding to the various electron orbits. In particular, this<br />
means that energy (or the Hamiltonian) can no longer be<br />
merely a function of the position and the moment, as it is in<br />
classical mechanics. Otherwise, by giving the positions and<br />
moments slightly different values, energy could be made to<br />
vary continuously. But as observation reveals, only discrete<br />
levels exist.
22'1<br />
REDISCOVERING TIME<br />
We therefore have to replace the conventional idea that the<br />
Hamiltonian is a function of position and momenta with something<br />
new; the basic idea of quantum mechanics is that the<br />
Hamiltonian as well as the other quantities of classical mechanics,<br />
such as coordinates q or momenta p, now become<br />
operators. This is one of the boldest ideas ever introduced in<br />
science, and we would like to discuss it in detail.<br />
It is a simple idea, even if at first it seems somewhat abstract.<br />
We have to distinguish the operator-a mathematical<br />
operation-and the object on which it operates-a function.<br />
As an example, take as the mathematical "operator" the derivative<br />
represented by d/dx and suppose it acts on a function-say,<br />
x2; the result of this operation is a new function, this<br />
time "2x." However, certain functions behave in a peculiar<br />
way with respect to derivation. For example, the derivative of<br />
"e3x" is "3e3x": here we return to the original function simply<br />
multiplied by some number-here, 3. Functions that are<br />
merely recovered by a given operator to them are known as the<br />
"eigenfunctions" of this operator, and the numbers by which<br />
the eigenfunction is multiplied after the application of the operator<br />
are the "eigenvalues" of the operator.<br />
To each operator there thus corresponds an ensemble, a "reservoir"<br />
of numerical values; this ensemble forms its "spectrum."<br />
This spectrum is "discrete" when the eigenvalues form<br />
a discrete series. There exists, for instance, an operator with<br />
all the integers 0, 1, 2 . . . as eigenvalues. A spectrum may<br />
also be continuous-for example, when it consists of all the<br />
numbers between 0 and 1.<br />
The basic concept of quantum mechanics may thus be expressed<br />
as follows: to all physical quantities in classical mechanics<br />
there corresponds in quantum mechanics an operator,<br />
and the numerical values that may be taken by this physical<br />
quantity are the eigenvalues of this operator. The essential<br />
point is that the concept of physical quantity (represented by<br />
an operator) is now distinct from that of its numerical values<br />
(represented by the eigenvalues of the operator). In particular,<br />
energy will now be represented by the Hamiltonian operator,<br />
and the energy levels-the observed values of the energywill<br />
be identified with the eigenvalues corresponding to this<br />
operator.<br />
The introduction of operators opened up to physics a micro-
ORDER OUT OF CHAOS 222<br />
scopic world of unsuspected richness, and we regret that we<br />
cannot devote more space to this fascinating subject, in which<br />
creative imagination and experimental observation are so successfully<br />
combined. Here we wish merely to stress that the<br />
microscopic world is governed by laws having a new structure,<br />
thereby putting an end once and for all to the hope of discovering<br />
a single conceptual scheme common to all levels of description.<br />
A new mathematical language invented to deal with a certain<br />
situation may actually open up fields of inquiry that are<br />
full of surprises, going far beyond the expectations of its originators.<br />
This was true for differential calculus, which lies at<br />
the root of the formulation of classical dynamics. It is true as<br />
well for operator calculus. Quantum theory, initiated as demanded<br />
by the result of unexpected experimental discoveries,<br />
was quick to reveal itself as pregnant with new content.<br />
Today, more than fifty years after the introduction of operators<br />
into quantum mechanics, their significance remains a subject<br />
of lively discussion. From the historical point of view, the<br />
introduction of operators is linked to the existence of energy<br />
levels, but today operators have applications even in classical<br />
physics. This implies that their significance has been extended<br />
beyond the expectations of the founders of quantum mechanics.<br />
Operators now come into play as soon as, for one reason<br />
or another, the notion of a dynamic trajectory has to be discarded,<br />
and with it, the deterministic description a trajectory<br />
implies.<br />
Heisenbergs Uncertainty Relation<br />
We have seen that in quantum mechanics to each physical<br />
quantity corresponds an operator that acts on functions. Of<br />
special importance are the eigenfunctions and the eigenvalues<br />
corresponding to the operator under consideration. The eigenvalues<br />
correspond precisely to the numerical values the physical<br />
quantity can now take. Let us take a closer look at the<br />
operators quantum mechanics associates with coordinates q<br />
and momenta p; their coordinates are, as we have seen in<br />
Chapter II, the canonical variables.<br />
In classical mechanics coordinates and momenta are inde-
223 REDISCOVERING TIME<br />
pendent in the sense that we can ascribe to a coordinate a<br />
numerical value quite independent of the value we have ascribed<br />
to the momentum. However, the existence of Planck's<br />
constant h implies the reduction in the number of independent<br />
variables. We could have guessed this right away from the<br />
Einstein-de Broglie relation A= hlp, which, as we have seen,<br />
connects wavelength to momentum. Planck's constant h expresses<br />
a relation between lengths (closely related to the concept<br />
of coordinates) and momenta. Therefore, positions and<br />
momenta can no longer be independent variables, as in classical<br />
mechanics. The operators corresponding to positions and<br />
momenta can be expressed in terms of the coordinate alone or<br />
in terms of the momentum, something explained in all textbooks<br />
dealing with quantum mechanics.<br />
The important point is that in all cases, only one type of<br />
quantity appears (either coordinate or momentum), but not<br />
both. In this sense we may say that the quantum mechanics<br />
divides the number of classical mechanical variables by a factor<br />
of two.<br />
One fundamental property results from the relation between<br />
operators in quantum mechanics: the two operators q0 P and<br />
Pop do not commute-that is, the results of q0pP0p and of<br />
Pop%p applied to the same function are different. This has profound<br />
implications, since only commuting operators admit<br />
common eigenfunctions. Thus we cannot identify a function<br />
that would be an eigenfunction of both coordinate and momentum.<br />
As a consequence of the definition of the coordinate and<br />
momentum operators in quantum mechanics, there can be no<br />
state in which the physical quantities, coordinate q and momentum<br />
p, both have a well-defined value. This situation, unknown<br />
in classical mechanics, is expressed by Heisenberg's<br />
famous uncertainty relations. We can measure a coordinate<br />
and a momentum, but the dispersions of the respective possible<br />
predictions as expressed by f::j,q,f::j,p are related by the<br />
Heisenberg inequality f::j,qf::j,p;;::.h. We can make f::j,q as small as<br />
we want, but then f::j,p goes to infinity, and vice versa.<br />
Much has been written about Heisenberg's uncertainty relations,<br />
and our discussion is admittedly oversimplified. But we<br />
wish to give our readers some understanding of the new problem<br />
that re:sult:s from the u:se of operators; Heisenberg's uncertainty<br />
relation necessarily leads to a revision of the concept of
ORDER OUT OF CHAOS 22 4<br />
causality. It is possible to determine the coordinate precisely.<br />
But the moment we do so, the momentum will acquire an arbitrary<br />
value, positive or negative. In other words, in an instant<br />
the position of the object will become arbitrarily distant.<br />
The meaning of localization becomes blurred: the concepts<br />
that form the basis of classical mechanics are profoundly altered.<br />
These consequences of quantum mechanics were unacceptable<br />
to many physicists, including Einstein; and many experiments<br />
were devised to demonstrate their absurdity. An<br />
attempt was also made to minimize the conceptual change involved.<br />
In particular, it was suggested that the foundation of<br />
quantum mechanics is in some way related to perturbations<br />
resulting from the process of observation. A system was<br />
thought to possess intrinsically well-defined mechanical parameters<br />
such as coordinates and momenta; but some of them<br />
would be made fuzzy by measurement, and Heisenberg's uncertainty<br />
relation would only express the perturbation created<br />
by the measurement process. Classical realism thus would remain<br />
intact on the fundamental level, and we would simply<br />
have to add a positivistic qualification. This interpretation<br />
seems too narrow. It is not the quantum measurement process<br />
that disturbs the results. Far from it: Planck's constant forces<br />
us to revise our concepts of coordinates and momenta. This<br />
conclusion has been confirmed by recent experiments designed<br />
to test the assumption of local hidden variables that<br />
were introduced to restore classical determinism. s The results<br />
of those experiments confirm the striking consequences of<br />
quantum mechanics.<br />
That quantum mechanics obliges us to speak less absolutely<br />
about the localization of an object implies, as Niels Bohr often<br />
emphasized, that we must give up the realism of classical<br />
physics. For Bohr, Planck's constant defines the interaction<br />
between a quantum system and the measurement device as<br />
nondecomposable. It is only to the quantum phenomenon as a<br />
whole, including the measurement interaction, that we can ascribe<br />
numerical values. All description thus implies a choice<br />
of the measurement device, a choice of the question asked. In<br />
this sense, the answer, the result of the measurement, does not<br />
give us access to a given reality. We have to decide which measurement<br />
we are going to perform and which question our ex-
225 REDISCOVERING TIME<br />
periments will ask the system. Thus there is an irreducible<br />
multiplicity of representations for a system, each connected<br />
with a determined set of operators.<br />
This implies a departure from the classical notion of objectivity,<br />
since in the classical view the only "objective" description<br />
is the complete description of the system as it is,<br />
independent of the choice of how it is observed.<br />
Bohr always emphasized the novelty of the positive choice<br />
introduced through measurement. The physicist has to choose<br />
his language, to choose the macroscopic experimental device.<br />
Bohr expressed this idea through the principle of complementarity,9<br />
which may be considered as an extension of Heisenberg's<br />
uncertainty relations. We can measure coordinates or<br />
momenta, but not both. No single theoretical language articulating<br />
the variables to which a well-defined value can be attributed<br />
can exhaust the physical conent of a system. Various<br />
possible languages and points of view about the system may<br />
be complementary. They all deal with the same reality, but it is<br />
impossible to reduce them to one single description. The irreducible<br />
plurality of perspectives on the same reality expresses<br />
the impossibility of a divine point of view from which the<br />
whole of reality is visible. However, the lesson of the principle<br />
of complementarity is not a lesson in resignation. Bohr used to<br />
say that the significance of quantum mechanics always made<br />
him dizzy, and we do indeed feel dizzy when we are torn from<br />
the comfortable routine of common sense.<br />
The real lesson to be learned from the principle of complementarity,<br />
a lesson that can perhaps be transferred to other<br />
fields of knowledge, consists in emphasizing the wealth of reality,<br />
which overflows any single language, any single logical<br />
structure. Each language can express only part of reality. Music,<br />
for example, has not been exhausted by any of its realizations,<br />
by any style of composition, from Bach to Schonberg.<br />
We have emphasized the importance of operators because<br />
they demonstrate that the reality studied by physics is also a<br />
mental construct; it is not merely given. We must distinguish<br />
between the abstract notion of a coordinate or of momentum,<br />
represented mathematically by operators, and their numerical<br />
realization, which can be reached through experiments. One<br />
of the reasons for the opposition between the ··two cultures"<br />
may have been the belief that literature corresponds to a con-
ORDER OUT OF CHAOS 226<br />
ceptualization of reality, to "fiction," while science seems to<br />
express objective "reality. " Quantum mechanics teaches us<br />
that the situation is not so simple. On all levels reality implies<br />
an essential element of conceptualization.<br />
The Temporal Evolution of Quantum Systems<br />
We shall now move on to discuss the temporal evolution of<br />
quantum systems. As in classical mechanics, the Hamiltonian<br />
plays a fundamental role. As we have seen, in quantum mechanics<br />
it is replaced by the Hamiltonian operator Hop· This<br />
energy operator plays a central role: on the one hand, its<br />
eigenvalues correspond to the energy levels; on the other<br />
hand, as in classical mechanics, the Hamiltonian operator determines<br />
the temporal evolution of the system. In quantum<br />
mechanics the role played by the canonical equation of classical<br />
mechanics is taken by the Schrodinger equation, which<br />
expresses the time evolution of the fu nction characterizing the<br />
quantum state as the result of the application of the operator<br />
Hop on the wave function \jJ (there are, of course, other formulations,<br />
which we cannot describe here). The term "wave<br />
function" has been chosen to emphasize once again the waveparticle<br />
duality so fu ndamental in all of quantum physics. \jJ is<br />
a wave amplitude that evolves according to a particle type of<br />
equation determined by the Hamiltonian. Schrodinger's equation,<br />
like the canonical equation of classical physics, expresses<br />
a reversible and deterministic evolution. The<br />
reversible change of wave function corresponds to a reversible<br />
motion along a trajectory. If the wave fu nction at a given instant<br />
is known, Schrodinger's equation allows it to be calculated<br />
for any previous or subsequent instant. From this viewpoint,<br />
the situation is strictly similar to that in classical mechanics.<br />
This is because the uncertainty relations of quantum mechanics<br />
do not include time. Time remains a number, not an operator,<br />
and only operators can appear in Heisenberg's uncertainty<br />
re lations.<br />
Quantum mechanics deals with only half of the variables of<br />
dassical mechanics. As a result, classical determinism becomes<br />
inapplicable, and in quantum physics statistical consid-
227 REDISCOVERING TIME<br />
erations play a central role. It is through the wave intensity ttJ2<br />
(the square of the amplitude) that we make contact with statistical<br />
considerations.<br />
The standard statistical interpretation of quantum mechanics<br />
runs as follows: consider the eigenfunctions of some operator-say,<br />
the energy operator H0 P -and the corresponding<br />
eigenvalues. In general the wave function tts will not be the<br />
eigenfunction of the energy operator, but it can be expressed<br />
as the superposition of these eigenfunctions. The respective<br />
importance of each eigenfunction in this superposition allows<br />
us to calculate the probability for the appearance of the various<br />
possible corresponding eigenvalues.<br />
Here again we notice a fundamental departure from classical<br />
theory. Only probabilities can be predicted, not single<br />
events. This was the second time in the history of science that<br />
probabilities were used to explain some basic features of nature.<br />
The first time was in Boltzmann's interpretation of entropy.<br />
There , however, a subjective point of view remained<br />
possible; in this view, "only" our ignorance in the face of the<br />
complexity of the systems considered prevented us from achieving<br />
a complete description. (We shall see that today it is possible<br />
to overcome this attitude.) Here, as before, the use of probabilities<br />
was unacceptable to many physicists-including Einstein-who<br />
wished to achieve a "complete" deterministic<br />
description. Just as with irreversibility, an appeal to our ignorance<br />
seemed to offer a way out: our inaptitude would make us<br />
responsible for statistical behavior in the quantum world, just<br />
as it makes us responsible for irreversibility.<br />
Once again we come to the problem of hidden variables.<br />
However, as we have said, there has been no experimental evidence<br />
to justify the introduction of such variables, and the rol<br />
of probabilities seems irreducible.<br />
There is only one case in which the Schrodinger equation<br />
leads to a deterministic prediction: that is when tlJ, instead of<br />
being a superposition of eigenfunctions, is reduced to a single<br />
one. In particular, in an ideal measurement process, a system<br />
may be prepared in such a way that the result of a given measurement<br />
may be predicted. We then know that the system is<br />
described by the corresponding eigenfunction. From then on,<br />
the system may be described with certainty as being in the<br />
eigenstate indicated by the measurement result.
ORDER OUT OF CHAOS 228<br />
The measurement process in quantum mechanics has a spe·<br />
cial significance that is attracting considerable interest today.<br />
Suppose we start with a wave function, which is indeed a superposition<br />
of eigenfunctions. As a result of the measurement<br />
process, this single collection of systems all represented by<br />
the same wave function is replaced by a collection of wave<br />
functions corresponding to the various eigenvalues that may<br />
be measured. Stated technically, a measurement leads from a<br />
single wave function (a "pure" state) to a mixture .<br />
As Bohr and Rosenfeld IO repeatedly pointed out, every<br />
measurement contains an element of irreversibility, an appeal<br />
made to irreversible phenomena, such as chemical processes<br />
corresponding to the recording of the "data." Recording is accompanied<br />
by an amplification whereby a microscopic event<br />
produces an effect on a macroscopic level-that is, a level at<br />
which we can read the measuring instruments. The measurement<br />
thus presupposes irreversibility.<br />
This was in a sense already true in classical physics. How·<br />
ever, the problem of the irreversible character of measurement<br />
is more urgent in quantum mechanics because it raises questions<br />
at the level of its formulation.<br />
The usual approach to this problem states that quantum mechanics<br />
has no choice but to postulate the coexistence of two<br />
mutually irreducible processes. the reversible and continuous<br />
evolution described by Schrodinger's equation and the irreversible<br />
and discontinuous reduction of the wave function to<br />
one of its eigenfunctions at the time of measurement. Thus the<br />
paradox: the reversible Schrodinger equation can be tested<br />
only by irreversible measurements that the equation is by definition<br />
unable to describe. It is thus impossible for quantum<br />
mechanics to set up a closed structure.<br />
In the face of these difficulties, some physicists have once<br />
more taken refuge in subjectivism, stating that we-our measurement<br />
and even, for some, our mind-determine the evolution<br />
of the system that breaks the law of natural, "objective"<br />
reversibility. 11 Others have concluded that Schrodinger's equation<br />
was not "complete" and that new terms must be added to<br />
account for the irreversibility of the measurement. Other more<br />
improbable "solutions" have also been proposed, such as<br />
Everett's many-world hypothesis (see d'Espagnat, ref. 8). For<br />
us, however, the coexistence in quantum mechanics of revers-
229 REDISCOVERING TIME<br />
ibility and irreversibility shows that the classical idealization<br />
that describes the dynamic world as self-contained is impossible<br />
at the microscopic level. This is what Bohr meant when he<br />
noted that the language we use to describe a quantum system<br />
cannot be separated from the macroscopic concepts that describe<br />
the functioning of our measurement instruments. Schrodinger's<br />
equation does not describe a separate level of reality;<br />
rather it presupposes the macroscopic world to which we belong.<br />
·<br />
The problem of measurement in quantum mechanics is thus<br />
an aspect of one of the problems to which this book is devoted-the<br />
connection between the simple world described by<br />
Hamiltonian trajectories and Schrodinger's equation, and the<br />
complex macroscopic world of irreversible processes.<br />
In Chapter IX, we shall see that irreversibility enters classical<br />
physics when the idealization involved in the concept of a<br />
trajectory becomes inadequate. The measurement problem in<br />
quantum mechanics is susceptible to the same type of solution.12<br />
Indeed, the wave function represents the maximum<br />
knowledge of a quantum system. As in classical physics, the<br />
object of this maximum knowledge satisfies a reversible evolution<br />
equation. In both cases, irreversibility enters when the<br />
ideal object corresponding to maximum knowledge has to be<br />
replaced by less idealized concepts. But when does this happen?<br />
This is the question of the physical mechanisms of irreversibility<br />
to which we shall turn in Chapter IX. But let us first<br />
summarize some other features of the renewal of contemporary<br />
science.<br />
A Nonequilibrium Universe<br />
The two scientific revolutions described in this chapter started<br />
as attempts to incorporate universal constants, c and h, into<br />
the framework of classical mechanics. This led to far-reaching<br />
consequences, some of which we have described here. From<br />
other perspectives, relativity and quantum mechanics seemed<br />
to adhere to the basic world view expressed in Newtonian mechanics.<br />
This is especially true regarding the role and meaning<br />
of time. In quantum mechanics, once the wave function at time
ORDER OUT OF CHAOS<br />
230<br />
zero is known, its value ljJ(t) both for future and past is deter·<br />
mined. Likewise, in relativity theory the static geometric charac·<br />
ter of time is often emphasized by the use of four-dimensional<br />
notation (three dimensions for space and one for time). As expressed<br />
concisely by Minkowski in 1908, "space by itself and<br />
time by itself are doomed to fade away into mere shadows, and<br />
only a kind of union of the two will preserve an independent<br />
reality . .. only a world in itself will subsist." 13<br />
But over the past five decades this situation has radically<br />
changed. Quantum mechanics has become the main tool for<br />
dealing with elementary particles and their transformations. It<br />
is outside the scope of this book to describe the bewildering<br />
variety of elementary particles that have appeared during the<br />
past few years.<br />
We want only to recall that, using both quantum mechanics<br />
and relativity, Dirac demonstrated that we have to associate to<br />
each particle of mass m and charge e an antiparticle of the<br />
same mass but of opposite charge. Positrons, the antiparticles<br />
of electrons, as well as antiprotons, are currently being produced<br />
in high-energy accelerators. Antimatter has become a<br />
common subject of study in particle physics. Particles and<br />
their corresponding antiparticles annihilate each other when they<br />
collide, producing photons, massless particles corresponding<br />
to light. The equations of quantum theory are symmetric in<br />
respect to the exchange particle-antiparticle, or more precisely,<br />
they are symmetric in respect to a weaker requirement<br />
known as the CPT symmetry. In spite of this symmetry, there<br />
exists a remarkable dissymmetry between particles and antiparticles<br />
in the world around us. We are made of particles<br />
(electrons, protons), while antiparticles remain rare laboratory<br />
products. If particles and antiparticles coexisted in equal<br />
amount, all matter would be annihilated. There is strong evidence<br />
that antimatter does not exist in our galaxy, but the possibility<br />
that it exists in distant galaxies cannot be excluded. We<br />
can imagine a mechanism in the universe that separates particles<br />
and antiparticles, hides antiparticles somewhere. However,<br />
it seems more likely that we live in a "nonsymmetrical"<br />
universe where matter completely dominates antimatter.<br />
How is this possible? A model explaining the situation was<br />
presented by Sakharov in 1966, and today much work is being<br />
done along these lines.14 One essential element of the model is
231 REDISCOVERING TIME<br />
that, at the time of the formation of matter, the universe had to<br />
be in n{)nequilibrium conditions, for at equilibrium the law of<br />
mass action discussed in Chapter V would have required equal<br />
amounts of matter and antimatter.<br />
What we want to emphasize here is that nonequilibrium has<br />
now acquired a new, cosmological dimension. Without nonequilibrium<br />
and without the irreversible processes linked to it,<br />
the universe would have a completely different structure.<br />
There would be no appreciable amount of matter, only some<br />
fluctuating local excesses of matter over antimatter, or vice<br />
versa.<br />
From a mechanistic theory that was modified to account for<br />
the existence of the universal constant h, quantum theory has<br />
evolved into a theory of mutual transformations of elementary<br />
particles. In recent attempts to formulate a "unified theory of<br />
elementary particles" it has even been suggested that all particles<br />
of matter, including the proton, are unstable (however, the<br />
lifetime of the proton would be enormous, of the order of 1 Q 3 0<br />
years). Mechanics, the science of motion, instead of corresponding<br />
to the fundamental level of description, becomes a<br />
mere approximation, useful only because of the long lifetime<br />
of elementary particles such as protons.<br />
Relativity theory has gone through the same transformations.<br />
As we mentioned, it started as a geometric theory that<br />
strongly emphasized timeless features. Today it is the main<br />
tool for investigating the thermal history of the universe, for<br />
providing clues to the mechanisms that led to the present<br />
structure of the universe. The problem of time, of irreversibility,<br />
has therefore acquired a new urgency. From the field of<br />
engineering, of applied chemistry, where it was first formulated,<br />
it has spread to the whole of physics, from elementary<br />
particles to cosmology.<br />
From the perspective of this book, the importance of quantum<br />
mechanics lies in its introduction of probability into microscopic<br />
physics. This should not be confused with the stochastic<br />
processes that describe chemical reactions as discussed in<br />
Chapter V. In quantum mechanics, the wave function evolves<br />
in a deterministic fashion, except in the measurement process.<br />
We have seen that in the fifty years since the formulation of<br />
quantum mechanics the study of nonequilibrium processes<br />
has revealed that fluctuations, stochastic elements, are impor-
ORDER OUT OF CHAOS 232<br />
tant even on the microscopic scale. We have repeatedly stated<br />
in this book that the reconceptualization of physics going on<br />
today leads from deterministic, reversible processes to stochastic<br />
and irreversible ones. We believe that quantum mechanics<br />
occupies a kind of intermediate position in this process. There<br />
probability appears, but not irreversibility. We expect, and we<br />
shall give some reasons for this in Chapter IX, that the next<br />
step will be the introduction of fundamental irreversibility on<br />
the microscopic level. In contrast with the attempts to restore<br />
classical orthodoxy through hidden variables or other means,<br />
we shall argue that it is necessary to move even farther away<br />
from deterministic descriptions of nature and adopt a statistical,<br />
stochastic description.
CHAPTER VIII<br />
THE CLASH OF<br />
DOCTRINES<br />
Probability and Irreversibility<br />
We shall see that nearly everywhere the physicist has<br />
purged from his science the use of one-way time, as<br />
though aware that this idea introduces an antrlropomorphlc<br />
element alien to the ideals of physics. Nevertheless,<br />
in several important cases unidirectional time<br />
and unidirectional causality have been invoked, but always,<br />
as we shall proceed to show. in support of some<br />
false doctrine.<br />
G. N. LEWIS'<br />
The law that entropy always increases-the second<br />
law of thermodynamics-holds, I think, the supreme<br />
position among the laws of Nature. If someone points<br />
out to you that your pet theory of the universe is in<br />
disagreement with Maxwell's equations-then so<br />
much the worse for Maxwells equations. If it is found to<br />
be contradicted by observation-well, these experimentalists<br />
do bungle things sometimes. But if your<br />
theory is found to be against the second law of thermodynamics<br />
I can give you no hope; there is nothing<br />
for it but to collapse in deepest humiliation.<br />
A S. EDDINGTON2<br />
With Clausius' formulation of the second law of thermodynam·<br />
ics, the conflict between thermodynamics and dynamics be·<br />
came obvious. There is hardly a single question in physis that<br />
has been more often and more actively discussed than the rela·<br />
233
ORDER OUT OF CHAOS<br />
234<br />
tion between thermodynamics and dynamics. Even now, a<br />
hundred and fifty years after Clausius, the question still<br />
arouses strong feelings. No one can remain neutral in this conflict,<br />
which involves the meaning of reality and time. Must<br />
dynamics, the mother of modern science, be abandoned in<br />
favor of some form of thermodynamics? That was the view of<br />
the "energeticists," who exerted great influence during the<br />
nineteenth century. Is there a way to "save" dynamics, to recoup<br />
the second law without giving up the formidable structure<br />
built by Newton and his successors? What role can<br />
entropy play in a world described by dynamics?<br />
We have already mentioned the answer proposed by Boltzmann.<br />
Boltzmann's famous equation S = k log P relates entropy<br />
and probability: entropy grows because probability grows.<br />
Let us immediately emphasize that in this perspective the second<br />
law would have great practical importance but would be of<br />
no fundamental significance. In his excellent book The Ambidextrous<br />
Universe, Martin Gardner writes: "Certain events go<br />
only one way not because they can't go the other way but because<br />
it is extremely unlikely that they go backward. "3 By improving<br />
our abilities to measure less and less unlikely events,<br />
we could reach a situation in which the second law would play<br />
as small a role as we want. This is the point of view that is<br />
often taken today. However, this was not Planck's point of<br />
view:<br />
It would be absurd to assume that the validity of the second<br />
law depends in any way on the skill of the physicist<br />
or chemist in observing or experimenting. The gist of the<br />
second law has nothing to do with experiment; the law<br />
asserts briefly that there exists in nature a quantity which<br />
changes always in the same sense in all natural processes.<br />
The proposition stated in this general form may<br />
be correct or incorrect; but whichever it may be, it will<br />
remain so, irrespective of whether thinking and measuring<br />
beings exist on the earth or not, and whether or not,<br />
assuming they do exist, they are able to measure the details<br />
of physical or chemical processes more accurately<br />
by one, two, or a hundred decimal places than we can.<br />
The limitation to the law, if any, must lie in the same<br />
province as its essential idea, in the observed Nature, and
235 THE CLASH OF DOCTRINES<br />
not in the Observer. That man's experience is called upon<br />
in the deduction of the law is of no consequence; for that<br />
is, in fact, our only way of arriving at a knowledge of<br />
natural law. 4<br />
However, Planck's views remained isolated . As we noted,<br />
most scientists considered the second law the result of approximations,<br />
the intrusion of subjective views into the exact world<br />
of physics. For example, in a celebrated sentence Born stated,<br />
"Irreversibility is the effect of the introduction of ignorance<br />
into the basic laws of physics. "5<br />
In the present chapter we wish to describe some of the basic<br />
steps in the development of the interpretation of the second<br />
law. We must first understand why this problem appeared to<br />
be so difficult. In Chapter IX we shall go on to present a new<br />
approach that, we hope, will clearly express both the radical<br />
originality and the objective meaning of the second law. Our<br />
conclusion will agree with Planck's view. We shall show that,<br />
far from destroying the formidable structure of dynamics, the<br />
second law adds an essential new element to it.<br />
First we wish to clarify Boltzmann's association of probability<br />
and entropy. We shall begin by describing the "urn<br />
model" proposed by P. and T. Ehrenfest. 6 Consider N objects<br />
(for example, balls) distributed between two containers A and<br />
B. At regular time intervals (for example, every second) a ball<br />
tioe n<br />
time n+1 1<br />
D EJ<br />
A<br />
A<br />
!-tottery<br />
B<br />
or N-k+1 N-k-1<br />
Figure 23. Ehrenfest's urn model. N balls are distributed between two containers<br />
A and B. At time n there are k balls in A and N- k balls in B. At regular<br />
time intervals a ball is taken at random from A and put in B.<br />
B
ORDER OUT OF CHAOS 236<br />
is chosen at random and moved from one container to the<br />
other. Suppose that at time n there are k balls in A and N- k<br />
balls in B. Then at time n + I there can be in A either k- I or<br />
k+ I balls. We have the transition probabilities kiN for k-+k - 1<br />
and 1-k/N for k-+k + 1. Suppose we continue the game. We<br />
expect that as a result of the exchanges of balls the most probable<br />
distribution in Boltzmann's sense will be reached. When the<br />
number N of balls is large, this distribution corresponds to an<br />
equal number N/2 of balls in each urn. This can be verified by<br />
elementary calculations or by performing the experiment.<br />
N<br />
k - -<br />
2<br />
t<br />
Figure 24. Approach to equilibrium (k = Nt2) in Ehrenfest's urn model<br />
(schematic representation).<br />
The Ehrenfest model is a simple example of a "Markov process"<br />
(or Markov "chain"), named after the great Russian<br />
mathematician Markov, who was one of the first to describe<br />
such processes (Poincare was another). In brief, their characteristic<br />
feature is the existence of well-defined transition probabilities<br />
independent of the previous history of the system.<br />
Markov chains have a remarkable property: they can be described<br />
in terms of entropy. Let us call P(k) the probability of<br />
finding k balls in A. We may then associate to it an "J-{ quantity,"<br />
which has the precise properties of entropy that we discussed<br />
in Chapter IV. Figure 25 gives an example of its<br />
evolution. The Jf quantity varies uniformly with time, as does<br />
the entropy of an isolated system. It is true that J-{ decreases<br />
with time, while the entropy S increases, but that is a matter of<br />
·<br />
definition: J-{ plays the role of -s.
