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ORDER OUT OF CHAOS
ORDER OUT OF CHAOS:
MAN'S NEW DIALOGUE WITH NATURE
A Bantam Book I April 1984
New Age and the accompanying figure design as well as the
statement "a search for meaning, growth and change" are
trademarks of Bantam Books, Inc.
All rights reserved.
Copyright© 1984 by llya Prigogine and Isabelle Stengers.
T he foreword "Science and Change" copyright© 1984 by
Alvin Tofjler.
Book design by Barbara N. Cohen
T his book may not be reproduced in whole or in part, by
mimeograph or tiny other means, without permission.
For information address: Bantam Books, Inc.
Library of Congress Cataloging in Publication Data
Prigogine. I. (llya)
Order out of chaos.
Based on the authors' Ia nouvelle alliance.
Includes bibliographical references and index.
I. Science-Philosophy. 2. Physics-Philosophy.
3. Thermodynamics. 4. Irreversible processes.
I. Stengers, Isabelle. II. Prigogine, I. (Ilya)
La nouvelle alliance. Ill. Title.
QI75.P8823 1984 510
83-21 403
ISBN 0-553-34082-4
Published simultaneously in the United States and Canada
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PRINTED IN THE UNITED STATES OF AMERICA
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ORDER OUT OF CHAOS
MAN'S NEW DIALOGUE
WITH NATURE
llya Prigogine
and
Isabelle Stengers
Foreword by
Alvin Toffler
TORONTO·
BANTAM BOOKS
NEW YORK LONDON SYDNEY
•
•
This book is dedicated to the memory of
Erich Jantsch
Aharon Katchalsky
Pierre Resibois
Leon Rosenfeld
TABLE OF CONTENTS
Science and Change by Alvin Toffier xi
FOREWORD:
PREFACE:
Man's New Dialogue with Nature
INTRODUCTION:
The Challenge to Science
xxvii
1
Book One: The Delusion of the Universal
CHAPTER I:
The Triumph of Reason 27
The New Moses 27
A Dehumanized World 30
The Newtonian Synthesis 37
4. The Experimental Dialogue 41
5. The Myth at the Origin of Science 44
6. The Limits of Classical Science 51
1.
2.
3.
CHAPTER n:
The Identification of the Real 57
Newton's Laws 57
Motion and Change 62
The Language of Dynamics
4. Laplace's Demon 75
1.
2.
3.
CHAPTER m:
1.
2.
3.
4.
5.
6.
The 1Wo Cultures
68
79
Diderot and the Discourse of the Living 79
Kant's Critical Ratification 86
A Philosophy of Nature? Hegel and Bergson 89
Process and Reality: Whitehead 93
"Ignoramus, lgnoramibus":
The Positivist's Strain 96
A New Start 98
Book Two: The Science of Complexity
CHAPTER IV:
Energy and the Industrial Age
103
1.
2.
3.
4.
5.
6.
7.
Heat, the Rival of Gravitation 1(}3
The Principle of the Conservation of Energy 107
Heat Engines and the Arrow of Time 111
From Technology to Cosmology 115
The Birth of Entropy 117
Boltzmann's Order Principle 122
Carnot and Darwin 127
CHAPTER v:
The Three Stages of Thermodynamics
131
1. Flux and Force 131
2. Linear Thermodynamics 137
3. Far from Equilibrium 140
4. Beyond the Threshold of Chemical Instability 146
5. The Encounter with Molecular Biology 153
6. Bifurcations and Symmetry-Breaking 160
7. Cascading Bifurcations and
the Transitions to Chaos 167
8. From Euclid to Aristotle 171
CHAPTER v1:
Order Through Fluctuations
177
1. Fluctuations and Chemistry 177
2. Fluctuations and Correlations 179
3. The Amplification of Fluctuations 181
4. Structural Stability 189
5. Logistic Evolution 192
6. Evolutionary Feedback 196
7. Modelizations of Complexity 203
8. An Open World 207
Book Three: From Being to Becoming
CHAPTER vn:
Rediscovering Time 213
A Change of Emphasis 213
The End of Universality 217
3. The R ise of Quantum Mechanics 218
4. Heisenberg's Uncertainty Relation 222
1.
2.
5. T h e Temporal Evolution of Quantum Systems
6. A Nonequilibrium Universe
229
226
CHAPTER vm:
The Clash of Doctrines 233
1.
2.
3.
4.
Probability and Irreversibility 233
Boltzmann's Breakthrough 240
Questioning Boltzmann's Interpretation 243
Dynamics and Thermodynamics: 1Wo Separate
Worlds 247
5. Boltzmann and the Arrow of Time 253
CHAPTER IX:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Irreversibility-the Entropy Barrier 257
Entropy and the Arrow of Time 257
Irreversibility as a Symmetry-Breaking Process 260
The Limits of Classical Concepts 261
The Renewal of Dynamics 264
From Randomness to Irreversibility 272
The Entropy Barrier 277
The Dynamics of Correlations 280
Entropy as a Selection Principle 285
Active Matter 286
coNcLusioNs:
From Earth to Heaventhe Reenchantment of Nature
An Open Science 291
Time and Times 293
The Entropy Barrier 295
The Evolutionary Paradigm 297
Actors and Spectators 298
A Whirlwind in a Turbulent Nature
Beyond Tautology 305
The Creative Course of Time 307
9. The Human Condition 311
10. The Renewal of Nature 312
291
1.
2.
3.
4.
5.
6.
7.
8.
NOTES
315
INDEX
335
301
FOREWORD
SCIENCE AND
CHANGE
by Alvin Toffler
One of the most highly developed skills in contemporary West
ern civilization is dissection: the split-up of problems into their
smallest possible components. We are good at it. So good, we
often forget to put the pieces back together again.
This skill is perhaps most finely honed in science. There we
not only routinely break problems down into bite-sized chunks
and mini-chunks, we then very often isolate each one from its
environment by means of a useful trick. We say ceteris par
ibus-all other things being equal. In this way we can ignore
the complex interactions between our problem and the rest of
the universe.
llya Prigogine, who won the Nobel Prize in 1977 for his
work on the thermodynamics of nonequilibrium systems , is
net satisfied, however, with merely taking things apart. He has
spent the better part of a lifetime trying to "put the pieces
back together again"-the pieces in this case being biology
and physics, necessity and chance, science and humanity.
Born in Russia in 1917 and raised in Belgium since the age
of ten, Prigogine is a compact man with gray hair, cleanly chis
eled features, and a laserlike intensity. Deeply interested in
archaeology, art, and history, he brings to science a remark
able polymathic mind. He lives with his engineer-wife, Ma
rina, and h i s son , Pascal, i n Brussels, where a cross
disciplinary team is busy exploring the i mplications of his
ideas in fields as disparate as the social behavior of ant colo
nies, diffusion reactions in chemical systems and dissipative
processes in quantum field theory.
He spends part of each year at the Ilya Prigogine Center for
Statistical Mechanics and Thermodynamics of the University
of Texas in Austin. To his evident delight and surprise, he was
xi
ORDER OUT OF CHAOS
xii
awarded the Nobel· Prize for his work on "dissipative struc
tures" arising out of nonlinear processes in nonequilibrium
systems. The coauthor of this volume, Isabelle Stengers, is a
philosopher, chemist, and historian of science who served for
a time as part of Prigogine 's Brussels team. She now lives in
Paris and is associated with the Musee de Ia Villette.
In Order Out of Chaos they have given us a landmark-a
work that is contentious and mind-energizing, a book filled
with flashing insights that subvert many of our most basic as
sumptions and suggest fresh ways to think about them.
Under the title La nouvelle alliance, its appearance in
France in 1979 triggered a marvelous scientific free-for-all
among prestigious intellectuals in fields as diverse as entomol
ogy and literary criticism.
It is a measure of America's insularity and cultural ar
rogance that this book, which is either published or about to
be published in twelve languages, has taken so long to cross
the Atlantic. The delay carries with it a silver lining, however,
in that this edition includes Prigogine 's newest findings, par
ticularly with respect to the Second Law of thermodynamics,
which he sets into a fresh perspective.
For all these reasons, Order Out of Chaos is more than just
another book: It is a lever for changing science itself, for com
pelling us to reexamine its goals, its methods, its epistemol
ogy-its world view. Indeed, this book can serve as a symbol
of today's historic transformation in science-one that no in
formed person can afford to ignore.
·
Some scholars picture science as driven by its own internal
logic, developing according to its own laws in splendid isola
tion from the world around it. Yet many scientific hypotheses ,
theories, metaphors, and models (not t o mention the choices
made by scientists either to study or to ignore various prob
lems) are shaped by economic, cultural, and political forces
operating outside the laboratory.
I do not mean to suggest too neat a parallel between the nature
of society and the reigning scientific world view or "paradigm."
,
Still less would I relegate science to some "superstructure .
mounted atop a socioeconomic "base," as Marxists are wont to
do. But science is not an "independent variable.'' It is an open
system embedded in society and linked to it by very dense feed
back loops. It is powerfully influenced by its external environ-
xiii
FOREWORD: SCIENCE AND CHANGE
ment, and, in a general way, its development is shaped by cultural
receptivity to its dominant ideas.
Take that body of ideas that came together in the seven
teenth and eighteenth centuries under the heading of "classi
cal science" or " Newtonianism." They pictured a world in
which every event was determined by initial conditions that
were, at least in principle, determinable with precision. It was
a world in which chance played no part, in which all the pieces
came together like cogs in a cosmic machine.
The acceptance of this mechanistic view coincided with the
rise of a factory civilization. And divine dice-shooting seems
hardly enough to account for the fact that the Age of the Ma
chine enthusiastically embraced scientific theories that pic
tured the entire universe as a machine.
This view of the world led Laplace to his famous claim that,
given enough facts, we could not merely predict the future but
retrodict the past. And this image of a simple, uniform; me
chanical universe not only shaped the development of science,
it also spilled over into many other fields. It influenced the
framers of the American Constitution to create a machine for
governing, its checks and balances clicking like parts of a
clock. Metternich, when he rode forth to create his balance of
power in Europe, carried a copy of Laplace's writings in his
baggage. And the dramatic spread of factory civilization, with
its vast clanking machine s , its heroic engineering break
throughs , the rise of the railroad , and new industries such as
steel, textile, and auto, seemed merely to confirm the image of
the universe as an engineer's Tinkertoy.
Today, however, the Age of the Machine is screeching to a
halt, if ages can screech-and ours certainly seems to. And
the decline of the industrial age forces us to confront the pain
ful limitations of the machine model of reality.
Of course, most of these limitations are not freshly dis
covered. The notion that the world is a clockwork, the planets
timelessly orbiting, all systems operating deterministically in
equilibrium, all subject to universal laws that an outside ob
server could discover-this model has come under withering
fire ever since it first arose.
In the early nineteenth century, thermod ynamics chal
lenged the timelessness implied i n the mechanistic image of
the universe . If the world was a big machine, the ther
modynamicists declared, it was running down , its useful en-
ORDER OUT OF CHAOS
xiv
ergy leaking out. It could not go on forever, and time ,
therefore, took on a new meaning. Darwin's followers soon
introduced a contradictory thought: The world-machine might
be running down, losing energy and organization, but biolog
ical systems, at least, were running up, becoming more, not
less, organized.
By the early twentieth century, Einstein had come along to
put the observer back into the system: The machine looked
different-indeed, for all practical purposes it was different
depending upon where you stood within it. But it was still a
deterministic machine, and God did not throw dice. Next, the
quantum people and the uncertainty folks attacked the model
with pickaxes, sledgehammers, and sticks of dynamite.
Nevertheless, despite all the ifs, ands, and buts, it remains
fair to say, as Prigogine and Stengers do, that the machine par
adigm is still the "reference point" for physics and the core
model of science in general. Indeed, so powerful is its con
tinuing influence that much of social science, and especially
economics, remains under its spell.
The importance of this book is not simply that it uses orig
inal arguments to challenge the Newtonian model, but also
that it shows how the still valid, though much limited, claims
of Newtonianism might fit compatibly into a larger scientific
image of reality. It argues that the old "universal laws" are not
universal at all, but apply only to local regions of reality. And
these happen to be the regions to which science has devoted
the most effort.
Thus, in broad-stroke terms, Prigogine and Stengers argue
that traditional science in the Age of the Machine tended to
emphasize stability, order, uniformity, and equilibrium. It con
cerned itself mostly with closed systems and linear relation
ships in which small inputs uniformly yield small results.
With the transition from an industrial society based on
heavy inputs of energy, capital, and labor to a high-technology
society in which information and innovation are the critical
resources, it is not surprising that new scientific world models
should appear.
What makes the Prigoginian paradigm especially interesting
is that it shifts attention to those aspects of reality that charac
terize today's accelerated social change: disorder, instability,
diversity, disequilibrium, nonlinear relationships (in which
xv
FOREWORD: SCIENCE AND CHANGE
small inputs can trigger massive consequences), and tem
porality-a heightened sensitivity to the flows of time.
The work of Ilya Prigogine and his colleagues in the so
called " Brussels school" may well represent the next revolu
tion in science as it enters into a new dialogue not merely with
nature, but with society itself.
The ideas of the Brussels school , based heavily on Pri
gogine's work, add up to a novel, comprehensive theory of
change.
Summed up and simplified, they hold that while some parts of
the universe may operate like machines, these are closed systems,
and closed systems, at best, form only a small part of the physical
universe. Most phenomena of interest to us are, in fact, open
systems, exchanging energy or matter (and, one might add, infor
mation) with their environment. Surely biological and social sys
tems are open, which means that the attempt to understand them
in mechanistic terms is doomed to failure.
This suggests , moreover, that most of reality, instead of
being orderly, stable, and equilibria!, is seething and bubbling
with change, disorder, and process.
In Prigoginian ter m s , all systems contain subs ystems,
which are continually "fluctuating." At times, a single fluctua
tion or a combination of them may become so powerful, as a
result of positive feedback, that it shatters the preexisting or
ganization. At this revolutionary moment-the authors call it
a "singular moment" or a "bifurcation point"-it is inherently
impossible to determine in advance which direction change
will take: whether the system will disintegrate into "chaos" or
leap to a new, more differentiated , higher level of "order" or
organization, which they call a "dissipative structure." (Such
physical or chemical structures are termed dissipative be
cause, compared with the simpler structures they replace , they
require more energy to sustain them.)
One of the key controversies surrounding this concept has
to do with Prigogine's insistence that order and organization
can actually arise "spontaneously" out of disorder and chaos
through a process of "self-organization . "
To grasp this extremely powerful idea. we first need to make
a distinction between systems that are in "equilibrium," sys-
ORDER OUT OF CHAOS
xvi
terns that are "near equilibrium," and systems that are "far
from equilibrium. "
Imagine a primitive tribe . If its birthrate and death rate are
equal, the size of the population remains stable. Assuming ad
equate food and other resources, the tribe forms part of a local
system in ecological equilibrium.
Now increase the birthrate . A few additional births (without
an equivalent number of deaths) might have little effect. The
system may move to a near-equilibria! state. Nothing much
happens. It takes a big jolt to produce big consequences in
systems that are in equilibria] or near-equilibria] states.
But if the birthrate should suddenly soar, the system is
pushed into a far-from-equilibrium condition, and here non
linear relationships prevail. In this state , systems do strange
things. They become inordinately sensitive to external influ
ences. Small inputs yield huge, startling effects. The entire
system may reorganize itself in ways that strike us as bizarre.
Examples of such self-reorganization abound in Order Out
of Chaos. Heat moving evenly through a liquid suddenly, at a
certain threshold, converts into a convection current that radi
cally reorganizes the liquid, and millions of molecules, as if on
cue, suddenly form themselves into hexagonal cells.
Even more spectacular are the "chemical clocks" described
by Prigogine and Stengers. Imagine a million white ping-pong
balls mixed at random with a million black ones, bouncing
around chaotically in a tank with a glass window in it. Most of
the time, the mass seen through the window would appear to
be gray, but now and then, at irregular moments, the sample
seen through the glass might seem black or white, depending
on the distribution of the balls at that moment in the vicinity of
the window.
Now imagine that suddenly the window goes all white , then
all black, then all white again, and on and on, changing its
color completely at fixed intervals-like a clock ticking.
Why do all the white balls and all the black ones suddenly
organize themselves to change color in time with one another?
By all the traditional rules, this should not happen at all. Yet, if
we leave ping-pong behind and look at molecules in certain chemi
cal reactions, we find that precisely such a self-organization or
ordering can and does occur--despite what classical physics and
the probability theories of Boltzmann tell us.
In far-from-equilibrium situations other seemingly spon-
xvii
FOREWORD: SCIENCE AND CHANGE
taneous, often dramatic reorganizations of matter within time
and space also take place. And if we begin thinking in terms of
two or three dimensions, the number and variety of such pos·
sible structures become very great.
Now add to this an additional discovery. Imagine a situation
in which a chemical or other reaction produces an enzyme
whose presence then encourages further production of the
same enzyme. This is an example of what computer scientists
would call a positive-feedback loop. In chemistry it is called
"auto-catalysis. " Such situations are rare in inorganic chemis·
try. But in recent decades the molecular biologists have found
that such loops (along with inhibitory or "negative" feedback
and more complicated "cross-catalytic" processes) are the
very stuff of life itself. Such processes help explain how we go
from little lumps of DNA to complex living organisms.
More generall y, therefore , in far-from-equilibrium condi
tions we find that very small perturbations or fluctuations can
become amplified into gigantic , structure-breaking waves .
And this sheds light o n all sorts of "qualitative" o r "revolu·
tionary" change processes. When one combines the new in
sights gained from studying far-from-equilibrium states and
nonlinear processes, along with these complicated feedback
systems, a whole new approach is opened that makes it possi
ble to relate the so-called hard sciences to the softer sciences
of life-and perhaps even to social processes as well.
(Such findings have at least analogical significance for so
cial, economic or political realities. Words like "revolution, "
"economic crash," "technological upheaval ," and "paradigm
shift" all take on new shades of meaning when we begin think
ing of them in terms of fluctuations, feedback amplification,
dissipative structures, bifurcations, and the rest of the Prigogi
nian conceptual vocabulary.) It is these panoramic vistas that
are opened to us by Order Out of Chaos.
Beyond this, there is the even more puzzling, pervasive is·
sue of time.
Part of today's vast revolution in both science and culture is
a reconsideration of time, and it is important enough to merit a
brief digression here before returning to Prigogine's role in it.
Take history, for example. One of the great contributions to
historiography has been Braudel's division of time into three
scales-"geographical time, " in which events occur over the
ORDER OUT OF CHAOS
xviii
course of aeons ; the much shorter "social time" scale by
which economies, states, and civilizations are measured; and
the even shorter scale of "individual time"-the history of
human events.
In social science, time remains a largely unmapped terrain.
Anthropology has taught us that cultures differ sharply in the
way they conceive of time. For some, time is cyclical-history
endlessly recurrent. For other cultures, our own included,
time is a highway stretched between past and future, and peo
ple or whole societies march along it. In still other cultures,
human lives are seen as stationary in time ; the future advances
toward us, instead of us toward it.
Each society, as I've written elsewhere, betrays its own
characteristic "time bias"-the degree to which it places em
phasis on past, present, or future. One lives in the past. An
other may be obsessed with the future.
Moreover, each culture and each person tends to think in
terms of "time horizons . " Some of us think only of the imme
diate-the now. Politicians, for example, are often criticized
for seeking only immediate, short-term results. Their time
horizon is said to be influenced by the date of the next elec
tion. Others among us plan for the long term. These differing
time horizons are an overlooked source of social and political
friction-perhaps among the most important.
B ut despite the growing recognition that cultural con
ceptions of time differ, the social sciences have developed little
in the way of a coherent theory of time. Such a theory might
reach across many disciplines, from politics to group dynam
ics and interpersonal psychology. It might, for example, take
account of what, in Future Shock, I called "durational expec
tancies"-our culturally induced assumptions about how long
certain processes are supposed to take.
We learn very early, for example, that brushing one's teeth
should last only a few minutes, not an entire morning, or that
when Daddy leaves for work, he is likely to be gone approx
imately eight hours, or that a "mealtime" may last a few min
utes or hours, but never a year. (Television, with its division of
the day into fixed thirty- or sixty-minute intervals , subtly
shapes our notions of duration. Thus we normally expect the
hero in a melodrama to get the girl or find the money or win
the war in the last five minutes. In the United States we expect
xix
FOREWORD: SCIENCE AND CHANGE
commercials to break in at certain intervals.) Our minds are
filled with such durational assumptions. Those of children are
much different from those of fully socialized adults, and here
again the differences are a source of conflict.
Moreover, ch ildren in an industrial society are " time
trained"-they learn to read the clock, and they learn to dis
tinguish even quite small slices of time, as when their parents
tell them, "You've only got three more minutes till bedtime!"
These s harply honed temporal skills are often absent in
slower-moving agrarian societies that require less precision in
daily scheduling than our time-obsessed society.
Such concepts , which fit within the social and individual
time scales of B raude!, have never been systematically de
veloped in the social sciences. Nor have they, in any signifi
cant way, been articulated with our scientific theories of time,
even though they are necessarily connected with our assump
tions about physical reality. And this brings us back to Pri
gogine, who has been fascinated by the concept of time since
boyhood. He once said to me that, as a young student, he was
struck by a grand contradiction in the way science viewed
time, and this contradiction has been the source of his life's
work ever since.
In the world model constructed by Newton and his fol
lowers , time was an afterthought. A moment, whether in the
present, past, or future, was assumed to be exactly like any
other moment. The endless cycling of the planets-indeed,
the operations of a clock or a simple machine-can, in princi
ple , go either backward or forward in time without altering the
basics of the system. For this reason, scientists refer to time in
Newtonian systems as "reversible . "
In the nineteenth century, however, a s the main focus of
physics shifted from dynamics to thermodynamics and the
Second Law of thermodynamics was proclaimed, time sud
denly became a central concern. For, according to the Second
Law, there is an inescapable loss of energy in the universe.
And , if the world machine is really running down and ap
proaching the heat death, then it follows that one moment is no
longer exactly like the last. You cannot run the universe back
ward to make up for entropy. Events over the long term cannot
replay themselve s . A nd this means that there is a direc
tionality or, as Eddington later called it, an "arrow" in time.
ORDER OUT OF CHAOS
xx
The whole universe is, in fact, aging. And, in turn, if this is
true, time is a one-way street. It is no longer reversible, but
irreversible.
In short, with the rise of thermodynamics, science split
down the middle with respect to time. Worse yet, even those
who saw time as irreversible soon also split into two camps.
After all, as energy leaked out of the system, its ability to
sustain organized structures weakened , and these, in turn,
broke down into less organized, hence more random ele
ments. But it is precisely organization that gives any system
internal diversity. Hence, as entropy drained the system of en
ergy, it also reduced the differences in it. Thus the Second
Law pointed toward an increasingly homogeneous-and, from
the human point of view, pessimistic-future.
Imagine the problems introduced by Darwin and his fol
lowers! For evolution, far from pointing toward reduced organ
ization and d iversity, points i n the opposite direction.
Evolution proceeds from simple to complex, from "lower" to
"higher" forms of life, from undifferentiated to differentiated
structures. And, from a human point of view, all this is quite
optimistic. The universe gets "better" organized as it ages,
continually advancing to a higher level as time sweeps by.
In this sense, scientific views of time may be summed up as
a contradiction within a contradiction.
It is these paradoxes that Prigogine and Stengers set out to
illuminate, asking, "What is the specific structure of dynamic
systems which permits them to 'distinguish' between past and
future? What is the minimum complexity involved?"
The answer, for them , is that time makes its appearance
with randomness: "Only when a system behaves in a suffi
ciently random way may the difference between past and fu
ture, and therefore irreversibility, enter its description. "
I n classical or mechanistic science, events begin with "ini
tial conditions , " and their atoms or particles follow "world
lines" or trajectories. These can be traced either backward
into the past or forward into the future. This is just the op
posite of certain chemical reactions, for example, in which two
liquids poured into the same pot diffuse until the mixture is
uniform or homogeneous . These liquids do not de-diffuse
themselves. At each moment of time the mixture is different,
the entire process is "time-oriented. "
For classical science, at least i n its early stages, such pro-
xxi
FOREWORD: SCIENCE AND CHANGE
cesses were regarded as anomalies, peculiarities that arose
from highly unlikely initial conditions.
It is Prigogine and Stengers' thesis that such time-depen
dent, one-way processes are not merely aberrations or devia
tions from a world in which time is irreversible. If anything,
the opposite might be true, and it is reversible time, associated
with "closed systems" (if such , indeed, exist in reality), that
may well be the rare or aberrant phenomenon.
What is more , irreversible processes are the source of
order-hence the title Order Out of Chaos. It is the processes
associated with randomness, openness, that lead to higher lev
els of organization, such as dissipative structures.
Indeed, one of the key themes of this book is its striking
reinterpretation of the Second Law of thermodynamics. For
according to the authors, entropy is not merely a downward
slide toward disorganization. Under certain conditions, en
tropy itself becomes the progenitor of order.
What the authors are proposing, therefore, is a vast syn
thesis that embraces both reversible and irreversible time, and
shows how they relate to one another, not merely at the level of
macroscopic phenomena, but at the most minute level as well.
It is a breathtaking attempt at "putting the pieces back to
gether again. " The argument is complex, and at times beyond
easy reach of the lay reader. But it flashes with fresh insight
and suggests a coherent way to relate seemingl y uncon
nected-even contradictory-philosophical concepts.
Here we begin to glimpse, in full richness, the monumental
synthesis proposed in these pages. By insisting that irrevers
ible time is not a mere aberration, but a characteristic of much
of the universe, they subvert classical dynamics. For Pri
gogine and Stengers, it is not a case of either/or. Of course,
reversibility still applies (at least for sufficiently long times)
but in closed systems only. Irreversibility applies to the rest of
the universe.
Prigogine and Stengers also undermine conventional views
of thermodynamics by showing that, under nonequilibrium
conditions , at least , entropy may produce , rather than de
grade, order, organization-and therefore life .
I f this i s so, then entropy, too, loses its either/or character.
Whi l e certai n systems run down , other s ystems simul
taneously evolve and grow more coherent. This mutualistic,
ORDER OUT OF CHAOS
xxii
nonexclusive view makes it possible for biology and physics to
coexist rather than merely contradict one another.
Finally, yet another profound synthesis is implied-a new
relationship between chance and necessity.
The role of happenstance in the affairs of the universe has
been debated, no doubt, since the first Paleolithic warrior ac
cidently tripped over a rock. In the Old Testament, God's will
is sovereign, and He not only controls the orbiting planets but
manipulates the will of each and every individual as He sees
fit. As Prime Mover, al l causality flows from Him, and all
events in the universe are foreordained. Sanguinary conflicts
raged over the precise meaning of predestination or free will,
from the time of Augustine through the Carolingian quarrels.
Wycliffe, Huss, Luther, Calvin-all contributed to the debate.
No end of interpreters attempted to reconcile determinism
with freedom of will. One ingenious view held that God did
indeed determine the affairs of the universe, but that with re
spect to the free will of the individual, He never demanded a
specific action. He merely preset the range of options avail
able to the human decision-maker. Free will downstairs oper
ated only within the limits of a menu determined upstairs.
In the secular culture of the Machine Age, hard-line deter
minism has more or less held sway even after the challenges of
Heisenberg and the "uncertaintists. " Even today, thinkers
such as Rene Thorn reject the idea of chance as illusory and
inherently unscientific.
Faced with such philosophical stonewalling, some defenders
of free will, spontaneity, and ultimate u ncertainty, especially
the existentialists, have taken equally uncompromising stands.
(For Sartre, the human being was "completely and always
free, " though even Sartre, in certain writings, recognized
practical limitations on this freedom.)
Two things seem to be happening to contemporary concepts
of chance and determinism. To begin with, they are becoming
more complex. As Edgar Morin, a leading French sociologist
turned-epistemologist, has written:
" Let us not forget that the problem of determinism has
changed over the course of a century. . . . In place of the idea
of sovereign, anonymous, permanent laws directing all things
in nature there has been substituted the idea of laws of interac
tion . . . . There is more: the problem of determinism has be-
xxiii
FOREWORD: SCIENCE AND CHANGE
come that of the order of the universe. Order means that there
are other things besides 'Jaws' : that there are constraints, in
variances, constancies, regularities in our universe . . . . In
place of the homogenizing and anonymous view of the old de
terminism, there has been substituted a diversifying and evo
lutive view of determinations."
And as the concept of determinism has grown richer, new
efforts have been made to recognize the co-presence of both
chance and necessity, not with one subordinate to the· other,
but as full partners in a universe that is simultaneously
organizing and de-organizing itself.
It is here that Prigogine and Stengers enter the arena. For they
have taken the argument a step farther. They not only demonstrate
(persuasively to me, though not to critics like the mathematician,
Rene Thorn) that both determinism and chance operate, they also
attempt to show how the two fit together.
Thus, according to the theory of change implied in the idea
of dissipative structures, when fluctuations force an existing
system into a far-from-equilibrium condition and threaten its
structure, it approaches a critical moment or bifurcation point.
At this point, according to the authors, it is inherently impos
sible to determine in advance the next state of the system.
Chance nudges what remains of the system down a new path
of development. And once that path is chosen (from among
many), determinism takes over again until the next bifurcation
point is reached.
Here, in short, we see chance and necessity not as irreconcil
able opposites, but each playing its role as a partner in destiny.
Yet another synthesis is achieved.
When we bring reversible time and irreversible time , disor
der and order, physics and biology, chance and necessity all
into the same novel frame, and stipulate their interrelation
ships, we have made a grand statement-arguable, no doubt,
but in this case both powerful and majestic.
Yet this accounts only in part for the excitement occasioned
by Order Out of Chaos. For this sweeping synthesis, as I have
suggested, has strong social and even political overtones . Just
as the Newtonian model gave rise to analogies in politics, di
plomacy, and other spheres seemingly remote from science,
so, too, does the Prigoginian model lend itself to analogical
extension.
ORDER OUT OF CHAOS·
xxfv
By offering rigorous ways of modeling qualitative change,
for example, they shed light on the concept of revolution. By
explaining how successive instabilities give rise to transforma
tory change, they illuminate organization theory. They throw a
fresh light, as well, on certain psychological processes-inno
vation, for example, which the authors see as associated with
" nonaverage" behavior of the kind that arises under nonequi
librium conditions.
Even more significant, perhaps, are the implications for the
study of collective behavior. Prigogine and Stengers caution
against leaping to genetic or sociobiological explanations for
puzzling social behavior. Many things that are attributed to
biological pre-wiring are not produced by selfish, determinist
genes, but rather by social interactions under nonequilibrium
conditions.
(In one recent study, for instance, ants were divided into
two categories: One consisted of hard workers, the other of
inactive or " lazy" ants. One might overhastily trace such
traits to genetic predisposition. Yet the study found that if the
system were shattered by separating the two groups from one
another, each in turn developed its own subgroups of hard
workers and idlers. A significant percentage of the "lazy" ants
suddenly turned into hardworking Stakhanovites.)
Not surprisingly, therefore, the ideas behind this remarkable
book are beginning to be researched in economics , urban
studies, human geography, ecology, and many other disci
plines.
No one-not even its authors-can appreciate the full im
plications of a work as crowded with ideas as Order Out of
Chaos. Each reader will no doubt come away puzzled by some
passages (a few are simply too technical for the reader without
scientific training); startled or stimulated by others (as their
implications strike home) ; occasionally skeptical; yet intellec
tually enriched by the whole. And if one measure of a book is
the degree to which it generates good questions, this one is
surely successful.
Here are just a couple that have haunted me.
How, outside a laboratory, might one define a .. fluctua
tion"? What, in Prigoginian ter ms, does one mean by ••cause"
or "effect"? And when the authors speak of molecules com
municating with one another to achieve coherent, synchro-
xxv
FOREWORD: SCIENCE AND CHANGE
nized change, one may assume they are not anthropomorphiz·
ing. But they raise for me a host of intriguing issues about
whether all parts of the environment are signaling all the time,
or only intermittently ; about the indirect, second , and nth
order communication that takes place, permitting a molecule
or an organism to respond to signals which it cannot sense for
lack of the necessary receptors. (A signal sent by the environ
ment that is undetectable by A may be received by B and con
verted into a different kind of signal that A is properly
equipped to receive-so that B serves as a relay/converter,
and A responds to an environmental change that has been sig
naled to it via second-order communication.)
In connection with time, what do the authors make of the
idea put forward by Harvard astronomer David Layzer, that
we might conceive of three distinct "arrows of time"-one
based on the continued expansion of the universe since the Big
Bang; one based on entropy; and one based on biological and
historical evolution?
Another question: How revolutionary was the Newtonian
revolution? Taking issue with some historians, Prigogine and
Stengers point out the continuity of Newton's ideas with al
chemy and religious notions of even earlier vintage . Some
readers might conclude from this that the rise of Newtoni
anism was neither abrupt nor revolutionary. Yet, to my mind ,
the Newtonian breakthrough should not be seen as a linear
outgrowth of these earlier ideas. Indeed, it seems to me that
the theory of change developed in Order Out of Chaos argues
against just such a "continuist" view.
Even if Newtonianism was derivative, this doesn't mean
that the intc;:rnal structure of the Newtonian world-model was
actually the same or that it stood in the same relationship to its
external environment.
The Newtonian system arose at a time when feudalism in
Western Europe was crumbling-when the social system was ,
s o to speak, far from equilibrium. The model of the universe
proposed by the classical scientists (even if partially deriva
tive) was applied analogously to new fields and disseminated
successfully, not just because of its scientific power or "right
ness," but also because an emergent industrial society based
on revolutionary principles provided a particularly receptive
environment for it.
As suggested earlier, machine civilization, in searching for
ORDER OUT OF CHAOS
.xxvi
an explanation of itself in the cosmic order of things, seized
upon the Newtonian model and rewarded those who further
developed it. It is not only in chemical beakers that we find
auto-catalysis, as the authors would be the first to contend.
For these reasons, it still makes sense to me to regard the
Newtonian knowledge system as, itself, a "cultural dissipative
structure" born of social fluctuation.
Ironically, as I've said, I believe their own ideas are central
to the latest revolution in science, and I cannot help but see
these ideas in relationship to the demise of the Machine Age
and the rise of what I have called a "Third Wave" civilization.
Applying their own terminology, we might characterize to
day's breakdown of industrial or "Second Wave" society as a
civilizational "bifurcation, " and the rise of a more differenti
ated, "Third Wave" society as a leap to a new "dissipative
structure" on a world scale. And if we accept this analogy,
might we not look upon the leap from Newtonianism to Pri
goginianism in the same way? Mere analogy, no doubt. But
illuminating, nevertheless.
Finally, we come once more to the ever-challenging issue of
chance and necessity. For if Prigogine and Stengers are right
and chance plays its role at or near the point of bifurcation,
after which deterministic processes take over once more until
the next bifurcation, are they not embedding chance, itself,
within a deterministic framework? By assigning a particular
role to chance, don't they de-chance it?
This question , however, I had the pleasure of discussing
with Prigogine , who smiled over dinner and replied, " Yes.
That would be true. But, of course, we can never determine
when the next bifurcation will arise. " Chance rises phoenix
like once more.
Order out of Chaos is a brilliant, demanding, dazzling book
challenging for all and richly rewarding for the attentive reader. It
is a book to study, to savor, to reread-and to question yet again. It
places science and humanity back in a world in which ceteris
paribus is a myth-a world in which other things are seldom held
steady, equal, or unchanging. In short, it projects science into
today's revolutionary world of instability, disequilibrium, and tur
bulence. In so doing, it serves the highest creative function-it
helps us create fresh order.
PREFACE
MAN'S NEW DIALOGUE
WITH NATURE
Our vision of nature is undergoing a radical change toward the
multiple, the temporal, and the complex. For a long time a
mechanistic world view dominated Western science. In this
view the world appeared as a vast automaton. We now under
stand that we live in a pluralistic world . It is true that there are
phenomena that appear to us as deterministic and reversible,
such as the motion of a frictionless pendulum or the motion of
the earth around the sun. Reversible processes do not know
any privileged direction of time. But there are also irreversible
processes that involve an arrow of time. If you bring together
two liquids such as water and alcohol, they tend to mix in the
forward direction of time as we experience it. We never ob
serve the reverse process, the spontaneous separation of the
mixture into pure water and pure alcohol. This is therefore an
irreversible process. All of chemistry involves such irrevers
ible processes.
Obviously, in addition to deterministic processes, there
must be an element of probability involved in some basic pro
cesses, such as, for example, biological evolution or the evolu
tion of human cultures. Even the scientist who is convinced of
the validity of deterministic descriptions would probably hesi
tate to imply that at the very moment of the Big Bang, the
moment of the creation of the universe as we know it, the date
of the publication of this book was already inscribed in the
laws of nature. In the classical view the basic processes of
nature were considered to be deterministic and reversible.
Processes involving randomness or irreversibility were consid
ered only exceptions. Today we see everywhere the role of
irreversible processes, of fluctuations_
Although Western science has stimulated an extremely fruitxxvii
ORDER OUT OF CHAOS
xxviii
ful dialogue between man and nature, some of its cultural con
sequences have been disastrous. The dichotomy between the
"two cultures" is to a large extent due to the conflict between
the atemporal view of classical science and the time-oriented
view that prevails in a large part of the social sciences and
humanities. But in the past few decades, something very dra
matic has been happening i n science, something as unex
pected as the birth of geometry or the grand vision of the
cosmos as expressed in Newton's work. We are becoming
more and more conscious of the fact that on all levels, from
elementary particles to cosmology, randomness and irrevers
ibility play an ever-increasing role. Science is rediscovering
time. It is this conceptual revolution that this book sets out to
describe.
This revolution is proceeding on all levels, on the level of
elementary particles, in cosmology, and on the level of so
called macroscopic physics, which comprises the physics and
chemistry of atoms and molecules either taken individually or
considered globally as, for example, in the study of liquids or
gases. It is perhaps particularly on this macroscopic level that
the reconceptualization of science is most easy to follow. Clas
sical dynamics and modern chemistry are going through a pe
riod of drastic change. If one asked a physicist a few years ago
what physics permits us to explain and which problems re
main open, he would have answered that we obviously do not
have an adequate understanding of elementary particles or of
cosmological evolution but that our knowledge of things in be
tween was pretty satisfactory. Today a growing minority, to
which we belong, would not share this optimism: we have only
begun to understand the level of nature on which we live, and
this is the level on which we have concentrated in this book.
To appreciate the reconceptualization of physics taking
place today, we must put it in proper historical perspective.
The history of science is far from being a linear unfolding that
corresponds to a series of successive approximations toward
some intrinsic truth. It is full of contradictions, of unexpected
turning points. We have devoted a large portion of this book to
the historical pattern followed by Western science, starting
with Newton three centuries ago. We have tried to place the
history of science in the frame of the history of ideas to inte
grate it in the evolution of Western culture during the past
xxfx
MAN'S NEW DIALOGUE WITH NATURE
three centuries. Only in this way can we appreciate the unique
moment in which we are presently living.
Our scientific her itage includes two basic questions to
which till now no answer was provided. One is the relation
between disorder and order. The famous law of increase of
entropy -describes the world as evolving from order to disor
der ; still, biological or social evolution shows us the compl�x
emerging from the simple . How is this possible? How can
structure arise from disorder ? Great progress has been real
ized in this question. We know now that nonequilibrium, the
flow of matter and energy, may be a source of order.
But there is the second question, even more basic: classical
or quantum physics describes the world as rever sible , as
static. In this description there is no evolution, neither to
order nor to disorder ; the "information, " as may be defined
from dynamics, remains constant in time. Therefore there is
an obvious contradiction between the static view of dynamics
and the evolutionary paradigm of thermodynamics. What is
irreversibility? What is entropy? Few questions have been dis
cussed more often in the course of the history of science. We
begin to be able to give some answers. Order and disorder are
complicated notions: the units involved in the static descrip
tion of dynamics are not the same as those that have to be
introduced to achieve the evolutionary paradigm as expressed
by the growth of entropy. This transition leads to a new con
cept of matter, matter that is "active," as matter leads to irre
ver sible processes and as irrever sible processes organize
matter.
The evolutionary paradigm, including the concept of en
tropy, has exerted a considerable fascination that goes far
beyond science proper. We hope that our unification of dynam
ics and thermodynamics will bring out clearly the radical nov
elty of the entropy concept in respect to the mechanistic world
view. Time and reality are closely related. For humans, reality
is embedded in the flow of time. As we shall see, the irrevers
ibility of time is itself closely connected to entropy. To make
time flow backward we would have to overcome an infinite
entropy barrier.
Traditionally science has dealt with universals, humanities
with particulars. The convergence of science and humanities
was emphasized in the French title of our book, La Nouvelle
ORDER OUT OF CHAOS
xxx
Alliance, published by Gallimard, Paris, in 1979. However, we
have not succeeded in finding a proper English equivalent of
this title. Furthermore, the text we present here differs from
the French edition, especially in Chapters VII through IX. Al
though the origin of structures as the result of nonequilibrium
processes was already adequately treated in the French edi
tion (as well as in the translations that followed), we had to
entirely rewrite the third part, which deals with our new re
sults concerning the roots of time as well as with the formula
tion of the evolutionary paradigm in the frame of the physical
sciences.
This is all quite recent. The reconceptualization of physics
is far from being achieved. We have decided, however, to pre
sent the situation as it seems to us today. We have a feeling of
great intellectual excitement: we begin to have a glimpse of the
road that leads from being to becoming. As one of us has de
voted most of his scientific life to this problem, he may per
haps be excused for expressing his feeling of satisfaction, of
aesthetic achievement, which he hopes the reader will share .
For too long there appeared a conflict between what seemed
to be eternal, to be out of time, and what was in time. We see
now that there is a more subtle form of reality involving both
time and eternity.
This book is the outcome of a collective effort in which
many colleagues and friends have been involved. We cannot
thank them all individually. We would like, however, to single
out what we owe to Erich Jantsch, Aharon Katchalsky, Pierre
Resibois , and Leon Rosenfeld, who unfortunately are no
longer with us. We have chosen to dedicate this book to their
memory.
We want also to acknowledge the continuous support we
have received from the Instituts Internationaux de Physique et
de Chimie, founded by E. Solvay, and from the Robert A .
Welch Foundation.
The human race is in a period of transition. Science is likely
to play an important role at this moment of demographic ex
plosion. It is therefore more important than ever to keep open
the channels of communication between science and society.
The present development of Western science has taken it out
side the cultural environment of the seventeenth centu.ry, in
which it was born. We believe that science today carries a uni-
xxxi
MAN'S NEW DIALOGUE WITH NATURE
versal message that is more acceptable to different cultural
traditions.
During the past decades Alvin Toffier's books have been im
portant in bringing to the attention of the public some features
of the "Third Wave" that characterizes our time. We are there
fore grateful to him for having written the Foreword to the
English-language version of our book. English is not our na
tive language. We believe that to some extent ever y language
provides a different way of describing the common reality in
which we are embedded. Some of these characteristics will
survive even the most careful translation. In any case, we are
most grateful to Joseph Ear ly, Ian MacG il vray, C arol
Thurston, and especially to Carl Rubino for their help in the
preparation of this English-language version. We would also
like to express our deep thanks to Pamela Pape for the careful
typing of the successive versions of the manuscript.
INTRODUCTION
THE CHALLENGE TO
SCIENCE
1
It is hardly an exaggeration to state that one of the greatest
dates in the history of mankind was April 28, 1686, when New
ton presented his Principia to the Royal Society of London. It
contained the basic laws of motion together with a clear for
mulation of some of the fundamental concepts we still use to
day, such as mass, acceleration , and inertia. The greatest
impact was probably made by Book III of the Principia, titled
The System of the World, which included the universal law of
gravitation . Newton's contemporaries immediately grasped
the unique importance of his work. Gravitation became a topic
of conversation both in London and Paris.
Three centuries have now elapsed since Newton's Principia.
Science has grown at an incredible speed, permeating the life
of all of us. Our scientific horizon has expanded to truly fan
tastic proportions. On the microscopic scale, elementary par
tide physics studies processes involving physical dimensions
of the order of w- ts em and times of the order of I0-22 sec
ond. On the other hand, cosmology leads us to times of the
order of 1010 years, the "age of the universe. " Science and
technology are closer than ever. Among other factors, new
biotechnologies and the progress in information techniques
promise to change our lives in a radical way.
Running parallel to this quantitative growth are deep quali
tative changes whose repercussions reach far beyond science
proper and affect the very image of nature. The great founders
of Western science stressed the universality and the eternal
character of natural laws. They set out to formulate general
schemes that would coincide with the very ideal of rationality.
ORDER OUT OF CHAOS
2
As Roger Hausheer says in his fine introduction to Isaiah
Berlin's Against the Current, " They sought all-embracing
schemas, universal unifying frameworks, within which every
thing that exists could be shown to be systematically-i.e. ,
logically or causally-interconnected, vast structures in which
there should be no gaps left open for spontaneous, unattended
developments , where everything that occurs should be, at
least in principle, wholly explicable in terms of immutable
general laws. " t
The story of this quest is indeed a dramatic one. There were
moments when this ambitious program seemed near comple
tion. A fundamental level from which all other properties of
matter could be deduced seemed to be in sight. Such moments
can be associated with the formulation of Bohr's celebrated
atomic model, which reduced matter to simple planetary sys
tems formed by electrons and protons. Another moment of
great suspense came when Einstein hoped to condense all the
laws of physics into a single "unified field theory. " Great prog
ress has indeed been realized in the unification of some of the
basic forces found in nature. Still, the fundamental level re
mains elusive. Wherever we look we find evolution, diver
sification, and instabilities. Curiously, this is true on all levels,
in the field of elementary particles, in biology, and in astro
physics, with the expanding universe and the formation of
black holes.
As we said in the Preface, our vision of nature is undergoing
a radical change toward the multiple, the temporal, and the
complex . Curiously, the unexpected complexity that has been
discovered in nature has not led to a slowdown in the progress
of science, but on the contrary to the emergence of new con
ceptual structures that now appear as essential to our under
standing of the physical world-the world that includes us. It
is this new situation, which has no precedent in the history of
science, that we wish to analyze in this book.
The story of the transformation of our conceptions about
science and nature can hardly be separated from another
story, that of the feelings aroused by science. With every new
intellectual program always come new hopes, fears, and ex
pectations. In classical science the emphasis was on time-in
dependent law s . As we shall see, once the particular state of a
system has been measured, the reversible laws of classical sci-
3
THE CHALLENGE TO SCIENCE
ence are supposed to determine its future, just as they had
determined its past. It is natural that this quest for an eternal
truth behind changing phenomena aroused enthusiasm. But it
also came as a shock that nature described in this way was in
fact debased : by the very success of science, nature was
shown to be an automaton, a robot.
The urge to reduce the diversity of nature to a web of illu
sions has been present in Western thought since the time of
Greek atomists. Lucretius, following his masters Democritus
and Epicurus, writes that the world is "just" atoms and void
and urges us to look for the hidden behind the obvious: "Still,
lest you happen to mistrust my words, because the eye cannot
perceive prime bodies, hear now of particles you must admit
exist in the world and yet cannot be seen. "2
Yet it is well known that the driving force behind the work of
the Greek atomists was not to debase nature but to free men
from fear, the fear of any supernatural being, of any order that
would transcend that of men and nature. Again and again Lu
cretius repeats that we have nothing to fear, that the essence of
the world is the ever-changing associations of atoms in the
void.
Modern science transmuted this fundamentall y ethical
stance into what seemed to be an established truth; and this
truth, the reduction of nature to atoms and void, in turn gave
rise to what Lenoble3 has called the "anxiety of modern
men." How can we recognize ourselves in the random world
of the atoms? Must science be defined in terms of rupture be
tween man and nature? '�JI bodies, the firmament, the stars,
the earth and its kingdoms are not equal to the lowest mind;
for mind knows all this in itself and these bodies nothing. "4
This "Pensee" by Pascal expresses the same feeling of alien
ation we find among contemporary scientists such as Jacques
Monod:
Man must at last finally awake from his millenary
dream ; and in doing so, awake to his total solitude, his
fundamental isolation. Now does he at last realize that,
like a gypsy, he lives on the boundary of an alien world. A
world that is deaf to his music. just as indifferent to his
hopes as it is to his suffering or his crimes.s
ORDER OUT OF CHAOS
4
This is a paradox. A brilliant breakthrough in molecular bi
ology, the deciphering of the genetic code, in which Monod
actively participated, ends upon a tragic note. This very prog
ress, we are told, makes us the gypsies ,0f the universe. How
can we explain this situation? Is not science a way of com
munication, a dialogue with nature?
In the past, strong distinctions were frequently made be
tween man's world and the supposedly alien natural world. A
famous passage by Vico in The New Science describes this
most vividly:
. . . in the night of thick darkness enveloping the earliest
antiquity, so remote from ourselves, there shines the eter
nal and never failing light of a truth beyond all question:
that the world of civil society has certainly been made by
men, and that its principles are therefore to be found
within the modifications of our own human mind.
Whoever reflects on this cannot but marvel that the phi
losophers should have bent all their energies to the study
of the world of nature, which, since God made it, He
alone knows; and that they should have neglected the
study of the world of nations, or civil world, which, since
men had made it, men could come to know. 6
Present-day research leads us farther and farther away from
the opposition between man and the natural world. It will be
one of the main purposes of this book to show, instead of rup
ture and opposition, the growing coherence of our knowledge
of man and nature.
In the past, the questioning of nature has taken the most di
verse forms. Sumer discovered writing; the Sumerian priests
speculated that the future might be written in some hidden
way in the events taking place around us in the present. They
even systematized this belief, mixing magical and rational ele
ments. 7 In this sense we may say that Western science, which
originated in the seventeenth century, only opened a new
chapter in the everlasting dialogue between man and nature.
5
THE CHALLENGE TO SCIENCI:
Alexandre Koyres has defined the innovation brought about
by modern science in terms of "experimentation . " Modern
science is based on the discovery of a new and specific form of
communication with nature-that is, on the conviction that
nature responds to experimental interrogation. How can we
define more precisely the experimental dialogue? Experimen
tation does not mean merely the faithful observation of facts
as they occur, nor the mere search for empirical connections
between phenomena, but presupposes a systematic interac
tion between theoretical concepts and observation.
In hundreds of different ways scientists have expressed ttieir
amazement when, on determining the right question, they dis
cover that they can see how the puzzle fits together. In this
sense, science is like a two-partner game ir;t which we have to
guess the behavior of a reality unrelated to our beliefs, our
ambitions, or our hopes. Nature cannot be forced to say any
thing we want it to. Scientific investigation is not a mono
logue. It is precisely the risk involved that makes this game
exciting.
But the uniqueness of Western science is far from being ex
hausted by such methodological considerations. When Karl
Popper discussed the normative description of scientific ra
tionality, he was forced to admit that in the final analysis ra
tional science owes its existence to its success; the scientific
method is applicable only by virtue of the astonishing points
of agreement between preconceived models and experimental
results.9 Science is a risky game, but it seems to have dis
covered questions to which nature provides consistent an
swers.
The success of Western science is an historical fact, unpre
dictable a priori , but which cannot be ignored. The surprising
success of modern science has led to an irreversible transfor
mation of our relations with nature. In this sense, the term
"scientific revolution" can legitimately be used. The history of
mankind has been marked by other turning points, by other
singular conjunctions of circumstances leading to irreversible
changes. One such crucial event is known as the "Neolithic
revolution. " But there, just as in the case of the "choices"
marking biological evolution, we can at present only proceed
by guesswork, while there is a wealth of information concern
ing decisive episodes in the evolution of science. The so-called
ORDER OUT OF CHAOS
6
"Neolithic revolution" took thousands of years. Simplifying
somewhat, we may say the scientific revolution started only
three ·Centuries ago. We have what is perhaps a unique oppor
tunity to apprehend the specific and intelligible mixture of
"chance" and "necessity" marking this revolution.
Science initiated a successful dialogue with nature. On the
other hand, the first outcome of this dialogue was the discov
ery of a silent world. This is the paradox of classical science.
It revealed to men a dead, passive nature, a nature that be
haves as an automaton which, once programmed, continues to
follow the rules inscribed in the program. In this sense the
dialogue with nature isolated man from nature instead of
bringing him closer to it. A triumph of human reason turned
into a sad truth. It seemed that science debased everything it
touched.
Modern science horrified both its opponents, for whom it
appeared as a deadly danger, and some of its supporters, who
saw in man's solitude as "discovered" by science the price we
had to pay for this new rationality.
The cultural tension associated with classical science can be
held at least partly responsible for the unstable position of sci
ence within society ; it led to an heroic assumption of the harsh
implications of rationality, but it led also to violent rejection.
We shall return later to present-day antiscience movements.
Let us take an earlier example-the irrationalist movement in
Germany in the 1 920s that formed the cultural background to
quantum mechanics. to In opposition to science , which was
identified with a set of concepts such as causality, determi
nism, reductionism, and rationality, there was a v iolent up
surge of ideas denied by science but seen as the embodiment
of the fundamental irrationality of nature. Life, destiny, free
dom, and spontaneity thus became manifestations of a shad
owy underworld impenetrable to reason. Without going into
the peculiar sociopolitical context to which it owed its vehe
ment nature , we can state that this rejection illustrates the
risks associated with classical science. By admitting only a
subjective meaning for a set of experiences men believe to be
significant, science runs the risk of transferring these into the
realm of the irrational , bestowing upon them a formidable
power.
As Joseph Needham has emphasized, Western thought has
7
THE CHALLENGE TO SCIENCE
always oscillated between the world as an automaton and a
theology in which God governs the universe. This is what
Needham c al l s t h e " c haracteristic E u ropean s c h i z o
phrenia. " I I In fact, these visions are connected. An automa
ton needs an external god .
Do we really have to make this tragic choice? Must we
choose between a science that leads to alienation and an anti
scientific metaphysical view of nature? We think such a choice
is no longer necessary, since the changes that science is under
going today lead to a radically new situation . This recent evo
lution of science gives us a unique opportunity to reconsider
its position in culture in general. Modern science originated in
the specific context of the European seventeenth century. We
are now approaching the end of the twentieth century, and it
seems that some more universal message is carried by sci
ence, a message that concerns the interaction of man and na
ture as well as of man with man.
What are the assumptions of classical science from which we
believe science has freed itself today? Generally those center
ing around the basic conviction that at some level the world is
simple and is governed by time-reversible fundamental laws.
Today this appears as an excessive simplification. We may
compare it to reducing buildings to piles of bricks. Yet out of
the same bricks we may construct a factory, a palace , or a
cathedral. It is on the level of the building as a whole that we
apprehend it as a creature of time, as a product of a culture, a
society, a style. B ut there is the additional and obvious prob
lem that, since there is no one to build nature, we must give to
its very "bricks"-that is, to its microscopic activity-a de
scription that accounts for this building process.
The quest of classical science is in itself an illustration of a
dichotomy that runs throughout the history of Western
thought . Only the immutable world of ideas was traditionally
recognized as "illuminated by the sun of the intelligible," to
use Plato's expression. In the same sense, only eternal laws
were seen to express scientific rationality. Temporality was
looked down upon as an illusion. This is no longer true today.
ORDER OUT OF CHAOS
8
We have discovered that far from being an illusion, irrevers·
ibility plays an essential role in nature and lies at the origin of
most processes of self-organization. We fi nd ourselves in a
world in which reversibility and determinism apply only to
limiting, simple cases, while irreversibility and randomness
are the rules.
.
The denial of time and complexity was central to the cultural
issues raised by the scientific enterprise in its classical defini
tion. The challenge of these concepts was also decisive for the
metamorphosis of science we wish to describe. In his great
book The Nature of the Physical World, Arthur Eddingtonl 2
introduced a distinction between primary and secondary laws.
"Primary laws" control the behavior of single particles, while
"secondary laws" are applicable to collections of atoms or
molecules. To insist on secondary laws is to emphasize that
the description of elementary behaviors is not sufficient for
understanding a system as a whole. An outstanding case of a
secondary law is, in Eddington's view, the second law of ther
modynamics, the law that introduces the "arrow of time" in
physics. Eddington writes: "From the point of view of phi
losophy of science the conception associated with entropy
must, I think, be ranked as the great contribution of the nine
teenth century to scientific thought. It marked a reaction from
the view that everything to which science need pay attention is
discovered by a microscopic dissection of objects. " 13 This
trend has been dramatically amplified today.
It is true that some of the greatest successes of modern sci
ence are discoveries at the microscopic level, that of mole
cules, atoms, or elementary particles. For example, molecular
biology has been immensely successful in isolating specific
molecules that play a central role in the mechanism of life. In
fact, this success has been so overwhelming that for many sci
entists the aim of research is identified with this "microscopic
dissection of objects," to use Eddington's expression. How
ever, the second law of thermodynamics presented the first
challenge to a concept of nature that would explain away the
complex and reduce it to the simplicity of some hidden world.
Today interest is shifting from substance to relation, to com
munication, to time.
This change of perspective is not the result of some arbi
trary decision. In physics it was forced upon us by new dis-
9
THE CHALLENGE TO SCIENCE
coveries no one could have fore see n . Who would have
expected that most (and perhaps all) elementary particles
would prove to be unstable? Who would have expected that
with the experimental confirmation of an expanding universe
we could conceive of the history of the world as a whole?
At the end of the twentieth century we have learned to un
derstand better the meaning of the two great revolutions that
gave shape to the physics of our time, quantum mechanics and
relativity. They started as attempts to correct classical me
chanics and to incorporate into it the newly found universal
constants . Today the situation has changed. Quantum me
chanics has given us the theoretical frame to describe the in
cessant transformations of particles into each other. Similarly,
general relativity has become the basic theory in terms of
which we can describe the thermal history of our universe in
its early stages.
Our universe has a pluralistic, complex character. Struc
tures may disappear, but also they may appear. Some pro
cesses are , as far as we know, well described by deterministic
equations, but others involve probabilistic processes.
How then can we overcome the apparent contradiction be
tween these concepts? We are living in a single universe. As
we shall see, we are beginning to appreciate the meaning of
these problems. Moreover, the importance we now give to the
various phenomena we observe and describe is quite different
from, even opposite to, what was suggested by classical physics.
There the basic processes , as we mentioned, are considered as
deterministic and reversible. Processes involving randomness
or irreversibility are considered to be exceptions . Today we
see everywhere the role of irreversible processes, of fluctua
tions. The models considered by classical physics seem to us
to occur only in limiting situations such as we can create ar
tificially by putting matter into a box and then waiting till it
reaches equilibrium.
The artificial may be deterministic and reversible. The natu
ral contains essential elements of randomness and irrevers
ibility. This leads to a new view of matter in which matter is no
longer the passive substance described in the mechanistic
world view but is associated with spontaneous activity. This
change is so profound that, as we stated in our Preface , we can
really speak about a new dialogue of man with nature.
ORDER OUT OF CHAOS
10
This book deals with the conceptual transformation of sci
ence from the Golden Age of classical science to the present.
To describe this transformation we could have chosen many
roads. We could have studied the problems of elementary par
ticles. We could have followed recent fascinating develop
ments in astrophysics. These are the subjects that seem to
delimit the frontiers of science. However, as we stated in our
Preface, over the past years so many new fea�ures of nature at
our level have been discovered that we decided to concentrate
on this intermediate level, on problems that belong mainly to
our macroscopic world, which includes atoms, molecules , and
especially biomolecules. Still it is important to emphasize that
the evolution of science proceeds on somewhat parallel lines at
every level, be it that of elementary particles, chemistry, biol
ogy, or cosmology. On every scale self-organization, complex
ity, and time play a new and unexpected role.
Therefore, our aim is to examine the significance of three
centuries of scientific progress from a definite viewpoint.
There is certainly a subjective element in the way we have
chosen our material. The problem of time is really the center
of the research that one of us has been pursuing all his life.
When as a young student at the University of Brussels he
came into contact with physics and chemistry for the first
time, he was astonished that science had so little to say about
time, especially since his earlier education had centered
mainly around history and archaeology. This surprise could
have led him to two attitudes, both of which we find ex
emplified in the past: one would have been to discard the prob
lem, since classical science seemed to have no place for time;
and the other would have been to look for some other way of
apprehending nature , in which time would play a different,
more basic role. This is the path Bergson and Whitehead, to
mention only two philosophers. of our century, chose. The first
position would be a "positivistic" one, the second a "meta
physical" one.
There was , however, a third path, which was to ask whether
the simplicity of the temporal evolution traditionally consid-
11
THE CHALLENGE TO SCIENCE
ered in physics and chemistry was due to the fact that atten
tion was paid mainly to some very simplified situations, to
heaps of bricks in contrast with the cathedral to which we have
alluded.
This book is divided into three parts. The first part deals
with the triumph of classical science and the cultural con
sequences of this triumph. Initially, science was greeted with
enthusiasm. We shall then describe the cultural polarization
that occurred as a result of the existence of classical science
and its astonishing success. Is this success to be accepted as
such, perhaps limiting its implications, or must the scientific
method itself be rejected as partial or illusory? Both choices
lead to the same result-the collision between what has often
been called the "two cultures," science and the humanities.
These questions have played a basic role in Western thought
since the formulation of classical science. Again and again we
come to the problem, " How to choose?" Isaiah Berlin has
rightly seen in this question the beginning of the schism be
tween the sciences and the humanities:
The specific and unique versus the repetitive and the uni
versal, the concrete versus the abstract, perpetual move
ment versus rest, the inner versus the outer, quality
versus quantity, culture-bound versus timeless principles,
mental strife and self-transformation as a permanent con
dition of man versus the possibility (and desirability) of
peace, order, final harmony and the satisfaction of all ra- tional human wishes-these are some of the aspects of
the contrast. 1 4
We have devoted much space t o classical mechanics. In
deed, in our view this is the best vantage point from which we
may contemplate the present-day transformation of science.
Classical dynamics seems to express in an especially clear and
striking way the static view of nature. Here time apparently i s
reduced t o a parameter, and future and past become equiv
alent. It is true that quantum theory has raised many new
problems not covered by classical dynamics but it has never
theless retained a number of the conceptual positions of classi
cal dynamics , particularly as far as time and process are
concerned.
ORDER OUT OF CHAOS
12
As early as at the beginning of the nineteenth century,
precisely when classical science was triumphant, when the
Newtonian program dominated French science and the latter
dominated Europe, the first threat to the Newtonian con
struction loomed into sight. In the second part of our study we
shall follow the development of the science of heat, this rival to
Newton's science of gravity, starting from the first gauntlet
thrown down when Fourier formulated the law governing the
propagation of heat. It was, in fact, the first quantitative de
scription of something inconceivable in classical dynamics
an irreversible process.
The two descendants of the science of heat, the science of
energy conversion and the science of heat engines, gave birth
to the first " nonclassical" science-thermodynamics . The
most original contribution of thermodynamics is the cele
brated second law, which introduced into physics the arrow of
time. This introduction was part of a more global intellectual
move. The nineteenth century was really the century of evolu
tion; biology, geology, and sociology emphasized processes of
becoming, of increasing complexity. As for thermodynamics ,
i t is based o n the distinction of two types of processes: revers
ible processes, which are independent of the direction of time,
and irreversible processes, which depend on the direction of
time. We shall see examples later. It was in order to distinguish
the two types of processes that the concept of entropy was
introduced, since entropy increases only because of the irre
versible processes.
During the nineteenth century the final state of thermody
namic evolution was at the center of scientific research. This
was equilibrium thermodynamics. Irreversible processes were
looked down on as nuisances, as disturbances, as subjects not
worthy of study. Today this situation has completely changed.
We now know that far from equilibrium, new types of struc
tures may originate spontaneously. In far-from-equilibrium
conditions we may have transformation from disorder, from
thermal chaos, into order. New dynamic states of matter may
originate, states that reflect the interaction of a given system
with its surroundings. We have called these new structures dis
sipative structures to emphasize the constructive role of dis
sipative processes in their formation.
This book describes some of the methods that have been
13
THE CHALLENGE TO SCIENCE
developed in recent years to deal with the appearance and evo
lution of dissipative structures. Here we find the key words
that run throughout this book like leitmotivs: nonlinearity, in
stability, fluctuations. They have begun to permeate our view
of nature even beyond the fields of physics and chemistry
proper.
We cited Isaiah Berlin when we discussed the opposition be
tween the sciences and the humanities. He opposed the specific
and unique to the repetitive and the universal. The remarkable
feature is that when we move from equilibrium to far-from
equilibrium conditions, we move away from the repetitive and
the universal to the specific and the unique. Indeed , the laws
of equilibrium are universal. Matter near equilibrium behaves
in a "repetitive" way. On the other hand, far from equilibrium
there appears a variety of mechanisms corresponding to the
possibility of occurrence of various types of dissipative struc
tures. For example, far from equilibrium we may witness the
appearance of chemical clocks, chemical reactions which be
have in a coherent, rhythmical fashion. We may also have pro
cesses of self-organization l eading to nonhomogeneous
structures to nonequilibrium crystals.
We would like to emphasize the unexpected character of this
behavior. Every one of us has an intuitive view of how a chemi
cal reaction takes place ; we imagine molecules floating
through space, colliding, and reappearing in new forms. We
see chaotic behavior similar to what the atomists described
when they spoke about dust dancing in the air. But in a chemi
cal clock the behavior is quite different. Oversimplifying some
what, we can say that in a chemical clock all molecules change
their chemical identity simultaneously, at regular time inter
vals. If the molecules can be imagined as blue or red, we
would see their change of color following the rhythm of the
chemical clock reaction.
Obviously such a situation can no longer be described in
terms of chaotic behavior. A new type of order has appeared.
We can speak of a new coherence, of a mechanism of "com
munication" among molecules. But this type of communica
tion can arise only in far-from-equilibrium conditions. It is
quite interesting that such communication seems to be the rule
in the world of biology. It may in fact be taken as the very basis
of the definition of a biological system.
ORDER OUT OF CHAOS
14
In addition, the type of dissipative structure depends crit
ically on the conditions in which the structure is formed. Ex
ternal fields such as the gravitational field of earth, as well as
the magnetic field, may play an essential role in the selection
mechanism of self-organization.
We begin to see how, starting from chemistry, we may build
complex structures, complex forms, some of which may have
been the precursors of life. What seems certain is that these
far-from-equilibrium phenomena illustrate an essential and un
expected property of matter: physics may henceforth describe
structures as adapted to outside conditions. We meet in rather
simple chemical systems a kind of prebiological adaptation
mechanism. To use somewhat anthropomorphic language: in
equilibrium matter is "blind," but in far-from-equilibrium con
ditions it begins to be able to perceive, to "take into account,"
in its way of functioning, differences in the external world
(such as weak gravitational or electrical fields).
Of course, the problem of the origin of life remains a diffi
cult one, and we do not think a simple solution is imminent.
Still, from this perspective life no longer appears to oppose the
"normal" laws of physics, struggling against them to avoid its
normal fate-its destruction. On the contrary, life seems to
express in a specific way the very conditions in which our bio
sphere is embedded, incorporating the nonlinearities of chemi
cal reactions and the far-from-equilibrium conditions imposed
on the biosphere by solar radiation.
We have discussed the concepts that allow us to describe the
formation of dissipative structures, such as the theory of bifur
cations. It is remarkable that near-bifurcations systems pre
sent large fluctuations. Such systems seem to " hesitate"
among various possible directions of evolution, and the fa
mous law of large numbers in its usual sense breaks down. A
small fluctuation may start an entirely new evolution that will
drastically change the whole behavior of the macroscopic sys
tem. The analogy with social phenomena, even with history, is
inescapable. Far from opposing "chance" and "necessity," we
now see both aspects as essential in the description of non
linear systems far from equilibrium.
The first two parts of this book thus deal with two conflict
ing views of th e physical universe: the static view of classical
dynamics , and the evolutionary view associated with entropy.
15
THE CHALLENGE TO SCIENCE
A confrontation between these views has become unavoid
able. For a long time this confrontation was postponed by con
sidering irreversibility as an illusion, as an approximation; it
was man who introduced time into a timeless universe. How
ever, this solution in which irreversibility is reduced to an illu
sion or to approximations can no longer be accepted, since we
know that irreversibility may be a source of order, of co
herence, of organization.
We can no longer avoid this confrontation. It is the subject
of the third part of this book. We describe traditional attempts
to approach the problem of irreversibility first in classical and
then in quantum mechanics. Pioneering work was done here,
especially by Boltzmann and Gibbs. However, we can state
that the problem was left largely unsolved. As Karl Popper
relates it, it is a dramatic story: first, Boltzmann thought he
had given an objective formulation to the new concept of time
implied in the second law. But as a result of his controversy
with Zermelo and others, he had to retreat.
In the light of history-or in the darkness of history
Boltzmann was defeated, according to all accepted stan
dards, though everybody accepts his eminence as a phys
icist. For he never succeeded in clearing up the status of
his Ji-theorem; nor did he explain entropy increase . . .
Such was the pressure that he lost faith in himself. . 15
.
.
.
The problem of irreversibility still remains a subject of lively
controversy. How is this possible one hundred fifty years after
the discovery of the second law of thermodynamics? There are
many aspects to this question, some cultural and some techni
cal. There is a cultural component in the mistrust of time. We
shall on several occasions cite the opinion of Einstein. His
judgment sounds final: time (as irreversibility) is an illusion . In
fact, Einstein was reiterating what Giordano Bruno had writ
ten in the sixteenth century and what had become for cen
turies the credo of science: "The universe is, therefore, one,
infinite, immobile . . . . It does not move itself locally. . . . It
does not generate itself. . . . It is not corruptible . . . . It is not
alterable . . . " 1 6 For a long time Bruno's vision dominated
the scientific view of the Western world. It is therefore not
surprising that the intrusion of irreversibility, coming mainly
.
ORDER OUT OF CHAOS
16
from the engineering sciences and physical chemistry, was re·
ceived with mistrust. But there are technical reasons in addi·
tion to cultural ones. Every attempt to "derive" irreversibility
from dynamics necessarily had to fail, because irreversibility
is not a universal phenomenon. We can imagine situations that
are strictly reversible, such as a pendulum in the absence of
friction, or planetary motion. This failure has led to dis·
couragement and to the feeling that, in the end, the whole con
cept of irreversibility has a subjective origin. We shall discuss
all these problems at some length. Let us say here that today
we can see this problem from a different point of view, since
we now know that there are different classes of dynamic sys
tems. The world is far from being homogeneous. Therefore the
question can be put in different terms: What is the specific
structure of dynamic systems that permits them to "dis
tinguish" past and future? What is the minimum complexity
involved?
Progress has been realized along these lines. We can now be
more precise about the roots of time in nature. This has far
reaching consequences. The second law of thermodynamics,
the law of entropy, introduced irreversibility into the mac
roscopic world. We now can understand its meaning on the
microscopic level as well. As we shall see, the second law cor
responds to a selection rule, to a restriction on initial condi
tions that i s then propagated by the laws of dynamics .
Therefore the second law introduces a new irreducible ele
ment into our description of nature. While it is consistent with
dynamics, it cannot be derived from dynamics.
Boltzmann already understood that probability and irrevers
ibility had to be closely related. Only when a system behaves
in a sufficiently random way may the difference between past
and future, and therefore irreversibility, enter into its descrip:
tion. Our analysis confirms this point of view. Indeed, what is
the meaning of an arrow of time in a deterministic description
of nature? If the future is already in some way contained in the
present, which also contains the past, what is the meaning of
an arrow of time? The arrow of time � a manifestation of the
fact that the future is not given, that, as the French poet Paul
Valery emphasized, "time is construction. " 17
The experience of our ever yday life manifests a radical dif
ference between time and space. We can move from one point
17
THE CHALLENGE TO SCIENCE
of space to another. However, we cannot turn time around. We
cannot exchange past and future. As we shall see, this feeling
of impossibility is now acquiring a precise scientific meaning.
Permitted states are separated from states that are prohibited
by the second law of thermodynamics by means of an infinite
entropy barrier. There are other barriers in physics. One is the
velocity of light, which in our present view limits the speed at
which signals may be transmitted. It is essential that this bar
rier exist; if not, causality would fall to pieces. Similarly, the
entropy barrier is the prerequisite for giving a meaning tQ com
munication. Imagine what would happen if our future would
become the past for other people ! We shall return to this later.
The recent evolution of physics has emphasized the reality
of time. I n the process new aspects of time have been u n
covered. A preoccupation with time runs all through our cen
tury. Think of Einstein, Proust, Freud , Teilhard, Peirce, or
Whitehead.
One of the most surprising results of Einstein's special the
ory of relativity, published in 1905, was the introduction of a
local time associated with each observer. However, this local
time remained a reversible time . Einstein's problem both in
the special and the general theories of relativity was mainly
that of the "communication" between observers, the way they
could compare time intervals. B ut we can now investigate time
in other conceptual contexts.
In classical mechanics time was a number characterizing the
position of a point on its trajectory. But time may have a dif
ferent meaning on a global level. When we look at a child and
guess his or her age, this age is not located in any special part
of the child's body. It is a global judgment. It has often been
stated that science spatializes time. But we now discover that
another point of view is possible. Consider a landscape and its
evolution: villages grow, bridges and roads connect different
regions and transform them. Space thus acquires a temporal
dimension; following the words of geographer B . Berry, we
have been led to study the "timing of space. "
But perhaps the most important progress i s that we now
may see the problem of structure, of order, from a different
perspective. As we shall show in Chapter VIII, from the point
of view of dynamics, be it classical or quantum, there can be
no one time-directed evolution. The "information" as it can be
ORDER OUT OF CHAOS
18
defined in terms of dynamics remains constant in time. This
sounds paradoxical. When we mix two liquids, there would
occur no "evolution" in spite of the fact that we cannot, with
out using some external device, undo the effect of the mixing.
On the contrary, the entropy law describes the mixing as the
evolution toward a "disorder, " toward the most probable
state. We can show now that there is no contradiction between
the two descriptions, but to speak about information, or order,
we have to redefine the units we are considering. The impor
tant new fact is that we now may establish precise rules to go
from one type of unit to the other. In other words, we have
achieved a microscopic formulation of the evolutionary para
digm expressed by the second law. As the evolutionary para
digm encompasses all of chemistry as well as essential parts of
biology and the social sciences, this seems to us an important
conclusion. This insight is quite recent. The process of recon
ceptualization occurring in physics is far from being complete.
However, our intention is not to shed light on the definitive
acquisitions of science, on its stable and well-established re
sults. What we wish to do is emphasize the conceptual cre
ativeness of scientific activity and the future prospects and
new problems it raises. In any case , we now know that we are
only at the beginning of this exploration. We shall not see the
end of uncertainty or risk. Thus we have chosen to present
things as we perceive them now, fully aware of how incomplete
our answers are.
Erwin Schrodinger once wrote, to the indignation of many phi
losophers of science:
l
. . . there is a tendency to forget that all science is bound
up with human culture in general, and that scientific find
ings, even those which at the moment appear the mostj
advanced and esoteric and difficult to grasp, are meaning-!
less outside their cultural context. A theoretical science !
unaware that those of its constructs considered relevant i
and momentou:s arc de:stined eventually to be framed in l
concepts and words that have a grip on the educated com- II
I
I
19
THE CHALLENGE TO SCIENCE
munity and become part and parcel of the general world
picture-a theoretical science, I say, where this is forgot
ten, and where the initiated continue musing to each
other in terms that are, at best, understood by a small
group of close fellow travellers , will necessarily be cut off
from the rest of cultural mankind ; in the long run it is
bound to atrophy and ossify however virulently esoteric
chat may continue within its joyfully isolated groups of
experts. t8
One of the main themes of this book is that of a strong inter
action of the issues proper to culture as a whole and the inter
nal conceptual problems of science in particular. We find
questions about time at the very heart of science. Becoming,
irreversibility-these are questions to which generations of
philosophers have also devoted their lives. Today, when his
tory-be it economic, demographic, or political-is moving at
an unprecedented pace, new questions and new interests re
·
quire us to enter into new dialogues, to look for a new co
herence.
However, we know the progress of science has often been
described in terms of rupture , as a shift away from concrete
experience toward a level of abstraction that is increasingly
difficult tc, grasp. We believe that this kind of interpretation is
only a reflection, at the epistemological level, of the historical
situation in which classical science found itself, a conse
quence of its inability to include in its theoretical frame vast
areas of the relationship between man and his environment.
There doubtless exists an abstract development of scientific
theories. However, the conceptual innovations that have been
decisive for the development of science are not necessarily of
this type. The rediscovery of time has roots both in the inter
nal history of science and in the social context in which sci
ence finds itself today. Discoveries such as those of unstable
elementary particles or of the expanding universe clearly be
long to the internal history of science, but the general interest
in nonequilibrium situations , in evolving systems, may reflect
our feeling that humanity as a whole is today in a transition
period. Many results we shall report in Chapters V and VI, for
example those on oscillating chemical reactions , could have
been discovered many years ago, but the study of these non-
ORDER OUT OF CHAOS
20
equilibrium problems was repressed in the cultural and ideo
logical context of those times.
We are aware that asserting this receptiveness to cultural
content runs counter to the traditional conception of science.
In this view science develops by freeing itself from outmoded
forms of understanding nature; it purifies itself in a process
that can be compared to an "ascesis" of reason. But this in
turn leads to the conclusion that science should be practiced
only by communities living apart, uninvolved in mundane
matters. In this view, the ideal scientific community should be
protected from the pressures, needs, and requirements of so
ciety. Scientific progress ought to be an essentially autono
mous proces s that any " outside" influence, such as the
scientists's participation in other cultural, social, or economic
activities, would merely disturb or delay.
This ideal of abstraction, of the scientist's withdrawal, finds
an ally in still another ideal, this one concerning the vocation
of a "true" researcher, namely, his desire to escape from
worldly vicissitudes. Einstein describes the type of scientist
who would find favor with the ·�ngel of the Lord" should the
latter be given the task of driving from the "Temple of Sci
ence" all those who are "unworthy''-it is not stated in what
respects. They are generally
. . . rather odd, uncommunicative, solitary fellows, who
despite these common characteristics resemble one an
other really less than the host of the banished.
What led them into the Temple? . . . one of the stron
gest motives that lead men to art and science is flight
from everyday life with its painful harshness and
wretched dreariness, and from the fetters of one's own
shifting desires. A person with a finer sensibility is driven
to escape from personal existence and to the world of
objective observing (Schauen) and understanding. This
motive can be compared with the longing that irresistibly
pulls the town-dweller away from his noisy, cramped
quarters and toward the silent, high mountains, where
the eye ranges freely through the still, pure air and traces
the calm contours that seem to be made for eternity.
