ilya_prigogine_isabelle_stengers_alvin_tofflerbookfi-org

I 

I 

I 

I

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. 

The foreword "Science and Change" copyright© 1984 by 

Alvin Tofjler. 

Book design by Barbara N. Cohen 

This 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. 

1 

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 50 83-21 403 

ISBN 0-553-34082-4 

Published simultaneously in the United States and Canada 

Bantam Books are published by Bantam Books.Inc.lts trademark. 

consisting of the words "Bantam Books" and the portrayal 

of a rooster, is Registered in the United States Patent 

and Trademark Office and in other countries. Marca Registrada. 

Bantam Books, Inc., 666 Fifth Avenue, New York, New 

York 10103. 

PRINTED IN THE UNITED STATES OF AMERICA 

FG 0987654321

ORDER OUT OF CHAOS 

MAN'S NEW DIALOGUE 

WITH NATURE 

llya Prigogine 

and 

Isabelle Stengers 

Foreword by 

Alvin Toffler 

BANTAM BOOKS 

TORONTO· NEW YORK • LONDON • SYDNEY

This book is dedicated to the memory of 

Erich Jantsch 

Aharon Katchalsky 

Pierre Resibois 

Leon Rosenfeld

TABLE OF CONTENTS 

FOREWORD: Science and Change by Alvin Toffier xi 

PREFACE: Man's New Dialogue with Nature 

xxvii 

INTRODUCTION: The Challenge to Science 1 

Book One: The Delusion of the Universal 

CHAPTER I: The Triumph of Reason 27 

1. The New Moses 27 

2. A Dehumanized World 30 

3. 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 

CHAPTER n: The Identification of the Real 57 

1. Newton's Laws 57 

2. Motion and Change 62 

3. The Language of Dynamics 68 

4. Laplace's Demon 75 

CHAPTER m: The 1Wo Cultures 79 

1. Diderot and the Discourse of the Living 79 

2. Kant's Critical Ratification 86 

3. A Philosophy of Nature? Hegel and Bergson 89 

4. Process and Reality: Whitehead 93 

5. "Ignoramus, lgnoramibus": 

The Positivist's Strain 96 

6. A New Start 98 

Book Two: The Science of Complexity 

CHAPTER IV: Energy and the Industrial Age 103

1. Heat, the Rival of Gravitation 1(}3 

2. The Principle of the Conservation of Energy 107 

3. Heat Engines and the Arrow of Time 111 

4. From Technology to Cosmology 115 

5. The Birth of Entropy 117 

6. Boltzmann's Order Principle 122 

7. 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 

1. A Change of Emphasis 213 

2. The End of Universality 217 

3. The Rise of Quantum Mechanics 218 

- 4. Heisenberg's Uncertainty Relation 222 

5. The Temporal Evolution of Quantum Systems 226 

6. A Nonequilibrium Universe 229

CHAPTER vm: The Clash of Doctrines 233 

1. Probability and Irreversibility 233 

2. Boltzmann's Breakthrough 240 

3. Questioning Boltzmann's Interpretation 243 

4. Dynamics and Thermodynamics: 1Wo Separate 

Worlds 247 

5. Boltzmann and the Arrow of Time 253 

CHAPTER IX: Irreversibility-the Entropy Barrier 257 

1. Entropy and the Arrow of Time 257 

2. Irreversibility as a Symmetry-Breaking Process 260 

3. The Limits of Classical Concepts 261 

4. The Renewal of Dynamics 264 

5. From Randomness to Irreversibility 272 

6. The Entropy Barrier 277 

7. The Dynamics of Correlations 280 

8. Entropy as a Selection Principle 285 

9. Active Matter 286 

coNcLusioNs: From Earth to Heaventhe 

Reenchantment of Nature 291 

1. An Open Science 291 

2. Time and Times 293 

3. The Entropy Barrier 295 

4. The Evolutionary Paradigm 297 

5. Actors and Spectators 298 

6. A Whirlwind in a Tu rbulent Nature 301 

7. Beyond Tautology 305 

8. The Creative Course of Time 307 

9. The Human Condition 311 

10. The Renewal of Nature 312 

NOTES 315 

INDEX 335

FOREWORD 

SCIENCE AND 

CHANGE 

by Alvin Toffler 

One of the most highly developed skills in contemporary We stern 

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 paribus-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 chiseled 

features, and a laserlike intensity. Deeply interested in 

archaeology, art, and history, he brings to science a remarkable 

polymathic mind. He lives with his engineer-wife, Marina, 

and his son, Pascal, in Brussels, where a crossdisciplinary 

team is busy exploring the implications of his 

ideas in fields as disparate as the social behavior of ant colonies, 

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


xii 

awarded the Nobel· Prize for his work on "dissipative structures" 

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 assumptions 

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 entomology 

and literary criticism. 

It is a measure of America's insularity and cultural arrogance 

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, particularly 

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 compelling 

us to reexamine its goals, its methods, its epistemology-its 

world view. Indeed, this book can serve as a symbol 

of today's historic transformation in science-one that no informed 

person can afford to ignore. 

Some scholars picture science as driven by its own internal 

logic, developing according to its own laws in splendid isolation 

from the world around it. Yet many scientific hypotheses, 

theories, metaphors, and models (not to mention the choices 

made by scientists either to study or to ignore various problems) 

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 feedback 

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 seventeenth 

and eighteenth centuries under the heading of "classical 

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 Machine 

enthusiastically embraced scientific theories that pictured 

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; mechanical 

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 machines, its heroic engineering breakthroughs, 

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 painful 

limitations of the machine model of reality. 

Of course, most of these limitations are not freshly discovered. 

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 observer 

could discover-this model has come under withering 

fire ever since it first arose. 

In the early nineteenth century, thermodynamics challenged 

the timelessness implied in the mechanistic image of 

the universe. If the world was a big machine, the thermodynamicists 

declared, it was running down, its useful en-


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 biological 

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 

diffe rent-indeed, for all practical purposes it was differentdepending 

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 paradigm 

is still the "reference point" for physics and the core 

model of science in general. Indeed, so powerful is its continuing 

influence that much of social science, and especially 

economics, remains under its spell. 

The importance of this book is not simply that it uses original 

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 concerned 

itself mostly with closed systems and linear relationships 

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 characterize 

today's accelerated social change: disorder, instability, 

diversity, disequilibrium, nonlinear relationships (in which

xv 


small inputs can trigger massive consequences), and temporality-a 

heightened sensitivity to the flows of time. 

The work of Ilya Prigogine and his colleagues in the socalled 

"Brussels school" may well represent the next revolution 

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 Prigogine'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, information) 

with their environment. Surely biological and social systems 

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 terms, all systems contain subsystems, 

which are continually "fluctuating." At times, a single fluctuation 

or a combination of them may become so powerful, as a 

result of positive feedback, that it shatters the preexisting organization. 

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 because, 

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-


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 adequate 

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 nonlinear 

relationships prevail. In this state, systems do strange 

things. They become inordinately sensitive to external influences. 

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 radically 

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 chemical 

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 


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 generally, therefore, in far-from-equilibrium conditions 

we find that very small perturbations or fluctuations can 

become amplified into gigantic, structure-breaking waves. 

And this sheds light on all sorts of "qualitative" or "revolu· 

tionary" change processes. When one combines the new insights 

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 possible 

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 social, 

economic or political realities. Words like "revolution," 

"economic crash," "technological upheaval ," and "paradigm 

shift" all take on new shades of meaning when we begin thinking 

of them in terms of fluctuations, feedback amplification, 

dissipative structures, bifurcations, and the rest of the Prigoginian 

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


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 people 

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 emphasis 

on past, present, or future. One lives in the past. Another 

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 immediate-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 election. 

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. 