237 THE CLASH OF DOCTRINES<br />
Figure 25. Time evolution of the J{ quantity (defined in the text) corresponding<br />
to the Ehrenfest model. This quantity decreases monotonously<br />
and vanishes for long times.<br />
The mathematical meaning of this "J-l quantity" is worth<br />
considering in more detail: it measures the difference between<br />
the probabilities at a given time and those that exist at the<br />
equilibrium state (where the number of balls in each urn is<br />
N/2). The argument used in the Ehrenfest urn model can be<br />
generalized. Let us consider the partition of a square-that is,<br />
we subdivide the square into a number of disjointed regions<br />
(see Figure 26). Then we consider the distribution of particles<br />
in the square and call P(k,t) the probability of finding a particle<br />
in the region k. Similarly, we call Peqm(k) this quantity when<br />
uniformity is reached. We assume that, as in the urn model,<br />
there exist well-defined transition probabilities. The definition<br />
of the J-l. quantity is<br />
t<br />
J{ = P(k,t) log J
ORDER OUT OF CHAOS 238<br />
sponding values of P(k,t) would be P( l ,t) = 1, all others zero. As<br />
the result we find .Jl= log (II[ 1/8]) =log 8. As time goes by, the<br />
particles become equally distributed and P(k,t) = Peqm(k) = 1/8.<br />
As the result the :I{ quantity vanishes. It can be shown that, in<br />
accordance with Figure 25, the decrease in the value of :H proceeds<br />
in a uniform fashion. (The demonstration is given in all<br />
textbooks dealing with the theory of stochastic processes.)<br />
This is why :H plays the role of -S, entropy. The uniform decrease<br />
of .H has a very simple meaning: it measures the progressive<br />
uniformization of the system. The initial information<br />
is lost, and the system evolves from "order" to "disorder."<br />
Note that a Markov process implies fluctuations, as clearly<br />
indicated in Figure 24. If we would wait long enough we would<br />
recover the initial state. However, we are dealing with averages.<br />
The :JiM quantity that decreases uniformly is expressed<br />
in terms of probability distributions and not in terms of individual<br />
events. It is the probability distribution that evolves irreversibly<br />
(in the Ehrenfest model, the distribution function<br />
tends uniformly to a binomial distribution). Therefore, on the<br />
level of distribution functions, Markov chains lead to a onewaynss<br />
in time.<br />
This arrow of time marks the difference between Markov<br />
chains and temporal evolution in quantum mechanics, where<br />
the wave function, though related to probabilities, evolves reversibly.<br />
It also illustrates the close relation between stochastic<br />
processes, such as Markov chains, and irreversibility.<br />
However, the increasing of entropy (or decreasing of Jf) is not<br />
based on an arrow of time present in the laws of nature but on<br />
our decision to use present knowledge to predict future (and<br />
not past) behavior. Gibbs states it in his usual lapidary manner:<br />
But while the distinction of prior and subsequent events<br />
may be immaterial with respect to mathematical fictions,<br />
it is quite otherwise with respect to the events of the real<br />
world. It should not be f<strong>org</strong>otten, when our ensembles<br />
are chosen to illustrate the probabilities of events in the<br />
real world, that while the probabilities of subsequent<br />
events may often be determined from the probabilities of<br />
prior events, it is rarely the case that probabilities of prior
239<br />
THE CLASH OF DOCTRINES<br />
events can be determined from those of subsequent<br />
events, for we are rarely justified in excluding the consideration<br />
of the antecedent probability of the prior events. 7<br />
It is an important point, which has led to a great deal of discussion.<br />
8 Probability calculus is indeed time-oriented. The prediction<br />
of the future is different from retrodiction. If this was<br />
the whole story, we would have to conclude that we are forced<br />
to accept a subjective interpretation of irreversibility, since the<br />
distinction between future and past would depend only on us.<br />
In other words, in the subjective interpretation of iri;"eversibility<br />
(further reinforced by the ambiguous analogy with information<br />
theory), the observer is responsible for the temporal<br />
asymmetry characterizing the system's development. Since<br />
the observer cannot in a single glance determine the positions<br />
and velocities of all the particles composing a complex system,<br />
he cannot know the instantaneous state that simultaneously<br />
contains its past and its future, nor can he grasp the<br />
reversible law that would allow him to predict its developments<br />
from one moment to the next. Neither can he manipulate the<br />
system like the demon invented by Maxwell, who can separate<br />
fast- and slow-moving particles and impose on a system an<br />
antithermodynamic evolution toward an increasingly less uniform<br />
temperature distribution.9<br />
Thermodynamics remains the science of complex systems;<br />
but, from this perspective, the only specific feature of complex<br />
systems is that our knowledge of them is limited and that our<br />
uncertainty increases with time. Instead of recognizing in irreversiblity<br />
something that links nature to the observer, the scientist<br />
is compelled to admit that nature merely mirrors his<br />
ignorance. Nature is silent; irreversibility, far from rooting us<br />
in the physical world, is merely the echo of human endeavor<br />
and of its limits.<br />
However, one immediate objection can be raised. According<br />
to such interpretations, thermodynamics ought to be as universal<br />
as our ignorance. There should exist only irreversible<br />
processes. This is the stumbling block for all universal interpretations<br />
of entropy that concentrate on our ignorance of initial<br />
(or boundary) conditions. Irreversibility is not a universal<br />
propercy. In order to link dynamics and thermodynamics, a
ORDER OUT OF CHAOS 240<br />
physical criterion is required to distinguish between reversible<br />
and irreversible processes.<br />
We shall take up this question in Chapter IX. Here let us<br />
return to the history of science and Boltzmann's pioneering<br />
work.<br />
Boltzn1anns Breakthrough<br />
Boltzmann's fundamental contribution dates from 1872, about<br />
thirty years before the discovery of Markov chains. His ambition<br />
was to derive a "mechanical" interpretation of entropy. In<br />
other words, while in Markov chains the transition probabilities<br />
are given from outside as, for example, in the Ehrenfest<br />
model, we now have to relate them to the dynamic behavior of<br />
the system. Boltzmann was so fascinated by this problem that<br />
he devoted most of his scientific life to it. In his Populiire<br />
Schriften10 he wrote: "If someone asked me what name we<br />
should give to this century, I would answer without hesitation<br />
that this is the century of Darwin." Boltzmann was deeply attracted<br />
by the idea of evolution, and his ambition was to become<br />
the "Darwin" of the evolution of matter.<br />
The first step toward the mechanistic interpretation of entropy<br />
was to reintroduce the concept of "collisions" of molecules<br />
or atoms into the physical description, and along with it<br />
the possibility of a statistical description. This step had been<br />
taken by Clausius and Maxwell. Since collisions are discrete<br />
events, we may count them and estimate their average frequency.<br />
We may also classify collisions-for example, distinguish<br />
between collisions producing a particle with a given<br />
velocity v and collisions destroying a particle with a velocity v,<br />
producing molecules with a different velocity (the "direct"<br />
and "inverse" collisions); l l<br />
The question Maxwell asked was whether it was possible to<br />
define a state of a gas such that the collisions that incessantly<br />
modify the velocities of the molecules no longer determine<br />
any evolution in the distribution of these velocities-that is, in<br />
the mean number of particles for each velocity value. What is<br />
the velocity distribution such that the effects of the different<br />
collisions compensate each other on the population scale?
241 THE CLASH OF DOCTRINES<br />
Maxwell demonstrated that this particular state, which is<br />
the thermodynamic equilibrium state, occurs when the velocity<br />
distribution becomes the well-known "bell-shaped curve,"<br />
the "gaussian," which Quetelet, the founder of "social physics,"<br />
had considered to be the very expression of randomness.<br />
Maxwell's theory permits us to give a simple interpretation of<br />
some of the basic laws describing the behavior of gases. An<br />
increase in temperature corresponds to an increase in the<br />
mean velocity of the molecules and thus of the energy associated<br />
with their motion. Experiments have verified Maxwell's<br />
law with great accuracy, and it still provides a basis for the<br />
solution of numerous problems in physical chemistry (for example,<br />
the calculation of the number of collisions in a reactive<br />
mixture).<br />
Boltzmann, however, wanted to go farther. He wanted to<br />
describe not only the state of equilibrium but also evolution<br />
toward equilibrium-that is, evolution toward the Maxwellian<br />
distribution. He wanted to discover the molecular mechanism<br />
that corresponds to the increase of entropy, the mechanism<br />
that drives a system from an arbitrary distribution of velocities<br />
toward equilibrium.<br />
Characteristically, Boltzmann approached the question of<br />
physical evolution not at the level of individual trajectories but<br />
at the level of a population of molecules. This, Boltzmann felt,<br />
was virtually tantamount to accomplishing Darwin's feat, but<br />
this time in physics: the driving force behind biological evolution-natural<br />
selection-cannot be defined for one individual<br />
but only for a large population. It is therefore a statistical concept.<br />
Boltzmann's result may be described in relatively simple<br />
terms. The evolution of the distribution function f ( v, t) of the<br />
velocities v in some region of space and at time t appears as<br />
the sum of two effects; the number of particles at any given<br />
time t having a velocity v varies both as the result of the free<br />
motion of the particles and as the result of collisions between<br />
particles. The first result can be easily calculated in the terms<br />
of classical dynamics. It is in the investigation of the second<br />
result, due to collisions, that the originality of Boltzmann's<br />
method lies. In the face of the difficulties involved in following<br />
the trajectories (including the interactions), Boltzmann came<br />
to use concepts similar to those outlined in Chapter V (in con-
ORDER OUT OF CHAOS 242<br />
nection with chemical reactions) and to calculate the average<br />
number of collisions creating or destroying a molecule corresponding<br />
to a velocity v.<br />
Here once again there are two processes with opposite<br />
effects-"direct" collisions, those producing a molecule with<br />
velocity v starting from two molecules with velocities v ' and<br />
v " , and "inverse" collisions, in which a molecule with velocity<br />
v is destroyed by collision with a molecule with velocity v "' .<br />
As with chemical reactions (see Chapter V, section 1), the frequency<br />
of such events is evaluated as being proportional to the<br />
product of the number of molecules taking part in these processes.<br />
(Of course, historically speaking, Boltzmann's method<br />
[1872] preceded that of chemical kinetics.)<br />
The results obtained by Boltzmann are quite similar to those<br />
obtained in Markov chains. Again we shall introduce an J{<br />
quantity, this time referring to the velocity distribution f. It<br />
may be written J{= f flog f dv. Once again, this quantity can<br />
only decrease in time until equilibrium is reached and the velocity<br />
distribution becomes the equilibrium Maxwellian distribution.<br />
In recent years there have been numerous numerical verifications<br />
of the uniform decrease of J{ with time. All of them<br />
confirm Boltzmann's prediction. Even today, his kinetic equation<br />
plays an important role in the physics of gases: transport<br />
coefficients such as those characterizing heat conductivity or<br />
diffusion can be calculated in good agreement with experimental<br />
data.<br />
However, it is from the conceptual standpoint that Boltzmann's<br />
achievement is greatest: the distinction between reversible<br />
and irreversible phenomena, which, as we have seen,<br />
underlies the second law, is now transposed onto the microscopic<br />
level. The change of the velocity distribution due to<br />
free motion corresponds to the reversible part, while the contribution<br />
due to collisions corresponds to the irreversible part.<br />
For Boltzmann this was the key to the microscopic interpretation<br />
of entropy. A principle of molecular evolution had been<br />
produced! It is easy to understand the fascination this discovery<br />
exerted on the physicists who followed Boltzmann, including<br />
Planck, Einstein, and Schrodinger. t2<br />
Boltzmann's breakthrough was a decisive step in the direc-
243 THE CLASH OF DOCTRINES<br />
tion of the physics of processes. What determines temporal<br />
evolution in Boltzmann's equation is no longer the Hamiltonian,<br />
depending on the type of forces; now, on the contrary,<br />
functions associated with the processes-for example, the<br />
cross section of scattering-will generate motion. Can we conclude<br />
that the problem of irreversibility has been solved, that<br />
Boltzmann's theory has reduced entropy to dynamics? The<br />
answer is clear: No, it has not. Let us have a closer look at this<br />
question.<br />
Questioning Boltzmanns Interpretation<br />
As soon as Boltzmann's fundamental paper appeared in 1872,<br />
objections were raised. Had Boltzmann really "deduced" irreversibility<br />
from dynamics? How could the reversible laws of<br />
trajectories lead to irreversible evolution? Is Boltzmann's kinetic<br />
equation in any way compatible with dynamics? It is<br />
easy to see that the symmetry present in Boltzmann's equation<br />
is in contradiction with the symmetry of classical mechanics.<br />
We have already seen that velocity inversion (v -v) produces<br />
in classical dynamics the same effect as time inversion<br />
(t-t). This is a basic symmetry of classical dynamics, and<br />
we would expect that Boltzmann's kinetic equation, which describes<br />
the time change of the distribution function, would<br />
share this symmetry. But this is not so. The collision term<br />
calculated by Boltzmann remains invariant with respect to velocity<br />
inversion. There is a simple physical reason for this.<br />
Nothing in Boltzmann's picture distinguishes a collision that<br />
proceeds toward the future from a collision proceeding toward<br />
the past. This is the basis of Poincare's objection to Boltzmann's<br />
derivation. A correct calculation can never lead to<br />
conclusions that contradict its premises.B· 14 As we have seen,<br />
the symmetry properties of the kinetic equation obtained by<br />
Boltzmann for the distribution function contradict those of dynamics.<br />
Boltzmann cannot, therefore, have "deduced" entropy<br />
from dynamics. He must have introduced something<br />
new, something foreign to dynamics. Thus his results ;an rep-
ORDER OUT OF CHAOS<br />
244<br />
resent at best only a phenomenological model that, however<br />
useful, has no direct relation with dynamics. This was also the<br />
objection that Zermelo (1896) brought against Boltzmann.<br />
Loschmidt's objection, on the other hand, makes it possible<br />
to determine the limits of validity of Boltzmann's kinetic<br />
model. In fact, Loschmidt observed (1876) that this model can<br />
no longer be valid after a reversal of the velocities corresponding<br />
to the transformation v-+-v.<br />
Let us explain this by means of a thought experiment. We<br />
start with a gas in a nonequilibrium condition and let it evolve<br />
till t0• We then invert the velocities. The system reverts to its<br />
past state. As a consequence, Boltzmann's entropy is the<br />
same at t=O and at t=2t0•<br />
We may multiply such thought experiments. Start with a<br />
mixture of hydrogen and oxygen; after some time water will<br />
appear. If we invert the velocities, we should go back to an<br />
initial state with hydrogen and oxygen and no water.<br />
It is interesting that in laboratory or computer experiments,<br />
we actually can perform a velocity inversion. For example, in<br />
Figures 26 and 27, Boltzmann's J{ quantity has been calculated<br />
for two-dimensional hard spheres (hard disks), starting<br />
first collision<br />
ae<br />
•<br />
•<br />
0 20 40 TIME 60<br />
Figure 26. Evolution of .Jf with time for N "hard spheres" by computer<br />
simulation; (a) corresponds to N=100, (b) to N=484, (c) to N=1225.
..<br />
245 THE CLASH OF DOCTRINES<br />
with disks on lattice sites with an isotropic velocity distribution.<br />
The results follow Boltzmann's predictions.<br />
If, after fifty or a hundred collisions, corresponding to about<br />
I0-6 sec in a dilute gas, the velocities are inverted, a new ensemble<br />
is obtained.15 Now, after the velocity inversion, Boltzmann's<br />
J{ quantity increases instead of decreasing.<br />
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... •<br />
. ...<br />
.<br />
..<br />
-<br />
t<br />
' -<br />
.<br />
-'"'<br />
.... , ·· ... .<br />
.... .,<br />
·11-------::-----------------<br />
equit .<br />
...<br />
:<br />
0 20 TIME 60<br />
Figure 27. Evolution of :H when velocities are inverted after 50 or 100<br />
collisions. Simulation with 100 "hard spheres."<br />
A similar situation can be produced in spin echo experiments<br />
or plasma echo experiments. There also, over limited<br />
periods of time, an "antithermodynamic" behavior in Boltzmann's<br />
sense may be observed.<br />
But it is important to note that the velocity inversion experiment<br />
becomes increasingly more difficult when the time interval<br />
t0 after which the inversion occurs is increased.<br />
To be able to retrace its past, the gas must remember everything<br />
that happened to it during the time interval from 0 to t0•<br />
There must be "storage" of information. We can express this<br />
storage in terms of correlations between particles. We shall<br />
come back to the question of correlations in Chapter IX. Let<br />
us only mention here that it is precisely this relation between<br />
correlations and collisions that is the element missing from
ORDER OUT OF CHAOS 246<br />
Boltzmann's considerations. When Loschmidt confronted him<br />
with this, Boltzmann had to accept that there was no way out:<br />
the collisions occurring in the opposite direction "undo" what<br />
was done previously, and the system has to revert to its initial<br />
state. Therefore, the function J{ must also increase until it<br />
again reaches its initial value. Velocity inversion thus calls for<br />
a distinction between the situations to which Boltzmann's reasoning<br />
applies and those to which it does not.<br />
Once the problem was stated (1894), it was easy to identify<br />
the nature of this limitation.l6, 17 The validity of Boltzmann's<br />
statistical procedure depends on the assumption that before<br />
they collide, the molecules behave independently of one another.<br />
This constitutes an assumption about the initial conditions,<br />
called the "molecular chaos" assumption. The initial<br />
conditions created by a velocity inversion do not conform to<br />
this assumption. If the system is made "to go backward in<br />
time," a new "anomalous" situation is created in the sense<br />
that certain molecules are then "destined" to meet at a predeterminable<br />
instant and to undergo a predetermined change<br />
of velocity at this time, however far apart they may be at the<br />
instant of velocity inversion.<br />
· Velocity inversion thus creates a highly <strong>org</strong>anized system,<br />
and thus the molecular chaos assumption fails. The various<br />
collisions produce, as if by a preestablished harmony, an apparently<br />
purposeful behavior.<br />
But there is more. What does the transition from order to<br />
disorder signify? In the Ehrenfest urn experiment, it is clearthe<br />
system will evolve till uniformity is reached. But other situations<br />
are not so clear; we may do computer experiments in<br />
which interacting particles are initially distributed at random.<br />
In time a lattice is formed. Do we still move from order to<br />
disorder? The answer is not obvious. To understand order and<br />
disorder we first have to define the objects in terms of which<br />
these concepts are used. Moving from dynamic to thermodynamic<br />
objects is easy in the case of dilute gases-as shown by<br />
the work of Boltzmann. However, it is not so easy in the case<br />
of dense systems whose molecules interact.<br />
Because of such difficulties, Boltzmann's creative and pioneering<br />
work remained incomplete.
247 THE CLASH OF DOCTRINES<br />
Dynarnics and Thermodynamics:<br />
Two Separate Worlds<br />
We already noted that trajectories are incompatible with the<br />
idea of irreversibility. However, the study of trajectories is not<br />
the only way in which we can give a formulation of dynamics.<br />
There is also the theory of ensembles introduced by Gibbs and<br />
Einstein,6. 18 which is of special interest in the case of systems<br />
formed by a large number of molecules. The essential new<br />
element in the Gibbs-Einstein ensemble theory is that we can<br />
formulate the dynamic theory independently of any precise<br />
specification of initial conditions.<br />
The theory of ensembles represents dynamic systems in<br />
"phase space." The dynamic state of a point particle is specified<br />
by position (a vector with three components) and by momentum<br />
(also a vector with three components). We may<br />
represent this state by two points, each in a three-dimensional<br />
space, or by a single point in the six-dimensional space formed<br />
by the coordinates and momenta. This is the phase space.<br />
This geometric representation can be extended to an arbitrary<br />
system formed by n particles. We then need n x 6 numbers to<br />
specify the state of the system, or alternatively we may specify<br />
this system by a single point in the 6n-dimensional phase<br />
space. The evolution in time of such a system will then be<br />
described by a trajectory in the phase space.<br />
It has already been stated that the exact initial conditions of<br />
a macroscopic system are never known. Nevertheless, nothing<br />
prevents us from representing this system by an "ensemble"<br />
of points-namely, the points corresponding to the various dynamic<br />
states ompatible with the information we have concerning<br />
the system. Each region of phase space may contain an<br />
infinite number of representative points, the density of which<br />
measures the probability of actually finding the system in this<br />
region. Instead of introducing an infinity of discrete points, it<br />
is more convenient to introduce a continuous density of representative<br />
points in the phase space. We shall call p (q1<br />
• • • q 3 0,<br />
p1 • • • p30) this density in phase space where q1 ,q2 • • • q3n are<br />
the coordinates of the n points; similarly, p1 ,p2 • • • p30 are the<br />
momenta (each point has three coordinates and three mo-
ORDER OUT OF CHAOS 248<br />
menta). This density measures the probability of finding a dynamic<br />
system around the point ql ... q3n,PI • • • P3n in phase<br />
space.<br />
Presented in such a way, the density function p may appear<br />
as an idealization, an artificial construct, whereas the trajectory<br />
of a point in phase space would correspond "directly" to<br />
the description of "natural" behavior. But in fact it is the<br />
point, not the density, that corresponds to an idealization. Indeed,<br />
we never know an initial state with the infinite degree of<br />
precision that would reduce a region in phase space to a single<br />
point; we can only determine an ensemble of trajectories starting<br />
from the ensemble of representative points corresponding<br />
to what we know about the initial state of the system. The<br />
density function p represents knowledge about a system, and<br />
the more accurate this knowledge, the smaller the region in the<br />
phase space where the density function is different from zero<br />
and where the system may be found. Should the density function<br />
everywhere have a uniform value, we would know nothing<br />
about the state of the system. It might be in any of the possible<br />
states compatible with its dynamic structure.<br />
From this perspective, a point thus represents the maximum<br />
knowledge we can have about a system. It is the result of a<br />
limiting process, the result of the ever-growing precision of our<br />
knowledge. As we shall see in Chapter IX, a fundamental<br />
problem will be to determine when such a limiting process is<br />
really possible. Through increased precision, this process<br />
means we go from a region where the density function p is<br />
different from zero to another, smaller region inside the first.<br />
We can continue this until the region containing the system<br />
becomes arbitrarily small. But as we shall see, we must be<br />
cautious: arbitrarily small does not mean zero, and it is not<br />
certain a priori that this limiting process will lead to the possibility<br />
of consistently predicting a single well-defined trajectory.<br />
The introduction of the theory of ensembles by Gibbs and<br />
Einstein was a natural continuation of Boltzmann's work. In<br />
this perspective the density function p in phase space replaces<br />
the velocity distribution function f used by Boltzmann. However,<br />
the physical content of p exceeds that off. Just like/, the<br />
density function p determines the velocity distribution, but it<br />
also contains other information, such as the probability of
29<br />
THE CLASH OF DOCTRINES<br />
meeting two particles a certain distance apart. In particular,<br />
correlations between particles, which we discussed in the preceding<br />
section, are now included in the density function p. In<br />
fact, this function contains the complete information about all<br />
statistical features of the n-b9dy system.<br />
We must now describe the evolution of the density function<br />
in phase space. At first sight, this appears to be an even more<br />
ambitious task than the one Boltzmann set himself for the velocity<br />
distribution function. But this is not the case. The Hamiltonian<br />
equations discussed in Chapter 11 allow us to obtain<br />
an exact evolution equation for p without any further approximations.<br />
This is the so-called Liouville equation, to which we<br />
shall return in Chapter IX. Here we wish merely to point out<br />
, that the properties of Hamiltonian dynamics imply that the<br />
evolution of the density function p in phase space is that of an<br />
incompressible fluid. Once the representative points occupy a<br />
region of volume V in phase space, this volume remains constant<br />
in time. The shape of the region may be deformed in an<br />
arbitrary way, but the value of the volume remains the same.<br />
Gibbs' theory of ensembles thus permits a rigorous combination<br />
of the statistical point of view (the study of the "population"<br />
described by p) and the laws of dynamics. It also permits<br />
a more accurate representation of the thermodynamic equilibrium<br />
state. Thus, in the case of an isolated system, the ensemp<br />
Figure 28. Time evolution in the phase space of a "volume" containing the<br />
representative points of a system: the volume is conserved while the shape<br />
is modified. The position in phase space is specified by coordinate& q and<br />
momentum p.<br />
q
ORDER OUT OF CHAOS<br />
250<br />
ble ci representative points corresponds to systems that all have<br />
the same energy E. The density p will differ from zero only on<br />
the "microcanonical surface" corresponding to the specified<br />
value of the energy in phase space. Initially, the density p may<br />
be distributed arbitrarily over this surface. At equilibrium, p<br />
must no longer vary with time and has to be independent of<br />
the specific initial state. Thus the approach to equilibrium has<br />
a simple meaning in terms of the evolution of p. The distribution<br />
function p becomes uniform over the microcanonical surface.<br />
Each ci the points on this surface has the same probability<br />
of actually representing the system. This corresponds to the<br />
"microcanonical ensemble."<br />
Does the theory of ensembles bring us any closer to the<br />
solution of the problem of irreversibility? Boltzmann's theory<br />
describes thermodynamic entropy in terms of the velocity distribution<br />
function f. He achieved this result through the introduction<br />
of his J{ quantity. As we have seen, the system evolves<br />
in time until the Maxwellian distribution is reached, while,<br />
during this evolution, the quantity J-{ decreases uniformly. Can<br />
we now, in a more general fashion, take the evolution of the<br />
distribution p in phase space toward the microcanonical ensemble<br />
as the basis for entropy increase? Would it be enough<br />
to replace Boltzmann's quantity J{ expressed in terms ofjby a<br />
"Gibbsian" quantity 3£0 defined in exactly the same way, but<br />
this time in terms of p? Unfortunately, the answer to both<br />
questions is "No." If we use the Liouville equation, which<br />
describes the evolution of the density phase space P. and take<br />
into account the conservation of volume in phase space we<br />
have mentioned, the conclusion is immediate: :Jf0 is a constant<br />
and thus cannot represent entropy. With respect to Boltzmann,<br />
this appears as a step backward rather than forward!<br />
Though it is negative, Gibbs' conclusion remains very important.<br />
We have already discussed the ambiguity of the ideas<br />
of order and disorder. What the constancy of 3£0 tells us is<br />
that there is no change of order whatsoever in the frame of<br />
dynamic theory! The "information" expressed by 3£0 remains<br />
constant. This can be understood as follows: we have seen that<br />
collisions introduce correlations. From the perspective of velocities,<br />
the result of collisions is randomization; therefore we<br />
can describe this process as a transition from order to disor-
251 THE CLASH OF DOCTRINES<br />
der, but the appearance of correlations as the result of collision<br />
points in the opposite direction, toward a transition from disorder<br />
to order! Gibbs' result shows that the two effects exactly<br />
cancel each other.<br />
We come, therefore, to an important conclusion. Whatever<br />
representation we use, be it the idea of trajectories or the<br />
Gibbs-Einstein ensemble theory, we will never be able to deduce<br />
a theory of irreversible processes that will be valid for<br />
every system that satisfies the laws of classical (or quantum)<br />
dynamics. There isn't even a way to speak of a transition from<br />
order to disorder! How should we understand these negative results?<br />
Is any theory of irreversible processes in absolute conflict<br />
with dynamics (classical or quantum)? It has often been<br />
proposed that we include some cosmological terms that would<br />
express the influence of the expanding universe on the equations<br />
of motion. Cosmological terms would ultimately provide<br />
the arrow of time. However, this is difficult to accept. On the<br />
one hand, it is not clear how we should add these cosmological<br />
terms; on the other, precise dynamic experiments seem to rule<br />
out the existence of such terms, at least on the terrestrial scale<br />
with which we are concerned here (think, for example, about<br />
the precision of space trip experiments, which confirm Newton's<br />
equations to a high degree). On the other hand, as we<br />
have already stated, we live in a pluralistic universe in which<br />
reversible and irreversible processes coexist, all embedded in<br />
the expanding universe.<br />
An even more radical conclusion is to affirm with Einstein<br />
that time as irreversibility is an illusion that will never find a<br />
place in the objective world of physics. Fortunately there is<br />
another way out, which we shall explore in Chapter IX. Irreversibility,<br />
as has been repeatedly stated, is not a universal<br />
property. Therefore , no general derivation of irreversibility from<br />
dynamics is to be expected.<br />
Gibbs' theory of ensembles introduces only one additional<br />
element with respect to trajectory dynamics, but a very important<br />
one-our ignorance of the precise initial conditions. It is<br />
unlikely that this ignorance alone leads to irreversibility.<br />
We should therefore not be astonished at our failure. We<br />
have not yet formulated the specific features that a dynamic<br />
system has to possess to lead to irreversible processes.
ORDER OUT OF CHAOS 252<br />
Why have so many scientists accepted so readily the subjective<br />
interpretation of irreversibility? Perhaps part of its attraction<br />
lies in the fact that, as we have seen, the irreversible<br />
increase of entropy was at first associated with imperfect manipulation,<br />
with our lack of control over operations that are<br />
ideally reversible.<br />
But this interpretation becomes absurd as soon as the irrelevant<br />
associations with technological problems are set aside. We<br />
must remember the context that gave the second law its significance<br />
as nature's arrow of time. According to the subjective<br />
interpretation, chemical affinity, heat conduction, viscosity,<br />
all the properties connected with irreversible entropy production<br />
would depend on the observer. Moreover, the extent to<br />
which phenomena of <strong>org</strong>anization originating in irreversibility<br />
play a role in biology makes it impossible to consider them as<br />
simple illusions due to our ignorance. Are we ourselves-living<br />
creatures capable of observing and manipulating-mere<br />
fictions produced by our imperfect senses? Is the distinction<br />
between life and death an illusion?<br />
Thus recent developments in thermodynamic theory have<br />
increased the violence of the conflict between dynamics and<br />
thermodynamics. Attempts to reduce the results of thermodynamics<br />
to mere approximations due to our imperfect knowledge<br />
seem wrong headed when the constructive role of entropy<br />
is understood and the possibility of an amplification of fluctuations<br />
is discovered. Conversely, it is difficult to reject dynamics<br />
in the name of irreversibility: there is no irreversibility in<br />
the motion of an ideal pendulum. Apparently there are two<br />
conflicting worlds, a world of trajectories and a world of processes,<br />
and there is no way of denying one by asserting the<br />
other.<br />
To a certain extent, there is an analogy between this conflict<br />
and the one that gave rise to dialectical materialism. We have<br />
described in Chapters V and VI a nature that might be called<br />
"historical"-that is, capable of development and innovation.<br />
The idea of a history of nature as an integral part of materialism<br />
was asserted by Mar x and, in greater detail, by Engels.<br />
Contemporary developments in physics, the discovery of the<br />
constructive role played by irreversibility, have thus raised<br />
within the natural sciences a question that has long been asked<br />
by materialists. For them, understanding nature meant under-
253<br />
THE CLASH OF DOCTRINES<br />
standing it as being capable of producing man and his societies.<br />
Moreover, at the time Engels wrote his Dialectics of Nature,<br />
the physical sciences seemed to have rejected the mechanistic<br />
world view and drawn closer to the idea of an historical<br />
development of nature. Engels mentions three fundamental<br />
discoveries: energy and the laws governing its qualitative<br />
transformations, the cell as the basic constituent of life, and<br />
Darwin's discovery of the evolution of species. In view of<br />
these great discoveries, Engels came to the conclusion that the<br />
mechanistic world view was dead.<br />
But mechanicism remained a basic difficulty facing dialectical<br />
materialism. What are the relations between the general<br />
laws of dialectics and the equally universal laws of mechanical<br />
motion? Do the latter "cease" to apply after a certain stage has<br />
been reached, or are they simply false or incomplete? To come<br />
back to our previous question, how can the world of processes<br />
and the world of trajectories ever be linked together?I9<br />
However, while it is easy to criticize the subjectivistic interpretation<br />
of irreversibility and to point out its weakness, it is<br />
not so easy to go beyond it and formulate an "objective" theory<br />
of irreversible processes. The history of this subject has<br />
some dramatic overtones. Many people believe that it is the<br />
recognition of the basic difficulties involved that may have led<br />
to Boltzmann's suicide in 1906.<br />
Boltzmann and the Arrow of Time<br />
As we have noted, Boltzmann at first thought that he could<br />
prove that the arrow of time was determined by the evolution<br />
of dynamic systems toward states of higher probability or a<br />
higher number of complexions: there would be a one-way increase<br />
of the number of complexions with time. We have also<br />
discussed the objections of Poincare and Zermelo. Poincare<br />
proved that every closed dynamic system reverts in time toward<br />
its previous state. Thus, all states are forever recurring.<br />
How could such a thing as an "arrow of time" be associated<br />
with entropy increase? This led to a dramatic change in Boltzmann's<br />
attitude. He abandoned his attempt to prove that an objective<br />
arrow of time exists and introduced instead an idea that,
ORDER OUT OF CHAOS<br />
254<br />
in a sense, reduced the law of entropy increase to a tautology.<br />
Now he argued that the arrow of time is only a convention that<br />
we (or perhaps all living beings) introduce into a world in<br />
which there is no objective distinction between past and future.<br />
Let us cite Boltzmann's reply to Zermelo:<br />
We have the choice of two kinds of picture. Either we<br />
assume that the whole universe is at the present moment<br />
in a very improbable state. Or else we assume that the<br />
aeons during which this improbable state lasts, and the<br />
distance from here to Sirius, are minute if compared with<br />
the age and size of the whole universe. In such a universe,<br />
which is in thermal equilibrium as a whole and<br />
therefore dead, relatively small regions of the size of our<br />
galaxy will be found here and there; regions (which we<br />
may call "worlds") which deviate significantly from thermal<br />
equilibrium for relatively short stretches of those<br />
"aeons" of time. Among these worlds the probabilities of<br />
their state (i.e. the entropy) will increase as often as they<br />
decrease. In the universe as a whole the two directions of<br />
time are indistinguishable, just as in space there is no up<br />
or down. However, just as at a certain place on the earth's<br />
surface we can call "down" the direction towards the<br />
centre of the earth, so a living <strong>org</strong>anism that finds itself in<br />
such a world at a certain period of time can define the<br />
"direction" of time as going from the less probable state<br />
to the more probable one (the former will be the "past"<br />
and the latter the "future"), and by virtue of this definition<br />
he will find that his own small region, isolated from<br />
the rest of the universe, is "initially" always in an improbable<br />
state. It seems to me that this way of looking at<br />
things is the only one which allows us to understand the<br />
validity of the second law, and the heat death of each individual<br />
world, without invoking a unidirectional change of<br />
the entire universe from a definite initial state to a final<br />
state.20<br />
Boltzmann's idea can be made clearer by referring to a diagram<br />
proposed by Karl Popper (Figure 29). The arrow of time<br />
would be as arbitrary as the vertical direction determined by<br />
the gravitational field.
255 THE CLASH OF DOCTRINES<br />
Arruw of tiM 1Jf c.hia<br />
•trett"h o£ tiM only<br />
ArrllW of ti• or this<br />
atrf'tch ,,, ti•• onlr<br />
t.qui liltri ... level<br />
tnt ropy c-urv• clettr•i ni.aa<br />
the dir•rtion of ti•<br />
Figure 29. Popper's schematic representation of Boltzmann's cosmological<br />
interpretation of the arrow of time (see text).<br />
Commenting on Boltzmann's text, Popper has written:<br />
I think that Boltzmann's idea is staggering in its boldness<br />
and beauty. But I also think that it is quite untenable, at<br />
least for a realist. It brands unidirectional change as an<br />
illusion. This makes the catastrophe of Hiroshima an illusion.<br />
Thus it makes our world an illusion, and with it all<br />
our attempts to find out more about our world. It is therefore<br />
self-defeating (like every idealism). Boltzmann's idealistic<br />
ad hoc hypothesis clashes with his own realistic<br />
and almost passionately maintained anti-idealistic philosophy,<br />
and with his passionate wish to know.21<br />
We fully agree with Popper's comments, and we believe that<br />
it is time to take up Boltzmann's task once again. As we have<br />
said, the twentieth century has seen a great conceptual revolution<br />
in theoretical physics, and this has produced new hopes<br />
for the unification of dynamics and thermodynamics. We are<br />
now entering a new era in the history of time, an era in which<br />
both being and becoming can be incorporated into a single<br />
noncontradictory vision.