With this negative motive there goes a positive one. Man
seeks to form for himself, in whatever manner is suitable
21
tHE CHALLENGE TO SCIENCE
for him, a simplified and. lucid image of the world (Bild
der Welt), and so to overcome the world of experience by
striving to replace it to some extent by this image. I9
The incompatibility between the ascetic beauty sought after
by science, on the one hand, and the petty swirl of worldly
experience so keenly felt by Einstein, on the other, is likely to
be reinforced by another incompatibility, this one openly Man
ichean, between science and society, or, more precisely, be·
tween free human creativity and political power. In this case, it
is not in an isolated community or in a temple that research
would have to be carried out, but in a fortress, or else·in a
madhouse, as Duerrenmatt imagined in his play The Physi
cists.20 There, three physicists discuss the ways and means of
advancing physics while at the same time safeguarding man
kind from the dire consequences that result when political
powers appropriate the results of its progress. The conclusion
they reach is that the only possible way is that which has al
ready been chosen by one of them; they all decide to pretend
to be mad, to hide in a lunatic asylum. At the end of the play,
as Fate would have it, this last refuge is discovered to be an
illusion. The director of the asylum, who has been spying on
her patient, steals his results and seizes world power.
Duerrenmatt's play leads to a third conception of scientific
activity: science progresses by reducing the complexity of re
ality to a hidden simplicity. What the physicist Moebius is try
i ng to conceal i n the madhouse i s the fact that he has
successfully solved the problem of gravitation, the unified the
ory of elementary particles, and, ultimately, the Principle of
Universal Discovery, the source of absolute power. Of course,
Duerrenmatt simplifies to make his point, yet it is commonly
held that what is being sought in the "Temple of Science" is
nothing less than the "formula" of the universe. The man of
science, already portrayed as an ascetic, now becomes a kind
of magician, a man apart, the potential holder of a universal
key to all physical phenomena,. thus endowed with a poten
tially omnipotent knowledge. This brings us back to an issue
we have already raised: it is only in a simple world (especially
in the world of classical science, where complexity merely
veils a fundamental simplicity) that a form of knowledge that
provides a universal key can exist.
ORDER OUT OF CHAOS
22
One of the problems of our time is to overcome attitudes
that tend to justify and reinforce the isolation of the scientific
community. We must open new channels of communication
between science and society. It is in this spirit that this book
has been written. We all know that man is altering his natural
environment on an unprecedented scale. As Serge Moscovici
puts it, he is creating a "new nature. "21 But to understand this
man-made world, we need a science that is not merely a tool
submissive to external interests, nor a cancerous tumor irre
sponsibly growing on a substrate society.
1\.vo thousand years ago Chuang Tsu wrote:
How [ceaselessly] Heaven revolves ! How [constantly]
Earth abides at rest ! Do the Sun and the Moon contend
about their respective places? Is there someone presiding
over and directing those things? Who binds and connects
them together? Who causes and maintains them without
trouble or exertion? Or is there perhaps some secret
mechanism in consequence of which they cannot but be
as they are ?22
We believe that we are heading toward a new synthesis, a
new naturalism. Perhaps we will eventually be able to combine
the Western tradition, with its emphasis on experimentation
and quantitative formulations, with a tradition such as the Chi
nese one, with its view of a spontaneous, self-organizing
world. Toward the beginning of this Introduction, we cited
Jacques Monod. His conclusion was: "The ancient alliance
has been destroyed; man knows at last that he is alone in the
universe's indifferent immensity out of which he emerged only
by chance. "23 Perhaps Monod was right. The ancient alliance
has been shattered. Our role is not to lament the past. It is to
try to discover in the midst of the extraordinary diversity of
the sciences some unifying thread. Each great period of sci
ence has led to some model of nature. For classical science it
was the clock; for nineteenth-century science, the period of
the Industrial Revolution, it was an engine running down.
What will be the symbol for us? What we have in mind may
perhaps be expressed best by a reference to sculpture, from
Indian or pre-Columbian art to our time. In some of the most
23
THE CHALLENGE TO SCIENCE
beautiful manifestations of sculpture , be it in the dancing
Shiva or in the miniature temples of Guerrero, there appears
very clearly the search for a junction between stillness and
motion, time arrested and time passing. We believe that this
confrontation will give our period its uniqueness.
I
I
BOOK ONE
THE DEWSION OF
THE UNIVERSAL
CHAPTER I
THE TRIUMPH OF
RE A S ON
The New Moses
Nature and Natures laws lay hid in night:
God said, let Newton be! and all was light.
-Alexander Pope,
Proposed Epitaph for Isaac ' Newton,
who died in 1 727
There is nothing odd in the dramatic tone employedc by Pope.
In the eyes of eighteenth-century England, Newton was the
"new Moses" who had been shown the "tables of the law. "
Poets, architects, and sculptors joined to propose monuments;
a whole nation assembled to celebrate this unique event: a
man had discovered the language that nature speaks-and
obeys.
Nature compelled, his piercing Mind obeys,
And gladly shows him all her secret Ways;
'Gainst Mathematicks she has no Defence,
And yields t'experimental Consequence. 1
Ethics and politics drew upon the Newtonian episode for ma
terial on which to " ground" their arguments . Thus De
saguliers transposed the meaning of the new natural order into
a political lesson: a constitutional monarchy is the best possi
ble system of government, since the King, like the Sun, has
his power limited by it.
Like Ministers attending ev'ry Glance
Worlds sweep round his Throne in Mystick Dance.
Six
27
ORDER OUT OF CHAOS
28
He turns their Motion from his Devious Course,
And bends their Orbits by Attractive Force ;
His Pow'r coerc'd by Laws, still leave them free,
Directs , but not Destroys, their Liberty;2
Although he himself did not encroach upon the domain of the
moral sciences, Newton had no hesitation regarding the uni
versal nature of the laws set out in his Principia. Nature is
"very consonant and conformable to herself," he asserts in
the celebrated Question 3 1 of his Opticks-and this strong and
elliptical statement conceals a vast c laim: combustion, fer
mentation, heat, cohesion, magnetism . . . there is no natural
process which would not be produced by these active forces
attractions and repulsions-that govern both the motion of the
stars and that of freely falling bodies.
Already a national hero before his death, nearly a century later
Newton was to become, mainly through the powerful influence
exerted by Laplace, the symbol of the scientific revolution in
Europe. Astronomers scanned a sky ruled by mathematics.
The Newtonian system succeeded in overcoming all obsta
cles. Furthermore, it opened the way to mathematical meth
ods by which apparent deviations could be accounted for and
even be used to infer the existence of a hitherto unknown
planet. The prediction of the existence of the planet Neptune
was the consecration of the prophetic power inherent in the
Newtonian vision.
At the dawn of the nineteenth century, Newton's name
tended to signify anything that claimed exemplarity. However,
conflicting interpretations of his method are given. Some saw
it as providing a blueprint for quantitative experimentation ex
pressible in mathematics. For them, chemistry found its New
ton in Lavoisier, who pioneered the systematic use of the
balance. This was indeed a decisive step in the definition of a
quantitative chemistry that took mass conservation as its
Ariadne's thread. According to others, the Newtonian strat
egy consisted in isolating some central, specific fact and then
using it as the basis for all further deductions concerning a
given set of phenomena. In this perspective Newton's genius
was located in his pragmatism. He did not try to explain grav
itation; he took it as a fact. Similarly, each discipline should
29
THE TRIUMPH OF REASON
then take some central unexplained fact as its startm!; point.
Physicians thus felt that they were authorized by Newton to
refashion the vitalist conception and to speak of a "vital
force" sui generis, the use of which would give the description
of living phenomena a hoped-for systematic consistency. This
is the same role that affinity, taken as the specificalJy chemical
force of interaction, was calJed upon to play.
Some "true Newtonians" took exception to this prolifera
tion of forces and reasserted the universality of the explana
tory power of gravitation . But it was too late. The term
Newtonian was now applied to everything that dealt with a
system of Jaws, with equilibrium, or even to all situations in
which natural order on one side and moral, social, and politi
cal order on the other could be expressed in terms of an all
embracing harmony. Romantic philosophers even discovered
in the Newtonian universe an enchanted world animated by
natural forces. More "orthodox" physicists saw in it a me
chanical world governed by mathematics. For the positivists it
meant the success of a procedure, a recipe to be identified
with the very definition of science.3
The rest is literature-often Newtonian literature: the har
mony that reigns in the society of stars, the elective affinities
and hostilities giving rise to the "social life" of chemical com
pounds appear as processes that can be transposed into the
world of human society. No wonder that this period appears as
the Golden Age of Classical Science.
Thday Newtonian science still occupies a unique position.
Some of the basic concepts it introduced represent a definitive
acquisition that has survived all the mutations science has
since undergone. However, today we know that the Golden
Age of Classical Science is gone, and with it also the convic
tion that Newtonian rationality, even with its various conflict
ing interpretations, forms a suitable basis for our dialogue with
nature.
A central subject of this book is that of the Newtonian tri
umph, the continual opening up of new fields of investigation
that have extended Newtonian thought right down to the pres
. ent day. It also deals with doubts and struggles that arose from
this triumph. Today we are beginning to see more clearly the
limits of Newtonian rationality. A more consistent conception
ORDER OUT OF CHAOS
30
of science and of nature seems to be emerging. This new con
ception paves the way for a new unity of knowledge and cul
ture.
A Dehumanized World
. . . May God us keep
From single Vision and Newtons sleep!
-Wil liam Blake,
in a letter to Thomas Butts
dated November 22, 1 802
There is no better illustration of the instability of the cultural
position of Newtonian science than the introduction to a
UNESCO colloquium on the relationship between science and
culture:
For more than a century the sector of scientific activity
has been growing to such an extent within the surround
ing cultural space that it seems to be replacing the totality
of the culture itself. Some believe that this is merely an
illusion due to its high growth rate and that the lines of
force of this culture will soon reassert themselves and
bring science back into the service of man. Others con
sider that the recent triumph of science entitles it at last
to rule over the whole of culture which, moreover, would
deserve to go on being known as such only because it was
transmitted through the scientific apparatus . Others
again, appalled by the danger of man and society being
manipulated if they come under the sway of science, per
ceive the spectre of cultural disaster looming in the dis
tance.4
In this statement science appears as a cancer in the body of
culture, a cancer whose proliferation threatens to destroy the
whole of cultural life. The question is whether we can domi
nate science and control its development, or whether we shall
be enslaved. In only one hundred fifty years , science has been
31
THE TRIUMPH OF REASON
downgraded from a source of inspiration for Western culture
to a threat. Not only does it threaten man's material existence,
but also, more subtly, it threatens to destroy the traditions and
experiences that are most deeply rooted in our cultural life. It
is not just the technological fallout of one or another scientific
breakthrough that is being accused, but "the spirit of science"
itself.
Whether the accusation refers to a global skepticism exuded
by scientific culture or to specific conclusions reached
through scientific theories, it is often asserted today that sci
ence is debasing our world. What for generations had been a
source ofjoy and amazement withers at its touch. Everything
it touches is dehumanized.
Oddly enough, the idea of a fatal disenchantment brought
about by scientific progress is an idea held not only by the
critics of science but often also by those who defend or glorify
it. Thus, in his book The Edge of Objectivity, historian C . C.
Gillispie expresses sympathy for those who criticize science
and constantly endeavor to blunt the "cutting edge of objec
tivity" :
Indeed, the renewals of the subjective approach to nature
make a pathetic theme. Its ruins lie strewn like good in
tentions aU along the ground traversed by science, until it
survives only in strange corners like Lysenkoism and an
throposophy, where nature is socialized or moralized.
Such survivals are relics of the perpetual attempt to es
cape the consequences of western man's most charac
teristic and successful campaign, which must doom to
conquer. So like any thrust in the face of the inevitable,
romantic natural philosophy has induced every nuance of
mood from desperation to heroism. At the ugliest, it is
sentimental or vulgar hostility to intellect. At the noblest,
it inspired Diderot's naturalistic and moralizing science,
Goethe's personification of nature, the poetry of Words
worth, and the philosophy of Alfred North Whitehead, or
of any other who would find a place in science for our
qualitative and aesthetic appreciation of nature. It is the
science of those who would make botany of blossoms and
meteorology of sunsets. 5
ORDER OUT OF CHAOS
32
Thus science leads to a tragic, metaphysical choice. Man has
to choose between the reassuring but irrational temptation to
seek in nature a guarantee of human values, or a sign pointing
to a fundamental corelatedness, and fidelity to a rationality
that isolates him in a silent world.
The echoes of another leitmotiv-domination-mingle with
that of disenchantment. A disenchanted world is, at the same
time, a world liable to control and manipulation. Any science
that conceives of the world as being governed according to a
universal theoretical plan that reduces its various riches to the
drab applications of general laws thereby becomes an instru
ment of domination. And man, a stranger to the world, sets
himself up as its master.
This disenchantment has taken various forms in recent de
cades. It is outside the aim of this book to study systematically
the various forms of antiscience. In Chapter III we shall pre
sent a fuller reaction of Western thought to the surprising tri
umph of Newtonian rationality. Here let us only note that at
present there is a shift of popular attitudes to nature associated
with a widespread but in our opinion erroneous belief that
there exists a fundamental antagonism between science and
" naturalism. " To illustrate at least some of the forms anti
scientific criticism has taken in recent years, we have chosen
three examples. First, Heidegger, whose philosophy holds a
deep fascination for contemporary thought. We shall also refer
to the criticisms stated by Arthur Koestler and by the great
historian of science, Alexander Koyre.
Martin Heidegger directs his criticism against the very core
of the scientific endeavor, which he sees as fundamentally re
lated to a permanent aim, the domination of nature. Therefore
Heidegger claims that scientific rationality is the final accom
plishment of something that has been implicitly present since
ancient Greece, namely, the will to dominate, which is at work
in any rational discussion or enterprise, the violence lurking in
all positive and communicable knowledge. Heidegger empha
sizes what he· calls the technological and scientific "framing"
(Geste/1), 6 which leads to the general setting to work of the
world and of men.
Thus Heidegger does not present a detailed analysis of any
particular technological or scientific product or process. What
he challenges is the essence of technology, the way each thing
33
THE TRIUMPH OF REASON
is taken into account. Each theory is part of the implementa
tion of the master plan that makes up Western history. What
we call a scientific "theory" implies, following Heidegger, a
way of questioning things by which they are reduced to en
slavement. The scientist, like the technologist, is a toy in the
hands of the will to power disguised as thirst for knowledge;
his very approach to things subjects them to systematic vio
lence.
Modern physics is not experimental physics because it
uses experimental devices in its questioning of nature.
Rather the reverse is true. Because physics, already as
pure theory, requests nature to manifest itself in terms of
predictable forces, it sets up the experiments precisely
for the sole purpose of asking whether and how nature
follows the scheme preconceived by science.7
Similarly, Heidegger is not concerned about the fact that in
dustrial pollution, for example, has destroyed all animal life in
the Rhine. What does concern him is that the river has been
put to man's service.
The hydroelectric plant is set into the current of the
Rhine. It sets the Rhine to supplying its hydraulic pres
sure, which then sets the turbines turning . . . . The hy
droelectric plant is not built into the Rhine river as was
the old bridge that joined bank wi'th bank for hundreds of
years. Rather the river is dammed up into the power
plant. What the river is now, namely, a water supplier,
derives from out of the essence of the power station.s
The old bridge over the Rhine is valued not as a proof of
soundly tested ability, of painstaking and accurate observa
tion, but because it does not "use" the river.
Heidegger's criticisms, taking the very ideal of a positive,
communicable knowledge as a threat, echo some themes of
the antiscience movement to which we referred in the Intro
duction. But the idea of an indissociable link between science
and the will to dominate also permeates some apparently very
different assessments of our present-day situation. For in
stance, under the very suggestive title "The Coming of the
ORDER OUT OF CHAOS
34
Golden Age, "9 Gunther Stent states that science is now reach
ing its limits. We are close to a point of diminishing returns,
where the questions we direct to things in order to master
them become more and more complicated and devoid of inter
est. This marks the end of progress, but it is the opportunity
for humanity to stop its frantic efforts, to end the age-old
struggle against nature, and to accept a static and comfortable
peace. We wish to show that the relative dissociation between
the scientific knowledge of an object and the possibility of
mastering it, far from marking the end of science, signals a
host of new perspectives and problems. Scientific understand
ing of the world around us is just beginning. There is yet
another idea of science that we feel is potentially just as detri
mental, namely, the fascination with a mysterious science
that, by paths of reasoning inaccessible to common mortals,
will lead to results that can, in one fell swoop, challenge the
meaning of basic concepts such as time, space, causality,
mind, or matter. This kind of " mystery science," the results of
which are imagined to be capable of shattering the framework
of any traditional conception, has actually been encouraged
by the successive "revelations" of relativity and quantum me
chanics. It is certainly true that some of the most imaginative
steps in the past, Einstein's interpretation of gravitation as a
space curvature or Dirac's antiparticles, for example, have
shaken some s eemingly well-established conceptions. Thus
there is a very delicate balance between the readiness to imag
ine that science can produce anything and a kind of down-to
earth realism. Today the balance is strongly shifting toward a
revival of mysticism, be it in the press media or even in science
itself, especially among cosmologists. IO It has even been sug
gested by certain physicists and popularizers of science that
mysterious relationships exist between parapsychology and
quantum physics. Let us cite Koestler:
We have heard a whole chorus of Nobel Laureates in
physics informing us that matter is dead, causality is
dead, determinism is dead. If that is so, let us give them a
decent burial, with a requiem of electronic music. It is
time for us to draw the lesson from twentieth-century
post-mechanistic science , and to get out of the strait-
36
THE TRIUMPH OF REASON
jacket which nineteenth-century materialism imposed on
our philosophical outlook. Paradoxically, had that out
look kept abreast with modern science itself, instead of
lagging a century behind it, we would have been liberated
from that strait-jacket long ago . . . . But once this is rec
ognized, we might become more receptive to phenomena
around us which one-sided emphasis on physical science
has made us ignore; might feel the draught that is blowing
through the chinks of the causal edifice; pay more atten
tion to confluential events ; include the paranormal phe
nomena in our concept of normality ; and realise that we
have been living in the "Country of the Blind." I I
We do not wish to judge or condemn a priori. There may be in
some of the apparently fantastic propositions we hear today
some seed of new knowledge. Nevertheless, we believe that
leaps into the unimaginable are far too simple escapes from
the concrete complexity of our world. We do not believe we
shall leave the "Country of the Blind" in a day, since con
ceptual blindness is not the main reason for the problems and
contradictions our society has failed to solve.
Our disagreement with certain criticisms or distortions of
science does not mean, however, that we wish to reject aJJ crit
icisms. Let us take, for instance, the position of Alexander
Koyre, who has made outstanding contributions to the under
standing of the development of modern science. In his study of
the significance and implications of the Newtonian synthesis,
Koyre wrote:
Yet there is something for which Newton-or better to
say not Newton alone, but modern science in general
can still be made responsible: it is the splitting of our
world in two. I have been saying that modern science
broke down the barriers that separated the heavens and
the earth, and that it united and unified the universe. And
that is true. But, as I have said, too, it did this by sub
stituting for our world of quality and sense perception,
the world in which we live, and love, and die, another
world-the world of quantity, of reified geometry, a world
in which, though there is a place for everything, there is
ORDER OUT OF CHAOS
36
no place for man. Thus the world of science-the real
world-became estranged and utterly divorced from the
world of life, which science has been unable to explain
not even to explain away by calling it "subjective. "
True, these worlds are everyday-and even more and
more-connected by the praxis. Yet for theory they are
divided by an abyss.
1\vo worlds: this means two truths. Or no truth at all.
This is the tragedy of the modern mind which "solved
the riddle of the universe," but only to replace it by an
other riddle: the riddle of itself. I2
However, we hear in the conclusions of Koyre the same
theme expressed by Pascal and Monod-this tragic feeling of
estrangement. Koyre 's criticism does not challenge scientific
thinking but rather classical science based on the Newtonian
perspective. We no longer have to settle for the previous di
lemma of choosing between a science that reduces man to
being a stranger in a disenchanted world and antiscientific ,
irrational protests. Koyre's criticism does not invoke the limits
of a "strait-jacket" rationality but only the incapacity of classi
cal science to deal with some fundamental aspects of the
world in which we live.
Our position in this book is that the science described by
Koyre is no longer our science. Not because we are concerned
today with new, unimaginable objects, closer to magic than to
logic, but because as scientists we are now beginning to find
our way toward the complex processes forming the world with
which we are most familiar, the natural world in which living
creatures and their societies develop. Indeed, today we are be
ginning to go beyond what Koyre called "the world of quan
tity" into the world of "qualities" and thus of "becoming."
This will be the main subject of Books One and 1\vo. We be
lieve it is precisely this transition to a new description that
makes this moment in the history of science so exciting. Per
haps it is not an exaggeration to say that it is a period like the
time of the Greek atomists or the Renaissance, periods in
which a new view of nature was being born. But let us first
return to Newtonian science, certainly one of the great mo
ments of human history.
37
THE TRIUMPH OF REASON
The Newtonian Synthesis
What lay behind the enthusiasm of Newton's contemporaries,
their conviction that the secret of the universe, the truth about
nature, had finally been revealed? Several lines of thought,
probably present from the very beginning of humanity, con
verge in Newton's synthesis: first of all, science as a way of
acting on our environment. Newtonian science is indeed an
active science ; one of its sources is the knowledge of the medi
eval craftsmen, the knowledge of the builders of machines.
This science provides the means for systematically acting on
the world, for predicting and modifying the course of natural
processes, for conceiving devices that can harness and exploit
the forces and material resources of nature.
In this sense, modern science is a continuation of the age
less.efforts of man to organize and exploit the world in which
he lives. We have very scanty knowledge about the early
stages of this endeavor. However, it is possible, in retrospect,
to assess the knowledge and skills required for the "Neolithic
Revolution" to take place, when man gradually began to orga
nize his natural and social environment, using new techniques
to exploit nature and to organize his society. We still use, or
have used until quite recently, Neolithic techniques-for ex
ample, animal and plant species either bred or selected, weav
ing, pottery, metalworking. Our social organization was for a
long time based on the same techniques of writing, geometry,
and arithmetic as those required to organize the hierarchically
differentiated and structured social groups of the Neolithic
city-states. Thus we cannot help acknowledging the continuity
that exists between Neolithic techniques and the scientific and
industrial revolutions. t3
Modern science has thus extended this ancient endeavor,
amplifying it and constantly speeding up its rhythm. Never
theless, this does not exhaust the significance of science in the
sense given to it by the Newtonian synthesis.
In addition to the various techniques used in a given society,
we find a number of beliefs and myths that seek to understand
man's place in nature. Like myths and cosmologies, science's
ORDER OUT OF CHAOS
38
endeavor is to understand the nature of the world, the way it is
organized, and man's place in it.
From our standpoint it is quite irrelevant that the early spec
ulations of the pre-Socratics appear to be adapted from the
Hesiodic myth of creation-that is, the initial polarization of
Heaven and Earth, the desire aroused by Eros, the birth of the
first generations of gods to form the differentiated cosmic
powers, discord and strife, alternating atrocities and vendet
tas, until stability is finally reached under the rule of Justice
(dike). What does matter is that, in the space of a few genera
tions, the pre-Socratics collected , discussed, and criticized
some of the concepts we are still trying to organize in order to
understand the relation between being and becoming, or the
appearance of order out of a hypothetically undifferentiated
initial environment.
Where does the instability of the homogeneous come from?
Why does it differentiate spontaneously? Why do things exist
at all? Are they the fragile and mortal result of an injustice, a
disequilibrium in the static equilibrium of forces between con
flicting natural powers? Or do the forces that create and drive
things exist autonomously-rival powers of love and hate lead
ing to birth, growth, decline, and dispersion? Is change an illu
sion or is it, on the contrary, the unceasing struggle between
opposites that constitutes things? Can qualitative change be
reduced to the motion in a vacuum, of atoms differing only in
their forms, or do atoms themselves consist of a multitude of
qualitatively different germs, each unlike the others? And last,
is the harmony of the world mathematical? Are numbers the
key to nature?
The numerical regularities among sounds that were dis
covered by the Pythagoreans are still part of our present theo
ries. The mathematical schemes worked out by the Greeks
form the first body of abstract thought in European history
that is, a thought whose results are communicable and re
producible for all reasoning human beings. The Greeks
achieved for the first time a form of deductive knowledge that
contained a degree of certainty unaffected by convictions, ex
pectations, or passions.
The most important aspect common to Greek thought and
to modern science , which contrasts with the religious and
39
THE TRIUMPH OF REASON
mythicaI form of inquiry, is thus the emphasis on criticaI dis
cussion and verification.14
Little is known about this pre-Socratic philosophy that grew
up in the lonian cities and the colonies of Magna Graecia.
Thus we can only speculate about the relationships that might
have existed between the development of theoretical and cos
mological hypotheses and the crafts and technological ac
tivities that tlourished in those cities. Tradition teUs that as a
result of a hostile religious and social reaction, philosophers
were accused of atheism and were either exiled or put to death.
This early "recall to order" may serve as a symbol of the im
por tance of social factors in the origin, and above alI the
growth, of conceptual innovations. To understand the success
of modem science we also have to explain why its fóunders
were as a rule not unduly persecuted and their theoretical ap
proach repressed in favor of a form of knowledge more consis
tent with social anticipations and convictions.
Be that as it may, from Plato and Aristotle onward, the limits
were set, and thought was channeled in socially acceptable
directions. ln particular, the distinction between theoretical
thinking and technological activity was established. The
words we still use today-machine, mechanical, engineer
have a similar meaning. They do not refer to rational knowl
edge but to cunning and expediency. The idea was not to leam
about natural processes in order to utilize them more effec
tively, but to deceive nature, to "machinate" against it-that
is, to work wonders and create effects extraneous to the "nat
ural order" of things. The fields of practical manipulation and
that of the rational understanding of nature were thus rigidly
separated. Archimedes' status is merely that of an engineer;
his mathematical analysis of the equilibrium of machines is not
considered to be applicable to the world of nature, at least
within the framework of traditional physics. ln contrast, the
Newtonian synthesis expresses a systematic alliance between
manipulation and theoretical understanding.
There is a third important element that found its expression
in the Newtonian revolution. There is a striking contrast,
which each of us has probably experienced, between the quiet
world of the stars and planets and thé ephemeral, turbulent
world around uso As Mircea Eliade has emphasized, in many
ORDER OUT OF CHAOS
40
ancient civilizations there is a separation between profane
space and sacred space, a division of the world into an ordi·
nary space that is subject to chance and degradation and a
sacred one that is meaningful, independent of contingency and
history. This was the very contrast Aristotle established be
tween the world of the stars and our sublunar world. This con
trast is crucial to the way in which Aristotle evaluated the
possibility of a quantitative description of nature. Since the
motion of the celestial bodies is not change but a "divine"
state that is eternally the same, it ntay be described by means
of mathematical idealizations. Mathematical precision and
rigor are not relevant to the sublunar world. Imprecise natural
processes can only be subjected to an approximate descrip
tion.
In any case, for an Aristotelian it is more interesting to
know why a process occurs than to describe how it occurs, or
rather, these two aspects are indivisible. One of the main
sources of Aristotle's thinking was the observation of em
bryonic growth, a highly organized process in which interlock
ing, although apparently independent, events participate in a
process that seems to be part of some global plan. Like the de
veloping embryo, the whole of Aristotelian nature is organized
according to final causes. The purpose of all change, if it is in
keeping with the nature of things, is to realize in each being the
perfection of its intelligible essence. Thus this essence, which,
in the case of living creatures, is at one and the same time their
final, formal, and effective cause, is the key to the understand
ing of nature. In this sense the "birth of modern science," the
clash between the Aristotelians and Galileo, is a clash be
tween two forms of rationality. I S
In Galileo 's view the question of "why," so dear to the Aris
totelians, was a very dangerous way of addressing nature, at
least for a scientist. The Aristotelians, on the other hand, con
sidered Galileo's attitude as a form of irrational fanaticism.
Thus, with the coming of the Newtonian system it was a
new universality that triumphed, and its emergence unified
what till then had appeared as divided.
41
THE TRIUMPH OF REASON
The Experimental Dialogue
We have already emphasized one of the essential elements of
modern science: the marriage between theory and practice,
the blending of the desire to shape the world and the desire to
understand it. For this to be possible, it was not enough, de
spite the empiricists' beliefs, merely to respect observed facts.
On certain points, including even the description of mechani
cal motion, it was in fact Aristotelian physics that was more
easily brought into contact with empirical facts. The experi
mental dialogue with nature discovered by modern science in
volves activity rather than passive observation. What must be
done is to manipulate physical reality, to "stage"it in such a
way that it conforms as closely as possible to a theoretical
description. The phenomenon studied must be prepared and
isolated until it approximates some ideal situation that may be
physically unattainable but that conforms to the conceptual
scheme adopted.
By way of example, let us take the description of a system of
pulleys, a classic since the time of Archimedes, whose reason
ing has been extended by modern scientists to cover all simple
machines. It is astonishing to find that the modern explanation
has eliminated, on the grounds that it is irrelevant, the very
thing that Aristotelian physics set out to explain, namely, the
fact that, using a typical image, a stone "resists" a horse's
efforts to pull it and that this resistance can be "overcome" by
applying traction through a system of pulleys. Nature, accord
ing to Galileo, never gives anything away, never does some
thing for nothing, and can never be tricked; it is absurd to
think that by cunning or by using some stratagem we can make
it perform extra work. I6 Since the work the horse is able to
perform is the same with or without the pulleys, the effect
produced must be the same. This then becomes the starting
point for a mechanical explanation, which thus refers to an
idealized world. In this world the "new"effect-the stone fi
nally set in motion-is of secondary importance; and the
stone's resistance is described only qualitatively, in terms of
friction and heating. Instead, what is described accurately is
the ideal situation, in which a relationship of equivalence links
ORDER OUT OF CHAOS
42
the cause, the work done by the horse, to the effect, the mo
tion of the stone. In this ideal world, the horse can, in any
case, shift the stone, and the system of pulleys has the sole
effect of modifying the way the pulling efforts are transmitted;
instead of moving the stone over a distance L, equal to the
distance it travels while pulling the rope, the horse only moves
it over a distance Lin, where n depends on the number of
pulleys. Like all simple machines, the pulleys form a passive
device that can only transmit motion without producing it.
The experimental dialogue thus corresponds to a highly spe
cific procedure. Nature is cross-examined through experimen
tation, as if in a court of law, in the name of a priori principles.
Nature's answers are recorded with the utmost accuracy, but
relevance of those answers is assessed in terms of the very
idealizations that guided the experiment. All the rest does not
count as information, but is idle chatter, negligible secondary
effects. It may well be that nature rejects the theoretical hy
pothesis in question. Nevertheless, the latter is still used as a
standard against which to measure the implications and the
significance of the response, whatever it may be. It is precisely
this imperative way of questioning nature that Heidegger re
fers to in his argument against scientific rationality.
For us the experimental method is truly an art that is, it is
based on special skills and not on general rules. As such there
are never any guarantees of success and one always remains at
the mercy of triviality or poor judgment. No methodological
principle can eliminate the risk, for instance, of persisting in a
blind alley of inquiry. The experimental method is the art of
choosing an interesting question and of scanning all the con
sequences of the theoretical framework thereby implied, all
the ways nature could answer in the theoretical language
chosen. Amid the concrete complexity of natural phenomena,
one phenomenon has to be selected as the most likely to em
body the theory's implications in an unambiguous way. This
phenomenon will then be abstracted from its environment and
"staged"to allow the theory to be tested in a reproducible and
communicable way.
Although this experimental procedure was criticized right
from the outset, ignored by the empiricists, and attacked by
others on the grounds that it was a kind of t orture, a way of
putting nature on the rack, it survived all the modifications of
43
THE TRIUMPH OF REASON
the theoretical content of scientific descriptions and ultimately
defined the new method of investigation introduced by mod
ern science.
Experimental procedure can even become a tool for purely
theoretical analysis. It i s then a "thought experiment, "the
imagining of experimental situations governed entirely by the
oretical principles, which permits the exploration of the con
sequences of these principles in a given situation. Such
thought experiments played a crucial role in Galileo's work,
and today they are at the center of investigations about the
consequences of the conceptual upheavals in contemporary
physics, namely, relativity and quantum mechanics. One of
the most famous of such thought experiments is Einstein's fa
mous train, from which an observer can measure the velocity
of propagation of a ray of light emitted along an embankment,
that is, moving at a velocity c in a reference system with re
spect to which the train is moving at a velocity v. According to
classical reasoning, the observer on the train should attribute
to the light, which is traveling in the same direction as he is, a
velocity of c- v. However, this classical conclusion represents
precisely the absurdity that the thought experiment was de
signed to expose. In relativity theory, the velocity of light ap
pears as a uni versal constant of nature. Whatever inertial
referer.ce system is used, the velocity of light is always the
same. And since then Einstein's train has gone on exploring
the physical consequences of this fundamental change.
The experimental method is central to the dialogue with na
ture established by modern science. Nature questioned in this
way is, of course, simplified and occasionally mutilated. This
does not deprive it of its capacity to refute most of the hypoth
eses we can imagine. Einstein used to say that nature says
"no" to most of the questions it is asked, and occasionally
"perhaps. " The scientist does not do as he pleases, and he
cannot force nature to say only what he wants to hear. He
cannot, at least in the long run, project upon it his most cher
ished desires and expectations. He actually runs a greater risk
and plays a more dangerous game the better his tactics suc
ceed in encircling nature, in setting it more squarely with its
back to the wall.l7 Moreover, it is true that, whether the an
swer is "yes" or "no," it will be expressed in the same theoret
ical language as the question. However, this language, too,
ORDER OUT OF CHAOS
44
develops according to a complex historical process involving
nature's replies in the past and its relations with other theoreti
cal languages. In addition, new questions arise corresponding
to the changing interests of each period. This sets up a com
plex relationship between the specific rules of the scientific
game-particularly the experimental method of reasoning
with nature , which places the greatest constraint on the
game-and a cultural network to which, sometimes unwit
tingly, the scientist belongs.
We believe that the experimental dialogue is an irreversible
acquisition of human culture. It actually provides a guarantee
that when nature is explored by man it is treated as an inde
pendent being. It forms the basis of the communicable and
reproducible nature of scientific results. However partially na
ture is allowed to speak, once it has expressed itself, there is
no further dissent: nature never lies.
The Myth at the Origin of Science
The dialogue between man and nature was accurately per
ceived by the founders of modern science as a basic step to
ward the intelligibility of nature. But their ambitions went even
farther. Galileo, and those who came after him, conceived of
science as being capable of discovering global truths about
nature. Nature not only would be written in a mathematical
language that can be deciphered by experimentation, but there
would actually exist only one such language. Following this
basic conviction, the world is seen as homogeneous, and local
experimentation can reveal global truth. The simplest phe
nomena studied by science can thus be interpreted as the key
to understanding nature as a whole; the complexity of the lat
ter is only apparent, and its diversity can be explained in terms
of the universal truth embodied, in Galileo's case, in the math
ematical laws of motion.
This conviction has survived centuries. In an excellent set
of lectures presented on the BBC several years ago, Richard
Feynman ts compared nature to a huge chess game. The com
plexity is only apparent; each move follows simple rules. In its
early days, modern science quite possibly needed this convic-
45
THE TRIUMPH OF REASON
tion of being able to reach global truth. Such a conviction
added an immense value to the experimental method and, to a
certain extent, inspired it. Perhaps a revolutionary conception
of the world, one as all-embracing as the "biological" con
ception of the Aristotelian world, was necessary to throw off
the yoke of tradition, to give the champions of experimentation
a strength of conviction and a power of argument that enabled
them to hold their own against the previous forms of rational
ism. Perhaps a metaphysical conviction was needed to trans
mute the craftsman's and machine builder's knowledge into a
new method for the rational exploration of nature. We may
also wonder what the implications of the existence of this kind
of "mythical"conviction are for explaining the way modern
science's first developments were accepted in the social con
text. On this highly controversial issue, we shall restrict our
selves to a few remarks of a quite general nature for the sole
purpose of pinpointing the problem-that is, the problem of a
science whose advance has been felt by some as the triumph
of reason, but by others as a disillusionment, as the painful
discovery of the robotlike stupidity of nature.
It seems hard to deny the fundamental importance of social
and economic factors-particularly the development of crafts
men's techniques in the monasteries, where the residual knowl
edge of a destroyed world was preserved, and later in the
bustling merchant cities-in the birth of experimental science,
which is a systematized form of part of the craftsmen's knowl
edge.
Moreover, a comparative analysis such as Needham'sl9 ex
poses the decisive importance of social structures at the close
of the Middle Ages. Not only was the class of craftsmen and
potential technical innovators not held in contempt, as it was
in ancient Greece, but, like the craftsmen, the intellectuals
were, in the main, independent of the authorities. They were
free entrepreneurs, craftsmen-inventors in search of patron
age, who tended to look for novelty and to exploit all the op
portunities it afforded, however dangerous they may have been
for the social order. On the other hand, as Needham points
out, Chinese men of science were officials, bound to observe
the rules of the bureaucracy. They formed an integral part of
the state, whose primar y objective was to keep law and order.
The compass, the printing press, and gunpowder, all of which
ORDER OUT OF CHAOS
46
were to contribute to undermining the foundations of medieval
society and to project Europe into the modern era, were dis
covered much earlier in China but had a much less destabiliz
ing effect on its society. The enterprising European merchant
society appears in contrast as particularly well suited to stim
ulate and sustain the dynamic and innovative growth of mod
ern science in its early stages.
However, the question remains. We know that the builders
of machines used mathematical concepts-gear ratios, the dis
placements of the various working parts, and the geometry of
their relative motions. But why was mathematization not re
stricted to machines? Why was natural motion conceived of in
the image of a rationalized machine? This question may also
be asked in connection with the clock, one of the triumphs of
medieval craftsmanship that was soon to set the rhythm of life
in the larger medieval towns. Why did the clock almost imme
diately become the very symbol of world order? In this last
question lies perhaps some elements of an answer. A watch is
a contrivance governed by a rationality that lies outside itself,
by a plan that is blindly executed by its inner workings. The
clock world is a metaphor suggestive of God the Watchmaker,
the rational master of a robotlike nature. At the origin of mod
ern science, a "resonance" appears to have been set up be
tween theological discourse and theoretical and experimental
activity-a resonance that was no doubt likely to amplify and
consolidate the claim that scientists were in the process of dis
covering the secret of the "great machine of the universe. "
Of course, the term resonance covers an extremely complex
problem. It is not our intention to state, nor are we in any
position to affirm, that religious discourse in any way deter
mined the birth of theoretical science, or of the "world view"
that happened to develop in conjunction with experimental ac
tivity. By using the term resonance-that is, mutual amplifica
tion of two di scourses-we have deliberately chosen an
expression that does not assume whether it was theological
discourse or the " scientific myth" that came first and trig
gered the other.
Let us note that to some philosophers the question of the
"Christian origin" of Western science is not only the question
of the sta bilization of the concept of nature as an auto maton,
but also the question of some "essential" link between experi-
47
THE TRIUMPH OF REASON
mental science as such and Western civilization in its Hebraic
and Greek components. For Alfred North Whitehead this link
is situated at the level of instinctive conviction. Such a convic
tion was " needed" to inspire the " scientific faith" of the
founders of modern science:
I mean the inexpugnable belief that every detailed occur
rence can be correlated with its antecedents in a perfectly
definite manner, exemplifying general principles. Without
this belief the incredible labours of scientists would be
without hope . It is this instinctive conviction, vividly
poised before the imagination, which is the motive power
of research: that there is a secret, a secret which can be
unveiled. How has this conviction been so vividly im
planted in the European mind?
When we compare this tone of thought in Europe with
the attitude of other civilizations when left to themselves,
there seems but one source for its origin. It must come
from the medieval insistence on the rationality of God,
conceived as with the personal energy of Jehovah and
with the rationality of a Greek philosopher. Every detail
was supervised and ordered: the search into nature could
only result in the vindication of the faith in rationality.
. Remember that I am not talking of the explicit beliefs of a
few individuals. What I mean is the impress on the Euro
pean mind arising from the unquestioned faith of cen
turies. By this I mean the instinctive tone of thought and
not a mere creed of words. 2o
We will not consider this matter further. It would be out of
the question to .. prove"that modern science could have orig
inated only in Christian Europe. It is not even necessary to ask
if the founders of modern science drew any real inspiration
from theological arguments. Whether or not they were sin
cere, the important point is that those arguments made the
speculations of modern science socially credible and accept
able, over a period of time varying from country to country.
Religious references were still frequent in English scientific
texts of the nineteenth century. Remarkably enough, in the
present-day revival of interest in mysticism, the direction of
the argument appears reversed. It is now science that appears
to lend credibility to mystical affirmation.
ORDER OUT OF CHAOS
48
The question we have confronted here obviously leads to·
ward a multitude of problems in which theological and scien·
tific issues are inextricably bound up with the "external"
history of science, that is, the description of the relationship
between the form and content of scientific knowledge on the
one hand, and on the other, the use to which it is put in its
social, economic, and institutional context. As we have al
ready said, the only point we are presently interested in is the
very particular character and implications of scientific dis·
course that was amplified by resonance with theological dis
courses.
Needham 2 t tells of the irony with which Chinese men of let·
ters of the eighteenth century greeted the Jesuits' announce
ment of the triumphs of modern science. The idea that nature
was governed by simple, knowable laws appeared to them as a
perfect example of anthropocentric foolishness. Needham be
lieves that this "foolishness" has deep cultural roots. In order
to illustrate the great differences between the Western and
Chinese conceptions , he cites the animal trials held in the
Middle Ages. On several occasions such freaks as a cock who
supposedly laid eggs were solemnly condemned to death and
burned for having infringed the laws of nature, which were
equated with the laws of God. Needham explains how, in
China, the same cock would, in all likelihood, merely have
disappeared discreetly. It was not guilty of any crime, but its
freakish behavior clashed with natural and social harmony.
The governor of the province or even the emperor might find
himself in a delicate situation if the misbehavior of the cock
became known. Needham comments that, according to a
philosophic conception dominant in China, the cosmos is in
spontaneous harmony and the regularity of phenomena is not
due to any external authority. On the contrary, this harmony in
nature, society, and the heavens originates from the equi
librium among these processes . Stable and interdependent,
they resonate with each other in a kind of nonconcerted har
mony. If any law were involved, it would be a law that no one,
neither God nor man, had ever conceived of. Such a law would
also have to be expressed in a language undecipherable by
man and not be a law established by a creator conceived in our
own image.
49
THE TRIUMPH OF REASON
Needham concludes by asking the following question:
In the outlook of modern science there is, of course, no
residue of the notions of command and duty in the
"Laws" of Nature. They are now thought of as statistical
regularities, valid only in given times and places, descrip
tions not prescriptions, as Karl Pearson put it in a famous
chapter. The exact degree of subjectivity in the formula
tions of scientific law has been hotly debated during the
whole period from Mach to Eddington, and such ques
tions cannot be followed further here. The problem i s
whether the recognition of such statistical regularities
and their mathematical expression could have been
reached by any other road than that which Western sci
ence actually travelled. Was perhaps the state of mind in
which an egg-laying cock could be prosecuted at law nec
essary in a culture which should later have the property
of producing a Kepler? 2 2
It must now be stressed that scientific discourse is in no way
mere transposition of traditional religious views. Obviously
the world described by classical physics is not the world of
Genesis, in which God created light, heaven, earth, and the
living species, the world ·where Providence has never ceased
to act, spurring man on toward a history where his salvation is
at stake. The world of classical physics is an atemporal world
which, if created, must have been created in one fell swoop,
somewhat as an engineer creates a robot before letting it func
tion alone. In this sense, physics has indeed developed in op
position to both religion and the traditional philosophies. And
yet we know that the Christian God was actually called upon
to provide a basis for the world's intelligibility. In fact, one can
speak here of a kind of "convergence"between the interests
of theologians, who held that the world had to acknowledge
God's omnipotence by its total submission to Him, and of
physicists seeking a world of mathematizable processes.
In any case, the Aristotelian world destroyed by modern sci
ence was unacceptable to both these theologians and physicists.
This ordered , harmonious , hierarchical , and rational world
a
was
too independent, the beings inhabiting it too powerful and
ORDER OUT OF CHAOS
50
active, and their subservience to the absolute sovereign too
suspect and limited for the needs of many theologians. 23 On
the other hand, it was too complex and qualitatively differenti
ated to be mathematized.
The "mechanized" nature of modern science, created and
ruled according to a plan that totally dominates it, but of which
it is unaware , glorifies its creator, and was thus admirably
suited to the needs of both theologians and the physicists. Al
though Leibniz had endeavored to demonstrate that mathema
tization is compatible with a world that can display active and
qualitatively differentiated behavior, scientists and theologians
joined forces to describe nature as a mindless, passive me
chanics that was basically alien to freedom and the purposes
of the human mind. 'i\ dull affair, soundless, scentless, colour
less, merely the hurrying of matter, endless, meaningless, "24
as Whitehead observes. This Christian nature, stripped of any
property that permits man to identify himself with the ancient
harmony of natural "becoming," leaving man alone, face to
face with God, thus converged with the nature that a single
language, and not the thousand mathematical voices heard by
Leibniz, was sufficient to describe.
Theology may also help comment on man's odd position
when he laboriously deciphers the laws governing the world.
Man is emphatically not part of the nature he objectively de
scribes ; he dominates it from the outside. Indeed, for Galileo,
the human soul , created in God's image, is capable of grasping
the intelligible truths underlying the plan of creation. It can
thus gradually approach a knowledge of the world that God
himself possessed intuitively, fully, and instantaneously. 25
Unlike the ancient atomists, who were persecuted on the
grounds of atheism, and unlike Leibniz, who was sometimes
suspected of denying the existence of grace or of human free
dom, modern scientists have managed to come up with a
culturally acceptable definition of their enterprise. The human
mind, incorporated in a body subject to the laws of nature,
can, by means of experimental devices, obtain access to the
vantage point from which God himself surveys the world, to
the divine plan of which this world is a tangible expression.
Nevertheless, the mind itself remains outside the results of its
achievement_ The scientist may descr ibe as secondary
qualities, not part of nature but projected onto it by the mind,
51
THE TRIUMPH OF REASON
everything that goes to make up the texture of nature, such as
its perfumes and its colors . The debasement of nature is paral
lel to the glorification of all that eludes it, God and man.
The Limits of Classical Science
We have tried to describe the unique historical situation in
which scientific practice and metaphysical conviction were
closely coupled. Galileo and those who came after him raised
the same problems as the medieval builders but broke away
from their empirical knowledge to assert, with the help of
God, the simplicity of the world and the universality of the
language the experimental method postulated and deciphered.
In this way, the basic myth underlying modern science can be
seen as a product of the peculiar complex which , at the close
of the Middle Ages, set up conditions of resonance and re
ciprocal amplification among economic, political, social, re
ligious, philosophic, and technical factors. However, the rapid
decomposition of this complex left classical science stranded
and isolated in a transformed culture.
Classical science was born in a culture dominated by the
alliance between man, situated midway between the divine
order and the natural order, and God, the rational and intelligi
ble legislator, the sovereign architect we have conceived in our
own image. It has outlived this moment of cultural consonance
that entitled philosophers and theologians to engage in science
and that entitled scientists to decipher and express opinions
on the divine wisdom and power at work in creation. With the
support of religion and philosophy, scientists had come to be
lieve their enterprise was self-sufficient, that it exhausted the
possibilities of a rational approach to natural phenomena. The
relationship between scientific description and natural phi
losophy did not, in this sense, have to be justified. It could be
seen as self-evident that science and philosophy were con
vergent and that science was discovering the principles of an
authentic natural philosophy. But, oddly enough, the self
sufficiency experienced by scientists was to outlive the departure
of the medieval God and the withdrawal of the epistemological
guarantee offered by theology. The originally bold bet had be
come the triumphant science of the eighteenth century, 26 the
ORDER OUT OF CHAOS
52
science that discovered the laws governing the motion of celes
tial and earthly bodies, a science that df\lembert and Euler
incorporated into a complete and consistent system and
whose history was defined by Lagrange as a logical achieve
ment tending toward perfection. It was the science honored by
the Academies founded by absolute monarchs such as Louis
XIV, Frederick II, and Catherine the Great,27 the science that
made Newton a national hero. In other words, it was a suc
cessful science, convinced that it had proved that nature is
transparent. "Je n 'ai pas besoin de cette hypothese" was
Laplace's reply to Napoleon, who had asked him God's place
in his world system.
The dualist implications of modern science were to survive
as well as its claims. For the science of Laplace which, in
many respects, is still the classical conception of science to
day, a description is objective to the extent to which the ob
server is excluded and the description itself is made from a
point lying de jure outside the world, that is, from the divine
viewpoint to which the human soul, created as it was in God's
image, had access at the beginning. Thus classical science still
aims at discovering the unique truth about the world, the one
language that will decipher the whole of nature-today we
would speak of the fundamental level of description from
which everything in existence can be deduced.
On this essential point let us cite Einstein, who has trans
lated into modern terms precisely what we may call the basic
myth underlying modern science:
What place does the theoretical physicist's picture of the
world occupy among all these possible pictures? It de
mands the highest possible standard of rigorous precision
in the description of relations, such as only the use of
mathematical language can give. In regard to his subject
matter, on the other hand, the physicist has to limit him
self very severely: he must content himself with describ
ing the most simple events which can be brought within
the domain of our experience; all events of a more com
plex order are beyond the power of the human intellect to
reconstruct with the subtle accuracy and logical perfec
tion which the theoretical physicist demands. Supreme
purity, clarity, and certainty at the cost of completeness.
·
53
THE TRIUMPH OF REASON
But what can be the attraction of getting to know such a
tiny section of nature thoroughly, while one leaves every
thing subtler and more complex shyly and timidly alone?
Does the product of such a modest effort deserve to be
called by the proud name of a theory of the universe?
In my belief the name is justified; for the general laws
on which the structure of theoretical physics is based
claim to be valid for any natural phenomenon what
soever. With them, it ought to be possible to arrive at the
description, that is to say, the theory, of every natural
process, including life , by means of pure deduction, if
that process of deduction were not far beyond the capac
ity of the human intellect. The physicist's renunciation of
completeness for his cosmos is therefore not a matter of
fundamental principle. 2s
For some time there were those who persisted in the illusion
that attraction in the form in which it is expressed in the law of
gravitation would justify attributing an intrinsic animation to
nature and that if it were generalized it would explain the ori
gins of increasingly specific forms of activity, including even
the interactions that compose human society. But this hope
was rapidly crushed, at least partly as a consequence of the
demands created by the political, economic, and institutional
setting where science developed. We shall not examine this
aspect of the problem, important though it is. Our point here is
to emphasize that this very failure seemed to establish the
consistency of the classical view and to prove that what had
once been an inspiring conviction was a sad truth. In fact, the
only interpretation apparently capable of rivaling this inter
pretation of science was henceforth the positivistic refusal of
the very project of understanding the world . For example,
Ernst Mach, the influential philosopher-scientist whose ideas
had a great impact on the young Einstein, defined the task of
scientific knowledge as arranging experience in as economical
an order as possible. Science has no other meaningful goal
than the simplest and most economical abstract expression of
facts:
Here we have a clue which strips science of all its mys
tery, and shows us what its power really is. With respect
ORDER OUT OF CHAOS
54
to specific results it yields us nothing that we could not
reach in a sufficiently long time without methods . . ..
Just as a single human being, restricted wholly to the
fruits of his own labor, could never amass a fortune, but
on the contrary the accumulation of the labor of many
men in the hands of one is the foundation of wealth and
power, so, also, no knowledge worthy of the name can be
gathered up in a single human mind limited to the span of
a human life and gifted only with finite powers, except by
the most exquisite economy of thought and by the careful
amassment of the economically ordered experience of
thousands of co-workers. 29
Thus science i s useful because it leads to economy of
thought. There may be some element of truth in such a state
ment, but does it tell the whole story? How far we have come
from Newton, Leibniz, and the other founders of Western sci
ence, whose ambition was to provide an intelligible frame to
the physical universe! Here science leads to interesting rules
of action, but no more.
This brings us back to our starting point, to the idea that it is
classical science , considered for a certain period of time as
the very symbol of cultural unity, and not science as such that
led to the cultural crisis we have described. Scientists found
themselves reduced to a blind oscillation between the thunder
ings of scientific myth" and the silence of "scientific serious
ness," between affirming the absolute and global nature of
scientific truth and retreating into a conception of scientific
theory as a pragmatic recipe for effective intervention in natu
ral processes.
As we have already stated, we subscribe to the view that
classical science has now reached its limit. One aspect of this
transformation is the discovery of the limitations of classical
concepts that imply that a knowledge of the world "as it is"
was possible . The omniscient beings, Laplace's or Maxwell's
demon, or Einstein's God, beings that play such an important
role in scientific reasoning, embody the kinds of extrapolation
physicists thought they were allowed to make . As random
ness, complexity, and irreversibility enter into physics as ob
jects of positive knowledge , we are moving away from this
rather naive assumption of a direct connection between our
..
55
THE TRIUMPH OF REASON
description of the world and the world itself. Objectivity in
theoretical physics takes on a more subtle meaning.
This evolution was forced upon us by unexpected supple
mental discoveries that have shown that the existence of uni
versal constants, such as the velocity of light, limit our power
to manipulate nature. (We shall discuss this unexpected situa
tion in Chapter VII.) As a result, physicists had to introduce
new mathematical tools that make the relation between percep
tion and interpretation more complex. Whatever reality may
mean, it always corresponds to an active intellectual construc
tion. The descriptions presented by science can no longer be
disentangled from our questioning activity and therefore can
no longer be attributed to some omniscient being.
On the eve of the Newtonian synthesis, John Donne la
mented the passing of the Aristotelian cosmos destroyed by
Copernicus:
And new Philosophy calls all in doubt,
The Element of fire is quite put out,
The Sun is lost, and th'earth, and no man's wit
Can well direct him where to look for it.
And freely men confess that this world's spent,
When in the Planets and the Firmament,
They seek so many new, then they see that this
Is crumbled out again to his Atomies
'Tis all in Pieces , all coherence gone.JO
The scattered bricks and stones of our present culture seem,
as in Donne's time, capable of being rebuilt into a new "co
herence. "Classical science, the mythical science of a simple,
passive world, belongs to the past, killed not by philosophical
criticism or empiricist resignation but by the internal develop
ment of science itself.
I
I
CHAPTER II
THE IDENTIFICATION
OFTHEREAL
Newtons Laws
We shall now take a closer look at the mechanistic world view
as it emerged from the work of Galileo, Newton, and their
successors. We wish to describe its strong points, the aspects
of nature it has succeeded in clarifying, but we also want to
expose its limitations.
·Ever since Galileo, one of the central problems of physics
has been the description of acceleration. The surprising fea
ture was that the change undergone by the state of motion of a
body could be formulated in simple mathematical terms. This
seems almost trivial to us today. Still, we should remember
that Chinese science, so successful in many areas, did not pro
duce a quantitative formulation of the laws of motion. Galileo
discovered that we do not need to ask for the cause of a state
of motion if the motion is uniform, any more than it is neces
sary to ask the reason for a state of rest. Both motion and rest
remain indefinitely stable unless something happens to upset
them. The central problem is the change from rest to motion,
and from motion to rest, as well as, more generally, all changes
of velocity. How do these changes occur? The formulation of
the Newtonian laws of motion made use of two converging
developments: one in physics, Kepler's laws for planetary mo
tion and Galileo's laws for falling bodies, and the other in
mathematics, the formulation of differential or "infinitesimal"
calculus.
How can a continuously varying speed be defined? How can
we describe the instantaneous changes in the various quan
tities, such as position, velocity, and acceleration? How can
we describe the state of a body at any given instant? To answer
57
ORDER OUT OF CHAOS
58
these questions, mathematicians have introduced the concept
of infinitesimal quantities. An infinitesimal quantity is the re
sult of a limiting process; it is typically the variation in a quan
tity occurring between two successive instants when the time
elapsing between these instants tends toward zero. In this way
the change is broken up into an infinite series of infinitely
small changes.
At each instant the state of a moving body can be defined by
its position r, by its velocity v, which expresses its "instanta
neous tendency" to modify this position, and by its accelera
tion a, again its "instantaneous tendency," but now to modify
its velocity. Instantaneous velocities and accelerations are lim
iting quantities that measure the ratio between two infinitesi
mal quantities : the variation of r (or v) during a temporal
interval 6.t, and this interval 6.t when 6.t tends to zero. Such
quantities are "derivatives with respect to time," and since
Leibniz they have been written as v=drldt and a=dv/dt.
Therefore, acceleration, the derivative of a derivative, a=d2r!
dt2, becomes a " second derivative." The problem on which
Newtonian physics concentrates is the calculation of this sec
ond derivative, that is, of the acceleration undergone at each
instant by the points that form a system. The motion of each of
these points over a finite interval of time can then be calcu
lated by integration, by adding up the infinitesimal velocity
changes occurring during this interval. The simplest case is
when a is constant (for example, for a freely falling body a is
the gravitational constant g). Generally speaking, acceleration
itself varies in time, and the physicist's task is to determine
precisely the nature of this variation.
In Newtonian language, to study acceleration means to de
termine the various "forces" acting on the points in the system
under examination. Newton's second law, F= ma, states that
the force applied at any point is proportional to the accelera
tion it produces. In the case of a system of material points, the
problem is more complicated , since the forces acting on a
given body are determined at each instant by the relative dis
tances between the bodies of the system, and thus vary at each
instant as a result of the motion they themselves produce.
A problem in dynamics is expressed in the form of a set of
"differential.. equations. The instantaneous state of each of
the bodies in a system is described as a point and defined by
59
THE IDENTIFICATION OF THE REAL
means of its position as well as by its velocity and accelera
tion, that is, by the first and second derivatives of the position.
At each instant, a set of forces, which is a function of the
distance between the points in the system (a function of r),
gives a precise acceleration to each point; the accelerations
then bring about changes in the distances separating these
points and therefore in the set of forces acting at the following
instant.
While the differential equations set up the dynamics prob
lem, their "integration" represents the solution of this prob
lem. It leads to the calculation of the trajectories, r(t). These
trajectories contain all the information acknowledged as rele
vant by dynamics ; it provides a complete description of the
dynamic system.
The description therefore implies two elements: the posi
tions and velocities of each of the points at one instant, often
called the "initial instant," and the equations of motion that
relate the dynamic forces to the accelerations. The integration
of the dynamic equations starting from the "initial state"un
fold the succession of states, that is, the set of trajectories of
its constitutive bodies.
The triumph of Newtonian science is the discovery that a
single force, gravity, determines both the motion of planets
and comets in the sky and the motions of bodies falling toward
the earth. Whatever pair of material bodies is considered, the
Newtonian system implies that they are linked by the same
force of attraction. Newtonian dynamics thus appears to be
doubly universal. The definition of the law of gravity that de
scribes how masses tend to approach one another contains no
reference to any scale of phenomena. It can be applied equally
well to the motion of atoms, of planets, or of the stars in a
galaxy. Every body, whatever its size, has a mass and acts as a
source of the Newtonian forces of interaction.
Since gravitational forces connect any two bodies (for two
bodies of mass m and m ' and separated by a distance r, the
gravitational force is kmm'fr2, where k is the Newtonian force
of attraction equal to 6. 67cm3g-1sec-2), the only true dy
namic system is the universe as a whole. Any local dynamic
system, such as our planetary system, can only be defined
approximately, by neglecting forces that are small in compari
son to those whose effect is being considered.
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60
It must be emphasized that whatever the dynamic system
chosen, the laws of motion can always be expressed in the
form F= ma. Other types of forces apart from those due to
gravity may be discovered (and actually have been discov
ered-for instance, electric forces of attraction and repulsion)
and would thereby modify the empirical content of the laws of
motion. They would not, however, modify the form of those
laws. In the world of dynamics, change is identified with acceler
ation or deceleration. The integration of the laws of motion
leads to the trajectories that the particles follow. Therefore the
laws of change, of time's impact on nature, are expressed in
terms of the characteristics of trajectories.
The basic characteristics of trajectories are lawfulness, de
terminism, and reversibility. We have seen that in order to cal
culate a trajectory we need, in addition to our knowledge of
the laws of motion, an empirical definition of a single in
stantaneous state of the system. The general law then deduces
from this "initial state"the series of states the system passes
through as time progresses, just as logic deduces a conclusion
from basic premises. The remarkable feature is that once the
forces are known, any single state is sufficient to define the
system completely, not only its future but also its past. At
each instant, therefore, everything is given. Dynamics defines
all states as equivalent: each of them allows all the others to be
calculated along with the trajectory which connects all states,
be they in the past or the future.
"Everything is given. "This conclusion of classical dynam
ics, which Bergson repeatedly emphasized, characterizes the
reality that dynamics describes. Everything is given, but ev
erything is also possible. A being who has the power to control
a dynamic system may calculate the right initial state in such a
way that the system " spontaneously" reaches any chosen
state at some chosen time. The generality of dynamic laws is .
matched by the arbitrariness of the initial conditions.
The reversibility of a dynamic trajectory was explicitly
stated by all the founders of dynamics. For instance, when
Galilee or Huyghens described the implications of the equiv
alence between cause and effect postulated as the basis of
their mathematization of motion, they staged thought experi
ments such as an elastic ball bouncing on the ground. As the
61
THE IDENTIFICATION OF THE REAL
result of its instantaneous velocity inversion, such a body
would return to its initial position. Dynamics assigns this
property of reversibility to all dynamic changes. This early
"thought experiment" illustrates a general mathematical prop
erty of dynamic equations. The structure of these equations
implies that if the velocities of all the points of a system are
reversed, the system will go "backward in time. " The system
would retrace all the states it went through during the previous
change . Dynamics defines as mathematically equivalent
changes such as t-+- t, time inversion, and v-+- v, velocity
reversal . What one dynamic change has achieved, another
change, defined by velocity inversion, can undo, and in this
way exactly restore the original conditions.
This property of reversibility in dynamics leads, however, to
a difficulty whose full significance was realized only with the
introduction of quantum mechanics. Manipulation and mea
surement are essentially irreversible. Active science is thus,
by definition, extraneous to the idealized, reversible world it is
describing. From a more general point of view, reversibility
may be taken as the very symbol of the "strangeness" of the
world described by dynamics. Everyone is familiar with the
absurd effects produced by projecting a film backward-the
sight of a match being regenerated by its flame, broken ink
pots that reassemble and return to a tabletop after the ink has
poured back into them, branches that grow young again and
turn into fresh shoots. In the world of classical dynamics, such
events are considered to be just as likely as the normal ones.
We are so accustomed to the laws of classical dynamics that
are taught to us early in school that we often fail to sense the
boldness of the assumptions on which they are based. A world
in which all trajectories are reversible is a strange world in
deed. Another astonishing assumption is that of the complete
independence of initial conditions from the laws of motion. It
is possible to take a stone and throw it with some initial ve
locity limited only by one's physical strength , but what about
a complex system SlJCh as a gas formed by many particles? It
is obvious that we can no longer impose arbitrary initial condi
tions. The initial conditions must be the outcome of the dy
namic evolution itself. This is an important point to which we
shall come back in the third part of this book. But whatever its
ORDER OUT OF CHAOS
62
limitations, today, three centuries later, we can only admire
the logical coherence and the power of the methods discovered
by the founders of classical dynamics.
Motion and Change
Aristotle made time the measure of change. But he was fully
aware of the qualitative multiplicity of change in nature. Still
there is only one type of change surviving in dynamics, one
"process," and that is motion. The qualitative diversity of
changes in nature is reduced to the study of the relative dis
placement of material bodies. Time is a parameter in terms of
which these displacements may be described. In this way
space and time are inextricabl y tied together in the world of
classical dynamics. (Also see Chapter IX.)
It is interesting to compare dynamic change with the atom
ists' conception of change, which enjoyed considerable favor
at the time Newton formulated his laws. Actually, it seems that
not only Descartes, Gassendi, and d'Alembert, but even New
ton himself believed that collisions between hard atoms were
the ultimate, and perhaps the only, sources of changes of mo
tion.t Nevertheless, the dynamic and the atomic descriptions
differ radically. Indeed, the continuous nature of the accelera
tion described by the dynamic equations is in sharp contrast
with the discontinuous, instantaneous collisions between hard
particles. Newton had already noticed that, in contradiction to
dynamics, an irreversible loss of motion is involved in each
hard collision. The only reversible collision-that is, the only
one in agreement with the laws of dynamics-is the "elastic,"
momentum-conserving collision. But how can the complex
property of "elasticity" be applied to atoms that are supposed
to be the fundamental elements of nature?
On the other hand , at a less technical level , the laws of dy
namic motion seem to contradict the randomness generally
attributed to collisions between atoms. The ancient philoso
phers had already pointed out that any natural process can be
interpreted in many different ways in terms of the motion of
and collisions between atoms. This was not a problem for the
atomists, since their main aim was to describe a godless, law-
63
THE IDENTIFICATION OF THE REAL
less world in which man is free and can expect to receive nei
ther punishment nor reward from any divine or natural order.
But classical science was a science of engineers and astrono
mers, a science of action and prediction. Speculations based
on hypothetical atoms could not satisfy its needs. In contrast,
Newton's law provided a means of predicting and manipulat
ing. Nature thus becomes law-abiding, docile, and predict
able, instead of being chaotic, unruly, and stochastic. But
what is the connection between the mortal, unstable world in
which atoms unceasingly combine and separate, and the im
mutable world of dynamics governed by Newton's law, a siagle
mathematical formula corresponding to an eternal truth un
folding toward a tautological future? In the twentieth century
we are again witnessing the clash between lawfulness and ran
dom events, which, as Koyre has shown, had already tor
mented Descartes. 2 Ever since the end of the nineteenth
century, with the kinetic theory of gases, the atomic chaos has
reintegrated physics, and the problem of the relationship be
tween dynamic law and statistical description has penetrated
to the very core of physics. It is one of the key elements in the
present renewal of dynamics (see Book III).
In the e ighteenth century, however, this contradiction
seemed to produce a deadlock. This may partly explain the
skepticism of some eighteenth-century physicists regarding
the significance of Newton's dynamic description. We have al
ready noted that collisions may lead to a loss of motion. They
thereby concluded that in such nonideal cases, "energy" is
not conserved but is irreversibly dissipated (see Chapter IV,
section 3). Therefore, the atomists could not help considering
dynamics as an idealization of limited value. Continental
physicists and mathematicians such as dl\lembert, Clairaut,
and Lagrange resisted the seductive charms of Newtonianism
for a long time.
Where do the roots of the Newtonian concept of change lie?
It appears to be a synthesis3 of the science of ideal machines,
where motion is transmitted without collision or friction be
tween parts already in contact, and the science of celestial
bodies interacting at a distance. We have seen that it appears
as the very antithesis of atomism, which is based on the con
cept of random collisions. Does this, however, vindicate the
view of those who believe that Newtonian dynamics repre-
ORDER OUT OF CHAOS
64
sents a rupture in the history of thinking, a revolutionary nov
elty? This is what positivist historians have claimed when they
described how Newton escaped the spell of preconceived no
tions and had the courage to infer from the mathematical study of
planetary motions and the laws of falling bodies the action of a
"universal" force. We know that on the contrary the eighteenth
century rationalists emphasized the apparent similarity be
tween his " mathematical" forces and traditional occult
qualities. Fortunately, these critics did not know the strange
story behind the Newtonian forces! For behind Newton's cau
tious declaration-"! frame no hypotheses"-concerning the
nature of the forces lurked the passion of an alchemist.4 We
now know that, side by side with his mathematical studies,
Newton had studied the ancient alchemists for thirty years
and had carried out painstaking laboratory experiments on
ways of achieving the master work, the synthesis of gold.
Recently some historians have gone so far as to propose that
the Newtonian synthesis of heaven and earth was the achieve
ment of a chemist, not an astronomer. The Newtonian force
"animating" matter and, in the stronger sense of the term,
making up the very activity of nature would then be the inheri
tor of the forces Newton the chemist observed and manipu
lated, the chemical "affinities" forming and disrupting ever
new combinations of matter.s The decisive role played by ce
lestial orbits of course remains. Still, at the start of his intense
astronomical studies-about 1679-Newton apparently ex
pected to find new forces of attraction only in the heavens,
forces similar to chemical forces and perhaps easier to study
mathematically. Six years later this mathematical study pro
duced an unexpected conclusion: the forces between the plan
ets and those accelerating freely falling bodies are not merely
similar but are the same. Attraction is not specific to each
planet; it is the same-for the moon circling the earth, for the
planets, and even for comets passing through the solar system.
Newton set out to discover in the sky forces similar to the
chemical forces : the specific affinities, different for each
chemical compound and giving each compound qualitatively
differentiated activities. What he actually found was a univer
sal law, which, as he emphasized, could be applied to all phe
nomena-whether chemical , mechanical , or celestial in
nature.
65
THE IDENTIFICATION OF THE REAL
The Newtonian synthesis is thus a surprise. It is an unex
pected, staggering discovery that the scientific world has com
memorated by making Newton the symbol of modern science.
What is particularly astonishing is that the basic code of na
ture appeared to have been cracked in a single creative act.
For a long time this sudden loquaciousness of nature, this
triumph of the English Moses, was a source of intellectual
scandal for continental rationalists. Newton 's work was
viewed as a purely empirical discovery that could thus equally
well be empirically disproved. In 1747 Euler, Clairaut, and
d�lembert, without doubt some of the greatest scientists of
the time, came to the same conclusion: Newton was wrong. In
order to describe the moon's motion, a more complex mathe
matical form must be given to the force of attraction, making it
the sum of two terms. For the following two years, each of
them believed that nature had proved Newton wrong, and this
belief was a source of excitement, not of dismay. Far from con
sidering Newton's discovery synonymous with physical sci
ence itself, physicists were blithely contemplating dropping it
altogether. D�lembert went so far as to express scruples
about seeking fresh evidence against Newton and giving him
"le coup de pied de l'iine. "6
Only one courageous voice against this verdict was raised in
France. In 1748, Buffon wrote:
A physical law is a law only by virtue of the fact that it is
easy to measure, and that the s�ale it represents is not
only always the same , but is actually unique . . .. M .
Clairaut has raised a n objection against Newton's sys
tem, but it is at best an objection and must not and cannot
become a principle; an attempt should be ,made to over
come it and not to turn it into a theory the entire con
sequences of which merely rest on a calculation; for, as I
have said, one may represent anything by means of cal
culation and achieve nothing; and if it is allowed to add
one or more terms to a physical law such as that of attrac
tion, we are only adding to arbitrariness instead of repre
senting reality. 7
Later Buffon was to announce what was to become, although
for only a short time, the research program for chemistry:
ORDER OUT OF CHAOS
66
The laws of affinity by means of which the constituent
parts of different substances separate from others to
combine together to form homogeneous substances are
the same as the general law governing the reciprocal ac
tion of all the celestial bodies on one another: they act in
(he same way and with the same ratios of mass and dis
tance ; a globule of water, of sand or metal acts upon an
other globule just as the terrestrial globe acts on the
moon, and if the laws of affinity have hitherto been re
garded as different from those of gravity, it is because
they have not been fully understood, not grasped com
pletely; it is because the whole extent of the problem has
not been taken in. The figure which, in the case of celes
tial bodies has little or no effect upon the law of interac
tion between bodies becau se of the great distance
involved, is, on the contrary, all important when the dis
tance is very small or zero . . . . Our nephews will be
able, by calculation, to gain access to this new field of
knowledge [that is, to deduce the law of interaction be
tween elementary bodies from their figures] .&
History was to vindicate the naturalist, for whom force was
not mer;ely a mathematical artifice but the very essence of the
new science of nature. The physicists were later compelled to
admit their mistake. Fifty years afterward, Laplace could
write his Systeme du Monde. The law of universal gravity had
stood all tests successfully: the numerous cases apparently
disproving it had been transformed into a brilliant demonstra
tion of its validity. At the same time, under Buffon's influence,
the French chemists rediscovered the odd analogy between
physical attraction and chemical affinity.9 Despite the sar
casms of d/\lembert, Condillac, and Condorcet, whose unbend
ing rationalism was quite incompatible with these obscure and
barren "analogies, " they trod Newton's path in the opposite
direction-from the stars to matter.
By the early nineteenth century, the Newtonian program
the reduction of all physicochemical phenomena to the action
of forces (in addition to gravitational attraction, this included
the repelling force of heat, which makes bodies expand and
favors dissolution, as well as electric and magnetic forces)
had become the official program of Laplace's school, which
67
THE IDENTIFICATION OF THE REAL
dominated the scientific world at the time when Napoleon
dominated Europe. JO
The early nineteenth century saw the rise of the great
French ecoles and the reorganization of the universities. This
is the time when scientists became teachers and professional
researchers and took up the ta�k of training their successors. • •
It is also the time of the first attempts to present a synthesis of
knowledge, to gather it together in textbooks and works of
popularization. Science was no longer discussed in the salons;
it was taught or popularized.1 2 It had become a matter of pro
fessional consensus and magistral authority. The first con
sensus centered around the Newtonian system: in France
Buffon's confidence finally triumphed over the rational skepti
cism of the Enlightenment.
One century after Newton's apotheosis in England, the
grandiloquence of these lines written by Ampere's son echoes
that of Pope's epitaph: 1 3
Announcing the coming of science's Messiah
Kepler had dispelled the clouds around the Arch.
Then the Word was made man, the Word of the seeing
God
Whom Plato revered, and He was called Newton.
He came, he revealed the principle supreme,
Eternal, universal, One and unique as God Himself.
The worlds were hushed, he spoke: ATTRACTION.
This word was the very word of creation. *
For a short time, which nevertheless left an indelible mark,
science was triumphant, acknowledged and honored by
powerful states and acclaimed as the possessor of a consistent
conception of the world. Worshiped by Laplace, Newton be
came the universal symbol of this golden age. It was a happy
moment, indeed, a moment in which scientists were regarded
both by themselves and others as the pioneers of progress,
achieving an enterprise sustained and fostered by society as a
whole.
What is the significance of the Newtonian synthesis today,
after the advent of field theory, relativity, and quantum me*Our translation-authors.
ORDER OUT OF CHAOS
68
chanics? This is a complex problem, to which we shall return.
We now know that nature is not always "comfortable and con
sonant with herself. " At the microscopic level, the laws of
classical mechanics have been replaced by those of quantum
mechanics. Likewise, at the level of the universe, relativistic
physics has displaced Newtonian physics. Classical physics
nevertheless remains the natural reference point. Moreover, in
the sense that we have defined it-that is, as the description of
deterministic, reversible, static trajectories-Newtonian dy
namics still may be said to form the core of physics .
O f course, since Newton the formulation of classical dy
namics has undergone great changes. This was a result of the
work of some of the greatest mathematicians and physicists,
such as Hamilton and Poincare. In brief, we may distinguish
two periods. First there was a period of clarification and of
generalization. During the second period, the very concepts
upon which classical dynamics rests, such as initial conditions
and the meaning of trajectories, have undergone a critical revi
sion even in the fields in which (in contrast to quantum me
chanics and relativity) classical dynamics remains valid. At
the moment this book is being written, at the end of the twen
tieth century, we are still in this second period. Let us turn
now to the general language of dynamics that was discovered
by nineteenth-century scientists. (In Chapter IX we shall de
scribe briefly the revival of classical dynamics in our time.)
The Language of Dynarnics
Today classical dy namics can be formulated in a compact and
elegant way. As we shall see, all the properties of a dynamic
system can be summarized in terms of a single function, the
Hamiltonian. The language of dynamics presents a remarkable
consistency and completeness. An unambiguous formulation
can be given to each "legitimate" problem. No wonder the
structure of dynamics has both fascinated and terrified the
imagination since the eighteenth century.
In dynamics,the same system can be studied from different
points of view. In classical dynamics all these points of view
are equivalent in the sense that we can go from one to another
by a transformation, a change of variables. We may speak of
69
THE IDENTIFICATION OF THE REAL
various equivalent representations in which the laws of dy
namics are valid. These various equivalent representations
form the general language of dynamics. This language can be
used to make explicit the static character classical dynamics
attributes to the systems it describes: for many classes of dy
namic systems, time appears merely as an accident, since
their description can be reduced to that of noninteracting me
chanical systems. To introduce these concepts in a simple way,
let us start with the principle of conservation of energy.
In the ideal world of dynamics, devoid of frictions and colli
sions, machines have an efficiency of one-the dynamic sys
tem comprising the machine merely transmits the whole of the
motion it receives. A machine receiving a certain quantity of
potential energy (for example, a compressed spring, a raised
weight, compressed air) can produce a motion corresponding
to an "equal" quantity of kinetic energy, exactly the quantity
that would be needed to restore the potential energy the ma
chine has used in producing the motion. The simplest case is
that in which the only force considered is gravity (which ap
plies to simple machines, pulleys, levers, capstans, etc.). In
this case it is easy to establish an overall relationship of equiv
alence between cause and effect. The height (h) through which
a body falls entirely determines the velocity acquired during
its fall. Whether a body of mass m falls vertically, runs down
an inclined plane, or follows a roller-coaster path, the acquired
velocity (v) and the kinetic energy (mv2/2) depend only on the
drop in level h (v = Vfiii) and enable the body to return to its
original height. The work done against the force of gravity im
plied in this upward motion restores the potential energy, mgh,
that the system lost during the fall. Another example is the
pendulum, in which kinetic energy and potential energy are
continuously transformed into one another.
Of course, if instead of a body falling toward the earth, we
are dealing with a system of interacting bodies, the situation is
less easily visualized. Still, at each instant the global variation
in kinetic energy compensates for the variation in potential
energy (bound to the variation in the distances between the
points in the system). Here also energy is conserved in an iso
lated system.
Potential energy (or ''potential," conventionally denoted as
V), which depends on the relative positions of the particles, is
ORDER OUT OF CHAOS
70
thus a generalization of the quantity that enabled builders of
machines to measure the motion a machine could produce as
the result of a change in its spatial configuration (for example,
the change in the height of a mass m, which is part of the
machine, gives it a potential energy mgh). Moreover, potential
energy allows us to calculate the set of forces applied at each
instant to the different points of the system to be described. At
each point the derivative of the potential with respect to the
space coordinate q measures the force applied at this point in
the direction of that coordinate. Newton's laws of motion thus
can be formulated using the potential function instead of force
as the main quantity: the variation in the velocity of a point
mass at each instant (or the momentum p, the product of the
mass and the velocity) is measured by the derivative of the
potential with respect to the coordinate q of the mass.