But despite the growing recognition that cultural conceptions 

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 dynamics 

and interpersonal psychology. It might, for example, take 

account of what, in Future Shock, I called "durational expectancies"-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 approximately 

eight hours, or that a "mealtime" may last a few minutes 

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 


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 diffe rences are a source of conflict. 

Moreover, children in an industrial society are "time 

trained"-they learn to read the clock, and they learn to distinguish 

even quite small slices of time, as when their parents 

tell them, "You've only got three more minutes till bedtime!" 

These sharply 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 Braude!, have never been systematically developed 

in the social sciences. Nor have they, in any significant 

way, been articulated with our scientific theories of time, 

even though they are necessarily connected with our assumptions 

about physical reality. And this brings us back to Prigogine, 

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 followers, 

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 principle, 

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, as the main focus of 

physics shifted from dynamics to thermodynamics and the 

Second Law of thermodynamics was proclaimed, time suddenly 

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 approaching 

the heat death, then it follows that one moment is no 

longer exactly like the last. Yo u cannot run the universe backward 

to make up for entropy. Events over the long term cannot 

replay themselves. And this means that there is a directionality 

or, as Eddington later called it, an "arrow" in time.


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 elements. 

But it is precisely organization that gives any system 

internal diversity. Hence, as entropy drained the system of energy, 

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 followers! 

For evolution, far from pointing toward reduced organization 

and diversity, points in 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 sufficiently 

random way may the difference between past and future, 

and therefore irreversibility, enter its description." 

In classical or mechanistic science, events begin with "initial 

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 opposite 

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 in its early stages, such pro-

xxi 


cesses were regarded as anomalies, peculiarities that arose 

from highly unlikely initial conditions. 

It is Prigogine and Stengers' thesis that such time-dependent, 

one-way processes are not merely aberrations or deviations 

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 levels 

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, entropy 

itself becomes the progenitor of order. 

What the authors are proposing, therefore, is a vast synthesis 

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 together 

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 seemingly unconnected-even 

contradictory-philosophical concepts. 

Here we begin to glimpse, in full richness, the monumental 

synthesis proposed in these pages. By insisting that irreversible 

time is not a mere aberration, but a characteristic of much 

of the universe, they subvert classical dynamics. For Prigogine 

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 degrade, 

order, organization-and therefore life. 

If this is so, then entropy, too, loses its either/or character. 

While certain systems run down, other systems simultaneously 

evolve and grow more coherent. This mutualistic,


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 accidently 

tripped over a rock. In the Old Te stament, 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, all 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 respect 

to the free will of the individual, He never demanded a 

specific action. He merely preset the range of options available 

to the human decision-maker. Free will downstairs operated 

only within the limits of a menu determined upstairs. 

In the secular culture of the Machine Age, hard-line determinism 

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 uncertainty, 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 sociologistturned-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 interaction 

. ... There is more: the problem of determinism has be-

xxiii 


come that of the order of the universe. Order means that there 

are other things besides 'Jaws': that there are constraints, invariances, 

constancies, regularities in our universe . ... In 

place of the homogenizing and anonymous view of the old determinism, 

there has been substituted a diversifying and evolutive 

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 impossible 

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 irreconcilable 

opposites, but each playing its role as a partner in destiny. 

Yet another synthesis is achieved. 

When we bring reversible time and irreversible time, disorder 

and order, physics and biology, chance and necessity all 

into the same novel frame, and stipulate their interrelationships, 

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, diplomacy, 

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 transformatory 

change, they illuminate organization theory. They throw a 

fresh light, as well, on certain psychological processes-innovation, 

for example, which the authors see as associated with 

"nonaverage" behavior of the kind that arises under nonequilibrium 

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 disciplines. 

No one-not even its authors-can appreciate the full implications 

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 intellectually 

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 .. fluctuation"? 

What, in Prigoginian terms, does one mean by ••cause" 

or "effect"? And when the authors speak of molecules communicating 

with one another to achieve coherent, synchro-

xxv 


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 environment 

that is undetectable by A may be received by B and converted 

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 signaled 

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 alchemy 

and religious notions of even earlier vintage . Some 

readers might conclude from this that the rise of Newtonianism 

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, 

so to speak, far from equilibrium. The model of the universe 

proposed by the classical scientists (even if partially derivative) 

was applied analogously to new fields and disseminated 

successfully, not just because of its scientific power or "rightness," 

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


.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 today's 

breakdown of industrial or "Second Wave" society as a 

civilizational "bifurcation," and the rise of a more differentiated, 

"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 Prigoginianism 

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 phoenixlike 

once more. 

Order out of Chaos is a brilliant, demanding, dazzling bookchallenging 

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 turbulence. 

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 understand 

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 observe 

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 irreversible 

processes. 

Obviously, in addition to deterministic processes, there 

must be an element of probability involved in some basic processes, 

such as, for example, biological evolution or the evolution 

of human cultures. Even the scientist who is convinced of 

the validity of deterministic descriptions would probably hesitate 

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 considered 

only exceptions. Today we see everywhere the role of 

irreversible processes, of fluctuations_ 

Although Western science has stimulated an extremely fruitxxvii


xxviii 

ful dialogue between man and nature, some of its cultural consequences 

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 dramatic 

has been happening in science, something as unexpected 

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 irreversibility 

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 socalled 

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. Classical 

dynamics and modern chemistry are going through a period 

of drastic change. If one asked a physicist a few years ago 

what physics permits us to explain and which problems remain 

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 between 

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 integrate 

it in the evolution of Western culture during the past

xxfx 


three centuries. Only in this way can we appreciate the unique 

moment in which we are presently living. 

Our scientific heritage 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 disorder 

; still, biological or social evolution shows us the complx 

emerging fr om the simple. How is this possible? How can 

structure arise from disorder? Great progress has been realized 

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 reversible, 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 discussed 

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 description 

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 concept 

of matter, matter that is "active," as matter leads to irreversible 

processes and as irreversible processes organize 

matter. 

The evolutionary paradigm, including the concept of entropy, 

has exerted a considerable fascination that goes far 

beyond science proper. We hope that our unification of dynamics 

and thermodynamics will bring out clearly the radical novelty 

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 irreversibility 

of time is itself closely connected to entropy. To make 

time flow backward we would have to overcome an infinite 

entropy barrier. 

Tr aditionally 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


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. Although 

the origin of structures as the result of nonequilibrium 

processes was already adequately treated in the French edition 

(as well as in the translations that followed), we had to 

entirely rewrite the third part, which deals with our new results 

concerning the roots of time as well as with the formulation 

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 present 

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 devoted 

most of his scientific life to this problem, he may perhaps 

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 unfo rtunately 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 explosion. 

It is therefore more important than ever to keep open 

the channels of communication between science and society. 

The present development of We stern science has taken it outside 

the cultural environment of the seventeenth centu.ry, in 

which it was born. We believe that science today carries a uni-

xxxi 


versal message that is more acceptable to different cultural 

traditions. 

During the past decades Alvin Toffier's books have been important 

in bringing to the attention of the public some features 

of the "Third Wave" that characterizes our time. We are therefore 

grateful to him for having written the Foreword to the 

English-language version of our book. English is not our native 

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 Early, Ian MacGilvray, Carol 

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 Newton 

presented his Principia to the Royal Society of London. It 

contained the basic laws of motion together with a clear formulation 

of some of the fundamental concepts we still use today, 

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 fantastic 

proportions. On the microscopic scale, elementary partide 

physics studies processes involving physical dimensions 

of the order of w- ts em and times of the order of I0-22 second. 

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 qualitative 

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 everything 

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 completion. 

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 systems 

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 progress 

has indeed been realized in the unification of some of the 

basic forces found in nature. Still, the fundamental level remains 

elusive. Wherever we look we fi nd evolution, diversification, 

and instabilities. Curiously, this is true on all levels, 

in the field of elementary particles, in biology, and in astrophysics, 

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 conceptual 

structures that now appear as essential to our understanding 

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 expectations. 