CHAPTER IX<br />
IRREVERSIBILITY<br />
THEENTROPY<br />
BARRIER<br />
Entropy and the Arrow of Time<br />
In the preceding chapter we described some difficulties in the<br />
microscopic theory of irreversible processes. Its relation with<br />
dynamics, either classical or quantum, cannot be simple, in<br />
the sense that irreversibility and its concomitant increase of<br />
entropy cannot be a general consequence of dynamics. A microscopic<br />
theory of irreversible processes will require additional,<br />
more specific conditions. We must accept a pluralistic<br />
world in which reversible and irreversible processes coexist.<br />
Yet such a pluralistic world is not easy to accept.<br />
In his Dictionnaire Philosophique Voltaire wrote the following<br />
about destiny:<br />
. . . everything is governed by immutable laws • . . everything<br />
is prearranged ... everything is a necessary<br />
effect . . .. There are some people who, frightened by<br />
this truth, allow half of it, like debtors who offer their<br />
creditors half their debt, asking for more time to pay the<br />
remainder. There are, they say, events which are necessary<br />
and others which are not. It would be strange if a<br />
part of what happens had to happen and another part did<br />
not . • . . I necessarily must have the passion to write this,<br />
and you must have the passion to condemn me; we are<br />
both equally foolish, both toys in the hands of destiny.<br />
Your nature is to do ill, mine is to love truth, and to publish<br />
it in spite of you.1<br />
257
ORDER OUT OF CHAOS 258<br />
However convincing they may sound, such a priori arguments<br />
can lead us astray. Voltaire's reasoning is Newtonian:<br />
nature always conforms to itself. But, curiously, today we find<br />
ourselves in the strange world mocked by Voltaire; we are astonished<br />
to discover the qualitative diversity presented by nature.<br />
It is not surprising that people have vacillated between the<br />
two extremes; between the elimination of irreversibility from<br />
physics, advocated by Einstein, as we have mentioned,2 and,<br />
on the contrary, the emphasis on the importance of irreversibility,<br />
as in Whitehead's concept of process. There can be no doubt<br />
that irreversibility exists on the macroscopic level and has an<br />
important constructive role, as we have shown in Chapters V<br />
and VI . Therefore there must be something in the microscopic<br />
world of which macroscopic irreversibility is the manifestation.<br />
The microscopic theory has to account for two closely<br />
related elements. First of all, we must follow Boltzmann in<br />
attempting to construct a microscopic model for entropy<br />
(Boltzmann's .J{ fu nction) that changes uniformly in time. This<br />
change has to define our arrow of time. The increase of entropy<br />
for isolated systems has to express the aging of the system.<br />
Often we may have an arrow of time without being able to<br />
associate entropy with the type of processes considered. Popper<br />
gives a simple example of a system presenting a unidirectional<br />
process and therefore an arrow of time.<br />
Suppose a film is taken of a large surface of water initially<br />
at rest into which a stone is dropped. The reversed<br />
film will show contracting, circular waves of increasing<br />
amplitude. Moreover, immediately behind the highest<br />
wave crest, a circular region of undisturbed water will<br />
close in towards the centre. This cannot be regarded as a<br />
possible classical process. It would demand a vast number<br />
of distant coherent generators of waves the coordination<br />
of which, to be explicable, would have to be shown,<br />
in the film, as originating from one centre. This, however,<br />
raises precisely the same difficulty again, if we try to reverse<br />
the amended film. 3
259<br />
IRREVERSIBILITY-THE ENTROPY BARRIER<br />
Indeed, whatever the technical means, there will always be<br />
a distance from the center beyond which we are unable to generate<br />
a contracting wave. There are unidirectional processes.<br />
Many other processes of the type presented by Popper can be<br />
imagined: we never see energy coming from all directions converge<br />
on a star, together with the backward-running nuclear<br />
reactions that would absorb that energy.<br />
In addition, there exist other arrows of time-for example,<br />
the cosmological arrow (see the excellent account by M.<br />
Gardner'). If we assume that the universe started with a Big<br />
Bang, this obviously implies a temporal order on the cosmological<br />
level. The size of the universe continues to increase,<br />
but we cannot identify the radius of the universe with entropy.<br />
Indeed, as we already mentioned, inside the expanding universe<br />
we find both reversible and irreversible processes. Similarly,<br />
in elementary-particle physics there exist processes that<br />
present the so-called T-violation. The T-violation implies that<br />
the equations describing the evolution of the system for +t are<br />
different from those describing the evolution for -t. However,<br />
this T-violation does not prevent us from including it in the<br />
usual (Hamiltonian) formulation of dynamics. No entropy<br />
fu nction can be defined as a result of the T-violation.<br />
We are reminded of the celebrated discussion between Einstein<br />
and Ritz published in 1909.5 This is a quite unusual paper,<br />
a very short one, less than a printed page long. It simply is<br />
a statement of disagreement. Einstein argued that irreversibility<br />
is a consequence of the probability concept introduced by<br />
Boltzmann. On the contrary, Ritz argued that the distinction<br />
between "retarded" and "advanced" waves plays an essential<br />
role. This distinction reminds us of Popper's argument. The<br />
waves we observe in the pond are retarded waves; they follow<br />
the dropping of the stone.<br />
Both Einstein and Ritz introduced essential elements into<br />
the discussion of irreversibility, but each of them emphasized<br />
only part of the story. We have already mentioned in Chapter<br />
VIII that probability presupposes a direction of time and<br />
therefore cannot be used to derive the arrow of time. We have<br />
also mentioned that the exclusion of processes such as advanced<br />
waves does not necessarily lead to a formulation of the<br />
second law. We need both types of arguments.
ORDER OUT OF CHAOS<br />
260<br />
Irreversibility as a Syrr1metry-Breaking Process<br />
Before discussing the problem of irreversibility, it is useful to<br />
remember how another type of symmetry-breaking, spatial<br />
symmetry-breaking, can be derived. In the equations describing<br />
reaction-diffusion systems, left and right play the same<br />
role (the diffusion equations remain invariant when we perform<br />
the space inversion r-+-r). Still, as we have seen, bifurca<br />
.tions may lead to solutions in which this symmetry is broken<br />
(see Chapter V) . For example, the concentration of some of<br />
the components may become higher on the left than on the<br />
right. The symmetry of the equations only requires that the symmetry-breaking<br />
solutions appear in pairs.<br />
There are, of course, many reaction-diffusion equations that<br />
do not present bifurcations and that therefore cannot break<br />
spatial symmetry. The breaking of spatial symmetry requires<br />
other highly specific conditions. This is valuable for understanding<br />
temporal symmetry-breaking, in which we are primarily<br />
interested here. We have to find systems in which the<br />
equations of motion may have realizations of lower symmetry.<br />
The equations are indeed invariant in respect to time inversion<br />
t-+-t. However, the realization of these equations may<br />
correspond to evolutions that lose this symmetry. The only<br />
condition imposed by the symmetry of equations is that such<br />
realizations appear in pairs. If, for example, we find one solution<br />
going to equilibrium in the far distant future (and not in<br />
the far distant past), we should also find a solution that goes to<br />
equilibrium in the far distant past (and not in the far distant<br />
future). Symmetry-broken solutions appear in pairs.<br />
Once we find such a situation we can express the intrinsic<br />
meaning of the second law. It becomes a selection principle<br />
stating that only one of the two types of solutions can be realized<br />
or may be observed in nature. Whenever applicable, the<br />
second law expresses an intrinsic polarization of nature. It can<br />
never be the outcome of dynamics itself. It has to appear as a<br />
supplementary selection principle that when realized is propagated<br />
by dynamics. Only a few years ago it seemed impossible<br />
to attempt such a program. However, over the past few de-
261 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
cades dynamics has made remarkable progress, and we can<br />
now understand in detail how these symmetry-breaking solutions<br />
emerge in dynamic systems "of sufficient complexity"<br />
and what the selection rule expressed by the second law of<br />
thermodynamics means on the microscopic level. This is what<br />
we want to show in the next part of this chapter.<br />
The Limits of Classical Concepts<br />
Let us start with classical mechanics. As we have already<br />
mentioned, if trajectory is to be the basic irreducible element,<br />
the world would be as reversible as the trajectories out of<br />
which it is formed. In this description there would be no entropy<br />
and no arrow of time; but, as a result of unexpected re·<br />
cent developments, the validity of the trajectory concept<br />
appears far more limited than we might have expected. Let us<br />
return to Gibbs' and Einstein's theory of ensembles, introduced<br />
in Chapter VIII. We have seen that Gibbs and Einstein<br />
introduced phase space into physics to account for the fact<br />
that we do not "know" the initial state of systems formed by a<br />
large number of particles. For them, the distribution function<br />
in phase space was only an auxiliary construction expressing<br />
our de fa cto ignorance of a situation that was well determined<br />
de jure. However, the entire problem takes on new dimensions<br />
once it can be shown that for certain types of systems an infi·<br />
nitely precise determination of initial conditions leads to a<br />
self-contradictory procedure. Once this is so, the fact that we<br />
never know a single trajectory but a group, an ensemble of<br />
trajectories in phase space, is not a mere expression of the<br />
limits of our knowledge. It becomes a starting point of a new<br />
way of investigating dynamics.<br />
It is true that in simple cases there is no problem. Let us<br />
take the example of a pendulum. It may oscillate or else rotate<br />
about its axis according to the initial conditions. For it to rotate,<br />
its kinetic energy must be large enough for it not to "fall<br />
back" before reaching a vertical position. These two types of<br />
motion define two disjointed regions of phase space. The reason<br />
for this is very simple: rotation requires more energy than<br />
oscillation (see Figure 30).
ORDER OUT OF CHAOS 262<br />
y<br />
0)<br />
a<br />
v<br />
b)<br />
Figure 30. Representation of a pendulum's motion in a space where Vis<br />
the velocity and e the angle of deflection. (a) typical trajectories in (V, e)<br />
space; (b) the shaded regions correspond to oscillations; the region outside<br />
corresponds to rotations.<br />
If our measurements allow us to establish that the system is<br />
initially in a given region, we may safely predict the type of<br />
motion displayed by the pendulum. We can increase the accu-
263<br />
IRREVERSIBILITY-THE ENTROPY BARRIER<br />
racy of our measurements and localize the initial state of the<br />
pendulum in a smaller region circumscribed by the first. In<br />
any case, we know the system's behavior for all time; nothing<br />
new or unexpected is likely to occur.<br />
One of the most surprising results achieved in the twentieth<br />
century is that such a description is not valid in general. On<br />
the contrary, "most" dynamic systems behave in a quite unstable<br />
way. 6 Let us indicate one kind of trajectory (for example,<br />
that of oscillation) by + and another kind (for example,<br />
that corresponding to rotation) by *· Instead of Figure 30,<br />
where the two regions were separated, we find, in general, a<br />
mixture of states that makes the transition to a single point<br />
ambiguous (see Figure 31). If we know only that the initial<br />
state of our system is in region A, we cannot deduce that its<br />
trajectory is of type +; it may equally well be of type *· We<br />
achieve nothing by increasing the accuracy by going from region<br />
A to a smaller region within it, for the uncertainty remains.<br />
In all regions, however small, there are always states<br />
belonging to each of the two types of trajectories.1<br />
v<br />
Figure 31. Schematic representation of any region, arbitrarily small, of the<br />
phase space V tor a system presenting dynamic instability. As in the case of<br />
the pendulum, there are two types of trajectories (represented here by +<br />
and •); however, in contrast with the pendulum, both motions appear in every<br />
region arbitrary small.
265 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
that at the end of the nineteenth century Bruns and Poincare<br />
demonstrated that most dynamic systems, starting with the<br />
famous "three body" problem, were not integrable.<br />
On the other hand, the very idea of approaching equilibrium<br />
in terms of the theory of ensembles requires that we go beyond<br />
the idealization of integrable systems. As we saw in Chapter<br />
VII I, according to the theory of ensembles, an isolated system<br />
is in equilibrium when it is represented by a "microcanonical<br />
ensemble," when all points on the surface of given energy<br />
have the same probability. This means that for a system to<br />
evolve to equilibrium, energy must be the only quantity conserved<br />
during its evolution. It must be the only "invariant."<br />
Whatever the initial conditions, the evolution of the system<br />
must allow it to reach all points on the surface of given energy.<br />
But for an integrable system, energy is far from being the only<br />
invariant. In fact, there are as many invariants as degrees of<br />
freedom, since each generalized momentum remains constant.<br />
Therefore we have to expect that such a system is "imprisoned"<br />
in a very small "fraction" of the constant-energy (see<br />
Figure 32) surface formed by the intersection of all these invar<br />
-iant surfaces.<br />
p<br />
Figure 32. Temporal evolution of a cell in phase space p, q. The "volume"<br />
of the cell and its form are maintained in time; moreover, most of the phase<br />
space is inaccessible to the system.<br />
q
265 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
that at the end of the nineteenth century Bmns and Poincare<br />
demonstrated that most dynamic systems, starting with the<br />
famous "three body" problem, were not integrable.<br />
On the other hand, the very idea of approaching equilibrium<br />
in terms of the theory of ensembles requires that we go beyond<br />
the idealization of integrable systems. As we saw in Chapter<br />
VIII, according to the theory of ensembles, an isolated system<br />
is in equilibrium when it is represented by a "microcanonical<br />
ensemble," when all points on the surface of given energy<br />
have the same probability. This means that for a system to<br />
evolve to equilibrium, energy must be the only quantity conserved<br />
during its evolution. It must be the only "invariant."<br />
Whatever the initial conditions, the evolution of the system<br />
must allow it to reach all points on the surface of given energy.<br />
But for an integrable system, energy is far from being the only<br />
invariant. In fact, there are as many invariants as degrees of<br />
freedom, since each generalized momentum remains constant.<br />
Therefore we have to expect that such a system is "imprisoned"<br />
in a very small "fraction" of the constant-energy (see<br />
Figure 32) surface formed by the intersection of all these invariant<br />
.<br />
surfaces.<br />
p<br />
..--- ...<br />
- ...<br />
1),-/<br />
. , ,<br />
<br />
," , ---""'04\<br />
"<br />
, I \ \<br />
I ,<br />
I ,<br />
, \<br />
I<br />
\<br />
\ , 'I<br />
, rs-'" ,<br />
I<br />
\ \ , "<br />
/<br />
, /<br />
,<br />
,/<br />
"<br />
'"<br />
..... _..-.,-.<br />
Figure 32. Temporal evolution of a cell in phase space p, q. The "volume"<br />
of the cell and its form are maintail1ed in time; moreover, most of the phase<br />
space is inaccessible to the system.<br />
q
ORDER OUT OF CHAOS 266<br />
To avoid these difficulties, Maxwell and Boltzmann introduced<br />
a new, quite different type of dynamic system. For these<br />
systems energy would be the only invariant. Such systems are<br />
called "ergodic" systems (see Figure 33).<br />
Great contributions to the theory of ergodic systems have<br />
been made by Birchoff, von Neumann, Hopf, Kolmogoroff,<br />
and Sinai, to mention only a few. s. 9. tO<br />
Today we know that<br />
there are large classes of dynamic (though non-Hamiltonian)<br />
systems that are ergodic. We also know that even relatively<br />
simple systems may have properties stronger than ergodicity.<br />
For these systems, motion in phase space becomes highly chaotic<br />
(while always preserving a volume that agrees with the<br />
Liouville equation mentioned in Chapter VII).<br />
p<br />
q<br />
Figure 33. Typical evolution in phase space of a cell corresponding to an<br />
ergodic system. Time going on, the "volume" and the form are conserved<br />
but the cell now spirals through the whole phase space.
267<br />
IRREVERSIBILITY-THE ENTROPY BARRIER<br />
Suppose that our knowledge of initial conditions permits us<br />
to localize a system in a small cell of the phase space. During<br />
its evolution, we shall see this initial cell twist and turn and,<br />
like an amoeba, send out "pseudopods" in all directions,<br />
spreading out in increasingly thinner and ever more twisted<br />
filaments until it finally invades the whole space. No sketch<br />
can do justice to the complexity of the actual situation. Indeed,<br />
during the dynamic evolution of a mixing system, two<br />
· points as close together in phase space as we might wish may<br />
head in different directions. Even if we possess a lot of information<br />
about the system so that the initial cell formed by its<br />
representative points is very small, dynamic evolution turns<br />
this cell into a true geometric "monster" stretching its network<br />
of filaments through phase space.<br />
p<br />
Figure 34. Typical evolution in phase space of a cell corresponding to a<br />
"mixing" system. The volume is still conserved but no more its form: the cell<br />
progressively spreads through the whole phase space.<br />
q
ORDER OUT OF CHAOS<br />
268<br />
We would like to illustrate the distinction between stable<br />
and unstable systems with a few simple examples. Consider a<br />
phase space with two dimensions. At regular time intervals,<br />
we shall replace these coordinates by new ones. The new point<br />
on the horizontal axis is p-q, the new ordinate p. Figure 35<br />
shows what happens when we apply this operation to a square.<br />
(0,-1) p<br />
Figure 35. Transformation of a volume in phase space generated by a<br />
discrete transformation: the abscissa p becomes p-q, the ordinate q becomes<br />
p. The transformation is cyclic: after six times the initial cell is recovered.<br />
The square is deformed, but after six transformations we return<br />
to the original square. The system is stable: neighboring<br />
points are transformed into neighboring points. Moreover, it<br />
corresponds to a cyclic transformation (after six operations<br />
the original square reappears).<br />
Let us now consider two examples of highly unstabl systems-the<br />
first mathematical, the second of obvious physical
11<br />
269 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
relevance. The first system consists of a transformation that,<br />
for obvious reasons, mathematicians call the "baker transformation.<br />
"9, to We take a square and flatten it into a rectangle,<br />
then we fold half of the rectangle over the other half to form a<br />
square again. This set of operations is shown in Figure 36 and<br />
may be repeated as many times as one likes.<br />
q:1<br />
n<br />
<br />
1<br />
q=1<br />
2 p:1<br />
B<br />
q=1<br />
)<br />
q:11J<br />
p:1<br />
112 p:1<br />
e-1<br />
Figure 36. Realization of the baker transformation (B) and of its inverse<br />
(B-1). The path of the two spots gives an idea of the transformations.<br />
Each time the surface of the square is broken up and redistributed.<br />
The square corresponds here to the phase space.<br />
The baker transformation transforms each point into a welldefined<br />
new point. Although the series of points obtained in<br />
this way is "deterministic," the system displays in addition<br />
irreducibly statistical aspects. Let us take , for instance, a system<br />
described by an initial condition such that a region A of<br />
the square is initially filled in a uniform way with representative<br />
points. It may be shown that after a sufficient number of<br />
repetitions of the transformation, this cell, whatever its size<br />
and localization, will be broken up into pieces (see Figure 37).<br />
The essential point is that any region, whatever its size, thus
ORDER OUT OF CHAOS 270<br />
p<br />
Figure 37. Time evolution of an unstable system. Time going on, region A<br />
splits into regions A' and A", which in turn will be divided.<br />
q<br />
always contains different trajectories diverging at each fragmentation.<br />
Although the evolution of a point is reversible and<br />
deterministic, the description of a region, however small, is<br />
basically statistical.<br />
A similar example involves the scattering of hard spheres.<br />
We may consider a small sphere rebounding on a collection of<br />
large, randomly distributed spheres. The latter are supposed<br />
to be fixed. This is the model physicists call the "Lorentz<br />
model," after the name of a great Dutch physicist, Hendrik<br />
Antoon Lorentz.<br />
The trajectory of the small mobile sphere is well defined.<br />
However, whenever we introduce the smallest uncertainty in<br />
the initial conditions, this uncertainty is amplified through<br />
successive collisions. As time passes, the probability of finding<br />
the small sphere in a given volume becomes uniform.<br />
Whatever the number of transformations, we never return to<br />
the original state.<br />
In the last two examples we have strongly unstable dynamic<br />
systems. The situation is reminiscent of instabilities as they<br />
appear in thermodynamic systems (see Chapter V). Arbitrarily<br />
small differences in initial conditions are amplified. As a<br />
result we can no longer perform the transition from ensembles
271 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
<br />
Q<br />
I<br />
I<br />
I<br />
I<br />
I<br />
'o<br />
I I<br />
I I<br />
I 1<br />
1 I<br />
1 I<br />
I<br />
' ----- <br />
- -o---- --:..<br />
...<br />
.... .. ,"<br />
... ..<br />
... ..<br />
... ...<br />
... "<br />
...<br />
"<br />
"<br />
,<br />
"<br />
/<br />
"<br />
"<br />
•<br />
<br />
Figure 38. Schematic representation of the instability of the trajectory of a<br />
small sphere rebounding on large ones. The least imprecision about the<br />
position of the small sphere makes it impossible to predict which large<br />
sphere it will hit after the first collision.<br />
in phase space to individual trajectories. The description in<br />
terms of ensembles has to be taken as the starting point. Statistical<br />
concepts are no longer merely an approximation with<br />
respect to some "objective truth." When faced with these unstable<br />
systems, Laplace's demon is just as powerless as we.<br />
Einstein's saying, "God does not play dice," is well known.<br />
In the same spirit Poincare stated that for a supreme mathematician<br />
there is no place for probabilities. However, Poincare<br />
himself mapped the path leading to the answer to this problem.11<br />
He noticed that when we throw dice and use probability<br />
calculus, it does not mean that we suppose dynamics to be<br />
wrong. It means something quite different. We apply the probability<br />
concept because in each interval of initial conditions,<br />
however small, there are as "many" trajectories that lead to<br />
each of the faces of the dice. This is precisely what happens<br />
with unstable dynamic systems. God could, if he wished to,
ORDER OUT OF CHAOS 272<br />
calculate the trajectories in an unstable dynamic world. He<br />
would obtain the same result as probability calculus permits<br />
us to reach. Of course, if he made use of his absolute knowledge,<br />
then he could get rid of all randomness.<br />
In conclusion, there is a close relationship between instability<br />
and probability. This is an important point, and we want to<br />
discuss it now.<br />
From Randomness to Irreversibility<br />
Consider a succession of squares to which we apply the baker<br />
transformation. This succession is represented in Figure 39. A<br />
shaded region may be imagined to be filled with ink, an unshaded<br />
region by water. At time zero we have what is called a<br />
generating partition. Out of this partition we form a series of<br />
either horizontal partitions when we go into the future or vertical<br />
partitions going into the past. These are the basic partitions.<br />
An arbitrary distribution of ink in the square can be<br />
written formally as a superposition of the basic partitions. To<br />
each basic partition we may associate an "internal" time that<br />
is simply the number of baker transformations we have to perform<br />
to go from the generating partition to the one under consideration.I2<br />
We therefore see that this type of system admits a<br />
kind of internal age.*<br />
The internal time Tis quite different from the usual mechanical<br />
time, since it depends on the global topology of the system.<br />
We may even speak of the "timing of space," thus coming<br />
close to ideas recently put forward by geographers, who have introduced<br />
the concept of "chronogeography." 13 When we look at<br />
the structure of a town , or of a landscape, we see temporal<br />
elements interacting and coexisting. Brasilia or Pompeii would<br />
correspond to a well-defined internal age, somewhat like one<br />
of the basic partitions in the baker transformation. On the contrary,<br />
modern Rome, whose buildings originated in quite dif-<br />
*It may be noticed that this internal time, which we shall denote by T. is in<br />
fact an operator like those introduced in quantum mechanics (see Chapter<br />
VII). Indeed, an arbitrary partition of the square does not have a welldefined<br />
time but only an "average" time corresponding to the superposition<br />
of the basic partitions out of which it is formed.
273 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
-m-<br />
-<br />
-=-<br />
past<br />
•1 0 z<br />
t<br />
genart1ting partition<br />
future<br />
Figure 39. Starting with the "generating partition" (see text) at time 0, we<br />
repeatedly apply the baker transformation. We generate horizontal stripes in<br />
this way. Similarly going into the past we obtain vertical stripes.<br />
ferent periods, would correspond to an average time exactly as<br />
an arbitrary partition may be decomposed into elements corresponding<br />
to different internal times.<br />
Let us again look at Figure 39. What happens if we go into<br />
the far distant future? The horizontal bands of ink will get<br />
closer and closer. Whatever the precision of our measurements,<br />
after some time we shall conclude that the ink is uniformly<br />
distributed over the volume. It is therefore not surprising that<br />
this kind of approach to "equilibrium" may be mapped into a<br />
stochastic process, such as the Markov chain we described in<br />
Chapter VIII. Recently this has been shown with full mathematical<br />
rigor, 14 but the result seems to us quite natural. As<br />
time passes, the distribution of ink reaches equilibrium, exactly<br />
like the distribution of balls in the urn experiment discussed<br />
in Chapter VIII. However, when we look into the past,<br />
again beginning from the generating partition at time zero, we<br />
see the same phenomenon. Now ink is distributed along shrinking<br />
vertical sections and, again, sufficiently far in the past we<br />
shall find a uniform distribution of ink. We may therefore conclude<br />
that we can also model this process in terms of a Markov<br />
chain, now, however, oriented toward the past. We see that out<br />
of the unstable dynamic processes we obtain two Markov<br />
chains, one reaching equilibrium in the future, one in the past.<br />
We believe that it is a very interesting result and we would<br />
like to commen t it. Internal time provides us with a new 'nonlocal'<br />
description.<br />
When we know the 'age' of the system, (that is, the corresponding<br />
partition), we can still not associate to it a well-defined<br />
local trajectory.
ORDER OUT OF CHAOS 274<br />
We know only that the system is in a shaded region (Figure<br />
39). Similarly, if we know some precise initial conditions corresponding<br />
to a point in the system, we don't know the partition<br />
to which it belongs, nor the age of the system. For such<br />
systems we know therefore two complementary descriptions,<br />
and the situation becomes somewhat reminiscent of the one<br />
we described in Chapter VII, when we discussed quantum mechanics.<br />
It is because of the existence of this new alternative, nonlocal<br />
description, that we can make the transition from dynamics<br />
to probabilities. We call the systems for which this is<br />
possible "intrinsically random systems".<br />
In classical deterministic systems, we may use transition<br />
probabilities to go from one point to another on a quite degenerate<br />
sense. This transition probability will be equal to one if<br />
the two points lie on the same dynamic trajectory, or zero if<br />
they are not.<br />
In contrast, in genuine probability theory, we need transition<br />
probabilities which are positive numbers between zero<br />
and one. How is this possible? Here we see in full light the<br />
conflict between subjectivistic views and objective interpretations<br />
of probability. The subjective interpretation corresponds<br />
to the situation where individual trajectories are not known.<br />
Probability (and, eventually, irreversibility, closely related to<br />
it) would originate from our ignorance. But fortunately, there<br />
is another objective interpretation: probability arises as a result<br />
of an alternative description of dynamics, a non-local description<br />
which arises in strongly unstable dynamical systems.<br />
Here, probability becomes an objective property generated<br />
from the inside of dynamics, so to speak, and which expresses<br />
a basic structure of the dynamical system. We have stressed<br />
the importance of Boltzmann's basic discovery: the connection<br />
between entropy and probability. For intrinsic random<br />
systems, the concept of probability acquires a dynamical<br />
meaning. We have now to make the transition from intrinsic<br />
random systems to irreversible systems. We have seen that out<br />
of unstable dynamical processes, we obtain two Markov<br />
chains.<br />
We may see this duality in a different way. Take a distribution<br />
concentrated on a line (instead of being distributed on a<br />
surface). This line may be vertical or horizontal. Let us look at
275 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
what will happen to this line when we apply the baker transformation<br />
going to the future. The result is represented in Figure<br />
40. The vertical line is cut successively into pieces and will<br />
reduce to a point in the far distant future. The horizontal line,<br />
on the contrary, is duplicated and will uniformly "cover" the<br />
surface in the far distant future. Obviously, just the opposite<br />
happens if we go to the past. For reasons that are easy to understand,<br />
the vertical line is called a contracting fiber, the<br />
horizontal line a dilating fiber.<br />
We see now the complete analogy with bifurcation theory. A<br />
contracting fiber and a dilating fiber correspond to two realizations<br />
of dynamics, each involving symmetry-breaking and appearing<br />
in pairs. The contracting tiber corresponds to equilibrium in<br />
the far distant past, the dilating fiber to the future. We therefore<br />
have two Markov chains oriented in opposite time directions.<br />
Now we have to make the transistion from intrinsically random<br />
to intrinsically irreversible systems. To do so we must<br />
understand more precisely the difference between contracting<br />
and dilating fibers. We have seen that another system as unsta-<br />
I<br />
Figure 40. Contracting and dilating fibers in the baker transformation; time<br />
going on, the contracting fiber A1 is shortened (sequence A1, 81, C1), while<br />
the dilating fibers are duplicated (sequences A2, 82, C2).<br />
ble as the baker transformation can describe the scattering of<br />
hard spheres. Here contracting and dilating fibers have a sim-
ORDER OUT OF CHAOS 276<br />
pie physical interpretation. A contracting fiber corresponds to<br />
a collection of hard spheres whose velocities are randomly distributed<br />
in the far distant past, and all become parallel in the<br />
far distant future. A dilating fiber corresponds to the inverse<br />
situation, in which we start with parallel velocities and go to a<br />
random distribution of velocities. Therefore the difference is<br />
very similar to the one between incoming waves and outgoing<br />
waves in the example given by Popper. The exclusion of the<br />
contracting fibers corresponds to the experimental fact that<br />
whatever the ingenuity of the experimenter, he will never be<br />
able to control the system to produce parallel velocities after<br />
an arbitrary number of collisions. Once we exclude con·<br />
tracting fibers we are left with only one of the two possible<br />
Markov chains we have introduced. In other words, the second<br />
law becomes a selection principle of initial conditions.<br />
Only initial conditions that go to equilibrium in the fu ture are<br />
retained.<br />
Obviously the validity of this selection principle is maintained<br />
by dynamics. It can easily be seen in the example of the<br />
baker transformation that the contracting fiber remains a contracting<br />
fiber for all times, and likewise for a dilating fiber. By<br />
suppressing one of the two Markov chains we go from an<br />
intrinsically random system to an intrinsically irreversible system.<br />
We find three basic elements in the description of irreversibility:<br />
instability<br />
f<br />
intrinsic randomness<br />
f<br />
·intrinsic irreversibility<br />
Intrinsic irreversibility is the strongest property: it implies<br />
randomness and instability.t4, ts<br />
How is this conclusion compatible with dynamics? As we<br />
have seen, in dynamics "information" is conserved, while in<br />
Markov chains information is lost (and entropy therefore increases;<br />
see Chapter VIII). There is, however , no contradiction;<br />
when we go from the dynamic description of the baker<br />
transformation to the thermodynamic description, we have to
277 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
modify our distribution function; the "objects" in terms of<br />
which entropy increases are different fro m the ones considered<br />
in dynamics. The new distribution function, p, corresponds<br />
to an intrinsically time-oriented description of the<br />
dynamic system. In this book we cannot dwell on the mathematical<br />
aspects of this transformation. Let us only emphasize<br />
that it has to be noncanonical (see Chapter II). We must abandon<br />
the usual formulations of dynamics to reach the thermodynamic<br />
description.<br />
It is quite remarkable that such a transformation exists and<br />
that as a result we can now unify dynamics and thermodynamics,<br />
the physics of being and the physics of becoming. We shall<br />
come back to these new thermodynamic objects later in this<br />
chapter as well as in our concluding chapter. Let us emphasize<br />
only that at equilibrium, whenever entropy reaches its maximum,<br />
these objects must behave randomly.<br />
It also seems quite remarkable that irreversibility emerges, so<br />
to speak, from instability, which introduces irreducible statistical<br />
features into our description. Indeed, what could an arrow<br />
of time mean in a deterministic world in which both future and<br />
past are contained in the present? It is because the future is<br />
not contained in the present and that we go from the present to<br />
the future that the ar row of time is associated with the transition<br />
from present to future. This construction of irreversibility out of<br />
randomness has, we believe, many consequences that go<br />
beyond science proper and to which we shall come back in our<br />
concluding chapter. Let us clarify the difference between the<br />
states permitted by the second law and those it prohibits.<br />
The Entropy Barrier<br />
Time flows in a single direction, from past to future. We cannot<br />
manipulate time, we cannot travel back to the past. Travel<br />
through time has preoccupied writers, from The Thousand<br />
and One Nights to H. G. Wells' The Time Machine. In our<br />
time, Nabokov's short novel, Look at the Harlequins!, 1' describes<br />
the torment of a narrator who finds himself as unable<br />
to switch from one spatial direction to the other as we are to
ORDER OUT OF CHAOS 278<br />
"twirl time." In the fifth volume of Science and Civilization in<br />
China, Needham describes the dream of the Chinese alchemists:<br />
their supreme aim was not to achieve the transmutation<br />
of metals into gold but to manipulate time, to reach immortality<br />
through a radical slowdown of natural decaying processes.<br />
I? We are now better able to understand why we cannot<br />
"twirl time," to use Nabokov's expression.<br />
An infinite entropy barrier separates possible initial conditions<br />
fr om prohibited ones. Because this barrier is infinite,<br />
technological progress never will be able to overcome it. We<br />
have to abandon the hope that one day we will be able to travel<br />
back into our past. The situation is somewhat analogous to the<br />
barrier presented by the velocity of light. Technological progress<br />
can bring us closer to the velocity of light, but in the present<br />
views of physics we will never cross it.<br />
To understand the origin of this barrier, let us return to the<br />
expression of the :H quantity as it appears in the theory of<br />
Markov chains (see Chapter VIII). To each distribution we can<br />
associate a number-the corresponding value of J-{. We can<br />
say that to each distribution corresponds a well-defined information<br />
content. The higher the information content, the more<br />
difficult it will be to realize the corresponding state. What we<br />
wish to show here is that the initial distribution prohibited by<br />
the second law would have an infinite information content.<br />
That is the reason why we can neither realize them nor find<br />
them in nature.<br />
Let us first come back to the meaning of :Has presented in<br />
Chapter VIII. We have to subdivide the relevant phase space<br />
into sectors or boxes. With each box k, we associate an probability<br />
Peqm(k) at equilibrium as well as a non-equilibrium probability<br />
P(k,t).<br />
The :H is a measure of the difference between P(k,t) and<br />
Peqm(k), and vanishes at equilibrium when this difference disappears.<br />
Therefore, to compare the Baker transformation with<br />
Markov chains, we have to make more precise the corresponding<br />
choice of boxes. Suppose we consider a system at time 2<br />
(see Figure 39), and suppose that this system originated at<br />
time ti. Then, a result of our dynamical theory is that the<br />
boxes correspond to all possible intersections among the partitions<br />
between timet; and t=2. If we consider now Figure 39.<br />
we see that when ti is receding towards the past, the boxes will
279 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
become steadily thinner as we have to introduce more and<br />
more vertical subdivisions. This is expressed in Figure 41, sequence<br />
B, where, going from top to bottom, we have ti = 1, 0,<br />
- 1, and finally ti = - 2. We see indeed that the number of<br />
boxes increa ses in this way from 4 to 32.<br />
Once we have the boxes, we can compare the non-equilibrium<br />
distribution with the equilibrium distribution for each<br />
box. In the present case, the non-equilibrium distribution is<br />
either a dilating fiber (sequence A) or a contracting fiber (sequence<br />
C). The important point to notice is that when ti is<br />
receding to the past, the dilating fiber occupies an increasing<br />
large number of boxes: for ti = -2 it occupies 4 boxes, for<br />
ti = - 2 it occupies 8 boxes, and so on.<br />
As a result, when we apply the fo rmula given in Chapter<br />
VIII, we obtain a finite result, even if the number of boxes<br />
goes to infinity for tr-+ -oo.<br />
In contrast, the contracting fiber remains always localized<br />
in 4 boxes whatever ti. As a result, .Jl, when applied to a con-<br />
A 8 c<br />
Figure 41. Dilating (sequence A) and contracting (sequence C) fibers<br />
cross various numbers of the boxes which subdivide a Baker transformation<br />
phase space. All "squares" on a given sequence refer to the same time, t=2,<br />
but the number of the boxes subdividing each square depends on the initial<br />
time of the system ti.