In the nineteenth century this formulation was generalized
through the introduction of a new function, the Hamiltonian
(H). This function is simply the total energy, the sum of the
system's potential and kinetic energy. However, this energy is
no longer expressed in terms of positions and velocities, con
ventionally denoted by q and dq/dt, but in terms of so-called
canonical variables-coordinates and momenta-for which
the standard notation is q and p. In simple cases, such as with
a free particle, there is a straightforward relation between ve
locity and momentum (p m dqldt), but in general the relation
is more complicated.
A single function, the Hamiltonian, H(p, q), describes the
dynamics of a system completely. All our empirical knowledge
is put into the form of H. Once this function is known, we may
solve, at least in principle, all possible problems. For example,
the time variation of the coordinate and of the momenta is
simply given by the derivatives of H in respect to p or q. This
Hamiltonian formulation of dynamics is one of the greatest
achievements in the history of science. It has been progres
sively extended to cover the theory of electricity and magne
tism. It has also been used in quantum mechanics. It is true
that in quantum mechanics, as we shall see later, the meaning
of the Hamiltonian H had to be generalized: here it is no
longer a simple function of the coordinates and momenta, but
=
it becomes a new kind of entity, an operator. (We shall return
to this question in Chapter VII.) In any case, the Hamiltonian
71
THE IDENTIFICATION OF THE REAL
description is still of the greatest importance today. The equa
tions which, through the derivatives of the Hamiltonian, give
the time variation of the coordinates and momenta are the so
called canonical equations. They contain the general proper
ties of all dynamic changes. Here we have the triumph of the
mathematization of nature. All dynamic change to which clas
sical dynamics applies can be reduced to these simple mathe
matical equations.
Using these equations , we can verify the above-mentioned
general properties implied by classical dynamics. The canoni
cal equations are reversible: time inversion is mathematically
the equivalent of velocity inversion. They are also conserva
tive: the Hamiltonian, which expressed the system's energy in
the canonical variables-coordinates and momenta-is itself
conserved by the changes it brings about in the course of time.
We have already noticed that there exist many points of view
or " representations" in which the Hamiltonian form of the
equations of motion is maintained. They correspond to various
choices of coordinates and momenta. One of the basic prob
lems of dynamics is to examine precisely how we can select
the pair of canonical variables q and p to obtain as simple a
description of dynamics as possible. For example, we could
look for canonical variables by which the Hamiltonian is re
duced to kinetic energy and depends only on the momenta
(and not on the coordinates). What is remarkable is that in this
case momenta become constants of motion. Indeed, as we
have seen, the time variation of the momenta depends, accord
ing to the canonical equation, on the derivative of the Hamilto
nian i n respect to the coordinates. When this derivative
vanishes, the momenta indeed become constants of motion.
This is similar to what happens in a "free particle" system.
What we have done when we go to a free particle system is
"eliminate" the interaction through a change of representa
tion. We will define systems for which this is possible as "inte
grable s y ste m s . " Any integrable s y st e m may thus b e
represented a s a set of units, each changing i n isolation, quite
independently of all the others, in that eternal and immutable
motion Aristotle attributed to the heavenly bodies (Figure 1).
We have already noted that in dynamics "everything is
given. " Here this means that, from the very first instant, the
value of the various invariants of the motion is fixed; nothing
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72
•
•
•
>
•
•
•
•
(a )
(b)
Figure 1 . Two representations of the same dynamic system : (a) as a set of
interacting points; the interaction between the points is represented by wavy
lines; (b) as a set where each point behaves independently from the others.
The potential energy being eliminated, their respective · motions are not ex
plicitly dependent on their relative positions.
may "happen" or "take place. " Here we reach one of those
dramatic moments in the history of science when the descrip
tion of nature was nearly reduced to a static picture. Indeed,
through a clever change of variables, all interaction could be
made to disappear. It was believed that integrable systems ,
reducible to free particles, were the prototype of dynamic sys
tems. Generations of physicists and mathematicians tried hard
to find for each kind of systems the "right" variables that
would eliminate the interactions. One widely studied example
was the three-body problem, perhaps the most important
problem in the history of dynamics. The moon's motion, influ
enced by both the earth and the sun, is one instance of this
problem. Countless attempts were made to express it in the
form of an integrable system until, at the end of the nineteenth
century, Bruns and Poincare showed that this was impossible.
This came as a surprise and, in fact, announced the end of all
simple extrapolations of dynamics based on integrable sys
tems . The discovery of Bruns and Poincare shows that dy
namic systems are not isomorphic. Simple, integrable systems
can indeed be reduced to noninteracting units, but in general,
interactions cannot be eliminated. Although this d iscovery
was not clearly understood at the time, it implied the demise of
the conviction that the dynamic world is homogeneous, re
ducible to the concept of integrable systems. Nature as an
73
THE IDENTIFICATION OF THE REAL
evolving, interactive multiplicity thus resisted its reduction to
a timeless and universal scheme.
There were other indications pointing in the same direction.
We have mentioned that trajectories correspond to determinis
tic laws; once an initial state is given, the dynamic laws of mo
tion permit the calculation of trajectories at each point in the
future or the past. However, a trajectory may become intrin
sically indeterminate at certain singular points. For instance, a
rigid pendulum may display two qualitatively different types of
behavior-it may either oscillate or swing around its points of
suspension. If the initial push is just enough to bring it into a
vertical position with zero velocity, the direction in which it
will fall, and therefore the nature of its motion, are indetermi
nate. An infinitesimal perturbation would be enough to set it
rotating or oscillating. (This problem of the "instability" of
motion will be discussed fully in Chapter IX.)
It is significant that Maxwell had already stressed the impor
tance of these singular points. After describing the explosion
of gun cotton, he goes on to say:
In all such cases there is one common circumstance
the system has a quantity of potential energy, which is
capable of being transformed into motion, but which can
not begin to be so transformed till the system has reached
a certain configuration, to attain which requires an expendi
ture of work, which in certain cases may be infinitesimally
small, and in general bears no definite proportion to the
energy developed in consequence thereof. For example,
the rock loosed by frost and balanced on a singular point
of the mountain-side, the little spark which kindles the
great forest, the little word which sets the world a fight
ing, the little scruple which prevents a man from doing his
will, the little spore which blights all the potatoes, the
little gemmule which makes us philosophers or idiots.
Every existence above a certain rank has its singular
points: the higher the rank, the more of them. At these
points, influences whose physical magnitude is too small
to be taken account of by a finite being, may produce
results of the greatest importance. All great results pro
duced by human endeavour depend on taking advantage
of these singular states when they occur. I4
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74
This conception received no further elaboration owing to the
absence of suitable mathematical techniques for identifying
systems containing such singular points and the absence of the
chemical and biological knowledge that today affords, as we
shall see later, a deeper insight into the truly essential role
played by such singular points.
Be that as it may, from the time of Leibniz' monads (see the
conclusion to section 4) down to the present day (for example,
the stationary states of the electrons in the Bohr model-see
Chapter VII), integrable systems have been the model par ex
cellence of dynamic systems, and physicists have attempted to
extend the properties of what is actually a very special class of
Hamiltonian equations to cover all natural processes. This is
quite understandable. The class of integrable systems is the
only one that, until recently, had been thoroughly explored.
Moreover, there is the fascination always associated with a
closed system capable of posing all problems, provided it does
not define them as meaningless. Dynamics is such a language;
being complete, it is by definition coextensive with the world
it is describing. It assumes that all problems, whether simple
or complex, resemble one another since it can always pose
them in the same general form. Thus the temptation to con
clude that all problems resemble one another from the point of
view of their solutions as well, and that nothing new can ap
pear as a result of the greater or lesser complexity of the inte
gration procedure. It is this intrinsic homogeneity that we now
know to be false. Moreover, the mechanical world view was
acceptable as long as all observables referred in one way or
another to motion. This is no longer the case. For example
unstable particles have an energy that can be related to motion
but that also has a lifetime that is a quite different type of ob
servable, more closely related to irreversible processes, as we
shall describe them in Chapters IV and V. The necessity of
introducing new observables into the theoretical sciences was,
and still is today, one of the driving forces that move us beyond
the mechanical world view.
,
75
THE IDENTIFICATION OF THE REAL
Laplaces Demon
Extrapolations from the dynamic description discussed above
have a symbol-the demon imagined by Laplace, capable at
any given instant of observing the position and velocity of
each mass that forms part of the universe and of inferring its
evolution, both toward the past and toward the future. Of
course, no one has ever dreamed that a physicist might one
day benefit from the knowledge possessed by Laplace's de
mon. Laplace himself only used this fiction to demonstrate the
extent of our ignorance and the need for a statistical descrip
tion of certain processes. The problematics of Laplace's de
mon are not related to the question of whether a deterministic
prediction of the course of events is actually possible, but
whether it is possible in principle, de jure. This possibility
seems to be implied in mechanistic description, with its
characteristic duality based on dynamic law and initial condi
tions.
Indeed, the fact that a dynamic system is governed by a
deterministic law, even though in practice our ignorance of the
initial state precludes any possibility of deterministic predic
tions, allows the "objective truth" of the system as it would be
seen by Laplace's demon to be distinguished from empirical
limitations due to our ignorance. In the context of classical
dynamics, a deterministic description may be unattainable in
practice ; nevertheless, it stands as a limit that defines a series
of increasingly accurate descriptions.
It is precisely the consistency of this duality formed by dy
namic law and initial conditions that is challenged in the re
vival of classical mechanics, which we will describe in Chapter
IX. We shall see that the motion may become so complex, the
trajectories so varied, that no observation, whatever its preci
sion, can lead us to the determination of the exact initial condi
tions. But at that point the duality on which classical mechanics
was constructed breaks down. We can predict only the average
behavior of bundles of trajectories.
Modern science was born out of the breakdown of the ani
mistic alliance with nature. Man seemed to possess a place in
the Aristotelian world as both a living and a knowing creature.
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76
The world was made to his measure. The first experimental
dialogue received part of its social and philosophic justifica
tion from another alliance, this time with the rational God of
Christianity. To the extent to which dynamics has become and
still is the model of science, certain implications of this histor
ical situation have persisted to our day.
Science is still the prophetic announcement of a description
of the world seen from a divine or demonic point of view. It is
the science of Newton, the new Moses to whom the truth of
the world was unveiled; it is a revealed science that seems
alien to any social and historical context identifying it as the
result of the activity of human society. This type of inspired
discourse is found throughout the history of physics. It has ac
companied each conceptual innovation, each occasion at
which physics seemed at the point of unification and the pru
dent mask of positivism was dropped. Each time physicists
repeated what Ampere's son stated so explicitly: this word
universal attraction, energy, field theory, or elementary parti
cles-is the word of creation. Each time-in Laplace's time,
at the end of the nineteenth century, or even today-physicists
announced that physics was a closed book or about to become
so. There was only one final stronghold where nature con
tinued to resist, the fall of which would leave it defenseless,
conquered, and subdued by our knowledge. They were thus
unwittingly repeating the ritual of the ancient faith. They were
announcing the coming of the new Moses, and with him a new
Messianic period in science.
Some might wish to disregard this prophetic claim, this
somewhat naive enthusiasm, and it is certainly true that di
alogue with nature has gone on all the same, together with a
search for new theoretical languages, new questions, and new
answers. But we do not accept a rigid separation between the
scientist's "actual" work and the way he judges, interprets,
and orientates this work. To accept it would be to reduce sci
ence to an ahistorical accumulation of results and to pay no
attention to what scientists are looking for, the ideal knowl
edge they try to attain, the reasons why they occasionally
quarrel or remain unable to communicate with each other. ts
Once again, it was Einstein who formulated the enigma pro
duced by the myth of modern science. He has stated that the
miracle, the only truly astonishing feature, is that science ex-
77
THE IDENTIFICATION OF THE REAL
ists at all , that we find a convergence between nature and the
human mind. Similarly, when, at the end of the nineteenth
century, du Bois Reymond made Laplace's demon the very
incarnation of the logic of modern science, he added, "Igno
ramus, ignorabimus" : we shall always be totally ignorant of
the relationship between the world of science and the mind
which knows, perceives, and creates this science. I6
Nature speaks with a thousand voices, and we have only be
gun to listen. Nevertheless, for nearly two centuries Laplace's
demon has plagued our imagination, bringing a nightmare in
which all things are insignificant. If it were really true that the
world is such that a demon-a being that is, after all, like us,
possessing the same science , but endowed with sharper
senses and greater powers of calculation-could, starting from
the observation of an instantaneous state, calculate its future
and past, if nothing qualitatively differentiates the simple sys
tems we can describe from the more complex ones for which a
demon is needed, then the world is nothing but an immense
tautology. This is the challenge of the science we have inher
ited from our predecessors, the spell we have to exorcise to
day.
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CHAPTER Ill
THE TWO CULTURES
Diderot and the Discourse of the Living
In his interesting book on the history of the idea of progress,
Nisbet writes:
No single idea has been more important than, perhaps as
important as, the idea of progress in Western civilization
for nearly three thousand years. •
There has been no stronger support for the idea of progress
than the accumulation of knowledge. The grandiose spectacle
of this gradual increase of knowledge is indeed a magnificent
example of a successful collective human endeavor.
Let us recall the remarkable discoveries achieved at the end
of the eighteenth century and the beginning of the nineteenth
century: the theories of heat, electricity, magnetism, and op
tics. It is not surprising that the idea of scientific progress,
already clearly formulated in the eighteenth century, domi
nated the nineteenth. Still, as we have pointed out, the posi
tion of science in Western culture remained unstable. This
lends a dramatic aspect to the history of ideas from the high
point of the Enlightenment.
We have already stated the alternative: to accept science
with what appears to be its alienating conclusions or to turn to
an antiscientific metaphysics. We have also emphasized the
solitude felt by modern men, the loneliness described by Pas
cal, Kierkegaard , or Monod. We have mentioned the anti
scientific implications of Heidegger's metaphysics. Now we
wish to discuss more fully some aspects of the intellectual his
tory of the West, from Diderot, Kant, and Hegel to Whitehead
and Bergson; all of them attempted to analyze and limit the
scope of modern science as well as to open new perspectives
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80
seen as radically alien to that science. Today it is u sually
agreed that those attempts have for the most part failed. Few
would accept, for example, Kant's division of the world into
phenomenal and noumenal spheres, or Bergson's "intuition"
as an alternative path to a knowledge whose significance
would parallel that of science. Still these attempts are part of
our heritage. The history of ideas cannot be understood with
out reference to them.
We shall also briefly discuss scientific positivism, which is
based on the separation of what is true from what is scientifi
cally useful. At the outset this positivistic view may seem to op
pose clearly the metaphysical views we have mentioned, views
that I . Berlin described as the " Counter-Enlightenment. "
However, their fundamental conclusion is the same: we must
reject science as a basis for true knowledge even if at the same
time we recognize its practical importance or we deny, as posi
tivists do, the possibility of any other cognitive enterprise.
We must remember all these developments to understand
what is at stake� To what extent is science a basis for the intel
ligibility of nature, including man? What is the meaning of the
idea of progress today?
Diderot, one of the towering figures of the Enlightenment, is
certainly no representative of antiscientific thought. On the
contrary, his confidence in science, in the possibilities of
knowledge, was total. Yet this is the very reason why science
had, following Diderot, to understand life before it could hope
to achieve any coherent vision of nature.
We have already mentioned that the birth of modern science
was marked by the abandonment of vitalist inspiration and, in
particular, of Aristotelian final causes. However, the issue of
the organization of living matter remained and became a chal
lenge for classical science. Diderot, at the height of the Newto
nian triumph, emphasizes that this problem was repressed by
physics. He imagines it as haunting the dreams of physicists
who cannot conceive of it while they are awake. The physicist
d /\lembert is dreaming:
·� living point . . . No, that's wrong. Nothing at all to
begin with, and then a living point. This living point is
joined by another, and then another, and from these suc
cessive joinings there results a unified being, for I am
a
81
THE TWO CULTURES
unity, of that I am certain. . . . (As he said this he felt
himself all over.) But how did this unity come about?"
"Now listen, Mr.Philosopher, I can understand an aggre
gate, a tissue of tiny sensitive bodies, but an animal! . . .
A whole, a system that is a unit, an individual conscious
of its own unity! I can't see it, no, I can't see it."2
In an imaginary conversation with dj\lembert, Oiderot speaks
in the first person, demonstrating the inadequacy of mechanis
tic explanation:
Look at this egg: with it you can overthrow all the schools
of theology and all the churches in the world. What is this
egg? An insensitive mass before the germ is put into
it . . . How does this mass evolve into a new organiza
tion, into sensitivity, into life? Through heat. What will
generate heat in it? Motion. What will the successive
effects of motion be? Instead of answering me, sit down
and let us follow out these effects with our eyes from one
moment to the next. First there is a speck which moves
about, a thread growing and taking colour, flesh being
formed, a beak, wing-tips, eyes, feet coming into view, a
yellowish substance which unwinds and turns into intes
tines-and you have a living creature. . . . Now the wall
is breached and the bird emerges, walks, flies, feels pain,
runs away, comes back again, complains, suffers, loves,
desires, enjoys, it experiences all your affections and
does all the things you do. And will you maintain, with
Descartes, that it is an imitating machine pure and sim
ple? Why, even little children will laugh at you, and phi
losophers will answer that if it is a machine you are one
too! If, however, you admit that the only difference be
tween you and an animal is one of organization, you will
be showing sense and reason and be acting in good faith;
but then it will be concluded, contrary to what you had
said, that from an inert substance arranged in a certain
way and impregnated by another inert substance, sub
jected to heat and motion, you will get sensitivity, life,
memory, consciousness, passions, thought . . . Just lis
ten to your own arguments and you will feel how pitiful
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82
they are. You will come to feel that by refusing to enter
tain a simple hypothesis that explains everything-sen
sitivity as a property common to all matter or as a result
of the organization of matter-you are flying in the face of
common sense and plunging into a chasm of mysteries,
contradictions and absurdities.3
In opposition to rational mechanics, to the claim that mate
rial nature is nothing but inert mass and motion, Diderot ap
peals to one of physics' most ancient sources of inspiration,
namely, the growth, differentiation, and organization of the
embryo. Flesh forms, and so does the beak, the eyes, and the
intestines; a gradual organization occurs in biological "space,"
out of an apparently homogeneous environment differentiated
forms appear at exactly the right time and place through the
effects of complex and coordinated processes.
How can an inert mass, even a Newtonian mass animated
by the forces of gravitational interaction, be the starting point
for organized active local structures? We have seen that the
Newtonian system is a world system: no local configuration of
·bodies can claim a particular identity; none is more than a
contingent proximity between bodies connected by general re
lations.
But Diderot does not despair. Science is only beginning; ra
tional mechanics is merely a first, overly abstract attempt. The
spectacle of the embryo is enough to refute its claims to uni
versality. This is why Diderot compares the work of great
"mathematicians" such as Euler, Bernoulli, and di\lembert to
the pyramids of the Egyptians , awe-inspiring witnesses to the
genius of their builders, now lifeless ruins, alone and forlorn.
True science, alive and fruitful, will be carried on elsewhere."
Moreover, it seems to him that this new science of organized
living matter has already begun. His friend d' Holbach is busy
studying chemistry, Diderot himself has chosen medicine. The
problem in chemistry as well as in medicine is to replace inert
matter with active matter capable of organizing itself and pro
ducing living beings. Diderot claims that matter has to be sen
sitive. Even a stone has sensation in the sense that the
molecules of which it is composed actively seek certain com
binations rather than others and thus are governed their likes
and dislikes. The sensitivity of the whole organism is then
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THE TWO CULTURES
simply the sum of that of its parts, just as a swarm of bees with
its globally coherent behavior is the result of interactions be
tween one bee and another; and, Diderot thereby concludes,
the human soul does not exist any more than the soul of the
beehive does. s
Diderot's vitalist protest against physics and the universal
laws of motion thus stems from his rejection of any form of
spiritualist dualism. Nature must be described in such a way
that man's very existence becomes understandable. Other
wise, and this is what happens in the mechanistic world view,
the scientific description of nature will have its counterpart in
man as an automaton endowed with a soul and thereby alien to
nature.
The twofold basis of materialistic naturalism, at once chemi
cal and medical, that Diderot employed to counter the physics
of his time is recurrent in the eighteenth century. While biolo
gists speculated about the animal-machine, the preexistence of
germs, and the chain of living creatures-all problems close to
theology 6-chemists and physicians had to face directly the
complexity of real processes in both chemistry and life.Chem
istry and medicine were, in the late eighteenth century, priv
ileged sciences for those who fought against the physicists'
esprit de systeme in favor of a science that would take into
account the diversity of natural processes. A physicist could
be pure esprit, a precocious child, but a physician or a chemist
must be a man of experience: he must be able to decipher the
signs, to spot the clues. In this sense, chemistry and medicine
are arts. They demand judgment, application, and tenacious
observation. Chemistry is a madman's passion, Venel con
cluded in the article he wrote for Diderot's Encyclopedie, an
eloquent defense of chemistry against the abstract imperialism
of the Newtonians.7 To emphasize the fact that protests raised
by chemists and physicians against the way physicists reduced
living processes to peaceful mechanisms and the quiet unfold
ing of universal laws were common in Diderot's day, we invoke
the eminent figure of Stahl, the father of vitalism and inventor
of the first consistent chemical systematics.
According to Stahl, universal laws apply to the living only in
the sense that these laws condemn them to death and corrup
tion; the matter of which living beings are composed is so frail,
so easily decomposed, that if it were governed solely by the
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common laws of matter, it would not withstand decay or dis
solution for a moment. If a living creature is to survive in spite
of the general laws of physics, however short its life when it is
compared to that of a stone or another inanimate object, it has
to possess in itself a "principle of conservation" that main
tains the harmonious equilibrium of the texture and structure
of its body. The astonishing longevity of a living body in view
of the extreme corruptibility of its constitutive matter is thus
indicative of the action of a "natural, permanent, immanent
principle," of a particular cause that is alien to the laws of
inanimate matter and that constantly struggles against the con
stantly active corruption whose inevitability these laws imply.s
To us this analysis of life sounds both near and remote. It is
close to us in its acute awareness of the singularity and the pre
cariousness of life. It is remote because, like Aristotle, Stahl
defined life in static terms, in terms of conservation, not of
becoming or evolution. Still, the terminology used by Stahl
can be found in recent biological literature, for example,
where we read that enzymes "combat" decay and allow the
body to ward off the death to which it is inexorably doomed by
physics. Here also, biological organization defies the laws of
nature, and the only "normal" trend is that which leads to
death (see Chapter V ).
Indeed, Stahl's vitalism is relevant as long as the laws of
physics are identified with evolution toward decay and disor
ganization. Today the "vitalist principle" has been superseded
by the succession of improbable mutations preserved in the
genetic message "governing" the living structure. Nonethe
less, some extrapolations starting from molecular biology rele
gate life to the confines of nature-that is, conclude life is
compatible with the basic laws of physics but purely contin
gent. This was explicitly stated by Monod: life does not "fol
low from the laws of physics, it is compatible with them. Life
is an event whose singularity we have to recognize."
But the transition from matter to life can also be viewed in a
different way. As we shall see, far from equilibrium, new self
organizational processes arise. (These questions will be stud
ied in detail in Chapters V and VI.) In this way biological
organization begins to appear as a natural process.
However, long before these recent developments, the prob
lematics of life had been transformed. In a politically trans-
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THE TWO CULTURES
formed Europe the intellectual landscape was remodeled as
the Romantic movement, closely linked with the Counter
Enlightenment, shows.
Stahl criticized the metaphor of the automaton because, un
like a living being, the purpose of an automaton does not lie
within itself; its organization is imposed upon it by its maker.
Diderot, far from situating the study of life outside the reach of
science, saw it as representing the future of a science he con
sidered to be still in its infancy. A few years later, such points
of view were to be challenged.9 Mechanical change, activity as
described by the laws of motion, had now become syn
onymous with the artificial and with death. Opposed to it,
united in a complex with which we are now quite familiar, were
the concepts of life, spontaneity, freedom, and spirit. This op
position was paralleled by the opposition between calculation
and manipulation on the one hand, and the free speculative
activity of the mind on the other. Through speculation the phi
losopher would reach the spiritual activity at the core of na
ture. As for the scientist, his concern with nature would be
reduced to taking it as a set of manipulable and measurable
objects; he would thus be able to take possession of nature, to
dominate and control it but not understand it. Thus the intel
ligibility of nature would lie beyond the grasp of science.
We are not concerned here with the history of philosophy
but merely with emphasizing the extent to which the philo
sophical criticism of science had at this time become harsher,
resembling certain modern forms of antiscience. It was no
longer a question of refuting rather naive and shortsighted
generalizations that only have to be repeated aloud-to use
Diderot's language-to make even children laugh, but of refut
ing the type of approach that produced experimental and
mathematical knowledge of nature. Scientific knowledge is not
being criticized for its limitations but for its nature, and a rival
knowledge, based on another approach, is being announced.
Knowledge is fragmented into two opposed modes of inquiry.
From a philosophical point of view, the transition from Di
derot to the Romantics and, more precisely, from one of these
two types of critical attitudes toward science to the other, can
be found in Kant's transcendental philosophy, the essential
point being that the Kantian critique identified science in gen
eral with its Newtonian realization. It thereby branded as im-
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possible any opposition to classical science that was not an
opposition to science itself. Any criticism against Newtonian
physics must then be seen as aimed at downgrading the ra
tional understanding of nature in favor of a different form of
knowledge. Kant's approach had immense repercussions,
which continue down to our day. Let us therefore summarize
his point of view as presented in Critique of Pure Reason,
which, in opposition to the progressist views of the Enlighten
ment, presents the closed and limiting conception of science
we have just defined.
Kants Critical Ratification
How to restore order in the intellectual landscape left in disar
ray with the disappearance of God conceived as the rational
principle that links science and nature? How could scientists
ever have access to global truth when it could no longer be
asserted, except metaphorically, that science deciphers the
word of creation? God was now silent or at least no longer
spoke the same language as human reason. Moreover, in a na
ture from which time was eliminated, what remained of our
subjective experience? What was the meaning of freedom,
destiny, or ethical values?
Kant argued that there were two levels of reality: a phenom
enal level that corresponds to science, and a noumenal level
corresponding to ethics. The phenomenal order is created by
the human mind. The noumenal level transcends man's intel
lect; it corresponds to a spiritual reality that supports his ethi
cal and religious life. In a way, Kant's solution is the only one
possible for those who assert both the reality of ethics and the
reality of the objective world as it is expressed by classical
science. Instead of God, it is now man himself who is the
source of the order he perceives in nature. Kant justifies both
scientific knowledge and man's alienation from the phenom
enal world described by science. From this perspective we can
see that Kantian philosophy explicitly spells out the philo
sophical content of classical science.
Kant defines the subject of critical philosophy as transcen
dental. It is not concerned with the objects of exp erience but
is based on the a priori fact that a systematic knowledge of
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THE TWO CULTURES
these objects is possible (this is for him proved by the exis
tence of physics), going on to state the a priori conditions of
possibility for this mode of knowledge.
To do so a distinction must be made between the direct sen
sations we receive from the outside world and the objective,
"rational" mode of knowledge. Objective knowledge is not
passive; it forms its objects. When we take a phenomenon as
the object of experience, we assume a priori before we actually
experience it that it obeys a given set of principles. Insofar as it
is perceived as a possible object of knowledge, it is the prod
uct of our mind's synthetic activity. We find ourselves in the
objects of our knowledge, and the scientist himself is thus the
source of the universal laws he discovers in nature.
The a priori conditions of experience are also the conditions
for the existence of the objects of experience. This celebrated
statement sums up the "Copernican revolution" achieved by
Kant's "transcendental" inquiry. The subject no longer "re
volves" around its object, seeking to discover the laws by
which it is governed or the language by which it may be de
ciphered. Now the subject itself is at the center, imposing its
laws, and the world perceived speaks the language of that sub
ject. No wonder, then, that Newtonian science is able to de
scribe the world from an external, almost divine point of view!
That all perceived phenomena are governed by the laws of
our mind does not mean that a concrete knowledge of these
objects is useless. According to Kant, science does not engage
in a dialogue with nature but imposes its own language upon it.
Still it must discover, in each case, the specific message ex
pressed in this general language. A knowledge of the a priori
concepts alone is vain and empty.
From the Kantian point of view Laplace's demon, the sym
bol of the scientific myth, is an illusion, but it is a rational
illusion. Although it is the result of a limiting process and, as
such, illegitimate, it is still the expression of a legitimate con
viction that is the driving force of science-the conviction
that, in its entirety, nature is rightfully subjected to the laws
that scientists succeed in deciphering. Wherever it goes, what
ever it questions, science will always obtain, if not the same
answer, at least the same kind of answer. There exists a single
universal syntax that includes all possible answers.
Transcendental philosophy thus ratified the physicist's
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88
claim to have found the definitive form of all positive knowl
edge. At the same time,however,it secured for philosophy a
dominant position in. respect to science. It was no longer nec
essary to look for the philosophic significance of the results of
scientific activity. From the transcendental standpoint,those
results cannot lead to anything really new. It is science,not its
results, that is the subject of philosophy; science taken as a
repetitive and closed enterprise provides a stable foundation
for transcendental reflection.
Therefore, while it ratifies all the claims of science,Kant's
critical philosophy actually limits scientific activity to prob
lems that can be considered both easy and futile. It condemns
science to the tedious task of deciphering the monotonous lan
guage of phenomena while keeping for itself questions of hu
man "destiny ": what man may know, what he must do , what
he may hope for. The world studied by science,the world acces
sible to positive knowledge is "only" the world of phenomena.
Not only is the scientist unable to know things in themselves,
but even the questions he asks are irrelevant to the real prob
lems of mankind. Beauty, freedom,and ethics cannot be ob
jects of positive knowledge. They belong to the noumenal
world,which is the domain of philosophy,and they are quite
unrelated to the phenomenal world.
We can accept Kant's starting point , his emphasis on the
active role man plays in scientific description. Much has al
ready been said about experimentation as the art of choosing
situations that are hypothetically governed by the law under
investigation and staging them to give clear,experimental an
swers. For each experiment certain principles are presup
posed and thus cannot be established by that experiment.
However, as we have seen,Kant goes much further. He denies
the diversity of possible scientific points of view,the diversity
of presupposed principles. In agreement with the myth of clas
sical science, Kant is after the unique language that science
deciphers in nature, the unique set of a priori principles on
which physics is based and that are thus to be identified with
the categories of human understanding. Thus Kant denies the
need for the scientist's active choice,the need for a selection
of a problematic situation corresponding to a particular theo
retical language in which definite questions may be asked and
experimental answers sought.
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THE TWO CULTURES
Kant's critical ratification defines scientific endeavor as
silent and systematic, closed within itself. By so doing, phi
losophy endorses and perpetuates the rift, debasing and sur
rendering the whole field of positive knowledge to science
while retaining for itself the field of freedom and ethics, con
ceived as alien to nature.
A Prlilosophy of Nature? Hegel and Bergson
The Kantian truce between science and philosophy was a frag
ile one.Post-Kantian philosophers disrupted this truce in favor of
a new philosophy of science, presupposing a new path to knowl
edge that was distinct from science and actually hostile to it.
Speculation released from the constraints of any experimental
dialogue reigned supreme, with disastrous consequences for
the dialogue between scientists and philosophers. For most
scientists, the philosophy of nature became synonymous with
arrogant, absurd speculation riding roughshod over facts, and
indeed regularly proven wrong by the facts. On the other side,
for most philosophers it has become a symbol of the dangers
involved in dealing with nature and in competing with science.
The rift among science, philosophy, and humanistic studies
was thus made greater by mutual disdain and fear.
As an example of this speculative approach to nature, let us
first consider Hegel. Hegel's philosophy has cosmic dimen
sions. In his system increasing levels of complexity are spec
ified, and nature's purpose is the eventual self-realization of its
spiritual element.Nature's history is fulfilled with the appear
ance of man-that is, with the coming of Spirit apprehending
itself.
The Hegelian philosophy of nature systematically incorpo
rates all that is denied by Newtonian science. In particular, it
rests on the qualitative difference between the simple behavior
described by mechanics and the behavior of more complex en
tities such as living beings. It denies the possibility of reducing
those levels, rejecting the idea that differences are merely ap
parent and that nature is basically homogeneous and simple. It
affirms the existence of a hierarchy, each level of which pre
supposes the preceding ones.
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90
Unlike the Newtonian authors of romans de Ia matiere, of
world-embracing panoramas ranging from gravitational inter
actions to human passions, Hegel knew perfectly well that his
distinctions among levels (which, quite apart from his own in
terpretation, we may acknowledge as corresponding to the
idea of an increasing complexity in nature and to a concept of
time whose significance would be richer on each new level )
ran counter to his day's mathematical science of nature. He
therefore set out to limit the significance of this science, to
show that mathematical description is restricted to the most
trivial situations. Mechanics can be mathematized because it
attributes only space-time properties to matter. ·� brick does
not kill a man merely because it is a brick, but solely because
of its acquired velocity; this means that the man is killed tzy
space and time." IO The man is killed by what we call kinetic
energy (mv2/2)-by an abstract quantity defining mass and ve
locity as interchangeable; the same murderous effect can be
achieved by reducing one and increasing the other.
It is precisely this interchangeability that Hegel sets as a
condition for mathematization that is no longer satisfied when
the mechanical level of description is abandoned for a "higher"
one involving a larger spectrum of physical properties.
In a sense Hegel's system provides a consistent philosophic
response to the crucial problems of time and complexity.
However, for generations of scientists it represented the epit
ome of abhorrence and contempt. In a few years, the intrinsic
difficulties of Hegel's philosophy of nature were aggravated by
the obsolescence of the scientific background on which his
system was based, for Hegel, of course, based his rejection of
the Newtonian system on the scientific conceptions of his
time.11 And it was precisely those conceptions that were to fall
into oblivion with astonishing speed.It is difficult to imagine a
less opportune time than the beginning of the nineteenth cen
tury for seeking experimental and theoretical support for an
alternative to classical science. Although this time was charac
terized by a remarkable extension of the experimental scope of
science (see Chapter IV) and by a proliferation of theories that
seemed to contradict Newtonian science, most of those theo
ries had to be given up only a few years after their appearance.
At the end of the nineteenth c:entury, when Bergson under-
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THE TWO CULTURES
took his search for an acceptable alternative to the science of
his time,he turned to intuition as a form of speculative knowl
edge, but he presented it as quite different from that of the
Romantics. He explicitly stated that intuition is unable to pro
duce a system but produces only results that are always partial
and nongeneralizable,results to be formulated with great cau
tion. In contrast, generalization is an attribute of "intel
ligence," the greatest achievement of which is classical
science. Bergsonian intuition is a concentrated attention, an
increasingly difficult attempt to penetrate deeper into the sin
gularity of things. Of course,to communicate,intuition must
have recourse to language-"in order to be transmitted,it will
have to use ideas as a conveyance." 12 This it does with infinite
patience and circumspection, at the same time accumulating
images and comparisons in order to "embrace reality," 13 thus
suggesting in an increasingly precise way what cannot be com
municated by means of general terms and abstract ideas.
Science and intuitive metaphysics "are or can become
equally precise and definite. They both bear upon reality it
self. But each one of them retains only half of it so that one
could see in them,if one wished,two subdivisions of science
or two departments of metaphysics,if they did not mark diver
gent directions of the activity of thought." 14
The definition of these two divergent directions may also be
considered as the historical consequence of scientific evolu
tion. For Bergson, it is no longer a question of finding scien
tific alternatives to the physics of his time. In his v iew,
chemistry and biology had definitely chosen mechanics as
their model. The hopes that Diderot had cherished for the fu
ture of chemistry and medicine had thus been dashed. In
Bergson's view, science is a whole and must therefore be
judged as a whole. And this is what he does when he presents
science as the product of a practical intelligence whose aim is
to dominate matter and that develops by abstraction and gen
eralization the intellectual categories needed to achieve this
domination. Science is the product of our vital need to exploit
the world,and its concepts are determined by the necessity of
manipulating objects,of making predictions,and of achieving
reproducible actions. This is why rational mechanics repre
sents the very essence of science, its actual embodiment. The
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92
other sciences are more vague,awkward manifestations of an
approach that is all the more successful the more inert and
disorganized the terrain it explores.
For Bergson all the limitations of scientific rationality can
be reduced to a single and decisive one: it is incapable of un
derstanding duration since it reduces time to a sequence of
instantaneous states linked by a deterministic law.
"Time is invention, or it is nothing at all." 15 Nature is
change,the continual elaboration of the new,a totality being
created in an essentially open process of development without
any preestablished model. "Life progresses and endures in
time." 1 6 The only part of this progression that intelligence can
grasp is what it succeeds in fixing in the form of manipulable
and calculable elements and in referring to a time seen as
sheer juxtaposition of instants.
Therefore, physics "is limited to coupling simultaneities
between the events that make up this time and the posi
tions of the mobile T on its trajectory. It detaches these
events from the whole,which at every moment puts on a
new form and which communicates to them something of
its novelty. It considers them in the abstract,such as they
would be outside of the living whole, that is to say,in a
time unrolled in space. It retains only the events or sys
tems of events that can be thus isolated without being
made to undergo too profound a deformation, because
only these lend themselves to the application of its
method. Our physics dates from the day when it was
known how to isolate such systems." 1 7
When it comes to understanding duration itself,science is
powerless. What is needed is intuition,a "direct vision of the
mind by the mind." 18 "Pure change, real duration, is some
thing spiritual. Intuition is what attains the spirit, duration,
pure change.I9
Can we say Bergson has failed in the same way that the
post-Kantian philosophy of nature failed? He has failed inso
far as the metaphysics based on intuition he wished to create
has not materialized. He has not failed in that,unlike Hegel,
he had the good fortune to pass judgment upon science that
was, on the whole, firmly established-that is, classical sci-
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THE TWO CULTURES
ence at its apotheosis, and thus identified problems which are
indeed still our problems. But, like the post-Kantian critics,
he identified the science of his time with science in general.
He thus attributed to science de jure limitations that were only
de facto. As a consequence he tried to define once and for all a
statu quo for the respective domains of science and other in
tellectual activities. Thus the only perspective remaining open
for him was to introduce a way in which antagonistic ap
proaches could at best merely coexist.
In conclusion, even if the way in which Bergson sums up the
achievement of classical science is still to some extent accept
able, we can no longer accept it as a statement of the eternal
limits of the scientific enterprise. We conceive of it more as a
program that is beginning to be implemented by the meta
morphosis science is now undergoing. In particular, we know
that time linked with motion does not exhaust the meaning of
time in physics. Thus the limitations Bergson criticized are
beginning to be overcome, not by abandoning the scientific
approach or abstract thinking but by perceiving the limitations
of the concepts of classical dynamics and by discovering new
formulations valid in more general situations.
Process and Reality: Whitehead
As we have emphasized, the element common to Kant, Hegel,
and Bergson is the search for an approach to reality that is
different from the approach of classical science. This is also
the fundamental aim of Whitehead's philosophy, which is reso
lutely pre-Kantian. In his most important book, Process and
Reality, he puts us back in touch with the great philosophies of
the Classical Age and their quest for rigorous conceptual ex
perimentation.
Whitehead sought to understand human experience as a pro
cess belonging to nature, as physical existence. This challenge
led him, on the one hand, to reject the philosophic tradition
that defined subjective experience in terms of consciousness,
thought, and sense perception, and, on the other, to conceive
of all physical existence in terms of enjoyment, feeling, urge,
appetite, and yearning-that is, to cross swords with what he
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94
calls "scientific materialism," born in the seventeenth cen
tury. Like Bergson, Whitehead was thus led to point out the
basic inadequacies of the theoretical scheme developed by
seventeenth-century science:
The seventeenth century had finally produced a scheme
of scientific thought framed by mathematicians, for the
use of mathematicians. The great characteristic of the
mathematical mind is its capacity for dealing with ab
stractions; and for eliciting from them clear-cut demon
strative trains of reasoning, entirely satisfactory so long
as it is those abstractions which you want to think about.
The enormous success of the scientific abstractions,
yielding on the one hand matter with its simple location
in space and time, on the other hand mind, perceiving,
suffering, reasoning, but not interfering, has foisted on to
philosophy the task of accepting them as the most con
crete rendering of fact.
Thereby, modern philosophy has been ruined. It has
oscillated in a complex manner between three extremes.
There are the dualists, who accept matter and mind as on
equal basis, and the two varieties of monists, those who
put mind inside matter, and those who put matter inside
mind. But this juggling with abstractions can never over
come the inherent confusion introduced by the ascription
of misplaced concreteness to the scientific scheme of the
seventeenth century.20
However, Whitehead considered this to be only a temporary
situation. Science is not doomed to remain a prisoner of con
fusion.
We have already raised the question of whether it is possible
to formulate a philosophy of nature that is not directed against
science. Whitehead's cosmology is the most ambitious at
tempt to do so. Whitehead saw no basic contradiction be
tween science and philosophy. His purpose was to define the
conceptual field within which the problem of human experi
ence and physical processes could be dealt with consistently
and to determine the conditions under which the problem
could be solved. What had to be done was to formulate: the:
principles necessary to characterize all forms of existence,
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THE TWO CULTURES
from that of stones to that of man. It is precisely this univer
sality that, in Whitehead's opinion, defines his enterprise as
"philosophy." While each scientific theory selects and ab
stracts from the world's complexity a peculiar set of relations,
philosophy cannot favor any particular region of human expe
rience. Through conceptual experimentation it must construct
a consistency that can accommodate all dimensions of experi
ence, whether they belong to physics, physiology, psychology,
biology, ethics, etc.
Whitehead understood perhaps more sharply than anyone
else that the creative evolution of nature could never be con
ceived if the elements composing it were defined as permanent,
individual entities that maintained their identity throughout all
changes and interactions. But he also understood that to make
all permanence illusory, to deny being in the name of becom
ing, to reject entities in favor of a continuous and ever-changing
flux meant falling once again into the trap always lying in wait
for philosophy-to "indulge in brilliant feats of explaining
away." 2 1
Thus for Whitehead the task of philosophy was to reconcile
permanence and change, to conceive of things as processes, to
demonstrate that becoming forms entities, individual identi
ties that are born and die. It is beyond the scope of this book to
give a detailed presentation of Whitehead's system. Let us
only emphasize that he demonstrated the connection between
a philosophy of rela tion-no element of nature is a permanent
support for changing relations; each receives its identity from
its relations with others-and a philosophy of innovating be
coming. In the process of its genesis, each existent unifies the
multiplicity of the world, since it adds to this multiplicity an
extra set of relations. At the creation of each new entity "the
many become one and are increased by one. " 22
In the conclusion of this book, we shall again encounter
Whitehead's question of permanence and change, this time as
it is raised in physics; we shall speak of entities formed by
their irreversible interaction with the world. Today physics has
discovered the need to assert both the distinction and interde
pendence between units and relations. It now recognizes that,
for an interaction to be real, the "nature" of the related things
must derive from these relations, while at the same time the re
lations must derive from the "nature" of the things (see Chap-
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96
ter X). This is the forerunner of "self-consistent " descriptions
as expressed, for instance, by the "bootstrap" philosophy in
elementary-particle physics, which asserts the universal con
nectedness of all particles. However, when Whitehead wrote
Process and Reality, the situation of physics was quite dif
ferent, and Whitehead's philosophy found an echo only in bi
ology.2 3
Whitehead's case as well as Bergson's convince us that only
an opening, a widening of science can end the dichotomy be
tween science and philosophy.This widening of science is pos
sible only if we revise our conception of time. To deny time
that is, to reduce it to a mere deployment of a reversible law
is to abandon the possibility of defining a conception of nature
coherent with the hypothesis that nature produced living
beings, particularly man. It dooms us to choosing between an
antiscientific philosophy and an alienating science.
"Ignoramus, lgnoramibus": The Positivists Strain
Another method of overcoming the difficulties of classical ra
tionality implied in classical science was to separate what was
scientifically most fruitful from what is "true." This is another
form of the Kantian cleavage. In his 1865 address "On the Goal
of the Natural Sciences," Kirchoff stated that the ultimate
goal of science is to reduce every phenomenon to motion, mo
tion that in turn is described by theoretical mechanics.A simi
lar statement was made by Helmholtz, a chemist, physician,
physicist, and physiologist who dominated the German uni
versities at the time when they were becoming the hub of Eu
ropean science. He stated: "the phenomena of nature are to be
referred back to motions of material particles possessing un
changeable moving forces, which are dependent upon condi
tions of space alone." 2 4
The aim of the natural sciences, therefore, was to reduce all
observations to the laws formulated by Newton and extended
by such illustrious physicists and mathematicians as Lagrange,
Hamilton, and others. We were not to ask why these forces
exist and enter Newton's equation. In any case, we could not
"understand" matter or forces even if we used these concepts
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THE TWO CULTURES
to formulate the laws of dynamics.The why, the basic nature
of forces and masses,remains hidden from us. Du Bois Rey
mond, as we already mentioned, expressed concisely the
limitations of our knowledge: "Ignoramus, ignoramibus." Sci
ence provides no access to the mysteries of the universe.What
then is science?
We have already referred to Mach's influential view: Science
is part of the Darwinian struggle for life. It helps us to organize
our experience.It leads to an economy of thought.Mathemati
cal laws are nothing more than conventions useful for sum
marizing the results of possible experiments.At the end of the
nineteenth century,scientific positivism exercised a great in
tellectual appeal.In France it influenced the work of eminent
thinkers such as Duhem and Poincare.
One more step in the elimination of "contemptible meta
physics" and we come to the Vienna school.Here science is
granted jurisdiction over all positive knowledge and philoso
phy needed to keep this positive knowledge in .order. This
meant a radical submission of all rational knowledge and ques
tions to science. When Reichenbach, a distinguished neo
positivist philosopher, wrote a book on the "direction of
time," he stated:
There is no other way to solve the problem of time than
the way through physics. More than any other science,
physics has been concerned with the nature of time. If
time is objective the physicist must have discovered the
fact.If there is Becoming,the physicist must know it; but
if time is merely subjective and Being is timeless, the
physicist must have been able to ignore time in his con
struction of reality and describe the world without the
help of time....It is a hopeless enterprise to search for
the nature of time without studying physics.If there is a
solution to the philosophical problem of time, it is written
down in the equations of mathematical physics.25
Reichenbach's work is of great interest to anyone wishing to
see what physics has to say on the subject of time, but it is not
so much a book on the philosophy of nature as an account of
the way in which the problem of time challenges scientists, not
philosophers.
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What then is the role of philosophy? It has often been said
that philosophy should become the science of science. Phi
losophy's objective would then be to analyze the methods of
science, to axiomatize and to clarify the concepts used. Such
a role would make of the former "queen of sciences" some
thing like their housemaid. Of course, there is the possibility
that this clarification of concepts would permit further prog
ress, that philosophy understood in this way would, through
the use of other methods-logic, semantics-produce new
knowledge comparable to that of science proper. It is this hope
that sustains the "analytic philosophy " so prevalent in Anglo
American circles. We do not want to minimize the interest of
such an inquiry. However, the problems that concern us here
are quite different. We do not aim to clarify or axiomatize ex
isting knowledge but rather to close some fundamental gaps in
this knowledge.
A New Start
In the first part of this book we described, on the one hand,
dialogue with nature that classical science made possible and,
on the other, the precarious cultural position of science. Is
there a way out? In this chapter we have discussed some at
tempts to reach alternative ways of knowledge. We have also
considered the positivist view, which separates science from
reality.
The moments of greatest excitement at scientific meetings
very often occur when scientists discuss questions that are
likely to have no practical utility whatsoever, no survival
value-topics such as possible interpretations of quantum me
chanics, or the role of the expanding universe in our concept
of time. If the positivistic view, which reduces science to a
symbolic calculus, was accepted , science would lose much of
its appeal. Newton's synthesis between theoretical concepts
and active knowledge would be shattered. We would be back
to the situation familiar from the time of Greece and Rome,
with an unbridgeable gap between technical, practical knowl
edge on one side and theoretical knowledge on the other.
For the ancients. nature was a source of wisdom. Medieval
nature spoke of God. In modern times nature has become so
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THE TWO CULTURES
silent that Kant considered that science and wisdom,science
and truth, ought to be completely separated. We have been
living with this dichotomy for the past two centuries. It is time
for it to come to an end. As far as science is concerned, the
time is ripe for this to happen. From our present perspective,
the first step toward a possible reunification of knowledge was
the discovery in the nineteenth century of the theory of heat,
of the laws of thermodynamics. Thermodynamics appears as
the first form of a "science of complexity." This is the science
we now wish to describe, from its formulation to recent de
velopments.
I
I
BOOK TWO
THE SCIENCE OF
COMPLEXITY
I
I
I
I
I
I
CHAPTER IV
ENERGY AND THE
INDUSTRIAL AGE
Heat, the Rival of Gravitation
Ignis mutat res . Ageless wisdom has always linked chemistry
to the "science of fire." Fire became part of experimental sci
ence during the eighteenth century, starting a conceptual
transformation that forced science to reconsider what it had
previously . rejected in the name of a mechanistic world view,
topics such as irreversibility and complexity.
Fire transforms matter; fire leads to chemical reactions, to
processes such as melting and evaporation. Fire makes fuel
burn and release heat. Out of all this common knowledge,
nineteenth-century science concentrated on the single fact
that combustion produces heat and that heat may lead to an
increase in volume; as a result, combustion produces work.
Fire leads, therefore, to a new kind of machine, the heat en
gine, the technological innovation on which industrial society
has been founded. I
It is interesting to note that Adam Smith was working on his
Wealth of Nations and collecting data on the prospects and
determinants of industrial growth at the same university at
which James Watt was putting the finishing touches on his
steam engine. Yet the only use for coal that Adam Smith could
find was to provide heat for workers. In the eighteenth cen
tury, w ind, water, and animals, and the simple machines
driven by them, were still the only conceivable sources of
power.
The rapid spread of the British steam engine brought about
a new interest in the mechanical effect of heat, and ther
modynamics. born out of this interest, was thus not so
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104
concerned with the nature of heat as with heat's possibilities
for producing "mechanical energy."
As for the birth of the "science of complexity," we propose
to date it in 1811, the year Baron Jean-Joseph Fourier, the pre
fect oflsere, won the prize of the French Academy of Sciences
for his mathematical description of the propagation of heat in
solids.
The result stated by Fourier was surprisingly simple and ele
gant: heat flow is proportional to the gradient of temperature.
It is remarkable that this simple law applies to matter, whether
its state is solid, liquid, or gaseous. Moreover, it remains valid
whatever the chemical composition of the body is, whether it
is iron or gold. It is only the coefficient of proportionality be
tween the heat flow and the gradient of temperature that is
specific to each substance.
Obviously, the universal character of Fourier's law is not
directly related to dynamic interactions as expressed by New
ton's law, and its formulation may thus be considered the start
ing point of a new type of science. Indeed, the simplicity of
Fourier's mathematical description of heat propagation stands
in sharp contrast to the complexity of matter considered from
the molecular point of view. A solid, a gas, or a liquid are
macroscopic systems formed by an immense number of mole
cules, and yet heat conductivity is described by a single law.
Fourier formulated his result at the time when Laplace's
school dominated European science. Laplace, Lagrange, and
their disciples vainly joined forces to criticize Fourier's the
ory, but they were forced to retreat. 2 At the peak of its glory,
the Laplacian dream met with its first setback. A physical the
ory had been created that was every bit as mathematically
rigorous as the mechanical laws of motion but that remained
completely alien to the Newtonian world. From this time on,
mathematics, physics, and Newtonian science ceased to be
synonymous.
The formulation of the law of heat conduction had a lasting
influence. Curiously, in France and Britain it was the starting
point of different historical paths leading to our time.
In France, the failure of Laplace's dream led to the positivist
classification of science into the well-defined compartments
introduced by Auguste Comte. The Comtean division of sci
ence has been well analyzed by Michel Serres3-heat and
105
ENERGY AND THE INDUST RIAL AGE
gravity, two universals, coexist in physics. Worse, as Comte
was to state later, they are antagonistic. Gravitation acts on an
inert mass that submits to it without being affected by it in any
other way than by the motion it acquires or transmits. Heat
transforms matter, determines changes of state, and leads to a
modification of intrinsic properties. This was, in a sense, a
confirmation of the protest made by the anti-Newtonian chem
ists of the eighteenth century and by all those who emphasized
the difference between the purely spatiotemporal behavior at. tributed to mass and the specific activity of matter. This dis
tinction was used as a foundation for the classification of the
sciences, all placed by Comte under the common sign of
order-that is, of equilibrium. To the mechanical equilibrium
between forces the positivist classification simply adds the
concept of thermal equilibrium.
In Britain, on the other hand, the theory of heat propagation
did not mean giving up the attempt to unite the fields of knowl
edge but opened a new line of inquiry, the progressive formula
tion of a theory of irreversible processes.
Fourier's law, when applied to an isolated body with an un
homogeneous temperature distribution, describes the gradual
onset of thermal equilibrium. The effect of heat propagation is
to equalize progressively the distribution of temperature until
homogeneity is reached. Everyone knew that this was an irre
versible process; a century before, Boerhave had stressed that
heat always spread and leveled out. The science of complex
phenomena-involving interaction among a large number of
par ticles-and the occurrence of temporal asymmetry thus
were linked from the outset. But heat conduction did not be
come the starting point of an investigation into the nature of
irreversibility before it was first linked with the notion of dis
sipation as seen from an engineering point of view.4
Let us go into some detail about the structure of the new
"science of heat" as it took shape in the early nineteenth cen
tury. Like mechanics, the science of heat implied both an orig
inal conception of the phy sical object and a definition of
machines or engines-that is, an identification of cause and
effect in a specific mode of production of mechanical work.
The study of the physical processes involving heat entails
defining a system, not, as in the case of dynamics, by the posi
tion and velocity of its constituents (there are some IQ23 mole-
ORDER OUT OF CHAOS
106
cutes in a volume of gas or a solid fragment of the order of a
cm3), but by a set of macroscopic parameters such as tem
perature, pressure, volume, and so on. In addition, we have to
take into account the boundary conditions that describe the
relation of the system to its environment.
Let us consider specific heat, one of the characteristic prop
erties of a macroscopic system, as an example. The specific
heat is a measure of the amount of heat required to raise the
temperature of a system by one degree while its volume or
pressure is kept constant. To study the specific heat-for in
stance, at constant volume-the system must be brought into
interaction with its environment; it must receive a certain
amount of heat while at the same time its volume is kept con
stant and its pressure is allowed to vary.
More generally, a system may be subjected to mechanical
action (for example, either the pressure or the volume may be
fi xed by using a piston device), thermal action (a certain
amount of heat may be given to or removed from the system,
or the system itself may be brought to a given temperature
through heat exchange), or chemical action (a flux of reactants
and reaction products between the system and the environ
ment). As we have already mentioned, pressure, volume,
chemical composition, and temperature are the classical phys
icochemical parameters in terms of which the properties of
macroscopic systems are defined. Thermodynamics is the sci
ence of the correlation among the variations in these proper
ties. In comparison with dynamic objects, thermodynamic
objects therefore lead to a new point of view. The aim of the
theory is not to predict the changes in the system in terms of
the interactions among par ticles; it aims instead to predict
how the system will react to modifications we may impose on
it from the outside.
A mechanical engine gives back in the form of work the
potential energy it has received from the outside world. Both
cause and effect are of the same nature and, at least ideally,
equivalent. In contrast, the heat engine implies material
changes of states, including the transformation of the system's
mechanical properties, dilatation, and expansion. The me
chanical work produced must be seen as the result of a true
process of transformation and not only as a transmission of
movement. Thus the heat engine is not merely a passive de-
107
ENERGY AND THE INDUSTRIAL AGE
vice; strictly speaking, it produces motion. This is the origin of
a new problem: in order to restore the system's capacity to
produce motion, the system must be brought back to its initial
state. Thus a second process is needed, a second change of
state that compensates for the change producing the motion.
In a heat engine, this second process, which is opposite to the
first, involves cooling the system until it regains its initial tem
perature, pressure, and volume.
The problem of the efficiency of heat engines, of the ratio
between the work done and the heat that must be supplied to
the system to produce the two mutually compensating pro
cesses, is the very point at which the concept of irreversible
process was introduced into physics. We shall return to the
importance of Fourier's law in this context. Let us first de
scribe the essential role played by the principle of energy con
servation.
The Principle of the Conservation of Energy
We have already emphasized the central place of energy in
classical dynamics. The Hamiltonian (the sum of the kinetic
and potential energies) is expressed in terms of canonical vari
ables-coordinates and momenta-and leads to changes in
these variables while itself remaining constant throughout the
motion. Dynamic change merely modifies the respective im
portance of potential and kinetic energy, conserving their to
tality.
The early nineteenth century was characterized by unprece
dented experimental ferment.5 Physicists realized that motion
does more than bring about changes in the relative position of
bodies in space. New processes identified in the laboratories
gradually formed a network that ultimately linked all the new
fields of physics with other, more traditional branches, such as
mechanics. One of these connections was accidentally dis
covered by Galvani. Before him, only static electric charges
were known. Galvani, using a frog's body, set up the first ex
perimental electric current. Volta soon recognized that the
··galvanic" contractions in the frog were actually the effect of
an electric current passing through it. In 1800, Volta con-
ORDER OUT OF C HAOS
108
structed a chemical battery; electricity could thus be pro
duced by chemical reactions. Then came electrolysis: electric
current can modify chemical affinities and produce chemical
reactions. But this current can also produce light and heat;
and, in 1820, Oersted discovered the magnetic effects pro
duced by electrical currents. In 1822, Seebeck showed that,
inversely, heat could produce electricity and, in 1834, how
matter could be cooled by electricity. Then, in 183 1, Faraday
induced an electric current by means of magnetic effects. A
whole network of new effects was gradually uncovered. The
scientific horizon was expanding at an unprecedented rate.
In 1847 a decisive step was taken by Joule: the links among
chemistry, the science of heat, electricity, magnetism, and bi
ology were recognized as a "conversion." The idea of con
version, which postulates that "something" is quantitatively
conserved while it is qualitatively transformed, generalizes
what occurs during mechanical motion. As we have seen, total
energy is conserved while potential energy is converted into
kinetic energy, or vice versa. Joule defined a general equiv
alent for physicochemical transformations, thus making it
possible to measure the quantity conserved. This quantity was
later6 to become known as "energy." He established the first
equivalence by measuring the mechanical work required to
raise the temperature of a given quantity of water by one de
gree. A unifying element had been discovered in the middle of
a bewildering variety of new discoveries. The conservation of
energy, throughout the various transformations undergone by
physical, chemical, and biological systems, was to provide a
guiding principle in the exploration of the new processes.
No wonder that the principle of the conservation of energy
was so important to nineteenth-century physicists. For many
of them it meant the unification of the whole of nature. Joule
expressed this conviction in an English context:
Indeed the phenomena of nature, whether mechanical,
chemical, or vital, consist almost entirely in a continual
conversion of attraction through space, living force
(N.B., kinetic energy) and heat into one another. Thus it
is that order is maintained in the universe-nothing is de
ranged, nothing ever lost, but the entire machinery, com-
109
ENERGY AND THE INDUSTRIAL AGE
plicated as it is, works smoothly and harmoniously. And
though, as in the awful vision of Ezekiel, "wheel may be
in the middle of wheel," and everything may appear com
plicated and involved in the apparent confusion and intri
cacy of an almost ·endless variety of causes, effects,
conversions, and arrangements, yet is the most perfect
regularity preserved-the whole being governed by the
sovereign will of God. 7
The case of the Germans Helmholtz, Mayer, and Liebig-all
three belonging to a culture that would have rejected Joule's
conviction on the grounds of strictly positivist practice-is
even more striking. At the time of their discoveries, none of
the three was, strictly speaking, a physicist. On the other
hand, all of them were interested in the physiology of respira
tion. This had become, since Lavoisier, a model problem in
which the functioning of a living being could be described in
precise physical and chemical terms, such as the combustion
of oxygen, the release of heat, and muscular work. It was thus
a question that would attract physiologists and chemists hos
tile to Romantic speculation and eager to contribute to experi
mental science. However, judging from the account of how
these three scientists came to the conclusion that respiration,
and then the whole of nature, was governed by some funda
mental "equivalence," we may state that the German philo
sophic tradition had imbued them with a conception that was
quite alien to a positivist position: without hesitation they all
concluded that the whole of nature, in each of its details, is
ruled by this single principle of conservation.
The case of Mayer is the most remarkable.s As a young doc
tor working in the Dutch colonies in Java, he noticed the bright
red color of the venous blood of one of his patients. This led
him to conclude that, in a warm, tropical climate, the inhabi
tants need to burn less o xygen to maintain body temperature;
this results in the bright color of their blood. Mayer went on to
establish the balance between oxygen consumption, which is
the source of energy, and the energy consumption involved in
maintaining body temperature despite heat losses and manual
work. This was quite a leap, since the color of the blood could
as well be due to the patient's ··taziness. ,, But Mayer went
ORDER OUT OF CHAOS
110
further and concluded that the balance between o xygen con
sumption and heat loss was merely the particular manifesta
tion of the existence of an indestructible "force" underlying all
phenomena.
This tendency to see natural phenomena as the products of
an underlying reality that remains constant throughout its
transformations is strikingly reminiscent of Kant. Kant's influ
ence can also be recognized in another idea held by some
physiologists, the distinction between vitalism as philosophi
cal speculation and the problem of scientific methodology. For
those physiologists, even if there was a "vital" force underly
ing the function of living beings, the object of physiology
would nonetheless be purely physicochemical in nature. From
the two points of view mentioned, Kantianism, which ratified
the systematic form taken by mathematical physics during the
eighteenth century, can also be identified as one of the roots of
the renewal of physics in the nineteenth century.9
Helmholtz quite openly acknowledged Kant's influence.
For Helmholtz, the principle of the conservation of energy was
merely the embodiment in physics of the general a priori re
quirement on which all science is based-the postulate that
there is a basic invariance underlying natural transformations:
The problem of the sciences is, in the first place, to seek
the laws by which the particular processes of nature may
be referred to, and deduced from, general rules.
We are justified, and indeed impelled in this proceed
ing, by the conviction that every change in nature must
have a sufficient cause. The proximate causes to which
we refer phenomena may, in themselves, be either vari
able or invariable; in the former case the above convic
tion impels us to seek for causes to account for the
change, and thus we proceed until we at length arrive at
final causes which are unchangeable, and which there
fore must, in all cases where the exterior conditions are
the same, produce the same invariable effects. The final
aim of the theoretic natural sciences is therefore to dis
cover the ultimate and unchangeable causes of natural
phenomena. tO
With the principle of the conservation of energy, the idea of
111
ENERGY AND THE INDUSTRIAL AGE
a new golden age of physics began to take shape, an age that
would lead to the ultimate generalization of mechanics.
The cultural implications were far-reaching, and they in
cluded a conception of society and men as energy-transforming
engines. But energy conversion cannot be the whole story. It
represents the aspects of nature that are peaceful and control
lable, but below there must be another, more "active" level.
Nietzsche was one of those who detected the echo of creations
and destructions that go far beyond mere conservation or con
version. Indeed, only difference, such as a difference of tem
perature or of potential energy, can produce results that are
also differences.11 Energy conversion is merely the destruc
tion of a difference, together with the creation of another dif
ference. The power of nature is thus concealed by the use of
equivalences. However, there is another aspect of nature that
involves the boilers of steam engines, chemical transforma
tions, life and death, and that goes beyond equivalences and
conservation of energy.12 Here we reach the most original con
tribution of thermodynamics, the concept of irreversibility.
Heat Engines and the Arrow of Time
When we compare mechanical devices to thermal engines, for
example, to the red-hot boilers of locomotives, we can see at a
glance the gap between the classical age and nineteenth
century technology. Still, physicists first thought that this gap
could be ignored, that thermal engines could be described like
mechanical ones, neglecting the crucial fact that fuel used by
the steam engine disappears forever. But such complacency
soon became impossible. For classical mechanics the symbol
of nature was the clock; for the Industrial Age, it became a
reservoir of energy that is always threatened with exhaustion.
The world is burning like a furnace; energy, although being
conserved, also is being dissipated.
The original formulation of the second law of thermo
dynamics, which would lead to the first quantitative expres
sion of irreversibility, was made by Sadi Carnot in 1824, before
the general formulation of the principle of �;onservation of en
ergy by Mayer (1842) and Helmholtz (1847).Carnot analyzed
ORDER OUT OF CHAOS
112
the heat engine, closely following the work of his father, Lazare
Carnot, who had produced an influential description of me..
chanical engines.
The description of mechanical engines assumes motion as a
given. In modern language this corresponds to conservation of
energy and momentum. Motion is merely converted and
transferred to other bodies. But the analogy between mechani..
cal and thermal engines was a natural one for Sadi Carnot,
since he assumed, with most of the scientists of his time, that
heat as well as mechanical energy are conserved.
Water falling from one level to another can drive a mill. Sim..
ilarly, Sadi Carnot assumed two sources, one of which gives
heat to the engine system, and the other, at a different tern..
perature, which absorbs the heat given by the former. It is the
motion of the heat through the engine, between the two
sources at different temperatures-that is, the driving force of
fire-that will make the engine work.
Carnot repeated his father's questions. l3 Which machine
will have the highest efficiency? What are the sources of loss?
What are the processes whereby heat propagates without pro
ducing work? Lazare Carnot had concluded that in order to
obtain maximum efficiency from a mechanical machine it
must be built and made to function to reduce to a minimum
shocks, friction, or discontinuous changes of speed-in short,
all that is caused by the sudden contact of bodies moving at
different speeds. In doing so he had merely applied the phys
ics of his time: only continuous phenomena are conservative;
all abrupt changes in motion cause an irreversible loss of the
"living force. " Similarly, the ideal heat engine, instead of hav
ing to avoid all contacts between bodies moving at different
speeds, will have to avoid all contact between bodies having
different temperatures.
The cycle therefore has to be designed so that no tempera
ture change results from direct heat flow between two bodies
at different temperatures. Since such flows have no mechani
cal effect, they would merely lead to a loss of efficiency.
The ideal Carnot cycle is thus a rather tricky device that
achieves the paradoxical result of a heat transfer between two
sources at different temperatures without any contact between
bodies of different temperatures. It is divided into four phases.
During each of the two isothermal phases, the system is in
113
ENERGY AND THE INDUSTRIAL AGE
contact with one of the two heat sources and is kept at the
temperature of this source. When in contact with the hot
source, it absorbs heat and expands; when in contact with the
cold source, it loses heat and contracts. The two isothermal
phases are linked up by two phases in which the system is
isolated from the sources-that is, heat no longer enters or
leaves the system, but the temperature of the latter changes as
a result, respectively, of expansion and compression. The vol
ume continues to change until the system has passed from the
temperature of one source to that of the other.
p
'
'
'
...
..
..
..
...
',Q
.. _
...
....
....
...
....
...
.....
_
_
T
H
...
......
----
c
T
L
v
Figure 2. Pressure-volume diagram of the Carnot cycle: a thermodynamic
engine, functioning between two sources, one "hot" at temperature TH, the
other "cold" at temperature TL. Between state a and state b, there is an
isothermal change: The system, kept at temperature TH, absorbs heat and
expands. Between b and c, the system is kept expanding while in thermal
isolation; its temperature goes down from TH to TL. Those two steps produce
mechanical energy. Between c and d, there is a second isothermal change:
the system is compressed and releases heat while being kept at temperature
TL. Between d and a , the system, again isolated, is compressed while its
temperature increases to temperature TH.
ORDER OUT OF CHAOS
114
It is quite remarkable that this description of an ideal ther
mal engine does not mention the irreversible processes that
are at the basis of its realization. No mention is made of the
furnace in which the coal is burning. The model is only con
cerned with the effect of the combustion, which permits the
maintenance of the temperature difference between the two
sources.
In 1850, Clausius described the Carnot cycle from the new
perspective provided by the conservation of energy. He dis
covered that the need for two sources and the formula for the
oretical efficiency stated by Carnot express a specific problem
with heat engines: the need for a process compensating for
conversion (in the present instance, cooling by contact with a
cold source) to restore the engine to its initial mechanical and
thermal conditions. Balance relations expressing energy con
version are now joined by new equivalence relations between
the effects of two processes on the state of the system, heat
flux between the sources and conversion of heat into work. A
new science, thermodynamics, which linked mechanical and
thermal effects, came into being.
The work of Clausius explicitly demonstrated that we can
not use without restriction the seemingly inexhaustible energy
reservoir that nature provides. Not all energy-conserving pro
cesses are possible. An energy difference, for instance, cannot
be created without the destruction of an at least equivalent
energy difference. Thus in the ideal Carnot cycle, the price for
the work produced is paid by the heat, which is transferred
from one source to the other. The outcome, as expressed by
the mechanical work produced on one side, and the transfer of
heat on the other, is linked by an equivalence. This equiv
alence is valid in both directions. By working in reverse, the
same machine can restore the initial temperature difference
while consuming the work produced. No heat engine can be
constructed using a single source of heat.
Clausius was no more concerned than Carnot with the
losses whereby all real engines have an efficiency lower than
the ideal value predicted by the theory. His description, like
that of Carnot, corresponds to an idealization. It leads to the'
definition of the limit nature imposes on the yield of thermal
engines.
115
ENERGY AND THE INDUSTRIAL AGE
However, since the eighteenth century, the status of idealiza
tions had changed. Based as it was on the principle of the con
servation of energy, the new science claimed to describe not
only idealizations, but also nature itself, including "losses."
This raised a new problem, whereby irreversibility entered
physics. How does one describe what happens in a real en
gine? How does one include losses in the energy balance?
How do they reduce efficiency? These questions paved the
way to the second law of thermodynamics.
From Technology to Cosmology
As we have seen, the question raised by Carnot and Clausius
led to a description of ideal engines that was based on conser
vation and compensation. In addition, it provided an oppor
tunity for presenting new problems, such as the dissipation of
energy. William Thomson, who had great respect for Fourier's
work, was quick to grasp the importance of the problem, and
in 1852 he was the first to formulate the second law of thermo
dynamics.
It was Fourier's heat propagation that Carnot had identified
as a possible cause for the power losses in a heat engine. Car
not's cycle, no longer the ideal cycle but the "real" cycle, thus
became the point of convergence of the two universalities dis
covered in the nineteenth century-energy conversion and
heat propagation. The combination of these two discoveries
led Thomson to formulate his new principle: the existence in
nature of a universal tendency toward the degradation of me
chanical energy. Note the word "universal," which has ob
vious cosmological connotations.
The world of Laplace was eternal, an ideal perpetual-motion
machine. Since Thomson's cosmology is not merely a reflec
tion of the new ideal heat engine but also incorporates the con
sequences of the irreversible propagation of heat in a world in
which energy is conserved. This world is described as an engine
in which heat is converted into motion only at the price of some
irreversible waste and useless dissipation. Effect-producing
differences in nature progressively diminish. The world uses
ORDER OUT OF CHAOS
116
up its differences as it goes from one conversion to another
and tends toward a final state of thermal equilibrium, "heat
death." In accordance with Fourier's law, in the end there will
no longer be any differences of temperature to produce a me
chanical effect.
Thomson thus made a dizzy leap from engine technology to
cosmology. H�s formulation of the second law was couched in
the scientific terminology of his time: the conservation of en
ergy, engines, and Fourier's law. It is clear, moreover, that the
part played by the cultural context was important. It is gener
ally accepted that the problem of time took on a new impor
tance during the nineteenth century. Indeed, the essential role
of time began to be noticed in all fields-in geology, in biology,
in language, as well as in the study of human social evolution
and ethics. But it is interesting that the specific form in which
time was introduced in physics, as a tendency toward homoge
neity and death, reminds us more of ancient mythological and
religious archetypes than of the progressive complexification
and diversification described by biology and the social sci
ences. The return of these ancient themes can be seen as a
cultural repercussion of the social and economic upheavals of
the time. The rapid transformation of the technological mode
of interaction with nature, the constantly accelerating pace of
change experienced by the nineteenth century, produced a
deep anxiety. This anxiety is still with us and takes various
forms, from the repeated proposals for a "zero growth" so
ciety or for a moratorium on scientific research to the
announcement of "scientific truths" concerning our disin
tegrating universe. Present knowledge in astrophysics is still
scanty and very problematic, since in this field gravitational
effects play an essential role and problems imply the simul
taneous use of thermodynamics and relativity. Yet most texts
in this field are unanimous in predicting final doom. The con
clusion of a recent book reads:
The unpalatable truth appears to be that the inexorable
disintegration of the universe as we know it seems as
sured, the organization w hich sustains all ordered ac
tivity, frem men to galaxies, is slowly but inevitably
running down, and may even be overtaken by total grav
itational collapse into oblivion.t4
117
ENERGY A N D THE I NDUSTRIAL AGE
Others are more optimistic. In an excellent article on the
energy of the universe, Freeman Dyson has written:
It is conceivable however that life may have a larger role
to play than we have yet imagined. Life may succeed
against all of the odds in molding the universe to its own
purpose. And the design of the inanimate universe may
not be as detached from the potentialities of life and intel
ligence as scientists of the twentieth century have tended
to suppose. 15
In spite of the important progress made by Hawking and oth
ers, our knowledge of large-scale transformations in our uni
verse remains inadequate.
The Birth of Entropy
In 1865, it was Clausius' turn to make the leap from technol
ogy to cosmology. At the outset he merely reformulated his
earlier conclusions, but in doing so he introduced a new con
cept, entropy. His first goal was to distinguish clearly between
the concepts of conservation and of reversibility. Unlike mechan
ical transformations, where reversibility and conservation
coincide, a physicochemical transformation may conserve en
ergy even though it cannot be reversed. This is true, for in
stance, in the case of friction, in which motion is converted
into heat, or in the case of heat conduction as it was described
by Fourier.
We are already familiar with energy, which is a function of
the state of a system-that is, a function dependent only on
the value of the parameters (pressure, volume, temperature)
by which that state may be defined.t6 But we must go beyond
the principle of energy conservation and find a way to express
the distinction between "useful" exchanges of energy in the
Carnot cycle and "dissipated" energy that is irreversibly
wasted.
This is precisely the role of Clausius' new function, entropy,
generally denoted by S.
Apparently Clausius merely wished to express in a new form
ORDER OUT OF CHAOS
118
the obvious requirement that an engine return to its initial
state at the end of its cycle. The first definition of entropy is
centered on conservation: at the end of each cycle, whether
ideal or not, the function of the system's state, entropy, returns
to its initial value. But the parallel between entropy and en
ergy ends as soon as we abandon idealizations.t7
Let us consider the variation of the entropy dS over a short
time interval dt. The situation is quite different for ideal and
real engines. In the first case, dS may be expressed completely
in terms of the exchanges between the engine and its environ
ment. We can set up experiments in which heat is given up by
the system instead of flowing into the system. The corre
sponding change in entropy would simply have its sign
changed. This kind of contribution to entropy, which we shall
call deS, is therefore reversible in the sense that it can have
either a positive or a negative sign. The situation is drastically
different in a real engine. Here, in addition to reversible ex
changes, we have irreversible processes inside the system,
such as heat losses, friction, and so on. These produce an en
tropy increase or "entropy production" inside the system.
This increase of entropy, which we shall call diS, cannot
change its sign through a reversal of the heat exchange with
the outside world. Like all irreversible processes (such as heat
conduction), entropy production always proceeds in the same
direction. In other words, diS can only be positive or vanish in
the absence of irreversible processes. Note that the positive
sign of diS is chosen merely by convention; it could just as
well have been negative. The point is that the variation is mo
notonous, that entropy production cannot change its sign as
time goes on.
The notations deS and diS have been chosen to remind the
reader that the first term refers to exchanges (e) with the out
side world, while the second refers to the irreversible pro
cesses inside (i) the system. The entropy variation dS is
therefore the sum of the two terms deS and diS, which have
quite different physical meanings. 18
To grasp the peculiar feature of this decomposition of en
tropy variation into two parts, it is useful to apply our formula
tion to energy. Let us denote energy by E and variation over a
short time dt by dE. Of course, we would still write that dE is
equal to the sum of a term deE due to the exchanges of energy
119
ENERGY AND THE INDUSTRIAL AGE
and a term diE linked to the "internal production" of energy.
However, the principle of the conservation of energy states
that energy is never "produced" but only transferred from one
place to another. The variation in energy dE is then reduced to
deE.On the other hand, if we take a nonconserved quantity,
such as the quantity of hydrogen molecules contained in a ves
sel, this quantity may vary both as the result of adding hydro
gen to the vessel or through chemical reactions occurring
inside the vessel. But in this case, the sign of the "production"
is not determined. Depending on the circumstances, we can
produce or destroy hydrogen molecules by transferring hydro
gen atoms to other chemical components. The peculiar feature
of the second law is the fact that the production term diS is
always positive. The production of entropy expresses the oc
currence of irreversible changes inside the system.
Clausius was able to express quantitatively the entropy flow
deS in terms of the heat received (or given up) by the system.
In a world dominated by the concepts of reversibility and con
servation, this was his main concern. Regarding the irrevers
ible processes involved in entropy production, he merely stated
the existence of the inequality diS/dt>O. Even so, important
progress had been made, for, if we leave the Carnot cycle and
consider other thermodynamic systems, the distinction be
tween entropy flow and entropy production can still be made.
For an isolated system .that has no exchanges with its environ
ment, the entropy flow is, by definition, zero. Only the pro
duction term remains, and the system's entropy can only
increase or remain constant. Here, then, it is no longer a ques
tion of irreversible transformations considered as approximations
of reversible transformations; increasing entropy corresponds
to the spontaneous evolution of the system. Entropy thus be
comes an "indicator of evolution," or an "arrow of time," as
Eddington aptly called it. For all isolated systems, the future
is the direction of increasing entropy.
What system would be better "isolated" than the universe
as a whole? This concept is the basis of the cosmological for
mulation of the two laws of thermodynamics given by Clausius
in 1865:
Die Energie der Welt ist konstant.