In classical science the emphasis was on time-independent 

laws. 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 illusions 

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 Lucretius 

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 fundamentally 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 between 

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 alienation 

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


This is a paradox. A brilliant breakthrough in molecular biology, 

the deciphering of the genetic code, in which Monod 

actively participated, ends upon a tragic note. This very progress, 

we are told, makes us the gypsies ,0f the universe. How 

can we explain this situation? Is not science a way of communication, 

a dialogue with nature? 

In the past, strong distinctions were frequently made between 

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 eternal 

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 philosophers 

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 rupture 

and opposition, the growing coherence of our knowledge 

of man and nature. 

In the past, the questioning of nature has taken the most diverse 

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 elements. 

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? Experimentation 

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 interaction 

between theoretical concepts and observation. 

In hundreds of different ways scientists have expressed ttieir 

amazement when, on determining the right question, they discover 

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 anything 

we want it to. Scientific investigation is not a monologue. 

It is precisely the risk involved that makes this game 

exciting. 

But the uniqueness of Western science is far from being exhausted 

by such methodological considerations. When Karl 

Popper discussed the normative description of scientific rationality, 

he was forced to admit that in the final analysis rational 

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 discovered 

questions to which nature provides consistent answers. 

The success of Western science is an historical fact, unpredictable 

a priori, but which cannot be ignored. The surprising 

success of modern science has led to an irreversible transformation 

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 concerning 

decisive episodes in the evolution of science. The so-called


"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 opportunity 

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 discovery 

of a silent world. This is the paradox of classical science. 

It revealed to men a dead, passive nature, a nature that behaves 

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 science 

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 1920s that formed the cultural background to 

quantum mechanics. to In opposition to science, which was 

identified with a set of concepts such as causality, determinism, 

reductionism, and rationality, there was a violent upsurge 

of ideas denied by science but seen as the embodiment 

of the fundamental irrationality of nature. Life, destiny, freedom, 

and spontaneity thus became manifestations of a shadowy 

underworld impenetrable to reason. Without going into 

the peculiar sociopolitical context to which it owed its vehement 

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


always oscillated between the world as an automaton and a 

theology in which God governs the universe. This is what 

Needham calls the "characteristic European schizophrenia. 

" II In fact, these visions are connected. An automaton 

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 antiscientific 

metaphysical view of nature? We think such a choice 

is no longer necessary, since the changes that science is undergoing 

today lead to a radically new situation. This recent evolution 

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 science, 

a message that concerns the interaction of man and nature 

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 centering 

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. Ye t 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. But there is the additional and obvious problem 

that, since there is no one to build nature, we must give to 

its very "bricks"-that is, to its microscopic activity-a description 

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 We stern 

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. Te mporality was 

looked down upon as an illusion. This is no longer true today.


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 find 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 definition. 

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 Eddingtonl2 

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 thermodynamics, 

the law that introduces the "arrow of time" in 

physics. Eddington writes: "From the point of view of philosophy 

of science the conception associated with entropy 

must, I think, be ranked as the great contribution of the nineteenth 

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 science 

are discoveries at the microscopic level, that of molecules, 

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 scientists 

the aim of research is identified with this "microscopic 

dissection of objects," to use Eddington's expression. However, 

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 communication, 

to time. 

This change of perspective is not the result of some arbitrary 

decision. In physics it was forced upon us by new dis-


coveries no one could have foreseen. 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 understand 

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 mechanics 

and to incorporate into it the newly found universal 

constants. Today the situation has changed. Quantum mechanics 

has given us the theoretical frame to describe the incessant 

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. Structures 

may disappear, but also they may appear. Some processes 

are , as far as we know, well described by deterministic 

equations, but others involve probabilistic processes. 

How then can we overcome the apparent contradiction between 

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 fluctuations. 

The models considered by classical physics seem to us 

to occur only in limiting situations such as we can create artificially 

by putting matter into a box and then waiting till it 

reaches equilibrium. 

The artificial may be deterministic and reversible. The natural 

contains essential elements of randomness and irreversibility. 

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.


This book deals with the conceptual transformation of science 

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 particles. 

We could have followed recent fascinating developments 

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 feaures 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, biology, 

or cosmology. On every scale self-organization, complexity, 

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 exemplified 

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 "metaphysical" 

one. 

There was, however, a third path, which was to ask whether 

the simplicity of the temporal evolution traditionally consid-


ered in physics and chemistry was due to the fact that attention 

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 consequences 

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 between 

the sciences and the humanities: 

The specific and unique versus the repetitive and the universal, 

the concrete versus the abstract, perpetual movement 

versus rest, the inner versus the outer, quality 

versus quantity, culture-bound versus timeless principles, 

mental strife and self-transformation as a permanent condition 

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.14 

We have devoted much space to classical mechanics. Indeed, 

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 is 

reduced to a parameter, and future and past become equivalent. 

It is true that quantum theory has raised many new 

problems not covered by classical dynamics but it has nevertheless 

retained a number of the conceptual positions of classical 

dynamics, particularly as far as time and process are 

concerned.


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 construction 

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 description 

of something inconceivable in classical dynamicsan 

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 celebrated 

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 evolution; 

biology, geology, and sociology emphasized processes of 

becoming, of increasing complexity. As for thermodynamics, 

it is based on the distinction of two types of processes: reversible 

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 irreversible 

processes. 

During the nineteenth century the final state of thermodynamic 

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 structures 

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 dissipative 

structures to emphasize the constructive role of dissipative 

processes in their fo rmation. 

This book describes some of the methods that have been


developed in recent years to deal with the appearance and evolution 

of dissipative structures. Here we find the key words 

that run throughout this book like leitmotivs: nonlinearity, instability, 

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 between 

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-fromequilibrium 

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 structures. 

For example, far from equilibrium we may witness the 

appearance of chemical clocks, chemical reactions which behave 

in a coherent, rhythmical fashion. We may also have processes 

of self-organization leading 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 chemical 

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 chemical 

clock the behavior is quite different. Oversimplifying somewhat, 

we can say that in a chemical clock all molecules change 

their chemical identity simultaneously, at regular time intervals. 

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 "communication" 

among molecules. But this type of communication 

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.


In addition, the type of dissipative structure depends critically 

on the conditions in which the structure is formed. External 

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 unexpected 

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 conditions 

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 difficult 

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 biosphere 

is embedded, incorporating the nonlinearities of chemical 

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 bifurcations. 

It is remarkable that near-bifurcations systems present 

large fluctuations. Such systems seem to "hesitate" 

among various possible directions of evolution, and the famous 

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 system. 

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 nonlinear 

systems far from equilibrium. 

The first two parts of this book thus deal with two conflicting 

views of the physical universe: the static view of classical 

dynamics, and the evolutionary view associated with entropy.


A confrontation between these views has become unavoidable. 

For a long time this confrontation was postponed by considering 

irreversibility as an illusion, as an approximation; it 

was man who introduced time into a timeless universe. However, 

this solution in which irreversibility is reduced to an illusion 

or to approximations can no longer be accepted, since we 

know that irreversibility may be a source of order, of coherence, 

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 standards, 

though everybody accepts his eminence as a physicist. 

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 technical. 

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 written 

in the sixteenth century and what had become for centuries 

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 ... . "16 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


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 concept 

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 systems. 

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 "distinguish" 

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 farreaching 

consequences. The second law of thermodynamics, 

the law of entropy, introduced irreversibility into the macroscopic 

world. We now can understand its meaning on the 

microscopic level as well. As we shall see, the second law corresponds 

to a selection rule, to a restriction on initial conditions 

that is then propagated by the laws of dynamics. 

Therefore the second law introduces a new irreducible element 

into our description of nature. While it is consistent with 

dynamics, it cannot be derived from dynamics. 

Boltzmann already understood that probability and irreversibility 

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 everyday life manifests a radical difference 

between time and space. We can move from one point


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 barrier 

exist; if not, causality would fall to pieces. Similarly, the 

entropy barrier is the prerequisite for giving a meaning tQ communication. 