ORDER OUT OF CHAOS<br />
280<br />
tracting fiber, diverges to infinity when ti recedes to the past.<br />
In summary, the difference between a dynamical system and<br />
the Markov chain is that the number of boxes to be considered<br />
in a dynamical system is infinite. It is this fact that leads to a<br />
selection principle. Only measures or probabilities, which in<br />
the limit of infinite number of boxes give a finite information<br />
or a finite J{ quantity, can be prepared or observed. This excludes<br />
contracting fibers.ts For the same reason we must also<br />
exclude distributions concentrated on a single point. Initial<br />
conditions corresponding to a single point in unstable systems<br />
would again correspond to infinite information and are therefore<br />
impossible to realize or observe. Again we see that the<br />
second law appears as a selection principle.<br />
In the classical scheme, initial conditions were arbitrary.<br />
This is no longer so for unstable systems. Here we can associate<br />
an information content to each initial condition, and this<br />
information content itself depends on the dynamics of the system<br />
(as in the baker transformation we used the successive<br />
fragmentation of the cells to calculate the information content).<br />
Initial conditions and dynamics are no longer independent.<br />
The second law as a selection rule seems to us so<br />
important that we would like to give another illustration based<br />
on the dynamics of correlations.<br />
The Dynamics of Correlations<br />
In Chapter VIII we discussed briefly the velocity inversion<br />
experiment. We may consider a dilute gas and follow its evolution<br />
in time. At time t0 we proceed to a velocity inversion of<br />
each molecule. The gas then returns to its initial state. We<br />
have already noted that for the gas to retrace its past there<br />
must be some storage of information. This storage can be described<br />
in terms of "correlations" between particles.t9<br />
Consider first a cloud of particles directed toward a target (a<br />
heavy, motionless particle). This situation is described in Figure<br />
42. In the far distant past, there were no correlations between<br />
particles. Now, scattering has two effects, already<br />
mentioned in Chapter VIII. It disperses the particles (it makes<br />
the velocity distribution more symmetrical) and, in addition, it
281 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
produces correlations between the scattered particles and the<br />
scatterer. The correlations can be made explicit by performing<br />
a velocity inversion (that is, by introducing a spherical mirror).<br />
Figure 43 represents this situation (the wavy lines represent<br />
the correlations). Therefore, the role of scattering is as follows:<br />
In the direct process, it makes the velocity distribution<br />
more symmetrical and creates correlations; in the inverse process,<br />
the velocity distribution becomes less symmetrical and<br />
correlations disappear. We see that the consideration of cor-<br />
•<br />
•<br />
.<br />
.. 0<br />
- .. ...<br />
• •<br />
Figure 42. Scattering of particles. Initially all particles have the same velocity.<br />
After the collision, the velocities are no more identical, and the scattered<br />
particles are correlated with the scatterer (correlations are always<br />
represented by wavy lines).<br />
relations introduces a basic distinction between the direct and<br />
the inverse processes.<br />
We can apply our conclusions to many-body systems. Here<br />
also we may consider two types of situations: in one, uncorrelated<br />
particles enter, are scattered, and correlated particles are<br />
produced (see Figure 44). In the opposite situation, correlated<br />
• •<br />
• ..<br />
<br />
0 • ..<br />
• ..<br />
Figure 43. Effect of a velocity inversion after a collision: after the new<br />
"inverted" collision, the correlations are suppressed and all particles have<br />
the same velocity.
ORDER OUT OF CHAOS 282<br />
particles enter, the correlations are destroyed through collisions,<br />
and uncorrelated particles re sult (see Figure 45).<br />
The two situations diffe r in the temporal order of collisions<br />
and correlations. In the first case , we have "postcollisional"<br />
correlations. With this distinction between pre- and postcollisional<br />
correlations in mind, let us re turn to the ve locity inversion<br />
experiment. We start at t = 0, with an initial state<br />
corresponding to no correlations between particles. During<br />
the time ot0 we have a "normal" evolution. Collisions bring<br />
the ve locity distribution closer to the Maxwellian equilibrium<br />
distribution. They also create postcollisional correlations be-<br />
0 0<br />
0 0<br />
before<br />
after<br />
Figure 44. Creation of postcollisional correlations represented by wavy<br />
lines; for details see text.<br />
tween the particles. At t0 after the velocity inversion, a completely<br />
new situation arises. Postcollisional correlations are<br />
now transformed into precollisional correlations. In the time<br />
interval between t0 and 2t0, these precollisional correlations<br />
disappear, the velocity distribution becomes less symmet-rical,<br />
and at time 2t0 we are back in the noncorrelational state. The<br />
history of this system therefore has two stage s. During the<br />
0 0<br />
0<br />
<br />
0<br />
I before<br />
after<br />
Figure 45. Destruction of precollisional correlations (wavy lines) through<br />
collisions.
283<br />
IRREVERSIBILITY-THE ENTROPY BARRIER<br />
first, collisions are transformed into correlations; in the second,<br />
correlations turn back into collisions. Both types of processes<br />
are compatible with the laws of dynamics. Moreover, as<br />
we have mentioned in Chapter VIII, the total "information"<br />
described by dynamics remains constant. We have also seen<br />
that in Boltzmann's description the evolution from time 0 till t0<br />
corresponds to the usual decrease of J{, while from t0 to 2t0 we<br />
have an abnormal situation: J{ would increase and entropy decrease.<br />
We would then be able to devise experiments in the<br />
laboratory or on computers in which the second Jaw would be<br />
violated! The irreversibility during time 0-t0 would be "compensated"<br />
by "anti-irreversibility" during time t0 - 2t0•<br />
This is quite unsatisfactory. All these difficulties disappear<br />
if we go, as in the baker transformation, to the new "thermodynamic<br />
representation" in terms of which dynamics becomes a<br />
probabilistic process like a Markov chain. We must also take<br />
into account that velocity inversion is not a "natural" process;<br />
it requires that "information" be given to molecules from the<br />
outside for them to invert their velocity. We need a kind of<br />
Maxwellian demon to perform the velocity inversions, and<br />
Maxwell's demon has a price. Let us represent the J{ quantity<br />
(for the probabilistic process) as a function of time. This is<br />
done in Figure 46. In this approach, in contrast with Boltzmann's,<br />
the effect of correlations is retained in the new definition<br />
of :H. Therefore at the velocity inversion point t0 the J{<br />
quantity will jump, since we abruptly create abnormal precollisional<br />
correlations that will have to be destroyed later. This<br />
jump corresponds to the entropy or information price we have<br />
to pay.<br />
Now we have a faithful representation of the second law: at<br />
every moment the J{ quantity decreases (or the entropy increases).<br />
There is one exception at time t0: J{ jumps upward,<br />
but that corresponds to the very moment at which the system<br />
is open. We can invert the velocities only by acting from the<br />
outside.<br />
There is another essential point: at time t0 the new J{ quantity<br />
has two different values, one for the system before velocity<br />
inversion and the other after a velocity inversion. These<br />
two situations have different entropies. This resembles what<br />
occurs in the baker transformation when the contracting and<br />
dilating fibers are velocity inversions of each other.
ORDER OUT OF CHAOS<br />
284<br />
Suppose we wait a sufficient time before making the velocity<br />
in-version. The postcollisional correlations wmdd have<br />
an arbitrary range, and the entropy price for velocity inversion<br />
would become too high. The velocity inversion would then<br />
require too high an entropy price and thus would be excluded.<br />
In physical terms this means that the second law excludes persistent<br />
long-range precollisional correlations.<br />
The analogy with the macroscopic description of the second<br />
law is striking. From the point of view of energy conservation<br />
(see Chapters IV and V), heat and work play the same role,<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
----·-------+ t<br />
t 0 2t 0<br />
Figure 46. Time variation of the .J l -function in the vlocity inversion experiment:<br />
at time t0, the velocities are inversed and J-l presents a discontinuity.<br />
At time 2t0 the system is in the same state as at ti_me 0, and .J l<br />
recovers the value it had initially. At all times (except at t0), J-l is decreasing.<br />
The important fact is that at time t0 the J-l-quantity takes two different values<br />
(see text).<br />
but no longer from the point of view of the second law. Briefly<br />
speaking, work is a more coherent form of energy and always<br />
can be converted into heat, but the inverse is not true. There is<br />
on the microscopic level a similar distinction between collisions<br />
and correlations. From the point of view of dynamics, collisions<br />
and correlations play equivalent roles. Collisions give<br />
rise to correlations, and correlations may destroy the effect of<br />
I
285 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
collisions. But there is an essential difference. We can control<br />
collisions and produce correlations, but we cannot control correlations<br />
in a way that will destroy the effects collisions have<br />
brought into the system. It is this essential difference that is<br />
missing in dynamics but that can be incorporated into thermodynamics.<br />
Note that thermodynamics does not enter into conflict<br />
with dynamics at any point. It adds an additional, essential<br />
element to our understanding of the physical world.<br />
Entropy as a Selection Principle<br />
It is amazing how closely the microscopic theory of irreversible<br />
processes resembles traditional macroscopic theory. In both<br />
cases entropy initially has a negative meaning. In its macroscopic<br />
aspect it prohibits some processes, such as heat flowing from<br />
cold to hot. In its microscopic aspect it prohibits certain<br />
classes of initial conditions. The distinction between what is<br />
permitted and what is prohibited is maintained in time by the<br />
laws of dynamics. It is from the negative aspect that the positive<br />
aspect emerges: the existence of entropy together with its<br />
probability interpretation. Irreversibility no longer emerges as<br />
if by a miracle at some macroscopic level. Macroscopic irreversibility<br />
only makes apparent the time-oriented polarized<br />
nature of the universe in which we live.<br />
As we have emphasized repeatedly, there exist in nature systems<br />
that behave reversibly and that may be fully described by<br />
the laws of classical or quantum mechanics. But most systems<br />
of interest to us, including all chemical systems and therefore<br />
all biological systems, are time-oriented on the macroscopic<br />
level. Far from being an "illusion," this expresses a broken<br />
time-symmetry on the microscopic level. Irreversibility is either<br />
true on all levels or on none. It cannot emerge as if by a<br />
miracle, by going from one level to another.<br />
Also we have already noticed, irreversibility is the starting<br />
point of other symmetry breakings. For example, it is generally<br />
accepted that the difference between particles and antiparticles<br />
could arise only in a nonequilibrium world. This may<br />
be extended to many other situations. It is likely that irreversibility<br />
also played a role in the appearance of chiral symmetry
ORDER OUT OF CHAOS 286<br />
through the selection of the appropriate bifurcation. One of<br />
the most active subjects of research now is the way in which<br />
irreversibility can be "inscribed" into the structure of matter.<br />
The reader may have noticed that in the derivation of microscopic<br />
irreversibility we have been concentrating on classical<br />
dynamics. However, the ideas of correlations and the distinction<br />
between pre- and postcollisional correlations apply to<br />
quantum systems as well. The study of quantum systems is<br />
more complicated than the study of classical systems. There is<br />
a reason for this: the difference between classical and quantum<br />
mechanics. Even small classical systems, such as those<br />
formed by a few hard spheres, may present intrinsic irreversibility.<br />
However, to reach irreversibility in quantum systems we<br />
need large systems, such as those realized in liquids, gases, or<br />
in field theory. The study of large systems is obviously more<br />
difficult mathematically, and that is why we will not go into the<br />
matter here. However, the situation remains essentially the<br />
same in quantum theory. There also irreversibility begins as<br />
the result of the limitation of the concept of wave function due<br />
to a form of quantum instability.<br />
Moreover, the idea of collisions and correlations may also be<br />
used in quantum theory. Therefore, as in classical theory, the<br />
second law prohibits long-range precollisional correlations.<br />
The transition to a probability process introduces new entities,<br />
and it is in terms of these new entities that the second<br />
law can be understood as an evolution from order to disorder.<br />
This is an important conclusion. The second law leads to a<br />
new concept of matter. We would like to describe this concept<br />
now.<br />
Active Matter<br />
Once we associate entropy with a dynamic system, we come<br />
back to Boltzmann's conception: the probability will be maximum<br />
at equilibrium. The units we use to describe thermo<br />
dynamic evolution will therefore behave in a chaotic way at<br />
equilibrium. In contrast , in near-equilibrium conditions correlations<br />
and coherence will appear.<br />
We come to one of our main conclusions; At all levels, be it
287 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
the level of macroscopic physics, the level of fluctuations, or<br />
the microscopic level, nonequilibrium is the source of order.<br />
Nonequilibrium brings "order out of chaos." But as we already<br />
mentioned, the concept of order (or disorder) is more<br />
complex than was thought. It is only in some limiting situations,<br />
such as with dilute gases, that it acquires a simple meaning<br />
in agreement with Boltzmann's pioneering work.<br />
Let us once more contrast the dynamic description of the<br />
physical world in terms of forces and fields with the thermo<br />
dynamic description. As we mentioned, we can construct<br />
computer experiments in which interacting particles initially<br />
distributed at random form a lattice. The dynamic interpretation<br />
would be the appearance of order through interparticle<br />
forces. The thermodynamic interpretation is, on the contrary,<br />
the approach to disorder (when the system is isolated), but to<br />
disorder expressed in quite different units, which are in this<br />
case collective modes involving a large number of particles. It<br />
seems to us worthwhile to reintroduce the neologism we used<br />
in Chapter VI to define the new units in terms of which the<br />
system is incoherent at equilibrium: we call them "hypnons,"<br />
sleepwalkers, since they ignore each other at equilibrium.<br />
Each of them may be as complex as you wish (think about<br />
molecules of the complexity of an enzyme), but at equilibrium<br />
their complexity is turned "inward." Again, inside a molecule<br />
there is an intensive electric field, but this field in a dilute gas<br />
is negligible as far as other molecules are concerned.<br />
One of the min subjects in present-day physics is the problem<br />
of elementary particles. However, we know that elementary<br />
particles are far from elementary. New layers of structure<br />
are disclosed at higher and higher energies. But what, after all,<br />
is an elementary particle? Is the planet earth an elementary<br />
particle? Certainly not, because part of this energy is in its<br />
interaction with the sun, the moon, and the other planets. The<br />
concept of elementary particles requires an "autonomy" that<br />
is very difficult to describe in terms of the usual concepts.<br />
Thke the case of electrons and photons. We are faced with a<br />
dilemma: either there are no well-defined particles (because<br />
the energy is partly between the electrons and protons), or<br />
there are noninteracting particles if we can eliminate the interaction.<br />
Even if we knew how to do that, it seems too radical a
ORDER OUT OF CHAOS 288<br />
procedure. Electrons absorb photons or emit photons. A way<br />
out may be to go to the physics of processes. The units, the<br />
elementary particles, would then be defined as hypnons, as<br />
the entities that evolve independently at equilibrium. We hope<br />
that there soon will be experiments available to test this hypothesis;<br />
it would be quite appealing if atoms interacting with<br />
photons (or unstable elementary particles) already carried the<br />
arrow of time that expresses the global evolution of nature.<br />
A subject widely discussed today is the problem of cosmic<br />
evolution. How could the world near the moment of the Big<br />
Bang be so "ordered"? Yet this order is necessary if we wish<br />
to understand cosmic evolution as the gradual movement from<br />
order to disorder.<br />
To give a satisfactory answer we need to know what "hypnons"<br />
could have been appropriate to the extreme conditions<br />
of temperature and density that characterized the early universe.<br />
Thermodynamics alone will not, of course, solve these<br />
problems; neither will dynamics, even in its most refined form<br />
field theory. That is why the unification of dynamics and thermodynamics<br />
opens new perspectives.<br />
In any case, it is striking how the situation has changed since<br />
the formulation of the second law of thermodynamics one hundred<br />
fifty years ago. At first it seemed that the atomistic view<br />
contradicted the concept of entropy. Boltzmann attempted to<br />
save the mechanistic world view at the cost of reducing the<br />
second law to a statement of probability with great practical<br />
importance but no fundamental significance. We do not know<br />
what the definitive solution will be; but today the situation is<br />
radically different. Matter is not given. In the present-day view<br />
it has to be constructed out of a more fundamental concept in<br />
terms of quantum fields. In this construction of matter, thermodynamic<br />
concepts (irreversibility, entropy) have a role to<br />
play.<br />
Let us summarize what has been achieved here. The central<br />
role of the second law (and of the correlative concept of irreversibility)<br />
at the level of macroscopic systems has already<br />
been emphasized in Books One and 1\vo.<br />
What we have tried to show in Book Three is that we now<br />
can go beyond the macroscopic level, and discover the microscopic<br />
meaning of irreversibility.<br />
However, this requires basic changes in the way in which we
289 IRREVERSIBILITY-THE ENTROPY BARRIER<br />
conceive the fundamental laws of physics. It is only when the<br />
classical point of view is lost-as it is in the case of sufficiently<br />
unstable systems-that we can speak of 'intrinsic ra ndomness'<br />
and 'intrinsic irreversibility. '<br />
It is for such systems that we may introduce a new enlarged<br />
description of time in terms of the operator time T. As we have<br />
shown in the example of the Baker transformation (Chapter IX<br />
"From randomness to irreversibility"), this operator has as<br />
eigenfunctions partitions of the phase space (see Figure 39).<br />
We come therefore to a situation quite reminiscent of that of<br />
quantum mechanics. We have indeed two possible descriptions.<br />
Either we give ourselves a point in phase space, and<br />
then we don't know to which partition it belongs, and therefore<br />
we don't know its internal age ; or we know its internal<br />
age, but then we know only the partition, but not the exact<br />
localization of the point.<br />
Once we have introduced the internal time T, we can use<br />
entropy as a selection principle to go from the initial description<br />
in terms of the distribution function p to a new one, p<br />
where the distribution p has an intrinsic arrow of time, satisfying<br />
the second law of thermodynamics. The basic difference<br />
between p and p appears when these functions are expanded in<br />
terms of the eigenfunction of the operator time T (see Chapter<br />
VII, "The rise of quantum mechanics"). In p, all internal ages,<br />
be they from past or future, appear symmetrically. In contrast,<br />
in p, past and future play different roles. The past is included,<br />
but the future remains uncertain. That is the meaning of the<br />
arrow of time. The fascinating aspect is that there appears now<br />
a relation betwen initial conditions and the laws of change. A<br />
state with an arrow of time emerges from a law, which has also<br />
an arrow of time, and which transforms the state, however<br />
keeping this arrow of time.<br />
We have concentrated mostly on the classical situation.2o<br />
However, our analysis applies as well to quantum mechanics,<br />
where the situation is more complicated, as the existence of<br />
Planck's constant destroys already the concept of a trajectory,<br />
and leads therefore also to a kind of delocalization in phase<br />
space. In quantum mechanics we have therefore to superpose<br />
the quantum delocalization with de localization due to irreversibility.<br />
As emphasized in Chapter VII, the two great revolutions in
ORDER OUT OF CHAOS<br />
290<br />
the physics of our century correspond to the incorporation, in<br />
the fundamental structure of physics, of impossibilities foreign<br />
to classical mechanics: the impossibility of signals propagating<br />
with a velocity larger than the velocity of light , and the impossibility<br />
of measuring simultaneously coordinates and momentum.<br />
It is not astonishing that the second principle, which as well<br />
limits our ability to manipulate matter, also leads to deep<br />
changes in the structure of the basic laws of physics.<br />
Let us conclude this part of our monograph wit h a word of<br />
caution. The phenomenological theory of irreversible processes<br />
is at present well established. In contrast , the basic microscopic<br />
theory of irreversible processes is quite new. At the<br />
time of correcting the proofs of this book, experiments are in<br />
preparation to test these views. As long as they have not been<br />
performed, a speculative element is unavoidable.
CONCLUSION<br />
FROM EARTH TO HEAVEN<br />
THE REENCHANTMENT<br />
OF NATURE<br />
In any attempt to bridge the domains of experience belonging<br />
to the spiritual and physical s1des of our nature, time occupies<br />
the key position.<br />
A. S. EDDINGTON1<br />
An Open Science<br />
Science certainly involves manipulating nature, but it is also<br />
an attempt to understand it, to dig deeper into questions that<br />
have been asked generation after generation. One of these questions<br />
runs like a leitmotiv, almost as an obsession, through this<br />
book, as it does through the history of science and philosophy.<br />
This is the question of the relation between being and becoming,<br />
between permanence and change.<br />
We have mentioned pre-Socratic speculations: Is change,<br />
whereby things are born and die, imposed from the outside on<br />
some kind of inert matter? Or is it the result of the intrinsic and<br />
independent activity of matter? Is an external driving force<br />
necessary, or is becoming inherent in matter? Seventeenthcentury<br />
science arose in opposition to the biological model of<br />
a spontaneous and autonomous <strong>org</strong>anization of natural beings.<br />
But it was confronted with another fundamental alternative. Is<br />
nature intrinsically random? Is ordered behavior merely the<br />
transient result of the chance collisions of atoms and of their<br />
unstable associations?<br />
One of the main sources of fascination in modern science<br />
was precisely the feeling that it had discovered eternal laws at<br />
291
ORDER OUT OF CHAOS 292<br />
the core of nature's transformations and thus had exorcised<br />
time and becoming. This discovery of an order in nature produced<br />
the fe eling of intellectual security described by French<br />
sociologist Levy-Bruhl:<br />
Our feeling of intellectual security is so deeply anchored<br />
in us that we even do not see how it could be shaken.<br />
Even if we suppose that we could observe some phenomenon<br />
seemingly quite mysterious, we still would remain<br />
persuaded that our ignorance is only provisional, that this<br />
phenomenon must satisfy the general laws of causality,<br />
and that the reasons for which it has appeared will be<br />
determined sooner or later. Nature around us is order<br />
and reason, exactly as is the human mind. Our everyday<br />
activity implies a perfect confidence in the universality of<br />
the laws of nature.2<br />
This feeling of confidence in the "reason" of nature has<br />
been shattered, partly as the result of the tumultuous growth<br />
of science in our time. As we stated in the Preface, our vision<br />
of nature is undergoing a radical change toward the multiple,<br />
the temporal, and the complex. Some of these changes have<br />
been described in this book.<br />
We were seeking general, all-embracing schemes that could<br />
be expressed in terms of eternal laws, but we have found time,<br />
events, evolving particles. We were also searching for symmetry,<br />
and here also we were surprised, since we discovered<br />
symmetry-breaking processes on all levels, from elementary<br />
particles up to biology and ecology. We have described in this<br />
book the clash between dynamics, with the temporal symmetry<br />
it implies, and the second law of thermodynamics, with its<br />
directed time.<br />
A new unity is emerging: irreversibility is a source of order<br />
at all levels. Irreversibility is the mechanism that brings order<br />
out of chaos. How could such a radical transformation of our<br />
views on nature occur in the relatively short time span of the<br />
past few decades? We believe that it shows the important role<br />
intellectual construction plays in our concept of reality. This<br />
was very well expressed by Bohr, when he said to Werner Heisenberg<br />
on the occasion of a visit at Kronberg Castle:
293<br />
. FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
Isn't it strange how this castle changes as soon as one<br />
imagines that Hamlet lived here? As scientists we believe<br />
that a castle consists only of stones, and admire the way<br />
the architect put them together. The stones, the green<br />
roof with its patina, the wood carvings in the church, constitute<br />
the whole castle. None of this should be changed by<br />
the fact that Hamlet lived here, and yet it is changed completely.<br />
Suddenly the walls and the ramparts speak a different<br />
language . ... Yet all we really know about Hamlet<br />
is that his name appears in a thirteenth-century chronicle<br />
. ... But everyone knows the questions Shakespeare<br />
had him ask, the human depths he was made to reveal,<br />
and so he too had to have a place on earth, here in Kronberg}<br />
The question of the meaning of reality was the central subject<br />
of a fascinating dialogue between Einstein and Tagore."'<br />
Einstein emphasized that science had to be independent of the<br />
existence of any observer. This led him to deny the reality of<br />
time as irreversibility, as evolution. On the contrary, Thgore<br />
maintained that even if absolute truth could exist, it would be<br />
inaccessible to the human mind. Curiously enough, the present<br />
evolution of science is running in the direction stated by<br />
the great Indian poet. Whatever we call reality, it is revealed to<br />
us only through the active construction in which we participate.<br />
As it is concisely expressed by D. S. Kothari, "The simple<br />
fact is that no measurement, no experiment or observation<br />
is possible without a relevant theoretical framework. "5<br />
Time and Times<br />
The statement that time is basically a geometric parameter<br />
that makes it possible to follow the unfolding of the succession<br />
of dynamical states has been asserted in physics for more than<br />
three centuries. Emile Meyerson6 tried to describe the history<br />
of modern science as the gradual implementation of what he<br />
regarded as a basic category of human reason: the different<br />
and the changing had to be reduced to the identical and the<br />
permanent. Time had to be eliminated.