Die Entropie der Welt strebt einem Maximum zu.19
The statement that the entropy of an isolated system in-
ORDER OUT OF CHAOS
120
creases to a maximum goes far beyond the technological prob
lem that gave rise to thermodynamics. Increasing entropy is
no longer synonymous with loss but now refers to the natural
processes within the system. These are the processes that ul
timately lead the system to thermodynamic "equilibrium"
corresponding to the state of maximum entropy.
In Chapter I we emphasized the element of surprise in
volved in the discovery of Newton's universal laws of dynam
ics. Here also the element of surprise is apparent. When Sadi
Carnot formulated the laws of ideal thermal engines, he was
far from imagining that his work would lead to a conceptual
revolution in physics.
Reversible transformations belong to classical science in the
sense that they define the possibility of acting on a system, of
controlling it. The dynamic object could be controlled through
its initial conditions. Similarly, when defined in terms of its
reversible transformations, the thermodynamic object may be
controlled through its boundary conditions: any system in
thermodynamic equilibrium whose temperature, volume, or
pressure are gradually changed passes through a series of
equilibrium states, and any reversal of the manipulation leads
to a return to its initial state. The reversible nature of such
change and controlling the object through its boundary condi
tions are interdependent processes. In this context irrevers
ibility is "negative"; it appears in the form of "uncontrolled"
changes that occur as soon as the system eludes control. But
inversely, irreversible processes may be considered as the last
remnants of the spontaneous and intrinsic activity displayed
by nature when experimental devices are employed to harness
it.
Thus the "negative" property of dissipation shows that, un
like dynamic objects, thermodynamic objects can only be par
tially controlled. Occasionally they "break loose" into
spontaneous change.
All changes are not equivalent for a thermodynamic sys
tem. This is the meaning of the expression dS =deS+ diS.
Spontaneous change toward equilibrium diS is different from
the change deS, which is determined and controlled by a mod
ification of the boundary conditions (for example, ambient
temperature). For an isolated system, equilibrium appears as
121
ENERGY AND THE INDUSTRIAL AGE
an ••attractor" of nonequilibrium states. Our initial assertion
may thus be generalized by saying that evolution toward an
attractor state differs from all other changes, especially from
changes determined by boundary conditions.
Max Planck often emphasized the difference between the
two types of change found in nature. Nature, wrote Planck,
seems to "favor" certain states. The irreversible increase in
entropy diS/dt describes a system's approach to a state which
"attracts" it, which the system prefers and from which it will
not move of its own "free will." "From this point of view, Na
ture does not permit processes whose final states she finds
less attractive than their initial states. Reversible processes are
limiting cases. In them, Nature has an equal propensity for
initial and final states; this is why the passage between them
can be made in both directions. "20
How foreign such language sounds when compared with the
language of dynamics! In dynamics, a system changes accord
ing to a trajectory that is given once and for all, whose starting
point is never forgotten (since initial conditions determine the
trajectory for all time). However, in an isolated system all non
equilibrium situations produce evolution toward the same kind
of equilibrium state. By the time equilibrium has been
reached, the system has forgotten its initial conditions-that
is, the way it had been prepared.
Thus specific heat or the compressibility of a system in
equilibrium are properties independent of the way the system
has been set up. This fortunate circumstance greatly simplifies
the study of the physical states of matter. Indeed, complex
systems consist of an immense number of particles.* From the
dynamic standpoint it is practically impossible to reproduce
any state of such systems in view of the infinite variety of dy
namic states that may occur.
We are now confronted with two ba�ically different descrip
tions: dynamics, which applies to the world of motion, and
*Physical chemistry often employs Avogadro's number-that is, the num
ber of molecules in a "mole" of matter (a mole always contains the same
number of particles, the number of atoms contained in one gram of hydro
gen). This number is of the order of 6.1023, and it is the characteristic order
of magnitude of the number of particles forming systems governed by the
laws of classical thermodynamics.
ORDER OUT OF CHAOS
122
thermodynamics, the science of complex systems with its in
trinsic direction of evolution toward increasing entropy. This
dichotomy immediately raises the question of how these de
scriptions are related, a problem that has been debated since
the laws of thermodynamics were formulated.
Boltzmanns Order Principle
The second law of thermodynamics contains two fundamental
elements: ( 1) a "negative " one that expresses the impossibility
of certain processes (heat flows from the hot source to the cold
and not vice versa) and (2) a "positive," constructive one. The
second is a consequence of the first; it is the impossibility of
certain processes that permits us to introduce a function, en
tropy, which increases uniformly for isolated systems. En
tropy behaves as an attractor for isolated systems.
How could the formulations of thermodynamics be recon
ciled with dynamics? At the end of the nineteenth century,
most scientists seemed to think this was impossible. The prin
ciples of thermodynamics were new laws forming the basis of a
new science that could not be reduced to traditional physics.
Both the qualitative diversity of energy and its tendency to
ward dissipation had to be accepted as new axioms. This was
the argument of the "energeticists" as opposed to the "atom
ists," who refused to abandon what they considered to be the
essential mission of physics-to reduce the complexity of nat
ural phenomena to the simplicity of elementary behavior ex
pressed by the laws of motion.
Actually, the problems of the transition from the micro
scopic to the macroscopic level were to prove exceptionally
fruitful for the development of physics as a whole. Boltzmann
was the first to take up the challenge. He felt that new con
cepts had to be developed to extend the physics of trajectories
to cover the situation described by thermodynamics. Follow
ing in Maxwell's footsteps, Boltzmann sought this conceptual
innovation in the theory of probability.
That probability could play a role in the description of com
plex phenomena was not surprising: Maxwell himself appears
123
ENERGY AND THE INDUSTRIAL AGE
to have been influenced by the work of Quetelet, the inventor
of the "average" man in sociology. The innovation was to in
troduce probability in physics not as a means of approxima
tion but rather as an explanatory principle, to use it to show
that a system could display a new type of behavior by virtue of
its being composed of a large population to which the laws of
probability could be applied.
Let us consider a simple example of the application of the
concept of probability in physics. An ensemble composed of
N particles is contained in a box divided into two equal com
par tments. The problem is to find the probability of the
various possible distributions of particles between the com
partments-that is, the probability of finding N1 particles in
the first compartment (and N2 = N-N1 in the second).
Using combinatorial analysis, it is easy to calculate the
number of ways in which each different distribution of N parti
cles can be achieved. Thus if N =8, there is only one way of
placing the eight particles in a single half. There are, however,
eight dif ferent ways of putting one particle in one half and
seven in the other half, if we suppose the particles to be dis
tinguishable, as is assumed in classical physics. Furthermore,
equal distribution of the eight particles between the two halves
can be carried out in 8!14!4! = 70 different ways (where
n! = 1·2·3 ...(n-l)·n). Likewise, whatever the value of N, a
number P of situations called complexions in physics may be
defined, giving the number of ways of achieving any given dis
tribution N1,N2• Its expression is P =N!IN1!N2!.
For any given population, the larger the number of complex
ions the smaller the difference between N1 and N2• It is max
imum when the population is equally distributed over the two
halves. Moreover, the larger the value of N, the greater the
difference between the number of complexions corresponding
to the different ways of distribution. For values of N of the
2
order of 1Q 3 values found in macroscopic systems, the over
whelming majority of possible distributions corresponds to
the distribution N1 =N2 =NI2. For systems composed of a
large number of particles, all states that differ from the state
corresponding to an equal distribution are thus highly im
probable.
Boltzmann was the .first to realize that irreversible increase
ORDER OUT OF CHAOS
124
in entropy could be considered as the expression of a growing
molecular disorder, of the gradual forgetting of any initial dis
symmetry, since dissymmetry decreases the number of com
plexions when compared to the state cor responding to the
maximum of P. Boltzmann thus aimed to identify entropy S
with the number of complexions: entropy characterizes each
macroscopic state in terms of the number of ways of achieving
this state. Boltzmann's famous equation S = k lg pt expresses
this idea in quantitative form. The proportionality factor k in
this formula is a universal constant, known as Boltzmann's
constant.
Boltzmann's results signify that irreversible thermodynamic
change is a change toward states of increasing probability and
that the attractor state is a macroscopic state corresponding to
maximum probability. This takes us far beyond Newton. For
the first time a physical concept has been explained in terms of
probability. Its utility is immediately apparent. Probability can
adequately explain a system's forgetting of all initial dissym
metry, of all special distributions (for example, the set of par
ticles concentrated in a subregion of the system, or the
distribution of velocities that is created when two gases of dif
ferent temperatures are mixed). This forgetting is possible be
cause, whatever the evolution peculiar to the system, it will
ultimately lead to one of the microscopic states corresponding
to the macroscopic state of disorder and maximum symmetry,
since these macroscopic states correspond to the overwhelm
ing majority of possible microscopic states. Once this state
has been reached, the system will move only short distances
from the state, and for short periods of time. In other words,
the system will merely fluctuate around the attractor state.
Boltzmann's order principle implies that the most probable
state available to a system is the one in which the multitude of
events taking place simultaneously in the system compensates
for one another statistically. In the case of our first example,
whatever the initial distribution, the system's evolution will
ultimately lead it to the equal distribution N1 = N2• This state
will put an end to the system's irreversible macroscopic evolutThe logarithmic expression indicates that entropy is an additive quantity
(S 1 +2 S 1 + S2), while the number of complexions is multiplicative
(PI +2 =PI·P2).
=
125
ENERGY AND THE INDUSTRIAL AGE
tion. Of course, the particles will go on moving from one half
to the other, but on the average, at any given instant, as many
willbe going in one direction as in the other. As a result, their
motion will cause only small, short-lived fluctuations around
the equilibrium state N1 =N2• Boltzmann's probabilistic inter
pretation thus makes it possible to understand the specificity
of the attractor studied by equilibrium thermodynamics.
This is not the whole story, and we shall devote the third
part of this book to a more detailed discussion. A few remarks
suffice here. In classical mechanics (and, as we shall see, in
quantum mechanics as well), everything is determined in
terms of initial states and the laws of motion. How then does
probability enter the description of nature? Here it is common
to invoke our ignorance of the exact dynamic state of the sys
tem. This is the subjectivistic interpretation of entropy. Such
an interpretation was acceptable when irreversible processes
were considered to be mere nuisances corresponding to fric
tion or, more generally, to losses in the functioning of thermal
engines. But today the situation has changed. As we shall see,
irreversible processes have an immense constructive impor
tance: life would not be possible without them. The subjec
tivistic interpretation is therefore highly questionable. Are we
ourselves merely the result of our ignorance, of the fact that
we only observe macroscopic states.
Moreover, both in thermodynamics as well as in its proba
bilistic interpretation, there appears a dissymmetry in time:
entropy increases in the direction of the future, not of the past.
This seems impossible when we consider dynamic equations that
are invariant in respect to time inversion. As we shall see, the
second law is a selection principle compatible with dynamics
but not deducible from it. It limits the possible initial condi
tions available to a dynamic system. The second law therefore
marks a radical departure from the mechanistic world of clas
sical or quantum dynamics. Let us now return to Boltzmann's
work.
So far we have discussed isolated systems in which the num
ber of particles as well as the total energy are fixed by the
boundary conditions. However, it is possible to extend
Boltzmann's explanation to open systems that interact with
their environment. In a closed system, defined by boundary
conditions such that its temperature Tis kept constant by heat
ORDER OUT OF C HAOS
126
exchange with the environment, equilibrium is not defined in
terms of maximum entropy but in terms of the minimum of a
similar function, free energy: F = E - TS, where E is the en
ergy of the system and Tis the temperature (measured on the
so-called Kelvin scale, where the freezing point of water is
273°C and its boiling point is 373°C).
This formula signifies that equilibrium is the result of com
petition between energy and entropy. Temperature is what
determines the relative weight of the two factors. At low
temperatures, energy prevails, and we have the formation of
ordered (weak-entropy) and low-energy structures such as
crystals. Inside these structures each molecule interacts with
its neighbors, and the kinetic energy involved is small com
pared with the potential energy that results from the interac
tions of each molecule with its neighbors. We can imagine
each particle as imprisoned by its interactions with its neigh
bors. At high temperatures, however, entropy is dominant and
so is molecular disorder. The importance of relative motion
increases, and the regularity of the crystal is disrupted; as the
temperature increases, we first have the liquid state, then the
gaseous state.
The entropy S of an isolated system and the free energy F of
a system at fi xed temperature are examples of "thermody
namic potentials . " The extremes of thermodynamic potentials
such as S or F define the attractor states toward which sys
tems whose boundary conditions correspond to the definition
of these potentials tend spontaneously.
Boltzmann's principle can also be used to study the coexis
tence of structures (such a s the liquid phase and the solid
phase) or the equilibrium between a crystallized product and
the same product in solution. It is impor tant to remember,
however, that equilibrium structures are defined on the mo
lecular level. It is the interaction between molecules acting
over a range of the order of some to-s em, the same order of
magnitude as the diameter of atoms in molecules, that makes a
crystal structure stable and endows it with its macroscopic
properties. Crystal size, on the other hand, is not an intrinsic
property of structure. It depends on the quantity of matter in
the crystalline phase at equilibrium.
127
ENERGY AND THE INDUSTRIAL AGE
Carnot and Darwin
Equilibrium thermodynamics provides a satisfactory explana
tion for a vast number of physicochemical phenomena. Yet it
may be asked whether the concept of equilibrium structures
encompasses the different structures we encounter in nature.
Obviously the answer is no.
Equilibrium structures can be seen as the results of statisti
cal compensation for the activity of microscopic elements
(molecules, atoms). By definition they are inert at the global
level. For this reason they are also "immortal." Once they
have been formed, they may be isolated and maintained indefi
nitely without further interaction with their environment.
When we examine a biological cell or a city, however, the situa
tion is quite different: not only are these systems open, but
also they exist only because they are open. They feed on the
flux of matter and energy coming to them from the outside
world. We can isolate a crystal, but cities and cells die when
cut off from their environment. They form an integral part of
the world from which they draw sustenance, and they cannot
be separated from the fluxes that they incessantly transform.
However, it is not only living nature that is profoundly alien
to the models of thermodynamic equilibrium. Hydrodynamics
and chemical reactions usually involve exchanges of matter
and energy with the outside world.
It is difficult to see how Boltzmann's order principle can be
applied to such situations. The fact that a system becomes
more uniform in the course of time can be understood in terms
of complexions; in a state of uniformity, when the "dif fer
ences" created by the initial conditions have been forgotten,
the number of complexions will be maximum. But it is impos
sible to understand spontaneous convection from this point of
view. The convection current calls for coherence, for the co
operation of a vast number of molecules. It is the opposite of
disorder, a privileged state to which only a comparatively
small number of complexions may correspond. In Boltz
mann's terms, it is an "improbable" state. If convection must
be considered a "miracle,·· what then is there to say about life,
ORDER OUT OF CHAOS
128
with its highly specific features present in the simplest orga
nisms?
The question of the relevance of equilibrium models can be re
versed. In order to produce equilibrium, a system must be
"protected" from the fluxes that compose nature. It must be
"canned," so to speak, or put in a bottle, like the homunculus
in Goethe's Faust, who addresses to the alchemist who cre
ated him: "Come, press me tenderly to your breast, but not
too hard, for fear the glass might break. This is the way things
are: something natural, the whole world hardly suffices what
is, but what is artificial demands a closed space." In the world
that we are familiar with, equilibrium is a rare and precarious
state. Even evolution toward equilibrium implies a world like
ours, far enough away from the sun for the partial isolation of a
system to be conceivable (no "canning" is possible at the tem
perature of the sun), but a world in which nonequilibrium re
mains the rule, a "lukewarm" world where equilibrium and
nonequilibrium coexist.
For a long time, however, physicists thought they could de
fine the inert structure of crystals as the only physical order
that is predictable and reproducible and approach equilibrium
as the only evolution that could be deduced from the funda
mental laws of physics. Thus any attempt at extrapolation
from thermodynamic descriptions was to define as rare and
unpredictable the kind of evolution described by biology and
the social sciences. How, for example, could Darwinian evolu
tion-the statistical selection of rare events-be reconciled
with the statistical disappearance of all peculiarities, of all rare
configurations, described by Boltzmann? As Roger Caillois21
asks: "Can Carnot and Darwin both be right?"
It is interesting to note how similar in essence the Darwinian
approach is to the path explored by Boltzmann. This may be
more than a coincidence. We know that Boltzmann had im
mense admiration for Darwin. Darwin's theory begins with an
assumption of the spontaneous fluctuations of species; then
selection leads to irreversible biological evolution. Therefore,
as with Boltzmann, a randomness leads to irreversibility. Yet
the result is very different. Boltzmann's interpretation implies
the forgetting of initial conditions, the "destruction" of initial
structures, while Darwinian evolution is associated with self
organization, ever-increasing complexity.
129
ENERGY AND THE INDUSTRIAL AGE
To sum up our argument so far, equilibrium thermodynam
ics was the first response of physics to the problem of nature's
complexity. This response was expressed in terms of the dissipa
tion of energy, the forgetting of initial conditions, and evolu
tion toward disorder. Classical dynamics, the science of eternal,
reversible trajectories, was alien to the problems facing the
nineteenth century, which was dominated by the concept of
evolution. Equilibrium thermodynamics was in a position to
oppose its view of time to that of other sciences: for thermody
namics, time implies degradation and death. As we have seen,
Diderot had already asked the question: Where do we, orga
nized beings endowed with sensations, fit in an inert world
subject to dynamics? There is another question, which has
plagued us for more than a century: What significance does
the evolution of a living being have in the world described by
thermodynamics, a world of ever-increasing disorder? What is
the relationship between thermodynamic time, a time headed
toward equilibrium, and the time in which evolution toward
increasing complexity is occurring?
Was Bergson right? Is time the very medium of innovation,
or is it nothing at all?
CHAPTERV
THE THREE STAGES
OF THERMODYNAMICS
Flux and Force
Let us return I to the description of the second law given in the
previous chapter. The concept of entropy plays a central role
in the description of evolution. As we have seen, its variation
can be written as the sum of two terms-the term deS, linked
to the exchanges between the system and the rest of the world,
and a production term, diS, resulting from irreversible phe
nomena inside the system. This term is always positive except
at thermodynamic equilibrium, when it becomes zero. For iso
lated systems (deS= 0), the equilibrium state corresponds to a
state of maximum entropy.
In order to appreciate the significance of the second law for
physics, we need a more detailed description of the various
irreversible phenomena involved in the entropy production diS
or in the entropy production per unit time P= diS/dt.
For us chemical reactions are of particular significance. To
gether with heat conduction, they form the prototype of irre
versible processes. In addition to their intrinsic importance,
chemical processes play a fundamental role in biology. The
living cell presents an incessant metabolic activity. There
thousands of chemical reactions take place simultaneously to
transform the matter the cell feeds on, to synthesize the fun
damental biomolecules, and to eliminate waste products. As
regards both the different reaction rates and the reaction sites
within the cell, this chemical activity is highly coordinated.
The biological structure thus combines order and activity. In
contrast, an equilibrium state remains inert even though it may
be structured, as, for example, with a crystal. Can chemical
131
ORDER OUT OF CHAOS
132
processes provide us with the key to the difference between
the behavior of a cry stal and that of a cell?
We will have to consider chemical reactions from a dual
point of view, both kinetic and thermodynamic.
From the kinetic point of view, the fundamental quantity is the
reaction rate. The classical theory of chemical kinetics is
based on the assumption that the rate of a chemical reaction is
proportional to the concentrations of the products taking part
in it. Indeed, it is through collisions between molecules that a
reaction takes place, and it is quite natural to assume that the
number of collisions is proportional to the product of the con
centrations of the reacting molecules.
For the sake of example, let us take a simple reaction such
as A + X�B + Y. This "reaction equation" means that when
ever a molecule of component A encounters a molecule of X,
there is a certain probability that a reaction will take place and
a molecule of B and a molecule of Ywill be produced. A colli
sion producing such a change in the molecules involved is a
"reactive collision." Only a usually very small fraction (for
example, 111 (6) of all collisions are of this kind. In most cases,
the molecules retain their original nature and merely exchange
energy.
Chemical kinetics deals with changes in the concentration
of the different products involved.in a reaction. This kinetics is
described by differential equations, just as motion is described
by the Newtonian equations. However, in this case, we are not
calculating acceleration but the rates of change of con
centration, and these rates are expressed as a function of the
concentrations of the reactants. The rate of change of con
centration of X, dXldt, is thus proportional to the product of
the concentrations of A and X in the solution-that is,
dXldt= -kA'X, where k is a proportionality factor that is
linked to quantities such as temperature and pressure and that
provides a measure for the fraction of reactive collisions tak
ing place and leading to the reaction A + X� Y+ B. Since, in
the example taken, whenever a molecule of X disappears, a
molecule of A disappears too, and a molecule of Yand one of
B are formed, the rates of change of concentration are related:
dXldt=dAldt= -dYldt= -dBldt.
But if the collision between a molecule of X and a molecule
133
THE THREE STAGES OF THERMODYNAMICS
of A can set off a chemical reaction, the collision between mol
ecules of Y and B can set off the opposite reaction. A second
reaction Y +B-.X +A thus occurs within the system de
scribed, bringing about a supplementary variation in the con
centration of X, dX/dt = k' YB. The total variation in
concentration of a chemical compound is given by the balance
between the forward and the reverse reaction. In our example,
dX/dt (= -dY/dt= . . . )= -kAX+k'YB.
If left to itself, a system in which chemical reactions occur
tends toward a state of chemical equilibrium. Chemical equi
librium is therefore a typical example of an "attractor" state.
Whatever its initial chemical composition, the system spon
taneously reaches this final stage, where the forward and re
verse reactions compensate one another statistically so that
there is no longer any overall variation in the concentrations
(dX/dt= O). This compensation implies that the ratio between
equilibrium concentrations is given by AXIYB= k'lk= K. This
result is known as the "law of mass action," or Guldberg and
Waage's law, and K is the equilibrium constant. The ratio be
tween concentrations determined by the law of mass action
corresponds to chemical equilibrium in the same way that uni
formity of temperature (in the case of an isolated system) cor
responds to thermal equilibrium. The corresponding entropy
production vanishes.
Before we deal with the thermodynamic description of
chemical reactions, let us briefly consider an additional aspect
of the kinetic description. The rate of chemical reactions is
affected not only by the concentrations of the reacting mole
cules and thermodynamic parameters (for example, pressure
and temperature) but also may be affected by the presence in
the system of chemical substances that modify the reaction
rate without themselves being changed in the process. Sub
stances of this kind are known as "catalysts. " Catalysts can,
for instance, modify the value of the kinetic constants k or k'
or even allow the system to follow a new "reaction path. " In
biology, this role is played by specific proteins, the "en
zymes. " These macromolecules have a spatial configuration
that allows them to modify the rate of a given reaction. Often
they are highly specific and affect only one reaction. A possi
ble mechanism for the catalytic effect of enzymes is to present
ORDER OUT OF CHAOS
134
different "reaction sites" to which the different molecules in
volwd in the reaction tend to attach themselves, thus increas
ing the likelihood of their coming into contact and reacting.
One very important type of catalysis, particularly in biol
ogy, is the one in which the presence of a product is required
for its own synthesis. In other words, in order to produce the
molecule X we must begin with a system already containing X.
Very frequently, for instance, the molecule X activates an en
zyme. By attaching itself to the enzyme it stabilizes that par
ticular configuration in which the reaction site is available. To
such an autocatalysis process correspond reaction schemes
such as A+ 2X�3X; in the presence of molecules X, a mole
cule A is converted into a molecule X. Therefore we need X to
produce more X. This reaction may be symbolized by the re
action "loop":
A
One important feature of systems involving such "reaction
loops" is that the kinetic equations describing the changes oc
curring in them are nonlinear differential equations.
. If we apply the same method as above, the kinetic equation
obtained for the reaction A+ 2X�3X is dX/dt = kA)(2, where
the rate of variation of the concentration of X is proportional
to the square of its concentration.
Another very important class of catalytic reactions in biol
ogy is that of crosscatalysis-for example, 2X+ Y�3X, B +X
� Y+ D, which may be represented by the loop of Figure 3.
This is a case of crosscatalysis, since X is produced from Y,
and simultaneously Y from X. Catalysis does not necessarily
increase the reaction rate; it may, on the contrary, lead to inhi
bition, which can also be represented by suitable feedback
loops.
The peculiar mathematical properties of the nonlinear dif
ferential equations describing chemical processes with cata
lytic steps are vitally important, as we shall see later, for the
thermodynamics of far-from-equilibrium chemical processes.
In addition, as we have already mentioned, molecular biology
135
THE THREE STAGES OF THERMODYNAMICS
,......
.. ��------£ a
A -.x
or
D
e.x -.v.o
2.X•V -..Jx
x-.E
Figure 3. This graph represents the reaction paths for the "Brusselator"
reactions, which are further described in the text.
has established that these loops play an essential role in meta
bolic functions. For example, the relation between nucleic
acids and proteins can be described in terms of a crosscata
lytic effect: nucleic acids contain the information to produce
proteins, which in turn produce nucleic acids.
In addition to the rates of chemical reactions, we must also
consider the rates of other irreversible processes, such as heat
transfer and the diffusion of matter. The rates of irreversible
processes are also called fluxes and are denoted by the symbol
J. There is no general theory from which we can derive the
form of the rates or fluxes. In chemical reactions the rate de
pends on the molecular mechanism, as can be verified by the
examples already indicated. The thermodynamics of irreversi
ble processes introduces a second type of quantity: in addition
to the rates, or fluxes, J, it uses "generalized forces," X, that
"cause" the fluxes. The simplest example is that of heat con
duction. Fourier's law tells us that the heat flux J is propor
tional to the temperature gradient. This temperature gradient
is the "force" causing the heat flux. By definition, flux and
forces both vanish at thermal equilibrium. As we shall see, the
production of entropy P= diS/dt can be calculated from the
flux and the forces.
Let us consider the definition of the generalized force corre
sponding to a chemical reaction. Recall the reaction A+ X
ORDER OUT OF CHAOS
136
-+ Y+ B. We have seen how, at equilibrium, the ratio between
concentrations is given by the law of mass action. As The
ophile De Donder has shown, a "chemical force" can be intro
duced, the "affinity" a that determines the direction of the
chemical reaction rate just as the temperature gradient deter
mines the direction in which heat will flow. In the case of the
reaction we are considering, the affinity is proportional to log
KB YIAX, where K is the equilibrium constant. It is immedi
ately apparent that the af finity a vanishes at equilibrium
where, following the law of mass action, we have AXIBY=K.
The affinity increases (in absolute value) when we drive the
system away from equilibrium. We can see this if we eliminate
from the system a fraction of the molecules B once they are
formed through the reaction A+ X-+ Y+ B. Affinity can be said
to measure the distance between the actual state of the system
and its equilibrium state. Moreover, as we have mentioned, its
sign determines the direction of the chemical reaction. If a is
positive, then there are "too many" molecules B and Y, and
the net reaction proceeds in the direction B+ Y-+A+ X. On the
contrary, if a is negative there are "too few" B and Y, and the
net reaction proceeds in the opposite direction.
Affinity as we have defined it is a way of rendering more
precise the ancient affinity described by the alchemists, who
deciphered the elective· relationships between chemical
bodies-that is, the "likes" and "dislikes" of molecules. The
idea that chemical activity cannot be reduced to mechanical
trajectories, to the calm domination of dynamic laws, has been
emphasized from the beginning. We could cite Diderot at
length. Later, Nietzsche, in a different context, asserted that it
was ridiculous to speak of "chemical laws," as though chemi
cal bodies were governed by laws similar to moral laws. In
chemistry, he protested, there is no constraint, and each body
does as it pleases. It is not a matter of "respect" but of a power
struggle, of the ruthless domination of the weaker by the
stronger.2 Chemical equilibrium, with vanishing affinity, corre
sponds to the resolution of this conflict. Seen from this point
of view, the specificity of thermodynamic affinity thus re
phrases an age-old problem in modern language,3 the problem
of the distinction between the legal and indifferent world of
dynamic law, and the world of spontaneous and productive
activity to which chemical reactions belong.
137
THE THREE STAGES OF THERMODYNAMICS
Let us emphasize the basic conceptual distinction between
physics and chemistry. In classical physics we can at least
conceive of reversible processes such as the motion of a fric
tionless pendulum. To neglect irreversible processes in dy
namics always corresponds to an idealization, but, at least in
some cases, it is a meaningful one. The situation in chemistry
is quite different. Here the processes that define chemistry
chemical transformations characterized by reaction rates
are irreversible. For this reason chemistry cannot be reduced
to the idealization that lies at the basis of classical or quantum
mechanics, in which past and future play equivalent roles.
As could be expected, all possible irreversible processes ap
pear in entropy production. Each of them enters through the
product of its rate or flux J multiplied by the corresponding
force X. The total entropy production per unit time, P= diS/dt,
is the sum of these contributions. Each of them appears
through the product JX.
We can divide thermodynamics into three large fields, the
study of which corresponds to three successive stages in its
development. Entropy production, the fluxes, and the forces
are all zero at equilibrium. In the close-to-equilibrium region,
where thermodynamic forces are "weak," the rates Jk are lin
ear functions of the forces. The third field is called the "non
linear" region, since in it the rates are in general more
complicated functions of the forces. Let us first emphasize
some general features of linear thermodynamics that apply to
close-to-equilibrium situations.
Linear Thermodynamics
In 1931, Lars Onsager discovered the first general relations
in nonequilibrium thermodynamics for the linear, near-to
equilibrium region. These are the famous "reciprocity rela
tions." In qualitative terms, they state that if a force-say,
"one" (corresponding, for example, to a temperature gra
dient)-may influence a flux "two" (for example, a diffusion
process), then force "two" (a concentration gradient) will also
influence the flux ..one.. (the heat flow). This has indeed been
verified. For example, in each case where a thermal gradient
ORDER OUT OF CHAOS
138
induces a process of diffusion of matter, we find that a con
centration gradient can set up a heat flux through the system.
The general nature of Onsager's relations has to be empha
sized. It is immaterial, for instance, whether the irreversible
processes take place in a gaseous, liquid, or solid medium.
The reciprocity expressions are valid independently of any mi
croscopic assumptions.
Reciprocity relations have been the first results in the ther
modynamics of irreversible processes to indicate that this was
not some ill-defined no-man's-tand but a worthwhile subject of
study whose fertility could be compared with that of equi
librium thermodynamics. Equilibrium thermodynamics was
an achievement of the nineteenth century, nonequilibrium
thermodynamics was developed in the twentieth century, and
Onsager's relations mark a crucial point in the shift of interest
away from equilibrium toward nonequilibrium.
A second general result in this field of linear, nonequilib
rium thermodynamics bears mention here. We have already
spoken of thermodynamic potentials whose extrema correspond
to the states of equilibrium toward which thermodynamic evo
lution tends irreversibly. Such are the entropy S for is<.'lated
systems, and the free energy F for closed systems at a given
temperature. The thermodynamics of close-to-equilibrium
systems also introduces such a potential function. It is quite
remarkable that this potential is the entropy-production P it
self. The theorem of minimum entropy production does, in
fact, show that in the range of validity of Onsager's relations
that is, the linear region-a system evolves toward a stationary
state characterized by the minimum entropy production com
patible with the constraints imposed upon the system. These
contraints are determined by the boundary conditions. They
may, for instance, correspond to two points in the system kept
at different temperatures, or to a flux of matter that continu
ously supports a reaction and eliminates its products.
The stationary state toward which the system evolves is
then necessarily a nonequilibrium state at which dissipative
processes with nonvanishing rates occur. But since it is a sta
tionary state, all the quantities that describe the system, such
as temperature concentrations, become time-independent.
Similarly, the entropy of the system now becomes independent
139
THE THREE STAGES OF THERMODYNAMICS
of time. Therefore its time variation dS= 0 vanishes. But we
have seen that the time variation of entropy is made up of two
terms-the entropy flow deS and the positive entropy produc
tion diS. Therefore, dS= O implies that deS= -diS<O. The
heat or matter flux coming from the environment determines a
negative flow of entropy deS, which is, however, matched by
the entropy production diS due to irreversible processes inside
the system. A negative flux deS means that the system trans
fers entropy to the outside world. Therefore at the stationary
state, the system's activity continuously increases the entropy
of its environment. This is true for all stationary states. But
the theorem of minimum entropy production says more. The
particular stationary state toward which the system tends is
the one in which this transfer of entropy to the environment is
as small as is compatible with the imposed boundary condi
tions. In this context, the equilibrium state corresponds to the
special case that occurs when the boundary conditions allow a
vanishing entropy production. In other words, the theory of
minimum entropy production expresses a kind of "inertia."
W hen the boundary conditions prevent the system from going
to equilibrium it does the next best thing; it goes to a state of
minimum entropy production-that is, to a state as close to
equilibrium as "possible."
Linear thermodynamics thus describes the stable, predict
able behavior of systems tending toward the minimum level of
activity compatible with the fluxes that feed them. The fact
that linear thermodynamics, like equilibrium thermodynam
ics, may be described in terms of a potential, the entropy pro
duction, implies that, both in evolution toward equilibrium and
in evolution toward a stationary state, initial conditions are
forgotten. W hatever the initial conditions, the system will fi
nally reach the state determined by the imposed boundary
conditions. As a result, the reaction of such a system to any
change in its boundary conditions is entirely predictable.
We see t hat in the linear range the situa tion remains
basically the same as at equilibrium. Although the entropy
production does not vanish, neither does it prevent the irrever
sible change from being identified as an evolution toward a
state that is wholly deducible from general laws. This "becom
ing.. inescapably leads to the destruction of any difference,
ORDER OUT OF CHAOS
140
any specificity. Carnot or Darwin? The paradox mentioned in
Chapter IV remains. There is still no connection between the
appearance of natural organized forms on one side, and on the
other the tendency toward "forgetting" of initial conditions,
along with the resulting disorganization.
Far "from Equilibrium
At the root of nonlinear thermodynamics lies something quite
surprising, something that first appeared to be a failure: in
spite of much effort, the generalization of the theorem of mini
mum entropy production for systems in which the fluxes are
no longer linear functions of the forces appeared impossible.
Far from equilibrium, the system may still evolve to some
steady state, but in general this state can no longer be charac
terized in terms of some suitably chosen potential (such as
entropy production for near-equilibrium states).
The absence of any potential function raises a new question:
W hat can we say about the stability of the states toward which
the system evolves? Indeed, as long as the attractor state is
defined by the minimum of a potential such as the entropy
production, its stability is guaranteed. It is true that a fluctua
tion may shift the system away from this minimum. The sec
ond law of thermodynamics, however, imposes the return
toward the attractor. The system is thus "immune" with re
spect to fluctuations. Thus whenever we define a potential, we
are describing a "stable world" in which systems follow an
evolution that leads them to a static situation that is estab
lished once and for all.
When the thermodynamic forces acting on a system become
such that the linear region is exceeded, however, the stability
of the stationary state, or its independence from fluctuations,
can no longer be taken for granted. Stability is no longer the
consequence of the general laws of physics. We must examine
the way a stationary state reacts to the different types of fluc
tuation produced by the system or its environment. In some
cases, the analysis leads to the conclusion that a state is "un
stable"-in such a state, certain fluctuations, instead of re-
141
THE THREE STAGES OF THERMODYNAMICS
gressing, may be amplified and invade the entire system,
compelling it to evolve toward a new regime that may be
qualitatively quite different from the stationary states corre
sponding to minimum entropy production.
Thermodynamics leads to an initial general conclusion con
cerning systems that are liable to escape the type of order gov
erning equilibrium. These systems have to be "far from
equilibrium." In cases where instability is possible, we have to
ascertain the threshold, the distance from equilibrium, at
which fluctuations may lead to new behavior, different from
the "normal" stable behavior characteristic of equilibrium or
near-equilibrium systems.
Why is this conclusion so interesting?
Phenomena of this kind are well known in the field of hydro
dynamics and fluid flow. For instance, it has long been known
that once a certain flow rate of flux has been reached, tur
bulence may occur in a fluid. Michel Serres has recently re
called4 that the early atomists were so concerned about
turbulent flow that it seems legitimate to consider turbulence
as a basic source of inspiration of Lucretian physics. Some
times, wrote Lucretius, at uncertain times and places, the
eternal, universal fall of the atoms is disturbed by a very slight
deviati0n-the "clinamen." The resulting vortex gives rise to
the world, to all natural things. The clinamen, this spon
taneous, unpredictable deviation, has often been criticized as
one of the main weaknesses of Lucretian physics, as being
something introduced ad hoc. In fact, the contrary is true
the clinamen attempts to explain events such as laminar flow
ceasing to be stable and spontaneously turning into turbulent
flow. Today hydrodynamic experts test the stability of fluid
flow by introducing a perturbation that expresses the effect of
molecular disorder added to the average flow. We are not so far
from the clinamen of Lucretius!
For a long time turbulence was identified with disorder or
noise. Today we know that this is not the case. Indeed, while
turbulent motion appears as irregular or chaotic on the mac
roscopic scale, it is, on the contrary, highly organized on the
microscopic scale. The multiple space and time scales in
volved in turbulence correspond to the coherent behavior of
millions and millions of molecules. Viewed in this way, the
transition from laminar flow to turbulence is a process of self-
ORDER OUT OF CHAOS
142
organization. Part of the energy of the system, which in lami
nar flow was in the thermal motion of the molecules, is being
transferred to macroscopic organized motion.
The "Benard instability" is another striking example of the
instability of a stationary state giving rise to a phenomenon of
spontaneous self-organization. The instability is due to a verti
cal temperature gradient set up in a horizontal liquid layer. The
lower surface of the latter is heated to a given temperature,
which is higher than that of the upper sur face. As a result of
these boundary conditions, a permanent heat flux is set up,
moving from the bottom to the top. When the imposed gra
dient reaches a threshold value, the fluid's state of rest-the
stationary state in which heat is conveyed by conduction
alone, without convection-becomes unstable. A convection
corresponding to the coherent motion of ensembles of mole
cules is produced, increasing the rate of heat transfer. There
fore, for given values of the constraints (the gradient of
temperature), the entropy production of the system is in
creased; this contrasts with the theorem of minimum entropy
production. The Benard instability is a spectacular phe
nomenon. The convection motion produced actually consists
of the complex spatial organization of the system. Millions of
molecules move coherently, forming hexagonal convection
cells of a characteristic size.
In Chapter IV we introduced Boltzmann's order principle,
which relates entropy to probability as expressed by the num
ber of complexions P. Can we apply this relation here? To each
distribution of the velocities of the molecules corresponds a
number of complexions. This number measures the number of
ways in which we can realize the velocity distribution by at
tributing some velocity to each molecule. The argument runs
parallel to that in Chapter IV, where we expressed the number
of complexions in terms of the distributions of molecules be
tween two boxes. Here also the number of complexions is
large when there is disorder-that is, a wide dispersion of ve
locities. In contrast, coherent motion means that many mole
cules travel with nearly the same speed (small dispersion of
velocities). To such a distribution corresponds a number of
complexions P so low that there seems almost no chance for
the phenomenon of self-organization to occur. Yet it occurs!
We see, therefore, that calculating the number of complexions,
1-43
THE THREE STAGES OF THERMODYNAMICS
which entails the hypothesis of an equal a priori probability for
each molecular state, is misleading. Its irrelevance is par
ticularly obvious as far as the genesis of the new behavior is
concerned. In the case of the Benard instability it is a fluctua
tion, a microscopic convection current, which would have
been doomed to regression by the application of Boltzmann's
order principle, but which on the contrary is amplified until it
invades the whole system. Beyond the critical value of the im
posed gradient, a new molecular order has thus been produced
spontaneously. It corresponds to a giant fluctuation stabilized
through energy exchanges with the outside world.
In far-from-equilibrium conditions, the concept of proba
bility that underlies Boltzmann's order principle is no longer
valid in that the structures we observe do not correspond to a
maximum of complexions. Neither can they be related to a
minimum of the free energy F = E- TS. The tendency toward
leveling out and forgetting initial conditions is no longer a gen
eral property. In this context, the age-old problem of the origin
cf life appears in a different perspective. It is certainly true that
life is incompatible with Boltzmann's order principle but not with
the kind of behavior that can occur in far-from-equilibrium
conditions.
Classical thermodynamics leads to the concept of "equi
librium structures" such as crystals. Benard cells are struc
tures too, but of a quite different nature. That is why we have
introduced the notion of "dissipative structures," to empha
size the close association, at first paradoxical, in such situa
tions between structure and order on the one side, and
dissipation or waste on the other. We have seen in Chapter IV
that heat transfer was considered a source of waste in classical
thermodynamics. In the Benard cell it becomes a source of
order.
The interaction of a system with the outside world, its em
bedding in nonequilibrium conditions, may become in this way
the starting point for the formation of new dynamic states of
matter-dissipative structures. Dissipative structures actually
correspond to a form of supramolecular organization. Al
though the parameters describing crystal structures may be
derived from the properties of the molecules of which they are
composed, and in particular from the range of their forces of
attraction and repulsion, Benard cells, like all dissipative
ORDER OUT OF CHAOS
144
structures, are essentially a reflection of the global situation of
nonequilibrium producing them. The parameters describing
them are macroscopic; they are not of the order of 10-8 cm,
like the distance between the molecules of a crystal, but of the
order of centimeters. Similarly, the time scales are different
they correspond not to molecular times (such as periods of
vibration of individual molecules, which may correspond to
about 10-15 sec) but to macroscopic times: seconds, minutes,
or hours.
Let us return to the case of chemical reactions. There are
some fundamental differences from the Benard problem. In
the Benard cell the instability has a simple mechanical origin.
When we heat the liquid layer from below, the lower part of the
fluid becomes less dense, and the center of gravity rises. It is
therefore not surprising that beyond a critical point the system
tilts and convection sets in.
But in chemical systems there are no mechanical features of
this type. Can we expect any self-organization? Our mental
image of chemical reactions corresponds to molecules speed
ing through space, colliding at random in a chaotic way. Such
an image leaves no place for self-organization, and this may be
one of the reasons why chemical instabilities have only re
cently become a subject of interest. There is also another dif
ference. All flows become turbulent at a "sufficiently" large
distance from equilibrium (the threshold is measured by di
mensionless numbers such as Reynolds' number). This is not
true for chemical reactions. Being far from equilibrium is a
necessary requirement but not a sufficient one. For many
chemical systems, whatever the constraints imposed and the
rate of the chemical changes produced, the stationary state
remains stable and arbitrary fluctuations are damped, as is the
case in the close-to-equilibrium range. This is true in particu
lar of systems in which we have a chain of transformations of
the type A-+B-+C-+D . . and that may be described by linear
differential equations.
The fate of the fluctuations perturbing a chemical system,
as well as the kinds of new situations to which it may evolve,
thus depend on the detailed mechanism of the chemical re
actions. In contrast with close-to-equilibrium situations, the
behavior of a far-from-equilibrium system becomes highly spe
cific. There is no longer any universally valid law from which
.
145
THE THREE STAGES OF THERMODYNAMICS
the overall behavior of the system can be deduced. Each sys
tem is a separate case; each set of chemical reactions must be
investigated and may well produce a qualitatively different be
havior.
Nevertheless, one general result has been obtained, namely
a necessary condition for chemical instability: in a chain of
chemical reactions occurring in the system, the only reaction
stages that, under certain conditions and circumstances, may
jeopardize the stability of the stationary state are precisely the
"catalytic loops"-stages in which the product of a chemical
reaction is involved in its own synthesis. This is an interesting
conclusion, since it brings us closer to some of the fundamen
tal achievements of modern molecular biology (see Figure 4).
X
Figure 4. Catalytic loops correspond to nonlinear terms. In the case of a
one-independent-variable problem, this means the occurrence of at least
one term where the independent variable appears with a power higher than
1; in this simple case, it is easy to see the relation between such nonlinear
terms and the potential instability of stationary states.
Let us take for the independent vari ab l e X the time evolution dXIdt= f(X). It
is always possible to decompose f(X) in two functions representing a gain
and a loss f+ ( X) and f (X) each of which is positive or 0, such that
f(X) f+(X)- f_(X). In this way, stationary states (dX!dt= 0) correspond to
values where f+(X)= f_(X).
Those states are graphically given by the intersections of the two graphs
plotting f+ and f If f+ and f are linear, there can only be one intersection.
In other cases, the type of the intersection permits us to infer the stability of
the stationary state.
Four cases are possible:
Sl: stable with respect to negative fluctuations, unstable with respect to
positive ones: If the system deviates slightly to the left of Sl , the positive
difference between f+ and f will reduce this deviation back to Sl; deviations
to the right will be amplified.
_
,
=
_.
_
_
SS: stable with respect to positive and negative fluctuations.
IS: stable only with respect to positive fluctuations.
II: unstable with respect to positive and negative fluctuations.
ORDER OUT 0� CHAOS
146
Beyond the Threshold of Chemical Instability
Today the study of chemical instabilities is common. Both the
oretical and experimental work are being pursued in a large
number of institutions and laboratories. Indeed, as will be
come clear, these investigations are of interest to a wide range
of scientists-not only to mathematicians, physicists, chem
ists, and biologists, but also to economists and sociologists.
In far-from-equilibrium conditions various new phenomena
appear beyond the threshold of chemical instability. To de
scribe them in a concrete fashion, it is useful to start with a
simplified theoretical model, one that has been developed at
Brussels during the past decade. American scientists have
called this model the "Brusselator," and this name is used in
scientific literature (Geographical associations seem to have
become the rule in this field; in addition to the Brusselator,
there is a n "Oregonator," and most recently a "Palo
altonator" !). Let us briefly describe the Brusselator. The steps
responsible for instability have already been noted (see Figure
3). The product X, synthetized from A and broken down into
the form of E, is linked by a relationship of crosscatalysis to
produce Y. X is produced from Y during a trimolecular step
but, conversely, Y is synthetized by a reaction between X and
a product B.
In this model, the concentrations of the products A, B, D, and
E are given parameters (the "control substances"). The behav
ior of the system is explored for increasing values of B, with A
remaining constant. The stationary state toward which such a
system is likely to evolve-the state for which dX/dt =dY/dt
0-corresponds to concentrations X0 =A and Y0 =BIA. This
can be easily verified by writing the kinetic equations and
looking for the stationary state. However, this stationary state
ceases to be stable as soon as the concentration of B exceeds a
critical threshold (everything else being kept equal). After the
critical threshold has been reached, the stationary state be
=
comes an unstable "focus" and the system leaves this focus to
reach a "limit cycle."
147
0
THE THREE STAGES OF THERMODYNAMICS
2
3
4
y
Figure 5. This scheme represents concentration of component X vs. con
centration of component Y. The cycle's focus (point S) is the stationary state,
which is unstable for 8>(1 + A2). All the trajectories (of which five are plot
ted), whatever their intitial state, lead to the same cycle.
Instead of remaining stationary, the concentrations of X and Y
begin to oscillate with a well-defined periodicity. The oscilla
tion period depends both on the kinetic constants characteriz
ing the reaction rates and the boundary conditions imposed on
the system as a whole (temperature, concentration of A., B,
etc.).
Beyond the critical threshold the system spontaneously
leaves the stationary state X0=A, Y0=BIA as the result of
fluctuations. W hatever the initial conditions, it approaches the
limit cycle, the periodic behavior of which is stable. We there
fore have a periodic chemical process-a chemical clock. Let
us pause a moment to emphasize how unexpected such a phe
nomenon is. Suppose we have two kinds of molecules, "red"
and "blue." Because of the chaotic motion of the molecules,
we would expect that at a given moment we would have more
red molecules, say, in the left part of a vessel. Then a bit later
more blue molecules would appear, and so on. The vessel
would appear to us as "violet," with occasional irregular
ORDER OUT OF CHAOS
148
flashes of red or blue. However, this is not what happens with
a chemical clock; here the system is all blue, then it abruptly
changes its color to red, then again to blue. Because all these
changes occur at regular time intervals, we have a coherent
process.
Such a degree of order stemming from the activity of biIlions
of molecules seems incredible, and indeed, if chemical clocks
had not been obser ved, no one would believe that such a pro
cess is possible. To change color all at once, molecules must
have a way to "communicate. " The system has to act as a
whole. We will return repeatedly to this key word, communi
cate, which is of obvious importance in so many fields, from
chemistry to neurophysiology. Dissipative structures intro
duce probably one of the simplest physical mechanisms for
communication.
There is an interesting difference between the simplest kind
of mechanical oscillator, the spring, and a chemical clock. The
chemical clock has a well-defined periodicity corresponding
to the limit cycle its trajectory is following. On the contrary, a
spring has a frequency that is amplitude-dependent. From this
point of view a chemical clock is more reliable as a timekeeper
than a spring.
But chemical clocks are not the only type of self-organization.
Until now diffusion has been neglected . All substances were
assumed to be evenly distributed over the reaction space. This
is an idealization; small fluctuations will always lead to differ
ences in concentrations and thus to diffusion. We therefore
have to add diffusion to the chemical reaction equations. The
diffusion-reaction equations of the Brusselator display an as
tonishing range of behaviors available to this system. Indeed,
whereas at equilibrium and near-equilibrium the system re
mains spatially homogeneous, the diffusion of the chemical
throughout the system induces, in the far-from-equilibrium re
gion, the possibility of new types of instability, including the
amplification of fluctuations breaking the initial spatial sym
metry. Oscillations in time, chemical clocks, thus cease to be
the only kind of dissipative structure available to the system.
Far from it; for example, oscillations may appear that are now
both time- and space-dependent. They correspond to chemical
waves of X and Y concentrations that periodically pass through
the system.
149
THE THREE STAGES OF THERMODYNAMICS
X hO
hO.S8
3
X
o
o
h 1.10
h 1.88
I--_�ror.-
o
X
____
1
�_...,
_ _____
o
h2.04
h
3.435
3
21....-- - - -
o
--
- --'"
1
0
Figure 6. Chemical waves simulated on computer: successive steps of
evolution of spatial profile of concentration of constituent X in the "Brussela
tor" trimolecular model. At time t= 3.435 we recover the same distribution as
at time t= O. Concentration of A and B: 2, 5.45 (B>[1 +A2]). Diffusion coeffi
cients for X and Y: 810-3,410-3•
In addition, especially when the values of the diffusion con
stants of X and Y are quite different from each other, the sys
tem may display a stationary, time-independent behavior, and
stable spatial structures may appear.
ORDER OUT OF CHAOS
150
Here we must pause once again, this time to emphasize how
much the spontaneous formation of spatial structures contra
dicts the laws of equilibrium physics and Boltzmann's order
principle. Again, the number of complexions corresponding to
such structures would be extremely small in comparison with
the number in a uniform distribution. Still , nonequilibrium
processes may lead to situations that would appear impossible
from the classical point of view.
The number of different dissipative structures compatible
with a given set of boundary conditions may be increased still
further when the problem is studied in two or three dimen
sions instead of one. In a circular, two-dimensional space, for
instance, the spatially structured stationary state may be
characterized by the occurrence of a privileged axis.
X
--·
--·
Figure 7. Stationary state with privileged axis obtained by computer sim
ulation. Concentration X is a function of geometrical coordinates p,a in the
horizontal plane. The location of the perturbation applied to the uniform un
stable solution (X , Y0) is indicated by an arrow.
Q
This corresponds to a new, extremely interesting symmetry
breaking process, especially when we recall that one of the
first stages in morphogenesis of the embryo is the formation of
a gradient in the system. We will return to these problems later
in this chapter and again in Chapter VI.
151
THE THREE STAGES OF THERMODYNAMICS
Up to now it has been assumed that the "control sub
stances" (A, B, D, and E) are uniformly distributed throughout
the reaction system. If this simplification is abandoned, addi
tional phenomena can occur. For example, the system takes
on a "natural size ," which is a function of the parameters de
scribing it. In this way the system determines its own intrinsic
size-that is, it determines the region that is spatially struc
tured or crossed by periodic concentration waves.
These results still give a very incomplete picture of the vari
ety of phenomena that may occur far from equilibrium. Let us
first mention the possibility of multiple states far from equi
librium. For given boundary conditions there may appear
more than one stationary state-'-for instance one rich in the
chemical X, the other poor. The shift from one state to another
plays an important role in control mechanisms as they have
been described in biological systems.
Since the classical work of Lyapounov and Poincare ,
characteristic points such as focus or lines such as limit cycles
Iii
y
Figure 8. (a) Bromide-ion concentration in the Belousov-Zhabotinsky reac
tion at times t1 and t1 + T (cf. R. H. Simoyi, A. Wolf, and H. L. Swinney,
Physics Review Letters, Vol. 49 (1982), p. 245; see J. Hirsch, "Condensed
Matter Physics," and on computers, Physics Today (May 1983), pp. 44-52).
(b) Attractor lines calculated by Hao Bai-lin for a Brusselator with external
periodic supply of component X (personal communication).
ORDER OUT OF CHAOS
152
were known to mathematicians as the "attractors" of stable
systems. What is new is their application to chemical systems.
It is worth noting that the first paper dealing with instabilities
in reaction-diffusion systems was published by Thring in 1952.
In recent years new types of attractors have been identified.
They appear only when the number of independent variables
increases (there are two independent variables in the Brussela
tor, the variables X and Y). In particular, we can get "strange
attractors" that do not correspond to periodic behavior.
Figure 8, which summarizes some calculations by Hao Bai
lin, gives an idea of such very complicated attractor lines cal
culated for a model generalizing the Brusselator through the
addition of an external periodic supply of X. What is remark
able is that most of the possibilities we have described have
been observed in inorganic chemistry as well as in a number of
biological situations.
In inorganic chemistry the best-known example is the
Belousov-Zhabotinsky reaction discovered in the early 1960s.
The corresponding reaction scheme, the Oregonator, intro
duced by Noyes and his colleagues, is in essence similar to the
Brusselator though more complex. The Belousov-Zhabotinsky
reaction consists of the oxidation of an organic acid (malonic
acid) by a potassium bromate in the presence of a suitable cat
alyst, cerium, manganese, or ferroin.
INFLOW
TEMPERATURE T
MALONIC �....:::
P.r U::-M::P:<---=-. �r--"-"
.
ACID =L...
::":
. I ...r
:": ----"----.
,
��IM
� P� • �
�
r 3 =-- �. :
I
--�� .
r:P::-:U-:-:"M=-P ....-. .--.
C&(50413
• .
KBO
==L..':I ...J
::":':" :-�""'I
.
P U MP
1
OUTFLOW
__
__
.
...... .�____
��
H SO "--". :...:
,...__
::. I'-.;...
2
4
PUMP CONTROL
BROMIDE
ION PROBE
1--0___•
TO
COMPUTER
Figure 9. Schematic representation of a chemical reactor used to study the
oscillations in the Belousov-Zhabotinsky reaction (there is a stirring device in
the reactor to keep the system homogeneous). The reaction has over thirty
products and intermediates. The evolution Of different reaction paths de
pends (among others factors) on the entries controlled by the pumps.
153
THE THREE STAGES OF THERMODYNAMICS
Various experimental conditions may be set up giving different
forms of autoorganization within the same system-a chemi
cal clock, a stable spatial differentiation, or the formation of
waves of chemical activity over macroscopic distances.5
Let us now turn to a matter of the greatest interest: the rele
vance of these results for the understanding of living systems.
The Encounter with Molecular Biology
Earlier in this chapter we showed that in far-from-equilibrium
conditions various types of self-organization processes may
occur. They may lead to the appearance of chemical oscilla
tions or to spatial structures . We have seen that the basic con
dition for the appearance of such phenomena is the existence
of catalytic effects.
Although the effects of "nonlinear" reactions (the presence
of the reaction product) have a feedback action on their
"cause" and are comparatively rare in the inorganic world,
molecular biology has discovered that they are virtually the
rule as far as living systems are concerned. Autocatalysis (the
presence of X accelerates its own synthesis), autoinhibition
(the presence of X blocks a catalysis needed to synthesize it),
and crosscatalysis (two products belonging to two different re
action chains activate each other's synthesis) provide the clas
sical regulation mechanism guaranteeing the coherence of the
metabolic function.
Let us emphasize an interesting difference. In the examples
known in inorganic chemistry, the molecules involved are
simple but t he reaction mechanisms are complex-in the
Belousov-Zhabotinsky reaction, about thirty compounds have
been identified. On the contrary, in the many biological exam
ples we have, the reaction scheme is simple but the molecules
(proteins, nucleic acids, etc.) are highly complex and specific.
This can hardly be an accident. Here we encounter an initial
element marking the difference between physics and biology.
Biological systems have a past. Their constitutive molecules
are the result of an evolution; they have been selected to take
part in the autocatalytic mechanisms to generate very specific
forms of organization processes.
A description of the network of metabolic activations and
ORDER OUT OF CHAOS
154
inhibitions is an essential step in understanding the functional
logic of biological systems. This includes the triggering of syn
theses the moment they are needed and the blocking of those
chemical reactions whose unused products would accumulate
in the cell.
The basic mechanism through which molecular biology ex
plains the transmission and exploitation of genetic information
is itself a feedback loop, a "nonlinear" mechanism. Deoxyri
bonucleic acid (DNA), which contains in sequential form all
the information required for the synthesis of the various basic
proteins needed in cell building and functioning, participates
in a sequence of reactions during which this information is trans
lated into the form of different protein sequences. Among the
proteins synthesized, some enzymes exert a feedback action
that activates or controls not only the different transformation
stages but also the autocatalytic mechanism of DNA replica
tion, by which genetic information is copied at the same rate
as the cells multiply.
Here we have a remarkable case of the convergence of two
sciences. The understanding attained here required comple
mentary developments in physics and biology, one toward the
complex and the other toward the elementary.
Indeed, from the point of view of physics, we now investigate
"complex" situations far removed from the ideal situations
that can be described in terms of equilibrium thermodynam
ics. On the other hand, molecular biology succeeded in relat
ing living structures to a relatively small number of basic
biomolecules. Investigating the diversity of chemical mecha
nisms, it discovered the intricacy of the metabolic reaction
chains, the subtle, complex logic of the control, inhibition, and
activation of the catalytic function of the enzymes associated
with the critical step of each of the metabolic chains. In this
way molecular biology provides the microscopic basis for the
instabilities that may occur in far-from-equilibrium conditions.
In a sense, living systems appear as a well-organized fac
tory: on the one hand, they are the site of multiple chemical
transformations ; on the other, they present a remarkable "space
time" organization with highly nonuniform distribution of bio
chemical material. We can now link function and structure.
Let us briefly consider two examples that have been studied
extensively in the past few years.
155
THE THREE STAGES OF THERMODYNAMICS
First we shall consider glycolysis, the chain of metabolic
reactions during which glucose is broken down and an energy
rich substance ATP (adenosine triphosphate) is synthetized,
providing an essential source of energy common to all living
cells . For each glucose molecule that is broken down , two
molecules of ADP (adenosine disphosphate) are transformed
into two molecules of ATP. Glycolysis provides a fine example
of how complemetary the analytical approach of biology and
the investigation of stability in far-from-equilibrium conditions
are.6
Biochemical experiments have discovered the existence of
temporal oscillations in concentrations related to the glyco
lytic cycle.7 It has been shown that these oscillations are deter
mined by a key step in the reaction sequence , a step activated
by ADP and inhibited by ATP. This is a typical nonlinear phe
nomenon well suited to regulate metabolic functioning. In
deed , each time the cell draws on its energy reserves, it is
exploiting the phosphate bonds, and ATP is converted into
ADP. ADP accumulation inside the cell thus signifies intensive
energy consumption and the need to replenish stocks. ATP
accumulation, on the other hand, means that glucose can be
broken down at a slower rate.
Theoretical investigation of this process has shown that this
mechanism is indeed liable to produce an oscillation phe
nomenon, a chemical clock. The theoretically calculated val
ues of the chemical concentrations necessary to produce
oscillation and the period of the cycle agree with the experi
mental data. Glycolytic oscillation produces a modulation of
all the cell's energy processes which are d ependent on ATP
concentration and therefore indirectly on numerous other met
abolic chains.
We may go farther and show that in the glycolytic pathway
the reactions controlled by some of the key enzymes are in far
from-equilibrium conditions. Such calculations have been re
ported by Benno Hesss and have since been extended to other
systems. Under usual conditions the glycolytic cycle corre
sponds to a chemical clock, but changing these conditions can
induce spatial pattern formations in complete agreement with
the predictions of existing theoretical models.
A living system appears very complex from the thermody
namic point of view. Certain reactions are close to e qui-
ORDER OUT OF CHAOS
156
librium, and others are not. Not everything in a living system
is "alive." The energy flow that crosses it somewhat resem
bles the flow of a river that generally moves smoothly but that
from time to time tumbles down a waterfall; which liberates
part of the energy it contains.
Let us consider another biological process that also has
been studied from the point of view of stability: the aggrega
tion of slime molds, the Acrasiales amoebas (Dictyostelium
discoideum). This process9A is an interesting case on the bor
derline between unicellular and pluricellular biology. When
The aggregation of cellular slime molds furnishes a particularly re
markable example of a self-organization phenomenon in a biological
sYltem in which a chemical clock plays an essential role. See Fi g ure A.
(Y:J
0
00
spores
fruiting
body
Q)�
I
?
�
\
growth
\
(��kx
(�t'�':; ;
\ ,
,
�
'
.. , I
'
....- -
aggregation
<::::J
migration
Figure A
When coming out of spores the amoebae grow and multiply as unicellular
organisms. This situation extends until food, principally furnished by bacte
ria, becomes scarce. Then the amoebae cease to reproduce and enter into
an interphase that lasts some eight hours. At the end of this period the
amoebae begin to aggregate around cells that behave as aggregation cen
ters. The aggregation occurs in response to chemotactic signals emitted by
the centers_ The aggregate thus formed migrates until the conditions for the
formation of a fruiting body are satisfied. Then the mass of cells differentiates
to form a stalk surmounted by a mass of spores.
157
THE THREE STAGES OF THERMODYNAMICS
In Dictyostelium discoideum, the aggregation proceeds in a periodic man
ner. Movies of aggregation process show the existence of concentric waves
of amoebae moving toward the center with a periodicity of several minutes.
The nature of the chemotactic factor is known: it is cyclic AMP (cAMP), a
substance involved in numerous biochemical processes such as hormonal
regulations. The aggregation centers release the signals of cAMP in a pe
riodic fashion. The other cells respond by moving toward the centers and by
relaying the signals to the periphery of the aggregation territory. The exis
tence of a mechanism of relay of the chemotactic signals allows each center
to control the aggregation of some 105 amoebae.
The analysis of a model of the process of aggregation reveals the exis
tence of two types of bifurcations. First the aggregation itself represents a
breaking of spatial symmetry. The second bifurcation breaks the temporal
symmetry.
Initially the amoebae are homogeneously distributed. When some of them
begin to secrete the chemotactic signals, there appear local fluctuations in
the concentration of cAMP. For a critical value of some parameter of the
system (diffusion coefficient of cAMp, motility of the amoebae, etc.), fluctua
tions are amplified: the homogeneous distribution becomes unstable and the
amoebae evolve toward an inhomogeneous. distribution in space. This new
distribution corresponds to the accumulation of amoebae around aggrega
tion centers.
To understand the origin of the periodicity in the aggregation of D. dis
coideum, it is necessary to study the mechanism of synthesis of the chem
otactic signal. On the basis of experimental observations one can describe
this mechanism by the scheme of Figure B.
_----+�:.--cAM P �
ATP
Figure B
Y
cAMP
On the surface of the cell, receptors (R) bind the molecules of cAMP. The
receptor faces the extracellular medium and is functionally linked to an en
zyme, adenylate cyclase (C), which transforms intracellular ATP into cAMP.
The cAMP thus synthesized is transported across the membrane into the
extracellular medium, where it is degraded by phosphodiesterase, an en
zyme that is secreted by the amoebae. The experiments show that binding of
extracellular cAMP to the membrane receptor activates adenylate cyclase
(positive feedback indicated by +).
On the basis of this autocatalytic regulation, the analysiS of a model for
ORDER OUT OF CH AOS
158
cAMP synthesis has permitted unification of different types of behavior ob
served during aggregation.se
Two key parameters of the model are the concentrations of adenylate
cyclase (s) and of phosphodiesterase (k). Figure C (redrawn from A. GoLD·
BETER and L. SEGEL, Differentiation, Vol. 17 [1980], pp. 127-35), shows the
behavior of the modelized system in the space formed by s and k.
cu
-
0
:>.
c
cu
"'Q
<
Figure C
A
Phosphodiesterase
,
k
T hree regions can be distinguished for different values of k and s. Region
A corresponds to a stable, nonexcitable stationary state; region B to a sta
tionary state stable but excitable: the system is capable of amplifying small
perturbations in the concentration of cAMP in a pulsatory manner (and thus
of relaying cAMP signals); region C corresponds to a regime of sustained
oscillations around an unstable stationary state.
The arrow indicates a possible "developmental path" corresponding to a
rise in phosphodiesterase (k) and adenylate cyclase (s), a rise that is ob
served to occur after the beginning of starvation. The crossing of regions A,
B and C corresponds to the observed change of behavior: cells are at first
incapable of responding to extracellular cAMP signals; thereafter they relay
these signals and, finally, they become capable of synthetizing them pe
riodically in an autonomous way. The aggregation centers would thus be the
cells for which the parameters s and k have reached the more rapidly a point
located inside region 0 after starvation has begun.
159
THE THREE STAGES OF THERMODYNAMICS
the environment in which these amoebas live and multiply be
comes poor in nutrients, they undergo a spectacular transfor
mation. (See Figure A. ) Starting as a population of isolated
cells, they join to form a mass composed of several tens of
thousands of cells. This "pseudoplasmodium" then undergoes
differentiation, all the while changing shape. A "foot" forms,
consisting of about one third of the cells and containing abun
dant cellulose. This foot supports a round mass of spores,
which will detach themselves and spread, multiplying as soon
as they come in contact with a suitable nutrient medium and
thus forming a new colony of amoebas. This is a spectacular
example of adaptation to the environment. The population
lives in one region until it has exhausted the available re
sources. It then goes through a metamorphosis by means of
which it acquires the mobility to invade other environments.
An investigation of the first stage of the aggregation process
reveals that it begins with the onset of displacement waves in
the amoeba population, with a pulsating motion of con
vergence of the a moebaes toward a "center of attraction,"
which appears to be produced spontaneously. Experimental
investigation and modelization have shown that this migration
is a response by the cells to the existence in the environment
of a concentration gradient in a key substance, cyclic AMP,
which is periodically produced by an amoeba which is the at
tractor center and later by other cells through a relay mecha
nism. Here we again see the remarkable role of chemical
clocks. They provide, as we have already stressed, new means
of communication. In the present case, the self-organization
mechanism leads to communication between cells.
There is another aspect we wish to emphasize. Slime mold
aggregation is a typical example of what may be termed "order
through fluctuations" : the setting up of the attractor center
giving off the AMP indicates that the metabolic regime corre
sponding to a normal nutritive environment has become un
stable -that is, the nutritive env ironment has become
exhausted. The fact that under such conditions of food short
age any given amoeba can be the first to emit cyclic AMP and
thus become an attractor center corresponds to the random
behavior of fluctuations. This fluctuation is then amplified and
organizes the medium.
ORDER OUT OF CHAOS
160
Bifurcations and Symmetry-Breaking
Let us take a closer look at the emergence of self-organization
and the processes that occur when we go beyond this thresh
old. At equilibrium or near-equilibrium, there is only one
steady state that will depend on the values of some control
parameters . We shall call A the control parameter, which, for
example, may be the concentration of substance B in the
Brusselator described in section 4. We now follow the change
in the state of the system as the value of B increases. In this
way the system is pushed farther and farther away from equi
librium. At some point we reach the threshold of the stability
of the "thermodynamic branch." Then we reach what is gener
ally called a "bifurcation point." (These are the points whose
role Maxwell emphasized in his thoughts on the relation be
tween determinism and free choice [see Chapter II, section
3].)
X
A
,.
,
,
I
I
\
\
\
\
--=--=-.�..·�-B\
E
t----
Figure 10. Bifurcation diagram. The diagram plots the steady-state values
of X as function of a bifurcation parameter �. Continuous lines are stable
stationary states; broken lines are unstable stationary states. The only way
to get to branch D is to start with some concentration X0 higher than the
value of X corresponding to branch E.
161
THE THREE STAGES OF THERMODYNAMICS
Let us consider some typical bifurcation diagrams. At bifur
cation point B, the thermodynamic branch becomes unstable
in respect to fluctuations. For the value Ac of the control pa
rameter A, the system may be in three different steady states:
C, E, D. 1\vo of these states are stable, one unstable. It is very
important to emphasize that the behavior of such systems de
pends on their history. Suppose we slowly increase the value
of the control parameter A; we are likely to follow the path A,
B, C in Figure 10. On the contrary, if we start with a large
value of the concentration X and maintain the value of the con
trol parameter constant, we are likely to come to point D. The
state we reach depends on the previous history of the system.
Until now history has been commonly used in the interpreta
tion of biological and social phenomena, but that it may play
an important role in simple chemical processes is quite unex
pected.
Consider the bifurcation diagram represented in Figure 11.
This differs from the previous diagram in that at the bifurca
tion point two new stable solutions emerge. Thus a new ques
tion: Where will the system go when we reach the bifurcation
point? We have here a "choice" between two possibilities;
X
Figure 11. Symmetrical bifurcation diagram. X is plotted as a function of A.
For A<Ac there is only one stationary state, which is stable. For A>Ac there
are two stable stationary states for each value of A (the formerly stable state
becomes unstable).
ORDER OUT OF CHAOS
162
they may represent either of the two nonuniform distributions
of chemical X in space, as represented in Figures 12 and 1 3.
X
�------�r
X
�------�
r
Figures 12 and 13. Two possible spatial distributions of the chemical com
po nent X, corresponding to each of the two branches i n Figure 11. Figure 12
corresponds to a "left" structure as component X has a higher concentration
in the left part; similarly, Figure 13 corresponds to a "right" structure.
The two structures are mirror images of one another. In Figure
12 the concentration of X is larger at the left; in Figure 13 it is
larger at the right. How will the system choose between left
and right? There is an irreducible random element; the mac
roscopic equation cannot predict the path the system will
take. Turning to a microscopic description will not help. There
is also no distinction between left and right. We are faced with
chance events very similar to the fall of dice.
163
THE THREE STAGES OF THERMODYNAMICS
We would expect that if we repeat the experiment many
times and lead the system beyond the bifurcation point , half of
the system will go into the left configuration, ha lf into the
right. Here another interesting question arises : In the world
around us, some basic simple symmetries seem to be bro
ken.to Everybody has obser ved that shells often have a prefer
ential chirality. Pasteur went so far as to see in dissymmetry, in
the breaking of symmetry, the very characteristic of life. We
know today that DNA, the most basic nucleic acid, takes the
form of a left-handed helix. How did this dissymmetry arise ?
One common answer is that it comes from a unique event that
has by chance favored one of the two possible outcomes; then
an autocatalytic process sets in, and the left-handed structure
produces other left-handed structures. Others imagine a
"war" between left- and right-handed structures in which one
of them has annihilated the other. These are problems for
which we have not yet found a satisfactory answer. To speak of
unique events is not satisfactory ; we need a more "system
atic" explanation.
We have recently discovered a striking example of the fun
damental new properties that matter acquires in far-from
equilibrium conditions: external fields, such as the gravitational
field, can be "perceived" by the system, creating the possibil
ity of pattern selection.
How would an external field-a gravitational field-change
an equilibrium situation ? The answer is provided by Boltz
mann's order principle: the basic quantity involved is the ratio
of potential energy/thermal energy. This is a small quantity for
the gravitational field of earth; we would have to climb a moun
tain to achieve an appreciable change in pressure or in the
composition of the atmosphere. But recall the Benard cell ;
from a mechanical perspective, its instability is the raising of
its center of gravity as the result of thermal dilatation. In other
words, gravitation plays an essential role here and leads to a
new structure in spite of the fact that the Benard cell may have
a thickness of only a few millimeters. The effect of gravitation
on such a thin layer would be negligible at equilibrium, but
because of the nonequilibrium induced by the difference in
temperature, macroscopic effects due to gravitation become
visible even in this thin layer. Nonequilibrium magnifies the
effect of gravitation.tl
ORDER OUT OF CHAOS
164
Gravitation obviously will modify the diffusion flow in a re
action diffusion equation. Detailed calculations show that this
can be quite dramatic near a bifurcation point of an unper
turbed system. In particular, we can conclude that very small
gravitational fields can lead to pattern selection.
Let us again consider a system with a bifurcation diagram
such as represented in Figure 1 1 . Suppose that for no gravita
tion, g=O, we have, as in Figures 1 2 and 13, an asymmetric
"up/down" pattern as well as its mirror image, "down/up. "
Both are equally probable, but when g is taken into account,
the bifurcation equations are modified because the diffusion
flow contains a term proportional to g. As a result, we now
obtain the bifurcation diagram represented in Figure 14. The
original bifurcation has disappeared-this is true whatever the
value of the field. One structure (a) now emerges continuously
as the bifurcation parameter grows, while the other (b) can be
attained only through a finite perturbation.
x
. .. .
..
.
.
.
.
.
�.
.
.
.• .
�
"
.
..
.
'.
,
••••••
••
.------
..
.
b)
. ..
.
. .
Figure 14. Phenomenon of assisted bifurcation in the presence of an exter
nal field. X is plotted as a function of parameter ". The symmetrical bifurca
tion that would occur in the absence of the field is indicated by the dotted
line. The bifurcation value is "0; the stable branch (b) is at finite di'stance from
branch (a).
165
THE THREE STAGES OF THERMODYNAMICS
Therefore, if we follow the path (a), we expect the system to
follow the continuous path. This expectation is correct as long
as the distance s between the two branches remains large in re
spect to thermal fluctuations in the concentration of X. There
occurs what we would like to call an "assisted" bifurcation.
As before, at about the value Ac a self-organization process
may occur. But now one of the two possible patterns is pre
ferred and will be selected.
The important point is that, depending on the chemical pro
cess responsible for the bifurcation, this mechanism expresses
an extraordinary sensitivity. Matter, as we mentioned earlier
in this chapter, perceives differences that would be insignifi
cant at equilibrium. Such possibilities lead us to think of the
simplest organisms, such as bacteria, which we know are able
to react to electric or magnetic fields . More generally they
show that far-from-equilibrium chemistry leads to possible
"adaptation" of chemical processes to outside conditions. This
contrasts strongly with equilibrium situations, in which large
perturbations or modifications of the boundary conditions are
necessary to determine a shift for one structure to another.
The sensitivity of far-from-equilibrium states to external fluc
tuations is another example of a system's spontaneous "adapta
tive organization" to its environment. Let us give an example 12
of self-organization as a function of fluctuating external condi
tions. The simplest conceivable chemical reaction is the isom
erization reaction where A�B. In our model the product A can
also enter into another reaction: A+ light-+A*-+A+ heat. A
absorbs light and gives it back as heat while leaving its excited
state A*. Consi der these two processes as taking place in a
closed system: only light and heat can be exchanged with the
outside. Nonlinearity exists in the system because the transfor
mation from B to A absorbs heat: the higher the temperature,
the faster the formation of A. But also the higher the concentra
tion of A, the higher the absorption of light by A and its trans
formation into heat, and the higher the temperature. A
catalyzes its own formation.
We expect to find that the concentration of A corresponding
to the stationary state increases with the light intensity. This is
indeed the case. But starting from a critical point, there ap
pears one of the standard far-from-equilibrium phenomena:
the coexistence of multiple stationary states. For the same val-
ORDER OUT OF CHAOS
166
ues of light intensity and temperature, the system can be
found in two different stable stationary states with different
concentrations of A. A third, unstable state marks the thresh
old between the first two. Such a coexistence of stationary
states gives birth to the well-known phenomenon of hystere
sis. But this is not the whole story. If the light intensity, in
stead of being constant, is taken as randomly fluctuating, the
situation is altered profoundly. The zone of coexistence be
tween the two stationary states increases, and for certain val
ues of the parameters coexistence among three stationary
stable states becomes possible.
In such a case, a random fluctuation in the external flux,
often termed "noise," far from being a nuisance, produces
new types of behavior, which would imply, under deterministic
fluxes, much more complex reaction schemes. It is important
to remember that random noise in the fluxes may be consid-
X
p
•
1\..
I
J
v
•
p':
b1
'
•
I
''
�
'
I
'I
�
Q
b2
b
Figure 15. This figure shows how a "hysteresis" phenomenon occurs if we
have the value of the bifurcation parameter b first growing and then diminish
ing. If the system is initially in a stationary state belonging to the lower
branch, it will stay there while b grows. But at b=b2, there will be a discon
tinuity: The system jumps from Q to Q', on the higher branch. Inversely,
starting from a state on the higher branch, the system will remain there till
b=b1, when it will jump down toP'. Such types of bistable behavior are
observed in many fields, such as lasers, chemical reactions or biological
membranes.
167
THE THREE STAGES OF THERMODYNAMICS
ered as unavoidable in any "natural system." For example, in
biological or ecological systems the parameters defining inter
action with the environment cannot generally be considered as
constants. Both the cell and the ecological niche draw their
sustenance from their env ironment ; and humidity, pH, salt
concentration, light, and nutrients form a permanently fluc
tuating environment. The sensitivity of nonequilibrium states,
not only to fluctuations produced by their internal activity but
also to those coming from their environment, suggests new
perspectives for biological inquiry.
Cascading Bifurcations
and the Transitions to Chaos
The preceding paragraph dealt only with the first bifurcation
or, as mathematicians put it, the primary bifurcation, which
occurs when we push a system beyond the threshold of sta
bility. Far from exhausting the new solutions that may appear,
this primary bifurcation introduces only a single characteristic
time (the period of the limit cycle) or a single characteristic
length. To generate the complex spatial temporal activity ob
ser ved in chemical or biological systems, we have to follow the
bifurcation diagram farther.
We have already alluded to phenomena arising from the
complex interplay of a multitude of frequences in hydrody
namical or chemical systems. Let us consider Benard struc
tures, which appear at a critical distance from equilibrium.