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. In the process new aspects of time have been uncovered. 

A preoccupation with time runs all through our century. 

Think of Einstein, Proust, Freud, Te ilhard, Peirce, or 

Whitehead. 

One of the most surprising results of Einstein's special theory 

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. But 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 different 

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 is 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


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, without 

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 important 

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 paradigm 

expressed by the second law. As the evolutionary paradigm 

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 reconceptualization 

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 results. 

What we wish to do is emphasize the conceptual creativeness 

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 philosophers 

of science: 

. . . there is a tendency to forget that all science is boundl 

up with human culture in general, and that scientific findings, 

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 relevanti 

and momentou:s arc de:stined eventually to be framed inl 

concepts and words that have a grip on the educated com-I 

I 

I 

I


munity and become part and parcel of the general world 

picture-a theoretical science, I say, where this is forgotten, 

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 interaction 

of the issues proper to culture as a whole and the internal 

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 history-be 

it economic, demographic, or political-is moving at 

an unprecedented pace, new questions and new interests require 

·us to enter into new dialogues, to look for a new coherence. 

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 consequence 

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 internal 

history of science and in the social context in which science 

finds itself today. Discoveries such as those of unstable 

elementary particles or of the expanding universe clearly belong 

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-


equilibrium problems was repressed in the cultural and ideological 

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 society. 

Scientific progress ought to be an essentially autonomous 

process 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 Science" 

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 another 

really less than the host of the banished. 

What led them into the Te mple? . .. one of the strongest 

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 Manichean, 

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 Physicists.20 

There, three physicists discuss the ways and means of 

advancing physics while at the same time safeguarding mankind 

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 already 

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 reality 

to a hidden simplicity. What the physicist Moebius is trying 

to conceal in the madhouse is the fact that he has 

successfully solved the problem of gravitation, the unified theory 

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 potentially 

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.


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 irresponsibly 

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 Chinese 

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 science 

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


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 ASON 

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 1727 

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 material 

on which to "ground" their arguments. Thus Desaguliers 

transposed the meaning of the new natural order into 

a political lesson: a constitutional monarchy is the best possible 

system of government, since the King, like the Sun, has 

his power limited by it. 

Like Ministers attending ev'ry Glance 

Six Worlds sweep round his Throne in Mystick Dance. 

27


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 universal 

nature of the laws set out in his Principia. Nature is 

"very consonant and conformable to herself," he asserts in 

the celebrated Question 31 of his Opticks-and this strong and 

elliptical statement conceals a vast claim: combustion, fermentation, 

heat, cohesion, magnetism . .. there is no natural 

process which would not be produced by these active forcesattractions 

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 obstacles. 

Furthermore, it opened the way to mathematical methods 

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 expressible 

in mathematics. For them, chemistry found its Newton 

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 strategy 

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 gravitation; 

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 proliferation 

of forces and reasserted the universality of the explanatory 

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 political 

order on the other could be expressed in terms of an allembracing 

harmony. Romantic philosophers even discovered 

in the Newtonian universe an enchanted world animated by 

natural forces. More "orthodox" physicists saw in it a mechanical 

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 harmony 

that reigns in the society of stars, the elective affinities 

and hostilities giving rise to the "social life" of chemical compounds 

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 conviction 

that Newtonian rationality, even with its various conflicting 

interpretations, forms a suitable basis for our dialogue with 

nature. 

A central subject of this book is that of the Newtonian triumph, 

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


of science and of nature seems to be emerging. This new conception 

paves the way for a new unity of knowledge and culture. 

A Dehumanized World 

... May God us keep 

From single Vision and Newtons sleep! 

-William Blake, 

in a letter to Thomas Butts 

dated November 22, 1802 

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 surrounding 

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 consider 

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, perceive 

the spectre of cultural disaster looming in the distance.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 dominate 

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 science 

is debasing our world. What for generations had been a 

source of joy 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 objectivity": 

Indeed, the renewals of the subjective approach to nature 

make a pathetic theme. Its ruins lie strewn like good intentions 

aU along the ground traversed by science, until it 

survives only in strange corners like Lysenkoism and anthroposophy, 

where nature is socialized or moralized. 

Such survivals are relics of the perpetual attempt to escape 

the consequences of western man's most characteristic 

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 Wordsworth, 

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


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 instrument 

of domination. And man, a stranger to the world, sets 

himself up as its master. 

This disenchantment has taken various forms in recent decades. 

It is outside the aim of this book to study systematically 

the various forms of antiscience. In Chapter III we shall present 

a fuller reaction of Western thought to the surprising triumph 

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 antiscientific 

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 related 

to a permanent aim, the domination of nature. Therefore 

Heidegger claims that scientific rationality is the final accomplishment 

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 emphasizes 

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


is taken into account. Each theory is part of the implementation 

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 enslavement. 

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 violence. 

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 industrial 

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 pressure, 

which then sets the turbines turning . ... The hydroelectric 

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 observation, 

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 Introduction. 

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 instance, 

under the very suggestive title "The Coming of the


34 

Golden Age, "9 Gunther Stent states that science is now reaching 

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 interest. 

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 understanding 

of the world around us is just beginning. There is yet 

another idea of science that we feel is potentially just as detrimental, 

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 mechanics. 

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 seemingly well-established conceptions. Thus 

there is a very delicate balance between the readiness to imagine 

that science can produce anything and a kind of down-toearth 

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 suggested 

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 


jacket which nineteenth-century materialism imposed on 

our philosophical outlook. Paradoxically, had that outlook 

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 recognized, 

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 attention 

to confluential events; include the paranormal phenomena 

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 conceptual 

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 criticisms. 

Let us take, for instance, the position of Alexander 

Koyre, who has made outstanding contributions to the understanding 

of the development of modern science. In his study of 

the significance and implications of the Newtonian synthesis, 

Koyre wrote: 

Ye t there is something for which Newton-or better to 

say not Newton alone, but modern science in generalcan 

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 substituting 

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


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 explainnot 

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 another 

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 dilemma 

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 classical 

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 beginning 

to go beyond what Koyre called "the world of quantity" 

into the world of "qualities" and thus of "becoming." 

This will be the main subject of Books One and 1\vo. We believe 

it is precisely this transition to a new description that 

makes this moment in the history of science so exciting. Perhaps 

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 moments 

of human history.


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, converge 

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 medieval 

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 ageless.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 organize 

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 example, 

animal and plant species either bred or selected, weaving, 

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. Nevertheless, 

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


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 speculations 

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 fo rm the differentiated cosmic 

powers, discord and strife, alternating atrocities and vendettas, 

until stability is finally reached under the rule of Justice 

(dike). What does matter is that, in the space of a few generations, 

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 conflicting 

natural powers? Or do the forces that create and drive 

things exist autonomously-rival powers of love and hate leading 

to birth, growth, decline, and dispersion? Is change an illusion 

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 discovered 

by the Pythagoreans are still part of our present theories. 

The mathematical schemes worked out by the Greeks 

form the first body of abstract thought in European historythat 

is, a thought whose results are communicable and reproducible 

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, expectations, 

or passions. 

The most important aspect common to Greek thought and 

to modern science , which contrasts with the religious and

39 


mythicaI form of inquiry, is thus the emphasis on criticaI discussion 

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 cosmological 

hypotheses and the crafts and technological activities 

that tlourished in those cities. Tr adition 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 importance 

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 approach 

repressed in favor of a form of knowledge more consistent 

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, engineerhave 

a similar meaning. They do not refer to rational knowledge 

but to cunning and expediency. The idea was not to leam 

about natural processes in order to utilize them more effectively, 

but to deceive nature, to "machinate" against it-that 

is, to work wonders and create effects extraneous to the "natural 

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


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 between 

the world of the stars and our sub lunar world. This contrast 

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 sub lunar world. Imprecise natural 

processes can only be subjected to an approximate description. 