ORDER OUT OF CHAOS<br />
294<br />
Nearer to our own time, Einstein appears as the incarnation<br />
of this drive toward a formulation of physics in which no reference<br />
to irreversibility would be made on the fundamental level.<br />
An historic scene took place at the Societe de Philosophie in<br />
Paris on April 6, 1922,1 when Henri Bergson attempted to defend<br />
the cause of the multiplicity of coexisting "lived" times<br />
against Einstein. Einstein's reply was absolute: he categorically<br />
rejected "philosophers' time." Lived experience cannot<br />
save what has been denied by science.<br />
Einstein's reaction was somewhat justified. Bergson had obviously<br />
misunderstood Einstein's theory of relativity. However,<br />
there also was some prejudice on Einstein's part: duree,<br />
Bergson's "lived time," refers to the basic dimensions of becoming,<br />
the irreversibility that Einstein was willing to admit<br />
only at the phenomenological level. We have already referred<br />
to Einstein's conversations with Carnap (see Chapter VII).<br />
For him distinctions among past, present, and future were outside<br />
the scope of physics.<br />
It is fascinating to follow the correspondence between Einstein<br />
and the closest friend of his young days in Zurich, Michele<br />
Besso. 8 Although he was an engineer and scientist, toward the<br />
end of his life Besso became increasingly concerned with philosophy,<br />
literature, and the problems that surround the core of<br />
human existence. Untiringly he kept asking the same questions:<br />
What is irreversibility? What is its relationship with the<br />
laws of physics? And untiringly Einstein would answer with a<br />
patience he showed only to this closest friend: irreversibility is<br />
merely an illusion produced by "improbable" initial conditions.<br />
This dialogue continued over many years until Besso, older<br />
than Einstein by eight years, passed away, only a few months<br />
before Einstein's death. In a last letter to Besso's sister and<br />
son, Einstein wrote: "Michele has left this strange world just<br />
before me. This is of no importance. For us convinced physicists<br />
the distinction between past, present and future is an illusion,<br />
although a persistent one." In Einstein's drive to perceive<br />
the basic laws of physics, the intelligible was identified with<br />
the immutable.<br />
Why was Einstein so strongly opposed to the introduction<br />
of irreversibility into physics? We can only guess. Einstein<br />
was a rather lonely man; he had few friends, few coworkers,
295<br />
FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
few students. He lived in a sad time: the two World Wars , the<br />
rise of anti-Semitism. It is not surprising that for Einstein science<br />
was the road that led to victory over the turmoil of time.<br />
What a contrast, however, with his scientific work. His world<br />
was full of observers, of scientists situated in various coordinate<br />
systems in motion with one another, situated on various<br />
stars differing by their gravitational fields. All these observers<br />
were exchanging information through signals all over the universe.<br />
What Einstein wanted to preserve above all was the objective<br />
meaning of this communication. However, we can<br />
perhaps state that Einstein stopped short of accepting that<br />
communication and irreversibility are closely related. Communication<br />
is at the base of what probably is the· most irreversible<br />
process accessible to the human mind, the progressive<br />
increase of knowledge.<br />
The Entropy Barrier<br />
In Chapter IX we described the second law as a selection principle:<br />
to each initial condition there corresponds an "information."<br />
All initial conditions for which this information is finite<br />
are permitted. However, to reverse the direction of time we<br />
would need infinite information; we cannot produce situations<br />
that would evolve into our past ! This is the entropy barrier we<br />
have introduced.<br />
There is an interesting analogy with the concept of the velocity<br />
of light as the maximum velocity of transmission of signals.<br />
As we have seen in Chapter VII, this is one of the basic<br />
postulates of Einstein's relativity theory. The existence of the<br />
velocity of light barrier is necessary to give meaning to causality.<br />
Suppose we could, in a science-fiction ship, leave the earth at<br />
a velocity greater than the velocity of light. We could overtake<br />
light signals and in this way precede our own past. Likewise,<br />
the entropy barrier is necessary to give meaning to communication.<br />
We have already mentioned that irreversibility and<br />
communication are closely related. Norbert Wiener has argued<br />
that the existence of two directions of time would have<br />
disastrous consequences. It is worthwhile to cite a passage<br />
from his well-known book Cybernetics:
ORDER OUT OF CHAOS 296<br />
Indeed, it is a very interesting intellectual experiment<br />
to make the fantasy of an intelligent being whose time<br />
should run the other way to our own. To such a being, all<br />
communication with us would be impossible. Any signal<br />
he might send would reach us with a logical stream of<br />
consequents from his point of view, antecedents from<br />
ours. These antecedents would already be in our experience,<br />
and would have served to us as the natural explanation<br />
of his signal, without presupposing an intelligent<br />
being to have sent it. If he drew us a square, we should<br />
see the remains of his figure as its precursors, and it would<br />
seem to be the curious crystallization-always perfectly<br />
explainable-of these remains. Its meaning would seem<br />
to be as fortuitous as the faces we read into mountains<br />
and cliffs. The drawing of the square would appear to us<br />
as a catastrophe-sudden indeed, but explainable by natural<br />
laws-by which that square wo_Id cease to exist.<br />
Our counterpart would have exactly similar ideas concerning<br />
us. Within any world with which we can communicate,<br />
the direction of time is uniform.9<br />
It is precisely the infinite entropy barrier that guarantees the<br />
uniqueness of the direction of time, the impossibility of switching<br />
from one direction of time to the opposite one.<br />
In the course of this book, we have stressed the importance<br />
of demonstrations of impossibility. In fact, Einstein was the<br />
first to grasp that importance when he based his concept of relative<br />
simultaneity on the impossibility of transmitting information<br />
at a velocity greater than that of light. The whole theory of<br />
relativity is built around the exclusion of "unobservable" simultaneities.<br />
Einstein considered this step as somewhat similar to<br />
the step taken in thermodynamics when perpetual motion was<br />
excluded. But some of his contemporaries-Heisenberg, for<br />
example-emphasized an important difference between these<br />
two impossibilities. In the case of thermodynamics, a certain<br />
situation is defined as being absent from nature; in the case of<br />
relativity, it is an observation that is defined as impossiblethat<br />
is, a type of dialogue, of communication between nature<br />
and the person who describes it. Thus Heisenberg saw himself<br />
as following Einstein's example, in spite of Einstein's skepticism,<br />
when he grounded quantum mechanics on the ex.clusion
297<br />
FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
of what the quantum uncertainty principle defines as unobservable.<br />
As long as the second law was considered to express only a<br />
practical improbability, it had little theoretical interest. You<br />
could always hope to overcome it by sufficient technical skill.<br />
But we have seen that this is not so. At its roots there is a<br />
selection of possible initial states. It is only after these states<br />
have been selected that the probability interpretation becomes<br />
possible. Indeed, as Boltzmann stated for the first time, the<br />
increase of entropy expresses the increase of probability, of<br />
disorder. However, his interpretation results from the conclusion<br />
that entropy is a selection principle breaking the time<br />
symmetry. It is only after this symmetry-breaking that any<br />
probabilistic interpretation becomes possible.<br />
In spite of the fact that we have recouped much of Boltzmann's<br />
interpretation of entropy, the basis of our interpretation<br />
of his second law is radically different, since we have in<br />
succession<br />
the second law as a symmetry-breaking selection principle<br />
<br />
probabilistic interpretation<br />
<br />
irreversibility as increase of disorder<br />
It is only the unification of dynamics and thermodynamics<br />
through the introduction of a new selection principle that gives<br />
the second law its fundamental importance as the evolutionary<br />
paradigm of the sciences. This point is of such importance that<br />
we shall dwell on it in more detail.<br />
The Evolutionary Paradigm<br />
f<br />
The world of dynamics, be it classical or quantum, is a reversible<br />
world. As we have emphasized in Chapter VIII, no<br />
evolution can be ascribed to this world; the "information" expressed<br />
in terms of dynamical units remains constant. It is<br />
therefore of great importance that the existence of an evolutionary<br />
paradigm can now be established in physics-not only
ORDER OUT OF CHAOS 298<br />
on the level of macroscopic description but also on all levels.<br />
Of course, there are conditions: as we have seen, a minimum<br />
complexity is necessary. But the immense importance of irreversible<br />
processes shows that this requirement is satisfied for<br />
most systems of interest. Remarkably, the perception of oriented<br />
time increases as the level of biological <strong>org</strong>anization increases<br />
and probably reaches its culminating point in human<br />
consciousness.<br />
How general is this evolutionary paradigm? It includes isolated<br />
systems that evolve to disorder and open systems that<br />
evolve to higher and higher forms of complexity. It is not surprising<br />
that the entropy metaphor has tempted a number of<br />
writers dealing with social or economic problems. Obviously<br />
here we have to be careful; human beings are not dynamic<br />
objects, and the transition to thermodynamics cannot be formulated<br />
as a selection principle maintained by dynamics. On<br />
the human level irreversibility is a more fundamental concept,<br />
which is for us inseparable from the meaning of our existence.<br />
Still it is essential that in this perspective we no longer see the<br />
internal feeling of irreversibility as a subjective impression that<br />
alienates us from the outside world, but as marking our participation<br />
in a world dominated by an evolutionary paradigm.<br />
Cosmological problems are notoriously difficult. We still do<br />
not know what the role of gravitation was in the early universe.<br />
Can gravitation be included in some form of the second law, or<br />
is there a kind of dialectical balance between thermodynamics<br />
and gravitation? Certainly irreversibility could not have appeared<br />
abruptly in a time-reversible world. The origin of irreversibility<br />
is a cosmological problem and would require an<br />
analysis of the universe in its first stages. Here our aim is more<br />
modest. What does irreversibility mean today? How does it<br />
relate to our position in the world we describe?<br />
Actors and Spectators<br />
The denial of becoming by physics created deep rifts within<br />
science and estranged science from philosophy. What had<br />
originally been a daring wager with the dominant Aristotelian
299<br />
FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
tradition gradually became a dogmatic assertion directed<br />
against all those (chemists, biologists, physicians) for whom a<br />
qualitative diversity existed in nature. At the end of the nineteenth<br />
century this conflict had shifted from inside science to<br />
the relation between "science" and the rest of culture, especially<br />
philosophy. We have described in Chapter III this aspect<br />
of the history of Western thought, with its continual<br />
struggle to achieve a new unity of knowledge. The "lived<br />
time" of the phenomenologists, the Lebenswelt opposed to<br />
the objective world of science, may be related to the need to<br />
erect bulwarks against the invasion of science.<br />
Today we believe that the epoch of certainties and absolute<br />
oppositions is over. Physicists have no privilege whatsoever to<br />
any kind of extraterritoriality. As scientists they belong to<br />
their culture, to which, in their turn, they make an essential<br />
contribution. We have reached a situation close to the one recognized<br />
long ago in sociology: Merleau-Ponty had already<br />
stressed the need to keep in mind what he termed a "truth<br />
within situations. "<br />
So long as I keep before me the ideal of an absolute observer,<br />
of knowledge in the absence of any viewpoint, I<br />
can only see my situation as being a source of error. But<br />
once I have acknowledged that through it I am geared to<br />
all actions and all knowledge that are meaningful to me,<br />
and that it is gradually filled with everything that may be<br />
for me, then my contact with the social in the finitude of<br />
my situation is revealed to me as the starting point of all<br />
truth, including that of science and, since we have some<br />
idea of the truth, since we are inside truth and cannot get<br />
outside it, all that I can do is define a truth within the<br />
situation. 10<br />
It is this conception of knowledge as both objective and participatory<br />
that we have explored through this book.<br />
In his Themesll Merleau-Ponty also asserted that the "philosophic"<br />
discoveries of science, its basic conceptual transformations,<br />
are often the result of negative discoveries. which<br />
provide the occasion and the starting point for a reversal of<br />
point of view. Demonstrations of impossibility, whether in rel-
ORDER OUT OF CHAOS<br />
300<br />
ativity, quantum mechanics, or thermodynamics, have shown<br />
us that nature cannot be described "from the outside," as if by<br />
a spectator. Description is dialogue, communication, and this<br />
communication is subject to constraints that demonstrate that<br />
we are macroscopic beings embedded in the physical world.<br />
We may summarize the situation as we see it today in the<br />
following diagram:<br />
observer ---+<br />
t<br />
dissipative structures<br />
t<br />
dynamics<br />
irreversibility +- randomness +- unstable dynamic systems<br />
We start from the observer, who measures coordinates and<br />
momenta and studies their change in time. This leads him to<br />
the discovery of unstable dynamic systems and other concepts<br />
of intrinsic randomness and intrinsic irreversibility, as we discussed<br />
them in Chapter IX. Once we have intrinsic irreversibility<br />
and entropy, we come in far-from-equilibrium systems<br />
to dissipative structures, and we can understand the timeoriented<br />
activity of the observer.<br />
There is no scientific activity that is not time-oriented. The<br />
preparation of an experiment calls for a distinction between<br />
"before.. and "after... It is only because we are aware of irreversibility<br />
that we can recognize reversible motion. Our diagram<br />
shows that we have now come full circle, that we can see<br />
ourselves as part of the universe we describe.<br />
The scheme we have presented is not an a priori schemededucible<br />
from some logical structure. There is, indeed, no<br />
logical necessity for dissipative structures actually to exist in<br />
nature; the "cosmological fact" of a universe far from equilib<br />
rium is needed for the macroscopic world to be a world inhabited<br />
by "observers"-that is, to be a living world. Our scheme thus<br />
does not correspond to a logical or epistemological truth but<br />
refers to our condition as macroscopic beings in a world far<br />
from equilibrium. Moreover, an essential characteristic of our<br />
scheme is that it does not suppose any fundamental mode of<br />
description; each level of description is implied by another and<br />
implies the other. We need a multiplicity of levels that are all<br />
connected, none of which may have a claim to preeminence.<br />
I
301 FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
We have already emphasized that irreversibility is not a universal<br />
phenomenon. We can perform experiments corresponding<br />
to thermodynamic equilibrium in limited portions of space.<br />
Moreover, the importance of time scales varies. A stone<br />
evolves according to the time scale of geological evolution;<br />
human societies, especially in our time, obviously have a<br />
much shorter time scale. We have already mentioned that irreversibility<br />
starts with a minimum complexity of the dynamic<br />
systems involved. It is interesting that with an increase of<br />
complexity, going from the stone to human societies, the role<br />
of the arrow of time, of evolutionary rhythms, increases. Molecular<br />
biology shows that everything in a cell is not alive in<br />
the same way. Some processes reach equilibrium, others are<br />
dominated by regulatory enzymes far from equilibrium. Similarly,<br />
the arrow of time plays quite different roles in the universe<br />
around us. From this point of view, in the sense of this<br />
time-oriented activity, the human condition seems unique. It<br />
seems to us, as we said in Chapter IX, quite important that<br />
irreversibility, the arrow of time, implies randomness. "Time<br />
is construction." This conclusion, one that Valery 12 reached<br />
quite independently, carries a message that goes beyond science<br />
proper.<br />
A Whirlwind in a Turbulent Nature<br />
In our society, with its wide spectrum of cognitive techniques,<br />
science occupies a peculiar position, that of a poetical interrogation<br />
of nature, in the etymological sense that the poet is a<br />
"maker"-active, manipulating, and exploring. Moreover, science<br />
is now capable of respecting the nature it investigates.<br />
Out of the dialogue with nature initiated by classical science,<br />
with its view of nature as an automaton, has grown a quite<br />
different view in which the activity of questioning nature is<br />
part of its intrinsic activity.<br />
As we have written at the start of this chapter, our feeling of<br />
intellectual security has been shattered. We can now appreciate<br />
in a nonpolemical fashion the relation between science and<br />
philosophy. We have already mentioned the Einstein-Bergson<br />
conflict. Bergson was certainly "wrong" on some technical<br />
points, but his task as a philosopher was to attempt to make
ORDER OUT OF CHAOS 302<br />
explicit inside physics the aspects of time he thought science<br />
was neglecting.<br />
Exploring the implications and the coherence of those fundamental<br />
concepts, which appear both scientific and philosophical,<br />
may be risky, but it can be very fruitful in the dialogue<br />
between science and philosophy. Let us illustrate this with<br />
some brief references to Leibniz, Peirce, Whitehead, and Lucretius.<br />
Leibniz introduced the strange concept of monads, noncommunicating<br />
physical entities that have "no windows through<br />
which something can get in or out. " His views have often been<br />
dismissed as mad, and still, as we have seen in Chapter 11, it is<br />
an essential property of all integrable systems that there exist<br />
a transformation that may be described in terms of noninteracting<br />
entities. These entities translate their own initial<br />
state throughout their motion, but at the same time, like monads,<br />
they coexist with all the others in a "preestablished" harmony:<br />
in this representation, the state of each entity, although<br />
perfectly self-determined, reflects the state of the whole system<br />
down to the smallest detail.<br />
All integrable systems thus can be viewed as "monadic" systems.<br />
Conversely, Leibnizian monadology can be translated into<br />
dynamic language: the universe is an integrable system.13<br />
Monadology thus becomes the most consequential formulation<br />
of a universe from which all becoming is eliminated. By<br />
considering Leibniz's efforts to understand the activity of matter,<br />
we can measure the gap that separates the seventeenth<br />
century from our time. The tools were not yet ready; it was<br />
impossible, on the basis of a purely mechanical universe, for<br />
Leibniz to give an account of the activity of matter. Still some<br />
of his ideas, that substance is activity, that the universe is an<br />
interrelated unit, remain with us and are today taking on a new<br />
form.<br />
We regret that we cannot devote sufficient space to the work<br />
of Charles S. Peirce. At least let us cite one remarkable passage:<br />
You have all heard of the dissipation of energy. It is found<br />
that in all transformations of energy a part is converted<br />
into heat and heat is always tending to equalize its tem-
303 FROM EARTH TO HEAVEN- THE REENCHANTMENT OF NATURE<br />
perature. The consequence is that the energy of the universe<br />
is tending by virtue of its necessary laws toward a<br />
death of the universe in which there shall be no force but<br />
heat and the temperature everywhere the same . . . .<br />
But although no force can counteract this tendency,<br />
chance may and will have the opposite influence. Force is<br />
in the long run dissipative; chance is in the long run concentrative.<br />
The dissipation of energy by the regular laws<br />
of nature is by these very laws accompanied by circumstances<br />
more and more favorable to its reconcentration<br />
by chance. There must therefore be a point at which the<br />
two tendencies are balanced and that is no doubt the actual<br />
condition of the whole universe at the present time. J4<br />
Once again, Peirce's metaphysics was considered as one more<br />
example of a philosophy alienated from reality. But, in fact,<br />
today Peirce's work appears to be a pioneering step toward the<br />
understanding of the pluralism involved in physical laws.<br />
Whitehead's philosophy takes us to the other end of the spectrum.<br />
For him, being is inseparable from becoming. Whitehead<br />
wrote: "The elucidation of the meaning of the sentence 'everything<br />
flows' is one of metaphysics' main tasks." 15 Physics and<br />
metaphysics are indeed coming together today in a conception<br />
of the world in which process, becoming, is taken as a primary<br />
constituent of physical existence and where, unlike Leibniz'<br />
monads, existing entities can interact and therefore also be<br />
born and die.<br />
The ordered world of classical physics, or a monadic theory<br />
of parallel changes, resembles the equally parallel, ordered,<br />
and eternal fall of Lucretius' atoms through infinite space. We<br />
have already mentioned the clinamen and the instability of<br />
laminar flows. But we can go farther. As Serresi6 points out,<br />
the infinite fall provides a model on which to base our conception<br />
of the natural genesis of the disturbance that causes<br />
things to be born. If the vertical fall were not disturbed "without<br />
reason" by the clinamen, which leads to encounters and<br />
associations between uniformly falling atoms, no nature could<br />
be created; all that would be reproduced would be the repetitive<br />
connection between equivalent causes and effects governed<br />
by the laws of fate (foedera fati).
ORDER OUT OF CHAOS 304<br />
Denique si semper motus conectitur omnls<br />
et uetere exoritur [semper] novus ordine certo<br />
nee declinando faciunt primordia motus<br />
principium quoddam quod fatl foedera rumpat,<br />
ex lnfinito ne causa m causa sequatur,<br />
libera per terras unde haec animantibus exstat . . . ?17<br />
Lucretius may be said to have invented the clinamen in the<br />
same way that archaeological remains are "invented": one<br />
"guesses" they are there before one begins to dig. If only uniformly<br />
reversible trajectories existed, where would the irreversible<br />
processes we produce and experience come from?<br />
The point where the trajectories cease to be determined,<br />
where thefoederafati governing the ordered and monotonous<br />
world of deterministic change break down, marks the beginning<br />
of nature. It also marks the beginning of a new science<br />
that describes the birth, proliferation, and death of natural<br />
beings. "The physics of falling, of repetition, of rigorous concatenation<br />
is replaced by the creative science of change and<br />
circumstances. "18 The foedera fati are replaced by the<br />
foedera naturae, which, as Serres emphasizes, denote both<br />
"laws" of nature-local, singular, historical relations-and an<br />
"alliance," a form of contract with nature.<br />
In Lucretian physics we thus again find the link we have<br />
discovered in modern knowledge between the choices underlying<br />
a physical description and a philosophic, ethical, or religious<br />
conception relating to man's situation in nature. The<br />
physics of universal connections is set against another science<br />
that in the name of law and domination no longer struggles<br />
with disturbance or randomness. Classical science from Archime<br />
des to Clausius was opposed to the science of turbulence<br />
and of bifurcating changes.<br />
It is here that Greek wisdom reaches one of its pinnacles.<br />
Where man is in the world, of the world, in matter, of<br />
matter, he is not a stranger, but a friend, a member of the<br />
family, and an equal. He has made a pact with things.<br />
Conversely, many other systems and many other sciences<br />
are based on breaking this pact. Man is a stranger<br />
to the world, to the dawn, to the sky, to things. He hates
305<br />
FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
them, and fights them. His environment is a dangerous<br />
enemy to be fought, to be kept enslaved . . • . Epicurus<br />
and Lucretius live in a reconciled universe. Where the<br />
science of things and the science of man coincide. I am a<br />
disturbance, a whirlwind in turbulent nature. 1<br />
Beyond Tautology<br />
The world of classical science was a world in which the only<br />
events that could occur were those deducible from the instantaneous<br />
state of the system. Curiously, this conception,<br />
which we have traced back to Galileo and Newton, was not<br />
new at that time. Indeed, it can be identified with Aristotle's<br />
conception of a divine and immut a ble heaven. In Aristotle's<br />
opinion, it was only the heavenly world to which we could<br />
hope to apply an exact mathematical description. In the Introduction,<br />
we echoed the complaint that science has "disenchanted"<br />
the world. But this disenchantment is paradoxically<br />
due to the glorification of the earthly world, henceforth<br />
worthy of the kind of intellectual pursuit Aristotle reserved for<br />
heaven. Classical science denied becoming, natural diversity,<br />
both considered by Aristotle as attributes of the sublunar, inferior<br />
world. In this sense, classical science brought heaven to<br />
earth. However, this apparently was not the intention of the<br />
fathers of modern science. In challenging Aristotle's claim that<br />
mathematics ends where nature begins, they did not seek to<br />
discover the immutable concealed behind the changing, but<br />
rather to extend changing, corruptible nature to the boundaries<br />
of the universe. In his Dialogue Concerning the Two<br />
Chief World Systems, Galileo is amazed at the notion that the<br />
world would be a nobler place if the great flood had left only a<br />
sea of ice behind, or if the earth had the incorruptible hardness<br />
of jasper; let those who think the earth would be more<br />
beautiful after being changed into a crystal ball be transformed<br />
by Medusa's stare into a diamond statue!<br />
But the objects chosen by the first physicists to explore the<br />
validity of a quantitative description-that is, the ideal pendulum<br />
with its conservative motion, simple machines, planetary<br />
orbits, etc.-were found to correspond to a unique mathemati-
ORDER OUT OF CHAOS<br />
306<br />
cal description that actually reproduced the divine ideality of<br />
Aristotle's heavenly bodies.<br />
Like Aristotle's gods, the objects of classical dynamics are<br />
concerned only with themselves. They can learn nothing from<br />
the outside. At any instant, each point in the system knows all<br />
it will ever need to know-that is, the distribution of masses in<br />
space and their velocities. Each state contains the whole truth<br />
concerning all possible other states, and each can be used to<br />
predict the others, whatever their respective positions on the<br />
time axis. In this sense, this description leads to a tautology,<br />
since both future and past are contained in the present.<br />
The radical change in the outlook of modern science, the<br />
transition toward the temporal, the multiple, may be viewed as<br />
the reversal of the movement that brought Aristotle's heaven to<br />
earth. Now we are bringing earth to heaven. We are discovering<br />
the primacy of time and change, from the level of elementary<br />
particles to cosmological models.<br />
Both at the macroscopic and microscopic levels, the natural<br />
sciences have thus rid themselves of a conception of objective<br />
reality that implied that novelty and diversity had to be denied<br />
in the name of immutable universal laws. They have rid themselves<br />
of a fascination with a rationality taken as closed and a<br />
knowledge seen as nearly achieved. They are now open to the<br />
unexpected, which they no longer define as the result of imperfect<br />
knowledge or insufficient<br />
·<br />
control.<br />
This opening up of science has been well defined by Serge<br />
Moscovici as the "Keplerian revolution," to distinguish it<br />
from the "Copernican revolution" in which the idea of an absolute<br />
point of view was maintained. In many of the passages<br />
cited in the Introduction to this book, science was likened to a<br />
"disenchantment" of the world. Let us quote Moscovici's description<br />
of the changes going on in the sciences today:<br />
Science has become involved in this adventure, our adventure,<br />
in order to renew everything it touches and<br />
warm all that it penetrates-the earth on which we live<br />
and the truths which enable us to live. At each turn it is<br />
not the echo of a demise, a bell tolling for a passing away<br />
that is heard, but the voice of rebirth and beginning, ever<br />
afresh, of mankind and materiality, fixed for an instant in<br />
their ephemeral permanence. That is why the great dis-
307<br />
FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
coveries are not revealed on a deathbed like that of<br />
Copernicus, but offered like Kepler's on the road of<br />
dreams and passion.2o<br />
The Creative Course of Time<br />
It is often said that without Bach we would not have had the<br />
"St. Matthew Passion" but that relativity would have been discovered<br />
without Einstein. Science is supposed to take a deterministic<br />
course, in contrast with the unpredictability involved<br />
in the history of the arts. When we look back on the strange<br />
history of science, three centuries of which we have tried to<br />
outline, we may doubt the validity of such assertions. There<br />
are striking examples of facts that have been ignored because<br />
the cultural climate was not ready to incorporate them into a<br />
consistent scheme. The discovery of chemical clocks probably<br />
goes back to the nineteenth century, but their result seemed to<br />
contradict the idea of uniform decay to equilibrium. Meteorites<br />
were thrown out of the Vienna museum because there<br />
was no place for them in the description of the solar system.<br />
Our cultural environment plays an active role in the questions<br />
we ask, but beyond matters of style and social acceptance, we<br />
can identify a number of questions to which each generation<br />
returns.<br />
The question of time is certainly one of those questions.<br />
Here we disagree somewhat with Thomas Kuhn's analysis of<br />
the formation of "normal" science.21 Scientific activity best<br />
corresponds to Kuhn's view when it is considered in the context<br />
of the contemporary university, in which research and the<br />
training of future researchers is combined. Kuhn's analysis, if<br />
it is taken as a description of science in general, leading to<br />
conclusions about what knowledge must be, can be reduced to<br />
a new psychosocial version of the positivist conception of scientific<br />
development, namely, the tendency to increasing specialization<br />
and compartmentalization; the identification of<br />
"normal" scientific behavior with.that of the "serious," "silent"<br />
researcher who wastes no time on "general" questions<br />
about the overall significance of his research but sticks to specialized<br />
problems; and the essential independence of scientific<br />
development from cultural, economic, and social problems.
ORDER OUT OF CHAOS<br />
306<br />
The academic structure in which the "normal science" de·<br />
scribed by Kuhn came into being took shape in the nineteenth<br />
century. Kuhn emphasizes that it is by repeating in the form of<br />
exercises solutions to the paradigmatic problems of previous<br />
generations that students learn the concepts upon which research<br />
is based. It is in this way that they are given the criteria<br />
that define a problem as interesting and a solution as acceptable.<br />
The transition from student to researcher takes place<br />
gradually; the scientist continues to solve problems using similar<br />
techniques.<br />
Even in our time, for which Kuhn's description has the<br />
greatest relevance, it refers to only one specific aspect of scientific<br />
activity. The importance of this aspect varies according<br />
to the individual researchers and the institutional context.<br />
In Kuhn's view the transformation of a paradigm appears as<br />
a crisis: instead of remaining a silent, almost invisible rule,<br />
instead of remaining unspoken, the paradigm is actually questioned.<br />
Instead of working in unison, the members of the community<br />
begin to ask "basic" questions and challenge the<br />
legitimacy of their methods. The group, which by training was<br />
homogeneous, now diversifies. Different points of view, cultural<br />
exper.iences, and philosophic convictions are now expressed<br />
and often play a decisive role in the discovery of a new<br />
paradigm. The emergence of the new paradigm further increases<br />
the vehemence of the debate. The rival paradigms are<br />
put to the test until the academic world determines the victor.<br />
With the appearance of a new generation of scientists, silence<br />
and unanimity take over again. New textbooks are written,<br />
and once again things "go without saying."<br />
In this view the driving force behind scientific innovation is<br />
the intensely conservative behavior of scientific communities,<br />
which stubbornly apply to nature the same techniques, the<br />
same concepts, and always end up by encountering an equally<br />
stubborn resistance from nature. When nature is eventually<br />
seen as refusing to express itself in the accepted language, the<br />
crisis explodes with the kind of violence that results from a<br />
breach of confidence. At this stage, all intellectual resources<br />
are concentrated on the search for a new language. Thus scientists<br />
have to deal with crises imposed upon them against<br />
their will.<br />
The questions we have investigated have led us to emphasize
309<br />
FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
aspects that differ considerably from those to which Kuhn's<br />
description applies. We have dwelled on continuities, not the<br />
"obvious" continuities but the hidden ones, those involving<br />
difficult questions rejected by many as illegitimate or false but<br />
that keep coming back generation after generation-questions<br />
such as the dynamics of complex systems, the relation of the<br />
irreversible world of chemistry and biology with the reversible<br />
description provided by classical physics. In fact, the interest<br />
of such questions is hardly surprising. To us, the problem is<br />
rather to understand how they could ever have been neglected<br />
after the work of Diderot, Stahl, Venel, and others.<br />
The past one hundred years have been marked by several<br />
crises that correspond closely to the description given by<br />
Kuhn-none of which were sought by scientists. Examples<br />
are the discovery of the instability of elementary particles, or<br />
of the evolving universe. However, the recent history of science<br />
is also characterized by a series of problems that are the<br />
consequences of deliberate and lucid questions asked by scientists<br />
who knew that the questions had both scientific and<br />
philosophical aspects. Thus scientists are not doomed to behave<br />
like "hypnons"!<br />
It is important to point out that the new scientific development<br />
we have described, the incorporation of irreversibility<br />
into physics, is not to be seen as some kind of "revelation,"<br />
the possession of which would set its possessor apart from the<br />
cultural world he lives in. On the contrary, this development<br />
clearly reflects both the internal logic of science and the<br />
cultural and social context of our time.<br />
In particular, how can we consider as accidental that the<br />
rediscovery of time in physics is occurring at a time of extreme<br />
acceleration in human history? Cultural context cannot be the<br />
complete answer, but it cannot be denied either. We have to<br />
incorporate the complex relations between "internal" and<br />
"external" determinations of the production of scientific concepts.<br />
In the preface of this book, we have emphasized that its<br />
French title (La nouvelle alliance) expresses the coming together<br />
of the "two cultures". Perhaps the confluence is nowhere<br />
as clear as in the problem of the microscopic<br />
foundations of irreversibility we have studied in Book Three.<br />
As mentioned repeatedly, both classical and quantum me-
ORDER OUT OF CHAOS 310<br />
chanics are based on arbitrary initial conditions and deterministic<br />
laws (for trajectories or wave functions). In a sense,<br />
laws made simply explicit what was already present in the initial<br />
conditions. This is no longer the case when irreversibilty is<br />
taken into account. In this perspective, initial conditions arise<br />
from previous evolution and are transformed into states of the<br />
same class through subsequent evolution.<br />
We come therefore close to the central problem of Western<br />
ontology: the relation between Being and Becoming. We have<br />
given a brief account of the problem in Chapter III. It is<br />
remarkable that two of the most influential works of the century<br />
were precisely devoted to this problem. We have in mind<br />
Whitehead's Process and Reality and Heidegger's Sein und<br />
Zeit. In both cases, the aim is to go beyond the identification<br />
of Being with timelessness, following the Voie Royale of western<br />
philosophy since Plato and Aristotle.22<br />
But obviously, we cannot reduce Being to Time, and we<br />
cannot deal with a Being devoid of any temporal connotation.<br />
The direction which the microscopic theory of irreversibility<br />
takes gives a new content to the speculations of Whitehead<br />
and Heidegger.<br />
It would go beyond the aim of this book to develop this problem<br />
in greater detail; we hope to do it elsewhere. Let us notice<br />
that initial conditions, as summarized in a state of the system,<br />
are associated with Being; in contrast, the laws involving temporal<br />
changes are associated with Becoming.<br />
In our view, Being and Becoming are not to be opposed one<br />
to the other: they express two related aspects of reality.<br />
A state with broken time symmetry arises from a law with<br />
broken time symmetry, which propagates it into a state belonging<br />
to the same category.<br />
In a recent monograph (From Being to Becoming), one of<br />
the authors concluded in the following terms: "For most of the<br />
founders of classical science-even for Einstein-science was<br />
an attempt to go beyond the world of appearances, to reach a<br />
timeless world of supreme rationality-the world of Spinoza.<br />
But perhaps there is a more subtle form of reality that involves<br />
both laws and games, time and eternity. "<br />
This is precisely the direction which the microscopic theory<br />
of irreversible processes is taking.
311 FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
The Human Condition<br />
We agree completely with Herman Weyl:<br />
Scientists would be wrong to ignore the fact that theo<br />
retical construction is not the only approach to the phenomena<br />
of life; another way, that of understanding from<br />
within (interpretation), is open to us . . . . Of myself, of<br />
my own acts of perception, thought, volition, feeling and<br />
doing, I have a direct knowledge entirely different from<br />
the theoretical knowledge that represents the "parallel"<br />
cerebral processes. in symbols. This inner awareness of<br />
myself is the basis for the understanding of my fellowmen<br />
whom I meet and acknowledge as beings of my own<br />
kind, with whom I communicate sometimes so intimately<br />
as to share joy and sorrow with them. 23<br />
Until recently, however, there was a striking contrast. The external<br />
universe appeared to be an automaton following deterministic<br />
causal laws, in contrast with the spontaneous activity<br />
and irreversibility we experience. The two worlds are now<br />
drawing closer together. Is this a loss for the natural sciences?<br />
Classical science aimed at a "transparent" view of the physical<br />
universe. In each case you would be able to identify a<br />
cause and an effect. Whenever a stochastic description becomes<br />
necessary, this is no longer so. We can no longer speak<br />
of causality in each individual experiment; we can only speak<br />
about statistical causality. This has, in fact, been the case ever<br />
since the advent of quantum mechanics, but it has been greatly<br />
amplified by recent developments in which randomness and<br />
probability play an essential role, even in classical dynamics<br />
or chemistry. Therefore, the modern trend as compared to the<br />
classical one leads to a kind of "opacity" as compared to the<br />
transparency of classical thought.<br />
Is this a defeat for the human mind? This is a difficult question.<br />
As scientists, we have no choice; we cannot describe for<br />
you the world as we would like to see it, but only as we are<br />
able to see it through the ombined impat of experimental
ORDER OUT OF CHAOS 312<br />
results and new theoretical concepts. Also, we believe that<br />
this new situation reflects the situation we seem to find in our<br />
own mental activity. Classical psychology centered around<br />
conscious, transparent activity; modern psychology attaches<br />
much weight to the opaque functioning of the unconscious.<br />
Perhaps this is an image of the basic features of human existence.<br />
Remember Oedipus, the lucidity of his mind in front of<br />
the sphinx and its opacity and darkness when confronted with<br />
his own origins. Perhaps the coming together of our in sights<br />
about the world around us and the world inside us is a satisfying<br />
feature of the recent evolution in science that we have tried<br />
to describe.<br />
It is hard to avoid the impression that the distinction between<br />
what exists in time, what is irreversible, and, on the<br />
other hand, what is outside of time, what is eternal, is at the<br />
origin of human symbolic activity. Perhaps this is especially so<br />
in artistic activity. Indeed, one aspect of the transformation of<br />
a natural object, a stone, to an object of art is closely related to<br />
our impact on matter. Artistic activity breaks the temporal<br />
symmetry of the object. It leaves a mark that translates our<br />
temporal dissymmetry into the temporal dissymmetry of the<br />
object. Out of the reversible, nearly cyclic noise level in which<br />
we live arises music that is both stochastic and time-oriented.<br />
The Renewal of Nature<br />
It is quite remarkable that we are at a moment both of profound<br />
change in the scientific concept of nature and of the<br />
structure of human society as a result of the demographic explosion.<br />
As a result, there is a need for new relations between<br />
man and nature and between man and man. We can no longer<br />
accept the old a priori distinction between scientific and ethical<br />
values. This was possible at a time when the external world<br />
and our internal world appeared to conflict, to be nearly<br />
orthogonal. Today we know that time is a construction and<br />
therefore carries an ethical responsibility.<br />
The ideas to which we have devoted much space in this<br />
book-the ideas of instability, of fluctuation-diffuse into the<br />
social sciences. We know now that societies are immensely
313 FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE<br />
complex systems involving a potentially enormous number of<br />
bifurcations exemplified by the variety of cultures that have<br />
evolved in the relatively short span of human history. We know<br />
that such systems are highly sensitive to fluctuations. This<br />
leads both to hope and a threat: hope, since even small fluctuations<br />
may grow and change the overall structure. As a result,<br />
individual activity is not doomed to insignificance. On<br />
the other hand, this is also a threat, since in our universe the<br />
security of stable, permanent rules seems gone forever. We are<br />
living in a dangerous and uncertain world that inspires no<br />
blind confidence, but perhaps only the same feeling of<br />
qualified hope that some Talmudic texts appear to have attributed<br />
to the God of Genesis:<br />
1\venty-six attempts preceded the present genesis, all of<br />
which were destined to fail. The world of man has arisen<br />
out of the chaotic heart of the preceding debris; he too is<br />
exposed to the risk of failure, and the return to nothing.<br />
"Let's hope it works" [Halway Sheyaamod] exclaimed<br />
God as he created the World, and this hope, which has<br />
accompanied all the subsequent history of the world and<br />
mankind, has emphasized right from the outset that this<br />
history is branded with the mark of radical uncertainty.24
NOTES<br />
Introduction<br />
I. I. BERLIN, Against the Current, selected writings ed. H. Hardy<br />
(New York: The Viking Press, 1980), p. xxvi.<br />
2. See TITUS LucRETIUS CARUS, De Natura Rerum, Book I, v.<br />
267-70. ed . and comm. C. Bailey (Oxford: Oxford University<br />
Press 1947, 3 vols.)<br />
3. R. LENOBLE, Histoire de /'idee de nature (Paris: Albin Michel,<br />
1969).<br />
4. B. PASCAL, "Pensees," frag. 792, in Oeuvres Completes (Paris:<br />
Brunschwig-Boutroux-Gazier, 1904-14).<br />
5. J. MoNOD, Chance and Necessity (New York: Vintage Books,<br />
1972), pp. 172-73.<br />
6. G. V1co, The New Science, trans. T. G. Bergin and M. H. Fisch<br />
(New York: 1968), par. 331.<br />
7. J. P. VERNANT et al., Divination et rationalite, esp. J. BOTTERO,<br />
"Symptomes, signes, ecritures" (Paris: Seuil, 1974).<br />
8. A. KoYRE , Galileo Studies (Hassocks, Eng.: The Harvester<br />
Press, 1978).<br />
9. K. PoPPER, Objective Knowledge (Oxford: Clarendon Press,<br />
1972).<br />
10. P. FoRMAN, "Weimar Culture, Causality and Quantum Theory,<br />
1918-1927; Adaptation by German Physicists and Mathematicians<br />
to an Hostile Intellectual Environment," Historical Studies<br />
in Physical Sciences, Vol. 3 (1971), pp. 1-1 15.<br />
11. J. NEEDHAM and C. A. RoNAN, A Shorter Science and Civilization<br />
in China, Vol. I (Cambridge: Cambridge University Press,<br />
1978), p. 170.<br />
12. A. EDDINGTON, The Nature of the Physical World (Ann Arbor:<br />
University of Michigan Press, 1958), pp. 68-80.<br />
13. Ibid., p. 103.<br />
14. BERLIN, op. cit., p. 109.<br />
15. K. POPPER, Unended Quest (La Salle, Ill.: Open Court Publishing<br />
Company, 1976), pp. 161-62.<br />
16. G. BRUNO, 5th dialogue, "De Ia causa," Opere ltaliane, I (Bari:<br />
1907); cf. I. LECLERC, The Nature of Physical Existence<br />
(London: Ge<strong>org</strong>e Allen & Unwin, 1972).<br />
315
ORDER OUT OF CHAOS 316<br />
17. P. VA LRY, Cahiers, (2 vols.) ed. Mrs. Robinson-Valery, (Paris:<br />
Gallimard, 1973-74).<br />
18. E. ScHRODINGER, "A re there Quantum Jumps?" The British<br />
Journal fo r the Philosophy of Science, Vol. III (1952), pp. 109-<br />
10; this text has been quoted with indignation by P. W. Bridgmann<br />
in his contribution to Determinism and Freedom in the<br />
Age of Modern Science, ed. S. Hook (New York: New York<br />
University Press, 1958).<br />
19 .. A. EINSTEIN, "Prinzipien der Forschung, Rede zur 60.<br />
Geburstag van Max Planck" (1918) in Mein Weltbild, Ullstein<br />
Verlag 1977, pp. 107-10, trans. Ideas and Opinions (New York:<br />
Crown, 1954), pp. 224-27.<br />
20. R DDRRENMATT, The Physicists. (New York: Grove, 1964).<br />
21. S. MoscoviCI, Essai sur /'histoire humaine de Ia nature, Collection<br />
Champs (Paris: Flammarion, 1977).<br />
22. Quoted in Ronan, op. cit., p. 87.<br />
23. MoNon , op. cit., p. 180.<br />
Chapter 1<br />
1. J. T. DESAGULIERS, "The Newtonian System of the World, The<br />
Best Model of Government: an Allegorical Poem," 1728, quoted<br />
in H. N. FA IRCHILD, Religious Trends in English Poetry, Vol. I<br />
(New York: Columbia University Press, 1939), p. 357.<br />
2. Ibid., p. 358.<br />
3. Gerd Buchdahl emphasized and illustrated this ambiguity of the<br />
cultural influence of the Newtonian model in its dimensions both<br />
empirical (Opticks) and systematic (Principia) in The Image of<br />
Newton and Locke in the Age of Reason, Newman History and<br />
Philosophy of Science Series (London: Sheed & Ward, 1961).<br />
4. La Science et Ia diversite des cultures, (Paris: UNESCO, PUF,<br />
1974), pp. 15-16.<br />
5. C. C. GILLISPIE, The, Edge of Objectivity (Princeton, N.J.:<br />
Princeton University Press, 1970), pp. 199-200.<br />
6. M. HEIDEGGER, The Question Concerning Te chnology (New<br />
York: Harper & Row, 1977), p. 20.<br />
7. Ibid., p. 21.<br />
8. Ibid., p. 16.<br />
9. "The Coming of the Golden Age," Paradoxes of Progress (San<br />
Francisco: Freeman & Company, 1978).<br />
10. See, for instance. P. DAVIES, Other Worlds (Toronto: J. M. Dent<br />
& Sons, 1980).