Farther away from thermal equilibrium the convection flow
begins to oscillate in time; as the distance from equilibrium is
increased still farther, more and more oscillation frequencies
appear, and eventually the transition to equilibrium is com
plete.13 The interplays among the frequencies produce pos
sibilities of large fluctuations; the "region" in the bifurcation
diagram defined by such values of the parameters is often
called "chaotic." In cases such as the Benard instability, order
or coherence is sandwiched between thermal chaos and non
equilibrium turbulent chaos. Indeed, if we continue to in
�rease the gradient of temperature, the �onvcction patterns
become more complex ; oscillations set in, and the ordered as-
ORDER OUT OF CHAOS '
168
TRACES OF Br- CONCENTRATION
Homogeneous Steady State
f\/l.J\NVVVV\
Sinusoidal Oscillations
Complex Periodic States
•
•
•
(Subharmonic b ifurcation)
Chaos
2:
o
0:
u..
Mix e d - Mode Oscillations
&£J
U
Z
Chaotic
�
!:!!
and
c
Periodic
Relaxation Oscillations
TIME
Figure 1 6. Temporal oscillations of the ion Br- in the Belousov
Z habotinski reaction. The figure represents a succession of regions corre
sponding to qualitative differences. This is a schematic representation. The
experimental data indicate the existence of much more complicated se
quences.
pect of the convection is largely destroyed. However, we
should not confuse "equilibrium thermal chaos" and "non
equilibrium turbulent chaos." In thermal chaos as realized in
equilibrium, all characteristic space and time scales are of mo
lecular range, while in turbulent chaos we have such an abun
dance of macroscopic time and length scales that the system
appears chaotic. In chemistry the relation between order and
chaos appears highly complex: successive regimes of ordered
(oscillatory) situations follow regimes of chaotic .behavior.
This has, for instance, been observed as a function of the flow
rate in the Belousov-Zhabotinsky reaction.
169
THE. THREE STAGES OF THERMODYNAMICS
In many cases it is difficult to disentangle the meaning of
words such as "order" and "chaos." Is a tropical forest an
ordered or a chaotic system? The history of any particular
animal species will appear very contingent, dependent on
other species and on environmental accidents. Nevertheless,
the feeling persists that, as such, the overall pattern of a tropi
cal forest ; as represented, for instance, by the diversity of spe
cies, corresponds to the very archetype of order. W hatever the
precise meaning we will eventually give to this terminology, it
is clear that in some cases the succession of bifurcations forms
an irreversible evolution where the determinism of charac
teristic frequencies produces an increasing randomness stem
ming from the multiplicity of those frequencies.
A remarkably simple road to "chaos" that has already at
tracted a lot of attention is the "Feigenbaum sequence ." It
concerns any system whose behavior is characterized by a
very general feature-that is, for a determined range of param
eter values the system's behavior is periodic, with a period T;
beyond this range, the period becomes 2T, and beyond yet an
other critical threshold, the system needs 4 Tin order to repeat
itself. The system is thus characterized by a succession of bi
furcations, with successive period doubling . This constitutes a
typical route going from simple periodic behavior to the com
plex aperiodic behavior occurring when the period has dou
bled ad infinitum. This route, as Feigenbaum discovered, is
characterized by universal numerical features independent of
the mechanism involved as long as the system possesses the
qualitative property of period doubling. "In fact, most mea
surable properties of any such system in this aperiodic limit
now can be determined in a way that essentially bypasses the
details of the equations governing each specific system.
" 14
In other cases, such as those represented in Figure 16, both
deterministic and stochastic elements characterize the history
of the system.
If we consider Figure 17 and a value of the control param
eter of the order of X6 , we see that the system already has a
wealth of possible stable and unstable behaviors. The "histori
cal" path along which the system evolves as the control parame
ter grows is characterized by a succession of stable regions,
where deterministic laws dominate, and of instable ones, near
the bifurcation points, where the system can "choose" be.
.
.
ORDER OUT OF CHAOS
170
tween or among more than one possible future. Both the deter
ministic character of the kinetic equations whereby the set of
possible states and their respective stability can be calculated,
and the random fluctuations "choosing" between or among
the states around bifurcation points are inextricably con
nected. This mixture of necessity and chance constitutes the
history of the system.
Solutions
I
,
\
' ,,
(c'I),,
'
(/ ..
, -
-
-
, ..__ : ; -- -t----<a-,----�·--------··1c)
....___
......
',
(1:))
..
..
-..�
'
' ...
Figure 17. Bifurcation diagram. Steady-state solutions are plotted against
each
value of ).; this set of states forms the branch a. For ).=A1 two other sets of
stationary states become possible (branches b and b').
The states of b' are unstable but become stable at A= A2 while the states
of branch a become unstable. For A=A3 the branch b' is unstable again, and
two other stable branches appear.
For A=A4 the unstable branch a attains a new bifurcation point where two
new branches become possible, which will be u nstable up to A= A5 and
).=A6.
bifurcation parameter A. For A <A1 there is only one stationary state for
171
THE THREE STAGES OF THERMODYNAMICS
From Euclid to Aristotle
One of the most interesting aspects of dissipative structures is
their coherence. The system behaves as a whole, as if it were
the site of long-range forces. In spite of the fact that interac
tions among molecules do not exceed a range of some I0-8
em, the system is structured as though each molecule were
"informed" about the overall state of the system.
It has often been said-and we have already repeated it
that modern science was born when Aristotelian space, for
which one source of inspiration was the organization and soli
darity of biological functions, was replaced by the homoge
neous and isotropic space of Euclid. However, the theory of
dissipative structures moves us closer to Aristotle 's concep
tion. W hether we are dealing with a chemical clock, concen
tration waves , or the inhomogeneous distribution of chemical
products , instability has the effect of breaking symmetry, both
temporal and spatial. In a limit cycle , no two instants are
equivalent; the chemical reaction acquires a phase similar to
that characterizing a light wave , for example. Again, when a
favored direction results from an instability, space ceases to be
isotropic. We move from Euclidian to Aristotelian space!
It is tempting to speculate that the breaking of space and
time symmetry plays an important part in the fascinating phe
nomena of morphogenesis. These phenomena have often led
to the conviction that some internal purpose must be involved,
a plan realized by the embryo when its growth is complete. At
the beginning of this century, German e mbryologist Hans
Driesch believed that some immaterial "entelechy" was re
sponsible for the embryo's development. He had discovered
that the embryo at an early stage was capable of withstanding
the severest perturbations and, in spite of them, of developing
into a normal, functional organism. On the other hand, when
we obser ve embryological development on film, we "see"
jumps corresponding to radical reorganizations followed by
periods of more "pacific" quantitative growth. There are, for
tunately, few mistakes. The jumps are performed in a re
producible fashion. We might speculate that the basic
mechanism of evolution is based on the play between bifurca-
ORDER OUT OF CHAOS
172
tions as mechanisms of exploration and the selection of chemi
cal interactions stabilizing a particular trajectory. Some forty
years ago, the biologist Waddington introduced such an idea.
The concept of "chreod" that he introduced to describe the
stabilized paths of development would correspond to possible
lines of development produced as a result of the double imper
ative of flexibility and security. 15 Obviously the problem is
very complex and can be dealt with only briefly here.
Many years ago embryologists introduced the concept of a
morphogenetic field and put forward the hypothesis that the
differentiation of a cell depends on its position in that field.
But how does a cell "recognize" its position? One idea that is
often debated is that of a "gradient" of a characteristic sub
stance, of one or more "morphogens." Such gradients could
actually be produced by symmetry-breaking instabilities in
far-from-equilibrium conditions. Once it has been produced, a
chemical gradient can provide each cell with a different chemi
cal environment and thus induce each of them to synthesize a
specific set of proteins . This model, which is now widely used,
seems to be in agreement with experimental evidence. In par
ticular, we may refer to Kauffman's work16 on drosophila. A
reaction-diffusion system is taken as responsible for the com
mitment to alternative development programs that appear to
occur in different groups of cells in the early embryo. Each
3
1
3
4 r-----�---r--���----�- 4
Figure 18. Schematic representation of the structure of the drosophila em
bryo as it results from successive binary choices. See text for more detail.
173
THE THREE STAGES OF THERMODYNAMICS
compartment would be specified by a unique combination of
binary choices, each of these choices being the result of a spa
tial symmetry-breaking bifurcation. The model leads to suc
cessful predictions about the result of transplantations as a
function of the "distance" between the original and final re
gions-that is , of the number of differences among the states
of the binary choices or "switches" that specify each of them.
Such ideas and models are especially important in biolog
ical systems where the embryo begins to develop in an appar
ently symmetrical state (for example , Fucus, Acetabularia).
We may ask if the embryo is really homogeneous at the begin
ning. And even if small inhomogeneities are present in the ini
tial environment, do they cause or channel evolution toward a
given structure? Precise answers to such questions are not
available at present. However, one thing seems established:
the instability connected with chemical reactions and trans
port appears as the only general mechanism capable of break
ing the symmetry of an initially homogeneous situation.
The very possibility of such a solution takes us far beyond
the age-old conflict between reductionists and antireduc
tionists. Ever since Aristotle (and we have cited Stahl, Hegel,
Bergson, and other antireductionists), the same conviction
has been expressed: a concept of complex organization is re
quired to connect the various levels of description and account
for the relationship between the whole and the behavior of the
parts. In answer to the reductionists , for whom the sole
"cause" or organization can lie only in the part, Aristotle with
his formal cause , Hegel with his emergence of Spirit in Na
ture, and Bergson with his simple , irrepressible , organization
creating act , assert that the whole is predominant. To cite
Bergson,
In general, when the same object appears in one aspect
as simple and in another as infinitely complex, the two
aspects have by no means the same importance, or rather
the same degree of reality. In such cases, the simplicity
belongs to the object itself, and the infinite complexity to
the views we take in turning around it, to the symbols by
which our senses or intellect represent it to us, or, more
gc::nc::rally, to c::l c::m c::nts
of a different order,
with which we:
try to imitate it artificially, but with which it remains in-
ORDER OUT OF CHAOS
174
commensurable, being of a different nature. An artist of
genius has painted a figure on his canvas. We can imitate
his picture with many-coloured squares of mosaic. And
we shall reproduce the curves and shades of the model so
much the better as our squares are smaller, more numer
ous and more varied in tone. But an infinity of elements
infinitely small, presenting an infinity of shades, would
be necessary to obtain the exact equivalent of the figure
that the artist has conceived as a simple thing, which he
has wished to transport as a whole to the canvas, and
which is the more complete the more it strikes us as the
projection of an indivisible intuition. 17
In biology, the conflict between reductionists and antireduc
tionists has often appeared as a conflict between the assertion
of an external and an internal purpose. The idea of an imma
nent organizing intelligence is thus often opposed by an orga
nizational model borrowed from the technology of the time
(mechanical , heat, cybernetic machines), which immediately
elicits the retort: " W ho" built the machine, the automaton
that obeys external purpose?
As Bergson emphasized at the beginning of this century,
both the technological model and the vitalist idea of an inter
nal organizing power are expressions of an inability to con
ceive evolutive organization without immediately referring it
to some preexisting goal. Today, in spite of the spectacular suc
cess of molecular biology, the conceptual situation remains
about the same : Bergson's argument could be applied to con
temporary metaphors such as "organizer," "regulator," and
"genetic program." Unorthodox biologists such as Paul Weiss
and Conrad Waddington 18 have rightly criticized the way this
kind of qualification attributes to individual molecules the
power to produce the global order biology aims to understand,
and, by so doing, mistakes the formulation of the problem for
its solution.
It must be recognized that technological analogies in biol
ogy are not without interest. However, the general validity of
such analogies would imply that, as in an electronic circuit, for
example, there is a basic homogeneity between the description
of molecular interaction and that of global behavior: The func
tioning of a circuit may be deduced from the nature and posi-
175
THE THREE STAGES OF THERMODYNAMICS
tion of its relays; both refer to the same scale, since the relays
were designed and installed by the same engineer who built
the whole machine. This cannot be the rule in biology.
It is true that when we come to a biological system such as
the bacterial chemotaxis, it is hard not to speak of a molecular
machine consisting of receptors, sensory and regulatory pro
cessing systems, and motor response . We know of approx
imately twenty or thirty receptors that can detect highly
specific classes of compounds and make a bacterium swim up
spatial gradients of attractants or down gradients of repellents.
This "behavior" is determined by the output of the processing
system-that is, the switching on or off of a tumble that gener
ates a change in the bacterium's direction.l9
But such cases, fascinating as they are, do not tell the whole
story. In fact it is tempting to see them as limiting cases, as the
end products of a specific kind of selective evolution, empha
sizing stability and reproducible behavior against openness
and adaptability. In such a perspective, the relevance of the
technological metaphor is not a matter of principle but of op
portunity.
The problem of biological order involves the transition from
the molecular activity to the supermolecular order of the cell.
This problem is far from being solved.
Often biological order is simply presented as an improbable
physical state created and maintained by enzymes resembling
Maxwell's demon, enzymes that maintain chemical differ
ences in the system in the same way as the demon maintains
temperature and pressure differences. If we accept this, biol
ogy would be in the position described by Stahl. The laws of
nature allow only death. Stahl's notion of the organizing action
of the soul is replaced by the genetic information contained in
the nucleic acids and expressed in the formation of enzymes
that permit life to be perpetuated. Enzymes postpone death
and the disappearance of life.
In the context of the physics of irreversible processes, the
results of biology obviously have a different meaning and dif
ferent implications. We know today that both the biosphere as
a whole as well as its components, living or dead, exist in far
from-equilibrium conditions. In this context life, far from
being outside the natural order, appears as the supreme ex
pression of the self-organizing processes that occur.
ORDER OUT OF CHAOS
176
We are tempted to go so far as to say that once the condi·
tions for self-organization are satisfied, life becomes as predic
table as the Benard instability or a falling stone. I t is a
. remarkable fact that recently discovered fossil forms of life
appear nearly simultaneously with the first rock formations
(the oldest microfossils known today date back 3.8. 109 years,
while the age of the earth is supposed to be 4. 6,109 years; the
formation of the first rocks is also dated back to 3.8. 109 years).
The early appearance of life is certainly an argument in favor of
the idea that life is the result of spontaneous seif-organization
that occurs whenever conditions for it permit. However, we
must admit that we remain far from any quantitative theory.
To return to our understanding of life and evolution, we are
now in a better position to avoid the risks implied by any de·
nunciation of reductionism. A system far from equilibrium may
be described as organized not because it realizes a plan alien
to elementary activities, or transcending them, but, on the
contrary, because the amplification of a microscopic fluctuation
occurring at the "right moment" resulted in favoring one reaction
path over a number of other equally possible paths. Under cer·
tain circumstances, therefore, the role played by individual be
havior can be decisive. More generally, the "overall" behavior
cannot in general be taken as dominating in any way the ele
mentary processes constituting it. Self-organization processes
in far-from-equilibrium conditions correspond to a delicate in
terplay between chance and necessity, between fluctuations
and deterministic laws. We expect that near a bifurcation, fluc
tuations or random elements would play an important role,
while between bifurcations the deterministic aspects would
become dominant. These are the questions we now need to
investigate in more detail.
CHAPTER VI
ORDER THROUGH
FWCTUATIONS
Fluctuations and Chemistry
In our Introduction we noted that a reconceptualization of the
physical sciences is occurring today. They are moving from
deterministic, reversible processes to stochastic and irrevers
ible ones. This change of perspective affects chemistry in a
striking way. As we have seen in Chapter V, chemical pro
cesses, in contrast to the trajectories of classical dynamics,
correspond to irreversible processes. Chemical reactions lead
to entropy production. On the other hand, classical chemistry
continues to rely on a deterministic description of chemical
evolution. As we have seen in Chapter V, it is necessary to
produce differential equations involving the concentration of
the various chemical components. Once we know these con
centrations at some initial time (as well as at appropriate
boundary conditions when space-dependent phenomena such as
diffusion are involved), we may calculate what the concenK-ation
will be at a later time. It is interesting to note that the deter
ministic view of chemistry fails when far-from-equilibrium
processes are involved.
We have repeatedly emphasized the role of fluctuations. Let
us summarize here some of the more striking features . W hen
ever we reach a bifurcation point, deterministic description
breaks down. The type of fluctuation present in the system
will lead to the choice of the branch it will follow. Crossing a
bifurcation is a stochastic process, such as the tossing of a
coin. Chemical chaos provides another example (see Chapter
V). Here we can no longer follow an individual chemical tra
jectory. We cannot predict the details of temporal evolution.
177
ORDER OUT OF CHAOS
178
Once again , only a statistical description is possible. The exis
tence of an instability may be viewed as the result of a fluctua
tion that is first localized in a small part of the system and then
spreads and leads to a new macroscopic state.
This situation alters the traditional view of the relation be
tween the microscopic level as described by molecules or
atoms and the macroscopic level described in ter ms of global
variables such as concentration. In many situations fluctua
tions correspond only to small corrections. As an example, let
us take a gas composed of N molecules enclosed in a vessel of
volume V. Let us divide this volume into two equal parts.
W hat is the number of particles X in one of these two parts?
Here the variable X is a "random" variable, and we would
expect it to have a value in the neighborhood of N/2.
A basic theorem in probability theory, the law of large num
bers, provides an estimate of the "error " due to fluctuations.
In essence, it states that if we measure X we have to expect a
value of the order N/2±v'Nii. If N is large, the difference
introduced by fluctuations v'Nfi may a lso be large (if
N= 1Q24, VN= 1012); however, the relative error introduced
by fluctuations is of the order of (v'N!i)/(N/2) or llYN and
thus tends toward zero for a sufficiently large value of N. As
soon as the system becomes large enough, the law of large
numbers enables us to make a clear distinction between mean
values and fluctuations, and the latter may be neglected.
Hewever, in nonequilibrium processes we may find just the
opposite situation. Fluctuations determine the global out
come. We could say that instead of being corrections in the
average values, fluctuations now modify those averages. This
is a new situation. For this reason we would like to introduce a
neologism and call situations resulting from fluctuation "order
through fluctuation." Before giving examples, let us make
some general remarks to illustl·ate the conceptual novelty of
this situation.
Readers may be familiar with the Heisenberg uncertainty
relations, which express in a striking way the probabilistic as
pects of quantum theory. Since we can no longer simultane
ously measure position and coordinates in quantum theory,
classical determinism is breaking down. This was believed to
be of no impor tance for the description of macroscopic objects
---
-- --
----
-----
1 79
ORDER THROUGH FLUCTUATIONS
such as living systems. But the role of fluctuations in nonequi
librium systems shows that this is not the case. Randomness
remains essential on the macroscopic level as well. It is inter
esting to note another analogy with quantum theory, which
assigns a wave behavior to all elementar y particles. As we
have seen, chemical systems far from equilibrium may also
lead to coherent wave behavior: these are the chemical clocks
discussed in Chapter V. Once again, some of the properties
quantum mechanics discovered on the microscopic level now
appear on the macroscopic level.
Chemistry is actively involved in the reconceptualization of
science. I We are probably only at the beginning of new direc
tions of research. It may well be, as some recent calculations
suggest, that the idea of reaction rate has to be replaced in
some cases by a statistical theory involving a distribution of
reaction probabilities.2
Fluctuations and Correlations
Let us go back to the types of chemical reaction discussed in
Chapter V. To take a specific example, consider a chain of
reactions such as APXPF. The kinetic equations in Chapter V
refer to the average concentrations. To emphasize this we shall
now write <X> instead of X. We can then ask what is the
probability at a given time of finding a number X for the con
centration of this component. Obviously this probability will
fluctuate, as do the number of collisions among the various
molecules involved. It is easy to write an equation that de
scribes the change in this probability distribution P(X,t) as a
result of processes that produce molecule X and of processes
that destroy that molecule. We may perform the calculation for
equilibrium systems or for steady-state systems. Let us first
mention the results obtained for equilibrium systems.
At equilibrium we virtually recover a classical probabilistic
distribution, the Poisson distribution, which is described in
every textbook on probabilities, since it is valid in a variety of
situations, such as the distribution of telephone calls, waiting
times in restaurants, or the fluctuation of the concentration of
ORDER OUT OF CHAOS
180
particles in a gas or a liquid. The mathematical form of this
distribution is of no importance here. We merely want to em
phasize two of its aspects. First, it leads to the law of large
numbers as formulated in the first section of this chapter. Thus
fluctuations indeed become negligible in a large system. More
over, this law enables us to calculate the correlation between
the number of particles X at two different points in space sepa
rated by some distance r. The calculation demonstrates that at
equilibrium there is no such correlation. The probability of
finding two molecules X and X' at two different points rand r'
is the product of finding X at r and X' at r' (we consi. der dis
tances that are large in respect to the range of intermolecular
forces).
One of the most unexpected results of recent research is that
this situation changes drastically when we move to nonequilib
rium situations. First, when we come close to bifurcation
points the fluctuations become abnormally high and the law of
large numbers is violated. This is to be expected, since the
system may then "choose" among various regimes. Fluctua
tions can even reach the same order of magnitude as the mean
macroscopic values. Then the distinction between fluctuations
and mean values breaks down. Moreover, in the case of a non
linear type of chemical reaction discussed in Chapter V, long
range correlations appear. Particles separated by macroscopic
distances become linked. Local events have repercussions
throughout the whole system. It is interesting to note3 that
such long-range correlations appear at the precise point of
transition from equilibrium to nonequilibrium. From this point
of view the transition resembles a phase transition. However,
the amplitudes of these long-range correlations are at first
small but increase with distance from equilibrium and may be
come infinite at the bifurcation points.
We believe that this type of behavior is quite interesting,
since it gives a molecular basis to the problem of communica
tion mentioned in our discussion of the chemical clock. Even
before the macroscopic bifurcation, the system is organized
through these long-range correlations. We come back to one of
the main ideas of this book: nonequilibrium as a source of or
der. Here the situation is especially clear. At equilibrium mole
cules behave as essentially independent entities; they· ignore
one another. We would like to call them "hypnons," "sleep-
181
ORDER THROUGH FLUCTUATIONS
walkers." Though each of them may be as complex as we like,
they ignore one another. However, nonequilibrium wakes them
up and introduces a coherence quite foreign to equilibrium.
The microscopic theory of irreversible processes that we shall
develop in Chapter IX will present a similar picture of matter.
Matter's activity is related to the nonequilibrium conditions
that it itself may generate. Just as in macroscopic behavior, the
laws of fluctuations and correlations are universal at equi
librium (when we find the Poisson type of distribution); they
become highly specific depending on the type of nonlinearity
involved when we cross the boundary between equilibrium
and nonequilibrium.
The Amplification of Fluctuations
Let us first take two examples wherein the growth of a fluctua
tion preceding the formation of a new structure can be fol
lowed in detail. The first is the aggregation of slime molds,
which when threatened with starvation coalesce into a single
supracellular mass. We have already mentioned this in Chap
ter V. Another illustration of the role of fluctuations is the first
stage in the construction of a termites' nest. This was first
described by Grasse, and Deneubourg has studied it from the
standpoint that interests us here."'
Self-Aggregation Process in an Insect Population
Larvae of a coleoptera (Dendroctonus micans [Scot.]). are initially dis
tributed at random between two horizontal sheets of glass, 2 mm apart. The
borders are open and the surface is equal to 400 cm2.
The aggregation process appears to result from the competition between
two factors: the random moves of the larvae, and their reaction to a chemical
product, a "pheromon" they synthetize from terpanes contained in the tree
on which they feed and that each of them emits at a rate depending on its
nutrition state. The pheromon diffuses in space, and the larvae move in the
direction of its concentration gradient. Such a reaction provides an auto
catalytic mechanism since, as they gather in a cluster, the larvae contribute
to enhance the attractiveness of the corresponding region. The higher the
local density of larvae in this region, the stronger the gradient and the more
intense the tendency to move toward the crowded point.
The experiment shows that the density of the larvae population deter
mines not only the rate of the aggregation process but its effectiveness as
well-that is, the number of larvae that will finally be part of the cluster. At
ORDER OUT OF CHAOS
182
high density (Figure A) a cluster appears and rapidly grows at the center of
the experimental setup. At very low density (Figure B), no stable cluster ap
pears.
Moreover, other experiments have explored the possibility for a cluster to
develop starting from a "nucleus" artificially created in a peripheral region of
the system. Different solutions appear depending on the number of larvae in
this initial nucleus.
J
,.
(
I'
t
\
'
I
Figure A. Self-aggregation at high density. The times are 0 minutes and 21
minutes.
183
ORDER THROUGH FLUCTUATIONS
If this number is small compared with the total number of larvae, the clus
ter fails to develop (Figure D). If it is large, the cluster grows (Figure E). For
intermediate values of the initial nucleus, new types of structure may de
velop: Two, three or four other clusters appear and coexist, with a time of life
at least greater than the time of observation (Figures F and G).
No such multicluster structure was ever observed in experiments with ho
mogeneous initial conditions. It would seem they correspond, in a bifurcation
4)
"'
\1
.,..
\
"-
I
.,.
'
,.
"
1
"'
..
A
....
�
...
-
(,l
•
l
Figure B. Self-aggregation at low density. The times are 0 minutes and 22
minutes.
184
ORDER OUT OF CH AOS
diagram, to stable states compatible with the value of the parameters
characterizing the system but that cannot be attained by this system starting
from homogeneous conditions. The nucleus would play the part of a finite
perturbation necessary to excite the system and deport it in a region of the
bifurcation diagram corresponding to such families of multicluster solutions.
•
100
Q
•
80
60
40
,--.. --- -�
20
10
20
�
-
---
f
--
.
-
-
r �-�-� t
. · ·· ·]f . ,-f
,-
---
.. ..
.. .
. . . ..
.. ......
.. ....
.
. .
30
. .. · · ······
.
40
Figure C. Percent of the total number of larvae in the central cluster
function of time at three different densities.
in
N
CRITICAl NUCLEUS dK�nc of e10 lot- nt... nud<ua
tetot jiiOpUiat•on ,.., •pcnmcnt 80 lor-
JO mon
Figure D. Fall of initial clusters of 10 larvae. Total population, 80 larvae. N:
number of larvae in clusters.
185
ORDER T;-iROUGH FWCTUATIONS
N
CR.tTiCAL NUCLEUS
!l'owlh of• :or�.-- inilinlnutk.. 1•··•1
growth of a 30 la<v• ntiot nuck.a ( -•
fetal population por -nmcnl 80 ..._
0 �----.
-�-80min
Figure E. Growth of initial clusters of 20 and 30 larvae. Total population, 80
larvae.
0
Chelk
1110
2
•
80 min.
Figure F. Multicluster solutions. Initial value of the cluster, 15 larvae. Total
population, 80 larvae.
1 86
ORDER OUT OF CHAOS
•
�
.,
�
>
, ,
.,
,.
"
�
...
I
I
I
...
�
"'
.,
"
.,
1
...
I
'
\
.,
...
I
;)
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.,
"
.I I
.,
,
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.,
..
')
")
_,
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)
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,'
'�'
,J
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•
Figure G. Growth of a cluster (I) introduced peripherally, which induce the
formation of a second little cluster (II).
The construction of a termites' nest is one of those coherent
activities that have led some scientists to speculate about a
.. collective mind" in insect communities. But curiously, it ap
pears that in fact the termites need very little information to
participate in the construction of such a huge and complex
edifice as the nest. The first stage in this activity, the con
struction of the base, has been shown by Grasse to be the
result of what appears to be disordered behavior among ter
mites. At this stage, they transport and drop lumps of earth in
a random fashion, but in doing so they impregnate the lumps
w ith a hormone that attracts other termites. The situation
could thus be represented as follows: the initial "fluctuation"
would be the slightly larger concentration of lumps of earth,
which inevitably occurs at one time or another at some point
in the area. The amplification of this event is produced by the
187
·
ORDER THROUGH FLUCTUATIONS
increased density of termites in the region, attracted by the
slightly higher hormone concentration. As termites become
more numerous in a region, the probability of their dropping
lumps of earth there increases, leading in turn to a still higher
concentration of the hormone. In this way " pillars" are
formed , separated by a distance related to the range over
which the hormone spreads. Similar examples have recently
been described.
Although Boltzmann's order principle enables us to de
scribe chemical or biological processes in which differences
are leveled out and initial conditions forgotten, it cannot ex
plain situations such as these , where a few "decisions" in an
unstable situation may channel a system formed by a large
number of interactive entities toward a global structure.
When a new structure results from a finite perturbation, the
fluctuation that leads from one regime to the other cannot pos
sibly overrun the initial state in a single move. It must first estab
lish itself in a limited region and then invade the whole space:
there is a nucl eation mechanism. Depending on whether the
size of the initial fluctuating region lies below or above some
critical value (in the case of chemical dissipative structures,
this threshold depends in particular on the kinetic constants
and diffusion coefficients), the fluctuation either regresses or
else spreads to the whole system. We are familiar with nuclea
tion phenomena in the classical theory of phase change: in a
gas, for example, condensation droplets incessantly form and
evaporate. That temperature and pressure reach a point where
the liquid state becomes stable means that a critical droplet
size can be defined (which is smaller the lower the temperature
and the higher the pressure). If the size of a droplet exceeds
this "nucleation threshold, " the gas almost instantaneously
transforms into a liquid.
Moreover, theoretical studies and numerical simulations
show that the critical nucleus size increases with the efficacy
of the diffusion mechanisms that link all the regions of sys
tems. In other words, the faster communication takes place
within a system, the greater the percentage of unsuccessful
fluctuations and thus the more stable the system. This aspect
of the critical-size problem means that in such situations the
"outside world,·· t he environment of t he fluctuat ing region,
always tends to damp fluctuations. These will be destroyed or
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(a)
188
( b)
Figure 19. Nucleation of a liquid droplet in a supersaturated vapor. (a)
droplet smaller than the critical size; (b) droplet larger than the critical size.
The existence of the threshold has been experimentally verified for dissipa
tive structures.
amplified according to the effectiveness of the communication
between the fluctuating region and the outside world. The crit
ical size is thus determined by the competition between the
system's "integrative power" and the chemical mechanisms
amplifying the fluctuation.
This model applies to the results obtained recently in in
vitro experimental studies of the onset of cancer tumors.s An
individual tumor cell is seen as a "fluctuation, " uncontrollably
and permanently able to appear and to develop through rep
lication. It is then confronted with the population of cytotoxic
cells that either succeeds in destroying it or fails. Following
the values of the different parameters characteristic of the rep
lication and destruction processes, we can predict a regression
or an amplification of the tumor. This kind of kinetic study has
led to the recognition of unexpected features in the interaction
between cytotoxic cells and the tumor. It seems that cytotoxic
cells can confuse dead tumor cells with living ones. As a re
sult, the destruction of the cancer cells becomes increasingly
difficult.
The question of the limits of complexity has often been
raised. Indeed, the more complex a system is, the more nu
merous are the types of fluctuations that threaten its stability.
How then, it has been asked, can systems as complex as eco
logical or human organizations possibly exist? How do they
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manage to avoid permanent chaos? The stabilizing effect of
communication, of diffusion processes, could be a partial an·
swer to these questions. In complex systems, where species
and individuals interact in many different ways, diffusion and
communication among various parts of the system are likely
to be efficient . There is competition between stabilization
through communication and instability through fluctuations.
The outcome of that competition determines the threshold of
stability.
Structural Stability
W hen can we begin to speak about "evolution" in its proper
sense? As we have seen, dissipative structures require far-from
equilibrium conditions. Yet the reaction diffusion equations con
tain parameters that can be shifted back to near-equilibrium
conditions. The system can explore the bifurcation diagram in
both directions. Similarly, a liquid can shift from laminar flow
to turbulence and back. There is no definite evolutionary pat·
tern involved.
The situation for models involving the size of the system as
a bifurcation parameter is quite different. Here, growth occur
ring irreversibly in time produces an irreversible evolution.
But this remains a special case, even if it can be relevant for
morphogenetic development.
Be it in biological, ecological, or social evolution, we cannot
take as given either a definite set of interacting units, or a defi
nite set of transformations of these units. The definition of the
system is thus liable to be modified by its evolution. The sim
plest example of this kind of evolution is associated with the
concept of structural stability. It concerns the reaction of a
given system to the introduction of new units able to multiply
by taking part in the system's processes.
The problem of the stability of a system vis-a-vis this kind of
change may be formulated as follows: the new constituents, in
troduced in small quantities, lead to a new set of reactions among
the system's components. This new set of reactions then en
ters into competition with the system·s previous mode of func
tioning. If the system is "structurally stable" as far as this
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190
intrusion is concerned, the new mode of functioning will be
unable to establish itself and the "innovators" will not survive.
If, however, the structural fluctuation successfully imposes it
self-if, for example, the kinetics whereby the "innovators"
multiply is fast enough for the latter to invade the system in
stead of being destroyed-the whole system will adopt a new
mode of functioning : its activity will be governed by a new
"syntax. "6
The simplest example of this situation is a population of
macromolecules reproduced by polymerization inside a sys
tem being fed with the monomers A and B. Let us assume the
polymerization process to be autocatalytic-that is, an already
synthesized polymer is used as a model to form a chain having
the same sequence. This kind of synthesis is much faster than
a synthesis in which there is no model to copy. Each type of
polymer, characterized by a particular sequence of A and B,
can be described by a set of parameters measuring the speed
of the synthesis of the copy it catalyzes, the accuracy of the
copying process, and the mean life of the macromolecule it
self. It may be shown that, under certain conditions, a single
type of polymer having a sequence, shall we say, ABABABA . . .
dominates the population, the other polymers being reduced
to mere "fluctuations" with respect to the first. The . problem
of structural stability arises each time that, as a result of a
copying "error," a new type of polymer characterized by a
hitherto unknown sequence and by a new set of parameters
appears in the system and begins to multiply, competing with
the dominant species for the available A and B monomers.
Here we encounter an elementary case of the classic Darwin
ian idea of the "survival of the fittest. "
Such ideas form the basis for the model of prebiotic evolu
tion developed by Eigen and his coworkers. The details of
Eigen's argument are easily accessible elsewhere. ? Let us
briefly state that it seems to show that there is only one type of
system that can resist the "errors" that autocatalytic popula
tions continually make-a polymer system structurally stable
for any possible "mutant polymer." This system is composed
of two sets of polymer molecules. The molecules of the first
set are of the "nucleic acid" type; each molecule is capable of
reproducing itself and act s as a catalyst in the synthesis of
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a molecule of the second set, which is of the proteic type: each
molecule of this second set catalyzes the self-reproduction of a
molecule of the first set. This transcatalytic association be
tween molecules of the two sets may turn into a cycle (each
"nucleic acid" reproduces itself with the help of a "protein").
It is then capable of stable survival, sheltered from the con
tinual emergence of new polymers with higher reproductive
efficiency: indeed, nothing can intrude into the self-replicating
cycle formed by "proteins" and "nucleic acids. " A new kind
of evolution may thus begin to grow on this stable foundation,
heralding the genetic code.
Eigen 's approach is certainly of great interest. Darwinian
selection for faithful self-reproduction is certainly important
in an environment with a limited capacity. But we tend to be
lieve that this is not the only aspect involved in prebiotic evolu
tion. The "far-from-equilibrium" conditions related to critical
amounts of flow of energy and matter are also important. It
seems reasonable to assume that some of the first stages mov
ing toward life were associated with the formation of mecha
nisms capable of absorbing and transforming chemical energy,
so as to push the system into "far-from-equilibrium" condi
tions. At this stage life, or "prelife, " probably was so diluted
that Darwinian selection did not play the essential role it did in
later stages.
Much of this book has centered around the relation between
the microscopic and the macroscopic. One of the most impor
tant problems in evolutionary theory is the eventual feedback
between macroscopic structures and microscopic events: mac
roscopic structures emerging from microscopic events would
in turn lead to a modification of the microscopic mechanisms.
Curiously, at present, the better understood cases concern so
cial situations. W hen we build a road or a bridge, we can pre
dict how this will affect the behavior of the population, and
this will in turn determine other modifications of the modes of
communication in the region. Such interrelated processes gen
erate very complex situations, the understanding of which is
needed before any kind of modelization. This is why what we
will now describe are only very simple cases.
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Logistic Evolution
In social cases, the problem of structural stability has a large
number of applications. But it must be emphasized that such
applications imply a drastic simplification of a situation de
fined simply in terms of competition between self-replicating
processes in an environment where only a limited amount of
the needed resources exists.
In ecology the classic equation for such a problem is called
the "logistic equation." This equation describes the evolution
of a population containing N individuals, taking into account
the birthrate, the death rate, and the amount of resources avail
able to the population. The logistic equation can be written
dN/dt = rN(K - N) - mN, w here r and m are characteristic
birth and death constants and K the "carrying capacity" of the
environment. W hatever the initial value of N, as time goes on
it will reach the steady-state value N K - mlr determined by
the differences of the carrying capacity and the ratio of death
and birth constants. W hen this value is reached, the environ=
N
K - ..!Il
r
t
Figure 20. Evolution of a population N as a function of time t according to
the logistic curve. The stationary state N = O is unstable while the stationary
state N K- mlr is stable with respect to fluctuations of N.
=
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ORDER THROUGH FLUCTUATIONS
ment is saturated, and at each instant as many individuals die
as are born.
The apparent simplicity of the logistic equation conceals to
some extent the complexity of the mechanisms involved. We
have already mentioned the effect of external noise, for exam
ple. Here it has an especially simple meaning. Obviously, if
only because of climatic fluctuations, the coefficients K, m,
and r cannot be taken as constant. We know that such fluctua
tions can completely upset the ecological equilibrium and
even drive the population to extinction. Of course, as a result,
new processes, such as the storage of food and the formation
of new colonies, will begin and eventually evolve so that some
effects of external fluctuation may be avoided.
But there is more. Instead of writing the logistic equation as
continuous in time, let us compare the population at fixed time
intervals (for example, separated by a year). This "discrete"
logistic equation can be written in the form Nt + 1 = Nt(l + r
[ 1 - N/KJ), wliere Nt and Nt+ 1 are the populations separated
by a one-year interval (we neglect here the death term). The
remarkable feature, noted by R. May,8 is that such equations,
in spite of their simplicity, admit a bewildering number of solu
tions. For values of the parameter O�r�2, we have, as in the
continuous case, a uniform approach to equilibrium. For val
ues of r lower than 2.444, a limit cycle sets in: we now have a
periodic behavior with a two-year period . This is followed by
four-, eight-, etc., year cycles, until the behavior can only be
described as chaotic (if r is larger than 2.57). Here we have a
transition to chaos as described in Chapter V. Does this chaos
arise in nature? Recent studies9 seem to indicate that the pa
rameters characterizing natural populations keep them from
the chaotic region. W hy is this so? Here we have one of the
very interesting problems created by the confluence of evolu
tionary problems with the mathematics produced by computer
simulation.
Up to now we have taken a static point of view. Let us now
move to mechanisms, whereby the parameters K, r, and m
may vary during biological or ecological evolution.
We have to expect that during evolution the values of the
ecological parameters K, r, and m will vary (as well as many
other parameters and variables, whether they are quantifiable
or not). Living societies continually introduce new ways of ex·
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194
ploiting exi.;ting resources or of discovering new ones (that is,
increases) and continually discover new ways of extending
their lives or of multiplying more quickly. Each ecological
equilibrium defined by the logistic equation is thus only tem
porary, and a logistically defined niche will be occupied suc
cessively by a series of species , each capable of ousting the
preceding one when its "aptitude" for exploiting the niche , as
measured by the quantity K - mlr, becomes greater. (See Fig
ure 2 1 .) Thus the logistic equation leads to the definition of a
very simple s ituation where we can give a quantitative for
mulation of the Darwinian idea of the "survival of the fittest. "
The "fittest" is the species for which at a given time the quan
tity K - mlr is the largest .
As restricted as the problem described by the logistic equa
tion is , it nonetheless leads to some mar velous examples of
nature's inventi veness .
Take the example of caterpillars, who m ust remain un
detected , since the slowness of their movement <makes escape
impossible.
The evolved strategies of using poisons and irritating hairs
and spines, as well as intimidating displays, are highly effec
tive in repelling birds and other potential predators. But none
K
X
l
'
,'
•
I
I
I
1'\
J
'·
I
I
I
I
/
,' X
•
\•
\
\
'
t
Figure 21 . Evolution of total population X as function of time; the popula
tion i:5 made up by species X1 , X2 and X3, which appear successively ana
are characterized by increasing values of K- mlr (see text).
195
ORDER THROUGH FLUCTUATIONS
of these strategies is effective against all predators at all times,
particularly if a predator is hungry enough. The ideal strategy
is to remain totally undetected. Some caterpillars approach
this ideal, and the variety and sophistication of the strategies
used by the hundreds of lepidopteran species to remain un
detected bring to mind the words of distinguished nineteenth
century naturalist Louis Agassiz: "The possibilities of exis
tence run so deeply into the extravagant that there is scarcely
any conception too extraordinary for Nature to realize. " I O
We cannot resist giving an example reported by Milton
Love. ' ' The sheep liver trematode has to pass from an ant to a
sheep, where it will finally reproduce itself. The chances of
sheep swallowing an infected ant are very small, but the ant
behaves in a remarkable way: it starts to maximize the proba
bility of its encounter with a sheep. The trematode has truly
"body snatched" its host. It has burrowed into the ant's brain,
compelling its victim to behave in a suicidal way: the pos
sessed ant, instead of staying on the ground, climbs to the tip
of a blade of grass and there, immobile, waits for a sheep. This
is indeed an incredibly "clever" solution to the parasites prob
lem. How it was selected remains a puzzle.
Other situations in biological evolution may be investigated
using models similar to the logistic equation. For instance, it is
possible to calculate the conditions of interspecies competi
tion under which it may be advantageous for a fraction of the
population to specialize in warlike and nonproductive activity
(for example , the "soldiers" among the social insects). We can
also determine the kind of environment in which a species that
has become specialized , that has restricted the range of its
food resources , will survive more easily than a nonspecialized
species that consumes a wider range of resources. t2 But here
we are approaching some very different problems , which con
cern the organization of internally differentiated populations.
Clear distinctions are absolutely necessary if we are to avoid
confusion. In populations where individuals are not inter
changeable and where each, with its own memory, character,
and experience , is called upon to play a singular role, the rele
vance of the logistic equation and, more generally, of any sim
ple Darwinian reasoning becomes quite relative. We shall
return to this problem.
It is interesting to note that the type of curve represented in
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196
Figure 21 showing the succession of growths and peaks de
fined by a given logistic equation's family with increasing
K mlr has also been used to describe the multiplication of
certain technical procedures or products. Here too, the dis
covery or introduction of a new technique or product breaks
some kind of social, technological, or economic equilibrium.
This equilibrium would correspond to the maximum reached
by the growth curve of the techniques or products with which
the innovation is going to have to compete and that play a simi
lar role in the situation described by the equation. 13 Thus, to
choose but one example, not only did the spread of the steam
ship lead to the disappearance of most sailing ships, but, by
reducing the cost of transportation and increasing its speed, it
caused an increase in the demand for sea transport ("K") and
consequently an increase in the population of ships. We are
obviously representing here an extremely simple situation,
supposedly governed by purely economic logic. Indeed, in
this case innovation seems merely to satisfy, albeit in a dif
ferent way, a preexisting need that remains unchanged. How
ever, in ecology as in human societies, many innovations are
successful without such a preexisting "niche." Such innova
tions transform the environment in which they appear, and as
they spread, they create the conditions necessary for their
own multiplication. their "niche." In social situations. in par
ticular, the creation of a "demand," and even of a "need" for
this demand to fulfill, often appears as correlated with the pro
duction of the goods or techniques that satisfy the demand.
-
Evolutionary Feedback
A first step toward accounting for this dimension of the evolu
tionary process can be achieved by making the "carrying ca
pacity" of a system a function of the way it is exploited instead
of taking it as given.
In this way some supplementary dimensions of economic
activities, and more particularly the "multiplying effects," can
be represented. Thus we can describe the self-accelerating
properties of systems and the spatial differentiation between
different levels of activity.
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Geographers have already constructed a model correlating
these processes, the Christaller model, defining the optimal
spatial distribution of centers of economic activity. Important
centers would be at the intersection of an hexagonal network,
each being surrounded by a ring of towns of the next smallest
size, each being, etc . . . . Obviously, in actual cases, such a
regular hierarchical distribution is very infrequent: historical,
political, and geographical factors abound, disrupting the spa
tial symmetry. But there is more. Even if all the important
sources of asymmetrical development were excluded and we
started from a homogeneous economic and geographical space,
the modeling of the genesis of a distribution such as defined by
Christaller establishes that the kind of static optimalization he
describes constitutes a possible but quite unlikely result of the
process.
The model in question14 stages only the minimal set of vari
ables implied by a calculation such as Christaller. A set of
equations extending the logistic equations is constructed ,
starting from the basic supposition that populations tend to
migrate as a function of local levels of economic activity,
which thus define a kind of local "carrying capacity," here
reduced to an "employment" capacity. But the local popula
tion is .also a potential consumer for locally produced goods.
We have, in fact, a double positive feedback, called the "urban
multiplier," for a local development: both the local population
and the economic infrastructute produced by the already at
tained level of activity accelerate the increase of this activity.
B ut each local level of activity is also determined by competi
tion wi th similar centers of activity located elsewhere . The
sale of produced goods or ser v ices depends on the cost of
transporting them to consumers and on the size of the "enter
prise." The expansion of each such enterprise depends on a
demand that this expansion itself helps to create and for which
it competes. Thus the respective growth of population and
manufacturing or service activities is linked by strong feed
back and nonlinearities.
The model starts with a hypothetical initial condition, where
"level 1 " activity (rural) exists at the different points; it then
permits us to follow successive launchings of activities corre
sponding to "superior" levels in C hristaller's hierarchy-that
is, implying exportation oo a greater range. Even if the initial
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65
69
66
•
64
•
•
•
62
•
67
65
65
•
•
•
67
62
•
66
•
66
66
61
•
60
•
•
•
66
65
•
63
•
67
•
67
•
63
•
67
62
•
66
•
65
•
•
62
63
•
•
•
•
&9
69
•
•
68
•
68
•
Figure 22. A possible history of "urbanization." • have only function 1;
A have functions 1, 2 and 3.
are the largest
centers, with functions 1, 2, 3, and 4. At t=O (not represented), all points
• have functions 1 and 2;
�
199
61
•
ORDER THROUGH FLUCTUATIONS
67
•
66
•
62
64
67
62
•
64
•
•
•
•
66
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63
•
67
•
64
•
69
•
have a "population" of 67 units. At C, the largest center is gOing through a
maximum (152 population units); this is followed by an "urban sprawl," with
creation of satellite cities; this also occurs around the second m�in center.
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62
•
200
65
•
65
•
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64
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ORDER OUT OF CHAOS
i7
•
71
•
202
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ORDER THROUGH FLUCTUATIONS
state is quite homogeneous, the model shows that the mere
play of chance factors-factors uncontrolled by the model,
such as the place and time where the different enterprises
start-is sufficient to produce symmetry breakings: the ap
pearance of highly concentrated zones of activity while others
suffer a reduction in economic activity and are depopulated.
The different computer simulations show growth and decay,
capture and domination, periods of opportunity for alternative
developments followed by solidification of the existing domi
nation structures.
Whereas Christaller's symmetrical distribution ignores his
tory, this scenario takes it into account, at least in a very mini
mal sense, as an interplay between .. laws," in this case of a
purely economic nature, and the ..chance" governing the se
quence of launchings.
Modelizations of Corrplexity
In spite of its simplicity, our model succeeds in showing some
properties of the evolution of complex systems, and in particu
lar, the difficulty of ..governing" a development determined by
multiple interacting elements. Each individual action or each
local intervention has a collective aspect that can result in
quite unanticipated global changes. As Waddington empha
sized, at present we have very little understanding of how a
complex system is likely to respond to a given change. Often
this response runs counter to our intuition. The term ..coun
terintuitive" was introduced at MIT to express our frustration:
..The damn thing just does not do what it should do !" To take
the classic example cited by Waddington, a program of slum
clearance results in a situation worse than before. New build
ings attract a larger number of people into the area, but if there
are not enough jobs for them , they remain poor, and their
dwellings become even more overcrowded.t5 We are trained to
think in terms of linear causality, but we need new "tools of
thought" : one of the greatest benefits of models is precisely to
help us discover these tools and learn how to use them.
As we have already emphasized, logistic equations are most
relevant when the crucial dimension is the growth of a popula-
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204
tion, be it of animals, activities, or habits. What is presup
posed is that each member of a given population can be taken
as the equivalent of any of the others. But this general equiv
alence can itself be seen not as a simple general fact but as an
approximation, the validity of which depends on the con
straints and pressures to which this population was submitted
and on the strategy it used to cope with them.
Take, for example, the distinction ecologists have proposed
between K and r strategies. K and r refer to the parameters in
logistic equations. Though this distinction is only relative, it is
especially clear when it characterizes the divergence resulting
from a systematic interaction between two populations, par
ticularly the prey-predator interaction. In this view, the typical
evolution for a prey population will be the increase in the re
production rate r. The predator will evolve toward more effec
tive ways of capturing its prey-that is, toward an amelioration
of K. But this amelioration, defined in a logistic frame, is liable
to have consequences that go beyond the situations defined by
logistic equations.
As Stephen 1. Gould remarked, 16 a K strategy implies indi
viduals becoming more and more able to learn from experi
ence and t o store m emor ies-that is, i nd i v id uals more
complex with a longer period of maturation and apprentice
ship. This in turn means individuals both more "valuable"
representing a larger biological investment-and charac
terized by a longer period of vulnerability. The development of
"social" and "family" ties thus appears as a logical counter
part of the K strategy. From that point on, other factors, be
sides the mere number of indiv iduals in the population,
become more and more relevant and the logistic equation mea
suring the success by the number of individuals becomes mis
leading. We have here a par ticular example of what makes
modelization so risky. In complex systems, both the definition
of entities and of the interactions among them can be modified
by evolution. Not only each state of a system but also the very
definition of the system as modelized is generally unstable, or
at least metastable.
We come to problems where methodology cannot be sepa
rated from the question of the nature of the object investi
gated. We cannot ask the same questions about a population of
flies that reproduce and die by millions without apparently
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learning from or enlarging their experience and about a popu
lation of primates where each individual is an entanglement of
its own experiences and the traditions of the populations in
which he lives.
We also find that, within anthropology itself, basic choices
must be made between various approaches to collective phe
nomena. It is well known, for example , that structural an
thropology privileges those aspects of society where the tools
of logic and finite mathematics can be used, aspects such as
the elementary structures of kinship or the analysis of myths,
whose transformations are often compared to crystalline
growth. Discrete elements are counted and combined. This
contrasts with approaches that analyze evolution in terms of
processes involving large, partially chaotic populations. We
are dealing with two different outlooks and two types of mod
els: Levi-Strauss defines them respectively as "mechanical"
and "statistical. " In the mechanical model "the elements are
of the same scale as the phenomena" and individual behavior
is based on prescriptions referring to the structural organiza
tion of society. The anthropologist makes the logic of this be
havior explicit. The sociologist, on the other hand, works with
statistical models for large populations and defines averages
and thresholds.I7
A society defined entirely in terms of a functional model
would correspond to the Aristotelian idea of natural hierarchy
and order. Each official would perform the duties for which he
has been appointed. These duties would translate at each level
the different aspects of the organization of the society as a
whole. The king gives orders to the architect, the architect to
the contractor, the contractor to the workers. Everywhere a
mastermind is at work. On the contrary, termites and other
social insects seem to approach the "statistical" model. As we
have seen, there seems to be no mastermind behind the con
struction of the termites' nest, when interactions among indi
viduals produce certain types of collective behavior in some
circumstances, but none of these interactions refer to any
global task, being all purely local. Such a description neces
sarily implies averages and reintroduces the question of sta
bility and bifurcations.
Which events will regress, and which arc likely to affect the
whole system? What are the situations of choice, and what are
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206
the regimes of stability? Since size or the system's density
may play the role of a bifurcation parameter, how may purely
quantitative growth lead to qualitatively new choices? Ques
tions such as these call for an ambitious program indeed. As
with the rand K strategies, they lead us to connect the choice
of a "good" model for social behavior and history. How does
the evolution of a population lead it to become more "mechan
ical"? This question seems parallel to questions we have al
ready met in biology. How, for example, does the selection of
the genetic information governing the rates and regulations of
metabolic reactions favor certain paths to such an extent that
development seems to be purposive or appear as the transla
tion of a "message"?
We believe that models inspired by the concept of "order
through fluctuations" will help us with these questions and
even permit us in some circumstances to give a more precise
formulation to the complex interplay between individual and
collective aspects of behavior. From the physicist's point of
view, this involves a distinction between states of the system
in which all individual initiative is doomed to insignificance on
the one hand, and on the other, bifurcation regions in which an
individual, an idea, or a new behavior can upset the global
state. Even in those regions, amplification obviously does not
occur with just any individual, idea, or behavior, but only with
those that are "dangerous"-that is, those that can exploit to
their advantage the nonlinear relations guaranteeing the sta
bility of the preceding regime. Thus we are led to conclude
that the same nonlinearities may produce an order out of the
chaos of elementary processes and still, under different cir
cumstances, be responsible for the destruction of this same
order, eventually producing a new coherence beyond another
bifurcation.
"Order through fluctuations" models introduce an unstable
world where small causes can have large effects, but this world
is not arbitrary. On the contrary, the reasons for the amplifica
tion of a small event are a legitimate matter for rational inquiry.
Fluctuations do not cause the transformation of a systefll's ac
tivity. Obviously, to use an image inspired by Maxwell, the
match is responsible for the forest fire, but reference to a
match does not suffice to understand the fire. Moreover, the
fact that a fluctuation evades control does not mean that we
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ORDER THROUGH FLUCTUATIONS
cannot locate the reasons for the instability its amplification
causes.
An Open World
In view of the complexity of the questions raised here, we can
hardly avoid stating that the way in which biological and social
evolution has traditionally been interpreted represents a par
ticularly unfortunate use of the concepts and methods bor
rowed from physics 1 8-unfor tunate because the area of
physics where these concepts and methods are valid was very
restricted, and thus the analogies between them and social or
economic phenomena are completely unjustified.
The foremost example of this is the paradigm of optimiza
tion. It is obvious that the management of human society as
well as the action of selective pressures tends to optimize
some aspects of behaviors or modes of connection, but to con
sider optimization as the key to understanding how popula
tions and individuals survive is to risk confusing causes with
effects.
Optimization models thus ignore both the possibility of radi
cal transformations-that is, transformations that change the
definition of a problem and thus the kind of solution sought
and the inertial constraints that may eventually force a system
into a disastrous way of functioning. Like doctrines such as
Adam Smith's invisible hand or other definitions of progress in
terms of maximization or minimization criteria, this gives a
reassuring representation of nature as an all-powerful and ra
tional calculator, and of a coherent history characterized by
global progress. To restore both inertia and the possibility of
unanticipated events-that is, restore the open character of
history-we must accept its fundamental uncertainty. Here we
could use as a symbol the apparently accidental character of
the great cretaceous extinction that cleared the path for the
development of mammals, a small group of ratlike creatures.'�
This has been a general presentation, a kind of "bird's-eye
view, " and thus has omitted many topics of great interest:
flames, plasmas, and lasers, for example, present nonequilib
rium instabilities of great theoretical and practical interest.
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208
Everywhere we look, we find a nature that is rich in diversity
and innovations. The conceptual evolution we have described
is itself embedded in a wider history, that of the progressive
rediscovery of time.
We have seen new aspects of time being progressively incor
porated into physics, while the ambitions to omniscience in
herent in classical science were progressively rejected. In this
chapter we have moved from physics through biology and
ecology to human society, but we could have proceeded in the
inverse order. Indeed, history began by concentrating mainly
on human societies, after which attention was given to the
temporal dimensions of life and of geology. The incorporation
of time into physics thus appears as the last stage of a pro
gressive reinsertion of history into the natural and social sci
ences.
Curiously, at every stage of the process, a decisive feature of
this "historicization" has been the discovery of some temporal
heterogeneity. Since the Renaissance , Western society has
come into contact with different populations that were seen as
corresponding to different stages of development; nineteenth
century biology and geology learned to discover and classify
fossils and to recognize in landscapes the memories of a past
with which we coexist; finally, twentieth-century physics has
also discovered a kind of fossil, residual black-body radiation,
which tells us about the beginnings of the universe. Today we
know that we live in a world where different interlocked times
and the fossils of many pasts coexist.
We must now proceed to another question. We have said
that life is starting to seem as "natural as a falling body. " What
has the natural process of self-organization to do with a falling
body? What possible link can there be between dynamics, the
science of force and trajectories, and the science of complex
ity and becoming, the science of living processes and of the
natural evolution of which they are part? At the end of the
nineteenth century, irreversibility was associated with the phe
nomena of friction, viscosity, and heating. Irreversibility lay at
the origin of energy losses and waste. At that time it was still
possible to subscribe to the fiction that irreversibility was only
a result of our ineptitude, of our unsophisticated machines,
and that nature remained fundamentally reversible. Now it is
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no longer possible: today even physics tells us that irreversible
processes play a constructive and indispensable role.
So we come to a question that can be avoided no longer.
What is the relation between this new science of complexity
and the science of simple, elementary behavior? What is the
relation between these two opposing views of nature ? Are
there two sciences, two truths for a single world? How is that
possible?
In a certain sense, we have come back to the beginning of
modern science. Now, as at Newton's time , two sciences
come face to face-the science of gravitation, which describes
an atemporal nature subject to laws, and the science of fire,
chemistry. We now understand why it was impossible for the
first synthesis produced by science, the Newtonian synthesis,
to be complete ; the forces of interaction described by dynam
ics cannot explain the complex and irreversible behavior of
matter. Ignis mutat res . According to this ancient saying,
chemical structures are the creatures of fire, the results of irre
versible processes. How can we bridge the gap between being
and becoming-two concepts in conflict, yet both necessary
to reach a coherent description of this strange world in which
we live?
BOOK THREE
FROM BEING TO
BECOMING
I
I
I
I
I
I
I
I
CHAPTER VII
REDISCOVERING TIME
A Change of Emphasis
Whitehead wrote that a "clash of doctrines is not a disaster, it
is an opportunity. " t If this statement is true, few opportunities
in the history of science have been so promising: two worlds
have come face to face, the world of dynamics and the world of
thermodynamics.
Newtonian science was the outcome , the crowning syn
thesis of centuries of experimentation as well as of converging
lines of theoretical research. The same is true for thermo
dynamics. The growth of science is quite different from the
uniform unfolding of scientific disciplines, each in turn divided
into an increasing number of watertight compartments. Quite
the contrary, the convergence of different problems and points
of view may break open the compartments and stir up scien
tific culture. These turning points have consequences that go
beyond their scientific context and influence the intellectual
scene as a whole. Inversely, global problems often have been
sources of inspiration to science.
The clash of doctrines, the conflict between being and be
coming, indicates that a new turning point has been reached,
that a new synthesis is needed. Such a synthesis is taking
shape today, every bit as unexpected as the preceding ones.
We again find a remarkable convergence of research, all of
which contributes to identifying the difficulties inherent in the
Newtonian concept of a scientific theory.
The ambition of Newtonian science was to present a vision
of nature that would be universal, deterministic, and objective
inasmuch as it contains no reference to the observer, complete
inasmuch as it attains a level of description that escapes the
clutches of time.
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214
We have reached the core of the problem. "What is time?''
Must we accept the opposition , traditional since Kant, be
tween the static time of classical physics and the existential
time we experience in our lives? According to Carnap:
Once Einstein said that the problem of the Now wor
ried him seriously. He explained that the experience of
the Now means something special for man, something
essentially different from the past and the future, but
that this important difference does not and cannot occur
within physics. That this experience cannot be grasped
by science seemed to him a matter of painful but inevita
ble resignation. I remarked that all that occurs objec
tively can be described in science; on the one hand the
temporal sequence of events is described in physics ; and,
on the other hand, the peculiarities of man's experiences
with respect to time, including his different attitude to
wards past, present and future, can be described and (in
principle) explained in psychology. But Einstein thought
that these scientific descriptions cannot possibly satisfy
our human needs; that there is something essential about
the Now which is just outside of the realm of science.2
It is interesting to note that Bergson, in a sense following an
opposite road, also reached a dualistic conclusion (see Chap
ter III). Like Einstein, Bergson started with a subjective time
and then moved to time in nature, time as objectified by phys
ics. However, for him this objectivization led to a debasement
of time. Internal existential time has qualitative features that
are lost in the process. It is for this reason that Bergson intro
duced the distinction between physical time and duration, a
concept referring to existential time.
But we cannot stop here. As J. T. Fraser says, "The result
ing dichotomy between time felt and time understood is a hall
mark of scientific-industrial civilization, a sort of collective
schizophrenia. "3 As we have already emphasized, where clas
sical science used to emphasize permanence , we now find
change and evolution ; we no longer see in the skies the trajec
tories that filled Kant's heart with the same admiration as the
moral law residing in him. We now see strange objects: qua
sars,· pulsars, galaxies exploding and being torn apart, stars
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REDISCOVERING TIME
that, we are told, collapse into "black holes" irreversibly de
vouring everything they manage to ensnare.
Time has penetrated not only biology, geology, and the so
cial sciences but also the two levels from which it has been
traditionally excluded, the microscopic and the cosmic. Not
only life, but also the universe as a whole has a history; this
has profound implications.
The first theoretical paper dealing with a cosmological
model from the point of view of general relativity was pub
lished by Einstein in 1 917. It presented a static, timeless view
of the universe, Spinoza's vision translated into physics. But
then comes the unexpected. It became immediately evident
that there were other, time-dependent solutions to Einstein's cos
mological equations. We owe this discovery to the Russian astro
physicist A. Friedmann and the Belgian G. Lemaitre. At the
same time Hubble and his coworkers were studying the mo
tions of galaxies, and they demonstrated that the velocity of
distant galaxies is proportional to their distance from earth.
The relation with the expanding universe discovered by Fried
mann and Lemaitre was obvious. Yet for many years physi
cists remained reluctant to accept such an "historical"
description of cosmic evolution. Einstein himself was wary of
it. Lemaitre often said that when he tried to discuss with Ein
stein the possibility of making the initial state of the universe
more precise and perhaps finding there the explanation of cos
mic rays, Einstein showed no interest.
Today there is new evidence, the famous residual black
body radiation, the light that illuminated the explosion of the
hyperdense fireball with which our universe began. The whole
story appears as another irony of history. In a sense, Einstein
has, against his will, become the Darwin of physics. Darwin
taught us that man is embedded in biological evolution; Ein
stein has taught us that we are embedded in an evolving universe.
Einstein's ideas led him to a new continent, as unexpected to
him as was America to Columbus. Einstein, like many physi
cists of his generation, was guided by a deep conviction that
there was a fundamental, simple level in nature. Yet today this
level is becoming less and less accessible to experiment. The
only objects whose behavior is truly "simple" exist in our own
world, at the macroscopic level. Classical science carefully
chose its objects from this intermediate range. The first ob-
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216
jects singled out by Newton-falling bodies, the pendulum,
planetary motion-were simple. We know now, however, that
this simplicity is not the hallmark of the fundamental: it cannot
be attributed to the rest of the world.
Does this suffice? We now know that stability and simplicity
are exceptions. Should we merely disregard the totalizing to
talitarian claims of a conceptualization that, in fact, applies
only to simple and stable objects? Why worry about the in
compatibility between dynamics and thermodynamics?
We must not forget the words of Whitehead, words con
stantly confirmed by the history of science: a clash of doc
trines is an opportunity, not a disaster. It has often been
suggested that we simply ignore certain issues for practical
reasons on the grounds that they are based on idealizations
that are difficult to implement. At the beginning of this cen
tury, several physicists suggested abandoning determinism on
the grounds that it was inaccessible in real experience. 4 In
deed, as we have already emphasized, we never know the ex
act positions and velocities of the molecules in a large system;
thus an exact prediction of the system's future evolution is im
possible. More recently, Brillouin hoped to destroy determi
nism by appealing to the commonsense truth that accurate
prediction requires an accurate knowledge of the initial condi
tions and that this knowledge must be paid for; the exact pre
diction necessary to make determinism work requires that an
"infinite" price be paid.
These objections, while reasonable, do not affect the con
ceptual world of dynamics. They shed no new light on reality.
Moreover, the improvements in technology could bring us
closer and closer to the idealization implied by classical dy
namics.
In contrast, demonstrations of "impossibility" have a fun
damental importance. They imply the discovery of an unex
pected intrinsic structure of reality that dooms an intellectual
enterprise to failure. Such discoveries will exclude the pos
sibility of an operation that previously could have been imag
ined as feasible, at least in principle. "No engine can have an
efficiency greater than one , " " no heat engine can produce
useful work unless it is in contact with two sources" are exam
ples of statements of impossibility which have led to profound
conceptual innovations.
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.REDISCOVERING TIME
Thermodynamics, relativity, and quantum mechanics are all
rooted in the discovery of impossibilities, of limits to the ambi
tions of classical physics. Thus they marked the end of an ex
ploration that had reached its limits. But we can now see these
scientific innovations in a different light, not as an end but a
beginning, as the opening up of new opportunities. We shall
see in Chapter IX that the second law of thermodynamics ex
presses an "impossibility," even on the microscopic level, but
even there the newly discovered impossibility becomes a start·
ing point for the emergence of new concepts.
The End of Universality
Scientific description must be consistent with the resources
available to an observer who belongs to the world he describes
and cannot refer to some being who contemplates the physical
world "from the outside. " This is one of the fundamental re
quirements of relativity theory. In connection with the prop
agation of signals a limit appears that cannot be transgressed
by any observer. Indeed, c, the velocity of light in vacuum
(c=300,000 km/sec), is the limiting velocity for the propaga
mental role. It limits the region in space that may influence the
point where an observer is located.
There is no universal constant in Newtonian physics. This is
the reason for its claim to universality, why it can be applied in
the same way whatever the scale of the objects: the motion of
atoms, planets, and stars are governed by a single law.
The discovery of universal constants signified a radical
change. Using the velocity of light as the comparison stan
dard, physics has established a distinction between low and
high velocities, those approaching the speed of light.
Likewise, Planck's constant, h; sets up a natural scale ac
cording to the object's mass. The atom can no longer be re
garded as a tiny planetary system. Electrons belong to a
different scale than planets and all other heavy, slow-moving,
macroscopic objects, including ourselves.
Universal constants not only destroy the homogeneity of the
universe by introducing physical scales in terms of which varition of all signals. Thus this limiting velocity plays a funda
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218
ous behaviors become qualitatively different, they also lead to
a new conception of objectivity. No observer can transmit sig
nals at a velocity higher than that of light in a vacuum. Hence
Einstein's remarkable conclusion: we can no longer define the
absolute simultaneity of two distant events; simultaneity can
be defined only in terms of a given reference frame. The scope
of this book does not permit an extensive account of relativity
theory. Let us merely point out that Newton's laws did not
assume that the observer was a "physical being." Objective
description was defined precisely as the absence of any refer
ence to its author. For "nonphysical" intelligent beings capa
ble of communicating at an infinite velocity, the laws of
relativity would be irrelevant. The fact that relativity is based
on a constraint that applies only to physically localized ob
servers, to beings who can be in only one place at a time and
not everywhere at once, gives this physics a "human" quality.
This does not mean, however, that it is a "subjective" physics,
the result of our preferences and convictions; it remains sub
ject to intrinsic constraints that identify us as part of the physi
cal world we are describing. It is a physics that presupposes an
observer situated within the observed world . .Our dialogue
with nature will be successful only if it is carried on from
within nature.
The Rise of Quantum Mechanics
Relativity altered the classical concept of objectivity. How
ever, it left unchanged another fundamental characteristic of
classical physics, namely, the ambition to achieve a " com
plete" description of nature. After relativity, physicists could
no longer appeal to a demon who observed the entire universe
from outside, but they could still conceive of a supreme math
ematician who, as Einstein claimed, neither cheats nor plays
dice. This mathematician would possess the formula of the
universe, which would include a complete description of na
ture. In this sense, relativity remains a continuation of classi
cal physics.
Quantum mechanics, on the other hand, is the first physical
theory truly to have broken with the past. Quantum mechan
ics not only situates us in nature, it also labels us as "heavy"
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REDISCOVERING TIME
beings composed of a macroscopic number of atoms. In order
to visualize more clearly the consequences of the velocity of
light as a universal constant, Einstein imagined himself riding
a photon. But quantum mechanics discovered that we are too
heavy to ride photons or electrons. We cannot possibly replace
such airy beings, identify ourselves with them, and describe
what they would think, if they were able to think, and what
they would experience, if they were able to feel anything.
The history of quantum mechanics, like that of all con
ceptual innovations, is complex, full of unexpected events; it is
the history of a logic whose implications were discovered long
after it was conceived in the urgency of experiment and in a
difficult political and cultural environment.5 This history can
not be related here; we only wish to emphasize its role in the
construction of the bridge from being to becoming, which is
our main subject.
The birth of quantum mechanics was in itself part of the
quest for this bridge. Planck was interested in the interaction
between matter and radiation. Underlying his work was the
ambition to accomplish for the matter-light interaction what
Boltzmann had achieved for the matter-matter interaction,
namely, to discover a kinetic model for irreversible processes
leading to equilibrium. 6 To his surprise, he was forced, in or
der to reach experimental results valid at thermal equilibrium,
to assume that an exchange of energy between matter and ra
diation occurred only in discrete steps involving a new univer
sal constant. This universal constant "h" measures the "size"
of each step.
In this case, as in many others, the challenge of irreversibil
ity led to decisive progress in physics.
This discovery remained isolated until Einstein presented
the first general interpretation of Planck's constant. He un
derstood that it had far-reaching implications for the nature of
light. He introduced a revolutionary concept: the wave
particle duality of light.
Since the beginning of the nineteenth century, light had
been associated with wave properties manifest in phenomena
such as diffraction or interference. However, at the end of the
nineteenth century, new phenomena were discovered, notably
the photoelectric effect-that is, the expulsion of electrons as
the result of the absorption of light. These new experimental
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220
results were difficult to explain in terms of the traditional wave
properties of light. Einstein solved the riddle by assuming that
light may be both wave and particle and that these two aspects
are related through Planck's constant. More precisely, a light
wave is characterized by its frequency u and its wavelength X;
h permits us to go from frequency and wavelength to mechani
cal quantities such as energy e and momentum p. The relations
between u and A on the one side and e and p on the other are
Very simple: B = hu, p = h/X, and both involve h . 1\.venty years
later, Louis de Broglie extended this wave-particle duality
from light to matter; thus the starting point for the modern
formulation of quantum mechanics.
In 1913 Niels Bohr had linked the new quantum physics to
the structure of atoms (and later of molecules). As a result of
the wave-particle duality, he showed that there exist discrete
sequences of electron orbits. When an atom is excited, the
electron jumps from one orbit to another. At this very instant
the atom emits or absorbs a photon the frequency of which
corresponds to the difference between the energies charac
terizing the electron's motion in each of the two orbits. This
difference is calculated in terms of Einstein's formula relating
energy to frequency.
Thus we reach the decisive years 1925-27, a "golden age" of
physics.7 During this short period, Heisenberg, Born, Jordan,
Schrodinger, and Dirac made quantum physics into a consis
tent new theory. This theory incorporates Einstein's and de
Broglie's wave-particle duality in the framework of a new gen
eralized form of dynamics: quantum mechanics. For our pur
poses here, the conceptual novelty of quantum mechanics is
essential.
First and foremost, a new formulation, unknown in classical
physics, had to be introduced to allow "quantitization" to be
incorporated into the theoretical language. The essential fact
is that an atom can be found only in discrete energy levels
corresponding to the various electron orbits. In particular, this
means that energy (or the Hamiltonian) can no longer be
merely a function of the position and the moment, as it is in
classical mechanics. Otherwise, by giving the positions and
moments slightly different values, energy could be made to
vary continuously. But as observation reveals, only discrete
levels exist.
22'1
REDISCOVERING TIME
We therefore have to replace the conventional idea that the
Hamiltonian is a function of position and momenta with some
thing new; the basic idea of quantum mechanics is that the
Hamiltonian as well as the other quantities of classical me
chanics , such as coordinates q or momenta p, now become
operators. This is one of the boldest ideas ever introduced in
science, and we would like to discuss it in detail.
It is a simple idea, even if at first it seems somewhat ab
stract. We have to distinguish the operator-a mathematical
operation-and the object on which it operates-a function.
As an example, take as the mathematical "operator" the de
rivative represented by d/dx and suppose it acts on a func
tion-say, x2; the result of this operation is a new function, this
time "2x." However, certain functions behave in a peculiar
way with respect to derivation. For example, the derivative of
" e3x " is "3e3x": here we return to the original function simply
multiplied by some n umber-here , 3. Functions that are
merely recovered by a given operator to them are known as the
"eigenfunctions" of this operator, and the numbers by which
the eigenfunction is multiplied after the application of the op
erator are the "eigenvalues" of the operator.
To each operator there thus corresponds an ensemble, a "res
ervoir" of numerical values; this ensemble forms its "spec
trum. " This spectrum is "discrete" when the eigenvalues form
a discrete series. There exists, for instance, an operator with
all the integers 0, 1 , 2 . . as eigenvalues. A spectrum may
also be continuous-for example, when it consists of all the
numbers between 0 and 1 .
The basic concept o f quantum mechanics may thus be ex
pressed as follows: to all physical quantities in classical me
chanics there corresponds in quantum mechanics an operator,
and the numerical values that may be taken by this physical
quantity are the eigenvalues of this operator. The e ssential
point is that the concept of physical quantity (represented by
an operator) i s now distinct from that of its numerical values
(represented by the eigenvalues of the operator). In particular,
energy will now be represented by the Hamiltonian operator,
and the energy levels-the observed values of the energy
will be identified with the eigenvalues corresponding to this
.
operator.
The introduction of operators opened up to physics a micro-
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222
scopic world of unsuspected richness, and we regret that we
cannot devote more space to this fascinating subject, in which
creative imagination and experimental observation are so suc
cessfully combined. Here we wish merely to stress that the
microscopic world is governed by laws having a new structure,
thereby putting an end once and for all to the hope of discover
ing a single conceptual scheme common to all levels of de
scription.
A new mathematical language invented to deal with a cer
tain situation may actually open up fields of inquiry that are
full of surprises, going far beyond the expectations of its orig
inators. This was true for differential calculus, which lies at
the root of the formulation of classical dynamics. It is true as
well for operator calculus. Quantum theory, initiated as de
manded by the result of unexpected experimental discoveries,
was quick to reveal itself as pregnant with new content.
Today, more than fifty years after the introduction of opera
tors into quantum mechanics, their significance remains a sub
ject of lively discussion. From the historical point of view, the
introduction of operators is linked to the existence of energy
levels, but today operators have applications even in classical
physics. This implies that their significance has been extended
beyond the expectations of the founders of quantum mechan
ics . Operators now come into play as soon as, for one reason
or another, the notion of a dynamic trajectory has to be dis
carded, and with it, the deterministic description a trajectory
implies.
Heisenbergs Uncertainty Relation
We have seen that in quantum mechanics to each physical
quantity corresponds an operator that acts on functions. Of
special importance are the eigenfunctions and the eigenvalues
corresponding to the operator under consideration. The eigen
values correspond precisely to the numerical values the physi
cal quantity can now take. Let us take a closer look at the
operators quantum mechanics associates with coordinates q
and momenta p ; their coordinates are, as we have seen in
Chapter II, the canonical variables.
In classical mechanics coordinates and momenta are inde-
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REDISCOVERING TIME
pendent in the sense that we can ascribe to a coordinate a
numerical value quite independent of the value we have as
cribed to the momentum. However, the existence of Planck's
constant h implies the reduction in the number of independent
variables. We could have guessed this right away from the
Einstein-de Broglie relation A= hlp, which, as we have seen,
connects wavelength to momentum. Planck's constant h ex
presses a relation between lengths (closely related to the con
cept of coordinates) and momenta. Therefore, positions and
momenta can no longer be independent variables, as in classi
cal mechanics. The operators corresponding to positions and
momenta can be expressed in terms of the coordinate alone or
in terms of the momentum, something explained in all text
books dealing with quantum mechanics.
The important point is that in all cases, only one type of
quantity appears (either coordinate or momentum), but not
both. In this sense we may say that the quantum mechanics
divides the number of classical mechanical variables by a fac
tor of two.
One fundamental property results from the relation between
operators in quantum mechanics: the two operators q0P and
Pop do not commute-that is, the results of q0pP0p and of
Pop%p applied to the same function are different. This has pro
found implications, since only commuting operators admit
common eigenfunctions. Thus we cannot identify a function
that would be an eigenfunction of both coordinate and momen
tum. As a consequence of the definition of the coordinate and
momentum operators in quantum mechanics, there can be no
state in which the physical quantities, coordinate q and mo
mentum p, both have a well-defined value. This situation, un
known in classical mechanics, is expressed by Heisenberg's
famous uncertainty relations. We can measure a coordinate
and a momentum, but the dispersions of the respective possi
ble predictions as expressed by f::j,q,f::j,p are related by the
Heisenberg inequality f::j,qf::j,p;;::.h. We can make f::j,q as small as
we want, but then f::j,p goes to infinity, and vice versa.
Much has been written about Heisenberg's uncertainty rela
tions, and our discussion is admittedly oversimplified. But we
wish to give our readers some understanding of the new prob
lem that re:sult:s from the u:se of operators; Heisenberg's uncer
tainty relation necessarily leads to a revision of the concept of
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224
causality. It is possible to determine the coordinate precisely.
But the moment we do so, the momentum will acquire an ar
bitrary value, positive or negative. In other words, in an in
stant the position of the object will become arbitrarily distant.
The meaning of localization becomes blurred: the concepts
that form the basis of classical mechanics are profoundly al
tered.
These consequences of quantum mechanics were unac
ceptable to many physicists, including Einstein; and many ex
periments were devised to demonstrate their absurdity. An
attempt was also made to minimize the conceptual change in
volved. In particular, it was suggested that the foundation of
quantum mechanics is in some way related to perturbations
resulting from the process of observation. A system was
thought to possess intrinsically well-defined mechanical pa
rameters such as coordinates and momenta; but some of them
would be made fuzzy by measurement, and Heisenberg's un
certainty relation would only express the perturbation created
by the measurement process. Classical realism thus would re
main intact on the fundamental level, and we would simply
have to add a positivistic qualification. This interpretation
seems too narrow. It is not the quantum measurement process
that disturbs the results. Far from it: Planck's constant forces
us to revise our concepts of coordinates and momenta. This
conclusion has been confirmed by recent experiments de
signed to test the assumption of local hidden variables that
were introduced to restore classical determinism. s The results
of those experiments confirm the striking consequences of
quantum mechanics.