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 embryonic 

growth, a highly organized process in which interlocking, 

although apparently independent, events participate in a 

process that seems to be part of some global plan. Like the developing 

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 understanding 

of nature. In this sense the "birth of modern science," the 

clash between the Aristotelians and Galileo, is a clash between 

two forms of rationality. IS 

In Galileo 's view the question of "why," so dear to the Aristotelians, 

was a very dangerous way of addressing nature, at 

least for a scientist. The Aristotelians, on the other hand, considered 

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.


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, despite 

the empiricists' beliefs, merely to respect observed facts. 

On certain points, including even the description of mechanical 

motion, it was in fact Aristotelian physics that was more 

easily brought into contact with empirical facts. The experimental 

dialogue with nature discovered by modern science involves 

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 reasoning 

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, according 

to Galileo, never gives anything away, never does something 

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 finally 

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


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 specific 

procedure. Nature is cross-examined through experimentation, 

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 hypothesis 

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 refers 

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 consequences 

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 embody 

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 torture, a way of 

putting nature on the rack, it survived all the modifications of

43 


the theoretical content of scientific descriptions and ultimately 

defined the new method of investigation introduced by modern 

science. 

Experimental procedure can even become a tool for purely 

theoretical analysis. It is then a "thought experiment," the 

imagining of experimental situations governed entirely by theoretical 

principles, which permits the exploration of the consequences 

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 famous 

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 respect 

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 designed 

to expose. In relativity theory, the velocity of light appears 

as a universal 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 nature 

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 hypotheses 

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 cherished 

desires and expectations. He actually runs a greater risk 

and plays a more dangerous game the better his tactics succeed 

in encircling nature, in setting it more squarely with its 

back to the wall.l7 Moreover, it is true that, whether the answer 

is "yes" or "no," it will be expressed in the same theoretical 

language as the question. However, this language, too,


develops according to a complex historical process involving 

nature's replies in the past and its relations with other theoretical 

languages. In addition, new questions arise corresponding 

to the changing interests of each period. This sets up a complex 

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 unwittingly, 

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 independent 

being. It forms the basis of the communicable and 

reproducible nature of scientific results. However partially nature 

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 perceived 

by the founders of modern science as a basic step toward 

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 phenomena 

studied by science can thus be interpreted as the key 

to understanding nature as a whole; the complexity of the latter 

is only apparent, and its diversity can be explained in terms 

of the universal truth embodied, in Galileo's case, in the mathematical 

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 complexity 

is only apparent; each move follows simple rules. In its 

early days, modern science quite possibly needed this convic-


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" conception 

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 rationalism. 

Perhaps a metaphysical conviction was needed to transmute 

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 context. 

On this highly controversial issue, we shall restrict ourselves 

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 craftsmen's 

techniques in the monasteries, where the residual knowledge 

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 knowledge. 

Moreover, a comparative analysis such as Needham'sl9 exposes 

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 patronage, 

who tended to look for novelty and to exploit all the opportunities 

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 primary objective was to keep law and order. 

The compass, the printing press, and gunpowder, all of which


were to contribute to undermining the foundations of medieval 

society and to project Europe into the modern era, were discovered 

much earlier in China but had a much less destabilizing 

effect on its society. The enterprising European merchant 

society appears in contrast as particularly well suited to stimulate 

and sustain the dynamic and innovative growth of modern 

science in its early stages. 

However, the question remains. We know that the builders 

of machines used mathematical concepts-gear ratios, the displacements 

of the various working parts, and the geometry of 

their relative motions. But why was mathematization not restricted 

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 immediately 

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 modern 

science, a "resonance" appears to have been set up between 

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 discovering 

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 determined 

the birth of theoretical science, or of the "world view" 

that happened to develop in conjunction with experimental activity. 

By using the term resonance-that is, mutual amplification 

of two discourses-we have deliberately chosen an 

expression that does not assume whether it was theological 

discourse or the "scientific myth" that came first and triggered 

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 automaton, 

but also the question of some "essential" link between experi-


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 conviction 

was "needed" to inspire the "scientific faith" of the 

founders of modern science: 

I mean the inexpugnable belief that every detailed occurrence 

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 implanted 

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 European 

mind arising from the unquestioned faith of centuries. 

By this I mean the instinctive tone of thought and 

not a mere creed of words. 2 o 

We will not consider this matter further. It would be out of 

the question to .. prove" that modern science could have originated 

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 sincere, 

the important point is that those arguments made the 

speculations of modern science socially credible and acceptable, 

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.


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 already 

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 discourses. 

Needham 2t tells of the irony with which Chinese men of let· 

ters of the eighteenth century greeted the Jesuits' announcement 

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 believes 

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 equilibrium 

among these processes. Stable and interdependent, 

they resonate with each other in a kind of nonconcerted harmony. 

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.


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, descriptions 

not prescriptions, as Karl Pearson put it in a famous 

chapter. The exact degree of subjectivity in the formulations 

of scientific law has been hotly debated during the 

whole period from Mach to Eddington, and such questions 

cannot be followed further here. The problem is 

whether the recognition of such statistical regularities 

and their mathematical expression could have been 

reached by any other road than that which Western science 

actually travelled. Was perhaps the state of mind in 

which an egg-laying cock could be prosecuted at law necessary 

in a culture which should later have the property 

of producing a Kepler? 22 

It must now be stressed that scientific discourse is in no way 

a 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 function 

alone. In this sense, physics has indeed developed in opposition 

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 science 

was unacceptable to both these theologians and physicists. 

This ordered, harmonious, hierarchical, and rational world 

was too independent, the beings inhabiting it too powerful and


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 differentiated 

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. Although 

Leibniz had endeavored to demonstrate that mathematization 

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 mechanics 

that was basically alien to freedom and the purposes 

of the human mind. 'i\ dull affair, soundless, scentless, colourless, 

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 describes; 

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 freedom, 

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,


everything that goes to make up the texture of nature, such as 

its perfumes and its colors. The debasement of nature is parallel 

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 reciprocal 

amplification among economic, political, social, religious, 

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 intelligible 

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 believe 

their enterprise was self-sufficient, that it exhausted the 

possibilities of a rational approach to natural phenomena. The 

relationship between scientific description and natural philosophy 

did not, in this sense, have to be justified. It could be 

seen as self-evident that science and philosophy were convergent 

and that science was discovering the principles of an 

authentic natural philosophy. But, oddly enough, the selfsufficiency 

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 become 

the triumphant science of the eighteenth century, 26 the


science that discovered the laws governing the motion of celestial 

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 achievement 

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 successful 

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 today, 

a description is objective to the extent to which the observer 

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 translated 

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 demands 

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 himself 

very severely: he must content himself with describing 

the most simple events which can be brought within 

the domain of our experience; all events of a more complex 

order are beyond the power of the human intellect to 

reconstruct with the subtle accuracy and logical perfection 

which the theoretical physicist demands. Supreme 

purity, clarity, and certainty at the cost of completeness.


But what can be the attraction of getting to know such a 

tiny section of nature thoroughly, while one leaves everything 

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 whatsoever. 

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 capacity 

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 origins 

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 interpretation 

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 mystery, 

and shows us what its power really is. With respect


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 is useful because it leads to economy of 

thought. There may be some element of truth in such a statement, 

but does it tell the whole story? How far we have come 

from Newton, Leibniz, and the other founders of Western science, 

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 thunderings 

of .. scientific myth" and the silence of "scientific seriousness," 

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 natural 

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 randomness, 

complexity, and irreversibility enter into physics as objects 

of positive knowledge, we are moving away from this 

rather naive assumption of a direct connection between our

55 


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 supplemental 

discoveries that have shown that the existence of universal 

constants, such as the velocity of light, limit our power 

to manipulate nature. (We shall discuss this unexpected situation 

in Chapter VII.) As a result, physicists had to introduce 

new mathematical tools that make the relation between perception 

and interpretation more complex. Whatever reality may 

mean, it always corresponds to an active intellectual construction. 