317 NOTES<br />
11. A. K oESTLER, The Roots of Coincidence (London: Hutchinson,<br />
1972), pp. 138-39.<br />
12. A. KoYRE, Newtonian Studies (Chicago: University of Chicago<br />
Press, 1968), pp. 23-24.<br />
13. In "Race and History" (Structural Anthropology II, New York:<br />
Basic Books, 1976), Claude Levi-Strauss discusses the conditions<br />
that lead to the Neolithic and Industrial revolutions. The<br />
model he introduces, involving chain reactions and catalysis (a<br />
process with kinetics characterized by threshold and amplification<br />
phenomena) attests to an affinity between the problems of<br />
stability and fluctuation we discuss in Chapter VI as well as certain<br />
themes of the "structural approach" in anthropology.<br />
14. "Inside each society, the order of myth excludes dialogue: the<br />
group's myths are not discussed, they are transformed when<br />
they are thought to be repeated." C. LEvi-STRAUSS, L'Homme<br />
Nu (Paris: Pion, 1971), p. 585. Thus mythical discourse is to be<br />
distinguished from critical (scientific and philosophic) dialogue<br />
more because of the practical conditions of its reproduction than<br />
because of an intrinsic inability of such or such emitter to think<br />
in a rational way. The practice of critical dialogue has given to<br />
the cosmological discourse claiming truthfulness its spectacular<br />
evolutive acceleration.<br />
15. This is, of course, one of the main themes of Alexandre Koyre.<br />
16. The definition of such an "absurdity" opposes the age-long idea<br />
that a sufficiently tricky device would permit one to cheat nature.<br />
See the efforts devoted by engineers till the twentieth century<br />
to the construction of perpetual-motion machines in A. Ord<br />
Hume, Perpetual Motion: The History of an Obsession (New<br />
York: St. Martin's Press, 1977).<br />
17. Popper translated into a norm this excitement born out of the<br />
risks involved in the experimental games. He affirms, in The<br />
Logic of Scientific Discovery, that the scientific must look for<br />
the most "improbable" hypothesis-that is, the most risky<br />
one-to try to refute it as well as the corresponding theories.<br />
18. R. FEYNMAN, The Character of Physical Law (Cambridge,<br />
Mass.: M.I.T. Press, 1967), second chapter.<br />
19. J. NEEDHAM, "Science and Society in East and West," The<br />
Grand Titration (London: Allen & Unwin, 1969).<br />
20. A. N. WHITEHEAD, Science and the Modern World (New York:<br />
The Free Press, 1967), p. 12.<br />
21. NEEDHAM, op. cit., p. 308.<br />
22. NEEDHAM, op. cit., p. 330.<br />
23. R. HooYKAAS emphasized this "dedivinization" of the world by<br />
the Christian metaphor of the world machine in Religion and the
ORDER OUT OF CHAOS 318<br />
Rise of Modern Science (Edinburgh and London: Scottish Academic<br />
Press, 1972), esp. pp. 14-16.<br />
24. WHITEHEAD, Op. cit., p. 54.<br />
25. The famous text about nature being written in mathematical<br />
signs is to be fo und in 11 Saggiatore. See also The Dialogue Concerning<br />
the Two Chief World Systems, 2nd rev. ed. (Berkeley:<br />
University of California Press, 1967).<br />
26. At least it was triumphant in the academies created in France,<br />
Prussia, and Russia by absolute sovereigns. In The Scientist's<br />
Role in Society (Englewood Cliffs, N.J.: Foundations of Modern<br />
Sociology Series, Prentice-Hall, 1971), Ben David emphasized<br />
the distinction between physicists of these countries, dedicated<br />
to physics as a glamorous and purely theoretical science, and<br />
the English physicists immerged in a wealth of empirical and<br />
technical problems. Ben David proposed a connection between<br />
the fascination for a theoretical science and the relegation far<br />
from political power of the social class supporting the "scientific<br />
movement. "<br />
27. In his biography of dlembert-Jean d'Alembert, Science and<br />
Enlightenment (Oxford: Clarendon Press, 1970)-Thomas<br />
Hankins emphasized how closed and small was the first true<br />
scientific community, in the modern sense of the term, namely,<br />
that of the eighteenth-century physicists and mathematicians,<br />
and how intimate were their relations with. the absolute sovereigns.<br />
28. EINSTEIN, Op. cit., pp. 225-26.<br />
29. E. MACH, "The Economical Nature of Physical Inquiry, " Popular<br />
Scientific Lectures (Chicago: Open Court Publishing Company,<br />
1895), pp. 197-98.<br />
30. J. DONNE, An Anatomy of the World wherein . .. the frailty and<br />
the decay of the whole world is represented (London, catalog of<br />
the British Museum, 161 1).<br />
Chapter 2<br />
I. On this point, see T. HANKINS, "The Reception of Newton's<br />
Second Law of Motion in the Eighteenth Century, " Archives Internationales<br />
d'Histoire des Sciences, Vol. XX (1967), pp. 42-<br />
65, and I. B. CoHEN, "Newton's Second Law and the Concept<br />
of Force in the Principia," The Annus Mirabilis of Sir Isaac<br />
Newton, Tricentennial Celebration, The Texas Quarterly,<br />
Vol. X, No. 3 (1967), pp. 25-157. The four following paragraphs<br />
rest, for what concerns atomism and the conservation theories,
319 NOTES<br />
on W. Scarr, The Conflict Between Atomism and Conservation<br />
Theory (London: Macdonald, 1970).<br />
2. A. KovRE: , Galileo Studies (Hassocks, Eng.: The Harvester<br />
Press, 1978), pp. 89-94.<br />
3. In his history of mechanics-The Science of Mechanics: A Critical<br />
and Historical Account of Its Development (La Salle, Ill.:<br />
Open Court Publishing Company, 1960)-Ernst Mach laid stress<br />
on this dual filiation of modern dynamics of both the trajectories<br />
science and the engineer's computations.<br />
4. This at least is the conclusion of today's historians who began<br />
the study of the impressive mass of Newton's ·tchemical Papers,"<br />
which till now were ignored or disdained as "nonscientific."<br />
See B. J. DoBBS, The Foundations of Newton's Alchemy<br />
(Cambridge: Cambridge University Press, 1975); R. WESTFALL,<br />
"Newton and the Hermetic Tradition" in Science, Medicine and<br />
Society, ed. A. G. DEBUS (London: Heinemann, 1972); and R.<br />
WESTFALL, "The Role of Alchemy in Newton's Career," Reason,<br />
Experiment and Mysticism, ed. M. L. RIGHINI BONELLI<br />
and W. R. SHEA (London: Macmillan, 1975). As Lord Keynes,<br />
who played a crucial part in the collection of these papers, summarized<br />
(quoted in DoBBS, op. cit., p. 13), "Newton was not the<br />
first of the age of reason. He was the last of the Babylonians and<br />
Sumerians, the last great mind which looked out on the visible<br />
and intellectual world with the same eyes as those who began to<br />
build our intellectual inheritance rather less than 10,000 years<br />
ago."<br />
5. DoBBS, op. cit., also examined the role of the "mediator" by<br />
which two substances are made "sociable." We may recall here<br />
the importance of the mediator in Goethe's Elective Affinities<br />
(Engl. trans. Greenwood 1976). For what concerns chemistry,<br />
Goethe was not far from Newton.<br />
6. The story of Newton's "mistake" is told in HANKINs's, Jean<br />
d'Alembert, pp. 29-35.<br />
7. G. L. BuFFON, "Reflexions sur Ia loi d'attraction," appendix to<br />
Introduction a I' histoire des minhaux (1774), To me IX of<br />
Oeuvres Completes (Paris: Garnier Frere s), pp. 75, 77.<br />
8. G. L. BuFFON, Histoire naturelle. De Ia Nature, Seconde Vue<br />
(1765), quoted in H. METZGER, Newton, Stahl, Boerhaave et Ia<br />
doctrine chimique (Paris: Blanchard, 1974), pp. 57-58.<br />
9. A. THACKRAY describes the way French chemistry became Buffonian<br />
in Atom and Power: An Essay on Newtonian Matter Th e<br />
ory and the Development of Chemistry (Cambridge, Mass.:<br />
Harvard University Press, 1 970). pp. 199-233. Berthollet's<br />
Statique chimique accomplished Buffon's program and also
ORDER OUT OF CHAOS 320<br />
closed it, since his disciples gave up the attempt to understand<br />
chemical reactions in terms compatible with Newtonian con·<br />
cepts.<br />
10. We do not wish to try to explain here the reasons of Newton's<br />
triumph in France, nor of its fall, but to emphasize the at least<br />
chronological connection between these events and the stages of<br />
the process of professionalization of science. See M. CROS·<br />
LAND, The Society of Arcueil: A View of French Science at the<br />
Time of Napoleon (London: Heinemann, 1960), as well as his<br />
Gay Lussac (Cambridge: Cambridge University Press, 1978).<br />
11. Thomas Kuhn made of this role of scientific institutions, taking<br />
over the formation of the future scientists-that is assuring their<br />
own reproduction, the main characteristic of scientific activity<br />
as we know it today. This problem has also been approached by<br />
M. Crosland, R. Hahn, and W. Farrar in The Emergence of Science<br />
in Western Europe, ed. M. CROSLAND (London: Mac·<br />
millan, 1975).<br />
12. The role of "mundane" interest so despised by philosophers<br />
such as Gaston Bachelard in France should be taken as the sign<br />
of the open character of eighteenth-century science. In a way,<br />
we can truly speak about a regression during the nineteenth century,<br />
at least for what concerns the scientific culture. And we<br />
could learn today from the multiplicity of local academies and<br />
circles where scientific matters were discussed by nonprofessionals.<br />
13. Quoted in J. ScHLANGER, Les metaphores de I' <strong>org</strong>anisme<br />
(Paris: Vrin, 1971), p. 108.<br />
14. J. C. MAXWELL, Science and Free Will, in CAMPBELL and GAR·<br />
NETT , op. cit., p. 443. L. CAMPBELL & W. GARNETT, The Life of<br />
James Clerk Maxwell (London, Macmillan, 1882).<br />
15. This problem is one of the main themes of French philosopher<br />
Michel Serres. See, for instance, "Conditions" in his La naissance<br />
de Ia physique dans le texte de Lucrece (Paris: Minuit,<br />
1977). Some texts by M. Serres are now available in English<br />
translation, thanks to the pious zeal of the French Studies Department<br />
of Johns Hopkins University. See M. SERRES,<br />
Hermes: Literature, Science, Philosophy. (Baltimore: The<br />
Johns Hopkins University Press, 1982.)<br />
16. See, about the fate of Laplace's demon, E. CASSIRER, Determinism<br />
and Interdeterminism in Modern Physics (New Haven,<br />
Conn.: Yale University Press, 1956), pp. 3-25.
321 NOTES<br />
Chapter 3<br />
1. R. NISBET, History of the Idea of Progress (New York: Basic<br />
Books, 1980), p. 4.<br />
2. D. DIDEROT, d'Alembert's Dream (Harmondsworth:, Eng.: Penguin<br />
Books, 1976), pp. 166-67.<br />
3. D. DIDEROT, "Conversation Between d'Alembert and Diderot,"<br />
d'Alembert's Dream, pp. 158-59.<br />
4. D. DIDEROT, Pensees sur /'Interpretation de Ia Nature (1754),<br />
Oeuvres Completes, Tome II (Paris: Garnier Freres, 1875), p. II.<br />
5. Diderot ascribes this opinion to the physician Bordeu in the<br />
Dream.<br />
6. See, for instance, A. LovEJOY, The Great Chain of Beings<br />
(Cambridge, Mass.: Harvard University Press, 1973).<br />
7. The historian Gillispie proposed a relation between the protest<br />
against mathematical physics, as popularized by Diderot in the<br />
Encyclopedie, and the revolutionaries' hostility against this official<br />
science, as manifested by the closure of the Academy and<br />
Lavoisier's death. This is a very controversial point, but what is<br />
sure is that the Newtonian triumph in France coincides with the<br />
Napoleonic institutions, spelling the final victory of state academy<br />
over craftsmen (see C. C. GILLISPIE, "The Encyclopedia<br />
and the Jacobin Philosophy of Science: A Study in Ideas and<br />
Consequences," Critical Problems in the History of Science, ed.<br />
M. CLAGETT (Madison, Wis.: University of Wisconsin Press,<br />
1959), pp. 255-89.<br />
8. G. E. STAHL, "Veritable Distinction a etablir entre le mixte et le<br />
vivant du corps humain," Oeuvres medicophilosophiques et<br />
pratiques, To me II (Montpellier: Pitrat et Fils, 1861), esp.<br />
pp. 279-82.<br />
9. See J. SCHLANGER, Les metaphores de /'<strong>org</strong>anisme, for a description<br />
of the transformation of the meaning of "<strong>org</strong>anization"<br />
between Stahl and the Romanticists.<br />
10. Philosophy of Nature, §261.<br />
11. This is Knight's conclusion in "The German Science in the Romantic<br />
Period," The Emergence of Science in Western Europe.<br />
12. H. BERGSON, La pensee et le mouvant in Oeuvres (Paris: E ditions<br />
du Centenaire, PUP, 1970), p. 1285 ; trans. The Creative<br />
Mind (Totowa, N.J.: Littlefield, Adams, 1975), p. 42.<br />
13. Ibid., p. 1287; trans., p. 44.<br />
14. Ibid., p. 1286; trans. , p. 44.
ORDER OUT OF CHAOS 322<br />
15. H. BERGSON, L'evolution creatrice in Oeuvres, p. 784; trans.<br />
Creative Evolution (London: Macmillan, 191 1), p. 361.<br />
16. Ibid., p. 538; trans., p. 54.<br />
17. Ibid., p. 784; trans., p. 361.<br />
18. BERGSON, La pensee et /e mouvant, p. 1273; trans., p. 32.<br />
19. Ibid:, p. 1274; trans., p. 33.<br />
20. A. N. WHITEHEAD, Science and the Modern World, p. 55.<br />
21. A. N. WHITEHEAD, Process and Reality: An Essay in Cosmology<br />
(New York: The Free Press, 1969), p. 20.<br />
22. Ibid., p. 26.<br />
23. Joseph Needham and C. H. Waddington both acknowledged the<br />
importance of Whitehead's influence for what concerns their endeavor<br />
to describe in a positive way the <strong>org</strong>anism as a whole.<br />
24. H. HELMHOLTZ, Uber die Erhaltung der Kraft (1847), trans. in<br />
S. BRUSH, Kinetic Theory, Vol. I, The Nature of Gases and<br />
Heat (Oxford: Pergamon Press, 1965), p. 92. See also Y.<br />
ELKANA, Th e Discovery of the Conservation of Energy<br />
(London: Hutchinson Educational, 1974) and P. M. HEIMANN,<br />
"Helmholtz and Kant: The Metaphysical Foundations of Uber<br />
die Erhaltung der Kraft," Studies in the History and Philosophy<br />
of Sciences, Vol. 5 (1974), pp. 205-38.<br />
25. H. REICHENBACH, The Direction of Time (Berkeley: University<br />
of California Press, 1956), pp. 16-17.<br />
Chapter 4<br />
I. W. ScoTT, About the novelty of these problems, see The Conflict<br />
Between Atomism and Conservation Theory, Book II, and about<br />
the industrial context where these concepts were created, D.<br />
CARDWELL, From Wa tt to Clausius (London, Heinemann,<br />
1971). Particularly interesting in this respect is the convergence<br />
between on one hand the need determined by industrial problems<br />
and on the other the positivist simplifications by operational<br />
definitions.<br />
2. J. HERIVEL, Joseph Fourier: The Man and the Physicist (Oxford:<br />
Clarendon Press, 1975). In this biography we learn the following<br />
curious information: Fourier would have brought back<br />
from his trip with Bonaparte to Egypt a sickness causing permanent<br />
deperditions of heat.<br />
3. See, more particularly, the introduction to Comte's Philosophie<br />
Premiere (Paris: Herman, 1975), ·uguste Comte auto-traduit<br />
dans l'encyclopedie" in La Traduction (Paris: Minuit, 1974) and<br />
"Nuage," La Distribution (Paris: Minuit, 1977).
323 NOTES<br />
4. C. SMITH, "Natural Philosophy and Thermodynamics: William<br />
Thomson and the Dynamical Theory of Heat," The British Journal<br />
for the Philosophy of Science, Vol . 9 (1976), pp. 293-3 19 and<br />
M. CROSLAND and C. SMITH, "The Transmission of Physics<br />
from France to Britain, 1800-1840," Historical Studies in the<br />
Physical Sciences, Vol . 9 (1978), pp. 1-61.<br />
5. For what follows, see Y. ELKANA, The Discovery of the Conservation<br />
of Energy Principle, as well as the famous paper by<br />
Thomas Kuhn, "Energy conservation as an Example of Simultaneous<br />
Discovery," originally published in Critical Problems in<br />
the History of Science and recently in T. KuHN, The Essential<br />
Tension (Chicago: University of Chicago Press, 1977).<br />
6. ELKANA followed the slow crystallization of the concept of energy;<br />
see his book and "Helmholtz's Kraft: An Illustration of<br />
Concepts in Flux," Historical Studies in the Physical Sciences,<br />
Vol . 2 (1970), pp. 263-98.<br />
7. J. JouLE, "Matter, Living Force and Heat," The Scientific Papers<br />
of James Prescott Joule, Vol. 1 (London: Taylor & Francis,<br />
1884), pp. 265-76 (quotation, p. 273).<br />
8. The English translations of Mayer's two great papers, "On the<br />
Forces of In<strong>org</strong>anic Nature" and "The Motions of Organisms<br />
and Their Relation to Metabolism," are in Energy: Historical<br />
Development of a Concept, ed. R. B. LINDSAY (Stroudsburg,<br />
Pa.: Benchmarks Papers on Energy 1, Dowden, Hutchinson &<br />
Ross, 1975).<br />
9. E. BENTON, "Vitalism in the Nineteenth Century Scientific<br />
Thought: A 'JYpology and Reassessment," Studies in History<br />
and Philosophy of Science, Vol. 5 (1974), pp. 17-48.<br />
10. H.H ELMHOLTZ, "Uber die Erhaltung der Kraft," op. cit.,<br />
pp. 90-91.<br />
11. G. DELEUZE, Nietzsche et Ia phi/osophie (Paris: PUF, 1973),<br />
pp. 48-55.<br />
12. In this study of Zola's "Docteur Pascal ," Feux et signaux de<br />
brume Paris: Grassel (1975), p. 109, Michel Serres wrote: "The<br />
century that was practically drawing to a close when the novel<br />
appeared had opened with the majestic stability of the solar system,<br />
and was now filled with dismay at the relentless degradations<br />
of fire. Hence the fierce, positive dilemma: perfect cycle<br />
without residue, eternal and positively valued, i.e., the cosmology<br />
of the sun; or else a missed cycle, losing its difference, irreversible,<br />
historical and despised-a cosmology, a thermogony of<br />
fire which must either be extinguished or destroyed, without alternative.<br />
One dreams of Laplace, whilst Carnot and the others<br />
have forever smashed the cubby-hole, the niche, where one
ORDER OUT OF CHAOS 324<br />
could sleep in peace; one is dreaming, that is certain: then<br />
cultural archaisms having returned through another door,<br />
through another opening of the same door, are powerfully reawakened:<br />
immortal flame, purifying blaze or evil fire?"<br />
13. The continuity between Carnot father and son has been emphasized<br />
by Cardwell (From Wa tt to Clausius) and Scott (The Conflict<br />
Between Atomism and Conservation Theory).<br />
14. P. DAVIES, The Runaway Universe (New York: Penguin Books,<br />
1980), p. 197.<br />
15. R DYSON, "Energy in the Universe," Scientific American, Vol.<br />
225 (197 1), pp. 50-59.<br />
1 6. What was particularly important was to grasp that, unlike what<br />
happens in mechanics, it is not just any situation of a thermodynamic<br />
system that can be characterized as a "state"; quite<br />
the contrary. See E. DAUB, "Entropy and Dissipation," Historical<br />
Studies in the Physical Sciences, Vol. 2 (1970), pp. 321-54.<br />
17. In his autobiography, Scientific Autobiography (London:<br />
Williams & N<strong>org</strong>ate, 1950), Max Planck recalled how isolated he<br />
had been when he first emphasized the peculiarity of heat and<br />
pointed out that it is the conversion of heat into another form of<br />
energy that raises the irreversibility problem. Energeticists such<br />
as Ostwald wanted all forms of energy to be given the same<br />
status. For them, the fall of a body between two altitude levels<br />
takes place by virtue of the same kind of productive difference<br />
as the passage of heat between two bodies at different temperatures.<br />
Thus, Ostwald's comparison did away with the crucial<br />
distinction between an ideally reversible process, such as the<br />
mechanical motion, and an intrinsically irreversible one, such as<br />
heat diffusion. By doing so, he was actually taking up a position<br />
similar to what we have attributed to Lagrange: where Lagrange<br />
considered conservation of energy as a property belonging only<br />
to ideal cases but also the only one capable of being treated<br />
rigorously, Ostwald held conservation of energy as the property<br />
of any natural transformation, but defined conservation of energy<br />
differences (required by all transformation since only a difference<br />
can produce another difference) as an abstract ideal, but<br />
the sole object for a rational science.<br />
18. The splitting of the entropy variation into two different terms<br />
was introduced in l. Prigogine, Etude thermodynamique des<br />
Phenomenes irreversibles, These d'agn!gation presentee a Ia<br />
faculte des sciences de l'Universite Libre de Bruxelles 1945<br />
(Paris: Dunod, 1947).<br />
19. R. CLAUSIUS, Ann. Phys., Vol. 125 (1865), p. 353.<br />
20. M. PLANCK, "The Unity of the Physical Universe," A Survey of
325 NOTES<br />
Physics, Collection of Lectures and Essays (New York: E. P.<br />
Dutton, 1925), p. 16.<br />
21. R. CAILLOIS, "La dissymetrie," Coherences aventureuses, Collection<br />
Idees (Paris: Gallimard, 1973), p. 198.<br />
Chapter 5<br />
1. For what concerns the content of this and the fol lowing chapter,<br />
see P. GLANSDORFF and I. PRIGOGINE, Thermodynamic Theory<br />
of Structure, Stability and Fluctuations (New York: John Wiley<br />
& Sons, 1971) and G. NICOLls and I. PRIGOGINE, Self-Organization<br />
in Non-Equilibrium Systems (New York: John Wiley &<br />
Sons, 1977), where further references may be found.<br />
2. R NIETZSCHE, Der Wille zur Macht, Siimtliche Werke (Stuttgart:<br />
Kroner, 1964), aphorism 630.<br />
3. Which precise content can be given to the general law of entropy<br />
growth? For a theoretician physicist such as de Donder, chemical<br />
activity, which appeared obscure and inaccessible to the rational<br />
approach of mechanics, was mysterious enough to<br />
become the synonym of the irreversible process. Thus chemistry,<br />
whose question physicists had never truly answered, and the<br />
new enigma of irreversibility came to join in a challenge not to be<br />
ignored anymore. See Th. De Donder, L' Affinite (Paris:<br />
Gauthier-Villars, 1962) and L. Onsag'er Phys. Rev. 37, 405 (193 1).<br />
4. M. SERRES, La naissance de Ia physique dans le texte de Lucrece,<br />
op. cit.<br />
5. For more details concerning chemical oscillations, see A.<br />
WINFREE, "Rotating Chemical Reactions," Scientific Amer·<br />
ican, Vol. 230 (1974), pp. 82-95.<br />
6. A. GoLDBETER and G. Nicous, ''An Allosteric Model with<br />
Positive Feedback Applied to Glycolytic Oscillations," Progress<br />
in Theoretical Biology, Vol. 4 (1976), pp. 65-160; A. GOLDBETER<br />
and S. R. CAPLAN, "Oscillatory Enzymes," Annual Review of<br />
Biophysics and Bioengineering, Vol. 5 (1976), pp. 449-73 ..<br />
7. B. HESS, A. BOITEUX, and J. KROG ER, "Cooperation of<br />
Glycolytic Enzymes," Advances in·Enzyme Regulation, Vol. 7<br />
(1969), pp. 149-67; see also B. HESS, A. GOLDBETER, and R.<br />
LEFEVER, "Temporal, Spatial and Functional Order in Regulated<br />
Biochemical Cellular Systems," Advances in Chemical<br />
Physics, Vol. XXXVIII (1978), pp. 363-413.<br />
8_ R HEss, Ciba Foundation Symposium. Vol. 31 (1975), p. 369.<br />
9A.G. GERESCH, "Cell Aggregation and Differentiation in Die-
ORDER OUT OF CHAOS 326<br />
tyostelium Discoideum," in Developmental Biology, Vol. 3<br />
(1968), pp. 157-197.<br />
98. A. GoLDBETER and L. A. SEGEL, "Unified Mechanism for Relay<br />
and Oscillation of Cyclic AMP in Dictyostelium Discoideum,"<br />
Proceedings of the National Academy of Sciences,<br />
Vol. 74 (1977), pp. 1543-47.<br />
10. See M. GARDNER, The Ambidextrous Universe (New Yo rk:<br />
Charles Scribner's Sons, 1979).<br />
II. O. K. KONDEPUDI and I. PRIGOGINE, Physica, Vol. l07A (1981),<br />
pp. 1-24; D. K. KoNDEPUDI , Physica, Vol. 115A (1982),<br />
pp. 552-66. It could even be that chemistry may bring to the<br />
macroscopic scale the violation of parity in weak fo rces; D. K.<br />
KoNDEPUDI and G. W. NELSON, Physical Review Letters, Vol.<br />
50, No. 14 (1983), pp. 1023-26.<br />
12. R. LEFEVER and W. HoRSTHEMKE, "Multiple Transitions Induced<br />
by Light Intensity Fluctuations in Illuminated Chemical<br />
Systems," Proceedings of the National Academy of Sciences,<br />
Vol. 76 (1979), pp. 2490-94. See also W. HoRSTHEMKE and M.<br />
MALEK MANSOUR, "Influence of External Noise on Nonequilibrium<br />
Phase Transitions," Zeitschrift fu r Physik B, Vo l. 24<br />
(1976), pp. 307-1 3; L. ARNOLD, W. HORSTHEMKE, and R.<br />
LEFEVER, "White and Coloured External Noise and Transition<br />
Phenomena in Nonlinear Systems," Zeitschrift fii r Physik B,<br />
Vol. 29 (1978), pp. 367-73; W. HORSTHEMKE, "Nonequilibrium<br />
Transitions Induced by External White and Coloured Noise,"<br />
Dynamics of Synergetic Systems, ed. H. HAKEN (Berlin:<br />
Springer Ve rlag, 1980); for an application to a biological problem,<br />
R. LEFEVER and W. HORSTHEMKE, "Bistability in Fluctuating<br />
Environments: Implication in Tu mor Immunology,"<br />
Bulletin of Mathematic Biology, Vol. 41 (1979).<br />
13. H. L. SwiNNEY and J. P. GoLLUB, "The Transition to Thrbulence,"<br />
Physics Today, Vol. 31 , No.8 (1978), pp. 41-49.<br />
14. M. J. FEIGENBAUM, "Universal Behavior in Nonlinear Systems,"<br />
Los Alamos Science, No I (Summer 1980), pp. 4-27.<br />
15. The concept of chreod is part of the qualitative description of<br />
embryological development Waddington proposed more than<br />
twenty years ago. It is truly a bifurcating evolution: a progressive<br />
exploration along which the embryo grows in an "epigenetic<br />
landscape" where coexist stable segments and segments<br />
where a choice among several developmental paths is possible.<br />
See C. H. WA DDINGTON, The Strategy of the Genes (London:<br />
Allen & Unwin, 1957). C. H. Waddington's chreods are also a<br />
central reference in Rene Thorn's biological thought. They could<br />
thus become a meeting point for two approaches: the one we are<br />
presenting, starting from local mechanisms and exploring the
327 NOTES<br />
spectrum of collective behaviors they can generate; and Thorn's,<br />
starting from global mathematical entities and connecting the<br />
qualitatively distinct forms and transformations they imply with<br />
the phenomenological description of morphogenesis.<br />
16. S. A. KAUFFMAN, R. M. SHYMKO, and K. TRABERT, "Control of<br />
Sequential Compartment Formation in Drosophila," Science,<br />
Vol. 199 (1978), pp. 259-69.<br />
17. H. BERGSON, Creative Evolution, pp. 94-95.<br />
18. See C. H. WADDINGTON, The Evolution of an Evolutionist<br />
(Edinburgh: Edinburgh University Press, 1975) and P. WEISS,<br />
"The Living System: Determinism Stratified," Beyond Reductionism,<br />
ed. A. KOESTLER and J. R. SMYTHIES (London:<br />
Hutchinson, 1969).<br />
19. D. E. KosHLAND, · Model Regulatory System: Bacterial<br />
Chemotaxis," Physiological Review, Vol. 59, No. 4, pp. 811-62.<br />
Chapter 6<br />
J. G. NICOLlS and I. PRIGOGINE, Self-Organization in Nonequilibrium<br />
Systems (New York: John Wiley & Sons, 1977).<br />
2. R BARAS, G. Nicous, and M. MALEK MANSOUR, "Stochastic<br />
Theory of Adiabatic Explosion," Journal of Statistical Physics,<br />
Vol. 32, No. 1 (1983), pp. I.<br />
3. See, for example, M. MALEK MANSOUR, C. VAN DEN BROECK,<br />
G. Nicous, and J. W. TURNER, Annals of Physics, Vol. 131, No.<br />
2 (1981), p. 283.<br />
4. J. L. DENEUBOURG, 'pplication de l'ordre par fluctuation a Ia<br />
description de certaines etapes de Ia construction du nid chez<br />
les termites," Insectes Sociaux, Journal International pour<br />
/'etude des Arthropodes sociaux, To me 24, No. 2 (1977),<br />
pp. 117-30. This first model is presently being extended in connection<br />
with new experimental studies; 0. H. BRUINSMA, ·n<br />
Analysis of Building Behaviour of the Termite macrotermes subhyaiinus,"<br />
Proceedings of the VIII Congress IUSSI (Waegeningen,<br />
1977).<br />
5. R. P. GARAY and R. LEFEVER, · Kinetic Approach to the Immunology<br />
of Cancer: Stationary States Properties of Effector<br />
Target Cell Reactions," Journal of Theoretical Biology, Vol. 73<br />
(1978), pp. 417-38, and private communication.<br />
6. P. M. ALLEN, "Darwinian Evolution and a Predator-Prey Ecology,<br />
" Bulletin of Mathematical Biology, Vol. 37 (1975),<br />
PP- 389-405; "Evolution, Population and Stability, " Proceedings<br />
of the National Academy of Sciences, Vol. 73, No. 3 (1976),<br />
pp. 665-68. See also R. CzAPLEWSKI, ']\_ Methodology for Eval-
ORDER OUT OF CHAOS 328<br />
uation of Parent-Mutant Competition," Journal for Theoretical<br />
Biology, Vol. 40 (1973), pp. 429-39.<br />
7. See, for the present state of this work, M . . EIGEN and P. ScHus<br />
TER, The Hypercycle (Berlin: Springer Verlag, 1979).<br />
8. R. MAY in Science, Vol. 186 (1974), pp. 645-47; see also R. MAY,<br />
"Simple Mathematical Models with very Complicated Dynamics,"<br />
Nature, Vol. 261 (1976), pp. 459-67.<br />
9. M. P. HASSELL, The Dynamics in Arthropod Predator-Prey Systems<br />
(Princeton, N.J.: Princeton University Press, 1978).<br />
10. B. HEINRICH, ·rtful Diners," Natural History, Vol. 89, No. 6<br />
(1980), pp. 42-5 1, esp. quote, p. 42.<br />
11. M. LovE, "The Alien Strategy," Natural History, Vol. 89, No. 5<br />
(1980), pp. 30-32.<br />
12. J. L. DENEUBOURG and P. M. ALLEN, "Modeles theoriques de<br />
Ia division du travail des les societes d'insectes," Academie<br />
Royale de Belgique, Bulletin de Ia Classe des Sciences, Tome<br />
LXII (1976), pp. 416-29; P. M. ALLEN, "Evolution in an Ecosystem<br />
with Limited Resources," op. cit., pp. 408-15.<br />
13. E. W. MoNTROLL, "Social Dynamics and the Quantifying of Social<br />
Forces," Proceedings of the National Academy of Sciences,<br />
Vol. 75, No. 10 (1978), pp. 4633-37.<br />
14. P. M. ALLEN and M. SANGLIER, "Dynamic Model of Urban<br />
Growth," Journal for Social and Biological Structures, Vol. 1<br />
(1978), pp. 265-80, and "Urban Evolution, Self-Organization<br />
and Decision-making," Environment and Planning A, Vol. 13<br />
(1981), pp. 167-83.<br />
15. C. H. WADDINGTON, To ols for Thought, (St. Albans, Eng.: Pal·<br />
adin, 1976), p. 228.<br />
16. S. J. GouLD, Ontogeny and Phylogeny, op. cit. Belknap Press<br />
Harvard University Press, 1977.<br />
17. C. L:Evi-STRAUSS, "Methodes et enseignement," Anthropologie<br />
structurale (Paris: Pion), pp. 311-17.<br />
18. See, for instance, C. E. RusSET, The Concept of Equilibrium in<br />
American Social Thought (New Haven, Conn.: Yale University<br />
Press, 1966).<br />
19. S. J. GouLD, "The Belt of an Asteroid," Natural History, Vol.<br />
89, No. 1 (1980), pp. 26-33.<br />
Chapter 7<br />
1. A.N. WHITEHEAD, Science and the Modern World, p. 186.<br />
2. The Philosophy of Rudolf Carnap, ed. P.A. ScHILP.P (Cambridge:<br />
Cambridge University Press, 1963).