That quantum mechanics obliges us to speak less absolutely
about the localization of an object implies, as Niels Bohr often
emphasized, that we must give up the realism of classical
physics. For Bohr, Planck's constant defines the interaction
between a quantum system and the measurement device as
nondecomposable. It is only to the quantum phenomenon as a
whole, including the measurement interaction, that we can as
cribe numerical values. All description thus implies a choice
of the measurement device, a choice of the question asked. In
this sense, the answer, the result of the measurement, does not
give us a ccess to a given reality. We have to decide which mea
surement we are going to perform and which question our ex-
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REDISCOVERING TIME
periments will ask the system. Thus there is an irreducible
multiplicity of representations for a system, each connected
with a determined set of operators.
This implies a departure from the classical notion of ob
jectivity, since in the classical view the only "objective" de
scription is the complete description of the system as it is,
independent of the choice of how it is observed.
Bohr always emphasized the novelty of the positive choice
introduced through measurement. The physicist has to choose
his language, to choose the macroscopic experimental device.
Bohr expressed this idea through the principle of complemen
tarity,9 which may be considered as an extension of Heisen
berg's uncertainty relations. We can measure coordinates or
momenta, but not both. No single theoretical language artic
ulating the variables to which a well-defined value can be at
tributed can exhaust the physical con�ent of a system. Various
possible languages and points of view about the system may
be complementary. They all deal with the same reality, but it is
impossible to reduce them to one single description. The irre
ducible plurality of perspectives on the same reality expresses
the impossibility of a divine point of view from which the
whole of reality is visible. However, the lesson of the principle
of complementarity is not a lesson in resignation. Bohr used to
say that the significance of quantum mechanics always made
him dizzy, and we do indeed feel dizzy when we are torn from
the comfortable routine of common sense.
The real lesson to be learned from the principle of comple
mentarity, a lesson that can perhaps be transferred to other
fields of knowledge, consists in emphasizing the wealth of re
ality, which overflows any single language, any single logical
structure. Each language can express only part of reality. Mu
sic, for example, has not been exhausted by any of its realiza
tions, by any style of composition, from Bach to Schonberg.
We have emphasized the importance of operators because
they demonstrate that the reality studied by physics is also a
mental construct; it is not merely given. We must distinguish
between the abstract notion of a coordinate or of momentum,
represented mathematically by operators, and their numerical
realization, which can be reached through experiments. One
of the reasons for the opposition between the ··two cultures"
may have been the belief that literature corresponds to a con-
ORDER OUT OF CHAOS
226
ceptualization of reality, to "fiction," while science seems to
express objective "reality. " Quantum mechanics teaches us
that the situation is not so simple. On all levels reality implies
an essential element of conceptualization.
The Temporal Evolution of Quantum Systems
We shall now move on to discuss the temporal evolution of
quantum systems. As in classical mechanics, the Hamiltonian
plays a fundamental role. As we have seen, in quantum me
chanics it is replaced by the Hamiltonian operator Hop· This
energy operator plays a central role: on the one hand , its
eigenvalues correspond to the energy levels; on the other
hand, as in classical mechanics, the Hamiltonian operator de
termines the temporal evolution of the system. In quantum
mechanics the role played by the canonical equation of classi
cal mechanics is taken by the Schrodinger equation, which
expresses the time evolution of the function characterizing the
quantum state as the result of the application of the operator
Hop on the wave function \jJ (there are, of course, other for
mulations, which we cannot describe here). The term "wave
function" has been chosen to emphasize once again the wave
particle duality so fundamental in all of quantum physics. \jJ is
a wave amplitude that evolves according to a particle type of
equation determined by the Hamiltonian. Schrodinger's equa
tion, like the canonical equation of classical physics , ex
presses a reversible and deterministic evo lution. The
reversible change of wave function corresponds to a reversible
motion along a trajectory. If the wave function at a given in
stant is known, Schrodinger's equation allows it to be calculated
for any previous or subsequent instant. From this viewpoint,
the situation is strictly similar to that in classical mechanics.
This is because the uncertainty relations of quantum mechan
ics do not include time. Time remains a number, not an opera
tor, and only operators can appear in Heisenberg's uncertainty
relations.
Quantum mechanics deals with only half of the variables of
dassical mechanics. As a result, classical determinism be
comes inapplicable, and in quantum physics statistical consid-
227
REDISCOVERING TIME
erations play a central role. It is through the wave intensity ttJ2
(the square of the amplitude) that we make contact with statis
tical considerations.
The standard statistical interpretation of quantum mechan
ics runs as follows: consider the eigenfunctions of some opera
tor-say, the energy operator H0P-and the corresponding
eigenvalues. In general the wave function tts will not be the
eigenfunction of the energy operator, but it can be expressed
as the superposition of these eigenfunctions . The respective
importance of each eigenfunction in this superposition allows
us to calculate the probability for the appearance of the vari
ous possible corresponding eigenvalues.
Here again we notice a fundamental departure from classi
cal theory. Only probabilities can be predicted , not single
events. This was the second time in the history of science that
probabilities were used to explain some basic features of na
ture. The first time was in Boltzmann's interpretation of en
tropy. There , however, a subjective point of view remained
possible; in this view, "only" our ignorance in the face of the
complexity of the systems considered prevented us from achiev
ing a complete description. (We shall see that today it is possible
to overcome this attitude.) Here, as before, the use of proba
bilities was unacceptable to many physicists-including Ein
stein-who wished to achieve a "complete" deterministic
description. Just as with irreversibility, an appeal to our igno
rance seemed to offer a way out: our inaptitude would make us
responsible for statistical behavior in the quantum world, just
as it makes us responsible for irreversibility.
Once again we come to the problem of hidden variables.
However, as we have said , there has been no experimental evi
dence to justify the introduction of such variables, and the rol�
of probabilities seems irreducible.
There is only one case in which the Schrodinger equation
leads to a deterministic prediction: that is when tlJ, instead of
being a superposition of eigenfunctions, is reduced to a single
one. In particular, in an ideal measurement process, a system
may be prepared in such a way that the result of a given mea
surement may be predicted. We then know that the system is
described by the corresponding eigenfunction. From then on,
the system may be described with certainty as being in the
eigenstate indicated by the measurement result.
ORDER OUT OF CHAOS
228
The measurement process in quantum mechanics has a spe·
cial significance that is attracting considerable interest today.
Suppose we start with a wave function, which is indeed a su
perposition of eigenfunctions. As a result of the measurement
process, this single collection of systems all represented by
the same wave function is replaced by a collection of wave
functions corresponding to the various eigenvalues that may
be measured. Stated technically, a measurement leads from a
single wave function (a "pure" state) to a mixture .
As Bohr and Rosenfeld IO repeatedly pointed out, every
measurement contains an element of irreversibility, an appeal
made to irreversible phenomena, such as chemical processes
corresponding to the recording of the "data. " Recording is ac
companied by an amplification whereby a microscopic event
produces an effect on a macroscopic level-that is, a level at
which we can read the measuring instruments. The measure
ment thus presupposes irreversibility.
This was in a sense already true in classical physics. How·
ever, the problem of the irreversible character of measurement
is more urgent in quantum mechanics because it raises ques
tions at the level of its formulation.
The usual approach to this problem states that quantum me
chanics has no choice but to postulate the coexistence of two
mutually irreducible processes. the reversible and continuous
evolution described by Schrodinger's equation and the irre
versible and discontinuous reduction of the wave function to
one of its eigenfunctions at the time of measurement. Thus the
paradox : the reversible Schrodinger equation can be tested
only by irreversible measurements that the equation is by defi
nition unable to describe. It is thus impossible for quantum
mechanics to set up a closed structure.
In the face of these difficulties, some physicists have once
more taken refuge in subjectivism, stating that we-our mea
surement and even, for some, our mind-determine the evolu
tion of the system that breaks the law of natural, "objective"
reversibility. 1 1 Others have concluded that Schrodinger's equa
tion was not "complete" and that new terms must be added to
account for the irreversibility of the measurement. Other more
improbable " solutions" have also been proposed, such as
Everett's many-world hypothesis (see d'Espagnat, ref. 8). For
us, however, the coexistence in quantum mechanics of revers-
229
REDISCOVERING TIME
ibility and irreversibility shows that the classical idealization
that describes the dynamic world as self-contained is impossi
ble at the microscopic level. This is what Bohr meant when he
noted that the language we use to describe a quantum system
cannot be separated from the macroscopic concepts that de
scribe the functioning of our measurement instruments. Schro
dinger's equation does not describe a separate level of reality;
rather it presupposes the macroscopic world to which we belong.
The problem of measurement in quantum mechanics is thus
an aspect of one of the problems to which this book is de
voted-the connection between the simple world described by
Hamiltonian trajectories and Schrodinger's equation, and the
complex macroscopic world of irreversible processes.
In Chapter IX, we shall see that irreversibility enters classi
cal physics when the idealization involved in the concept of a
trajectory becomes inadequate. The measurement problem in
quantum mechanics is susceptible to the same type of solu
tion.12 Indeed, the wave function represents the maximum
knowledge of a quantum system. As in classical physics, the
object of this maximum knowledge satisfies a reversible evolu
tion equation. In both cases, irreversibility enters when the
ideal object corresponding to maximum knowledge has to be
replaced by less idealized concepts. But when does this hap
pen? This is the question of the physical mechanisms of irre
versibility to which we shall turn in Chapter IX. But let us first
summarize some other features of the renewal of contempo
rary science.
·
A Nonequilibrium Universe
The two scientific revolutions described in this chapter started
as attempts to incorporate universal constants, c and h, into
the framework of classical mechanics. This led to far-reaching
consequences, some of which we have described here. From
other perspectives, relativity and quantum mechanics seemed
to adhere to the basic world view expressed in Newtonian me
chanics. This is especially true regarding the role and meaning
of time. In quantum mechanics, once the wave function at time
ORDER OUT OF CHAOS
230
zero is known, its value ljJ(t) both for future and past is deter·
mined. Likewise, in relativity theory the static geometric charac·
ter of time is often emphasized by the use of four-dimensional
notation (three dimensions for space and one for time). As ex
pressed concisely by Minkowski in 1908, "space by itself and
time by itself are doomed to fade away into mere shadows, and
only a kind of union of the two will preserve an independent
reality . . . only a world in itself will subsist." 13
But over the past five decades this situation has radically
changed. Quantum mechanics has become the main tool for
dealing with elementary particles and their transformations. It
is outside the scope of this book to describe the bewildering
variety of elementary particles that have appeared during the
past few years.
We want only to recall that, using both quantum mechanics
and relativity, Dirac demonstrated that we have to associate to
each particle of mass m and charge e an antiparticle of the
same mass but of opposite charge. Positrons, the antiparticles
of electrons, as well as antiprotons, are currently being pro
duced in high-energy accelerators. Antimatter has become a
common subject of study in particle physics. Particles and
their corresponding antiparticles annihilate each other when they
collide, producing photons, massless particles corresponding
to light. The equations of quantum theory are symmetric in
respect to the exchange particle-antiparticle , or more pre
cisely, they are symmetric in respect to a weaker requirement
known as the CPT symmetry. In spite of this symmetry, there
exists a remarkable dissymmetry between particles and anti
particles in the world around us. We are made of particles
(electrons, protons), while antiparticles remain rare laboratory
products. If particles and antiparticles coexisted in equal
amount, all matter would be annihilated. There is strong evi
dence that antimatter does not exist in our galaxy, but the pos
sibility that it exists in distant galaxies cannot be excluded. We
can imagine a mechanism in the universe that separates parti
cles and antiparticles, hides antiparticles somewhere. How
ever, it seems more likely that we live in a "nonsymmetrical"
universe where matter completely dominates antimatter.
How is this possible? A model explaining the situation was
presented by Sa kharov in 1 966, a nd today much work is being
done along these lines. 14 One essential element of the model is
231
REDISCOVERING TIME
that, at the time of the formation of matter, the universe had to
be in n{)nequilibrium conditions, for at equilibrium the law of
mass action discussed in Chapter V would have required equal
amounts of matter and antimatter.
What we want to emphasize here is that nonequilibrium has
now acquired a new, cosmological dimension. Without non
equilibrium and without the irreversible processes linked to it,
the universe would have a completely different structure.
There would be no appreciable amount of matter, only some
fluctuating local excesses of matter over antimatter, or vice
versa.
From a mechanistic theory that was modified to account for
the existence of the universal constant h, quantum theory has
evolved into a theory of mutual transformations of elementary
particles. In recent attempts to formulate a "unified theory of
elementary particles" it has even been suggested that all parti
cles of matter, including the proton, are unstable (however, the
lifetime of the proton would be enormous, of the order of 1 Q30
years). Mechanics, the science of motion, instead of corre
sponding to the fundamental level of description, becomes a
mere approximation, useful only because of the long lifetime
of elementary particles such as protons.
Relativity theory has gone through the same transforma
tions. As we mentioned, it started as a geometric theory that
strongly emphasized timeless features. Today it is the main
tool for investigating the thermal history of the universe, for
providing clues to the mechanisms that led to the present
structure of the universe. The problem of time, of irreversibil
ity, has therefore acquired a new urgency. From the field of
engineering, of applied chemistry, where it was first formu
lated, it has spread to the whole of physics, from elementary
particles to cosmology.
From the perspective of this book, the importance of quan
tum mechanics lies in its introduction of probability into micro
scopic physics. This should not be confused with the stochastic
processes that describe chemical reactions as discussed in
Chapter V. In quantum mechanics, the wave function evolves
in a deterministic fashion, except in the measurement process.
We have seen that in the fifty years since the formulation of
quantum mechanics the study of nonequilibrium processes
has revealed that fluctuations, stochastic elements, are impor-
ORDER OUT OF CHAOS
232
tant even on the microscopic scale. We have repeatedly stated
in this book that the reconceptualization of physics going on
today leads from deterministic, reversible processes to stochas
tic and irreversible ones. We believe that quantum mechanics
occupies a kind of intermediate position in this process. There
probability appears, but not irreversibility. We expect, and we
shall give some reasons for this in Chapter IX, that the next
step will be the introduction of fundamental irreversibility on
the microscopic level. In contrast with the attempts to restore
classical orthodoxy through hidden variables or other means ,
we shall argue that i t i s necessary to move even farther away
from deterministic descriptions of nature and adopt a statisti
cal, stochastic description.
CHAPTER VIII
THE CLASH OF
DOCTRINES
Probability and Irreversibility
We shall see that nearly everywhere the physicist has
purged from his science the use of one-way time, as
though aware that this idea introduces an antrlropo
morphlc element alien to the ideals of physics. Never
theless, in several important cases unidirectional time
and unidirectional causality have been invoked, but al
ways, as we shall proceed to show. in support of some
false doctrine.
G. N. LEWIS'
The law that entropy always increases-the second
law of thermodynamics-holds, I think, the supreme
position among the laws of Nature. If someone points
out to you that your pet theory of the universe is in
d isagreement with Maxwe l l 's equations-then so
much the worse for Maxwells equations. If it is found to
be contradicted by observation-well, these expe
rimentalists do bungle things sometimes. But if your
theory is found to be against the second law of ther
modynamics I can give you no hope; there is nothing
for it but to collapse in deepest humiliation.
A S.
EDDINGTON2
With Clausius' formulation of the second law of thermodynam·
ics, the conflict between thermodynamics and dynamics be·
came obvious. There is hardly a single question in physi�s that
has been more often and more actively discussed than the rela·
233
ORDER OUT OF CHAOS
234
tion between thermodynamics and dynamics. Even now, a
hundred and fifty years after Clausius , the question still
arouses strong feelings. No one can remain neutral in this con
flict, which involves the meaning of reality and time. Must
dynamics , the mother of modern science , be abandoned in
favor of some form of thermodynamics? That was the view of
the "energeticists ," who exerted great influence during the
nineteenth century. Is there a way to "save" dynamics, to re
coup the second law without giving up the formidable struc
ture built by Newton and his successors? What role can
entropy play in a world described by dynamics?
We have already mentioned the answer proposed by Boltz
mann. Boltzmann's famous equation S = k log P relates en
tropy and probability: entropy grows because probability grows.
Let us immediately emphasize that in this perspective the sec
ond law would have great practical importance but would be of
no fundamental significance. In his excellent book The Ambi
dextrous Universe, Martin Gardner writes: "Certain events go
only one way not because they can't go the other way but be
cause it is extremely unlikely that they go backward. " 3 By im
proving our abilities to measure less and less unlikely events,
we could reach a situation in which the second law would play
as small a role as we want. This is the point of view that is
often taken today. However, this was not Planck's point of
view:
It would be absurd to assume that the validity of the sec
ond law depends in any way on the skill of the physicist
or chemist in observing or experimenting. The gist of the
second law has nothing to do with experiment; the law
asserts briefly that there exists in nature a quantity which
changes always in the same sense in all natural pro
cesses. The proposition stated in this general form may
be correct or incorrect; but whichever it may be, it will
remain so, irrespective of whether thinking and measur
ing beings exist on the earth or not, and whether or not,
assuming they do exist, they are able to measure the de
tails of physical or chemical processes more accurately
by one, two, or a hundred decimal places than we can.
The limitation to the law, if any, must lie in the same
province as its essential idea, in the observed Nature, and
THE CLASH OF DOCTRINES
235
not in the Observer. That man's experience is called upon
in the deduction of the law is of no consequence ; for that
is, in fact, our only way of arriving at a knowledge of
natural law. 4
However, Planck's views remained isolated . As we noted,
most scientists considered the second law the result of approx
imations, the intrusion of subjective views into the exact world
of physics. For example, in a celebrated sentence Born stated,
"Irreversibility is the effect of the introduction of ignorance
into the basic laws of physics. "5
In the present chapter we wish to describe some of the basic
steps in the development of the interpretation of the second
law. We must first understand why this problem appeared to
be so difficult. In Chapter IX we shall go on to present a new
approach that, we hope, will clearly express both the radical
originality and the objective meaning of the second law. Our
conclusion will agree with Planck's view. We shall show that,
far from destroying the formidable structure of dynamics, the
second law adds an essential new element to it.
First we wish to clarify Boltzmann's association of proba
bility and entropy. We shall begin by describing the " urn
model" proposed by P. and T. Ehrenfest. 6 Consider N objects
(for example, balls) distributed between two containers A and
B. At regular time intervals (for example, every second) a ball
tioe n
D EJ
!-tot ery
orN-k+1
time n+1
N-k-1
A
A
B
1
B
tainers A and B. At time n there are k balls in A and N- k balls in B. At regular
time intervals a ball is taken at random from A and put in B.
Figure 23. Ehrenfest's urn model. N balls are distributed between two con
ORDER OUT OF CHAOS
236
is chosen at random and moved from one container to the
other. Suppose that at time n there are k balls in A and N- k
balls in B. Then at time n + I there can be in A either k - I or
k + I balls. We have the transition probabilities kiN for k-+k - 1
and 1 - k/N for k-+k + 1 . Suppose we continue the game. We
expect that as a result of the exchanges of balls the most proba
ble distribution in Boltzmann's sense will be reached. When the
number N of balls is large, this distribution corresponds to an
equal number N/2 of balls in each urn. This can be verified by
elementary calculations or by performing the experiment.
N
k- 2
t
Figure 24. Approach to equilibrium (k = Nt2) in Ehrenfest's urn model
(schematic representation).
The Ehrenfest model is a simple example of a "Markov pro
cess" (or Markov " chain "), named after the great Russian
mathematician Markov, who was one of the first to describe
such processes (Poincare was another). In brief, their charac
teristic feature is the existence of well-defined transition prob
abilities independent of the previous history of the system.
Markov chains have a remarkable property: they can be de
scribed in terms of entropy. Let us call P(k) the probability of
finding k balls in A. We may then associate to it an "J-{ quan
tity," which has the precise properties of entropy that we dis
cussed in Chapter IV. Figure 25 gives an example of its
evolution. The Jf quantity varies uniformly with time, as does
the entropy of an isolated system. It is true that J-{ decreases
with time, while the entropy S increases, but that is a matter of
definition: J-{ plays the role of -s.
·
237
THE CLASH OF DOCTRINES
t
Figure 25. Time evolution of the J{ quantity (defined in the text) corre
sponding to the Ehrenfest model. This quantity decreases monotonously
and vanishes for long times.
The mathematical meaning of this "J-l quantity" is worth
considering in more detail: it measures the difference between
the probabilities at a given time and those that exist at the
equilibrium state (where the number of balls in each urn is
N/2). The argument used in the Ehrenfest urn model can be
generalized. Let us consider the partition of a square-that is,
we subdivide the square into a number of disjointed regions
(see Figure 26). Then we consider the distribution of particles
in the square and call P(k,t) the probability of finding a particle
in the region k. Similarly, we call Peqm(k) this quantity when
uniformity is reached. We assume that, as in the urn model,
there exist well-defined transition probabilities. The definition
of the J-l. quantity is
<k
'
J{ = � P(k, t) log
)
qm
e
k
Note the ratio P(k,t)IPeqm(k) that appears in this formula. Sup
J (k
pose there arc eight boxes and that Peqm(k)
=
1 /8 . For example,
we may start with all the particles in the first box. The corre-
ORDER OUT OF CHAOS
238
sponding values of P(k, t) would be P( l ,t) = 1 , all others zero. As
the result we find .Jl= log ( II[ 1/8]) = log 8. As time goes by, the
particles become equally distributed and P(k,t) = Peqm(k) = 1 /8.
As the result the :I{ quantity vanishes. It can be shown that, in
accordance with Figure 25 , the decrease in the value of :H pro
ceeds in a uniform fashion. (The demonstration is given in all
textbooks dealing with the theory of stochastic processes.)
This is why :H plays the role of -S, entropy. The uniform de
crease of .H has a very simple meaning: it measures the pro
gressive uniformization of the system. The initial information
is lost, and the system evolves from "order" to "disorder. "
Note that a Markov process implies fluctuations, as clearly
indicated in Figure 24. If we would wait long enough we would
recover the initial state. However, we are dealing with aver
ages. The :JiM quantity that decreases uniformly is expressed
in terms of probability distributions and not in terms of indi
vidual events. It is the probability distribution that evolves ir
reversibly (in the Ehrenfest model, the distribution function
tends uniformly to a binomial distribution). Therefore, on the
level of distribution functions, Markov chains lead to a one
wayn�ss in time.
This arrow of time marks the difference between Markov
chains and temporal evolution in quantum mechanics, where
the wave function, though related to probabilities, evolves re
versibly. It also illustrates the close relation between stochas
tic processes, such as Markov chains, and irreversibility.
However, the increasing of entropy (or decreasing of Jf) is not
based on an arrow of time present in the laws of nature but on
our decision to use present knowledge to predict future (and
not past) behavior. Gibbs states it in his usual lapidary man
ner:
But while the distinction of prior and subsequent events
may be immaterial with respect to mathematical fictions,
it is quite otherwise with respect to the events of the real
world. It should not be forgotten, when our ensembles
are chosen to illustrate the probabilities of events in the
real world , that while the probabilities of subsequent
events may often be determined from the probabilities of
prior events, it is rarely the case that probabilities of prior
239
THE CLASH OF DOCTRINES
events can be determined from those of subsequent
events, for we are rarely justified in excluding the consid
eration of the antecedent probability of the prior events. 7
It is an important point, which has led to a great deal of discus
sion. 8 Probability calculus is indeed time-oriented. The pre
diction of the future is different from retrodiction. If this was
the whole story, we would have to conclude that we are forced
to accept a subjective interpretation of irreversibility, since the
distinction between future and past would depend only on us.
In other words, in the subjective interpretation of iri;"eversibil
ity (further reinforced by the ambiguous analogy with infor
mation theory), the observer is responsible for the temporal
asymmetry characterizing the system's development. Since
the observer cannot in a single glance determine the positions
and velocities of all the particles composing a complex sys
tem, he cannot know the instantaneous state that simul
taneously contains its past and its future, nor can he grasp the
reversible law that would allow him to predict its developments
from one moment to the next. Neither can he manipulate the
system like the demon invented by Maxwell, who can separate
fast- and slow-moving particles and impose on a system an
antithermodynamic evolution toward an increasingly less uni
form temperature distribution.9
Thermodynamics remains the science of complex systems;
but, from this perspective, the only specific feature of complex
systems is that our knowledge of them is limited and that our
uncertainty increases with time. Instead of recognizing in irre
versiblity something that links nature to the observer, the sci
entist is compelled to admit that nature merely mirrors his
ignorance. Nature is silent; irreversibility, far from rooting us
in the physical world, is merely the echo of human endeavor
and of its limits.
However, one immediate objection can be raised. According
to such interpretations, thermodynamics ought to be as uni
versal as our ignorance. There should exist only irreversible
processes. This is the stumbling block for all universal inter
pretations of entropy that concentrate on our ignorance of ini
tial (or boundary) conditions. Irreversibility is not a universal
propercy. In order to link dynamics and thermodynamics, a
ORDER OUT OF CHAOS
240
physical criterion is required to distinguish between reversible
and irreversible processes.
We shall take up this question in Chapter IX. Here let us
return to the history of science and Boltzmann's pioneering
work.
Boltzn1anns Breakthrough
Boltzmann's fundamental contribution dates from 1 872, about
thirty years before the discovery of Markov chains. His ambi
tion was to derive a "mechanical" interpretation of entropy. In
other words, while in Markov chains the transition probabili
ties are given from outside as, for example, in the Ehrenfest
model, we now have to relate them to the dynamic behavior of
the system. Boltzmann was so fascinated by this problem that
he devoted most of his scientific life to it. In his Populiire
Schriften10 he wrote: "If someone asked me what name we
should give to this century, I would answer without hesitation
that this is the century of Darwin." Boltzmann was deeply at
tracted by the idea of evolution, and his ambition was to be
come the "Darwin" of the evolution of matter.
The first step toward the mechanistic interpretation of en
tropy was to reintroduce the concept of "collisions" of mole
cules or atoms into the physical description, and along with it
the possibility of a statistical description. This step had been
taken by Clausius and Maxwell. Since collisions are discrete
events, we may count them and estimate their average fre
quency. We may also classify collisions-for example, dis
tinguish between collisions producing a particle with a given
velocity v and collisions destroying a particle with a velocity v,
producing molecules with a different velocity (the " direct"
and "inverse" collisions); l l
The question Maxwell asked was whether it was possible to
define a state of a gas such that the collisions that incessantly
modify the velocities of the molecules no longer determine
any evolution in the distribution of these velocities-that is, in
the mean number of particles for each velocity value. What is
the velocity distribution such that the effects of the different
collisions compensate each other on the population scale?
241
THE CLASH OF DOCTRINES
Maxwell demonstrated that this particular state, which is
the thermodynamic equilibrium state, occurs when the veloc
ity distribution becomes the well-known "bell-shaped curve,"
the "gaussian," which Quetelet, the founder of "social phys
ics," had considered to be the very expression of randomness.
Maxwell's theory permits us to give a simple interpretation of
some of the basic laws describing the behavior of gases. An
increase in temperature corresponds to an increase in the
mean velocity of the molecules and thus of the energy associ
ated with their motion. Experiments have verified Maxwell's
law with great accuracy, and it still provides a basis for the
solution of numerous problems in physical chemistry (for ex
ample, the calculation of the number of collisions in a reactive
mixture).
Boltzmann, however, wanted to go farther. He wanted to
describe not only the state of equilibrium but also evolution
toward equilibrium-that is, evolution toward the Maxwellian
distribution. He wanted to discover the molecular mechanism
that corresponds to the increase of entropy, the mechanism
that drives a system from an arbitrary distribution of velocities
toward equilibrium.
Characteristically, Boltzmann approached the question of
physical evolution not at the level of individual trajectories but
at the level of a population of molecules. This, Boltzmann felt,
was virtually tantamount to accomplishing Darwin's feat, but
this time in physics: the driving force behind biological evolu
tion-natural selection-cannot be defined for one individual
but only for a large population. It is therefore a statistical con
cept.
Boltzmann's result may be described in relatively simple
terms. The evolution of the distribution function f ( v, t) of the
velocities v in some region of space and at time t appears as
the sum of two effects; the number of particles at any given
time t having a velocity v varies both as the result of the free
motion of the particles and as the result of collisions between
particles. The first result can be easily calculated in the terms
of classical dynamics. It is in the investigation of the second
result, due to collisions, that the originality of Boltzmann's
method lies. In the face of the difficulties involved in following
the trajectories (including the interactions), Boltzmann came
to use concepts similar to those outlined in Chapter V (in con-
ORDER OUT OF CHAOS
242
nection with chemical reactions) and to calculate the average
number of collisions creating or destroying a molecule corre
sponding to a velocity v.
Here once again there are two processes with opposite
effects-"direct" collisions, those producing a molecule with
velocity v starting from two molecules with velocities v ' and
"
v , and "inverse" collisions, in which a molecule with velocity
"'
v is destroyed by collision with a molecule with velocity v .
As with chemical reactions (see Chapter V, section 1 ), the fre
quency of such events is evaluated as being proportional to the
product of the number of molecules taking part in these pro
cesses. (Of course, historically speaking, Boltzmann's method
[1872] preceded that of chemical kinetics.)
The results obtained by Boltzmann are quite similar to those
obtained in Markov chains. Again we shall introduce an J{
quantity, this time referring to the velocity distribution f. It
may be written J{= f flog f dv. Once again, this quantity can
only decrease in time until equilibrium is reached and the ve
locity distribution becomes the equilibrium Maxwellian dis
tribution.
In recent years there have been numerous numerical ver
ifications of the uniform decrease of J{ with time. All of them
confirm Boltzmann's prediction. Even today, his kinetic equa
tion plays an important role in the physics of gases: transport
coefficients such as those characterizing heat conductivity or
diffusion can be calculated in good agreement with experimen
tal data.
However, it is from the conceptual standpoint that Boltz
mann's achievement is greatest: the distinction between re
versible and irreversible phenomena, which, as we have seen,
underlies the second law, is now transposed onto the micro
scopic level. The change of the velocity distribution due to
free motion corresponds to the reversible part, while the con
tribution due to collisions corresponds to the irreversible part.
For Boltzmann this was the key to the microscopic interpreta
tion of entropy. A principle of molecular evolution had been
produced ! It is easy to understand the fascination this discov
ery exerted on the physicists who followed Boltzmann, includ
ing Planck, Einstein, and Schrodinger. t2
Boltzmann's breakthrough was a decisive step in the direc-
243
THE CLASH OF DOCTRINES
tion of the physics of processes. What determines temporal
evolution in Boltzmann's equation is no longer the Hamilto
nian, depending on the type of forces; now, on the contrary,
functions associated with the processes-for example, the
cross section of scattering-will generate motion. Can we con
clude that the problem of irreversibility has been solved, that
Boltzmann's theory has reduced entropy to dynamics? The
answer is clear: No, it has not. Let us have a closer look at this
question.
Questioning Boltzmanns Interpretation
As soon as Boltzmann's fundamental paper appeared in 1 872,
objections were raised. Had Boltzmann really "deduced" irre
versibility from dynamics? How could the reversible laws of
trajectories lead to irreversible evolution? Is Boltzmann's ki
netic equation in any way compatible with dynamics? It is
easy to see that the symmetry present in Boltzmann's equa
tion is in contradiction with the symmetry of classical me
chanics.
We have already seen that velocity inversion (v� - v) pro
duces in classical dynamics the same effect as time inversion
(t� - t). This is a basic symmetry of classical dynamics, and
we would expect that Boltzmann's kinetic equation, which de
scribes the time change of the distribution function, would
share this symmetry. But this is not so. The collision term
calculated by Boltzmann remains invariant with respect to ve
locity inversion. There is a simple physical reason for this.
Nothing in Boltzmann's picture distinguishes a collision that
proceeds toward the future from a collision proceeding toward
the past. This is the basis of Poincare's objection to Boltz
mann's derivation. A correct calculation can never lead to
conclusions that contradict its premises.B· 14 As we have seen,
the symmetry properties of the kinetic equation obtained by
Boltzmann for the distribution function contradict those of dy
namics . Boltzmann cannot, therefore, have "deduced" en
tropy from dynamics . He must have introduced something
new, something foreign to dynamics. Thus his results �;an rep-
ORDER OUT OF CHAOS
244
resent at best only a phenomenological model that, however
useful, has no direct relation with dynamics. This was also the
objection that Zermelo (1896) brought against Boltzmann.
Loschmidt's objection, on the other hand, makes it possible
to determine the limits of validity of Boltzmann's kinetic
model. In fact, Loschmidt observed ( 1 876) that this model can
no longer be valid after a reversal of the velocities correspond
ing to the transformation v-+ - v.
Let us explain this by means of a thought experiment. We
start with a gas in a nonequilibrium condition and let it evolve
till t0 • We then invert the velocities. The system reverts to its
past state. As a consequence, Boltzmann's entropy is the
same at t = O and at t = 2t0•
We may multiply such thought experiments. Start with a
mixture of hydrogen and oxygen; after some time water will
appear. If we invert the velocities, we should go back to an
initial state with hydrogen and oxygen and no water.
It is interesting that in laboratory or computer experiments,
we actually can perform a velocity inversion. For example, in
Figures 26 and 27, Boltzmann's J{ quantity has been calcu
lated for two-dimensional hard spheres (hard disks), starting
first collision
ae
•
•
0
20
40
TIME
60
Figure 26. Evolution of .Jf with time for N "hard spheres" by computer
simulation; (a) corresponds to N= 1 00, (b) to N= 484, (c) to N = 1 225.
THE CLASH OF DOCTRINES
245
with disks on lattice sites with an isotropic velocity distribu
tion. The results follow Boltzmann's predictions.
If, after fifty or a hundred collisions, corresponding to about
I 0 - 6 sec in a dilute gas, the velocities are inverted, a new en
semble is obtained. 1 5 Now, after the velocity inversion, Boltz
mann's J{ quantity increases instead of decreasing.
I
•
...
•
.
•
•
•
•
.
-.
0
on
.
:
•
y
•
....
·:
:"
..... ...
.
•••
, ..t
..
•
•
u
•
•
0
N : IOO
-
•
�2
t
.
'
•
'\
....
-... .
..
•
_,.
••
•••
.
•
�
....
•
•
•
.
·�
'"·
.
....
..
�.
··� .. .
..,
.....
·11-------::------....
.. -�-------...
equit .
0
-'"'�
20
:
,
..
.....
TIME
60
Figure 27. Evolution of :H when velocities are inverted after 50 or 1 00
collisions. Simulation with 1 00 "hard spheres."
A similar situation can be produced in spin echo experi
ments or plasma echo experiments. There also, over limited
periods of time, an "antithermodynamic" behavior in Boltz
mann's sense may be observed.
But it is important to note that the velocity inversion experi
ment becomes increasingly more difficult when the time inter
val t0 after which the inversion occurs is increased.
To be able to retrace its past, the gas must remember every
thing that happened to it during the time interval from 0 to t0•
There must be "storage" of information. We can express this
storage in terms of correlations between particles. We shall
come back to the question of correlations in Chapter IX. Let
us only mention here that it is precisely this relation between
correlations and collisions that is the element missing from
ORDER OUT OF CHAOS
246
Boltzmann's considerations. When Loschmidt confronted him
with this, Boltzmann had to accept that there was no way out:
the collisions occurring in the opposite direction "undo" what
was done previously, and the system has to revert to its initial
state. Therefore , the function J{ must also increase until it
again reaches its initial value. Velocity inversion thus calls for
a distinction between the situations to which Boltzmann's rea
soning applies and those to which it does not.
Once the problem was stated ( 1 894), it was easy to identify
the nature of this limitation. l 6, 17 The validity of Boltzmann's
statistical procedure depends on the assumption that before
they collide, the molecules behave independently of one an
other. This constitutes an assumption about the initial condi
tions, called the "molecular chaos" assumption. The initial
conditions created by a velocity inversion do not conform to
this assumption. If the system is made "to go backward in
time," a new "anomalous" situation is created in the sense
that certain molecules are then "destined" to meet at a pre
determinable instant and to undergo a predetermined change
of velocity at this time, however far apart they may be at the
instant of velocity inversion.
Velocity inversion thus creates a highly organized system,
and thus the molecular chaos assumption fails. The various
collisions produce, as if by a preestablished harmony, an ap
parently purposeful behavior.
But there is more. What does the transition from order to
disorder signify? In the Ehrenfest urn experiment, it is clear
the system will evolve till uniformity is reached. But other sit
uations are not so clear; we may do computer ex periments in
which interacting particles are initially distributed at random.
In time a lattice is formed. Do we still move from order to
disorder? The answer is not obvious. To understand order and
disorder we first have to define the objects in terms of which
these concepts are used. Moving from dynamic to thermody
namic objects is easy in the case of dilute gases-as shown by
the work of Boltzmann. However, it is not so easy in the case
of dense systems whose molecules interact.
Because of such difficulties, Boltzmann's creative and pi
oneering work remained incomplete.
·
247
THE CLASH OF DOCTRINES
Dynarnics and T hermodynamics:
Two Separate Worlds
We already noted that trajectories are incompatible with the
idea of irreversibility. However, the study of trajectories is not
the only way in which we can give a formulation of dynamics.
There is also the theory of ensembles introduced by Gibbs and
Einstein,6. 1 8 which is of special interest in the case of systems
formed by a large number of molecules. The essential new
element in the Gibbs-Einstein ensemble theory is that we can
formulate the dynamic theory independently of any precise
specification of initial conditions.
The theory of ensembles represents dynamic systems in
"phase space. " The dynamic state of a point particle is spec
ified by position (a vector with three components) and by mo
mentum (also a vector with three components). We may
represent this state by two points, each in a three-dimensional
space, or by a single point in the six-dimensional space formed
by the coordinates and momenta. This is the phase space.
This geometric representation can be extended to an arbitrary
system formed by n particles. We then need n x 6 numbers to
specify the state of the system, or alternatively we may specify
this system by a single point in the 6n-dimensional phase
space. The evolution in time of such a system will then be
described by a trajectory in the phase space.
It has already been stated that the exact initial conditions of
a macroscopic system are never known. Nevertheless, nothing
prevents us from representing this system by an "ensemble"
of points-namely, the points corresponding to the various dy
namic states �ompatible with the information we have con
cerning the system. Each region of phase space may contain an
infinite number of representative points, the density of which
measures the probability of actually finding the system in this
region. Instead of introducing an infinity of discrete points, it
is more convenient to introduce a continuous density of repre
sentative points in the phase space. We shall call p (q 1
q30,
p1
p30) this density in phase space where q1 ,q2
q3n are
the coordinates of the n points; similarly, p 1 ,p2
p30 are the
momenta (each point has three coordinates and three mo•
•
•
•
•
•
•
•
•
•
•
•
ORDER OUT OF CHAOS
248
menta) . This density measures the probability of finding a dy
namic system around the point ql ... q3n,PI
P3n in phase
space.
Presented in such a way, the density function p may appear
as an idealization, an artificial construct, whereas the trajec
tory of a point in phase space would correspond "directly" to
the description of "natural" behavior. But in fact it is the
point, not the density, that corresponds to an idealization. In
deed, we never know an initial state with the infinite degree of
precision that would reduce a region in phase space to a single
point; we can only determine an ensemble of trajectories start
ing from the ensemble of representative points corresponding
to what we know about the initial state of the system. The
density function p represents knowledge about a system, and
the more accurate this knowledge, the smaller the region in the
phase space where the density function is different from zero
and where the system may be found. Should the density func
tion everywhere have a uniform value, we would know nothing
about the state of the system. It might be in any of the possible
states compatible with its dynamic structure.
From this perspective, a point thus represents the maximum
knowledge we can have about a system. It is the result of a
limiting process, the result of the ever-growing precision of our
knowledge. As we shall see in Chapter IX, a fundamental
problem will be to determine when such a limiting process is
really possible. Through increased precision, this process
means we go from a region where the density function p is
different from zero to another, smaller region inside the first.
We can continue this until the region containing the system
becomes arbitrarily small. But as we shall see, we must be
cautious: arbitrarily small does not mean zero, and it is not
certain a priori that this limiting process will lead to the pos
sibility of consistently predicting a single well-defined trajec
tory.
The introduction of the theory of ensembles by Gibbs and
Einstein was a natural continuation of Boltzmann's work. In
this perspective the density function p in phase space replaces
the velocity distribution function f used by Boltzmann. How
ever, the physical content of p exceeds that off. Just like/, the
density function p determines the velocity distribution, but it
also contains other information, such as the probability of
•
•
•
2�9
THE CLASH OF DOCTRINES
meeting two particles a certain distance apart. In particular,
correlations between particles, which we discussed in the pre
ceding section, are now included in the density function p. In
fact, this function contains the complete information about all
statistical features of the n -b9dy system.
We must now describe the evolution of the density function
in phase space. At first sight, this appears to be an even more
ambitious task than the one Boltzmann set himself for the ve
locity distribution function. But this is not the case. The Ham
iltonian equations discussed in Chapter 11 allow us to obtain
an exact evolution equation for p without any further approx
imations. This is the so-called Liouville equation, to which we
shall return in Chapter IX. Here we wish merely to point out
, that the properties of Hamiltonian dynamics imply that the
evolution of the density function p in phase space is that of an
incompressible fluid. Once the representative points occupy a
region of volume V in phase space, this volume remains con
stant in time. The shape of the region may be deformed in an
arbitrary way, but the value of the volume remains the same.
Gibbs' theory of ensembles thus permits a rigorous combi
nation of the statistical point of view (the study of the "popula
tion" described by p) and the laws of dynamics. It also permits
a more accurate representation of the thermodynamic equilib
rium state. Thus, in the case of an isolated system, the ensem-
p
q
Figure 28. Time evolution in the phase space of a "volume" containing the
representative points of a system: the volume is conserved while the shape
is modified. The position in phase space is specified by coordinate& q and
momentum p.
ORDER OUT OF CHAOS
250
ble ci representative points corresponds to systems that all have
the same energy E. The density p will differ from zero only on
the "microcanonical surface" corresponding to the specified
value of the energy in phase space. Initially, the density p may
be distributed arbitrarily over this surface. At equilibrium, p
must no longer vary with time and has to be independent of
the specific initial state. Thus the approach to equilibrium has
a simple meaning in terms of the evolution of p. The distribu
tion function p becomes uniform over the microcanonical sur
face. Each ci the points on this surface has the same probability
of actually representing the system. This corresponds to the
"microcanonical ensemble. "
Does the theory of ensembles bring u s any closer to the
solution of the problem of irreversibility? Boltzmann's theory
describes thermodynamic entropy in terms of the velocity dis
tribution function f. He achieved this result through the intro
duction of his J{ quantity. As we have seen, the system evolves
in time until the Maxwellian distribution is reached, while,
during this evolution, the quantity J-{ decreases uniformly. Can
we now, in a more general fashion, take the evolution of the
distribution p in phase space toward the microcanonical en
semble as the basis for entropy increase? Would it be enough
to replace Boltzmann's quantity J{ expressed in terms ofjby a
"Gibbsian" quantity 3£0 defined in exactly the same way, but
this time in terms of p? Unfortunately, the answer to both
questions is "No." If we use the Liouville equation, which
describes the evolution of the density phase space P. and take
into account the conservation of volume in phase space we
have mentioned, the conc lusion is immediate: :Jf0 is a con
stant and thus cannot represent entropy. With respect to Boltz
mann, this appears as a step backward rather than forward!
Though it is negative, Gibbs' conclusion remains very im
portant. We have already discussed the ambiguity of the ideas
of order and disorder. What the constancy of 3£0 tells us is
that there is no change of order whatsoever in the frame of
dynamic theory ! The "information" expressed by 3£0 remains
constant. This can be understood as follows: we have seen that
collisions introduce correlations. From the perspective of ve
locities, the result of collisions is randomization; therefore we
can describe this process as a transition from order to disor-
251
THE CLASH OF DOCTRINES
der, but the appearance of correlations as the result of collision
points in the opposite direction, toward a transition from dis
order to order! Gibbs' result shows that the two effects exactly
cancel each other.
We come, therefore, to an important conclusion. Whatever
representation we use, be it the idea of trajectories or the
Gibbs-Einstein ensemble theory, we will never be able to de
duce a theory of irreversible processes that will be valid for
every system that satisfies the laws of classical (or quantum)
dynamics. There isn't even a way to speak of a transition from
order to disorder! How should we understand these negative re
sults? Is any theory of irreversible processes in absolute con
flict with dynamics (classical or quantum)? It has often been
proposed that we include some cosmological terms that would
express the influence of the expanding universe on the equa
tions of motion. Cosmological terms would ultimately provide
the arrow of time. However, this is difficult to accept. On the
one hand, it is not clear how we should add these cosmological
terms; on the other, precise dynamic experiments seem to rule
out the existence of such terms, at least on the terrestrial scale
with which we are concerned here (think, for example, about
the precision of space trip experiments, which confirm New
ton's equations to a high degree). On the other hand, as we
have already stated, we live in a pluralistic universe in which
reversible and irreversible processes coexist, all embedded in
the expanding universe.
An even more radical conclusion is to affirm with Einstein
that time as irreversibility is an illusion that will never find a
place in the objective world of physics. Fortunately there is
another way out, which we shall explore in Chapter IX. Irre
versibility, as has been repeatedly stated, is not a universal
property. Therefore , no general derivation of irreversibility from
dynamics is to be expected.
Gibbs' theory of ensembles introduces only one additional
element with respect to trajectory dynamics, but a very impor
tant one-our ignorance of the precise initial conditions. It is
unlikely that this ignorance alone leads to irreversibility.
We should therefore not be astonished at our failure. We
have not yet formulated the specific features that a dynamic
system has to possess to lead to irreversible processes.
ORDER OUT OF CHAOS
252
Why have so many scientists accepted so readily the sub
jective interpretation of irreversibility? Perhaps part of its at
traction lies in the fact that, as we have seen, the irreversible
increase of entropy was at first associated with imperfect ma
nipulation, with our lack of control over operations that are
ideally reversible.
But this interpretation becomes absurd as soon as the irrele
vant associations with technological problems are set aside. We
must remember the context that gave the second law its signifi
cance as nature's arrow of time. According to the subjective
interpretation, chemical affinity, heat conduction, viscosity,
all the properties connected with irreversible entropy produc
tion would depend on the obser ver. Moreover, the extent to
which phenomena of organization originating in irreversibility
play a role in biology makes it impossible to consider them as
simple illusions due to our ignorance. Are we ourselves-liv
ing creatures capable of obser ving and manipulating-mere
fictions produced by our imperfect senses? Is the distinction
between life and death an illusion?
Thus recent developments in thermodynamic theory have
increased the violence of the conflict between dynamics and
thermodynamics. Attempts to reduce the results of thermody
namics to mere approximations due to our imperfect knowl
edge seem wrongheaded when the constructive role of entropy
is understood and the possibility of an amplification of fluctua
tions is discovered. Conversely, it is difficult to reject dynam
ics in the name of irreversibility: there is no irreversibility in
the motion of an ideal pendulum. Apparently there are two
conflicting worlds, a world of trajectories and a world of pro
cesses, and there is no way of denying one by asserting the
other.
To a certain extent, there is an analogy between this conflict
and the one that gave rise to dialectical materialism. We have
described in Chapters V and VI a nature that might be called
"historical"-that is, capable of development and innovation.
The idea of a history of nature as an integral part of material
ism was asserted by Mar x and, in greater detail, by Engels.
Contemporary developments in physics, the discovery of the
constructive role played by irreversibility, have thus raised
within the natural sciences a question that has long been asked
by materialists. For them, understanding nature meant under-
253
THE CLASH OF DOCTRINES
standing it as being capable of producing man and his so
cieties.
Moreover, at the time Engels wrote his Dialectics ofNature,
the physical sciences seemed to have rejected the mechanistic
world view and drawn closer to t he idea of an historical
development of nature . Engels mentions three fundamental
discoveries: energy and the laws governing its qualitative
transformations, the cell as the basic constituent of life, and
Darwin's discovery of the evolution of species. In v iew of
these great discoveries, Engels came to the conclusion that the
mechanistic world view was dead.
But mechanicism remained a basic difficulty facing dialecti
cal materialism. What are the relations between the general
laws of dialectics and the equally universal laws of mechanical
motion? Do the latter "cease" to apply after a certain stage has
been reached, or are they simply false or incomplete? To come
back to our previous question, how can the world of processes
and the world of trajectories ever be linked together?I9
However, while it is easy to criticize the subjectivistic inter
pretation of irreversibility and to point out its weakness, it is
not so easy to go beyond it and formulate an "objective" the
ory of irreversible processes. The history of this subject has
some dramatic overtones. Many people believe that it is the
recognition of the basic difficulties involved that may have led
to Boltzmann's suicide in 1906.
Boltzmann and the Arrow of Time
As we have noted, Boltzmann at first thought that he could
prove that the arrow of time was determined by the evolution
of dynamic systems toward states of higher probability or a
higher number of complexions: there would be a one-way in
crease of the number of complexions with time. We have also
discussed the objections of Poincare and Zermelo. Poincare
proved that every closed dynamic system reverts in time to
ward its previous state. Thus, all states are forever recurring.
How could such a thing as an "arrow of time" be associated
with entropy increase? This led to a dramatic change in Boltz
mann's attitude. He abandoned his attempt to prove that an ob
jective arrow of time exists and introduced instead an idea that,
ORDER OUT OF CHAOS
254
in a sense, reduced the law of entropy increase to a tautology.
Now he argued that the arrow of time is only a convention that
we (or perhaps all living beings) introduce into a world in
which there is no objective distinction between past and fu
ture. Let us cite Boltzmann's reply to Zermelo:
We have the choice of two kinds of picture. Either we
assume that the whole universe is at the present moment
in a very improbable state. Or else we assume that the
aeons during which this improbable state lasts, and the
distance from here to Sirius, are minute if compared with
the age and size of the whole universe. In such a uni
verse, which is in thermal equilibrium as a whole and
therefore dead, relatively small regions of the size of our
galaxy will be found here and there; regions (which we
may call "worlds") which deviate significantly from ther
mal equilibrium for relatively short stretches of those
"aeons" of time. Among these worlds the probabilities of
their state (i.e. the entropy) will increase as often as they
decrease. In the universe as a whole the two directions of
time are indistinguishable, just as in space there is no up
or down. However, just as at a certain place on the earth's
surface we can call "down " the direction towards the
centre of the earth, so a living organism that finds itself in
such a world at a certain period of time can define the
"direction" of time as going from the less probable state
to the more probable one (the former will be the "past"
and the latter the "future"), and by virtue of this defini
tion he will find that his own small region, isolated from
the rest of the universe, is "initially" always in an im
probable state. It seems to me that this way of looking at
things is the only one which allows us to understand the
validity of the second law, and the heat death of each indi
vidual world, without invoking a unidirectional change of
the entire universe from a definite initial state to a final
state.20
Boltzmann's idea can be made clearer by referring to a dia
gram proposed by Karl Popper (Figure 29). The arrow of time
would be as arbitrary as the vertical direction determined by
the gravitational field.
255
Arruw
THE CLASH OF DOCTRINES
of tiM 1Jf c.hia
o£ tiM only
•trett"h
ArrllW of ti• or this
atrf'tch ,,, ti•• onlr
t.qui liltri... level
tnt ropy c-urv• clettr•i ni.aa
the
dir•rtion of ti•
Figure 29. Popper's schematic representation of Boltzmann's cosmologi
cal interpretation of the arrow of time (see text).
Commenting on Boltzmann's text, Popper has written:
I think that Boltzmann's idea is staggering in its boldness
and beauty. But I also think that it is quite untenable, at
least for a realist. It brands unidirectional change as an
illusion. This makes the catastrophe of Hiroshima an illu
sion. Thus it makes our world an illusion, and with it all
our attempts to find out more about our world. It is there
fore self-defeating (like every idealism). Boltzmann's ide
alistic ad hoc hypothesis clashes with his own realistic
and almost passionately maintained anti-idealistic philos
ophy, and with his passionate wish to know. 21
We fully agree with Popper's comments, and we believe that
it is time to take up Boltzmann's task once again. As we have
said, the twentieth century has seen a great conceptual revolu
tion in theoretical physics, and this has produced new hopes
for the unification of dynamics and thermodynamics. We are
now entering a new era in the history of time, an era in which
both being and becoming can be incorporated into a single
noncontradictory vision.
CHAPTER IX
IRREVERSIBILITY
THEENTROPY
BARRIER
Entropy and the Arrow of Time
In the preceding chapter we described some difficulties in the
microscopic theory of irreversible processes. Its relation with
dynamics, either classical or quantum, cannot be simple, in
the sense that irreversibility and its concomitant increase of
entropy cannot be a general consequence of dynamics. A mi
croscopic theory of irreversible processes will require addi
tional, more specific conditions. We must accept a pluralistic
world in which reversible and irreversible processes coexist.
Yet such a pluralistic world is not easy to accept.
In his Dictionnaire Philosophique Voltaire wrote the follow
ing about destiny:
. . . everything is governed by immutable laws
ev
erything is prearranged ...everything is a necessary
effect . . .. There are some people who, frightened by
this truth, allow half of it, like debtors who offer their
creditors half their debt, asking for more time to pay the
remainder. There are, they say, events which are neces
sary and others which are not. It would be strange if a
part of what happens had to happen and another part did
not
I necessarily must have the passion to write this,
and you must have the passion to condemn me; we are
both equally foolish, both toys in the hands of destiny.
Your nature is to do ill, mine is to love truth, and to pub
lish it in spite of you.1
•
.
•
.
.
257
.
.
ORDER OUT OF CHAOS
258
However convincing they may sound, such a priori argu
ments can lead us astray. Voltaire's reasoning is Newtonian:
nature always conforms to itself. But, curiously, today we find
ourselves in the strange world mocked by Voltaire; we are as
tonished to discover the qualitative diversity presented by na
ture.
It is not surprising that people have vacillated between the
two extremes; between the elimination of irreversibility from
physics, advocated by Einstein, as we have mentioned,2 and,
on the contrary, the emphasis on the importance of irreversibil
ity, as in Whitehead's concept of process. There can be no doubt
that irreversibility exists on the macroscopic level and has an
important constructive role, as we have shown in Chapters V
and VI. Therefore there must be something in the microscopic
world of which macroscopic irreversibility is the manifesta
tion.
The microscopic theory has to account for two closely
related elements. First of all, we must follow Boltzmann in
attempting to constru ct a microscopic model for entropy
(Boltzmann's .J{ function) that changes uniformly in time. This
change has to define our arrow of time. The increase of en
tropy for isolated systems has to express the aging of the sys
tem.
Often we may have an arrow of time without being able to
associate entropy with the type of processes considered. Pop
per gives a simple example of a system presenting a unidirec
tional process and therefore an arrow of time.
Suppose a film is taken of a large surface of water ini
tially at rest into which a stone is dropped. The reversed
film will show contracting, circular waves of increasing
amplitude. Moreover, immediately behind the highest
wave crest, a circular region of undisturbed water will
close in towards the centre. This cannot be regarded as a
possible classical process. It would demand a vast num
ber of distant coherent generators of waves the coordina
tion of which, to be explicable, would have to be shown,
in the film, as originating from one centre. This, however,
raises precisely the same difficulty again, if we try to re
verse the amended film. 3
259
IRREVERSIBILITY-THE ENTROPY BARRIER
Indeed, whatever the technical means, there will always be
a distance from the center beyond which we are unable to gen
erate a contracting wave. There are unidirectional processes.
Many other processes of the type presented by Popper can be
imagined: we never see energy coming from all directions con
verge on a star, together with the backward-running nuclear
reactions that would absorb that energy.
In addition, there exist other arrows of time-for example,
the cosmological arrow (see the excellent account by M.
Gardner'). If we assume that the universe started with a Big
Bang, this obviously implies a temporal order on the cos
mological level. The size of the universe continues to increase,
but we cannot identify the radius of the universe with entropy.
Indeed, as we already mentioned, inside the expanding uni
verse we find both reversible and irreversible processes. Sim
ilarly, in elementary-particle physics there exist processes that
present the so-called T-violation. The T-violation implies that
the equations describing the evolution of the system for +t are
different from those describing the evolution for -t . However,
this T-violation does not prevent us from including it in the
u sual (Hamiltonian) formulation of dynamics. No entropy
function can be defined as a result of the T-violation.
We are reminded of the celebrated discussion between Ein
stein and Ritz published in 1909.5 This is a quite unusual pa
per, a very short one, less than a printed page long. It simply is
a statement of disagreement. Einstein argued that irreversibil
ity is a consequence of the probability concept introduced by
Boltzmann. On the contrary, Ritz argued that the distinction
between "retarded" and "advanced" waves plays an essential
role. This distinction reminds us of Popper's argument. The
waves we observe in the pond are retarded waves; they follow
the dropping of the stone.
Both Einstein and Ritz introduced essential elements into
the discussion of irreversibility, but each of them emphasized
only part of the story. We have already mentioned in Chapter
VIII that probability presupposes a direction of time and
therefore cannot be used to derive the arrow of time. We have
also mentioned that the exclusion of processes such as ad
vanced waves does not necessarily lead to a formulation of the
second law. We need both types of arguments.
ORDER OUT OF CHAOS
260
Irreversibility as a Syrr1metry-Breaking Process
Before discussing the problem of irreversibility, it is useful to
remember how another type of symmetry-breaking, spatial
symmetry-breaking, can be derived. In the equations describ
ing reaction-diffu sion systems , left and right play the same
role (the diffusion equations remain invariant when we per
form the space inversion r-+ -r). Still, as we have seen, bifurca
.tions may lead to solutions in which this symmetry is broken
(see Chapter V) . For example, the concentration of some of
the components may become higher on the left than on the
right. The symmetry of the equations only requires that the sym
metry-breaking solutions appear in pairs.
There are, of course, many reaction-diffusion equations that
do not present bifurcations and that therefore cannot break
spatial symmetry. The breaking of spatial symmetry requires
other highly specific conditions. This is valuable for under
standing temporal symmetry-breaking, in which we are pri
marily interested here. We have to find systems in which the
equations of motion may have realizations of lower symmetry.
The equations are indeed invariant in respect to time inver
sion t-+-t. However, the realization of these equations may
correspond to evolutions that lose this symmetry. The only
condition imposed by the symmetry of equations is that such
realizations appear in pairs. If, for example, we find one solu
tion going to equilibrium in the far distant future (and not in
the far distant past), we should also find a solution that goes to
equilibrium in the far distant past (and not in the far distant
future). Symmetry-broken solutions appear in pairs.
Once we find such a situation we can express the intrinsic
meaning of the second law. It becomes a selection principle
stating that only one of the two types of solutions can be real
ized or may be observed in nature. Whenever applicable, the
second law expresses an intrinsic polarization of nature. It can
never be the outcome of dynamics itself. It has to appear as a
supplementary selection principle that when realized is propa
gated by dynamics. Only a few years ago it seemed impossible
to attempt such a program. However, over the past few de-
261
IRREVERSIBILITY-THE ENTROPY BARRIER
cades dynamics has made remarkable progress, and we can
now understand in detail how these symmetry-breaking solu
tions emerge in dynamic systems "of sufficient complexity"
and what the selection rule expressed by the second law of
thermodynamics means on the microscopic level. This is what
we want to show in the next part of this chapter.
The Limits of Classical Concepts
Let u s start with classical mechanics. As we have already
mentioned, if trajectory is to be the basic irreducible element,
the world wou ld be as reversible as the trajectories out of
which it is formed. In this description there would be no en
tropy and no arrow of time; but, as a result of unexpected re·
cent developments, the validity of the trajectory concept
appears far more limited than we might have expected. Let us
return to Gibbs' and Einstein's theory of ensembles , intro
duced in Chapter VIII. We have seen that Gibbs and Einstein
introduced phase space into physics to account for the fact
that we do not "know" the initial state of systems formed by a
large number of particles. For them, the distribution function
in phase space was only an auxiliary construction expressing
our de facto ignorance of a situation that was well determined
de jure. However, the entire problem takes on new dimensions
once it can be shown that for certain types of systems an infi·
nitely precise determination of initial conditions leads to a
self-contradictory procedure. Once this is so, the fact that we
never know a single trajectory but a group, an ensemble of
trajectories in phase space, is not a mere expression of the
limits of our knowledge. It becomes a starting point of a new
way of investigating dynamics.
It is true that in simple cases there is no problem. Let us
take the example of a pendulum. It may oscillate or else rotate
about its axis according to the initial conditions. For it to ro
tate, its kinetic energy must be large enough for it not to "fall
back" before reaching a vertical position. These two types of
motion define two disjointed regions of phase space. The rea
son for this is very simple: rotation requires more energy than
oscillation (see F igure 30).
ORDER OUT OF CHAOS
262
y
0)
a
v
b)
Figure 30. Representation of a pendulum's motion in a space where Vis
the velocity and e the angle of deflection. (a) typical trajectories in (V, e)
space; (b) the shaded regions correspond to oscillations; the region outside
corresponds to rotations.
If our measurements allow us to establish that the system is
initially in a given region, we may safely predict the type of
motion displayed by the pendulum. We can increase the accu-
263
IRREVERSIBILITY-THE ENTROPY BARRIER
racy of our measurements and localize the initial state of the
pendulum in a smaller region circumscribed by the first. In
any case, we know the system's behavior for all time; nothing
new or unexpected is likely to occur.
One of the most surprising results achieved in the twentieth
century is that such a description is not valid in general. On
the contrary, "most" dynamic systems behave in a quite un
stable way. 6 Let us indicate one kind of trajectory (for exam
ple, that of oscillation) by + and another kind (for example,
that corresponding to rotation) by *· Instead of Figure 30 ,
where the two regions were separated, we find, in general , a
mixture of states that makes the transition to a single point
ambiguous (see Figure 31). If we know only that the initial
state of our system is in region A, we cannot deduce that its
trajectory is of type +; it may equally well be of type *· We
achieve nothing by increasing the accuracy by going from re
gion A to a smaller region within it, for the uncertainty re
mains. In all regions, however small, there are always states
belonging to each of the two types of trajectories.1
v
Figure 31. Schematic representation of any region, arbitrarily small, of the
phase space V tor a system presenting dynamic instability. As in the case of
the pendulum, there are two types of trajectories (represented here by +
and •); however, in contrast with the pendulum, both motions appear in every
region arbitrary small.
265
IRREVERSIBILITY-THE ENTROPY BARRIER
that at the end of the nineteenth century Bruns and Poincare
demonstrated that most dynamic systems, starting with the
famous "three body" problem, were not integrable.
On the other hand, the very idea of approaching equilibrium
in terms of the theory of ensembles requires that we go beyond
the idealization of integrable systems. As we saw in Chapter
VII I, according to the theory of ensembles, an isolated system
is in equilibrium when it is represented by a "microcanonical
ensemble, " when all points on the surface of given energy
have the same probability. This means that for a system to
evolve to equilibrium, energy must be the only quantity con
served during its evolution. It must be the only "invariant."
Whatever the initial conditions, the evolution of the system
must allow it to reach all points on the surface of given energy.
But for an integrable system, energy is far from being the only
invariant. In fact, there are as many invariants as degrees of
freedom, since each generalized momentum remains constant.
Therefore we have to expect that such a system is "impris
oned" in a very small "fraction" of the constant-energy (see
Figure 32) surface formed by the intersection of all these invar
-iant surfaces.
p
q
Figure 32. Temporal evolution of a cell in phase space p, q. The "volume"
of the cell and its form are maintained in time; moreover, most of the phase
space is inaccessible to the system.
265
IRREVERSIBILITY-THE ENTROPY BARRIER
that at the end of the nineteenth century Bmns and Poincare
demonstrated that most dynamic systems, starting with the
famous "three body" problem, were not integrable.
On the other hand, the very idea of approaching equilibrium
in terms of the theory of ensembles requires that we go beyond
the idealization of integrable systems. As we saw in Chapter
VIII, according to the theory of ensembles, an isolated system
is in equilibrium when it is represented by a "microcanonical
ensemble, " when all points on the surface of given energy
have the same probability. This means that for a system to
evolve to equilibrium, energy must be the only quantity con
served during its evolution. It must be the only "invariant. "
Whatever the initial conditions, the evolution of the system
must allow it to reach all points on the surface of given energy.
But for an integrable system, energy is far from being the only
invariant. In fact, there are as many invariants as degrees of
freedom, since each generalized momentum remains constant.
Therefore we have to expect that such a system is "impris
oned" in a very small "fraction" of the constant-energy (see
Figure 32) surface formed by the intersection of all these invar
.
iant surfaces.
p
- ..--- ..........
1),/
," , ---""'04\
.,
,
�
I
,
,
I
"
\
, ,\ rs-'" 'I ,
I
I
\
\
\
\
,
,
"
,
,
"
......... _..-.,-.
'"
,/
\
,
I
/
/
q
Figure 32. Temporal evolution of a cell in phase space p, q. The "volume"
of the cell and its form are maintail1ed in time; moreover, most of the phase
space is inaccessible to the system.
ORDER OUT OF CHAOS
266
To avoid these difficulties, Maxwell and Boltzmann intro
duced a new, quite different type of dynamic system. For these
systems energy would be the only invariant. Such systems are
called "ergodic" systems (see Figure 33).
Great contributions to the theory of ergodic systems have
been made by Birchoff, von Neumann, Hopf, Kolmogoroff,
and Sinai, to mention only a few. s. 9. tO Today we know that
there are large classes of dynamic (though non-Hamiltonian)
systems that are ergodic. We also know that even relatively
simple systems may have properties stronger than ergodicity.
For these systems , motion in phase space becomes highly cha
otic (while always preserving a volume that agrees with the
Liouville equation mentioned in Chapter VII).
p
q
Figure 33. Typical evolution in phase space of a cell corresponding to an
ergodic system. Time going on, the "volume" and the form are conserved
but the cell now spirals through the whole phase space.
267
·
IRREVERSIBILITY-THE ENTROPY BARRIER
Suppose that our knowledge of initial conditions permits us
to localize a system in a small cell of the phase space. During
its evolutio n, we shall see this initial cell twist and turn and,
like an amoeba, send out " pseudopods" in all directions,
spreading out in increasingly thinner and ever more twisted
filaments until it finally invades the whole space. No sketch
can do justice to the complexity of the actual situation. In
deed, during the dynamic evolution of a mixing system, two
points as close together in phase space as we might wish may
head in different directions. Even if we possess a lot of infor
mation about the system so that the initial cell formed by its
representative points is very small, dynamic evolution turns
this cell into a true geometric "monster" stretching its net
work of filaments through phase space.
p
q
Figure 34. Typical evolution in phase space of a cell corresponding to a
"mixing" system. The volume is still conserved but no more its form: the cell
progressively spreads through the whole phase space.
ORDER OUT OF CHAOS
268
We would like to illustrate the distinction between stable
and unstable systems with a few simple examples. Consider a
phase space with two dimensions. At regular time intervals ,
we shall replace these coordinates b y new ones. The new point
on the horizontal axis is p-q, the new ordinate p. Figure 35
shows what happens when we apply this operation to a square.
(0,-1)
p
Figure 35. Transformation of a volume in phase space generated by a
discrete transformation: the abscissa p becomes p-q, the ordinate q be
comes p. The transformation is cyclic: after six times the initial cell is re
covered.
The square is deformed, but after six transformations we re
turn to the original square. The system is stable: neighboring
points are transformed into neighboring points. Moreover, it
corresponds to a cyclic transformation (after six operations
the original square reappears).
Let us now consider two examples of highly unstabl� sys
tems-the first mathematical, the second of obvious physical
269
IRREVERSIBILITY-THE ENTROPY BARRIER
relevance. The first system consists of a transformation that,
for obvious reasons , mathematicians call the "baker transfor
mation. "9, to We take a square and flatten it into a rectangle ,
then we fold half of the rectangle over the other half t o form a
square again. This set of operations is shown in Figure 36 and
may be repeated as many times as one likes.
n
�
q:1
q=1
2
1
�
1111
p:1
B
q=1
�
p:1
)
112
q:1
1J
p :1
e-1
Figure 36. Realization of the baker transformation (B) and of its inverse
(B-1). The path of the two spots gives an idea of the transformations.
Each time the surface of the square is broken up and re
distributed. The square corresponds here to the phase space.
The baker transformation transforms each point into a well
defined new point. Although the series of points obtained in
this way is "deterministic," the system displays in addition
irreducibly statistical aspects. Let us take , for instance, a sys
tem described by an initial condition such that a region A of
the square is initially filled in a uniform way with representa
tive points. It may be shown that after a sufficient number of
repetitions of the transformation, this cell, whatever its size
and localization, will be broken up into pieces (see Figure 37).
The essential point is that any region, whatever its size, thus
ORDER OUT OF CHAOS
270
p
q
Figure 37. Time evolution of an unstable system. Time going on, region A
splits into regions A' and A", which in turn will be divided.
always contains different trajectories diverging at each frag
mentation. Although the evolution of a point is reversible and
deterministic, the description of a region, however small, is
basically statistical.
A similar example involves the scattering of hard spheres.
We may consider a small sphere rebounding on a collection of
large, randomly distributed spheres. The latter are supposed
to be fixed. This is the model physicists call the " Lorentz
model, " after the name of a great Dutch physicist, Hendrik
Antoon Lorentz.
The trajectory of the small mobile sphere is well defined.
However, whenever we introduce the smallest uncertainty in
the initial conditions, this uncertainty is amplified through
successive collisions. As time passes, the probability of find
ing the small sphere in a given volume becomes uniform.
Whatever the number of transformations, we never return to
the original state.
In the last two examples we have strongly unstable dynamic
systems. The situation is reminiscent of instabilities as they
appear in thermodynamic systems (see Chapter V). Arbitrar
ily small differences in initial conditions are amplified. As a
result we can no longer perform the transition from ensembles
271
IRREVERSIBILITY-THE ENTROPY BARRIER
�
Q
I
I
I
I
'
�............ ... ....
.......
o
----
... ... ...
� --
...
'o
I
"
"
"
,
"
,"
1
1
---
�
/
-
•
"
--
:..
I
I
I
I
I
I
1
I
I
��.....
"
Figure 38. Schematic representation of the instability of the trajectory of a
small sphere rebounding on large ones. The least imprecision about the
position of the small sphere makes it impossible to predict which large
sphere it will hit after the first collision.
in phase space to individual trajectories. The description in
terms of ensembles has to be taken as the starting point. Sta
tistical concepts are no longer merely an approximation with
respect to some "objective truth. " When faced with these un
stable systems, Laplace's demon is just as powerless as we.
Einstein's saying, "God does not play dice," is well known.
In the same spirit Poincare stated that for a supreme mathe
matician there is no place for probabilities. However, Poincare
himself mapped the path leading to the answer to this prob
lem.11 He noticed that when we throw dice and use probability
calculus, it does not mean that we suppose dynamics to be
wrong. It means something quite different. We apply the prob
ability concept because in each interval of initial conditions,
however small, there are as "many" trajectories that lead to
each of the faces of the dice. This is precisely what happens
with unstable dynamic systems. God could, if he wished to,
ORDER OUT OF CHAOS
272
calculate the trajectories in an unstable dynamic world. He
would obtain the same result as probability calculus permits
us to reach. Of course, if he made use of his absolute knowl
edge, then he could get rid of all randomness.
In conclusion, there is a close relationship between instabil
ity and probability. This is an important point, and we want to
discuss it now.
From Randomness to Irreversibility
Consider a succession of squares to which we apply the baker
transformation. This succession is represented in Figure 39. A
shaded region may be imagined to be filled with ink, an un
shaded region by water. At time zero we have what is called a
generating partition. Out of this partition we form a series of
either horizontal partitions when we go into the future or ver
tical partitions going into the past. These are the basic parti
tions. An arbitrary distribution of ink in the square can be
written formally as a superposition of the basic partitions. To
each basic partition we may associate an "internal" time that
is simply the number of baker transformations we have to per
form to go from the generating partition to the one under con
sideration.I2 We therefore see that this type of system admits a
kind of internal age.*
The internal time T is quite different from the usual mechan
ical time, since it depends on the global topology of the sys
tem. We may even speak of the "timing of space, " thus coming
close to ideas recently put forward by geographers, who have in
troduced the concept of "chronogeography. " 13 When we look at
the structure of a town , or of a landscape, we see temporal
elements interacting and coexisting. Brasilia or Pompeii would
correspond to a well-defined internal age, somewhat like one
of the basic partitions in the baker transformation. On the con
trary, modern Rome, whose buildings originated in quite dif*It may be noticed that this internal time, which we shall denote by T. is in
fact an operator like those introduced in quantum mechanics (see Chapter
VII). Indeed, an arbitrary partition of the square does not have a well
defined time but only an "average" time corresponding to the superposition
of the basic partitions out of which it is formed.
273
IRREVERSIBILITY-THE ENTROPY BARRIER
-m•1
past
-
-
0
t
genart1ting partition
=
-
z
future
Figure 39. Starting with the "generating partition" (see text) at time 0, we
repeatedly apply the baker transformation. We generate horizontal stripes in
this way. Similarly going into the past we obtain vertical stripes.
ferent periods, would correspond to an average time exactly as
an arbitrary partition may be decomposed into elements cor
responding to different internal times.
Let us again look at Figure 39. What happens if we go into
the far distant future ? The horizontal bands of ink will get
closer and closer. Whatever the precision of our measurements,
after some time we shall conclude that the ink is uniformly
distributed over the volume. It is therefore not surprising that
this kind of approach to "equilibrium" may be mapped into a
stochastic process, such as the Markov chain we described in
Chapter VIII. Recently this has been shown with full mathe
matical rigor,14 but the result seems to us quite natural. As
time passes, the distribution of ink reaches equilibrium, ex
actly like the distribution of balls in the urn experiment dis
cussed in Chapter VIII. However, when we look into the past,
again beginning from the generating partition at time zero, we
see the same phenomenon. Now ink is distributed along shrink
ing vertical sections and, again, sufficiently far in the past we
shall find a uniform distribution of ink. We may therefore con
clude that we can also model this process in terms of a Markov
chain, now, however, oriented toward the past. We see that out
of the unstable dynamic processes we obtain two Markov
chains, one reaching equilibrium in the future, one in the past.
We believe that it is a very interesting result � and we would
like to comment it. Internal time provides us with a new 'non
local' description.
When we know the 'age' of the system, (that is, the corre
sponding partition) , we can still not associate to it a well-de
fined local trajectory.
ORDER OUT OF CHAOS
274
We know only that the system is in a shaded region (Figure
39). Similarly, if we know some precise initial conditions cor
responding to a point in the system, we don't know the parti
tion to which it belongs, nor the age of the system. For such
systems we know therefore two complementary descriptions,
and the situation becomes somewhat reminiscent of the one
we described in Chapter VII, when we discussed quantum me
chanics.
It is because of the existence of this new alternative, non
local description, that we can make the transition from dy
namics to probabilities. We call the systems for which this is
possible "intrinsically random systems".
In classical deterministic systems, we may use transition
probabilities to go from one point to another on a quite degen
erate sense. This transition probability will be equal to one if
the two points lie on the same dynamic trajectory, or zero if
they are not.
In contrast, in genuine probability theory, we need transi
tion probabilities which are positive numbers between zero
and one. How is this possible? Here we see in full light the
conflict between subjectivistic views and objective interpreta
tions of probability. The subjective interpretation corresponds
to the situation where individual trajectories are not known.
Probability (and, eventually, irreversibility, closely related to
it) would originate from our ignorance. But fortunately, there
is another objective interpretation: probability arises as a re
sult of an alternative description of dynamics, a non-local de
scription which arises in strongly unstable dynamical systems.
Here, probability becomes an objective property generated
from the inside of dynamics, so to speak, and which expresses
a basic structure of the dynamical system. We have stressed
the importance of Boltzmann's basic discovery: the connec
tion between entropy and probability. For intrinsic random
systems, the concept of probability acquires a dynamical
meaning. We have now to make the transition from intrinsic
random systems to irreversible systems. We have seen that out
of unstable dynamical processes , we obtain two Markov
chains.
We may see this duality in a different way. Take a distribu
tion concentrated on a line (instead of being distributed on a
surface). This line may be vertical or horizontal. Let us look at
275
IRREVERSIBILITY-THE ENTROPY BARRIER
what will happen to this line when we apply the baker transfor
mation going to the future. The result is represented in Figure
40. The ver tical line is cut successively into pieces and will
reduce to a point in the far distant future. The horizontal line,
on the contrary, is duplicated and will uniformly "cover" the
surface in the far distant future. Obviously, just the opposite
happens if we go to the past. For reasons that are easy to un
derstand, the ver tical line is called a contracting fiber, the
horizontal line a dilating fiber.
We see now the complete analogy with bifurcation theory. A
contracting fiber and a dilating fiber correspond to t wo realiza
tions of dynamics, each involving symmetry-breaking and appear
ing in pairs. The contracting tiber corresponds to equilibrium in
the far distant past, the dilating fiber to the future. We therefore
have two Markov chains oriented in opposite time directions.
Now we have to make the transistion from intrinsically ran
dom to intrinsically irreversible systems. To do so we must
understand more precisely the difference between contracting
and dilating fibers. We have seen that another system as unsta-
I
Figure 40. Contracting and dilating fibers in the baker transformation; time
going on, the contracting fiber A1 is shortened (sequence A1, 81, C1), while
the dilating fibers are duplicated (sequences A2, 82, C2).
ble as the baker transformation can describe the scattering of
hard spheres. Here contracting and dilating fibers have a sim-
ORDER OUT OF CHAOS
276
pie physical interpretation. A contracting fiber corresponds to
a collection of hard spheres whose velocities are randomly dis
tributed in the far distant past, and all become parallel in the
far distant future. A dilating fiber corresponds to the inverse
situation, in which we start with parallel velocities and go to a
random distribution of velocities. Therefore the difference is
very similar to the one between incoming waves and outgoing
waves in the example given by Popper. The exclusion of the
contracting fibers corresponds to the experimental fact that
whatever the ingenuity of the experimenter, he will never be
able to control the system to produce parallel velocities after
an arbitrary number of collision s . Once we exclude con·
tracting fibers we are left with only one of the two possible
Markov chains we have introduced. In other words, the sec
ond law becomes a selection principle of initial conditions.
Only initial conditions that go to equilibrium in the future are
retained.