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 lamented 

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 "coherence." 

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 development 

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 feature 

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 produce 

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 necessary 

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 motion 

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 quantities, 

such as position, velocity, and acceleration? How can 

we describe the state of a body at any given instant? To answer 

57


these questions, mathematicians have introduced the concept 

of infinitesimal quantities. An infinitesimal quantity is the result 

of a limiting process; it is typically the variation in a quantity 

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 "instantaneous 

tendency" to modify this position, and by its acceleration 

a, again its "instantaneous tendency," but now to modify 

its velocity. Instantaneous velocities and accelerations are limiting 

quantities that measure the ratio between two infinitesimal 

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 second 

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 calculated 

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 determine 

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 acceleration 

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 distances 

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 acceleration, 

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 problem, 

their "integration" represents the solution of this problem. 

It leads to the calculation of the trajectories, r(t). These 

trajectories contain all the information acknowledged as relevant 

by dynamics; it provides a complete description of the 

dynamic system. 

The description therefore implies two elements: the positions 

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" unfold 

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 describes 

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 dynamic 

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 comparison 

to those whose effect is being considered.


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 discovered-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 acceleration 

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, determinism, 

and reversibility. We have seen that in order to calculate 

a trajectory we need, in addition to our knowledge of 

the laws of motion, an empirical definition of a single instantaneous 

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 dynamics, 

which Bergson repeatedly emphasized, characterizes the 

reality that dynamics describes. Everything is given, but everything 

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 equivalence 

between cause and effect postulated as the basis of 

their mathematization of motion, they staged thought experiments 

such as an elastic ball bouncing on the ground. As the


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 property 

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 measurement 

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 indeed. 

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 velocity 

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 conditions. 

The initial conditions must be the outcome of the dynamic 

evolution itself. This is an important point to which we 

shall come back in the third part of this book. But whatever its


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 displacement 

of material bodies. Time is a parameter in terms of 

which these displacements may be described. In this way 

space and time are inextricably tied together in the world of 

classical dynamics. (Also see Chapter IX.) 

It is interesting to compare dynamic change with the atomists' 

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 Newton 

himself believed that collisions between hard atoms were 

the ultimate, and perhaps the only, sources of changes of motion.t 

Nevertheless, the dynamic and the atomic descriptions 

differ radically. Indeed, the continuous nature of the acceleration 

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 dynamic 

motion seem to contradict the randomness generally 

attributed to collisions between atoms. The ancient philosophers 

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 neither 

punishment nor reward from any divine or natural order. 

But classical science was a science of engineers and astronomers, 

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 manipulating. 

Nature thus becomes law-abiding, docile, and predictable, 

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 immutable 

world of dynamics governed by Newton's law, a siagle 

mathematical formula corresponding to an eternal truth unfolding 

toward a tautological future? In the twentieth century 

we are again witnessing the clash between lawfulness and random 

events, which, as Koyre has shown, had already tormented 

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 between 

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 eighteenth 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 already 

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 between 

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 concept 

of random collisions. Does this, however, vindicate the 

view of those who believe that Newtonian dynamics repre-


64 

sents a rupture in the history of thinking, a revolutionary novelty? 

This is what positivist historians have claimed when they 

described how Newton escaped the spell of preconceived notions 

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 eighteenthcentury 

rationalists emphasized the apparent similarity between 

his "mathematical" forces and traditional occult 

qualities. Fortunately, these critics did not know the strange 

story behind the Newtonian forces! For behind Newton's cautious 

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 achievement 

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 inheritor 

of the forces Newton the chemist observed and manipulated, 

the chemical "affinities" forming and disrupting ever 

new combinations of matter.s The decisive role played by celestial 

orbits of course remains. Still, at the start of his intense 

astronomical studies-about 1679-Newton apparently expected 

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 produced 

an unexpected conclusion: the forces between the planets 

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 universal 

law, which, as he emphasized, could be applied to all phenomena-whether 

chemical, mechanical, or celestial in 

nature.


The Newtonian synthesis is thus a surprise. It is an unexpected, 

staggering discovery that the scientific world has commemorated 

by making Newton the symbol of modern science. 

What is particularly astonishing is that the basic code of nature 

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 

dlembert, 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 mathematical 

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 considering 

Newton's discovery synonymous with physical science 

itself, physicists were blithely contemplating dropping it 

altogether. Dlembert 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 sale it represents is not 

only always the same, but is actually unique . ... M. 

Clairaut has raised an objection against Newton's system, 

but it is at best an objection and must not and cannot 

become a principle; an attempt should be ,made to overcome 

it and not to turn it into a theory the entire consequences 

of which merely rest on a calculation; for, as I 

have said, one may represent anything by means of calculation 

and achieve nothing; and if it is allowed to add 

one or more terms to a physical law such as that of attraction, 

we are only adding to arbitrariness instead of representing 

reality. 7 

Later Buffon was to announce what was to become, although 

for only a short time, the research program for chemistry:


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 action 

of all the celestial bodies on one another: they act in 

(he same way and with the same ratios of mass and distance; 

a globule of water, of sand or metal acts upon another 

globule just as the terrestrial globe acts on the 

moon, and if the laws of affinity have hitherto been regarded 

as different from those of gravity, it is because 

they have not been fully understood, not grasped completely; 

it is because the whole extent of the problem has 

not been taken in. The figure which, in the case of celestial 

bodies has little or no effect upon the law of interaction 

between bodies because of the great distance 

involved, is, on the contrary, all important when the distance 

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 between 

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 demonstration 

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 sarcasms 

of d/\lembert, Condillac, and Condorcet, whose unbending 

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 programthe 

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


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 tak 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 professional 

consensus and magistral authority. The first consensus 

centered around the Newtonian system: in France 

Buffon's confidence finally triumphed over the rational skepticism 

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: 13 

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 became 

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.


68 

chanics? This is a complex problem, to which we shall return. 

We now know that nature is not always "comfortable and consonant 

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 dynamics 

still may be said to form the core of physics. 

Of course, since Newton the formulation of classical dynamics 

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 revision 

even in the fields in which (in contrast to quantum mechanics 

and relativity) classical dynamics remains valid. At 

the moment this book is being written, at the end of the twentieth 

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 describe 

briefly the revival of classical dynamics in our time.) 

The Language of Dynarnics 

Today classical dynamics 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


various equivalent representations in which the laws of dynamics 

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 dynamic 

systems, time appears merely as an accident, since 

their description can be reduced to that of noninteracting mechanical 

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 collisions, 

machines have an efficiency of one-the dynamic system 

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 machine 

has used in producing the motion. The simplest case is 

that in which the only force considered is gravity (which applies 

to simple machines, pulleys, levers, capstans, etc.). In 

this case it is easy to establish an overall relationship of equivalence 

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 implied 

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 isolated 

system. 

Potential energy (or ''potential," conventionally denoted as 

V), which depends on the relative positions of the particles, is


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, conventionally 

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 velocity 

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 progressively 

extended to cover the theory of electricity and magnetism. 

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


description is still of the greatest importance today. The equations 

which, through the derivatives of the Hamiltonian, give 

the time variation of the coordinates and momenta are the socalled 

canonical equations. They contain the general properties 

of all dynamic changes. Here we have the triumph of the 

mathematization of nature. All dynamic change to which classical 

dynamics applies can be reduced to these simple mathematical 

equations. 

Using these equations, we can verify the above-mentioned 

general properties implied by classical dynamics. The canonical 

equations are reversible: time inversion is mathematically 

the equivalent of velocity inversion. They are also conservative: 

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 problems 

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 reduced 

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, according 

to the canonical equation, on the derivative of the Hamiltonian 

in 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 representation. 