329 NOTES<br />
3. J. FRASER, "The Principle of Temporal Levels: A Framework for<br />
the Dialogue?" communication at the conference Scientific<br />
Concepts of Time in Humanistic and Social Perspectives (Bellagio,<br />
July 1981).<br />
4. See on this point S. BRUSH, Statistical Physics and Irreversible<br />
Processes, esp. pp. 616-25.<br />
5. Feuer has rather convincingly shown how the cultural context of<br />
Bohr's youth could have helped his decision to look for a nonmechanistic<br />
model of the atom; Einstein and the Generation of<br />
Science (New York: Basic Books, 1974). See also W. HEISEN<br />
BERG, Physics and Beyond (New York: Harper & Row, 1971)<br />
and D. SERWER, "Unmechanischer Zwang: Pauli, Heisenberg<br />
and the Rejection of the Mechanical Atom 1923-1925," Historical<br />
Studies in the Physical Sciences, Vol. 8 (1977), pp. 189-256.<br />
6. In Black-Body Theory and the Quantum Discontinuity, 1894-<br />
1912 (Oxford: Clarendon Press and New York: Oxford University<br />
Press, 1978), Thomas Kuhn has beautifully shown how<br />
closely Planck followed Boltzmann's statistical treatment of irreversibility<br />
in his own work.<br />
7. J. MEHRA and H. RECHENBERG, The Historical Development of<br />
Quantum Theory, Vols. 1-4 (New York: Springer Verlag, 1982).<br />
8. See, about the conceptual framework of the experimental tests<br />
recently conceived for hidden variables in quantum mechanics,<br />
B. o'EsPAGNAT, Conceptual Foundations of Quantum Mechanics,<br />
2nd aug. ed. (Reading, Mass.: Benjamin, 1976). See also B.<br />
o'EsPAGNAT, "The Quantum Theory and Reality," Scientific<br />
American, Vol. 241 (1979), pp. 128-40.<br />
9. See, for the complementarity principle, B. o'EsPAGNAT, op. cit.;<br />
M. JAMMER, The Philosophy of Quantum Mechanics (New<br />
York: John Wiley & Sons, 1974); and A. PETERSEN, Quantum<br />
Mechanics and the Philosophical Tradition (Cambridge, Mass.:<br />
MIT Press, 1968). C. GEORGE and I. PRIGOGINE, "Coherence<br />
and Randomness in Quantum Theory, " Physica, Vol. 99A<br />
(1979), pp. 369-82.<br />
10. L. RosENFELD, "The Measuring Process in Quantum Mechanics,"<br />
Supplement of the Progress of Theoretical Physics .(1965),<br />
p. 222.<br />
11. About the quantum mechanics paradoxes, which can truly be<br />
said to be the nightmares of the classical mind, since they all<br />
(SchrOdinger's cat, Wigner's friend, multiple universes) call to<br />
life again the phoenix idea of a closed objective theory (this time<br />
in the guise of SchrOdinger's equation), see the books by d'Espagnat<br />
and Jammer.<br />
12. B. MISRA, I. PRIGOGINE, and M. COURBAGE, "Lyapounov Vari-
ORDER OUT OF CHAOS<br />
330<br />
able; Entropy and Measurement in Quantum Mechanics," Pro<br />
ceedings of the National Academy of Sciences, Vol. 76 (1979),<br />
pp. 4768-4772. I. PRIGOGINE and C. GEORGE, "The Second<br />
Law as a Selection Principle: The Microscopic Theory of Dissipative<br />
Processes in Quantum Systems," to appear in Proceedings<br />
of the National Academy of Sciences. Vol 80 (1983)<br />
4590-94.<br />
13. H. MINKOWSKI, "Space and Time," The Principles of Relativity<br />
(New York: Dover Publications, 1923).<br />
14. A. D. SAKHAROV, Pisma Zh. Eksp. Teor. Fiz., Vol. 5, No. 23<br />
(1%7).<br />
Chapter 8<br />
I. G. N. LEWIS, "The Symmetry of Time in Physics," Science,<br />
Vol. 71 (1930), p. 570.<br />
2. A. S. EDDINGTON, The Nature of the Physical World (New<br />
York: Macmillan, 1948), p. 74.<br />
3. M. GARDNER, The Ambidextrous Universe: Mirror Asymmetry<br />
and Time-Reversed Worlds (New York: Charles Scribner's Sons,<br />
1979), p. 243.<br />
4. M. PLANCK, Treatise on Thermodynamics (New York: Dover<br />
Publications, 1945), p. 106.<br />
5. Quote by K. DENBIGH, "How Subjective Is Entropy?" Chemistry<br />
in Britain, Vol. 17 (198 1), pp. 168-85.<br />
6. See, for instance, M. KAC, Probability and Related Topics in<br />
Physical Sciences (London: Interscience Publications, 1959).<br />
7. J. W. GIBBS, Elementary Principles in Statistical Mechanics<br />
(New York: Dover Publications, 1960), Chap. XII.<br />
8. For instance, S. Watanabe introduces a strong distinction between<br />
the world to be contemplated and the world upon which<br />
we, as active agents, work; he states there is no consistent way<br />
of speaking about entropy increase if it is not in connection with<br />
our actions on the world. However, all our physics is in fact<br />
about the world to be acted on, and Watanabe's distinction thus<br />
does not help to clarify the relation between "microscopic deterministic<br />
symmetry" and "macroscopic probabilistic asymmetry."<br />
The question is left without an answer. How can we<br />
meaningfully say that the sun is irreversibly burning? See S.<br />
WATANABE, "Time and the Probabilistic View of the World,"<br />
The Voices of Time, ed. J. FRASER (New York: Braziller, 1966).<br />
9. Maxwell's demon appears in J. C. MAXWELL. TheorY of Heat<br />
(London: Longmans, 1871), Chap. XXII; see also E. DAUB,
331<br />
NOTES<br />
"Maxwell's Demon" and P. HEIMANN, "Molecular Forces, Statistical<br />
Representation and Maxwell's Demon," both in Studies<br />
in History and Philosophy of Science, Vol. I (1970); this volume<br />
is entirely devoted to Maxwell.<br />
10. L. BoLTZMANN, Populiire Schriften, new ed. (Braunschweig<br />
Wiesbaden: Vieweg, 1979). As Elkana emphasizes in "Boltzmann's<br />
Scientific Research Program and Its Alternatives," Interaction<br />
Between Science and Philosophy (Atlantic,<br />
Highlands, N.J.: Humanities Press, 1974), the Darwinian idea of<br />
evolution is explicitly expressed mostly in Boltzmann's view<br />
about scientific knowledge-that is, in his defense of mechanistic<br />
models against energeticists. See, for instance, his 1886 lecture<br />
"The Second Law of Thermodynamics," Theoretical<br />
Physics and Philosophical Problems, ed. B. McGuiNNESS (Dordrecht,<br />
Netherlands: D. Reidel , 1974).<br />
11. For a recent account see I. PRIGOGINE, From Being to Becoming-Time<br />
and Complexity in the Physical Sciences (San Francisco:<br />
W. H. Freeman & Company, 1980).<br />
12. In his Scientific Autobiography, Planck describes his changing<br />
relationship with Boltzmann (who was first hostile to the phenomenological<br />
distinction introduced by Planck between reversible<br />
and irreversible processes). See also on this point Y.<br />
ELKANA, op. cit., and S. BRUSH, Statistical Physics and Irreversible<br />
Processes, pp. 640-5 1; for Einstein, op. cit., pp. 672-74;<br />
for Schrodinger, E. SCHR6DINGER, Science, Theory and Man<br />
(New York: Dover Publications, 1957).<br />
13. H. POINCARE, "La mecanique et I' experience," Revue de Metaphysique<br />
et de Morale, Vol. 1 (1893), pp. 534-37. H. POINCARE,<br />
Lefons de Thermodynamique, ed. J. Blondin (1892; Paris: Hermann<br />
1923).<br />
14. See for a study of the controversies around Boltzmann's entropy,<br />
see on this point S. BRUSH, The Kind of Motion We Call Heat,<br />
op. cit., and Planck's remarks in his biography (Loschmidt was<br />
Planck's student).<br />
15. I. PRIGOGINE, C. GEORGE, R HENIN, and L. ROSENFELD, ·<br />
Unified Formulation of Dynamics and Thermodynamics,"<br />
Chemica Scripta, Vol. 4 (1973), pp. 5-32.<br />
16. D. PARK, The Image of Eternity: Roots of Time in the Physical<br />
World (Amherst, Mass.: University of Massachusetts Press,<br />
1980).<br />
17. See also on this point S. BRUSH, The Kind of Motion We Call<br />
Heat-Book I, Physics and the Atomists; Book II, Statistical<br />
Physics and Irre versible Processes (Amsterdam: North Holland<br />
Publishing Company, 1976), as well as his commented anthology,
ORDER OUT OF CHAOS<br />
332<br />
Kinetic Theory: Vol. I, The Nature of Gases and Heat: Vol. II.<br />
Irreversible Processes (Oxford: Pergamon Press, 1965 and 1966).<br />
18. J. W. GIBBS, Elementary Principles in Statistical Mechanics<br />
(New Yo rk: Dover Publications, 1%0), Chap XII. For an historical<br />
account, see J. MEHRA, "Einstein and the Foundation of<br />
Statistical Mechanics, Physica, Vol. 79A, No. 5 (1974), p. 17.<br />
19. Many Marxist nature philosophers seem to take inspiration from<br />
Engels (quoted by Lenin in his Philosophic Notebooks) when he<br />
wrote in Anti-Diihring (Moscow: Foreign Languages Publishing<br />
House, 1954), p. 167, "Motion is a contradiction: even simple<br />
mechanical change of a position can only come about through a<br />
body being at one and the same moment of time both in one<br />
place and in another place, being in one and the same place and<br />
also not in it. And the continuous and simultaneous solution of<br />
this contradiction is precisely what motion is."<br />
20. L. BoLTZMANN, Lectures on Gas Theory (Berkeley: University<br />
of California Press, 1964), p. 446f, quoted in K. POPPER, Unended<br />
Quest (La Salle, Ill.: Open Court Publishing Company,<br />
1976), p. 160.<br />
21. POPPER, op. cit., p. 160.<br />
Chapter 9<br />
1. VoLTAIRE, Dictionnaire Philosophique. (Paris: Garnier, 1954.)<br />
2. See note 2, Chapter VII.<br />
3. K. PoPPER, "The Arrow of Time," Nature, Vol. 177 (1956),<br />
p. 538.<br />
4. See M. GARDNER, The Ambidextrous Universe, pp. 271-72.<br />
5. A. EINSTEIN and W RITZ, Phys. Zsch., Vol. lO (1909), p. 323.<br />
6. H. POINCARE, Les methodes nouvelles de Ia mecanique celeste<br />
(New York: Dover Publications, 1957); E. T. WHITTAKER, A<br />
Treatise on the Analytical Dynamics of Pa rticles and Rigid<br />
Bodies (Cambridge: Cambridge University Press, 1965).<br />
7. J. MosER, Stable and Random Motions in Dynamical Systems<br />
(Princeton, N.J. : Princeton University Press, 1974).<br />
8. For a general review, see J. LEBOWITZ and 0. PENROSE, "Modern<br />
Ergodic Theory," Physics To day (Feb. 1973), pp. 23-29.<br />
9. For a more detailed study, see R. BALESCU, Equilibrium and<br />
Non-Equilibrium Statistical Mechanics (New York: John Wiley<br />
& Sons, 1975).<br />
10. V. ARNOLD and A. Av Ez, Ergodic Problems of Classical Mechanics<br />
(New York: Benjamin, 1968).
333 NOTES<br />
11. H. PoiNCARE, "Le Hasard," Science et Methode (Paris: Flammarion,<br />
1914), p. 65.<br />
12. B. MISRA, I. PRIGOGINE and M. CouRBAGE, "From Deterministic<br />
Dynamics to Probabilistic Descriptions," Physica, Vol. 98A<br />
(1979), pp. 1-26.<br />
13. D. N. PA RKS and N. J. THRIFf, Times, Spaces and Places: A<br />
Chronogeographic Perspective (New York: John Wiley & Sons,<br />
1980).<br />
14. M. COURBAGE and I. PRIGOGINE, "Intrinsic Randomness and<br />
Intrinsic Irreversibility in Classical Dynamical Systems," Proceedings<br />
of the National Academy of Sciences, 80 (April 1983).<br />
15. I. PRIGOGINE and C. GEORGE, "The Second Law as a Selection<br />
Principle: The Microscopic Theory of Dissipative Processes in<br />
Quantum Systems," Proceedings of the National Academy of<br />
Sciences, Vol. 80 (1983), pp. 4590-4594.<br />
16. V. NABOKOV, Look at the Harlequins! (McGraw-Hill 1974).<br />
17. J. NEEDHAM, "Science and Society in East and West," The<br />
Grand Titration (London: Allen & Unwin, 1%9).<br />
18. See for more details B. MISRA, I. PRIGOGINE and M. CouR<br />
BAGE, "From deterministic Dynamics to probabilistic Description",<br />
Physica 98A (1979) 1-26.; B. MISRA and I. PRIGOGINE<br />
"Time, Probability and Dynamics", in Long-time Prediction in<br />
Dynamics, eds. C. W. Horton, L. E. Recihl and A. G.<br />
Szebehely, (New York, Wiley 1983).<br />
19. I. PRIGOGINE, C. GEORGE, R HENIN, and L. ROSENFELD, ·<br />
Unified Formulation of Dynamics and Thermodynamics,"<br />
Chemica Scripta, Vol. 4 (1973), pp. 5-32.<br />
20. M. CouRBAGE "Intrinsic irreversibility of Kolmogorov dynamical<br />
systems," Physica 1983; B. Misra and I. Prigogine, Letters<br />
in Mathematical Physics, September 1983.<br />
Conclusion<br />
1. A. S. EDDINGTON, The Nature of the Physical World (N_ew<br />
York: Macmillan, 1948).<br />
2. L. LEVY-BRUHL, La Mentalite Primitif (Paris: PUF, 1922).<br />
3. G. MILLS, Hamlet's Castle (Austin: University of Texas Press,<br />
1976).<br />
4. R. TAGORE, "The Nature of Reality" (Calcutta: Modern Review<br />
XLIX, 1931), pp. 42-43.<br />
5. D. S. KOTHARI, Some Thoughts on Truth (New Delhi: Anniversary<br />
Address, Indian National Science Academy, Bahadur Shah<br />
Zafar Marg, 1975), p. 5.
ORDER OUT OF CHAOS<br />
334<br />
6. E. MEYERSON, Identity and Reality (New York: Dover Publications,<br />
1962).<br />
7. Described in H. BERGSON, Melanges (Paris: PUF, 1972),<br />
pp. 1340-46.<br />
8. Correspondence, Albert Einstein-Michele Besso, 1903-1955<br />
(Paris: Herman, 1972).<br />
9. N. WIENER, Cybernetics (Cambridge, Mass.: M.I.T. Press and<br />
New Yo rk: John Wiley & Sons, 1961).<br />
10. M. MERLEAU-PONTY, "Le philosophe et Ia sociologie," Eloge<br />
de Ia Philosophie, Collection Idees (Paris: Gallimard, 1960),<br />
pp. 136-37.<br />
11. M. MERLEAu-PoNTY, Resumes de Cours /952-/960 (Paris: Gallimard,<br />
1968), p. 119.<br />
12. P. VA LRY, Cahiers, La Pleiade (Paris: Gallimard, 1973), p. 1303.<br />
13. For what follows see also I. PRIGOGINE, I. STENGERS, and S.<br />
PA HAUT, "La dynamique de Leibniz a Lucrece," Critique "Special<br />
Serres," Vol. 35 (Jan. 1979), pp. 34-55. Engl. trans.: "Dynamics<br />
from Leibniz to Lucretius," Afterword to M. SERRES,<br />
Hermes: Literature, Science, Philosophy (Baltimore: Johns<br />
Hopkins Univ. Pr. , 1982), pp. 137-55.<br />
14. C. S. PEIRCE, The Monist Vo l. 2 (1892), pp. 321-337.<br />
15. A. N. WHITEHEAD, Process and Reality, pp. 240-41. On this<br />
subject, see I. LECLERC, Whitehead's Metaphysics (Bloomington:<br />
Indiana University Press, 1975).<br />
16. La naissance de Ia physique dans le texte de Lucrece, p. 139.<br />
17. LuCRETIUS, De Natura Rerum, Book II. ·gain, if all movement<br />
is always interconnected, the new arising from the old in a<br />
determinate order-if the atoms never swerve so as to originate<br />
some new movement that will snap the bonds of fate, the everlasting<br />
sequence of cause and effect-what is the source of the<br />
free will possessed by living things throughout the earth?"<br />
18. M. SERRES, op. cit., p. 136.<br />
19. M. SERRES, op. cit., p. 162; also pp. 85-86 and "Roumain et<br />
Faulkner traduisent l' Ecriture," La traduction (Paris: Minuit,<br />
1974).<br />
20. S. MoscoviCI, Hommes domestiques et hommes sauvages,<br />
pp. 297-98.<br />
21. T. KuHN, The Structure of Scientific Revolutions, 2nd ed. incr.<br />
(Chicago: Chicago University Press, 1970).<br />
22. See A. N. WHITEHEAD, Process and Reality, op. cit. and M.<br />
HEIDEGGER Sein und Zeit (Tiibingen: Niemeyer 1977).<br />
23. H. WEYL, Philosophy of Mathematics and Natural Science<br />
(Princeton, N.J.: Princeton University Press, 1949).<br />
24. A. NEHER, "Vision du temps et de l'histoire dans Ia culture<br />
juive," Les cultures et le temps (Paris: Payot, 1975), p. 179.
INDEX<br />
Note: Page numbers given in boldface indicate location of definitions<br />
or discussions of terms or concepts indexed here.<br />
Acceleration, 57-59<br />
Affinity, 29, 136<br />
Against the Current (Berlin), 2<br />
Agassiz, Louis, 195<br />
Alchemy, 64; affinity in, 136;<br />
Chinese, 278<br />
Alembert, Jean Le Rond d', 52;<br />
Diderot and, 80-82;<br />
opposition to Newtonian<br />
science of, 62, 63, 65, 66<br />
Ambidextrous Universe, The<br />
(Gardner), 234<br />
Amoebas, 156-60<br />
Ampere, Andre Marie, 67, 76<br />
Anaxagoras, 264<br />
Antireductionists, 173-74<br />
Archimedes, 39, 41, 304<br />
Aristotle, 39, 40, 71, 85, 173,<br />
305, 306; on change, 62;<br />
notion of space of, 171;<br />
physics of, 39-4 1; and<br />
theology, 49-50<br />
Arrow of time, xx, xxvii, 8, 16;<br />
Boltzmann on, 253-55; and<br />
elementary particles, 288;<br />
and entropy, 119, 257-59; and<br />
heat engines, 111-15; Layzer<br />
on, xxv; meaning of, 289;<br />
and probability, 238-39; roles<br />
played by, 30 I<br />
Atomists, 3, 36; conception of<br />
change of, 62, 63; on<br />
turbulence, 141<br />
Attractor, 121, 133, 140, 152<br />
Bach, J. S., 307<br />
Bachelard, Gaston, 320n<br />
Bacterial chemotaxis, I 75<br />
Baker transformation, 269,<br />
272-76, 278-79, 283, 289<br />
Being and Becoming, 310<br />
Belousov-Zhabotinsky reaction,<br />
151-53, 168<br />
Benard instability, 142-44;<br />
transition to chaos in, 167-68<br />
Bergson, Henri, 10, 79, 80,<br />
90-94, 96, 129, 173-74,<br />
301-2; on dynamics, 60; on<br />
time, 214, 294<br />
Berlin, Isaiah, 2, 11, 13, 80<br />
Bernoulli, Daniel, 82<br />
Berry, B., 17<br />
Berthollet, Claude Louis,<br />
Comte, 319n<br />
Besso, Michele, 294<br />
Bifurcations, xv, 160-61, 176,<br />
275; cascading, 167-70; in<br />
evolution, 171-72;<br />
fluctuations and, 177, 180; in<br />
reaction-diffusion systems,<br />
260; role of chance in, xxvi,<br />
170, 176; social, 313; and<br />
335
ORDER OUT OF CHAOS 336<br />
Bifurcations ( cont' d)<br />
statistical model, 205-6;<br />
theory of, 14<br />
Big Bang, xxvii, 288; and arrow<br />
of time, xxv, 259<br />
Biology, 2, 10; catalysts in,<br />
133-34; chemical reactions<br />
in, 131-32; "communication"<br />
among molecules in, 13;<br />
concepts from physics<br />
applied to, 207; and<br />
conversion, 108; evolution<br />
and, 12, 128; logistic<br />
equation in, 193-96;<br />
molecular, see Molecular<br />
biology; reductionistantireductionist<br />
conflict in,<br />
174; technological analogies<br />
in, 174-75; time in, 116;<br />
Whitehead on, 96<br />
Birchoff, 266<br />
Blake, William, 30<br />
Boerhave, Hermann, 105<br />
Bohr, Niels, 2, 74, 220, 224-25,<br />
228, 229, 292-93<br />
Boltzmann, Ludwig, xvii, 15,<br />
16, 122-27, 219, 227, 234-36,<br />
258, 259, 274, 286-87, 297,<br />
329n, 33ln; and arrow of<br />
time, 253-55; on ergodic<br />
systems, 266; on evolution<br />
toward equilibrium, 240-43;<br />
objections to theories of,<br />
243-46; and theory of<br />
ensembles, 248, 250<br />
Boltzmann's constant, 124<br />
Boltzmann's order principle,<br />
122-28, 142, 143, 150, 163,<br />
187<br />
Bordeu, 321n<br />
Born, Max, 220, 235<br />
Boundary conditions, 106,<br />
120-2 1, 125, 126, 138-39,<br />
142, 147, 151<br />
Braude!, xviii, xix<br />
Bridgmann, P. W., 316n<br />
Brillouin, 216<br />
Broglie, Louis de, 220<br />
Bruno, Giordano, 15<br />
Bruns, 72, 265<br />
Brusselator, 146, 148, 151, 152,<br />
160<br />
"Brussels school," xv<br />
Buchdahl, Gerd, 316n<br />
Buffon, Ge<strong>org</strong>es Louis Leclerc<br />
de, Comte, 65-67, 319n<br />
Butts, Thomas, 30<br />
Caillois, Roger, 128<br />
C<strong>alvin</strong>, John, xxii<br />
Cancer tumors, onset of, 188<br />
Canonical equation, 226<br />
Canonical variables, 70, 71,<br />
107, 222<br />
Cardwell, D., 322n<br />
Carnap, Rudolf, 214, 294<br />
Carnot, Lazare, 112<br />
Carnot, Sadi, 111-15, 117, 120,<br />
128, 140, 323n, 324n ; Carnot<br />
cycle, 112- 114, 117<br />
"Carrying capacity" of<br />
systems, 192-97<br />
Catalysis, 133-35, 145, 153<br />
Caterpillars, strategies for<br />
repelling predators of, 194-95<br />
Catherine the Great, 52<br />
Cells: Benard, 143; chemical<br />
reactions within, 131-32<br />
Chance, concepts of, xxii-xxiii,<br />
14, 170, 176, 203 ; see<br />
Randomness<br />
Change: motion and, 62-68;<br />
nature of, 29 1; of state, 106;<br />
in thermodynamic system,<br />
120-2 1; Whitehead on, 95<br />
Chemical clock, xvi, 13,<br />
147-48, 179, 307;<br />
communication in, 180; in<br />
glycolysis, 155; in slime mold<br />
aggregation, 159<br />
Chemical reactions, 127; in<br />
biology, 131-32; diffusion in,<br />
148-49; fluctuations and<br />
correlations in, 179-8 1;<br />
kinetic description of,<br />
132-34; self-<strong>org</strong>anization in,
337<br />
144-45; thermodynamic<br />
descriptions of, 133-37; see<br />
also specific reactions<br />
Chemistry, xi, I 0; Bergson on,<br />
91; and Buffo n, 65, 66;<br />
conceptual distinction<br />
between physics and, 137;<br />
and conversion, 108; Diderot<br />
on, 82, 83; fluctuations and,<br />
177-79; in<strong>org</strong>anic, 152, 153;<br />
irreversibility in, 209;<br />
Newtonian method in, 28;<br />
relation between order and<br />
chaos in, 168; and "science<br />
of fire," 103; temporal<br />
evolution in, 10-1 1<br />
China, 57; alchemy in, 278;<br />
social role of scientists in,<br />
45-46, 48<br />
Chiral symmetry, 285<br />
"Chreod," 172<br />
Chris taller model, 197, 203<br />
Christianity, 46, 47, 50, 76<br />
Chronogeography, 272<br />
Chuang Tsu, 22<br />
Clairaut, Alexis Claude, 62, 65<br />
Clausius, Rudolf Julius<br />
Emanuel, 114, 115, 233, 234,<br />
240, 304; entropy described<br />
by, 117- 19<br />
Clinamen, 141, 303 , 304<br />
Clocks: invention of, 46; as<br />
symbol of nature, Ill; see<br />
also Chemical clocks<br />
Closed systems, xv, 125<br />
Collective phenomena, xxiv; in<br />
amoebas, 156-160; in insects,<br />
181-86; in human geography,<br />
197-203 ; in social<br />
anthropology, 205, 317n<br />
Collisions, 63, 69, 132, 240-42,<br />
270-7 1 ' 280-85<br />
Combinatorial analysis, 123<br />
Communication: description as,<br />
300; in dissipative structures,<br />
13, 148; and entropy barrier,<br />
295-96; and fluctuations,<br />
187-88; and irreversibility,<br />
INDEX<br />
295; molecular basis to, xxv,<br />
180; stabilizing effects of, 189<br />
Compensation, 107; Clausiuson<br />
Carnot cycle, 114; statistical,<br />
124, 133, 240<br />
Complementarity, principle of,<br />
225<br />
Complexions, 123, 124, 127,<br />
150; in Benard instability,<br />
142-43<br />
Complexity: dynamics and<br />
science of, 208-9; limits of,<br />
188-89; modelizations of,<br />
203-7<br />
Comte, Auguste, 104-5<br />
Condillac, Etienne Bonnot de,<br />
66<br />
Condorcet, Marie Jean Antoine<br />
Nicolas Caritat, marquis de,<br />
66<br />
Conservation, life defined in<br />
terms of, 84<br />
Conservation of energy, I 07- 1 1 ;<br />
and Carnot cycle, 114, 115;<br />
and entropy, 117, 118;<br />
principle of, 69-7 1<br />
Convection, 127; in Benard<br />
instability, 142<br />
Copernicus, 307<br />
Correlations: dynamics of,<br />
280-85 ; fluctuations and,<br />
179-8 1<br />
Cosmology, xxviii, 1, 10;<br />
entropy and, 117; mysticism<br />
and, 34; and<br />
thermodynamics, 115-17;<br />
time and, 215, 259;<br />
Whitehead's, 94<br />
Counterintuitive responses, 203<br />
Critical threshold see Instability<br />
threshold<br />
Critique of Pure Reason (Karit),<br />
86<br />
Cybernetics (Weiner), 295-96<br />
Darwin, Charles, xiv, xx, 128,<br />
140, 215, 240, 24 1, 25 1
ORDER OUT OF CHAOS 338<br />
Darwinian selection, 190, 191,<br />
194, 195<br />
David, Ben, 318n<br />
Democritus, 3<br />
Deneubourg, J. L., 181<br />
Density function p, 247-50,<br />
261, 264; with arrow of time,<br />
277, 289; or distribution<br />
function, 289; in phase space,<br />
265-72, 274, 279<br />
Deoxyribonucleic acid (DNA),<br />
154; dissymetry of, 163<br />
Desaguliers, J. T. , 27<br />
Descartes, Re ne , 62, 63 , 81<br />
Destiny, 6, 257<br />
Determinism, xxv, 9, 60, 75 ,<br />
169-70, 176, 177, 216, 226,<br />
23 1 '<br />
264, 269-72, 304;<br />
concepts of, xxii-xxiii<br />
Dialectics of Nature (Engels),<br />
253<br />
Dialogue Concerning the Two<br />
Chief World Systems<br />
(Galileo), 305<br />
Dictionnaire Philosophique<br />
(Voltaire), 257<br />
Dictyostelium discoideum, 156,<br />
157<br />
Diderot, Denis, 79-85, 91, 136,<br />
309, 321n<br />
Differential calculus, 57, 222<br />
Diffusion, 148-49, 177<br />
Dirac, Paul, 34, 220, 230<br />
Disorder, xxvii, 18, 124, 126,<br />
142, 238, 246, 250, 286-87,<br />
293<br />
Dissipation (or loss), 63, 112,<br />
I 15, 117, 120, 125, 129,<br />
302-03<br />
Dissipative structures, xii, xv,<br />
xxiii, 12-14, 142-43, 189,<br />
300; coherence of, 170;<br />
communication in, 148;<br />
cultural , xxvi<br />
Dissymmetry, 124; 163; in time,<br />
125; see also Symmetrybreaking<br />
Distribution function se<br />
Density fu nction p<br />
Dobbs, B. J. , 319n<br />
Dander, Theophile de, 136, 325<br />
Donne, John, 55<br />
Driesch, Hans, 171<br />
Drosophila, 172<br />
du Bois Reymond, 77, 97<br />
Diierrenmatt, E, 21<br />
Duhem, Pierre Maurice Marie,<br />
97<br />
Duration, xxviii-xix; Bergson's<br />
concept of, 92, 294<br />
Dynamics, II, 14-15, 58-62,<br />
107; baker transformation in,<br />
276-77; basic symmetry of,<br />
243 ; change in, 62-68;<br />
concept of order in, 287; of<br />
correlations, 280-85;<br />
incompatibility of<br />
thermodynamics and, 216,<br />
233-34, 252-53; language of,<br />
68-74; and Laplace's demon,<br />
75-77; objects of, 306;<br />
operators in, 222; probability<br />
generated in, 274;<br />
reconciliation of<br />
thermodynamics and, 122;<br />
reversibility in, 120; and<br />
science of complexity, 208-9;<br />
static view of, xxix;<br />
symmetry-breaking in,<br />
260-6 1; and theories of<br />
irreversibility, 25 1; theory of<br />
ensembles in, 247-5 1;<br />
twentieth-century renewal of,<br />
264-72<br />
Dyson, Freeman, 1 17<br />
Ecology, logistic equation in,<br />
192-93, 196, 204<br />
Eddington, Arthur Stanley, xx,<br />
8, 49, 119, 233, 291<br />
Edge of Objectivity, Th<br />
(Gillespie), 31<br />
Ehrenfest model, 235-38, 240,<br />
246
339<br />
Eigen, M., 190-91<br />
Eigenfunctions, 22 1-23; of<br />
Hamiltonian operator, 227; of<br />