Obviously the validity of this selection principle is main
tained by dynamics. It can easily be seen in the example of the
baker transformation that the contracting fiber remains a con
tracting fiber for all times, and likewise for a dilating fiber. By
suppressing one of the two Markov chains we go from an
i ntrinsically random system to an intrinsically irreversible sys
tem. We find three basic elements in the description of irre
versibility:
instability
f
intrinsic randomness
f
·intrinsic irreversibility
Intrinsic irreversibility is the strongest property: it implies
randomness and instability. t4, ts
How is this conclusion compatible with dynamics? As we
have seen, in dynamics "information" is conserved, while in
Markov chains information is lost (and entropy therefore in
creases; see Chapter VIII). There is, however , no contradic
tion; when we go from the dynamic description of the baker
transformation to the thermodynamic description, we have to
277
IRREVERSIBILITY-THE ENTROPY BARRIER
modify our distribution function; the "obj ects" in terms of
which entropy increases are different from the ones consid
ered in dynamics. The new distribution function, p, corre
sponds to an intrinsically time-oriented description of the
dynamic system. In this book we cannot dwell on the mathe
matical aspects of this transformation. Let us only emphasize
that it has to be noncanonical (see Chapter II). We must aban
don the usual formulations of dynamics to reach the ther
modynamic description.
It is quite remarkable that such a transformation exists and
that as a result we can now unify dynamics and thermodynam
ics, the physics of being and the physics of becoming. We shall
come back to these new thermodynamic objects later in this
chapter as well as in our concluding chapter. Let us e mphasize
only that at equilibrium, whenever entropy reaches its max
imum, these objects must behave randomly.
It also seems quite remarkable that irreversibility emerges, so
to speak, from instability, which introduces irreducible statisti
cal features into our description. Indeed, what could an arrow
of time mean in a deterministic world in which both future and
past are contained in the present? It is because the future is
not contained in the present and that we go from the present to
the future that the arrow of time is associated with the transition
from present to future. This construction of irreversibility out of
randomness has, we believe , many consequences that go
beyond science proper and to which we shall come back in our
concluding chapter. Let us clarify the difference between the
states permitted by the second law and those it prohibits.
The Entropy Barrier
Time flows in a single direction, from past to future. We can
not manipulate time, we cannot travel back to the past. Travel
through time has preoccupied writers, from The Thousand
and One Nights to H. G. Wells' The Time Machine. In our
time, Nabokov's short novel, Look at the Harlequins!, 1' de
scribes the torment of a narrator who finds himself as unable
to switch from one spatial direction to the other as we are to
ORDER OUT OF CHAOS
278
"twirl time. " In the fifth volume of Science and Civilization in
China, Needham describes the dream of the Chinese alche
mists: their supreme aim was not to achieve the transmutation
of metals into gold but to manipulate time, to reach immor
tality through a radical slowdown of natural decaying pro
cesses. I? We are now better able to understand why we cannot
"twirl time," to use Nabokov's expression.
An infinite entropy barrier separates possible initial condi
tions from prohibited ones. Because this barrier is infinite,
technological progress never will be able to overcome it. We
have to abandon the hope that one day we will be able to travel
back into our past. The situation is somewhat analogous to the
barrier presented by the velocity of light. Technological prog
ress can bring us closer to the velocity of light, but in the pres
ent views of physics we will never cross it.
To understand the origin of this barrier, let us return to the
expression of the :H quantity as it appears in the theory of
Markov chains (see Chapter VIII). To each distribution we can
associate a number-the corresponding value of J-{. We can
say that to each distribution corresponds a well-defined infor
mation content. The higher the information content, the more
difficult it will be to realize the corresponding state. What we
wish to show here is that the initial distribution prohibited by
the second law would have an infinite information content.
That is the reason why we can neither realize them nor find
them in nature.
Let us first come back to the meaning of :Has presented in
Chapter VIII. We have to subdivide the relevant phase space
into sectors or boxes. With each box k, we associate an proba
bility Peqm(k) at equilibrium as well as a non-equilibrium prob
ability P(k,t).
The :H is a measure of the difference between P(k,t) and
Peqm(k), and vanishes at equilibrium when this difference dis
appears. Therefore, to compare the Baker transformation with
Markov chains, we have to make more precise the correspond
ing choice of boxes. Suppose we consider a system at time 2
(see Figure 39), and suppose that this system originated at
time ti. Then, a result of our dynamical theory is that the
boxes correspond to all possible intersections among the par
titions between timet; andt=2. If we consider now Figure 39.
we see that when
ti
is receding towards the past, the boxes will
279
IRREVERSIBILITY-THE ENTROPY BARRIER
become steadily thinner as we have to introduce more a nd
more vertical subdivisions. This is expressed in Figure 41 , se
quence B, where, going from top to bottom, we have ti = 1, 0 ,
- 1, and finally t i = - 2. We see indeed that the number of
boxes increa ses in this way from 4 to 32.
Once we have the boxes, we can compare the non-equi
librium distribution with the equilibrium distribution for each
box. In the present case, the non-equilibrium distribution is
either a dilating fiber (sequence A) or a contracting fiber (se
quence C). The important point to notice is that when ti is
receding to the past, the dilating fiber occupies an increasing
large number of boxes: for ti = -2 it occupies 4 boxes, for
ti = - 2 it occupies 8 boxes, and so on.
As a result, when we apply the formula given in Chapter
VIII, we obtain a finite result, even if the number of boxes
goes to infinity for tr-+ -oo
In contrast, the contracting fiber remains always localized
in 4 boxes whatever ti. As a result, .Jl, when applied to a con.
A
8
c
Figure 41. Dilating (sequence A) and contracting (sequence C) fibers
cross various numbers of the boxes which subdivide a Baker transformation
phase space. All "squares" on a given sequence refer to the same time, t=2,
but the number of the boxes subdividing each square depends on the initial
time of the system ti.
ORDER OUT OF CHAOS
280
tracting fiber, diverges to infinity when ti recedes to the past.
In summary, the difference between a dynamical system and
the Markov chain is that the number of boxes to be considered
in a dynamical system is infinite. It is this fact that leads to a
selection principle. Only measures or probabilities, which in
the limit of infinite number of boxes give a finite information
or a finite J{ quantity, can be prepared or observed. This ex
cludes contracting fibers.ts For the same reason we must also
exclude distributions concentrated on a single point. Initial
conditions corresponding to a single point in unstable systems
would again correspond to infinite information and are there
fore impossible to realize or observe. Again we see that the
second law appears as a selection principle.
In the classical scheme , initial conditions were arbitrary.
This is no longer so for unstable systems. Here we can associ
ate an information content to each initial condition, and this
information content itself depends on the dynamics of the sys
tem (as in the baker transformation we used the successive
fragmentation of the cells to calculate the information con
tent). Initial conditions and dynamics are no longer indepen
dent. The second law as a selection rule seems to us so
important that we would like to give another illustration based
on the dynamics of correlations.
The Dynamics of Correlations
In Chapter VIII we discussed briefly the velocity inversion
experiment. We may consider a dilute gas and follow its evolu
tion in time. At time t0 we proceed to a velocity inversion of
each molecule. The gas then returns to its initial state. We
have already noted that for the gas to retrace its past there
must be some storage of information. This storage can be de
scribed in terms of "correlations" between particles.t9
Consider first a cloud of particles directed toward a target (a
heavy, motionless particle). This situation is described in Fig
ure 42. In the far distant past, there were no correlations be
tween particles. Now, scattering has two effect s , already
mentioned in Chapter VIII. It disperses the particles (it makes
the velocity distribution more symmetrical) and, in addition, it
281
IRREVERSIBILITY-THE ENTROPY BARRIER
produces correlations between the scattered particles and the
scatterer. The correlations can be made explicit by performing
a velocity inversion (that is, by introducing a spherical mirror).
Figure 43 represents this situation (the wavy lines represent
the correlations). Therefore, the role of scattering is as fol
lows: In the direct process, it makes the velocity distribution
more symmetrical and creates correlations; in the inverse pro
cess, the velocity distribution becomes less symmetrical and
correlations disappear. We see that the consideration of cor-
•
•
.. 0
.
.-....
.. .
•
•
Figure 42. Scattering of particles. Initially all particles have the same ve
locity. After the collision, the velocities are no more identical, and the scat
tered particles are correlated with the scatterer (correlations are always
represented by wavy lines).
relations introduces a basic distinction between the direct and
the inverse processes.
We can apply our conclusions to many-body systems. Here
also we may consider two types of situations: in one, uncorre
lated particles enter, are scattered, and correlated particles are
produced (see Figure 44). In the opposite situation, correlated
�
0
•
•
•
..
•
..
•
..
Figure 43. Effect of a velocity inversion after a collision: after the new
"inverted" collision, the correlations are suppressed and all particles have
the same velocity.
ORDER OUT OF CHAOS
282
particles enter, the correlations are destroyed through colli
sions, and uncorrelated particles result (see Figure 45).
The two situations differ in the temporal order of collisions
and correlations. In the first case , we have "postcollisional"
correlations. With this distinction between pre- and postcolli
sional corre lations in mind, let us return to the velocity inver
sion expe rime n t . We start at t = 0 , with an ini tial state
corre sponding to no corre lations between particle s . During
the time o�t0 we have a "normal" evolution. Collisions bring
the velocity distribution closer to the Maxwellian equilibrium
distribution. They also create postcollisional correlations be0
0
0
0
after
before
Figure 44. Creation of postcollisional correlations represented by wavy
lines; for details see text.
tween the particles . At t0 after the velocity inversion, a com
pletely new situation arise s . Postcollisional corre lations a re
now transformed into precollisional corre lations. In the time
interval between t0 and 2t0, the se precollisional corre lations
disappear, the velocity distribution becomes less symmet-rical,
and at time 2t0 we are back in the noncorrelational state . The
history of this system therefore has two stages. During the
0
0
before
0
I
�
0
after
Figure 45. Destruction of precollisional correlations (wavy lines) through
collisions.
283
IRREVERSIBILITY-THE ENTROPY BARRIER
first, collisions are transformed into correlations ; in the sec
ond, correlations turn back into collisions. Both types of pro
cesses are compatible with the laws of dynamics. Moreover, as
we have mentioned in Chapter VIII, the total "information"
described by dynamics remains constant. We have also seen
that in Boltzmann's description the evolution from time 0 till t0
corresponds to the usual decrease of J{, while from t0 to 2t0 we
have an abnormal situation: J{ would increase and entropy de
crease. We would then be able to devise experiments in the
laboratory or on computers in which the second Jaw would be
violated ! The irreversibility during time 0 - t0 would be "com
pensated" by "anti-irreversibility" during time t0 - 2t0•
This is quite unsatisfactory. All these difficulties disappear
if we go, as in the baker transformation, to the new "thermody
namic representation" in terms of which dynamics becomes a
probabilistic process like a Markov chain. We must also take
into account that velocity inversion is not a "natural" process;
it requires that "information" be given to molecules from the
outside for them to invert their velocity. We need a kind of
Maxwellian demon to perform the velocity inversions, and
Maxwell's demon has a price. Let us represent the J{ quantity
(for the probabilistic process) as a function of time. This is
done in Figure 46. In this approach, in contrast with Boltz
mann's, the effect of correlations is retained in the new defini
tion of :H. Therefore at the velocity inversion point t0 the J{
quantity will jump, since we abruptly create abnormal precolli
sional correlations that will have to be destroyed later. This
jump corresponds to the entropy or information price we have
to pay.
Now we have a faithful representation of the second law: at
every moment the J{ quantity decreases (or the entropy in
creases). There is one exception at time t0: J{ jumps upward,
but that corresponds to the very moment at which the system
is open. We can invert the velocities only by acting from the
outside.
There is another essential point: at time t0 the new J{ quan
tity has two different values, one for the system before ve
locity inversion and the other after a velocity inversion. These
two situations have different entropies. This resembles what
occurs in the baker transformation when the contracting and
dilating fibers are velocity inversions of each other.
ORDER OUT OF CHAOS
284
Suppose we wait a sufficient time before making the ve
locity in-version. The postcollisional correlations wmdd have
an arbitrary range, and the entropy price for velocity inversion
would become too high. The velocity inversion would then
require too high an entropy price and thus would be excluded.
In physical terms this means that the second law excludes per
sistent long-range precollisional correlations.
The analogy with the macroscopic description of the second
law is striking. From the point of view of energy conservation
(see Chapters IV and V), heat and work play the same role,
I
I
I
I
I
I
I
I
�------·�--�---+t
2 t0
t0
I
l
Figure 46. Time variation of the .J -function in the v�locity inversion ex
periment: at time t0, the velocities are inversed and J-l presents a disconl
tinuity. At time 2t0 the system is in the same state as at ti_me 0, and .J
recovers the value it had initially. At all times (except at t0), J-l is decreasing.
The important fact is that at time t0 the J-l-quantity takes two different values
(see text).
but no longer from the point of view of the second law. Briefly
speaking, work is a more coherent form of energy and always
can be converted into heat, but the inverse is not true. There is
on the microscopic level a similar distinction between colli
sions and correlations. From the point of view of dynamics, colli
sions and correlations play equivalent roles. Collisions give
rise to correlations, and correlations may destroy the effect of
285
IRREVERSIBILITY-THE ENTROPY BARRIER
collisions. But there is an essential difference. We can control
collisions and produce correlations, but we cannot control cor
relations in a way that will destroy the effects collisions have
brought into the system. It is this essential difference that is
missing in dynamics but that can be incorporated into thermody
namics. Note that thermodynamics does not enter into conflict
with dynamics at any point. It adds an additional, essential
element to our understanding of the physical world.
Entropy as a Selection Principle
It is amazing how closely the microscopic theory of irrevers
ible processes resembles traditional macroscopic theory. In both
cases entropy initially has a negative meaning. In its macroscopic
aspect it prohibits some processes, such as heat flowing from
cold to hot . In its microscopic aspect it prohibits certain
classes of initial conditions. The distinction between what is
permitted and what is prohibited is maintained in time by the
laws of dynamics . It is from the negative aspect that the posi
tive aspect emerges: the existence of entropy together with its
probability interpretation. Irreversibility no longer emerges as
if by a miracle at some macroscopic level. Macroscopic irre
versibility only makes apparent the time-oriented polarized
nature of the universe in which we live.
As we have emphasized repeatedly, there exist in nature sys
tems that behave reversibly and that may be fully described by
the laws of classical or quantum mechanics. But most systems
of interest to us, including all chemical systems and therefore
all biological systems, are time-oriented on the macroscopic
level. Far from being an "illusion, " this expresses a broken
time-symmetry on the microscopic level. Irreversibility is ei
ther true on all levels or on none. It cannot emerge as if by a
miracle, by going from one level to another.
Also we have already noticed, irreversibility is the starting
point of other symmetry breakings. For example, it is gener
ally accepted that the difference between particles and anti
particles could arise only in a nonequilibrium world. This may
be extended to many other situations. It is likely that irrevers
ibility also played a role in the appearance of chiral symmetry
ORDER OUT OF CHAOS
286
through the selection of the appropriate bifurcation. One of
the most active subjects of research now is the way in which
irreversibility can be "inscribed" into the structure of matter.
The reader may have noticed that in the derivation of micro
scopic irreversibility we have been concentrating on classical
dynamics . However, the ideas of correlations and the distinc
tion between pre- and postcollisional correlations apply to
quantum systems as well . The study of quantum systems is
more complicated than the study of classical systems. There is
a reason for this: the difference between classical and quan
tum mechanics . Even small classical systems, such as those
formed by a few hard spheres, may present intrinsic irrevers
ibility. However, to reach irreversibility in quantum systems we
need large systems, such as those realized in liquids, gases, or
in field theory. The study of large systems is obviously more
difficult mathematically, and that is why we will not go into the
matter here. However, the situation remains essentially the
same in quantum theory. There also irreversibility begins as
the result of the limitation of the concept of wave function due
to a form of quantum instability.
Moreover, the idea of collisions and correlations may also be
used in quantum theory. Therefore, as in classical theory, the
second law prohibits long-range precollisional correlations.
The transition to a probability process introduces new en
tities, and it is in terms of these new entities that the second
law can be understood as an evolution from order to disorder.
This is an important conclusion. The second law leads to a
new concept of matter. We would like to describe this concept
now.
Active Matter
Once we associate entropy with a dynamic system, we come
back to Boltzmann's conception: the probability will be max
imum at equilibrium. The units we use to describe thermo
dynamic evolution will therefore behave in a chaotic way at
equilibrium. In contrast , in near-equilibrium conditions cor
relations and coherence will appear.
We come to one of our main conclusions; At all levels, be it
287
IRREVERSIBILITY-THE ENTROPY BARRIER
the level of macroscopic physics, the level of fluctuations, or
the microscopic level, nonequilibrium is the source of order.
Nonequilibrium brings "order out of chaos. " But as we al
ready mentioned, the concept of order (or disorder) is more
complex than was thought. It is only in some limiting situa
tions, such as with dilute gases, that it acquires a simple mean
ing in agreement with Boltzmann's pioneering work.
Let us once more contrast the dynamic description of the
physical world in terms of forces and fields with the thermo
dynamic description. As we mentioned, we can construct
computer experiments in which interacting particles initially
distributed at random form a lattice. The dynamic interpreta
tion would be the appearance of order through interparticle
forces. The thermodynamic interpretation is, on the contrary,
the approach to disorder (when the system is isolated), but to
disorder expressed in quite different units, which are in this
case collective modes involving a large number of particles. It
seems to us worthwhile to reintroduce the neologism we used
in Chapter VI to define the new units in terms of which the
system is incoherent at equilibrium: we call them "hypnons,"
sleepwalkers, since they ignore each other at equilibrium .
Each of them may b e as complex a s you wish (think about
molecules of the complexity of an enzyme), but at equilibrium
their complexity is turned "inward . " Again, inside a molecule
there is an intensive electric field, but this field in a dilute gas
is negligible as far as other molecules are concerned.
One of the m�in subjects in present-day physics is the prob
lem of elementary particles. However, we know that elemen
tary particles are far from elementary. New layers of structure
are disclosed at higher and higher energies. But what, after all,
is an elementary particle? Is the planet earth an elementary
particle? Certainly not, because part of this energy is in its
interaction with the sun, the moon, and the other planets. The
concept of elementary particles requires an "autonomy" that
is very difficult to describe in terms of the usual concepts .
Thke the case of electrons and photons. We are faced with a
dilemma: either there are no well-defined particles (because
the energy is partly between the electrons and protons), or
there are noninteracting particles if we can eliminate the inter
action. Even if we knew how to do that, it seems too radical a
ORDER OUT OF CHAOS
288
procedure. Electrons absorb photons or emit photons. A way
out may be to go to the physics of processes. The units, the
elementary particles, would then be defined as hypnons, as
the entities that evolve independently at equilibrium. We hope
that there soon will be experiments available to test this hy
pothesis; it would be quite appealing if atoms interacting with
photons (or unstable elementary particles) already carried the
arrow of time that expresses the global evolution of nature.
A subject widely discussed today is the problem of cosmic
evolution. How could the world near the moment of the Big
Bang be so "ordered"? Yet this order is necessary if we wish
to understand cosmic evolution as the gradual movement from
order to disorder.
To give a satisfactory answer we need to know what "hyp
nons" could have been appropriate to the extreme conditions
of temperature and density that characterized the early uni
verse. Thermodynamics alone will not, of course, solve these
problems ; neither will dynamics, even in its most refined form
field theory. That is why the unification of dynamics and ther
modynamics opens new perspectives.
In any case, it is striking how the situation has changed since
the formulation of the second law of thermodynamics one hun
dred fifty years ago. At first it seemed that the atomistic view
contradicted the concept of entropy. Boltzmann attempted to
save the mechanistic world view at the cost of reducing the
second law to a statement of probability with great practical
importance but no fundamental significance. We do not know
what the definitive solution will be; but today the situation is
radically different. Matter is not given. In the present-day view
it has to be constructed out of a more fundamental concept in
terms of quantum fields. In this construction of matter, ther
modynamic concepts (irreversibility, entropy) have a role to
play.
Let us summarize what has been achieved here. The central
role of the second law (and of the correlative concept of irre
versibility) at the level of macroscopic systems has already
been emphasized in Books One and 1\vo.
What we have tried to show in Book Three is that we now
can go beyond the macroscopic level, and discover the micro
scopic meaning of irreversibility.
However, this requires basic changes in the way in which we
289
IRREVERSIBILITY-THE ENTROPY BARRIER
conceive the fundamental laws of physics. It is only when the
classical point of view is lost-as it is in the case of sufficiently
unstable systems-that we can speak of 'intrinsic ra ndom
ness' and 'intrinsic irreversibility. '
It is for such systems that we may introduce a new enlarged
description of time in terms of the operator time T. As we have
shown in the example of the Baker transformation (Chapter IX
" From randomness to irreversibility"), this operator has a s
eigenfunctions partitions of the phase space (see Figure 39).
We come therefore to a situation quite reminiscent of that of
quantum mechanics . We have indeed two possible descrip
tions. Either we give ourselves a point in phase space, and
then we don't know to which partition it belongs, and there
fore we don't know its internal age ; or we know its internal
age, but then we know only the partition, but not the exact
localization of the point.
Once we have introduced the internal time T, we can use
entropy as a selection principle to go from the initial descrip
tion in terms of the distribution function p to a new one, p
where the distribution p has an intrinsic arrow of time, satisfy
ing the second law of thermodynamics. The basic difference
between p and p appears when these functions are expanded in
terms of the eigenfunction of the operator time T (see Chapter
VII , "The rise of quantum mechanics"). In p, all internal ages,
be they from past or future, appear symmetrically. In contrast,
in p, past and future play different roles. The past is included,
but the future remains uncertain. That is the meaning of the
arrow of time. The fascinating aspect is that there appears now
a relation betw�en initial conditions and the laws of change. A
state with an arrow of time emerges from a law, which has also
an arrow of time, and which transforms the state, however
keeping this arrow of time.
We have concentrated mostly on the classical situation.2o
However, our analysis applies as well to quantum mechanics,
where the situation is more complicated , as the existence of
Planck's constant destroys already the concept of a trajectory,
and leads therefore also to a kind of delocalization in phase
space. In quantum mechanics we have therefore to superpose
the quantum delocalization with delocalization due to irrevers
ibility.
As emphasized in Chapter VII, the two great revolutions in
ORDER OUT OF CHAOS
290
the physics of our century correspond to the incorporation, in
the fundamental structure of physics, of impossibilities foreign
to classical mechanics: the impossibility of signals propagating
with a velocity larger than the velocity of light , and the impos
sibility of measuring simultaneously coordinates and momen
tum.
It is not astonishing that the second principle, which as well
limits our abilit y to manipulate matter, also leads to deep
changes in t he structure of the basic laws of physics.
Let us conclude this part of our monograph wit h a word of
caution. The phenomenological theory of irreversible pro
cesses is at present well established. In contrast , the basic mi
croscopic theory of irreversible processes is quite new. At the
time of correcting the proofs of th i s book, experiments are in
preparation to test these views. As long as they have not been
performed, a speculative element is unavoidable.
CONCLUSION
FROM EARTH TO HEAVEN
THE REENCHANTMENT
OF NATURE
I n any attempt to bridge the domains of experience belonging
to the spiritual and physical s1des of our nature, time occupies
the key position.
A. S. EDDINGTON1
An Open Science
Science certainly involves manipulating nature, but it is also
an attempt to understand it, to dig deeper into questions that
have been asked generation after generation. One of these ques
tions runs like a leitmotiv, almost as an obsession, through this
book, as it does through the history of science and philosophy.
This is the question of the relation between being and becom
ing, between permanence and change.
We have mentioned pre-Socratic speculations: Is change,
whereby things are born and die, imposed from the outside on
some kind of inert matter? Or is it the result of the intrinsic and
independent activity of matter? Is an external driving force
necessary, or is becoming inherent in matter? Seventeenth
century science arose in opposition to the biological model of
a spontaneous and autonomous organization of natural beings.
But it was confronted with another fundamental alternative. Is
nature intrinsically random? Is ordered behavior merely the
transient result of the chance collisions of atoms and of their
unstable associations?
One of the main sources of fascination in modern science
was precisely the feeling that it had discovered eternal laws at
291
ORDER OUT OF CHAOS
292
the core of nature's transformations and thus had exorcised
time and becoming. This discovery of an order in nature pro
duced the feeling of intellectual security described by French
sociologist Levy-Bruhl:
Our feeling of intellectual security is so deeply anchored
in us that we even do not see how it could be shaken.
Even if we suppose that we could observe some phenom
enon seemingly quite mysterious, we still would remain
persuaded that our ignorance is only provisional, that this
phenomenon must satisfy the general laws of causality,
and that the reasons for which it has appeared will be
determined sooner or later. Nature around us is order
and reason, exactly as is the human mind. Our everyday
activity implies a perfect confidence in the universality of
the laws of nature. 2
This feeling of confidence in the " reason" of nature has
been shattered, partly as the result of the tumultuous growth
of science in our time. As we stated in the Preface , our vision
of nature is undergoing a radical change toward the multiple,
the temporal, and the complex. Some of these changes have
been described in this book.
We were seeking general, all-embracing schemes that could
be expressed in terms of eternal laws, but we have found time,
events, evolving particles. We were also searching for sym
metry, and here also we were surprised, since we discovered
symmetry-breaking processes on all levels, from elementary
particles up to biology and ecology. We have described in this
book the clash between dynamics, with the temporal symme
try it implies, and the second law of thermodynamics, with its
directed time.
A new unity is emerging: irreversibility is a source of order
at all levels. Irreversibility is the mechanism that brings order
out of chaos. How could such a radical transformation of our
views on nature occur in the relatively short time span of the
past few decades? We believe that it shows the important role
intellectual construction plays in our concept of reality. This
was very well expressed by Bohr, when he said to Werner Hei
senberg on the occasion of a visit at Kronberg Castle:
293
.
FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
Isn't it strange how this castle changes as soon as one
imagines that Hamlet lived here? As scientists we believe
that a castle consists only of stones, and admire the way
the architect put them together. The stones, the green
roofwith its patina, the wood carvings in the church, con
stitute the whole castle. None of this should be changed by
the fact that Hamlet lived here, and yet it is changed com
pletely. Suddenly the walls and the ramparts speak a dif
ferent language . . . . Yet all we really know about Hamlet
is that his name appears in a thirteenth-century chroni
cle . . . . But everyone knows the questions Shakespeare
had him ask, the human depths he was made to reveal,
and so he too had to have a place on earth, here in Kron
berg}
The question of the meaning of reality was the central sub
ject of a fascinating dialogue between Einstein and Tagore."'
Einstein emphasized that science had to be independent of the
existence of any observer. This led him to deny the reality of
time as irreversibility, as evolution. On the contrary, Thgore
maintained that even if absolute truth could exist, it would be
inaccessible to the human mind. Curiously enough, the pres
ent evolution of science is running in the direction stated by
the great Indian poet. Whatever we call reality, it is revealed to
us only through the active construction in which we partici
pate. As it is concisely expressed by D. S. Kothari, "The sim
ple fact is that no measurement, no experiment or observation
is possible without a relevant theoretical framework. "5
Time and Times
The statement that time is basically a geometric parameter
that makes it possible to follow the unfolding of the succession
of dynamical states has been asserted in physics for more than
three centuries. Emile Meyerson6 tried to describe the history
of modern science as the gradual implementation of what he
regarded as a basic category of human reason: the different
and the changing had to be reduced to the identical and the
permanent. Time had to be eliminated.
ORDER OUT OF CHAOS
294
Nearer to our own time, Einstein appears as the incarnation
of this drive toward a formulation of physics in which no refer
ence to irreversibility would be made on the fundamental level.
An historic scene took place at the Societe de Philosophie in
Paris on April 6, 1922,1 when Henri Bergson attempted to de
fend the cause of the multiplicity of coexisting "lived" times
against Einstein. Einstein's reply was absolute: he categori
cally rejected " philosophers' time." Lived experience cannot
save what has been denied by science.
Einstein's reaction was somewhat justified. Bergson had ob
viously misunderstood Einstein's theory of relativity. How
ever, there also was some prejudice on Einstein's part: duree,
Bergson's "lived time, " refers to the basic dimensions of be
coming, the irreversibility that Einstein was willing to admit
only at the phenomenological level. We have already referred
to Einstein's conversations with Carnap (see Chapter VII).
For him distinctions among past, present, and future were out
side the scope of physics.
It is fascinating to follow the correspondence between Ein
stein and the closest friend of his young days in Zurich, Michele
Besso.8 Although he was an engineer and scientist, toward the
end of his life Besso became increasingly concerned with phi
losophy, literature, and the problems that surround the core of
human existence. Untiringly he kept asking the same ques
tions: What is irreversibility? What is its relationship with the
laws of physics? And untiringly Einstein would answer with a
patience he showed only to this closest friend: irreversibility is
merely an illusion produced by "improbable" initial conditions.
This dialogue continued over many years until Besso, older
than Einstein by eight years, passed away, only a few months
before Einstein's death. In a last letter to Besso's sister and
son, Einstein wrote: "Michele has left this strange world just
before me. This is of no importance. For us convinced physi
cists the distinction between past, present and future is an illu
sion, although a persistent one." In Einstein's drive to perceive
the basic laws of physics, the intelligible was identified with
the immutable.
Why was Einstein so strongly opposed to the introduction
of irreversibility into physics? We can only guess. Einstein
was a rather lonely man; he had few friends, few coworkers,
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FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
few students. He lived in a sad time: the two World Wars , the
rise of anti-Semitism. It is not surprising that for Einstein sci
ence was the road that led to victory over the turmoil of time.
What a contrast, however, with his scientific work. His world
was full of observers, of scientists situated in various coordi
nate systems in motion with one another, situated on various
stars differing by their gravitational fields. All these observers
were exchanging information through signals all over the uni
verse. What Einstein wanted to preserve above all was the ob
jective meaning of this communication. However, we can
perhaps state that Einstein stopped short of accepting that
communication and irreversibility are closely related. Com
munication is at the base of what probably is the· most irrevers
ible process accessible to the human mind, the progressive
increase of knowledge.
The Entropy Barrier
In Chapter IX we described the second law as a selection prin
ciple: to each initial condition there corresponds an "informa
tion. " All initial conditions for which this information is finite
are permitted. However, to reverse the direction of time we
would need infinite information ; we cannot produce situations
that would evolve into our past ! This is the entropy barrier we
have introduced.
There is an interesting analogy with the concept of the ve
locity of light as the maximum velocity of transmission of sig
nals. As we have seen in Chapter VII, this is one of the basic
postulates of Einstein's relativity theory. The existence of the
velocity of light barrier is necessary to give meaning to causal
ity. Suppose we could, in a science-fiction ship, leave the earth at
a velocity greater than the velocity of light. We could overtake
light signals and in this way precede our own past. Likewise,
the entropy barrier is necessary to give meaning to communi
cation. We have already mentioned that irreversibility and
communication are closely related. Norbert Wiener has ar
gued that the existence of two directions of time would have
disastrous consequences. It is worthwhile to cite a passage
from his well-known book Cybernetics:
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296
Indeed, it is a very interesting intellectual experiment
to make the fantasy of an intelligent being whose time
should run the other way to our own. To such a being, all
communication with us would be impossible. Any signal
he might send would reach us with a logical stream of
consequents from his point of view, antecedents from
ours. These antecedents would already be in our experi
ence, and would have served to us as the natural explana
tion of his signal, without presupposing an intelligent
being to have sent it. If he drew us a square, we should
see the remains of his figure as its precursors, and it would
seem to be the curious crystallization-always perfectly
explainable-of these remains. Its meaning would seem
to be as fortuitous as the faces we read into mountains
and cliffs. The drawing of the square would appear to us
as a catastrophe-sudden indeed, but explainable by nat
ural laws-by which that square wo_Id cease to exist.
Our counterpart would have exactly similar ideas con
cerning us. Within any world with which we can commu
nicate, the direction of time is uniform.9
It is precisely the infinite entropy barrier that guarantees the
uniqueness of the direction of time, the impossibility of switching
from one direction of time to the opposite one.
In the course of this book, we have stressed the importance
of demonstrations of impossibility. In fact, Einstein was the
first to grasp that importance when he based his concept of rela
tive simultaneity on the impossibility of transmitting informa
tion at a velocity greater than that of light. The whole theory of
relativity is built around the exclusion of "unobservable" simul
taneities. Einstein considered this step as somewhat similar to
the step taken in thermodynamics when perpetual motion was
excluded. But some of his contemporaries-Heisenberg, for
example-emphasized an important difference bet ween these
two impossibilities. In the case of thermodynamics, a certain
situation is defined as being absent from nature; in the case of
relativity, it is an observation that is defined as impossible
that is, a type of dialogue, of communication between nature
and the person who describes it. Thus Heisenberg saw himself
as following Einstein's example, in spite of Einstein's skepti
cism, when he grounded quantum mechanics on the ex.clusion
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FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
of what the quantum uncertainty principle defines as unobser
vable.
As long as the second law was considered to express only a
practical improbability, it had little theoretical interest. You
could always hope to overcome it by sufficient technical skill.
But we have seen that this is not so. At its roots there is a
selection of possible initial states. It is only after these states
have been selected that the probability interpretation becomes
possible. Indeed, as Boltzmann stated for the first time, the
increase of entropy expresses the increase of probability, of
disorder. However, his interpretation results from the con
clusion that entropy is a selection principle breaking the time
symmetry. It is only after this symmetry-breaking that any
probabilistic interpretation becomes possible.
In spite of the fact that we have recouped much of Boltz
mann's interpretation of entropy, the basis of our interpreta
tion of his second law is radically different, since we have in
succession
the second law as a symmetry-breaking selection principle
�
probabilistic interpretation
�
irreversibility as increase of disorder
It is only the unification of dynamics and thermodynamics
through the introduction of a new selection principle that gives
the second law its fundamental importance as the evolutionary
paradigm of the sciences. This point is of such importance that
we shall dwell on it in more detail.
The Evolutionary Paradigm
f
The world of dynamics, be it classical or quantum, is a reversible world. As we have emphasized in Chapter VIII, no
evolution can be ascribed to this world ; the "information" ex
pressed in terms of dynamical units remains constant. It is
therefore of great importance that the existence of an evolu
tionary paradigm can now be established in physics-not only
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298
on the level of macroscopic description but also on all levels.
Of course, there are conditions: as we have seen, a minimum
complexity is necessary. But the immense importance of irre
versible processes shows that this requirement is satisfied for
most systems of interest. Remarkably, the perception of ori
ented time increases as the level of biological organization in
creases and probably reaches its culminating point in human
consciousness.
How general is this evolutionary paradigm? It includes iso
lated systems that evolve to disorder and open systems that
evolve to higher and higher forms of complexity. It is not sur
prising that the entropy metaphor has tempted a number of
writers dealing with social or economic problems. Obviously
here we have to be careful; human beings are not dynamic
objects, and the transition to thermodynamics cannot be for
mulated as a selection principle maintained by dynamics. On
the human level irreversibility is a more fundamental concept,
which is for us inseparable from the meaning of our existence.
Still it is essential that in this perspective we no longer see the
internal feeling of irreversibility as a subjective impression that
alienates us from the outside world, but as marking our partic
ipation in a world dominated by an evolutionary paradigm.
Cosmological problems are notoriously difficult. We still do
not know what the role of gravitation was in the early universe.
Can gravitation be included in some form of the second law, or
is there a kind of dialectical balance between thermodynamics
and gravitation? Certainly irreversibility could not have ap
peared abruptly in a time-reversible world. The origin of irre
versibility is a cosmological problem and would require an
analysis of the universe in its first stages. Here our aim is more
modest. What does irreversibility mean today? How does it
relate to our position in the world we describe?
Actors and Spectators
The denial of becoming by physics created deep rifts within
science and e stranged science from philosophy. What had
originally been a daring wager with the dominant Aristotelian
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FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
tradition gradually became a dogmatic asser tion directed
against all those (chemists, biologists, physicians) for whom a
qualitative diversity existed in nature. At the end of the nine
teenth century this conflict had shifted from inside science to
the relation between "science" and the rest of culture, es
pecially philosophy. We have described in Chapter III this as
pect of the history of Western thought, with its continual
struggle to achieve a new unity of knowledge. T he "lived
time" of the phenomenologists, the Lebenswelt opposed to
the objective world of science, may be related to the need to
erect bulwarks against the invasion of science.
Today we believe that the epoch of certainties and absolute
oppositions is over. P hysicists have no privilege whatsoever to
any kind of extraterritor iality. As scientists they belong to
their culture, to which, in their turn, they make an essential
contribution. We have reached a situation close to the one rec
ognized long ago in sociology: Merleau-Ponty had already
stressed the need to keep in mind what he termed a "truth
within situations. "
So long as I keep before me the ideal of an absolute ob
ser ver, of knowledge in the absence of any viewpoint, I
can only see my situation as being a source of error. But
once I have acknowledged that through it I am geared to
all actions and all knowledge that are meaningful to me,
and that it is gradually filled with everything that may be
for me, then my contact with the social in the finitude of
my situation is revealed to me as the starting point of all
truth, including that of science and, since we have some
idea of the truth, since we are inside truth and cannot get
outside it, all that I can do is define a truth within the
situation. 10
It is this conception of knowledge as both objective and par
ticipatory that we have explored through this book.
In his Themesll Merleau-Ponty also asserted that the "phil
osophic" discoveries of science, its basic conceptual transfor
mations, are often the result of negative discoveries. which
provide the occasion and the starting point for a reversal of
point of view. Demonstrations of impossibility, whether in rel-
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ativity, quantum mechanics, or thermodynamics, have shown
us that nature cannot be described "from the outside," as if by
a spectator. Description is dialogue, communication, and this
communication is subject to constraints that demonstrate that
we are macroscopic beings embedded in the physical world.
We may summarize the situation as we see it today in the
following diagram:
observer
-----+
t
dissipative structures
t
irreversibility
+-
randomness
+-
dynamics
I
unstable dynamic systems
We start from the observer, who measures coordinates and
momenta and studies their change in time. This leads him to
the discovery of unstable dynamic systems and other concepts
of intrinsic randomness and intrinsic irreversibility, as we dis
cussed them in Chapter IX. Once we have intrinsic irrevers
ibility and entropy, we come in far-from-equilibrium systems
to dissipative structures, and we can understand the time
oriented activity of the observer.
There is no scientific activity that is not time-oriented. The
preparation of an experiment calls for a distinction between
"before .. and "after... It is only because we are aware of irre
versibility that we can recognize reversible motion. Our dia
gram shows that we have now come full circle, that we can see
ourselves as part of the universe we describe.
The scheme we have presented is not an a priori scheme
deducible from some logical structure. There is, indeed, no
logical necessity for dissipative structures actually to exist in
nature; the "cosmological fact" of a universe far from equilib
rium is needed for the macroscopic world to be a world inhabited
by "observers"-that is, to be a living world. Our scheme thus
does not correspond to a logical or epistemological truth but
refers to our condition as macroscopic beings in a world far
from equilibrium. Moreover, an essential characteristic of our
scheme is that it does not suppose any fundamental mode of
description; each level of description is implied by another and
implies the other. We need a multiplicity of levels that are all
connected, none of which may have a claim to preeminence.
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FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
We have already emphasized that irreversibility is not a uni
versal phenomenon. We can perform experiments correspond
ing to thermodynamic equilibrium in limited portions of space.
Moreover, the impor tance of time scales varies. A stone
evolves according to the time scale of geological evolution;
human societies, especially in our time, obviously have a
much shorter time scale. We have already mentioned that irre
versibility starts with a minimum complexity of the dynamic
systems involved. It is interesting that with an increase of
complexity, going from the stone to human societies, the role
of the arrow of time, of evolutionary rhythms, increases. Mo
lecular biology shows that everything in a cell is not alive in
the same way. Some processes reach equilibrium, others are
dominated by regulatory enzymes far from equilibrium. Sim
ilarly, the arrow of time plays quite different roles in the uni
verse around us. From this point of view, in the sense of this
time-oriented activity, the human condition seems unique. It
seems to us, as we said in Chapter IX, quite important that
irreversibility, the arrow of time, implies randomness. "Time
is construction." This conclusion, one that Valery 12 reached
quite independently, carries a message that goes beyond sci
ence proper.
A Whirlwind in a Turbulent Nature
In our society, with its wide spectrum of cognitive techniques,
science occupies a peculiar position, that of a poetical inter
rogation of nature, in the etymological sense that the poet is a
"maker"-active, manipulating, and exploring. Moreover, sci
ence is now capable of respecting the nature it investigates.
Out of the dialogue with nature initiated by classical science,
with its view of nature as an automaton, has grown a quite
different view in which the activity of questioning nature is
part of its intrinsic activity.
As we have written at the start of this chapter, our feeling of
intellectual security has been shattered. We can now appreci
ate in a nonpolemical fashion the relation between science and
philosophy. We have already mentioned the Einstein-Bergson
conflict. Bergson was certainly "wrong" on some technical
points, but his task as a philosopher was to attempt to make
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302
explicit inside physics the aspects of time he thought science
was neglecting.
Exploring the implications and the coherence of those funda
mental concepts, which appear both scientific and philosoph
ical, may be risky, but it can be very fruitful in the dialogue
between science and philosophy. Let us illustrate this with
some brief references to Leibniz, Peirce, Whitehead, and Lu
cretius.
Leibniz introduced the strange concept of monads, non
communicating physical entities that have "no windows through
which something can get in or out. " His views have often been
dismissed as mad, and still, as we have seen in Chapter 11, it is
an essential property of all integrable systems that there exist
a transformation that may be described in terms of nonin
teracting entities. These entities translate their own initial
state throughout their motion, but at the same time, like mon
ads, they coexist with all the others in a "preestablished" har
mony: in this representation, the state of each entity, although
perfectly self-determined, reflects the state of the whole sys
tem down to the smallest detail.
All integrable systems thus can be viewed as "monadic" sys
tems. Conversely, Leibnizian monadology can be translated into
dynamic language: the universe is an integrable system.13
Monadology thus becomes the most consequential formula
tion of a universe from which all becoming is eliminated. By
considering Leibniz's efforts to understand the activity of mat
ter, we can measure the gap that separates the seventeenth
century from our time. The tools were not yet ready; it was
impossible, on the basis of a purely mechanical universe, for
Leibniz to give an account of the activity of matter. Still some
of his ideas, that substance is activity, that the universe is an
interrelated unit, remain with us and are today taking on a new
form.
We regret that we cannot devote sufficient space to the work
of Charles S. Peirce. At least let us cite one remarkable pas
sage:
You have all heard of the dissipation of energy. It is found
that in all transformations of energy a part is converted
into heat and heat is always tending to equalize its tem-
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FROM EARTH TO HEAVEN- THE REENCHANTMENT OF NATURE
perature. The consequence is that the energy of the uni
verse is tending by virtue of its necessary laws toward a
death of the universe in which there shall be no force but
heat and the temperature everywhere the same. . . .
But although no force can counteract this tendency,
chance may and will have the opposite influence. Force is
in the long run dissipative; chance is in the long run con
centrative. The dissipation of energy by the regular laws
of nature is by these very laws accompanied by circum
stances more and more favorable to its reconcentration
by chance. There must therefore be a point at which the
two tendencies are balanced and that is no doubt the ac
tual condition of the whole universe at the present time. 4
J
Once again, Peirce's metaphysics was considered as one more
example of a philosophy alienated from reality. But, in fact,
today Peirce's work appears to be a pioneering step toward the
understanding of the pluralism involved in physical laws.
Whitehead's philosophy takes us to the other end of the spec
trum. For him, being is inseparable from becoming. Whitehead
wrote: "The elucidation of the meaning of the sentence 'every
thing flows' is one of metaphysics' main tasks. " 15 Physics and
metaphysics are indeed coming together today in a conception
of the world in which process, becoming, is taken as a primary
constituent of physical existence and where, unlike Leibniz'
monads, existing entities can interact and therefore also be
born and die.
The ordered world of classical physics, or a monadic theory
of parallel changes, resembles the equally parallel, ordered,
and eternal fall of Lucretius' atoms through infinite space. We
have already mentioned the clinamen and the instability of
laminar flows . But we can go farther. As Serresi6 points out,
the infinite fall provides a model on which to base our con
ception of the natural genesis of the disturbance that causes
things to be born. If the vertical fall were not disturbed "with
out reason" by the clinamen, which leads to encounters and
associations between uniformly falling atoms, no nature could
be created; all that would be reproduced would be the repeti
tive connection between equivalent causes and effects gov
erned by the laws of fate (foedera fati).
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304
Denique si semper motus conectitur omnls
et uetere exoritur [semper] novus ordine certo
nee declinandofaciunt primordia motus
principium quoddam quod fatl foedera rumpat,
ex lnfinito ne causa m causa sequatur,
libera per terras unde haec animantibus exstat
. . .
?17
Lucretius may be said to have invented the clinamen in the
same way that archaeological remains are "invented": one
"guesses" they are there before one begins to dig. If only uni
formly reversible trajectories existed, where would the irre
versible processes we produce and experience come from?
The point where the trajectories cease to be determined,
where thefoederafati governing the ordered and monotonous
world of deterministic change break down, marks the begin
ning of nature. It also marks the beginning of a new science
that describes the birth, proliferation, and death of natural
beings. "The physics of falling, of repetition, of rigorous con
catenation is replaced by the creative science of change and
circumstances. "18 The foedera fati are replaced by the
foedera naturae , which, as Serres emphasizes, denote both
"laws" of nature-local, singular, historical relations-and an
"alliance," a form of contract with nature.
In Lucretian physics we thus again find the link we have
discovered in modern knowledge between the choices under
lying a physical description and a philosophic, ethical, or re
ligious conception relating to man's situation in nature. The
physics of universal connections is set against another science
that in the name of law and domination no longer struggles
with disturbance or randomness. Classical science from Ar
chime des to Clausius was opposed to the science of turbu
lence and of bifurcating changes.
It is here that Greek wisdom reaches one of its pinnacles.
Where man is in the world, of the world, in matter, of
matter, he is not a stranger, but a friend, a member of the
family, and an equal. He has made a pact with things.
Conversely, many other systems and many other sci
ences are based on breaking this pact. Man is a stranger
to the world, to the dawn, to the sky, to things. He hates
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FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
them, and fights them. His environment is a dangerous
enemy to be fought, to be kept enslaved
Epicurus
and Lucretius live in a reconciled universe. Where the
science of things and the science of man coincide. I am a
disturbance, a whirlwind in turbulent nature. 1�
.
.
•
.
Beyond Tautology
The world of classical science was a world in which the only
events that could occur were those deducible from the in
stantaneous state of the system. Curiously, this conception,
which we have traced back to Galileo and Newton, was not
new at that time. Indeed, it can be identified with Aristotle's
conception of a divine and immutable heaven. In Aristotle's
opinion, it was only the heavenly world to which we could
hope to apply an exact mathematical description. In the Intro
duction, we echoed the complaint that science has "disen
chanted" the world. But this disenchantment is paradoxically
due to the glorification of the ear thly world, henceforth
worthy of the kind of intellectual pursuit Aristotle reserved for
heaven. Classical science denied becoming, natural diversity,
both considered by Aristotle as attributes of the sublunar, in
ferior world. In this sense, classical science brought heaven to
earth. However, this apparently was not the intention of the
fathers of modern science. In challenging Aristotle's claim that
mathematics ends where nature begins, they did not seek to
discover the immutable concealed behind the changing, but
rather to extend changing, corruptible nature to the bound
aries of the universe. In his Dialogue Concerning the Two
Chief World Systems, Galileo is amazed at the notion that the
world would be a nobler place if the great flood had left only a
sea of ice behind, or if the earth had the incorruptible hard
ness of jasper; let those who think the earth would be more
beautiful after being changed into a crystal ball be trans
formed by Medusa's stare into a diamond statue!
But the objects chosen by the first physicists to explore the
validity of a quantitative description-that is, the ideal pendu
lum with its conservative motion, simple machines, planetary
orbits, etc.-were found to correspond to a unique mathemati-
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306
cal description that actually reproduced the divine ideality of
Aristotle's heavenly bodies.
Like Aristotle's gods, the objects of classical dynamics are
concerned only with themselves. They can learn nothing from
the outside. At any instant, each point in the system knows all
it will ever need to know-that is, the distribution of masses in
space and their velocities. Each state contains the whole truth
concerning all possible other states, and each can be used to
predict the others, whatever their respective positions on the
time a xis. In this sense, this description leads to a tautology,
since both future and past are contained in the present.
The radical change in the outlook of modern science, the
transition toward the temporal, the multiple, may be viewed as
the reversal of the movement that brought Aristotle's heaven to
earth. Now we are bringing earth to heaven. We are discover
ing the primacy of time and change, from the level of elemen
tary particles to cosmological models.
Both at the macroscopic and microscopic levels, the natural
sciences have thus rid themselves of a conception of objective
reality that implied that novelty and diversity had to be denied
in the name of immutable universal laws. They have rid them
selves of a fascination with a rationality taken as closed and a
knowledge seen as nearly achieved. They are now open to the
unexpected, which they no longer define as the result of imperfect knowledge or insufficient control.
This opening up of science has been well defined by Serge
Moscovici as the "Keplerian revolution," to distinguish it
from the "Copernican revolution" in which the idea of an ab
solute point of view was maintained. In many of the passages
cited in the Introduction to this book, science was likened to a
"disenchantment" of the world. Let us quote Moscovici's de
scription of the changes going on in the sciences today:
·
Science has become involved in this adventure, our ad
venture, in order to renew everything it touches and
warm all that it penetrates-the earth on which we live
and the truths which enable us to live. At each turn it is
not the echo of a demise, a bell tolling for a passing away
that is heard, but the voice of rebirth and beginning, ever
afresh, of mankind and materiality, fi xed for an instant in
their ephemeral permanence. That is why the great dis-
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FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
coveries are not revealed on a deathbed like that of
Copernicus, but offered like Kepler's on the road of
dreams and passion.2o
The Creative Course of Time
It is often said that without Bach we would not have had the
"St. Matthew Passion" but that relativity would have been dis
covered without Einstein. Science is supposed to take a deter
ministic course, in contrast with the unpredictability involved
in the history of the arts. W hen we look back on the strange
history of science, three centuries of which we have tried to
outline, we may doubt the validity of such assertions. There
are striking examples of facts that have been ignored because
the cultural climate was not ready to incorporate them into a
consistent scheme. The discovery of chemical clocks probably
goes back to the nineteenth century, but their result seemed to
contradict the idea of uniform decay to equilibrium. Mete
orites were thrown out of the Vienna museum because there
was no place for them in the description of the solar system.
Our cultural environment plays an active role in the questions
we ask, but beyond matters of style and social acceptance, we
can identify a number of questions to which each generation
returns.
The question of time is certainly one of those questions.
Here we disagree somewhat with Thomas Kuhn's analysis of
the formation of "normal" science.21 Scientific activity best
corresponds to Kuhn's view when it is considered in the con
text of the contemporary university, in which research and the
training of future researchers is combined. Kuhn's analysis, if
it is taken as a description of science in general, leading to
conclusions about what knowledge must be, can be reduced to
a new psychosocial version of the positivist conception of sci
entific development, namely, the tendency to increasing spe
cialization and compartmentalization; the identification of
"normal" scientific behavior with.that of the "serious," "si
lent" researcher who wastes no time on "general" questions
about the overall significance of his research but sticks to spe
cialized problems; and the essential independence of scientific
development from cultural, economic, and social problems.
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306
The academic structure in which the "normal science" de·
scribed by Kuhn came into being took shape in the nineteenth
century. Kuhn emphasizes that it is by repeating in the form of
exercises solutions to the paradigmatic problems of previous
generations that students learn the concepts upon which re
search is based. It is in this way that they are given the criteria
that define a problem as interesting and a solution as accept
able. The transition from student to researcher takes place
gradually; the scientist continues to solve problems using sim
ilar techniques.
Even in our time, for which Kuhn's description has the
greatest relevance, it refers to only one specific aspect of sci
entific activity. The importance of this aspect varies according
to the individual researchers and the institutional context.
In Kuhn's view the transformation of a paradigm appears as
a crisis: instead of remaining a silent, almost invisible rule,
instead of remaining unspoken, the paradigm is actually ques
tioned. Instead of working in unison, the members of the com
munity begin to ask "basic" questions and challenge the
legitimacy of their methods. The group, which by training was
homogeneous, now diversifies. Different points of view, cul
tural exper.iences, and philosophic convictions are now ex
pressed and often play a decisive role in the discovery of a new
paradigm. The emergence of the new paradigm further in
creases the vehemence of the debate. The rival paradigms are
put to the test until the academic world determines the victor.
W ith the appearance of a new generation of scientists, silence
and unanimity take over again. New textbooks are written,
and once again things "go without saying. "
In this view the driving force behind scientific innovation is
the intensely conservative behavior of scientific communities,
which stubbornly apply to nature the same techniques, the
same concepts, and always end up by encountering an equally
stubborn resistance from nature. W hen nature is eventually
seen as refusing to express itself in the accepted language, the
crisis explodes with the kind of violence that results from a
breach of confidence. At this stage, all intellectual resources
are concentrated on the search for a new language. Thus sci
entists have to deal with crises imposed upon them against
their will.
The questions we have investigated have led us to emphasize
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FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
aspects that differ considerably from those to which Kuhn's
description applies. We have dwelled on continuities, not the
"obvious" continuities but the hidden ones, those involving
difficult questions rejected by many as illegitimate or false but
that keep coming back generation after generation-questions
such as the dynamics of complex systems, the relation of the
irreversible world of chemistry and biology with the reversible
description provided by classical physics. In fact, the interest
of such questions is hardly surprising. To us, the problem is
rather to understand how they could ever have been neglected
after the work of Diderot, Stahl, Venel, and others.
The past one hundred years have been marked by several
crises that correspond closely to the description given by
Kuhn-none of which were sought by scientists. Examples
are the discovery of the instability of elementary particles, or
of the evolving universe. However, the recent history of sci
ence is also characterized by a series of problems that are the
consequences of deliberate and lucid questions asked by sci
entists who knew that the questions had both scientific and
philosophical aspects. Thus scientists are not doomed to be
have like "hypnons"!
It is important to point out that the new scientific develop
ment we have described, the incorporation of irreversibility
into physics, is not to be seen as some kind of "revelation,"
the possession of which would set its possessor apart from the
cultural world he lives in. On the contrary, this development
clearly reflects both the internal logic of science and the
cultural and social context of our time.
In particular, how can we consider as accidental that the
rediscovery of time in physics is occurring at a time of extreme
acceleration in human history? Cultural context cannot be the
complete answer, but it cannot be denied either. We have to
incorporate the complex relations between "internal" and
"external" determinations of the production of scientific con
cepts.
In the preface of this book, we have emphasized that its
French title (La nouvelle alliance) expresses the coming to
gether of the "two cultures". Perhaps the confluence is no
w h e r e a s c l e a r a s i n t h e p r o b l e m of the microscopic
foundations of irreversibility we have studied in Book Three.
As mentioned repeatedly, both classical and quantum me-
ORDER OUT OF CHAOS
310
chanics are based on arbitrary initial conditions and deter
ministic laws (for trajectories or wave functions). In a sense,
laws made simply explicit what was already present in the ini
tial conditions. This is no longer the case when irreversibilty is
taken into account. In this perspective, initial conditions arise
from previous evolution and are transformed into states of the
same class through subsequent evolution.
We come therefore close to the central problem of Western
ontology: the relation between Being and Becoming. We have
given a brief account of the problem in Chapter III. It is
remarkable that two of the most influential works of the cen
tury were precisely devoted to this problem. We have in mind
W hitehead's Process and Reality and Heidegger's Sein und
Zeit. In both cases, the aim is to go beyond the identification
of Being with timelessness, following the Voie Royale of west
ern philosophy since Plato and Aristotle.22
But obviously, we cannot reduce Being to Time, and we
cannot deal with a Being devoid of any temporal connotation.
The direction which the microscopic theory of irreversibility
takes gives a new content to the speculations of W hitehead
and Heidegger.
It would go beyond the aim of this book to develop this prob
lem in greater detail; we hope to do it elsewhere. Let us notice
that initial conditions, as summarized in a state of the system,
are associated with Being; in contrast, the laws involving tem
poral changes are associated with Becoming.
In our view, Being and Becoming are not to be opposed one
to the other: they express two related aspects of reality.
A state with broken time symmetry arises from a law with
broken time symmetry, which propagates it into a state be
longing to the same category.
In a recent monograph (From Being to Becoming), one of
the authors concluded in the following terms: "For most of the
founders of classical science-even for Einstein-science was
an attempt to go beyond the world of appearances, to reach a
timeless world of supreme rationality-the world of Spinoza.
But perhaps there is a more subtle form of reality that involves
both laws and games, time and eternity. "
This is precisely the direction which the microscopic theory
of irreversible processes is taking.
311
FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
The Human Condition
We agree completely with Herman Weyl:
Scientists would be wrong to ignore the fact that theo
retical construction is not the only approach to the phe
nomena of life; another way, that of understanding from
within (interpretation), is open to us. . . . Of myself, of
my own acts of perception, thought, volition, feeling and
doing, I have a direct knowledge entirely different from
the theoretical knowledge that represents the "parallel"
cerebral processes. in symbols. This inner awareness of
myself is the basis for the understanding of my fellow
men whom I meet and acknowledge as beings of my own
kind, with whom I communicate sometimes so intimately
as to share joy and sorrow with them. 23
Until recently, however, there was a striking contrast. The ex
ternal universe appeared to be an automaton following deter
ministic causal laws, in contrast with the spontaneous activity
and irreversibility we experience. The two worlds are now
drawing closer together. Is this a loss for the natural sciences?
Classical science aimed at a "transparent" view of the phys
ical universe. In each case you would be able to identify a
cause and an effect. W henever a stochastic description be
comes necessary, this is no longer so. We can no longer speak
of causality in each individual experiment; we can only speak
about statistical causality. This has, in fact, been the case ever
since the advent of quantum mechanics, but it has been greatly
amplified by recent developments in which randomness and
probability play an essential role, even in classical dynamics
or chemistry. Therefore, the modern trend as compared to the
classical one leads to a kind of "opacity" as compared to the
transparency of classical thought.
Is this a defeat for the human mind? This is a difficult ques
tion. As scientists, we have no choice; we cannot describe for
you the world as we would like to see it, but only as we are
able to see it through the �ombined impa�t of experimental
ORDER OUT OF CHAOS
312
results and new theoretical concepts. Also, we believe that
this new situation reflects the situation we seem to find in our
own mental activity. Classical psychology centered around
conscious, transparent activity; modern psychology attaches
much weight to the opaque functioning of the unconscious.
Perhaps this is an image of the basic features of human exis
tence. Remember Oedipus, the lucidity of his mind in front of
the sphinx and its opacity and darkness when confronted with
his own origins. Perhaps the coming together of our insights
about the world around us and the world inside us is a satisfy
ing feature of the recent evolution in science that we have tried
to describe.
It is hard to avoid the impression that the distinction be
tween what exists in time, what is irreversible, and, on the
other hand, what is outside of time, what is eternal, is at the
origin of human symbolic activity. Perhaps this is especially so
in artistic activity. Indeed, one aspect of the transformation of
a natural object, a stone, to an object of art is closely related to
our impact on matter. Artistic activity breaks the temporal
symmetry of the object. It leaves a mark that translates our
temporal dissymmetry into the temporal dissymmetry of the
object. Out of the reversible, nearly cyclic noise level in which
we live arises music that is both stochastic and time-oriented.
The Renewal of Nature
It is quite remarkable that we are at a moment both of pro
found change in the scientific concept of nature and of the
structure of human society as a result of the demographic ex
plosion. As a result, there is a need for new relations between
man and nature and between man and man. We can no longer
accept the old a priori distinction between scientific and ethi
cal values. This was possible at a time when the external world
and our internal world appeared to conflict, to be nearly
orthogonal. Today we know that time is a construction and
therefore carries an ethical responsibility.
The ideas to which we have devoted much space in this
book-the ideas of instability, of fluctuation-diffuse into the
social sciences. We know now that societies are immensely
313
FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE
complex systems involving a potentially enormous number of
bifurcations exemplified by the variety of cultures that have
evolved in the relatively short span of human history. We know
that such systems are highly sensitive to fluctuations. This
leads both to hope and a threat: hope, since even small fluc
tuations may grow and change the overall structure. As a re
sult, individual activity is not doomed to insignificance. On
the other hand, this is also a threat, since in our universe the
security of stable, permanent rules seems gone forever. We are
living in a dangerous and uncer tain world that inspires no
blind con fidence, but perhaps only the same feeling of
qualified hope that some Talmudic texts appear to have at
tributed to the God of Genesis:
1\venty-six attempts preceded the present genesis, all of
which were destined to fail. The world of man has arisen
out of the chaotic heart of the preceding debris; he too is
exposed to the risk of failure, and the return to nothing.
"Let's hope it works" [Halway Sheyaamod] exclaimed
God as he created the World, and this hope, which has
accompanied all the subsequent history of the world and
mankind, has emphasized right from the outset that this
history is branded with the mark of radical uncertainty.24
NOTES
Introduction
I. I . BERLIN, Against the Current, selected writings ed. H . Hardy
(New York: The Viking Press, 1980), p. xxvi.
2. See TITUS LucRETIUS CARUS, De Natura Rerum, Book I, v.
267-70. ed . and comm. C. Bailey (Oxford: Oxford University
Press 1947, 3 vols.)
3 . R. LENOBLE, Histoire de /'idee de nature (Paris: Albin Michel,
1969).
4. B . PASCAL, "Pensees ," frag. 792, in Oeuvres Completes (Paris:
Brunschwig-Boutroux-Gazier, 1904-14).
5. J. MoNOD, Chance and Necessity (New York: Vintage Books,
1972), pp. 172-73.
6. G. V1co, The New Science, trans. T. G. Bergin and M. H. Fisch
(New York: 1968), par. 33 1 .
7 . J. P. VERNANT e t al. , Divination e t rationalite, esp. J. BOTTERO,
"Symptomes, signes, ecritures" (Paris: Seuil, 1974).
8. A. KoYRE , Galileo Studies (Hassocks, Eng . : The Harvester
Press, 1 978).
9. K. PoPPER , Objective Knowledge (Oxford : Clarendon Press,
1972).
10. P. FoRMAN, "Weimar Culture, Causality and Quantum Theory,
1918-1927; Adaptation by German Physicists and Mathemati
cians to an Hostile Intellectual Environment," Historical Stud
ies in Physical Sciences, Vol . 3 ( 1 97 1 ), pp. 1-1 15.
1 1 . J. NEEDHAM and C . A. RoNAN, A Shorter Science and Civiliza
tion in China, Vol . I (Cambridge: Cambridge University Press,
1 978), p. 1 70.
12. A. EDDINGTON, The Nature of the Physical World (Ann Arbor:
University of Michigan Press, 1958), pp. 68-80.
1 3. Ibid. , p. 103.
14. BERLIN, op. cit . , p. 1 09.
15. K. POPPER, Unended Quest (La Salle, Ill . : Open Court Publish
ing Company, 1976), pp. 161-62.
16. G. BRUNO, 5th dialogue, "De Ia causa," Opere ltaliane, I (Bari:
1907); cf. I. LECLERC, The Nature of Physical Existence
(London: George Allen & Unwin, 1 972).
315
ORDER OUT OF CHAOS
316
1 7. P. VA L�RY, Cahiers, (2 vols.) ed. Mrs. Robinson-Valery, (Paris:
Gallimard, 1 973-74).
1 8. E . ScHRODINGER, "Are there Quantum Jumps?" The British
Journal for the Philosophy of Science, Vol. I II ( 1952), pp. 1 091 0 ; this text has been quoted with indignation by P. W. Bridg
mann in his contribution to Determinism and Freedom in the
Age of Modern Science, ed. S. Hook (New York: N ew York
University Press, 1 958).
19 A. E INSTE I N , "Pri n zi p i e n d er Forsc h un g , Red e zur 60.
Geburstag van Max Planck" (1918) in Mein Weltbild, U llstein
Verlag 1 977, pp. 1 07-10, trans. Ideas and Opinions (New York:
Crown, 1 954), pp. 224-27.
20. R DDRRENM ATT, The Physicists. (New York: Grove, 1 964).
2 1 . S. MoscoviCI, Essai sur /'histoire humaine de Ia nature, Collec
tion Champs (Paris: Flammarion, 1 977).
22. Quoted in Ronan, op. cit . , p. 87.
23. MoNon , op. cit . , p. 1 80.
..
Chapter 1
1. J. T. DESAGULIERS, "The Newtonian System of the World, The
Best Model of Government: an Allegorical Poem," 1 728, quoted
in H . N . FA IRCHILD, Religious Trends in English Poetry, Vol. I
(New York: Columbia University Press, 1 939), p. 357.
2. Ibid., p. 358.
3. Gerd Buchdahl emphasized and illustrated this ambiguity of the
cultural influence of the Newtonian model in its dimensions both
empirical (Opticks) and systematic (Principia) in The Image of
Newton and Locke in the Age of Reason, Newman History and
Philosophy of Science Series (London: Sheed & Ward, 1 96 1 ).
4. La Science et Ia diversite des cultures, (Paris: UNESCO, PUF,
1 974), pp. 1 5-16.
5. C . C . GILLISPIE, The, Edge of Objectivity (Princeton, N .J. :
Princeton University Press, 1 970), pp. 1 99-200.
6. M . HEIDEGGER , The Question Concerning Technology (New
York: Harper & Row, 1 977), p. 20.
7. Ibid. , p. 2 1 .
8. Ibid . , p. 1 6.
9. "The Coming of the Golden Age," Paradoxes of Progress (San
Francisco: Freeman & Company, 1 978).
10. See, for instance. P. DAVIES, Other Worlds (Toronto: J. M. Dent
& Sons, 1 980).
317
NOTES
11. A. K oESTLER, The Roots of Coincidence (London: Hutchinson,
1972), pp. 1 38-39.
12. A. KoYRE, Newtonian Studies (Chicago: University of Chicago
Press, 1 968), pp. 23-24.
13. In "Race and History" (Structural Anthropology II, New York:
Basic Books, 1 976), Claude Levi-Strauss discusses the condi
tions that lead to the Neolithic and Industrial revolutions. The
model he introduces, involving chain reactions and catalysis (a
process with kinetics characterized by threshold and amplifica
tion phenomena) attests to an affinity between the problems of
stability and fluctuation we discuss in Chapter VI as well as cer
tain themes of the "structural approach" in anthropology.
1 4. "Inside each society, the order of myth excludes dialogue: the
group's myths are not discussed, they are transformed when
they are thought to be repeated." C . LEvi-STRAUSS, L 'Homme
Nu (Paris: Pion, 1 97 1 ), p. 585. Thus mythical discourse is to be
distinguished from critical (scientific and philosophic) dialogue
more because of the practical conditions of its reproduction than
because of an intrinsic inability of such or such emitter to think
in a rational way. The practice of critical dialogue has given to
the cosmological discourse claiming truthfulness its spectacular
evolutive acceleration.
15. This is, of course, one of the main themes of Alexandre Koyre.
16. The definition of such an "absurdity" opposes the age-long idea
that a sufficiently tricky device would permit one to cheat na
ture. See the efforts devoted by engineers till the twentieth cen
tury to the construction of perpetual-motion machines in A. Ord
Hume, Perpetual Motion: The History of an Obsession (New
York: St. Martin's Press, 1 977).
17. Popper translated into a norm this excitement born out of the
risks involved in the experimental games. He affirms, in The
Logic of Scientific Discovery, that the scientific must look for
the most " improbable" hypothesis-that is, the most risky
one-to try to refute it as well as the corresponding theories.
1 8 . R. FEYNMAN, The Character of Physical Law (Cambridge,
Mass.: M.I.T. Press, 1 967), second chapter.
1 9 . J. NEEDHAM, " Science and Society in East and West, " The
Grand Titration (London: Allen & Unwin, 1969).
20. A. N. WHITEHEAD, Science and the Modern World (New York:
The Free Press, 1 967), p. 1 2.
2 1 . NEEDHAM, op. cit. , p. 308.
22. NEEDHAM, op. cit., p. 330.
23. R. HooYKAAS emphasized this "dedivinization" of the world by
the Christian metaphor of the world machine in Religion and the
ORDER OUT OF CHAOS
24.
25.
26.
27.
28.
29.
30.
318
Rise of Modern Science (Edinburgh and London: Scottish Aca
demic Press, 1972), esp. pp. 14-1 6.
WHITEHEAD, Op. cit. , p. 54.
The famous text about nature being written in mathematical
signs is to be found in 11 Saggiatore. See also The Dialogue Con
cerning the Two Chief World Systems, 2nd rev. ed. (Berkeley:
University of California Press, 1967).
At least it was triumphant in the academies created in France,
Prussia, and Russia by absolute sovereigns. In The Scientist's
Role in Society (Englewood Cliffs, N.J. : Foundations of Modern
Sociology Series, Prentice-Hall, 197 1 ), Ben David emphasized
the distinction between physicists of these countries, dedicated
to physics as a glamorous and purely theoretical science, and
the English physicists immerged in a wealth of empirical and
technical problems. Ben David proposed a connection between
the fascination for a theoretical science and the relegation far
from political power of the social class supporting the "scientific
movement. "
In his biography of d �lembert-Jean d'Alembert, Science and
Enlightenment (Oxford : Clarendon Press , 1 970)-Thomas
Hankins emphasized how closed and small was the first true
scientific community, in the modern sense of the term, namely,
that of the eighteenth-century physicists and mathematicians,
and how intimate were thei r relations with. the absolute sov
ereigns.
EINSTEIN, Op. cit . , pp. 225-26.
E. MACH, "The Economical Nature of Physical Inquiry, " Popu
lar Scientific Lectures (Chicago: Open Court Publishing Com
pany, 1 895), pp. 197-98.
J. DONNE, An Anatomy of the World wherein . . . the frailty and
the decay of the whole world is represented (London, catalog of
the British Museum, 161 1 ).
Chapter 2
I. On this point, see T. HANKINS, "The Reception of Newton's
Second Law of Motion in the Eighteenth Century, " Archives In
ternationales d'Histoire des Sciences, Vol . XX ( 1 967), pp. 4265, and I. B. CoHEN, "Newton's Second Law and the Concept
of Force in the Principia," The Annus Mirabilis of Sir Isaac
Newton , Tricentennial Celebration, The Texas Quarterly,
Vol. X, N o . 3 ( 1 967), pp. 25- 1 57. The four following paragraphs
rest, for what concerns atomism and the conservation theories,
319
2.
3.
4.
5.
6.
7.
8.
9.
NOTES
on W. Scarr, The Conflict Between Atomism and Conservation
Theory (London: Macdonald, 1970).
A. KovRE: , Galileo Studies ( Hassocks , Eng. : The Harvester
Press, 1978), pp. 89-94.
In his history of mechanics-The Science ofMechanics: A Crit
ical and Historical Account of Its Development (La Salle, Ill. :
Open Court Publishing Company, 1960)-Ernst Mach laid stress
on this dual filiation of modern dynamics of both the trajectories
science and the engineer's computations.
This at least is the conclusion of today's historians who began
the study of the impressive mass of Newton's ·�tchemical Pa
pers," which till now were ignored or disdained as "nonscien
tific . " See B. J. DoBBS, The Foundations of Newton's Alchemy
(Cambridge: Cambridge University Press, 1 975); R. WESTFALL,
"Newton and the Hermetic Tradition" in Science, Medicine and
Society, ed. A. G. DEBUS (London: Heinemann, 1 972); and R .
WESTFALL, "The Role of Alchemy in Newton's Career," Rea
son, Experiment and Mysticism, ed. M. L. RIGHINI BONELLI
and W. R. SHEA (London: Macmillan, 1975). As Lord Keynes,
who played a crucial part in the collection of these papers, sum
marized (quoted in DoBBS, op. cit . , p. 1 3), "Newton was not the
first of the age of reason. He was the last of the Babylonians and
Sumerians, the last great mind which looked out on the visible
and intellectual world with the same eyes as those who began to
build our intellectual inheritance rather less than 1 0,000 years
ago . "
DoBBS, op. cit . , also examined the role of the "mediator" by
which two substances are made "sociable. " We may recall here
the importance of the mediator in Goethe's Elective Affinities
(Engl. trans. Greenwood 1976). For what concerns chemistry,
Goethe was not far from Newton.
The story of Newton's "mistake" is told in H ANK I Ns 's, Jean
d'Alembert, pp. 29-35.
G . L. BuFFON, "Reflexions sur Ia loi d'attraction," appendix to
Introduction a I' histoire des minhaux ( 1 774), To me IX of
Oeuvres Completes ( Paris: Garnier Freres), pp. 75, 77.
G. L. BuFFON, Histoire naturelle. De Ia Nature, Seconde Vue
( 1 765), quoted in H. M ETZG ER , Newton, Stahl, Boerhaave et Ia
doctrine chimique (Paris: Blanchard, 1974), pp. 57-58.
A. THACKRAY describes the way French chemistry became Buf
fonian in Atom and Power: An Essay on Newtonian Matter The
ory and the Development of Chemistry (Cambridge, Mass . :
Harvard University Press, 1 970 ). pp. 199-233. Ber t hollet's
Statique chimique accomplished Buffon's program and also
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10.
11.
12.
13.
14.
15.
16.
320
closed it, since his disciples gave up the attempt to understand
chemical reactions in terms compatible with Newtonian con·
cepts.
We do not wish to try to explain here the reasons of Newton's
triumph in France, nor of its fall, but to emphasize the at least
chronological connection between these events and the stages of
the process of professionalization of science. See M . CROS·
LAND, The Society of Arcueil: A View of French Science at the
Time of Napoleon (London: Heinemann, 1960), as well as his
Gay Lussac (Cambridge: Cambridge University Press, 1978).
Thomas Kuhn made of this role of scientific institutions, taking
over the formation of the future scientists-that is assuring their
own reproduction, the main characteristic of scientific activity
as we know it today. This problem has also been approached by
M. Crosland, R. Hahn, and W. Farrar in The Emergence of Sci
ence in Western Europe, ed . M . CROSLAND (London: Mac·
millan, 1975).
The role of " mundane" interest so despised by philosophers
such as Gaston Bachelard in France should be taken as the sign
of the open character of eighteenth-century science. In a way,
we can truly speak about a regression during the nineteenth cen
tury, at least for what concerns the scientific culture. And we
could learn today from the multiplicity of local academies and
circles where scientific matters were discussed by nonprofessio
nals.
Q uoted i n J. ScHLANGER, Les metaphores de I' organism e
(Paris: Vrin, 197 1 ), p. 108.
J. C . MAXWELL, Science and Free Will, in CAMPBELL and GAR·
NETT , op. cit . , p. 443. L. CAMPBELL & W. GARNETT , The Life of
James Clerk Maxwell (London, Macmillan, 1 882).
This problem is one of the main themes of French philosopher
Michel Serres. See, for instance, "Conditions" in his La nais
sance de Ia physique dans le texte de Lucrece (Paris: Minuit,
1977). Some texts by M. Serres are now available in English
translation, thanks to the pious zeal of the French Studies De
partment of Johns H o p k i n s U n i v ersity. See M . S ERRES,
Hermes: Literature, Science , Philosophy. (Baltimore: The
Johns Hopkins University Press, 1982.)
See, about the fate of Laplace's demon, E. CASSIRER, Determi
nism and Interdeterminism in Modern Physics (New Haven,
Conn. : Yale University Press, 1956), pp. 3 -25.
321
NOTES
Chapter 3
1. R. NISBET, History of the Idea of Progress (New York: Basic
Books, 1 980), p. 4.
2. D. DIDEROT, d'Alembert's Dream (Harmondsworth : , Eng.: Pen
guin Books, 1 976), pp. 1 66-67.
3 . D. DIDEROT, "Conversation Between d'Alembert and Diderot,"
d'Alembert's Dream, pp. 1 58-59.
4. D. DIDEROT, Pensees sur /'Interpretation de Ia Nature ( 1754),
Oeuvres Completes, Tome II (Paris: Garnier Freres, 1 875), p. II.
5 . Diderot ascribes this opinion to the physician Bordeu in the
Dream.
6. See, for i nstance, A. LovEJOY, The Great Chain of Beings
(Cambridge, Mass. : Harvard University Press, 1973).
7. The historian Gillispie proposed a relation between the protest
against mathematical physics, as popularized by Diderot in the
Encyclopedie, and the revolutionaries' hostility against this offi
cial science, as manifested by the closure of the Academy and
Lavoisier's death. This is a very controversial point, but what is
sure is that the Newtonian triumph in France coincides with the
Napoleonic institutions, spelling the final victory of state acad
emy over craftsmen (see C. C. GILLISPIE, "The Encyclopedia
and the Jacobin Philosophy of Science: A Study in Ideas and
Consequences," Critical Problems in the History ofScience, ed.
M . CLAGETT (Madison, Wis . : U niversity of Wisconsin Press,
1959), pp. 255-89.
8. G. E. STAHL , " Veritable Distinction a etablir entre le mixte et le
vivant du corps humai n , " Oeuvres medicophilosophiques et
pratiques, Tome II (Montpellier: Pitrat et Fils, 1 86 1 ) , esp.
pp. 279-82.
9. See J. SCHLANGER, Les metaphores de /'organisme, for a de
scription of the transformation of the meaning of "organization"
between Stahl and the Romanticists.
10. Philosophy of Nature, §261 .
1 1 . This is Knight's conclusion in "The German Science i n the Ro
mantic Period," The Emergence of Science in Western Europe.
1 2. H. BERGSON, La pensee et le mouvant in Oeuvres (Paris: Edi
tions du Centenaire, PUP, 1 970), p. 1 285 ; tran s . The Creative
Mind (Totowa, N.J. : Littlefield, Adams, 1 975), p. 42.
13. Ibid . , p. 1 287; trans. , p. 44.
14. Ibid., p. 1286; trans. , p. 44.
ORDER OUT OF CHAOS
322
15. H . BERGSON, L 'evolution creatrice in Oeuvres, p. 784; trans.
Creative Evolution (London: Macmillan, 191 1), p. 36 1 .
16. Ibid . , p. 538; trans. , p. 54.
17. Ibid . , p. 784; trans. , p. 36 1 .
18. BERGSON, La pensee et /e mouvant, p. 1 273; trans. , p. 32.
19. Ibid:, p. 1 274; trans. , p. 33.
20. A . N. WHITEHEAD, Science and the Modern World, p. 55.
2 1 . A. N. WHITEHEAD, Process and Reality: An Essay in Cosmol
ogy (New York: The Free Press, 1969), p. 20.