We will define systems for which this is possible as "integrable 

systems." Any integrable system may thus be 

represented as a set of units, each changing in 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


• 

> 

• 

• 

• 

• 

• 

• 

(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 explicitly 

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 description 

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 systems. 

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, influenced 

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 systems. 

The discovery of Bruns and Poincare shows that dynamic 

systems are not isomorphic. Simple, integrable systems 

can indeed be reduced to noninteracting units, but in general, 

interactions cannot be eliminated. Although this discovery 

was not clearly understood at the time, it implied the demise of 

the conviction that the dynamic world is homogeneous, reducible 

to the concept of integrable systems. Nature as an


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 deterministic 

laws; once an initial state is given, the dynamic laws of motion 

permit the calculation of trajectories at each point in the 

future or the past. However, a trajectory may become intrinsically 

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 indeterminate. 

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 importance 

of these singular points. After describing the explosion 

of gun cotton, he goes on to say: 

In all such cases there is one common circumstancethe 

system has a quantity of potential energy, which is 

capable of being transformed into motion, but which cannot 

begin to be so transformed till the system has reached 

a certain configuration, to attain which requires an expenditure 

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 fighting, 

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 produced 

by human endeavour depend on taking advantage 

of these singular states when they occur. I4


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 excellence 

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 conclude 

that all problems resemble one another from the point of 

view of their solutions as well, and that nothing new can appear 

as a result of the greater or lesser complexity of the integration 

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 observable, 

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.


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 demon. 

Laplace himself only used this fiction to demonstrate the 

extent of our ignorance and the need for a statistical description 

of certain processes. The problematics of Laplace's demon 

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 conditions. 

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 predictions, 

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 dynamic 

law and initial conditions that is challenged in the revival 

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 precision, 

can lead us to the determination of the exact initial conditions. 

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 animistic 

alliance with nature. Man seemed to possess a place in 

the Aristotelian world as both a living and a knowing creature.


The world was made to his measure. The first experimental 

dialogue received part of its social and philosophic justification 

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 historical 

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 accompanied 

each conceptual innovation, each occasion at 

which physics seemed at the point of unification and the prudent 

mask of positivism was dropped. Each time physicists 

repeated what Ampere's son stated so explicitly: this worduniversal 

attraction, energy, field theory, or elementary particles-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 continued 

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 dialogue 

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 science 

to an ahistorical accumulation of results and to pay no 

attention to what scientists are looking for, the ideal knowledge 

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 produced 

by the myth of modern science. He has stated that the 

miracle, the only truly astonishing feature, is that science ex-


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, "Ignoramus, 

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 begun 

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 systems 

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 inherited 

from our predecessors, the spell we have to exorcise today.

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I

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 optics. 

It is not surprising that the idea of scientific progress, 

already clearly formulated in the eighteenth century, dominated 

the nineteenth. Still, as we have pointed out, the position 

of science in We stern 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 Pascal, 

Kierkegaard, or Monod. We have mentioned the antiscientific 

implications of Heidegger's metaphysics. Now we 

wish to discuss more fully some aspects of the intellectual history 

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 

79


80 

seen as radically alien to that science. Today it is usually 

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 without 

reference to them. 

We shall also briefly discuss scientific positivism, which is 

based on the separation of what is true from what is scientifically 

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 positivists 

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 intelligibility 

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 challenge 

for classical science. Diderot, at the height of the Newtonian 

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 successive 

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 aggregate, 

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 mechanistic 

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 organization, 

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 intestines-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 simple? 

Why, even little children will laugh at you, and philosophers 

will answer that if it is a machine you are one 

too! If, however, you admit that the only difference between 

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, subjected 

to heat and motion, you will get sensitivity, life, 

memory, consciousness, passions, thought ... Just listen 

to your own arguments and you will feel how pitiful


they are. You will come to feel that by refusing to entertain 

a simple hypothesis that explains everything-sensitivity 

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 material 

nature is nothing but inert mass and motion, Diderot appeals 

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 relations. 

But Diderot does not despair. Science is only beginning; rational 

mechanics is merely a first, overly abstract attempt. The 

spectacle of the embryo is enough to refute its claims to universality. 

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 producing 

living beings. Diderot claims that matter has to be sensitive. 

Even a stone has sensation in the sense that the 

molecules of which it is composed actively seek certain combinations 

rather than others and thus are governed their likes 

and dislikes. The sensitivity of the whole organism is then

83 


simply the sum of that of its parts, just as a swarm of bees with 

its globally coherent behavior is the result of interactions between 

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. Otherwise, 

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 chemical 

and medical, that Diderot employed to counter the physics 

of his time is recurrent in the eighteenth century. While biologists 

speculated about the animal-machine, the preexistence of 

germs, and the chain of living creatures-all problems close to 

theology6-chemists and physicians had to face directly the 

complexity of real processes in both chemistry and life. Chemistry 

and medicine were, in the late eighteenth century, privileged 

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 concluded 

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 unfolding 

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 corruption; 

the matter of which living beings are composed is so frail, 

so easily decomposed, that if it were governed solely by the


84 

common laws of matter, it would not withstand decay or dissolution 

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 maintains 

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 constantly 

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 precariousness 

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 disorganization. 

Today the "vitalist principle" has been superseded 

by the succession of improbable mutations preserved in the 

genetic message "governing" the living structure. Nonetheless, 

some extrapolations starting from molecular biology relegate 

life to the confines of nature-that is, conclude life is 

compatible with the basic laws of physics but purely contingent. 

This was explicitly stated by Monod: life does not "follow 

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 selforganizational 

processes arise. (These questions will be studied 

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 problematics 

of life had been transformed. In a politically trans-


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, unlike 

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 considered 

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 synonymous 

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 opposition 

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 philosopher 

would reach the spiritual activity at the core of nature. 

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 intelligibility 

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 philosophical 

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 refuting 

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 Diderot 

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 general 

with its Newtonian realization. It thereby branded as im-


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 rational 

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 Enlightenment, 

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 disarray 

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 nature 

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 phenomenal 

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 intellect; 

it corresponds to a spiritual reality that supports his ethical 

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 phenomenal 

world described by science. From this perspective we can 

see that Kantian philosophy explicitly spells out the philosophical 

content of classical science. 

Kant defines the subject of critical philosophy as transcendental. 

It is not concerned with the objects of experience but 

is based on the a priori fact that a systematic knowledge of


these objects is possible (this is for him proved by the existence 

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 sensations 

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 product 

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 "revolves" 

around its object, seeking to discover the laws by 

which it is governed or the language by which it may be deciphered. 

Now the subject itself is at the center, imposing its 

laws, and the world perceived speaks the language of that subject. 

No wonder, then, that Newtonian science is able to describe 

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 expressed 

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 symbol 

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 conviction 

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, whatever 

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


claim to have found the definitive form of all positive knowledge. 

At the same time, however, it secured for philosophy a 

dominant position in. respect to science. It was no longer necessary 

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 problems 

that can be considered both easy and futile. It condemns 

science to the tedious task of deciphering the monotonous language 

of phenomena while keeping for itself questions of human 

"destiny": what man may know, what he must do, what 

he may hope for. The world studied by science, the world accessible 

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 problems 

of mankind. Beauty, freedom, and ethics cannot be objects 

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 already 

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 answers. 

For each experiment certain principles are presupposed 

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 classical 

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 theoretical 

language in which definite questions may be asked and 

experimental answers sought.


Kant's critical ratification defines scientific endeavor as 

silent and systematic, closed within itself. By so doing, philosophy 

endorses and perpetuates the rift, debasing and surrendering 

the whole field of positive knowledge to science 

while retaining for itself the field of freedom and ethics, conceived 

as alien to nature. 

A Prlilosophy of Nature? Hegel and Bergson 

The Kantian truce between science and philosophy was a fragile 

one. Post-Kantian philosophers disrupted this truce in favor of 

a new philosophy of science, presupposing a new path to knowledge 

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 dimensions. 