operator time T, 289;<br />
superposition of, 227-28<br />
Eigenvalues, 22 1, 222; of<br />
Hamiltonian operator, 226,<br />
227<br />
Einstein, Albert, xiv, 76, 242,<br />
271, 301, 307, 310; on basic<br />
myth of science, 52-53;<br />
demonstration of<br />
impossibility by, 296;<br />
dialogue with Tagore, 293;<br />
ensemble theory of, 247-5 1,<br />
26 1; God of, 54; on<br />
gravitation, 34; on<br />
irreversibility, 15, 258, 259,<br />
294-95 ; Mach's influence on,<br />
53; and quantum mechanics,<br />
218-20, 224; on scientific<br />
asceticism, 20-2 1; on<br />
simultaneity, 218; special<br />
theory of relativity of, 17;<br />
thought experiments of, 43;<br />
on time, 214-15, 251 ;<br />
"unified field theory" of, 2;<br />
use of probabilities rejected<br />
by, 227<br />
Electrons, 287, 288; stationary<br />
states of, 74<br />
Elementary particle physics,<br />
xxviii, l, 2, 9, 10, 19, 34,<br />
230, 285-88; "bootstrap"<br />
philosophy in, 96; quantum<br />
mechanics and, 230-3 1;<br />
T-violation in, 259; wave<br />
behavior in, 179<br />
Eliade, Mircea, 39-40<br />
Elkana, Y. , 323n, 33 1 n<br />
Embryo: development of,<br />
81-82; formation of gradient<br />
system in morphogenesis of,<br />
150; internal purpose of,<br />
171-73<br />
Encyclopedie, 83<br />
Energeticists, 234<br />
INDEX<br />
Energy, 107; dissipation of,<br />
302-3; and elementary<br />
particles, 287; and entropy,<br />
118- 19; exhaustible, Ill, 114;<br />
as invariant, 265; for living<br />
cells, 155; in quantum<br />
mechanics, 220-2 1; of<br />
unstable particles, 74; of<br />
universe, 117; see also<br />
Conservation of energy<br />
Energy conversion, 12, 108,<br />
114<br />
Engels, Friedrich, 252-53, 332n<br />
Engines, 12, 103, 105-07,<br />
111-15<br />
Enlightenment, the, 67, 79, 80,<br />
86<br />
Ensemble theory (Einstein<br />
Gibbs), 247-5 1, 26 1;<br />
equilibrium and, 265<br />
Entelechy, 171<br />
Entropy, xix-xx, 12, 14-18,<br />
117-22, 227; and arrow of<br />
time, XXV, 253-54, 257-59;<br />
and atomism, 288; as barrier,<br />
277-80, 295-97; in evolution,<br />
131; flux and force and, 135,<br />
137; law of increase of, xxix;<br />
in linear thermodynamics,<br />
138-39; mechanistic<br />
interpretation of, 240-43 ;<br />
probability and, 124, 126,<br />
142, 234, 235, 237-38, 274;<br />
production of, 119, 131, 133,<br />
135, 137-39, 142; as<br />
progenitor of order, xxi-xxii;<br />
as selection principle,<br />
285-86; subjective<br />
interpretation of, 125, 235,<br />
25 1-52; thought experiment<br />
on, 244; universal<br />
interpretations of, 239<br />
Enzymes, 133-34; feedback<br />
action of, 154; in glycolysis,<br />
155; resembling Maxwell's<br />
demon, 175<br />
Epicurus, 3, 305
ORDER OUT OF CHAOS 340<br />
Equilibrium, xvi; and baker<br />
transformation, 273;<br />
chemical, 133; chemical<br />
reactions in, 179-80; and<br />
entropy, 120, 131; evolution<br />
toward, 24 1-43; flux and<br />
force at, 135-37; in future,<br />
275, 276; and matter-light<br />
interaction, 219; maximum<br />
probability at, 286; and<br />
theory of ensembles, 265;<br />
thermal, 105, 116; thermal<br />
chaos in, 168; in<br />
thermodynamics, 12, 13,<br />
125-29, 138; velocity<br />
distribution in state of, 241;<br />
see also Far-fromequilibrium;<br />
Nonequilibrium<br />
Ergodic systems, 266<br />
Espagnat, B. d', 329n<br />
Esprit de systeme, 83<br />
Euclid, 171<br />
Euler, Leonhard, 52, 65, 82<br />
Everett, 228<br />
Evolution, xx, 12, 128; and<br />
arrow of time, xxv;<br />
bifurcations in, 171-72;<br />
biological, 153; Boltzmann .<br />
on, 240; chemical, 177;<br />
concepts from physics<br />
applied to, 207-9; cosmic,<br />
215, 288; Darwinian, 128;<br />
from disorder to order, xxix;<br />
entropy in, 119, 131; toward<br />
equilibrium, 241-43;<br />
feedback in, 196-203 ;<br />
logistic, 192-96, 204;<br />
·paradigm of, 297-98; in<br />
quantum mechanics,<br />
226-228, 238; toward<br />
stationary state, 138-39;<br />
structural stability in, 189-91<br />
Existentialism, xxii<br />
Expanding universe, 2, 19, 215,<br />
259<br />
Experimentation, 5, 41-44; and<br />
global truth, 44-45 ; Kant<br />
and, 88; universality of<br />
language postulated by, 51;<br />
Whitehead on, 93, 95 ; see<br />
also Thought experiments<br />
Falling bodies, Galileo's laws<br />
for, 57, 64<br />
Far-from-equjlibrium, xxvi,<br />
xxvii, 13-14, 140-45, 300;<br />
chemical instability in,<br />
146-53; in chemistry, 177;<br />
dissipative structures in, 189;<br />
in molecular biology, 153-59;<br />
prebiotic evolution in, 191;<br />
self-<strong>org</strong>anization in, 176<br />
Faraday, Michael, 108<br />
Faust (Goethe), 128<br />
Feedback, 153; in biological<br />
systems, 154; in evolution,<br />
191, 196-203 ; between<br />
science and society, xiii<br />
Feigenbaum sequence, 169<br />
Feuer, 329n<br />
Feynman, Richard, 44<br />
Fluctuations, xv, xxiv-xxv,<br />
xxvii, 124-25, 140-41, 143;<br />
amplification of, 141, 143<br />
181-89; and chemistry,<br />
177-79; and correlations,<br />
179-8 1; in Markov process,<br />
238; on microscopic scale,<br />
23 1-32<br />
Fluid flow, 141<br />
Fluxes, 135-37; random noise<br />
in, 166-67; in reciprocity<br />
relations, 137-38<br />
Forces: generalized, 135-37; in<br />
reciprocity relations, 137-38<br />
Fourier, Baron Jean-Joseph, 12,<br />
104, 105, 107, 115-17<br />
Fraser, J. T. , 214<br />
Frederick II, King of Prussia,<br />
52<br />
Free particles, 70-72<br />
Free will, xxii<br />
Freud, Sigmund, 17<br />
Friedmann, Alexander, 215<br />
Fundamental level of<br />
description, 252-53
341<br />
Galileo, 40, 41, 50, 51, 305; on<br />
cause and effect, 60; and<br />
global truths, 44; and<br />
mechanistic world view, 57;<br />
thought experiments of, 43<br />
Galvani, Luigi, 107<br />
Gardner, Martin, 234, 259<br />
Gassendi, Pierre, 62<br />
Generalized forces, 135-37<br />
Geographical time, xviii<br />
Geography, 197; internal time<br />
in, 272<br />
Geology, 121; time in, 116, 208<br />
Gibbs, J. W., 15, 238, 247-5 1,<br />
261<br />
Gillispie, C. C., 31, 321n<br />
Glycolysis, 155-56<br />
Goethe, Johann Wolfgang von,<br />
128<br />
Gould, Stephen J. , 204<br />
Grasse, 181, 186<br />
Gravitation: Comte on, 105;<br />
and determination of motion,<br />
59; in early universe, 298;<br />
Einstein's interpretation of,<br />
34; explanatory power of, 28,<br />
29; in far-from-equilibrium<br />
conditions, 163-64; universal<br />
law of, I, 12, 66<br />
Guldberg and Waage's law (also<br />
Mass action, law oO, 133<br />
Hamilton, William Rowan, 68,<br />
%; see Hamiltonian<br />
Hamiltonian: equation, 249;<br />
function, 68, 70-7 1, 74, 107,<br />
220-2 1; operator, 221,<br />
226-27; and T-violation, 259<br />
Hankins, Thomas, 318n<br />
Hao Bai-lin, 151, 152<br />
Hausheer, Roger, 2<br />
Hawking, 117<br />
Heat, 12, 79, 103; conduction<br />
of, 104, 135; electricity<br />
produced by, 108; and heat<br />
engines, 12, 103, 106-7; heat<br />
engines, arrow of time and,<br />
111-15; propagation of,<br />
INDEX<br />
104-5; repelling force of, 66;<br />
specific, 106; transformation<br />
of matter by, 105<br />
Hegel, G. W. R, 79, 89-90, 92,<br />
93, 173<br />
Heidegger, Martin, 32-33, 42,<br />
79, 310<br />
Heisenberg, Werner, xxii, 220,<br />
292, 2%<br />
Heisenberg uncertainty<br />
relations, 178, 222-26<br />
Helmholtz, Hermann Ludwig<br />
Ferdinand von, %, 109-11<br />
Herivel, J. , 322n<br />
Hess, Benno, 155<br />
Hirsch, J. , 151<br />
History: of ideas, 79; open<br />
character of, 207; reinsertion<br />
of, into natural and social<br />
sciences, 208; of science, 307<br />
(cosmology) 208, 215,<br />
(geography) 197, 272,<br />
(geology) 116, 121, 208<br />
Holbach, Paul Henri Thiry,<br />
baron d ' , 82<br />
Hooykaas, R., 317n<br />
Hopf, 266<br />
Hubble, Edwin Powell , 215<br />
Humanities, schism between<br />
science and, 11, 13<br />
Hume, A. Ord, 317n<br />
Huss, John, xxii<br />
Huyghens, Christiaan, 60<br />
Hydrodynamics, 127; far-fromequilibrium<br />
phenomena in,<br />
141<br />
Hypnons, 180, 287, 288<br />
Hysteresis, 166<br />
Idealization, 41-43, 69,<br />
112-1 14, 115, 120, 216, 248,<br />
252, 305-06<br />
Impossibility, demonstrations<br />
of, 17, 216- 17, 296, 299-300<br />
Individual time, xviii<br />
Industrial Age , 111; combustion<br />
and, 103
ORDER OUT OF CHAOS<br />
342<br />
Information, 17-18, 250,<br />
278-79, 283, 295, 297-98<br />
Initial conditions (or state), 61 ,<br />
68, 75, 121, 124, 128, 129,<br />
139-40, 142, 147, 248,<br />
261 -67, 270-7 1, 276, 278-79,<br />
295, 310<br />
Innovation, psychological<br />
process of, xxiv<br />
Innovative becoming,<br />
philosophy of, 94<br />
Instability, dynamic, 73,<br />
268-72, 276, 300; chemical,<br />
144-53; thermodynamic,<br />
141-42; threshold, 146, 147,<br />
160<br />
Insects, self-aggregation of,<br />
181-86<br />
Integrable systems, 71-72, 74,<br />
264-65, 302<br />
Internal time, 272-73, 289<br />
Intrinsically irreversible<br />
systems, 275-77, 289<br />
Intrinsically random systems,<br />
274-76, 289<br />
Intuition, 80, 91, 92<br />
Irreversibility, xx, xxi, xxvii,<br />
xxviii, 7-9, 63, 115;<br />
acceptance by physics of,<br />
208-9; and biology, 128, 175;<br />
in chemistry, 131, 137, 177;<br />
controversy over, 15-16;<br />
cultural context of<br />
incorporation into physics of,<br />
309- 10; and dynamics of<br />
correlations, 280-85 ; Einstein<br />
on, 294-95; and ensemble<br />
theory, 250; in evolution, 128,<br />
189; formulation of theory of,<br />
105, 107, 117-2l; and limits<br />
of classical concepts, 261-64;<br />
and matter-light interaction,<br />
219; measurement and, 228;<br />
microscopic theory of, 242,<br />
257-59, 285-86, 288-90, 310;<br />
probability and, 16, 124, 125,<br />
233-40; quantitative<br />
expression of, Ill; from<br />
randomness to, 272-77; rate<br />
of, 135; in reciprocity<br />
relations, 138; as source of<br />
order, 15, 292; subjective<br />
interpretation of, 25 1-52; as<br />
symmetry-breaking process,<br />
260-6 1; in thermodynamics,<br />
12; see also Arrow of time;<br />
Entropy<br />
Isomerization reaction, 165<br />
Jammer, M., 329n<br />
Jordan, 220<br />
Joule, James Prescott, 108-9<br />
Kant, Immanuel, 79, 80, 85-89,<br />
93, 99, 214<br />
Kauffman, S. A., 172<br />
Kepler, Johannes, 49, 57, 67,<br />
307<br />
Keynes, Lord, 319n<br />
Kierkegaard, Soren, 79<br />
Kinetic energy, 69-7 1, 90, 107,<br />
261<br />
Kinetics, chemical , 132-34<br />
Kirchoff, Gustav, %<br />
Knight, 321 n<br />
Koestler, Arthur, 32, 34-35<br />
Kolmogoroff, 266<br />
Kothari, D. S., 293<br />
Koyre, Alexandre, 5, 32, 35-36,<br />
62, 317n, 319n<br />
Kuhn, Thomas, 307-9, 320n,<br />
329n<br />
Lagrange, Comte Joseph Louis,<br />
52, %, 104, 324n<br />
Laminar flow, 141-42, 303<br />
Laplace, Marquis Pierre Simon<br />
de, xiii, 28, 52, 54, 66, 67,<br />
115, 323n; Fourier criticized<br />
by, 104<br />
Laplace's demon, 75-77, 87,<br />
27 1<br />
Large numbers, law of, 14, 178,<br />
180<br />
Lavoisier, Antoine Laurent, 2&,<br />
109
343 INDEX<br />
Layzer, David, xxv<br />
Lebenswe/t, 299<br />
Leibniz, Gottfried von, 50, 54,<br />
302, 303; formulae for<br />
velocity and acceleration, 58;<br />
monads of, 74<br />
Lemaitre, Ge<strong>org</strong>es, 215<br />
Lenoble, R., 3<br />
Levi-Strauss, Claude, 205, 317n<br />
Levy-Bruhl, L., 292<br />
Lewis, G. N., 233<br />
Liebig, Baron Justus von, 109<br />
Life: Bergson on, 92;<br />
compatibility with far-fromequilibrium<br />
conditions, 143;<br />
as expression of self<strong>org</strong>anization,<br />
175-76; and<br />
order principle, 127-28;<br />
origin of, 14; Romantic<br />
concepts of, 85; Stahl's<br />
definition of, 84; symmetrybreaking<br />
as characteristic of,<br />
163; temporal dimensions of,<br />
208; see also Molecular<br />
biology<br />
Light: velocity of, 17, 55,<br />
217-19, 278, 295, 296; waveparticle<br />
duality of, 219-20<br />
Limit cycle, 146-47<br />
Linear thermodynamics,<br />
137-40<br />
Liouville equation, 249, 250,<br />
266<br />
Logistic evolution, 192-96,<br />
203-04<br />
Look at the Harlequins<br />
(Nabokov), 277<br />
Lorentz, Hendrik Antoon, 270<br />
Loschmidt, 244, 246<br />
Louis XIV, King of France, 52<br />
Love, Milton, 195<br />
Lucretius, 3, 141, 302-5, 315n,<br />
334n<br />
Luther, Martin, xxii<br />
Lyapounov, 151<br />
Mach. Ernst. 49. 53-54. 97.<br />
318n<br />
Machines: Archimedes's, 41;<br />
ideal, 63, 69-70; mathematics<br />
and, 46; using heat, 103<br />
Macroscopic system, 106-07<br />
Many-worlds hypothesis, 228<br />
Markov chains, 236, 238, 240,<br />
242, 273-76; and dynamics of<br />
correlations, 283 ; and entropy<br />
barrier, 278<br />
Marx, Karl, 252<br />
Mass action, law of, 133, 136,<br />
23 1<br />
Materialistic naturalism, 83<br />
Mathematization, 46; in<br />
Hamiltonian function, 71;<br />
Hegel's critique of, 90;<br />
Leibniz on, 50; of motion, 60<br />
Matter: active, 9, 286-90, 302;<br />
anti-matter, 230-3 1; Diderot<br />
on, 82; effect of heat on, 105;<br />
in far-from-equilibrium<br />
conditions, 14; interaction of<br />
radiation and, 219; new view<br />
of, 9; nonequilibrium<br />
generated by, 181; perception<br />
of differences by, 163, 165;<br />
properties of, 2; Stahl on,<br />
84-85; transition to life from,<br />
84; wave-particle duality of,<br />
221<br />
Maxwell, James Clerk, 54, 73,<br />
122, 160, 240, 24 1, 266<br />
Maxwell's demon, 175, 239<br />
Mayer, Julius Robert von, 109,<br />
Ill<br />
Measurement, irreversible<br />
character of, 228-29<br />
Mechanics, II, 15;<br />
generalization of, Ill; Hegel<br />
on, 90; and probability, 125;<br />
see also Dynamics; Quantum<br />
mechanics<br />
Medicine: Bergson on, 91;<br />
Diderot on, 82, 83<br />
Merleau-Ponty, M., 299<br />
Metternich, Clemens We nzel<br />
Nepomuk Lothar, Fiirst von,<br />
xiii
ORDER OUT OF CHAOS 344<br />
Meyerson, Emile, 293<br />
Microcanonical ensemble, 265<br />
Minimum entropy production,<br />
theorem of, 138-4 1<br />
Minkowski, H., 230<br />
Mole, 121n<br />
Molecular biology, 4, 8; farfrom-equilibrium<br />
conditions<br />
in, 153-59; vitalism and, 84<br />
Molecular chaos assumption,<br />
246<br />
Monads, 74, 302-03<br />
Monod, Jacques, 3-4, 22, 36,<br />
79, 84<br />
Morin, Edgar, xxii-xxiii<br />
Morphogenesis, 172, 189<br />
Moscovici, Serge, 22, 306-7<br />
Motion: and change, 62-68;<br />
complexity of, 75 ; instability<br />
of, 73; in mechanical engine<br />
vs. heat engine, 112;<br />
positivist notion of, 96;<br />
productton of, in heat engine,<br />
107; universal laws of, 57-62,<br />
83; see also Dynamics<br />
Nabokov, Vladimir, 277-78<br />
Napoleon, 52, 67<br />
Natural laws: belief in<br />
universality of, 1-2;<br />
mathematical concepts of, 46;<br />
Newton on, 28; primary and<br />
secondary, 8; social structure<br />
and views of, 48-49; timeindependent,<br />
2, 7; trials of<br />
animals for infringements of,<br />
48<br />
Nature of the Physical World,<br />
The (Eddington), 8<br />
Needham, Joseph, 6-7, 45, 48,<br />
49, 278, 322n<br />
"Neolithic Revolution," 5-6, 37<br />
Neumann, von, 266<br />
New Science, The (Vico), 4<br />
Newton, Isaac, xv, xxviii, 12,<br />
27-29, 76, 98, 104, 120, 124,<br />
234, 305, 319n; alchemy and,<br />
64; on change, 62, 63 ;<br />
eighteenth-century opposition<br />
to, 65 ; laws of motion, 70;<br />
and mechanistic world view,<br />
57; objectivity defined by,<br />
218; objects chosen for study<br />
by, 216; presentation of<br />
Principia to Royal Society, 1;<br />
second law, 58; see also<br />
Newtonian science<br />
Newtonian science, xiii, xiv,<br />
xix, xxv, xxvi, 37-40, 213;<br />
absence of universal constant<br />
in, 217; concept of change in,<br />
63-68; Diderot and, 80, 82;<br />
Koyre on, 35-36;<br />
incompleteness of, 209;<br />
instability and, 264;<br />
instability of cultural position<br />
of, 30; Kantian critique of,<br />
85-87; laws of motion of,<br />
57-59; limits of, 29-30;<br />
positivism and, 96; prophetic<br />
power of, 28; spread of,<br />
28-29; Voltaire and, 258;<br />
world view of, 229<br />
. Nietzsche, Friedrich, Ill, 136<br />
Nisbet, R., 79<br />
N onequilibrium: cosmological<br />
dimension of, 23 1; difference<br />
between particles and<br />
antiparticles in, 285 ;<br />
fluctuations in, 178-80;<br />
innovation and, xxiv; and<br />
origin of structures, xxix; as<br />
source of order, 287; see also<br />
Far-from-equilibrium<br />
Non-linearity, 14, 134, 153,<br />
154-55, 197 see Catalysis<br />
Non-linear thermodynamics,<br />
137, 140<br />
Noyes, 152<br />
Nucleation, 187, 188<br />
Oersted, Hans Christian, 108<br />
Old Te stament, xxii<br />
Onsager, Lars, 137, 138<br />
Operators, 22 1-22, 225;<br />
commuting, 223
345<br />
Opticks (Newton), 28<br />
Optimization, 197, 207<br />
Order, 12, 18, 126, 131, 143,<br />
171-75, 238, 246, 250-5 1,<br />
286-87<br />
Order through fluctuation, 159,<br />
178; models based on<br />
concept of, 206<br />
Organization theory, xxiv<br />
Oscillating chemical reactions,<br />
19, 147-49<br />
Oscillations: glycolytic, 155;<br />
time- and space-dependent,<br />
148<br />
Ostwald, Wilhelm, 324n<br />
Pascal, Blaise, 3, 36, 79<br />
Pasteur, Louis, 163<br />
Pattern selection, 163, 164<br />
Pearson, Karl, 49<br />
Peirce, Charles S., 17, 302-3<br />
Pendulum, 16, 73, 216, 261-62<br />
Phase changes, 187<br />
Phase space, 247-50, 261, 264;<br />
delocalization in, 289;<br />
unstable systems in, 266-72<br />
Photons, 230, 288<br />
Physicists, The (DOerrenmatt),<br />
21<br />
Physics: application of concepts<br />
to evolution, 207-9; Bergson<br />
on, 91, 92; changing<br />
perspective in, 8-9;<br />
complementary developments<br />
in biology and, 154; and<br />
concepts of change, 63;<br />
conceptual distinction<br />
between chemistry and, 137;<br />
Diderot on, 80-83;<br />
evolutionary paradigm in,<br />
297-98; inspired discourse of,<br />
76; introduction of<br />
probability in, 123; and laws<br />
of motion, 57; Lucretian,<br />
141 ; macroscopic, xii;<br />
objectivity in, 55 ; positivist<br />
view of, 97; of processes,<br />
INDEX<br />
243 ; and theology, 49; time<br />
in, 116; vitalism and, 84;<br />
Whitehead on, 95, 96<br />
Planck, Max, 121, 219, 242,<br />
324n, 329n, 33 ln; on second<br />
law of thermodynamics,<br />
234-35<br />
Planck's constant, 217, 219,<br />
220, 223, 224<br />
Planetary motion: Kepler's laws<br />
for, 57; in Newtonian<br />
dynamics, 59, 64<br />
Plato, 7, 39, 67<br />
Poincare, Jules Henn, 68, t2,<br />
97, 151, 236, 243, 253, 265<br />
27 1<br />
Poisson distribution, 179-8 1<br />
Pope, Alexander, 27, 67<br />
Popper, Karl, 5, 15, 254-55,<br />
258-59, 276, 317n<br />
Populiire Schriften<br />
(Boltzmann), 240<br />
Positive-feedback loops, xvit<br />
Positivism, 80, 96-98; Comte<br />
and, 104-5; German<br />
philosophy and, 109<br />
Potential: dynamic, 69-70;<br />
thermodynamic, 126, 138-40<br />
Potential energy, 69-70, 73, 107<br />
Prebiotic evolution, 190-91<br />
Pre-Socratics, 38-39<br />
Principia (Newton), l, 28<br />
Probability, 122-24; Einstein<br />
on, 259; at equilibrium, 286;<br />
in far-from-equilibrium<br />
conditions, 143; entropy and<br />
142, 274, 297; and<br />
fluctuations, 178, 179; and<br />
irreversibility, 233-40; in<br />
quantum mechanics, 227;<br />
subjective vs. objective<br />
interpretations of, 274; in<br />
unstable systems, 271-72<br />
Process: physics of, 12, 105,<br />
107, 243; Whitehead's<br />
concept of, 258, 303<br />
Process and Reality<br />
(Whitehead), 93, 96, 310
ORDER OUT OF CHAOS 346<br />
Proust, Marcel , 17<br />
Pulley systems, 41-42<br />
Quantization, 220<br />
Quantum mechanics, 9, II, 15,<br />
34, 218-22; causality in, 31 1;<br />
correlations in, 286; cultural<br />
background to, 6;<br />
delocalization in, 289;<br />
demonstrations of<br />
impossibility in, 217, 296-97;<br />
Hamiltonian function in, 70;<br />
Heisenberg's uncertainty<br />
relations in, 222-26; and<br />
Newtonian synthesis, 67-68;<br />
and probability, 125, 178-79;<br />
and reversibility, 61; temporal<br />
evolution in, 226-29, 238;<br />
thought experiments in, 43<br />
Quetelet, Adolphe, 123, 241<br />
Radiation: black-body, 209, 215;<br />
interaction of matter and, 219<br />
Randomness, xx, 8, 9, 126,<br />
23 1-32, 236; to irreversibility<br />
from, 272-77<br />
Rationality, 1, 29, 32, 36, 40,<br />
42, 92, 306<br />
Reaction-diffusion systems, 260<br />
Real ity, conceptualization of,<br />
225-26<br />
Reciprocity relations, 137-38<br />
Reduction of the wave function,<br />
227-28<br />
Reductionism, 173-74<br />
Reichenbach, H., 97<br />
Relation, philosophy of, 95<br />
Relativity, 9, 34, 215, 307; in<br />
astrophysics, 116; Bergson's<br />
misunderstanding of, 294;<br />
demonstrations of<br />
impossibility in, 217, 296;<br />
Einstein's special theory of,<br />
17; and elementary particles,<br />
230; and Newtonian<br />
synthesis, 67, 68, 229; static<br />
geometric character of time<br />
and , 230; and thermal history<br />
of universe, 23 1; thought<br />
experiments in, 43 ; and<br />
universal constants, 217 -18;<br />
and velocity of light, 295<br />
Religion: ancient Greek, 38, 39;<br />
resonance between science<br />
and, 46-5 1<br />
Residual black-body radiation,<br />
209, 215<br />
Respiration, physiology of, 109<br />
Reversibility: of canonical<br />
equations, 71; of<br />
thermodynamic<br />
transformation, 12, 112-13,<br />
120; of trajectories, 60-61<br />
Revolution, concept of, xxiv<br />
Reynolds' number, 144<br />
Ritz, W, 259<br />
Rosenfeld, Leon, 264, 329<br />
Sakharov, A. D., 230<br />
Sartre, Jean-Paul, xxii<br />
Schlanger, J. , 321n<br />
Schrodinger, Erwin, 18-19 220,<br />
242, 329n<br />
SchrOdinger equation, 226-29<br />
Science and Civilization in<br />
China (Needham), 278<br />
"Scientific revolution," 5, 6<br />
Scott, W, 322n<br />
Sein und Zeit (Heidegger), 310<br />
Self-<strong>org</strong>anization, xv; in Benard<br />
instability, 142; and<br />
bifurcations, 160-67; in<br />
chemical clock, 148; in<br />
chemical reactions, 144-45;<br />
and dynamics, 208; as<br />
function of fluctuating<br />
external conditions, 165-67;<br />
life as expression of, 175-76;<br />
in slime-mold aggregation,<br />
156; in turbulence, 141-42<br />
Serres, Michel, 104, \41, 303,<br />
304, 320n, 323n<br />
Shakespeare, William, 293<br />
Signals, propagation of, 217,<br />
218
3
ORDER OUT OF CHAOS 348<br />
Time (cont'd)<br />
277-78, 295-%; in everyday<br />
life, 16-17; global judgments<br />
of, 17; in Hamiltonian<br />
function, 70, 71; Hegel on,<br />
90; and human symbolic<br />
activity, 312; internal,<br />
272-73, 289; involved in<br />
turbulence, 141; meaning of,<br />
in physics, 93 ; as measure of<br />
change, 62; in nineteenthcentury<br />
physics, 117;<br />
oscillations dependent on,<br />
148; positivist view of, 97;<br />
progressive rediscovery of,<br />
208; in quantum mechanics.<br />
229-30; revision of<br />
conception of, 96; roots of, in<br />
nature, 18; social context of<br />
rediscovery of, 19; in<br />
thermodynamics, 12, 129;<br />
varying importance of scales<br />
of, 301; see also Arrow of<br />
time; Irreversibility<br />
Time Machine, The (Wells), 277<br />
Toftler, Alvin, xi-xxvi, xxxi<br />
Trajectories, 59-60, 68, 121,<br />
177; intrinsically<br />
indeterminate, 73 ; limits of<br />
concept of, 261-64; in phase<br />
space, 247-48; and<br />
probability, 122; in unstable<br />
systems, 270-72; variations<br />
in, 75<br />
Transition probabilities, 274<br />
Thrbulence, 141<br />
Turbulent chaos, 167-68<br />
Turing, Alan M., 152<br />
Unidirectional processes,<br />
258-59<br />
"Unified field theory," 2<br />
Universal constants, 217-19,<br />
229<br />
Universe: age of, 1; aging of,<br />
xix-xx; disintegration of,<br />
116; energy of, 117; entropy<br />
of, 118; expanding, 2, 19,<br />
215, 259; history of, 215; in<br />
Newtonian dynamics, 59;<br />
nonequilibrium, 22932;<br />
Pierce on, 302-3; pluralistic<br />
character of, 9; thermal<br />
history of, 9; time-oriented<br />
polarized nature of, 285<br />
Unstable particles, 74, 23 1, 288<br />
Urbanization, model of,<br />
198-202<br />
Urn model, 235-38, 246, 273<br />
Valery, Paul, 16, 301<br />
Velocity, 57-59; distribution of,<br />
240-42; instantaneous<br />
inversion of, 61, 243-46,<br />
280-85; of light, 17, 55,<br />
217-19, 278, 295, 2%<br />
Velocity distribution function,<br />
242-46, 248-50<br />
Velocity inversion experiment,<br />
280-84<br />
Venel, 83, 309<br />
Vico, G., 4<br />
Vienna school, 97<br />
Vitalism, 80, 83-84; vs.<br />
scientific methodology, llO<br />
Volta, Count Alessandro, 107-8<br />
Voltaire, 257-58<br />
Waddington, Conrad H., 172,<br />
174, 207, 322n, 326n<br />
Watanabe, S., 330n<br />
Watt, James, 103<br />
Wave behavior, 179; see also<br />
Chemical clocks<br />
Wave functions, 226-28; time<br />
and, 229-30<br />
Wave-particle duality, 219-20,<br />
226<br />
Wealth of Nations (Smith). 103<br />
Weiss, Paul, 174<br />
Wells, H. G., 277<br />
Wey l, Herman, 311
34Q<br />
Whitehead, Alfred North, 10,<br />
17, 47, 50, 79, 93-96, 212,<br />
216, 258, 302, 303, 310, 322n<br />
Wiener. Norbert, 295-96<br />
Wycliffe , John, xxii<br />
INDEX<br />
Zermelo, 15, 244, 253, 254<br />
"Zero growth" society, 116<br />
Zhang Shu-yu, 151<br />
Zola, Emile, 323n