22. Ibid . , p. 26.
23. Joseph Needham and C. H. Waddington both acknowledged the
importance of Whitehead's influence for what concerns their en
deavor to describe in a positive way the organism as a whole.
24. H. HELMHOLTZ, Uber die Erhaltung der Kraft ( 1 847), trans. in
S . B RUSH, Kinetic Theory, Vol . I , The Nature of Gases and
Heat (Oxford: Pergamon Pre s s , 1 965 ), p. 92. See also Y.
E LK AN A , Th e Discovery of the Conservation of Energy
(London: Hutchinson Educational, 1974) and P. M. HEIM AN N,
" Helmholtz and Kant: The Metaphysical Foundations of Uber
die Erhaltung der Kraft," Studies in the History and Philosophy
of Sciences, Vol. 5 ( 1 974), pp. 205-38.
25. H . REICHEN BACH, The Direction of Time (Berkeley: University
of California Press, 1956), pp. 16-17.
Chapter 4
I. W. ScoTT, About the novelty of these problems, see The Conflict
Between Atomism and Conservation Theory, Book II, and about
the industrial context where these concepts were created, D .
CARDWELL, From Watt t o Clausius ( London, Heinemann,
197 1 ). Particularly interesting in this respect is the convergence
between on one hand the need determined by industrial prob
lems and on the other the positivist simplifications by opera
tional definitions.
2. J. HERIVEL, Joseph Fourier: The Man and the Physicist (Ox
ford: Clarendon Press, 1975). In this biography we learn the fol
lowing curious information: Fourier would have brought back
from his trip with Bonaparte to Egypt a sickness causing perma
nent deperditions of heat.
3. See, more particularly, the introduction to Comte's Philosophie
Premiere (Paris: Herman, 1975), ·�uguste Comte auto-traduit
dans l'encyclopedie" in La Traduction (Paris: Minuit, 1974) and
"Nuage," La Distribution (Paris: Minuit, 1977).
323
NOTES
4. C . SMITH, "Natural Philosophy and Thermodynamics: William
Thomson and the Dynamical Theory of Heat, " The British Jour
nalfor the Philosophy ofScience, Vol . 9 ( 1976), pp. 293-3 1 9 and
M . CROSLAND and C. SMITH, "The Transmission of Physics
from France to Britain, 1800-1 840," Historical Studies in the
Physical Sciences, Vol . 9 ( 1 978), pp. 1-6 1 .
5. For what follows, see Y. ELKANA , The Discovery of the Con
servation of Energy Principle, as well as the famous paper by
Thomas Kuhn, "Energy conservation as an Example of Simul
taneous Discovery, " originally published in Critical Problems in
the History of Science and recently in T. KuHN, The Essential
Tension (Chicago: University of Chicago Press, 1977).
6. ELKANA followed the slow crystallization of the concept of en
ergy; see his book and "Helmholtz's Kraft: An Illustration of
Concepts in Flux," Historical Studies in the Physical Sciences,
Vol . 2 ( 1 970), pp. 263-98.
7. J. JouLE, "Matter, Living Force and Heat, " The Scientific Pa
pers ofJames Prescott Joule, Vol . 1 (London: Taylor & Francis,
1 884), pp. 265-76 (quotation, p. 273).
8. The English translations of Mayer's two great papers, " On the
Forces of Inorganic Nature" and "The Motions of Organisms
and Their Relation to Metabolism," are in Energy: Historical
Development of a Concept, ed. R . B . LINDSAY (Stroudsburg,
Pa . : Benchmarks Papers on Energy 1 , Dowden, Hutchinson &
Ross, 1975).
9. E . BENTON, " V italism in the Nineteenth Century Scientific
Thought: A 'JYpology and Reassessment, " Studies in History
and Philosophy of Science, Vol . 5 ( 1 974), pp. 1 7-48.
10. H. H ELMHOLTZ, " Uber die E rhalt ung der Kraft, " op. c i t . ,
pp. 90-9 1 .
1 1 . G . DELEUZE, Nietzsche et Ia phi/osophie (Paris: PUF, 1 973),
pp. 48-55.
12. In this study of Zola's "Docteur Pascal , " Feux et signaux de
brume Paris: Grassel ( 1 975), p. 109, Michel Serres wrote: "The
century that was practically drawing to a close when the novel
appeared had opened with the majestic stability of the solar sys
tem, and was now filled with dismay at the relentless degrada
tions of fire. Hence the fierce, positive dilemma: perfect cycle
without residue, eternal and positively valued, i.e . , the cosmol
ogy of the sun; or else a missed cycle, losing its difference, irre
versible, historical and despised-a cosmology, a thermogony of
fire which must either be extinguished or destroyed, without al
ternative. One dreams of Laplace, whilst Carnot and the others
have forever smashed the cubby-hole, the niche, where one
ORDER OUT OF CHAOS
13.
14.
15.
1 6.
17.
18.
19.
20.
324
could sleep in peace; one is dreaming, that is certain: then
cultural archaisms hav ing re turned through another door,
through another opening of the same door, are powerfully re
awakened: immortal flame, purifying blaze or evil fire?"
The continuity between Carnot father and son has been empha
sized by Cardwell (From Watt to Clausius) and Scott (The Con
flict Between Atomism and Conservation Theory).
P. DAVIES, The Runaway Universe (New York: Penguin Books,
1 980), p. 1 97.
R DYSON, "Energy in the Universe, " Scientific American, Vol.
225 (197 1 ), pp. 50-59.
What was particularly important was to grasp that, unlike what
happens in mec hanics, it is not j ust any situation of a ther
modynamic system that can be characterized as a "state"; quite
the contrary. See E. DAUB, "Entropy and Dissipation, " Histor
ical Studies in the Physical Sciences, Vol . 2 (1 970), pp. 321-54.
In his autobiography, Scientific Autobiography (London:
Williams & Norgate , 1 950), Max Planck recalled how isolated he
had been when he first emphasized the peculiarity of heat and
pointed out that it is the conversion of heat into another form of
energy that raises the irreversibility problem. Energeticists such
as Ostwald wanted all forms of energy to be given the same
status. For them, the fall of a body between two altitude levels
takes place by virtue of the same kind of productive difference
as the passage of heat between two bodies at different tempera
tures. T hus, Ostwald's comparison did away with the crucial
distinction between an ideally reversible process, such as the
mechanical motion, and an intrinsically irreversible one, such as
heat diffusion. By doing so, he was actually taking up a position
similar to what we have attributed to Lagrange: where Lagrange
considered conservation of energy as a property belonging only
to ideal cases but also the only one capable of being treated
rigorously, Ostwald held conservation of energy as the property
of any natural transformation, but defined conservation of en
ergy differences (required by all transformation since only a dif
ference can produce another difference) as an abstract ideal, but
the sole object for a rational science .
The splitting of the entropy variation into two different terms
was introduced in l. Prigogine, Etude thermodynamique des
Phenomenes irreversibles, These d 'agn!gation presentee a Ia
faculte des sciences de l ' Universite Libre de Bruxelles 1 945
(Paris: Dunod, 1 947).
R. CLAUSIUS, Ann. Phys., Vol. 125 (1865), p. 353.
M. PLANCK, "The Unity of the Physical Universe, " A Survey of
325
NOTES
Physics, Collection of Lectures and Essays (New York: E. P.
Dutton, 1925), p. 16.
2 1 . R. CAILLOIS, " La dissymetrie, " Coherences aventureuses, Col
lection Idees (Paris: Gallimard, 1973), p. 198.
Chapter 5
1. For what concerns the content of this and the following chapter,
see P. GLANSDORFF and I. PRIGOGINE, Thermodynamic Theory
of Structure, Stability and Fluctuations (New York: John Wiley
& Sons, 197 1 ) and G. NICOLls and I. PRIGOGINE, Self-Organiza
tion in Non-Equilibrium Systems (New York: John Wiley &
Sons, 1 977), where further references may be found.
2. R NIETZSCHE, Der Wille zur Macht, Siimtliche Werke (Stutt
gart: Kroner, 1 964), aphorism 630.
3. Which precise content can be given to the general law of entropy
growth? For a theoretician physicist such as de Donder, chemi
cal activity, which appeared obscure and inaccessible to the ra
tional approach of mechan i cs , was mysterious enough to
become the synonym of the irreversible process. Thus chemis
try, whose question physicists had never truly answered, and the
new enigma of irreversibility came to join in a challenge not to be
ignored anymore . See T h . De Donder, L' Affinite (Pari s :
Gauthier-Villars, 1 962) and L. Onsag'er Phys. Rev. 37, 405 (193 1 ).
4. M. SERRES, La naissance de Ia physique dans le texte de Lu
crece, op. cit.
5 . For more detai l s con cerning chemical o s ci ll ations, see A .
WINFREE, " Rotating Chemical Reactions," Scientific Amer·
ican, Vol . 230 (1974), pp. 82-95.
6. A. G oLDBETER and G . Nic ous, ''An Allosteric Model with
Positive Feedback Applied to Glycolytic Oscillations, " Progress
in Theoretical Biology, Vol . 4 ( 1976), pp. 65-1 60; A. GOLDBETER
and S. R. CAPLAN, "Oscillatory Enzymes," Annual Review of
Biophysics and Bioengineering, Vol . 5 (1 976), pp. 449-73 ..
7. B . H ESS , A . BOITE UX, and J. K ROGER, " Cooperation of
Glycolytic Enzymes," Advances in·Enzyme Regulation, Vol . 7
(1969), pp. 149-67 ; see also B. HESS, A. GOLDBETER, and R .
LEFEVER, "Temporal, Spatial and Fun ctional Order in Regu
lated B iochemical Cellular Systems , " Advances in Chemical
Physics, Vol . XXXVIII ( 1978), pp. 363-4 1 3 .
8_ R HEss, Ciba Foundation Symposium. Vol. 3 1 ( 1975), p . 369.
9A. G . GERESCH, "Cell Aggregation and Differentiation in Die-
ORDER OUT OF CHAOS
326
tyostelium Discoideu m , " in Developmental Biology, Vol . 3
( 1 968), pp. 1 57- 1 97.
98. A. G oLDBETER and L. A. SEGEL, "Unified Mechanism for Re
lay and O s c i l lation of C yc l ic AMP in Dictyostelium Dis
coideum," Proceedings of the National Academy of Sciences,
Vol. 74 ( 1 977), pp. 1 543-47.
10. See M . GARDNER, The Ambidextrous Universe (New Yo rk:
Charles Scribner's Sons, 1 979).
I I. O. K. KONDEPUDI and I. PRIGOGINE, Physica, Vol . l07A ( 1 98 1),
pp. 1-24 ; D . K . K oNDEPUDI , Physic a , Vol . 1 1 5A ( 1 982),
pp. 552-66. It could even be that chemistry may bring to the
macroscopic scale the violation of parity in weak forces ; D. K .
KoNDEPUDI and G. W. NELSON, Physical Review Letters, Vol .
50, No. 1 4 ( 1 983), pp. 1 023-26.
12. R. LEFEVER and W. H oRSTHEMKE, "Multiple Transitions In
duced by Light Intensity Fluctuations in Illuminated Chemical
Systems, " Proceedings of the National Academy of Sciences,
Vol. 76 ( 1 979), pp. 2490-94. See also W. H oRSTHEMKE and M.
MALEK MANSOUR, " Influence of External Noise on Nonequi
librium Phase Transitions," Zeitschrift fur Physik B, Vol . 24
(1976), pp. 307- 1 3 ; L . A RNOLD, W. HORSTHEMKE, and R .
LEFEVER, "White and Coloured External Noise and Transition
Phenomena in Nonlinear Systems , " Zeitschrift fiir Physik B,
Vol. 29 (1978), pp. 367-73; W. HORSTHEMKE, "Nonequilibrium
Transitions Induced by External White and Coloured Noise, "
Dynamics of Synergetic Systems , e d . H . H A K E N (Berlin:
Springer Verlag, 1 980); for an application to a biological prob
lem, R. LEFEVER and W. HORSTHEMKE, "Bistability in Fluc
tuating Environments: Implication in Tumor Immunology,"
Bulletin of Mathematic Biology, Vol. 4 1 (1 979).
1 3 . H . L . SwiNNEY and J. P. G oLLUB, "The Transition to Thr
bulence," Physics Today, Vol. 31 , No. 8 (1 978), pp. 4 1-49.
1 4. M . J. FEIGENBAUM , "Universal B ehavior i n Nonlinear Sys
tems," Los Alamos Science, No I (Summer 1980), pp. 4-27.
1 5 . The concept of chreod is part of the qualitative d escription of
embryological development Waddington proposed more than
twenty years ago. It is truly a bifurcating evolution: a pro
gressive exploration along which the embryo grows in an "epi
genetic l and scape" where coexist stable segments and segments
where a choice among several d evelopmental paths is possible.
See C. H. WADDINGTON, The Strategy of the Genes (London:
All en & Unwin , 1957). C. H. Waddington's chreods are also a
central reference in Rene Thorn's biological thought. They could
thus b ecome a meeting point for two approaches: the one we are
presenting, starting from local m echanisms and exploring the
327
16.
17.
18.
1 9.
NOTES
spectrum of collective behaviors they can generate; and Thorn's,
starting from global mathematical entities and connecting the
qualitatively distinct forms and transformations they imply with
the phenomenological description of morphogenesis.
S. A. KAUFFMAN, R. M. SHYMKO, and K. TRABERT, "Control of
Sequential Compartment Formation in Drosophila," Science,
Vol . 199 (1978), pp. 259-69.
H. BERGSON , Creative Evolution, pp. 94-95.
See C . H . WADDINGTO N , The Evolution of an Evolutionist
(Edinburgh: Edinburgh University Press, 1975) and P. WEISS,
"The Living System: Determinism Stratified," Beyond Reduc
tionism, ed. A . KOESTLER and J. R . S MYTHIES ( London:
Hutchinson, 1969).
D. E. KosHLAND, ·� Model Regulatory System: Bacterial
Chemotaxis," Physiological Review, Vol. 59, No. 4, pp. 8 1 1-62.
Chapter 6
J . G. NICOLlS and I. PRIGOGINE, Self-Organization in Nonequilib
rium Systems (New York: John Wiley & Sons, 1977).
2. R BARAS, G. Nicous, and M. MALEK MANSOUR , "Stochastic
Theory of Adiabatic Explosion," Journal of Statistical Physics,
Vol . 32, No. 1 ( 1 983), pp. I .
3 . See, for example, M . MALEK MANSOUR, C . VAN DEN BROECK,
G. Nicous, and J. W. TURNER, Annals ofPhysics, Vol . 1 3 1 , No.
2 (198 1 ), p. 283.
4. J. L. DENEUBOURG, '�pplication de l'ordre par fluctuation a Ia
description de certaines etapes de Ia construction du nid chez
les termite s , " Insectes Sociaux, Journal International pour
/'etude des Arthropodes sociaux, Tome 24, No. 2 ( 1 977),
pp. 1 1 7-30. This first model is presently being extended in con
nection with new experimental studies; 0. H. BRUINSMA, ·�n
Analysis of Building Behaviour of the Termite macrotermes sub
hyaiinus, " Proceedings of the VIII Congress IUSSI (Waegenin
gen, 1977).
5. R. P. GARAY and R. LEFEVER, ·� Kinetic Approach to the Im
munology of Cancer: Stationary States Properties of Effector
Target Cell Reactions," Journal of Theoretical Biology, Vol . 73
(1978), pp. 417-38, and private communication.
6. P. M. ALLEN , "Darwinian Evolution and a Predator-Prey Ecol
ogy, " Bulle tin of Mathem atical Biology, Vol . 37 ( 1 975),
PP- 389-405 ; " Ev oluti o n , Populat i o n and Stability, " Proceed
ings of the National Academy of Sciences, Vol. 73 , No. 3 (1 976),
pp. 665-68. See also R. CzAPLEWSKI, ']\_ Methodology for Eval-
ORDER OUT OF CHAOS
328
uation of Parent-Mutant Competition," Journal for Theoretical
Biology, Vol . 40 (1973), pp. 429-39.
7. See, for the present state of this work, M EIGEN and P. ScHus
TER, The Hypercycle (Berlin: Springer Verlag, 1979).
8. R. MAY in Science, Vol. 1 86 ( 1974), pp. 645-47; see also R. MAY,
"Simple Mathematical Models with very Complicated Dynam
ics," Nature, Vol. 261 ( 1 976), pp. 459-67.
9. M. P. HASSELL, The Dynamics in Arthropod Predator-Prey Sys
tems (Princeton, N.J. : Princeton University Press, 1978).
10. B. HEINRICH, ·�rtful Diners," Natural History, Vol. 89, No. 6
( 1 980), pp. 42-5 1 , esp. quote, p. 42.
1 1 . M. LovE, "T he Alien Strategy," Natural History, Vol. 89, No. 5
( 1980), pp. 30-32.
1 2 . J. L. DENEUBOURG and P. M. ALLEN, "Modeles theoriques de
Ia division du travail des les societes d ' insecte s , " Academie
Royale de Belgique, Bulletin de Ia Classe des Sciences, Tome
LXII ( 1 976), pp. 4 16-29; P. M. ALLEN, "Evolution in an Eco
system with Limited Resources," op. cit. , pp. 408-15.
13. E. W. MoNTROLL , "Social Dynamics and the Quantifying of So
cial Forces," Proceedings of the National Academy ofSciences,
Vol . 75, No. 10 ( 1 978), pp. 4633-37.
1 4 . P. M . ALLEN and M. SANGLIER, "Dynami c Model of Urban
Growth, " Journal for Social and Biological Structures, Vol. 1
( 1 978), pp. 265-80, and " Urban Evolution, Self-Organization
and Decision-making," Environment and Planning A, Vol. 1 3
(198 1 ), pp. 1 67-83.
15. C. H . WADDINGTON, Tools for Thought, (St. Albans, Eng. : Pal·
adin, 1976), p. 228.
16. S. J. GouLD , Ontogeny and Phylogeny, op. cit. Belknap Press
Harvard University Press, 1977.
17. C. L:Evi-STRAUSS, "Methodes et enseignement, " Anthropologie
structurale (Paris: Pion), pp. 3 1 1-17.
18. See, for instance , C. E. RusSET, The Concept of Equilibrium in
American Social Thought (New Haven, Conn.: Yale University
Press, 1966).
19. S. J. GouLD, "The Belt of an Asteroid, " Natural History, Vol.
89, No. 1 ( 1980), pp. 26-33.
. .
Chapter 7
1. A.N. WHITEHEAD, Science and the Modern World, p. 1 86.
2. The Philosophy of Rudolf Carnap, ed. P. A. ScHILP.P (Cam
bridge: Cambridge University Press, 1963).
329
NOTES
3. J. FRASER, "The Principle of Temporal Levels: A Framework for
the Dialogue?" communication at the conference Scientific
Concepts of Time in Humanistic and Social Perspectives (Bell
agio, July 198 1 ).
4. See on this point S. BRUSH, Statistical Physics and Irreversible
Processes, esp. pp. 6 16-25.
5. Feuer has rather convincingly shown how the cultural context of
Bohr's youth could have helped his decision to look for a non
mechanistic model of the atom; Einstein and the Generation of
Science (New York: Basic Books, 1974). See also W. HEISEN
BERG, Physics and Beyond (New York: Harper & Row, 1 97 1 )
and D. SERWER, " Unmechanischer Zwang: Pauli, Heisenberg
and the Rejection of the Mechanical Atom 1923-1 925," Histor
ical Studies in the Physical Sciences, Vol . 8 ( 1 977), pp. 1 89-256.
6. In Black-Body Theory and the Quantum Discontinuity, 18941912 (Oxford: Clarendon Press and New York: Oxford Univer
sity Press, 1 978), Thomas Kuhn has beautifully shown how
closely Planck followed Boltzmann's statistical treatment of irre
versibility in his own work.
7. J. MEHRA and H. RECHEN BERG, The Historical Development of
Quantum Theory, Vols. 1-4 (New York: Springer Verlag, 1 982).
8. See, about the conceptual framework of the experimental tests
recently conceived for hidden variables in quantum mechanics,
B. o'EsPAGNAT, Conceptual Foundations of Quantum Mechan
ics, 2nd aug. ed. (Reading, Mass. : Benjamin, 1 976). See also B .
o' EsPAGNAT, "The Quantum Theory and Reality," Scientific
American, Vol. 241 ( 1979), pp. 1 28-40.
9. See, for the complementarity principle, B . o'EsPAGNAT, op. cit . ;
M . JAMMER, The Philosophy of Quantum Mechanics (New
York: John Wiley & Sons, 1974); and A. PETERSEN, Quantum
Mechanics and the Philosophical Tradition (Cambridge, Mass. :
MIT Press, 1 968). C. GEORGE and I . PRIGOGINE, "Coherence
and Randomness in Quantum Theory, " Physica, Vol . 99A
( 1 979), pp. 369-82.
10. L. RosENFELD, "The Measuring Process in Quantum Mechan
ics," Supplement of the Progress of Theoretical Physics .(1965),
p. 222.
1 1. About the quantum mechanics paradoxes, which can truly be
said to be the nightmares of the classical mind, since they all
(SchrOdinger's cat, Wigner's friend, multiple universes) call to
life again the phoenix idea of a closed objective theory (this time
in the guise of SchrOdinger's equation), see the books by d'Es
pagnat and Jammer.
1 2 . B. MISRA, I. PRIGOGINE, and M . COURBAGE, "Lyapounov Vari-
ORDER OUT OF CHAOS
330
able; Entropy and Measurement in Quantum Mechanics," Pro
ceedings of the National Academy of Sciences, Vol. 76 ( 1979),
pp. 4768-4772. I. PRIGOGINE and C . GEORGE, "The Second
Law as a Selection Principle: The Microscopic Theory of Dis
sipative Processes in Quantum Systems, " to appear in Proceed
ings of th e National A cademy of Sciences. Vol 80 ( 1 983)
4590-94.
1 3 . H . MINKOWSKI, "Space and Time, " The Principles ofRelativity
(New York: Dover Publications, 1923).
14. A. D. SAKHAROV, Pisma Zh. Eksp. Teor. Fiz., Vol. 5, No. 23
( 1 %7).
Chapter 8
I . G. N. LEWIS, "The Symmetry of Time in Physics," Science,
Vol. 7 1 ( 1 930), p. 570.
2. A. S. EDDINGTON , The Nature of the Physical World (New
York: Macmillan, 1948), p. 74.
3 . M. GARDNER, The Ambidextrous Universe: Mirror Asymmetry
and Time-Reversed Worlds (New York: Charles Scribner's Sons,
1979), p. 243.
4. M. PLA N CK, Treatise on Thermodynamics (New York: Dover
Publications, 1 945), p. 106.
5. Quote by K. DENBIGH, "How Subjective Is Entropy?" Chemis
try in Britain, Vol. 1 7 ( 1 98 1 ), pp. 168-85.
6. See, for instance, M. KAC, Probability and Related Topics in
Physical Sciences (London: Interscience Publications, 1959).
7. J. W. GIBBS, Elementary Principles in Statistical Mechanics
(New York: Dover Publications, 1960), Chap. XII.
8. For instance, S. Watanabe introduces a strong distinction be
tween the world to be contemplated and the world upon which
we, as active agents, work; he states there is no consistent way
of speaking about entropy increase if it is not in connection with
our actions on t he world . However, all our physics is in fact
about the world to be acted on, and Watanabe's distinction thus
does not help to clarify the relation between "microscopic deter
ministic symmetry" and "macroscopic probabilistic asymme
try. " The question i s left without an answer. How can we
meaningfully say that the sun is irreversibly burning? See S .
WATANABE, "Time and the Probabilistic View of the World,"
The Voices of Time, ed. J. FRASER (New York: Braziller, 1966).
9. Maxwell's demon appears in J. C . MAXWELL. TheorY of Heat
(London: Longmans , 1 87 1 ), Chap. XXII ; see also E . DAU B ,
331
10.
11.
12.
13.
14.
15.
1 6.
17.
NOTES
"Maxwell's Demon" and P. HEIMANN, "Molecular Forces, Sta
tistical Representation and Maxwell's Demon," both in Studies
in History and Philosophy of Science, Vol. I ( 1 970) ; this volume
is entirely devoted to Maxwell.
L. BoLTZMAN N , Populiire Schriften, new ed. (Braunschweig
Wiesbaden: Vieweg, 1 979). As Elkana emphasizes in " Boltz
mann's Scientific Research Program and Its Alternatives," In
teraction B e t ween Scie n ce a n d Ph ilosophy ( A t l ant i c ,
Highlands, N.J. : Humanities Press, 1974), the Darwinian idea of
evolution is expl icitly expressed mostly in Boltzmann's view
about scientific knowledge-that is, in his defense of mechanis
tic models against energeticists. See, for instance, his 1886 lec
ture "The Second Law of Thermodynamics , " Theoretical
Physics and Philosophical Problems, ed. B. McGuiNNESS (Dor
drecht, Netherlands: D. Reidel , 1974).
For a recent account see I. PRIGOGINE, From Being to Becom
ing-Time and Complexity in the Physical Sciences (San Fran
cisco: W. H . Freeman & Company, 1980).
In his Scientific Autobiography, Planck describes his changing
relationship with Boltzmann (who was first hostile to the phe
nomenological distinction introduced by Planck between revers
ible and i rreversible p rocesses). See al so o n this point Y.
ELKANA, op. cit . , and S. B RUSH, Statistical Physics and Irre
versible Processes, pp. 640-5 1 ; for Einstein, op. cit., pp. 672-74;
for Schrodinger, E. SCHR6DINGER, Science, Theory and Man
(New York: Dover Publications, 1957).
H . POINCARE, "La mecanique et I' experience ," Revue de Meta
physique et de Morale, Vol. 1 ( 1 893), pp. 534-37. H . POINCARE,
Lefons de Thermodynamique, ed. J. Blondin ( 1 892; Paris: Her
mann 1923).
See for a study of the controversies around Boltzmann's entropy,
see on this point S. B RUSH , The Kind of Motion We Call Heat,
op. cit . , and Planck's remarks in his biography (Loschmidt was
Planck's student).
I. PRIGOGINE, C. GEORGE, R HENIN, and L. ROSENFELD, ·�
U nified Formulat ion of Dynamics and Thermodynamics , "
Chemica Scripta, Vol . 4 (1973), pp. 5-32.
D. PARK, The Image of Eternity: Roots of Time in the Physical
World (Amherst, Mass . : University of Massachusetts Press,
1980).
See also on this point S. BRUSH, The Kind of Motion We Call
Heat-Book I , Physics and the Atomists; Book II, Statistical
Physics and Irreversible Processes (Amsterdam: North Holland
Publishing Company, 1976), as well as his commented anthology,
ORDER OUT OF CHAOS
18.
19.
20.
21.
332
Kinetic Theory: Vol. I, The Nature of Gases and Heat: Vol . II.
Irreversible Processes (Oxford: Pergamon Press, 1965 and 1966).
J. W. GIBBS, Elementary Principles in Statistical Mechanics
(New York: Dover Publications, 1%0), Chap XII. For an histor
ical account, see J. MEHRA, "Einstein and the Foundation of
Statistical Mechanics, Physica, Vol. 79A, No. 5 ( 1974), p. 17.
Many Marxist nature philosophers seem to take inspiration from
Engels (quoted by Lenin in his Philosophic Notebooks) when he
wrote in Anti-Diihring (Moscow: Foreign Languages Publishing
House, 1954), p. 1 67, "Motion is a contradiction: even simple
mechanical change of a position can only come about through a
body being at one and the same moment of time both in one
place and in another place, being in one and the same place and
also not in it. And the continuous and simultaneous solution of
this contradiction is precisely what motion is."
L. BoLTZMANN, Lectures on Gas Theory (Berkeley: University
of California Press, 1 964), p. 446f, quoted in K. POPPER, Un
ended Quest (La Salle, Ill . : Open Court Publishing Company,
1976), p. 1 60.
POPPER , op. cit. , p. 1 60.
Chapter 9
1 . VoLTAIRE, Dictionnaire Philosophique. (Paris: Garnier, 1954.)
2. See note 2, Chapter VII.
3 . K . PoPPER, "The Arrow of Time, " Nature, Vol . 1 77 ( 1 956),
p. 538.
4. See M. GARDNER, The Ambidextrous Universe, pp. 27 1-72.
5. A. EINSTEIN and W RITZ, Phys. Zsch., Vol. lO (1909), p. 323.
6. H. POINCARE, Les methodes nouvelles de Ia mecanique celeste
(New York: Dover Publications, 1 957); E. T. WHITTAKER, A
Treatise on the Analytical Dynamics of Particles and Rigid
Bodies (Cambridge: Cambridge University Press, 1 965).
7. J. MosER, Stable and Random Motions in Dynamical Systems
(Princeton, N.J. : Princeton University Press, 1974).
8. For a general review, see J. LEBOWITZ and 0. PENROSE, "Mod
ern Ergodic Theory," Physics Today (Feb. 1 973), pp. 23-29.
9. For a more detailed study, see R. BALESCU, Equilibrium and
Non-Equilibrium Statistical Mechanics (New York: John Wiley
& Sons, 1 975).
10. V. ARNOLD and A. AvEz, Ergodic Problems of Classical Me
chanics (New York: Benjamin, 1968).
333
NOTES
1 1 . H . PoiNCARE, "Le Hasard, " Science et Methode (Paris: Flam
marion, 19 14), p. 65.
12. B. MISRA, I. PRIGOGINE and M. CouRBAGE, "From Determinis
tic Dynamics to Probabilistic Descriptions," Physica, Vol. 98A
( 1979), pp. 1-26.
1 3 . D. N. PARKS and N. J. THRIFf, Times, Spaces and Places: A
Chronogeographic Perspective (New York: John Wiley & Sons,
1980).
14. M. COURBAGE and I . PRIGOGINE, " Intrinsic Randomness and
Intrinsic Irreversibility in Classical Dynamical Systems," Pro
ceedings of the National Academy of Sciences, 80 (April 1983).
1 5 . I. PRIGOGINE and C. GEORGE, "The Second Law as a Selection
Principle: The Microscopic Theory of Dissipative Processes in
Quantum Systems," Proceedings of the National Academy of
Sciences, Vol. 80 ( 1 983), pp. 4590-4594.
16. V. NABOKOV, Look at the Harlequins! (McGraw-Hill 1974).
1 7 . J. NEEDHAM, "Science and Society in East and West," The
Grand Titration (London: Allen & Unwin, 1%9).
1 8 . See for more details B . MISRA, I . PRIGOGINE and M . CouR
BAGE, "From deterministic Dynamics to probabilistic Descrip
tion " , Physica 98A ( 1 979) 1-26. ; B. MISRA and I. PRIGOGINE
"Time, Probability and Dynamics" , in Long-time Prediction in
Dyn amic s , e d s . C . W. Horton , L . E . Rec i h l and A . G .
Szebehely, (New York, Wiley 1983).
19. I. PRIGOGINE, C. GEORGE, R HENIN, and L. ROSENFELD, ·�
Unified Formulation of Dynamics and Thermodynamic s , "
Chemica Scripta, Vol. 4 ( 1 973), pp. 5-32.
20. M. CouRBAGE "Intrinsic irreversibility of Kolmogorov dynam
ical systems," Physica 1983 ; B. Misra and I. Prigogine, Letters
in Mathematical Physics, September 1983.
Conclusion
1 . A. S. EDDINGTON , The Nature of the Physical World (N_ew
York: Macmillan, 1 948).
2. L. LEVY-BRUHL, La Mentalite Primitif (Paris: PUF, 1922).
3 . G . MILLS, Hamlet's Castle (Austin: University of Texas Press,
1976).
4. R. TAGORE, "The Nature of Reality" (Calcutta: Modern Review
XLIX, 193 1 ), pp. 42-43.
5. D. S. KOTHARI , Some Thoughts on Truth (New Delhi: Anniver
sary Address , Indian National Science Academy, Bahadur Shah
Zafar Marg, 1 975), p. 5 .
ORDER OUT OF CHAOS
334
6. E. MEYERSON, Identity and Reality (New York: Dover Publica
tions, 1 962).
7. Described in H . B ERGSON, Melanges (Paris : PUF, 1 972),
pp. 1 340-46.
8. Correspondence, Albert Einstein-Michele Besso, 1903-1955
(Paris: Herman, 1972).
9. N. WIENER, Cybernetics (Cambridge, Mass. : M.I.T. Press and
New York: John Wiley & Sons, 196 1 ).
10. M. MERLEAU-PONTY, "Le philosophe et Ia sociologie," Eloge
de Ia Philosophie, Collection Idees (Paris: Gallimard, 1 960),
pp. 136-37.
1 1 . M. MERLEAu-PoNTY, Resumes de Cours /952-/960 (Paris: Gal
limard, 1968), p. 1 19.
12. P. VAL�RY, Cahiers, La Pleiade (Paris: Gallimard, 1973), p. 1303 .
13. For what follows see also I. PRIGOGINE, I. STENGERS, and S .
PAHAUT, " L a dynamique d e Leibniz a Lucrece ," Critique "Spe
cial Serres, " Vol . 35 (Jan. 1979), pp. 34-55. Engl. trans. : Dy
namics from Leibniz to Lucretius, " Afterword to M. SERRES,
Hermes: Literature, Science , Philosophy (Baltimore: Johns
Hopkins Univ. Pr. , 1982), pp. 1 37-55.
14. C. S. PEIRCE, The Monist Vol. 2 ( 1 892), pp. 321 -337.
15. A. N. WHITEHEAD, Process and Reality, pp. 240-4 1 . On this
subject, see I. LECLERC, Whitehead's Metaphysics (Bloom
ington: Indiana University Press, 1975).
16. La naissance de Ia physique dans le texte de Lucrece, p. 1 39.
17. LuCRETIUS, De Natura Rerum, Book II. ·�gain, if all move
ment is always interconnected, the new arising from the old in a
determinate order-if the atoms never swerve so as to originate
some new movement that will snap the bonds of fate, the ever
lasting sequence of cause and effect-what is the source of the
free will possessed by living things throughout the earth?"
18. M. SERRES, op. cit . , p. 136.
19. M . S ERRES , op. cit. , p. 1 62 ; also pp. 85-86 and " Roumain et
Faulkner traduisent l' Ecriture," La traduction (Paris: Minuit,
1974).
20. S . MoscoviCI , Hommes domestiques et hommes sauvages,
pp. 297-98.
2 1 . T. KuHN, The Structure of Scientific Revolutions, 2nd ed. incr.
(Chicago: Chicago University Press, 1970).
22. See A. N. WHITEHEAD, Process and Reality, op. cit. and M.
HEIDEGGER Sein und Zeit (Tiibingen: Niemeyer 1977).
23 . H . WEYL, Philosophy of Mathematics and Natural Science
(Princeton, N .J.: Princeton University Press, 1 949).
24. A . NEHER, "Vision du temps et de l ' histoire dans Ia culture
juive," Les cultures et le temps (Paris: Payot, 1 975), p. 179.
"
INDEX
Note: Page numbers given in boldface indicate location of definitions
or discussions of terms or concepts indexed here.
change of, 62, 63; on
Acceleration, 57-59
Affinity, 29, 1 36
Against the Current (Berlin), 2
Agassiz, Louis, 1 95
Alchemy, 64; affinity in, 1 36;
Chinese, 278
Alembert, Jean Le Rond d ' , 52 ;
Diderot and, 80-82;
opposition to Newtonian
science of, 62, 63 , 65, 66
Ambidextrous Universe, The
(Gardner), 234
Amoebas, 1 56-60
Ampere, Andre Marie, 67, 76
Anaxagoras, 264
Antireductionists, 1 73-74
Archimedes, 39, 4 1 , 304
Aristotle, 39, 40, 7 1 , 85 , 1 73,
305, 306; on change, 62;
notion of space of, 1 7 1 ;
physics of, 39-4 1 ; and
theology, 49-50
Arrow of time, xx, xxvii, 8 , 16;
Boltzmann on, 253-55; and
elementary particles, 288;
and entropy, 1 1 9, 257-59; and
heat engines , 1 1 1- 1 5 ; Layzer
on, xxv ; meaning of, 289;
and probability, 238-39; roles
played by, 30 I
Atomists, 3, 36; conception of
turbulence, 1 4 1
Attractor, 1 2 1 , 1 33 , 140, 152
Bach, J. S., 307
Bachelard, Gaston, 320n
Bacterial chemotaxis, I 75
Baker transformation, 269,
272-76, 278-79, 283, 289
Being and Becoming, 3 1 0
Belousov-Zhabotinsky reaction,
1 5 1 -53 , 1 68
Benard instability, 1 42-44;
transition to chaos in, 1 67-68
Bergson, Henri, 10, 79, 80,
90-94, 96, 1 29, 1 73-74,
30 1 -2 ; on dynamics, 60; on
time, 2 14, 294
Berlin, Isaiah, 2 , 1 1 , 1 3 , 80
Bernoulli, Daniel, 82
Berry, B . , 1 7
Berthollet, Claude Louis ,
Comte, 3 19n
Besso, Michele, 294
B ifurcations, xv, 160-61 , 1 76,
275 ; cascading, 1 67-70; in
evolution, 1 7 1 -72;
fluctuations and, 1 77, 180; in
reaction-diffusion systems,
260; role of chance in, xxvi,
170, 1 76; social, 3 1 3 ; and
335
ORDER OUT OF CHAOS
Bifurcations (cont'd)
statistical model, 205-6;
theory of, 1 4
Big Bang, xxvii , 288; and arrow
of time, xxv, 259
Biology, 2, 10; catalysts in,
1 33-34; chemical reactions
in, 1 3 1 -32; "communication"
among molecules in, 1 3 ;
concepts from physics
applied to, 207 ; and
conversion , 1 08 ; evolution
and, 1 2 , 1 28 ; logistic
equation in, 1 93-96;
molecular, see Molecular
biology ; reductionist
antireductionist conflict in,
1 74; technological analogies
in, 1 74-75; time in, 1 1 6;
Whitehead on, 96
Birchoff, 266
Blake, William, 30
Boerhave, Hermann, 1 05
Bohr, Niels, 2, 74, 220, 224-25,
228, 229, 292-93
Boltzmann, Ludwig, xvii, 15,
1 6, 1 22-27, 2 1 9, 227, 234-36,
258, 259, 274, 286-87, 297,
329n, 33 l n ; and arrow of
time, 253-55; on ergodic
systems, 266; on evolution
toward equilibrium, 240-43;
objections to theories of,
243-46; and theory of
ensembles, 248, 250
Boltzmann's constant, 1 24
Boltzmann's order principle ,
122-28, 142, 1 43 , 1 50, 163 ,
1 87
Bordeu, 32 1 n
Born, Max, 220, 235
Boundary conditions, 106,
1 20-2 1 , 1 25, 1 26, 1 38-39,
1 42, 147, 1 5 1
Braude!, xviii, xix
Bridgmann, P. W. , 3 16n
Brillouin, 2 1 6
Broglie, Louis de, 220
336
Bruno, Giordano, 1 5
Bruns, 72, 265
Brusselator, 1 46, 1 48, 1 5 1 , 1 52,
1 60
"Brussels school , " xv
Buchdahl, Gerd, 3 1 6n
Buffon, Georges Louis Leclerc
de, Comte, 65-67, 3 1 9n
Butts, Thomas, 30
Caillois, Roger, 1 28
Calvin, John, xxii
Cancer tumors, onset of, 1 88
Canonical equation, 226
Canonical variables, 70, 7 1 ,
1 07, 222
Cardwell, D . , 322n
Carnap, Rudolf, 2 1 4, 294
Carnot, Lazare, 1 1 2
Carnot, Sadi, 1 1 1 - 1 5, 1 1 7, 1 20,
128, 140, 323n, 324n ; Carnot
cycle, 1 1 2- 1 1 4, 1 1 7
"Carrying capacity" of
systems, 1 92-97
Catalysis, 1 33-35, 1 45 , 1 53
Caterpillars, strategies for
repelling predators of, 1 94-95
Catherine the Great, 52
Cells: Benard , 143 ; chemical
reactions within, 1 3 1 -32
Chance, concepts of, xxii-xxiii,
14, 170, 1 76, 203 ; see
Randomness
Change: motion and, 62-68 ;
nature of, 29 1 ; of state, 1 06;
in thermodynamic system,
1 20-2 1 ; Whitehead on, 95
Chemical clock, xvi, 1 3 ,
1 47-48, 1 79, 307;
communication in, 1 80; in
glycolysis, 155; in slime mold
aggregation, 159
Chemical reactions, 1 27; in
biology, 1 3 1 -32; diffusion in,
1 48-49; fluctuations and
correlations in, 1 79-8 1 ;
kinetic description of,
1 32-34; self-organization in,
337
1 44-45; thermodynamic
descriptions of, 1 33-37; see
also specific reactions
Chemistry, xi, I 0; Bergson on,
9 1 ; and Buffon, 65 , 66;
conceptual distinction
between physics and, 1 37 ;
and conversion, 1 08; Diderot
on, 82, 83 ; fluctuations and,
1 77-79; inorganic, 1 52, 153;
irreversibility in, 209;
Newtonian method in, 28;
relation between order and
chaos in, 168 ; and "science
of fire," 1 03 ; temporal
evolution in, 10- 1 1
China, 57; alchemy in, 278;
social role of scientists in,
45-46, 48
Chiral symmetry, 285
"Chreod," 1 72
Chris taller model, 1 97, 203
Christianity, 46, 47, 50, 76
Chronogeography, 272
Chuang Tsu, 22
Clairaut, Alexis Claude, 62, 65
Clausius, Rudolf Julius
Emanuel, 1 14, 1 1 5 , 233 , 234,
240, 304 ; entropy described
by, 1 1 7- 19
Clinamen, 1 4 1 , 303 , 304
Clocks: invention of, 46; as
symbol of nature, I l l ; see
also Chemical clocks
Closed systems, xv, 1 25
Collective phenomena, xxiv; in
amoebas, 156-160; in insects,
1 8 1 -86; in human geography,
1 97-203 ; in social
anthropology, 205, 3 1 7n
Collisions, 63 , 69, 1 32, 240-42,
270-7 1 ' 280-85
Combinatorial analysis, 1 23
Communication: description as,
300; in dissipative structures,
1 3 , 148; and entropy barrier,
295-96; and fl uctuations,
1 87-88; and irreversibility,
INDEX
295 ; molecular basis to, xxv,
1 80; stabilizing effects of, 1 89
Compensation, 107; Clausiuson
Carnot cycle, 114; statistical ,
1 24, 1 3 3 , 240
Complementarity, principle of,
225
Complexions, 1 23 , 1 24, 1 27 ,
150; in Benard instability,
142-43
Complexity: dynamics and
science of, 208-9 ; limits of,
1 88-89; modelizations of,
203-7
Comte, Auguste , 1 04-5
Condillac, Etienne Bonnot de,
66
Condorcet, Marie Jean Antoine
Nicolas Caritat, marquis de,
66
Conservation, life defined in
terms of, 84
Conservation of energy, I 07- 1 1 ;
and Carnot cycle, 1 1 4, 1 1 5 ;
and entropy, 1 1 7, 1 1 8 ;
principle of, 69-7 1
Convection , 1 27; in Benard
instability, 1 42
Copernicus, 307
Correlations: dynamics of,
280-85 ; fluctuations and,
1 79-8 1
Cosmology, xxviii , 1 , 10;
entropy and, 1 1 7; mysticism
and, 34; and
thermodynamics, 1 1 5 - 1 7 ;
time and, 2 1 5 , 259;
Whitehead's, 94
Counterintuitive responses, 203
Critical threshold see Instability
threshold
Critique of Pure Reason (Karit),
86
Cybernetics (Weiner), 295-96
Darwi n , Charl e s , xiv, x x , 1 2 8 ,
1 40, 2 1 5 , 240, 24 1 , 25 1
ORDER OUT OF CHAOS
Darwinian selection, 1 90, 1 9 1 ,
1 94, 1 95
David, Ben, 3 1 8n
Democritus, 3
Deneubourg, J. L. , 1 8 1
Density function p , 247-50,
26 1 , 264 ; with arrow of time,
277, 289; or distribution
function, 289; in phase space,
265-72, 274, 279
Deoxyribonucleic acid (DNA),
1 54; dissymetry of, 1 63
Desaguliers, J. T. , 27
Descartes, Rene , 62, 63 , 8 1
Destiny, 6, 257
Determinism, xxv, 9, 60, 75 ,
1 69-70, 176, 1 77, 2 1 6, 226,
23 1 264, 269-72, 304;
'
concepts of, xxii-xxiii
Dialectics of Nature (Engels),
253
Dialogue Concerning the Two
Chief World Systems
(Galileo), 305
Dictionnaire Philosophique
(Voltaire), 257
Dictyostelium discoideum, 1 56,
1 57
Diderot, Denis, 79-85 , 9 1 , 1 36,
309, 32 1 n
Differential calculus, 57, 222
Diffusion, 1 48-49, 1 77
Dirac, Paul, 34, 220, 230
Disorder, xxvii , 1 8 , 1 24, 1 26,
142, 238, 246, 250, 286-87,
293
Dissipation (or loss), 63, 1 1 2,
I 15, 1 1 7, 120, 1 25 , 1 29,
302-03
Dissipative structures, xii, xv,
xxiii, 1 2- 14, 1 42-43, 1 89,
300; coherence of, 1 70;
communication in, 1 48 ;
cultural , xxvi
Dissymmetry, 1 24; 1 63 ; in time,
125; see also Symmetry
breaking
338
Distribution function se�
Density function p
Dobbs, B. J. , 3 1 9n
Dander, Theophile de, 1 36, 325
Donne , John, 55
Driesc h, Hans, 1 7 1
Drosophila, 1 72
du Bois Reymond , 77, 97
Diierrenmatt, E , 2 1
Duhem, Pierre Maurice Marie,
97
Duration, xxviii-xix ; Bergson's
concept of, 92, 294
Dynamics, I I , 14- 1 5, 58-62,
107; baker transformation in,
276-77 ; basic symmetry of,
243 ; change in, 62-68 ;
concept of order in, 287 ; of
correlations, 280-85 ;
incompatibility of
thermodynamics and, 2 1 6,
233-34, 252-53; language of,
68-74 ; and Laplace's demon,
75-77; objects of, 306;
operators in, 222; probability
generated in, 274;
reconciliation of
thermodynamics and, 1 22 ;
reversibi l ity i n , 1 20; and
science of complexity, 208-9;
static view of, xxix;
symmetry-breaking in,
260-6 1 ; and theories of
irreversibil ity, 25 1 ; theory of
ensembles in, 247-5 1 ;
twentieth-century renewal of,
264-72
Dyson, Freeman, 1 1 7
Ecology, logistic equation in,
1 92-93, 196, 204
Eddington, Arthur Stanley, xx,
8, 49, 1 1 9, 233 , 29 1
Edge of Objectivity, Th�
(Gillespie), 3 1
Ehrenfest model, 235-38, 240,
246
339
Eigen, M., 190-91
Eigenfunctions, 22 1 -23; of
Hamiltonian operator, 227; of
operator time T, 289;
superposition of, 227-28
Eigenvalues, 22 1 , 222; of
Hamiltonian operator, 226,
227
Einstein, Albert, xiv, 76, 242,
27 1 , 30 1 , 307, 3 1 0; on basic
myth of science, 52-53;
demonstration of
impossibility by, 296;
dialogue with Tagore, 293;
ensemble theory of, 247-5 1 ,
26 1 ; God of, 54; on
gravitation , 34 ; on
irreversibility, 15, 258, 259,
294-95 ; Mach's influence on,
53; and quantum mechanics,
2 1 8-20, 224; on scientific
asceticism, 20-2 1 ; on
simultaneity, 2 1 8 ; special
theory of relativity of, 1 7 ;
thought experiments of, 43 ;
on time, 2 14- 1 5 , 251 ;
"unified field theory" of, 2 ;
use of probabilities rejected
by, 227
Electrons, 287, 288; stationary
states of, 74
Elementary particle physics,
xxviii, l , 2, 9, 10, 19, 34,
230, 285-88 ; "bootstrap"
philosophy in, 96; quantum
mechanics and, 230-3 1 ;
T-violation in, 259; wave
behavior in, 1 79
Eliade, Mircea, 39-40
Elkana, Y. , 323n, 33 1 n
Embryo: development of,
8 1 -82; formation of gradient
system in morphogenesis of,
1 50; internal purpose of,
1 7 1 -73
Encyclopedie, 83
Energeticists, 234
INDEX
Energy, 107 ; dissipation of,
302-3 ; and elementary
particles, 287; and entropy,
1 1 8- 19; exhaustible, I l l , 1 14 ;
a s invariant, 265; for living
cells, 155; in quantum
mechanics, 220-2 1 ; of
unstable particles, 74; of
universe, 1 1 7 ; see also
Conservation of energy
Energy conversion, 12, 1 08,
1 14
Engels , Friedrich, 252-53, 332n
Engines, 12, 103, 105-07,
1 1 1- 1 5
Enlightenment, the, 67, 79, 80,
86
Ensemble theory (Einstein
G ibbs), 247-5 1 , 26 1 ;
equilibrium and, 265
Entelechy, 1 7 1
Entropy, xix-xx, 1 2, 1 4- 1 8,
1 1 7-22, 227; and arrow of
time, XXV, 253-54, 257-59;
and atomism, 288; as barrier,
277-80, 295-97; in evol ution,
1 3 1 ; flux and force and, 1 35 ,
1 37 ; law of increase of, xxix;
in linear thermodynamics,
1 38-39; mechanistic
interpretation of, 240-43 ;
probability and, 124, 1 26,
1 42, 234, 235 , 237-38, 274;
production of, 1 1 9, 1 3 1 , 1 3 3 ,
1 35, 1 37-39, 1 42; as
progenitor of order, xxi-xxii ;
as selection principle,
285-86; subjective
interpretation of, 1 25 , 235,
25 1 -52; thought experiment
on, 244; universal
interpretations of, 239
Enzymes, 1 33-34; feedback
action of, 154; in glycolysis,
155; resembling Maxwell's
demon, 1 75
Epicurus, 3 , 305
ORDER OUT OF CHAOS
Equilibrium, xvi; and baker
transformation, 273 ;
chemical , 1 33 ; chemical
reactions in, 1 79-80; and
entropy, 1 20, 1 3 1 ; evolution
toward , 24 1 -43 ; flux and
force at, 1 35-37 ; in future,
275, 276; and matter-light
interaction, 2 1 9 ; maximum
probability at, 286; and
theory of ensembles, 265 ;
thermal, 1 05, 1 1 6; thermal
chaos in, 168 ; in
thermodynamics, 12, 1 3 ,
1 25-29, 1 38; velocity
distribution in state of, 24 1 ;
see also Far-from
equilibriu m ; Nonequilibrium
Ergodic systems, 266
Espagnat, B. d', 329n
Esprit de systeme, 83
Euclid, 1 7 1
Euler, Leonhard , 5 2 , 65, 82
Everett, 228
Evolution, xx, 1 2 , 1 28 ; and
arrow of time, xxv;
bifurcations in, 1 7 1 -72;
biological, 153; Boltzmann .
on, 240; chemical, 177 ;
concepts from physics
applied to, 207-9; cosmic,
2 1 5 , 288; Darwinian, 1 28 ;
from disorder t o order, xxix;
entropy in, 1 1 9, 1 3 1 ; toward
equilibrium, 24 1 -43;
feedback in, 1 96-203 ;
logistic, 192-96, 204;
·paradigm of, 297-98; in
quantum mechanics,
226-228, 238; toward
stationary state , 1 38-39;
structural stability in, 1 89-91
Existentialism, xxii
Expanding universe, 2, 19, 2 1 5 ,
259
Experimentation, 5, 4 1 -44; and
global truth, 44-45 ; Kant
and, 88; universality of
340
language postulated by, 5 1 ;
Whitehead on, 93 , 95 ; see
also Thought experiments
Falling bodies, Galileo's laws
for, 57, 64
Far-from-equjlibrium, xxvi,
xxvii, 1 3- 14, 1 40-45, 300;
chemical instability in,
146-53 ; in chemistry, 1 77;
dissipative structures in, 1 89;
in molecular biology, 1 53-59;
prebiotic evolution in, 1 9 1 ;
self-organization in, 176
Faraday, Michael, 108
Faust (Goethe), 1 28
Feedback, 153; in biological
systems, 1 54 ; in evolution,
1 9 1 , 1 96-203 ; between
science and society, xiii
Feigenbaum sequence, 169
Feuer, 329n
Feynman, Richard, 44
Fluctuations, xv, xxiv-xxv,
xxvii, 1 24-25, 1 40-41 , 143 ;
amplification of, 1 4 1 , 1 43
1 8 1 -89; and chemistry,
1 77-79; and correlations,
1 79-8 1 ; in Markov process,
238; on microscopic scale,
23 1 -32
Fluid flow, 1 4 1
Fluxes, 1 35-37; random noise
in, 1 66-67; in reciprocity
relations, 1 37-38
Forces: generalized, 135-37; in
reciprocity relations, 1 37-38
Fourier, Baron Jean-Joseph, 1 2 ,
104, 1 05, 1 07, 1 15-17
Fraser, J. T. , 2 1 4
Frederick I I , King of Prussia,
52
Free particles, 70-72
Free will, xxii
Freud, Sigmund, 1 7
Friedmann, Alexander, 2 1 5
Fundamental level of
description, 252-53
341
Galileo, 40, 4 1 , 50, 5 1 , 305; on
cause and effect , 60; and
global truths, 44; and
mechanistic world view, 57;
thought experiments of, 43
Galvani, Luigi, 1 07
Gardner, Martin, 234, 259
Gassendi, Pierre, 62
Generalized forces, 1 35-37
Geographical time, xviii
Geography, 197; internal time
in, 272
Geology, 1 2 1 ; time in, 1 1 6, 208
Gibbs, J. W. , 1 5 , 238, 247-5 1 ,
26 1
Gillispie, C. C . , 3 1 , 3 2 1 n
Glycolysis, 155-56
Goethe, Johann Wolfgang von,
1 28
Gould, Stephen J. , 204
Grasse, 1 8 1 , 1 86
Gravitation: Comte on, 105;
and determination of motion,
59; in early universe, 298;
Einstein's interpretation of,
34; explanatory power of, 28,
29; in far-from-equilibrium
conditions, 163-64; universal
law of, I , 1 2, 66
Guldberg and Waage's law (also
Mass action, law oO, 1 33
Hamilton, William Rowan , 68,
%; see Hamiltonian
Hamiltonian: equation, 249;
function, 68, 70-7 1 , 74, 1 07,
220-2 1 ; operator, 22 1 ,
226-27 ; and T-violation, 259
Hankins, Thomas, 3 1 8n
Hao Bai-lin, 1 5 1 , 152
Hausheer, Roger, 2
Hawking, 1 1 7
Heat, 12, 79, 103 ; conduction
of, 104, 1 35 ; electricity
produced by, 108; and heat
engines, 12, 103, 1 06-7 ; heat
engine s , arrow of time and ,
1 1 1- 1 5 ; propagation of,
INDEX
1 04-5; repelling force of, 66;
specific, 106; transformation
of matter by, 1 05
Hegel, G. W. R , 79, 89-90, 92,
93 , 1 73
Heidegger, Martin, 32-33, 42,
79, 3 1 0
Heisenberg, Werner, xxii, 220,
292, 2%
Heisenberg uncertainty
relations, 1 78 , 222-26
Helmholtz, Hermann Ludwig
Ferdinand von, %, 1 09- 1 1
Herivel, J. , 322n
Hess, Benno, 155
Hirsch, J. , 1 5 1
History: of ideas, 79; open
character of, 207 ; reinsertion
of, into natural and social
sciences, 208; of science, 307
(cosmology) 208, 2 1 5 ,
(geography) 1 97, 272,
(geology) 1 1 6, 1 2 1 , 208
Holbach, Paul Henri Thiry,
baron d ' , 82
Hooykaas, R . , 3 17n
Hopf, 266
Hubble, Edwin Powell , 2 1 5
Humanities, schism between
science and, 1 1 , 1 3
Hume, A . Ord, 3 1 7n
Huss, John, xxii
Huyghens, Christiaan, 60
Hydrodynamics, 1 27; far-fromequilibrium phenomena in,
141
Hypnons, 1 80, 287, 288
Hysteresis, 1 66
Idealization, 4 1 -43, 69,
1 1 2- 1 14, 1 1 5 , 1 20, 2 16, 248,
252, 305-06
Impossibility, demonstrations
of, 17, 2 1 6- 17, 296, 299-300
Individual time, xviii
Industrial Age , 1 1 1 ; combusti on
and, 1 03
ORDER OUT OF CHAOS
Information, 1 7- 1 8, 250,
278-79, 283, 295, 297-98
Initial conditions (or state), 61 ,
68, 75, 1 2 1 , 1 24, 128, 1 29,
1 39-40, 142, 147, 248,
26 1 -67, 270-7 1 , 276, 278-79,
295, 3 1 0
Innovation, psychological
process of, xxiv
Innovative becoming,
philosophy of, 94
Instability, dynamic, 73,
268-72, 276, 300; chemical,
1 44-53; thermodynamic,
1 4 1 -42; threshold, 146, 1 47,
1 60
Insects, self-aggregation of,
1 8 1 -86
Integrable systems, 7 1 -72, 74,
264-65, 302
Internal time, 272-73, 289
Intrinsically irreversible
systems, 275-77, 289
Intrinsically random systems,
274-76, 289
Intuition, 80, 9 1 , 92
Irreversibility, xx, xxi, xxvii,
xxviii, 7-9, 63 , 1 1 5 ;
acceptance b y physics of,
208-9; and biology, 1 28, 1 75 ;
in chemistry, 1 3 1 , 1 37, 177;
controversy over, 1 5- 1 6;
cultural context of
incorporation into physics of,
309- 10; and dynamics of
correlations, 280-85 ; Einstein
on, 294-95 ; and ensemble
theory, 250; in evolution, 1 28,
1 89; formulation of theory of,
1 05 , 1 07, 1 1 7-2 l ; and limits
of classical concepts, 26 1 -64;
and matter-light interaction,
2 1 9; measurement and, 228;
microscopic theory of, 242,
257-59, 285-86, 288-90, 3 1 0;
probability and, 16, 1 24, 1 25 ,
233-40; quantitative
expression of, I l l ; from
342
randomness to, 272-77; rate
of, 1 35 ; in reciprocity
relations, 1 38; as source of
order, 15, 292; subjective
interpretation of, 25 1-52; as
symmetry-breaking process,
260-6 1 ; in thermodynamics,
12; see also Arrow of time;
Entropy
Isomerization reaction, 1 65
Jammer, M . , 329n
Jordan, 220
Joule, James Prescott, 1 08-9
Kant, Immanuel , 79, 80, 85-89,
93 , 99, 2 1 4
Kauffman, S . A . , 1 72
Kepler, Johannes, 49, 57, 67,
307
Keynes, Lord, 3 1 9n
Kierkegaard , Soren, 79
Kinetic energy, 69-7 1 , 90, 1 07,
26 1
Kinetics, chemical , 1 32-34
Kirchoff, Gustav, %
Knight, 32 1 n
Koestler, Arthur, 32, 34-35
Kolmogoroff, 266
Kothari , D. S . , 293
Koyre , Alexandre, 5, 32, 35-36,
62, 3 1 7n , 3 1 9n
Kuhn, Thomas, 307-9, 320n,
329n
Lagrange, Comte Joseph Louis,
52, %, 1 04, 324n
Laminar flow, 1 4 1 -42, 303
Laplace, Marquis Pierre Simon
de, xiii , 28, 52, 54, 66, 67,
1 1 5 , 323n ; Fourier criticized
by, 1 04
Laplace's demon, 75-77, 87,
27 1
Large numbers, law of, 14, 1 78,
1 80
Lavoisier, Antoine Laurent, 2&,
1 09
343
Layzer, David , xxv
Lebenswe/t, 299
Leibniz, Gottfried von, 50, 54,
302, 303 ; formulae for
velocity and acceleration, 58;
monads of, 74
Lemaitre, Georges, 2 1 5
Lenoble, R. , 3
Levi-Strauss, Claude, 205, 3 1 7n
Levy-Bruhl , L., 292
Lewis, G. N . , 233
Liebig, Baron Justus von, 1 09
Life: Bergson on, 92;
compatibility with far-from
equilibrium conditions, 143 ;
as expression of self
organization, 1 75-76; and
order principle, 1 27-28;
origin of, 14; Romantic
concepts of, 85 ; Stahl's
definition of, 84; symmetry
breaking as characteristic of,
1 63 ; temporal dimensions of,
208; see also Molecular
biology
Light: velocity of, 1 7, 55,
2 17- 19, 278, 295, 296; wave
particle duality of, 2 1 9-20
Limit cycle, 146-47
Linear thermodynamics,
1 37-40
Liouville equation , 249, 250,
266
Logistic evolution, 1 92-96,
203-04
Look at the Harlequins
(Nabokov), 277
Lorentz, Hendrik Antoon, 270
Loschmidt, 244, 246
Louis XIV, King of France, 52
Love, Milton, 1 95
Lucretius, 3 , 1 4 1 , 302-5 , 3 15n,
334n
Luther, Martin, xxii
Lyapounov, 1 5 1
Mach. Ernst. 49. 53-54. 97.
3 1 8n
INDEX
Machines: Archimedes's, 4 1 ;
ideal, 63, 69-70; mathematics
and, 46; using heat, 1 03
Macroscopic system, 1 06-07
Many-worlds hypothesis, 228
Markov chains, 236, 238, 240,
242, 273-76; and dynamics of
correlations, 283 ; and entropy
barrier, 278
Marx, Karl, 252
Mass action, law of, 1 33 , 1 36,
23 1
Materialistic naturalism, 83
Mathematization, 46; in
Hamiltonian function, 7 1 ;
Hegel's critique of, 90;
Leibniz on, 50; of motion, 60
Matter: active, 9, 286-90, 302;
anti-matter, 230-3 1 ; Diderot
on, 82; effect of heat on, 1 05;
in far-from-equilibrium
conditions, 14; interaction of
radiation and, 2 1 9 ; new view
of, 9; nonequilibrium
generated by, 1 8 1 ; perception
of differences by, 1 63 , 1 65 ;
properties of, 2 ; Stahl on,
84-85 ; transition to life from ,
84; wave-particle duality of,
221
Maxwell, James Clerk, 54, 73,
1 22, 1 60, 240, 24 1 , 266
Maxwell's demon, 1 75 , 239
Mayer, Julius Robert von, 1 09,
Il l
Measurement, irreversible
character of, 228-29
Mechanics, II , 1 5 ;
generalization of, I l l ; Hegel
on, 90; and probability, 125 ;
see also Dynamics; Quantum
mechanics
Medicine: Bergson on, 9 1 ;
Diderot on, 82, 83
Merleau-Ponty, M . , 299
Metternich, Clemens Wenzel
Nepomuk Lothar, Fiirst von,
xiii
O RDER OUT OF CHAOS
344
eighteenth-century opposition
to, 65 ; laws of motion, 70;
and mechanistic world view,
57; objectivity defined by,
2 1 8; objects chosen for study
by, 2 1 6; presentation of
Principia to Royal Society, 1 ;
second law, 58; see also
Newtonian science
Newtonian science, xiii, xiv,
xix, xxv, xxvi, 37-40, 2 1 3 ;
absence of universal constant
in, 2 1 7 ; concept of change in,
63-68; Diderot and, 80, 82;
Koyre on, 35-36;
incompleteness of, 209;
instability and, 264;
instability of cultural position
of, 30; Kantian critique of,
85-87; laws of motion of,
57-59; limits of, 29-30;
positivism and, 96; prophetic
power of, 28 ; spread of,
28-29; Voltaire and, 258;
world view of, 229
. Nietzsche, Friedrich, I l l , 1 36
Nisbet, R . , 79
Nabokov, Vladimir, 277-78
N onequilibrium: cosmological
Napoleon, 52, 67
dimension of, 23 1 ; difference
Natural laws: belief in
between particles and
universality of, 1 -2;
antiparticles in, 285 ;
mathematical concepts of, 46;
fluctuations in, 1 78-80;
Newton on, 28; primary and
innovation and, xxiv; and
secondary, 8; social structure
origin of structures, xxix; as
and views of, 48-49; time
source of order, 287 ; see also
independent, 2, 7; trials of
Far-from-equilibrium
animals for infringements of,
Non-linearity, 14, 1 34, 1 5 3 ,
48
1 54-55, 1 97 see Catalysis
Nature of the Physical World,
Non-linear thermodynamics,
The (Eddington), 8
1 37, 140
Needham, Joseph, 6-7, 45, 48,
Noyes, 152
49, 278, 322n
"Neolithic Revolution," 5-6, 37 Nucleation, 1 87, 1 88
Neumann, von, 266
New Science, The (Vico), 4
Oersted, Hans Christian, 108
Newton, Isaac, xv, xxviii, 1 2 ,
Old Testament, xxii
Onsager, Lars, 1 37, 1 38
27-29, 76, 98, 1 04, 1 20, 1 24,
Operators, 22 1 -22, 225 ;
234, 305, 3 19n ; alchemy and,
commuting, 223
64; on change, 62, 63 ;
Meyerson, Emile, 293
Microcanonical ensemble, 265
Minimum entropy production,
theorem of, 138-4 1
Minkowski, H . , 230
Mole, 1 2 1 n
Molecular biology, 4 , 8 ; far
from-equilibrium conditions
in, 1 53-59; vitalism and, 84
Molecular chaos assumption,
246
Monads, 74, 302-03
Monod, Jacques, 3-4, 22, 36,
79, 84
Morin, Edgar, xxii-xxiii
Morphogenesis, 172, 1 89
Moscovici, Serge, 22, 306-7
Motion: and change, 62-68;
complexity of, 75 ; instability
of, 73 ; in mechanical engine
vs. heat engine, 1 1 2;
positivist notion of, 96 ;
productton of, i n heat engine,
1 07; universal laws of, 57-62,
83 ; see also Dynamics
345
Opticks (Newton), 28
Optimization , 197, 207
Order, 12, 18, 126, 1 3 1 , 143 ,
1 7 1 -75 , 238, 246, 250-5 1 ,
286-87
Order through fluctuation, 159,
178; models based on
concept of, 206
Organization theory, xxiv
Oscillating chemical reactions,
19, 147-49
Oscillations: glycol ytic, 155;
time- and space-dependent,
148
Ostwald, Wilhelm, 324n
Pascal, Blaise, 3 , 36, 79
Pasteur, Louis, 163
Pattern selection, 1 63 , 1 64
Pearson, Karl, 49
Peirce , Charles S . , 17, 302-3
Pendulum, 16, 73, 2 1 6, 261 -62
Phase changes, 1 87
Phase space, 247-50, 261 , 264;
delocalization in, 289;
unstable systems in, 266-72
Photons, 230, 288
Physicists, The (DOerrenmatt),
21
Physics: application of concepts
to evolution, 207-9; Bergson
on, 9 1 , 92; changing
perspective in, 8-9;
complementary developments
in biology and, 1 54; and
concepts of change, 63;
conceptual distinction
between chemistry and, 1 37;
Diderot on, 80-83 ;
evolutionary paradigm in ,
297-98; inspired discourse of,
76; introduction of
probability in, 123; and laws
of motion, 57; Lucretian,
1 4 1 ; macroscopic, xii ;
objectivity in, 55 ; positivist
view of, 97; of processes,
INDEX
243 ; and theology, 49; time
in, 1 1 6; vitalism and, 84;
Whitehead on, 95, 96
Planck, Max, 1 2 1 , 2 19, 242,
324n, 329n, 33 l n ; on second
law of thermodynamics,
234-35
Planck's constant, 2 17, 2 1 9 ,
220, 223 , 224
Planetary motion: Kepler's laws
for, 57 ; in Newtonian
dynamics, 59, 64
Plato, 7, 39, 67
Poincare , Jules Henn, 68, t2,
97, 15 1 , 236, 243 , 253 , 265
27 1
Poisson distribution , 1 79-8 1
Pope, Alexander, 27, 67
Popper, Karl , 5 , 1 5 , 254-55,
258-59, 276, 3 17n
Populiire Schriften
(Boltzmann), 240
Positive-feedback loops , xvit
Positivism, 80, 96-98 ; Comte
and, 1 04-5 ; German
philosophy and, 109
Potential: dynamic, 69-70;
thermodynamic, 1 26, 138-40
Potential energy, 69-70, 73 , 107
Prebiotic evolution, 1 90-91
Pre-Socratics, 38-39
Principia (Newton), l, 28
Probability, 1 22-24; Einstein
on, 259; at equilibrium, 286;
in far-from-equilibrium
conditions, 1 43 ; entropy and�
142, 274, 297; and
fluctuations, 1 78, 1 79 ; and
irreversibility, 233-40; in
quantum mechanics , 227;
subjective vs. objective
interpretations of, 274; in
unstable systems, 27 1 -72
Process: physics of, 12, 105,
1 07, 243 ; Whitehead's
concept of, 258, 303
Process and Reality
(Whitehead), 93, 96, 3 1 0
ORDER OUT OF CHAOS
Proust, Marcel , 1 7
Pulley systems, 4 1 -42
Quantization, 220
Quantum mechanics, 9, I I , 15,
34, 2 1 8-22 ; causality in, 3 1 1 ;
correlations in, 286; cultural
background to, 6;
delocalization in, 289;
demonstrations of
impossibility in, 2 1 7, 296-97;
Hamiltonian function in, 70;
Heisenberg's uncertainty
relations in, 222-26; and
Newtonian synthesis, 67-68;
and probability, 125, 178-79;
and reversibi lity, 6 1 ; temporal
evolution in, 226-29, 238;
thought experiments in, 43
Quetelet, Adolphe, 1 23 , 241
Radiation: black-body, 209, 2 1 5 ;
interaction of matter and, 2 1 9
Randomness, xx, 8, 9, 1 26,
23 1-32, 236; to irreversibility
from, 272-77
Rationality, 1 , 29, 32, 36, 40,
42, 92, 306
Reaction-diffusion systems, 260
Real ity, conceptualization of,
225-26
Reciprocity relations, 1 37-38
Reduction of the wave function,
227-28
Reductionism, 1 73-74
Reichenbach, H . , 97
Relation, philosophy of, 95
Relativity, 9, 34, 2 1 5 , 307; in
astrophysics, 1 1 6; Bergson's
misunderstanding of, 294;
demonstrations of
impossibility in, 2 1 7 , 296;
Einstein's special theory of,
1 7 ; and elementary particles,
230; and Newtonian
synthesis, 67, 68, 229; static
geometric c haracter of time
and , 230; and thermal history
346
of universe, 23 1 ; thought
experiments in, 43 ; and
universal constants, 2 1 7- 1 8;
and velocity of light, 295
Religion: ancient Greek, 38, 39;
resonance between science
and, 46-5 1
Residual black-body radiation ,
209, 2 1 5
Respiration, physiology of, 1 09
Reversibi lity : of canonical
equations , 7 1 ; of
thermodynamic
transformation , 12, 1 1 2- 1 3 ,
120; of trajectories, 60-61
Revolution, concept of, xxiv
Reynolds' number, 144
Ritz, W, 259
Rosenfeld, Leon, 264, 329
Sakharov, A. D., 230
Sartre, Jean-Paul, xxii
Schlanger, J. , 3 2 1 n
Schrodinger, Erwin, 1 8- 1 9 220,
242, 329n
SchrOdinger equation, 226-29
Science and Civilization in
China (Need ham), 278
"Scientific revolution, " 5 , 6
Scott, W , 322n
Sein und Zeit (Heidegger), 3 10
Self-organization, xv; in Benard
instability, 142; and
bifurcations, 1 60-67 ; in
chemical clock, 148; in
chemical reactions, 144-45 ;
and dynamics, 208 ; as
function of fluctuating
external conditions, 165-67;
life as expression of, 1 75-76;
in slime-mold aggregation,
1 56; in turbulence, 1 4 1 -42
Serres, Michel, 104, \ 4 1 , 303,
304, 320n, 323n
Shakespeare, William, 293
Signals, propagation of, 2 17,
218
3<47
Simplicity, and classical
science, 7, 2 1 , 48, 5 1
Simultaneity, definition of, 2 1 8
Sinai , 266
Singular points, Maxwell on, 73
Smith, Adam, 103 , 207
Social evolution: concepts from
physics applied to, 207;
feedback in, 197-203 ; logistic
equation in, 192-94, 1%;
models of, 204-6
Social sciences, 3 1 2- 1 3 ;
evolution in, 128; time in, 1 1 6
Social time, xviii
Space: Euclidian vs.
Aristotelian, 1 7 1 ; involved in
turbulence, 141 ; oscillations
dependent on, 148 ; sacred vs.
profane, 40; temporal
dimension of, 17; see also
Phase space
Spatial symmetry-breaking, 260
Specific heat, 106
Spinoza, Baruch , 2 1 5, 3 1 0
Sterility, 139, 140, 144-45
Stahl, Georg Ernst, 83-85, 1 73 ,
175, 309
Stationary state: evolution
toward , 138-39; instability of,
1 40-42
Stent, Gunther, 34
Structural stability, 189-91
Subl unar world, 40, 305
"Survival of the fittest ," 192,
1 94
Symmetry-breaking, 1 60-67; in
embryo development, 1 73 ;
irreversibility and, 260-6 1
Systeme du Monde (Laplace),
66
T-violation, 259
Tagore, Rabindranath, 293
Technology, I ; analogies in
biology based on, 1 74-75;
and dynamics. 2 1 6 : Greek.
39; Heidegger on, 32;
INDEX
nineteenth-century, I l l ; see
Engines
Teil hard du Chardin, Pierre, 17
Temporal ity, 7; see also Time
Termites: role of fluctuations in
construction of nest of, 1 8 1 ,
186-87; and statistical model,
205
Thackray, A., 3 1 9n
Themes (Merleau-Ponty), 299
Thermal chaos, 167-68
Thermodynamics, xiv, xix-xx,
1 2 , 99, 103-7, 2 1 3 ; in
astrophysics, 1 1 6; and baker
transformation, 276-77;
bifurcation in, 160-67 ;
conservation of energy in,
1 07- I I ; and cosmology,
1 1 5- 17; discovery of
impossibilities in, 2 1 7, 2%;
and dynamics of correlations,
283-85 ; equilibrium, 125-29,
1 38; far-from-equilibrium,
1 40-45 ; flux and force in,
1 3 1 -37; incompatibility of
dynamics and, 2 16, 233-34,
252-53; linear, 1 37-40; of
living system, 155-56; non
linear, 1 37, 1 40 ; order
principle in, 1 22-26, 287;
Planck on, 234-35 ; second
law of, see Entropy ; see also
Heat
Thorn, Rene , xxii , xxiii , 326n
Thomson, William, 1 1 5-16
Thought experiments, 43 ; on
Boltzmann's entropy, 244; in
dynamics, 6 1 -62
Thousand and One Nights,
The, 277
Three-body problem, 72, 265
Time, xvii-xxi, to; Bergson on,
92 ; creative course of,
307- 10; "derivatives with
respect to," 58; in dissipative
structures, 144; dissymmetry
i n 1 25 : in dynamics. 6 1 . 69;
Einstein on, 2 1 4- 1 5 , 25 1 ;
.
ORDER OUT OF CHAOS
Time (cont'd)
277-78, 295-%; in everyday
life, 16- 1 7 ; global judgments
of, 17; in Hamiltonian
function, 70, 7 1 ; Hegel on,
90; and human symbolic
activity, 3 1 2; internal ,
272-73, 289; involved in
turbulence, 141 ; meaning of,
in physics, 93 ; as measure of
change, 62; in nineteenth
century physics, 1 1 7 ;
oscillations dependent on,
148 ; positivist view of, 97;
progressive rediscovery of,
208 ; in quantum mechanics.
229-30; revision of
conception of, 96; roots of, i n
nature, 18; social context of
rediscovery of, 19; in
thermodynamics, 12, 129;
varying importance of scales
of, 30 1 ; see also Arrow of
time; Irreversibility
Time Machine, The (Wells), 277
Toftler, Alvin, xi-xxvi, xxxi
Trajectories, 59-60, 68, 12 1 ,
177 ; intrinsically
indeterminate, 73 ; limits of
concept of, 261 -64; in phase
space, 247-48 ; and
probability, 122; in unstable
systems, 270-72 ; variations
in, 75
Transition probabilities, 274
Thrbulence, 141
Turbulent chaos, 1 67-68
Turing, Alan M . , 152
Unidirectional processes,
258-59
"Unified field theory, " 2
Universal constants, 2 1 7- 19,
229
Universe: age of, 1 ; aging of,
xix-xx; disintegration of,
1 1 6; energy of, 1 17 ; entropy
348
of, 1 1 8; expanding, 2, 19,
2 1 5 , 259; history of, 215; i n
Newtonian dynamics, 59;
nonequilibrium, 229�32;
Pierce on, 302-3 ; pluralistic
character of, 9; thermal
history of, 9; time-oriented
polarized nature of, 285
Unstable particles, 74, 23 1 , 288
Urbanization, model of,
198-202
Urn model, 235-38, 246, 273
Valery, Paul , 1 6, 301
Velocity, 57-59; distribution of,
240-42 ; instantaneous
inversion of, 61 , 243-46,
280-85; of light, 17, 55,
2 1 7- 19, 278, 295, 2%
Velocity distribution function,
242-46, 248-50
Velocity inversion experiment,
280-84
Venel, 83, 309
Vico, G. , 4
Vienna school, 97
Vitalism, 80, 83-84; vs.
scientific methodology, l lO
Volta, Count Alessandro, 107-8
Voltaire, 257-58
Waddington, Conrad H . , 172,
174, 207, 322n, 326n
Watanabe, S . , 330n
Watt, James, 103
Wave behavior, 179; see also
Chemical clocks
Wave functions, 226-28; time
and, 229-30
Wave-particle duality, 2 1 9-20,
226
Wealth of Nations (Smith). 103
Weiss, Paul, 174
Wells, H. G . , 277
Weyl , Herman, 3 1 1
34Q
Whitehead, Alfred North, 10,
17, 47, 50, 79, 93-96, 2 1 2,
2 1 6, 258, 302, 303 , 3 10, 322n
Wiener. Norbert, 295-96
Wycliffe, John, xxii
INDEX
Zermelo, 1 5 , 244, 253, 254
"Zero growt h" soc iety, 1 1 6
Zhang Shu-yu, 1 5 1
Zola, Emile, 323n