In his system increasing levels of complexity are specified, 

and nature's purpose is the eventual self-realization of its 

spiritual element. Nature's history is fulfilled with the appearance 

of man-that is, with the coming of Spirit apprehending 

itself. 

The Hegelian philosophy of nature systematically incorporates 

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 entities 

such as living beings. It denies the possibility of reducing 

those levels, rejecting the idea that differences are merely apparent 

and that nature is basically homogeneous and simple. It 

affirms the existence of a hierarchy, each level of which presupposes 

the preceding ones.


90 

Unlike the Newtonian authors of romans de Ia matiere, of 

world-embracing panoramas ranging from gravitational interactions 

to human passions, Hegel knew perfectly well that his 

distinctions among levels (which, quite apart from his own interpretation, 

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 velocity 

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 epitome 

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 century 

for seeking experimental and theoretical support for an 

alternative to classical science. Although this time was characterized 

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 theories 

had to be given up only a few years after their appearance. 

At the end of the nineteenth c:entury, when Bergson under-


took his search for an acceptable alternative to the science of 

his time, he turned to intuition as a form of speculative knowledge, 

but he presented it as quite different from that of the 

Romantics. He explicitly stated that intuition is unable to produce 

a system but produces only results that are always partial 

and nongeneralizable, results to be formulated with great caution. 

In contrast, generalization is an attribute of "intelligence," 

the greatest achievement of which is classical 

science. Bergsonian intuition is a concentrated attention, an 

increasingly difficult attempt to penetrate deeper into the singularity 

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 communicated 

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 itself. 

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 divergent 

directions of the activity of thought." 14 

The definition of these two divergent directions may also be 

considered as the historical consequence of scientific evolution. 

For Bergson, it is no longer a question of finding scientific 

alternatives to the physics of his time. In his view, 

chemistry and biology had definitely chosen mechanics as 

their model. The hopes that Diderot had cherished for the future 

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 generalization 

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 represents 

the very essence of science, its actual embodiment. The


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 understanding 

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." 16 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 positions 

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 systems 

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 something 

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 insofar 

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-


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 intellectual 

activities. Thus the only perspective remaining open 

for him was to introduce a way in which antagonistic approaches 

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 acceptable, 

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 metamorphosis 

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 resolutely 

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 experimentation. 

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


calls "scientific materialism," born in the seventeenth century. 

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 abstractions; 

and for eliciting from them clear-cut demonstrative 

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 concrete 

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 overcome 

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 confusion. 

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 attempt 

to do so. Whitehead saw no basic contradiction between 

science and philosophy. His purpose was to define the 

conceptual field within which the problem of human experience 

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,

95 


from that of stones to that of man. It is precisely this universality 

that, in Whitehead's opinion, defines his enterprise as 

"philosophy." While each scientific theory selects and abstracts 

from the world's complexity a peculiar set of relations, 

philosophy cannot favor any particular region of human experience. 

Through conceptual experimentation it must construct 

a consistency that can accommodate all dimensions of experience, 

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 conceived 

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 becoming, 

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." 21 

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 identities 

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 relation-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 becoming. 

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 interdependence 

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 relations 

must derive from the "nature" of the things (see Chap-

ORDER OUI OF CHAOS 

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 connectedness 

of all particles. However, when Whitehead wrote 

Process and Reality, the situation of physics was quite different, 

and Whitehead's philosophy found an echo only in biology.2 

3 

Whitehead's case as well as Bergson's convince us that only 

an opening, a widening of science can end the dichotomy between 

science and philosophy. This widening of science is possible 

only if we revise our conception of time. To deny timethat 

is, to reduce it to a mere deployment of a reversible lawis 

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 rationality 

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 similar 

statement was made by Helmholtz, a chemist, physician, 

physicist, and physiologist who dominated the German universities 

at the time when they were becoming the hub of European 

science. He stated: "the phenomena of nature are to be 

referred back to motions of material particles possessing unchangeable 

moving forces, which are dependent upon conditions 

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


to formulate the laws of dynamics. The why, the basic nature 

of forces and masses, remains hidden from us. Du Bois Reymond, 

as we already mentioned, expressed concisely the 

limitations of our knowledge: "Ignoramus, ignoramibus." Science 

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. Mathematical 

laws are nothing more than conventions useful for summarizing 

the results of possible experiments. At the end of the 

nineteenth century, scientific positivism exercised a great intellectual 

appeal. In France it influenced the work of eminent 

thinkers such as Duhem and Poincare. 

One more step in the elimination of "contemptible metaphysics" 

and we come to the Vienna school. Here science is 

granted jurisdiction over all positive knowledge and philosophy 

needed to keep this positive knowledge in .order. This 

meant a radical submission of all rational knowledge and questions 

to science. When Reichenbach, a distinguished neopositivist 

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 construction 

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.


What then is the role of philosophy? It has often been said 

that philosophy should become the science of science. Philosophy'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" something 

like their housemaid. Of course, there is the possibility 

that this clarification of concepts would permit further progress, 

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 existing 

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 attempts 

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 mechanics, 

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 knowledge 

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

99 


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 developments.

I 

I

BOOK TWO 

THE SCIENCE OF 

COMPLEXITY

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 science 

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 engine, 

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 century, 

wind, 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 thermodynamics. 

born out of this interest, was thus not so much 

103


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 prefect 

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 elegant: 

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 between 

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 Newton's 

law, and its formulation may thus be considered the starting 

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 molecules, 

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 theory, 

but they were forced to retreat. 2 At the peak of its glory, 

the Laplacian dream met with its first setback. A physical theory 

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 science 

has been well analyzed by Michel Serres3-heat and

105 ENERGY AND THE INDUSTRIAL 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 chemists 

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 distinction 

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 knowledge 

but opened a new line of inquiry, the progressive formulation 

of a theory of irreversible processes. 

Fourier's law, when applied to an isolated body with an unhomogeneous 

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 irreversible 

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 

particles-and the occurrence of temporal asymmetry thus 

were linked from the outset. But heat conduction did not become 

the starting point of an investigation into the nature of 

irreversibility before it was first linked with the notion of dissipation 

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 century. 

Like mechanics, the science of heat implied both an original 

conception of the physical 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 position 

and velocity of its constituents (there are some IQ23 mole-


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 temperature, 

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 properties 

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 instance, 

at constant volume-the system must be brought into 

interaction with its environment; it mu s t receive a certain 

amount of heat while at the same time its volume is kept constant 

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 

fixed 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 environment). 

As we have already mentioned, pressure, volume, 

chemical composition, and temperature are the classical physicochemical 

parameters in terms of which the properties of 

macroscopic systems are defined. Thermodynamics is the science 

of the correlation among the variations in these properties. 

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 particles; 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 mechanical 

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-


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 temperature, 

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 processes, 

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 describe 

the essential role played by the principle of energy conservation. 

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 variables-coordinates 

and momenta-and leads to changes in 

these variables while itself remaining constant throughout the 

motion. Dynamic change merely modifies the respective importance 

of potential and kinetic energy, conserving their totality. 

The early nineteenth century was characterized by unprecedented 

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 discovered 

by Galvani. Before him, only static electric charges 

were known. Galvani, using a frog's body, set up the first experimental 

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-


108 

structed a chemical battery; electricity could thus be produced 

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 produced 

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 biology 

were recognized as a "conversion." The idea of conversion, 

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 equivalent 

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 degree. 

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 deranged, 

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 complicated 

and involved in the apparent confusion and intricacy 

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 Lie big-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 respiration. 

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 hostile 

to Romantic speculation and eager to contribute to experimental 

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 fundamental 

"equivalence," we may state that the German philosophic 

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 doctor 

working in the Dutch colonie s 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 inhabitants 

need to burn less oxygen 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


further and concluded that the balance between oxygen consumption 

and heat loss was merely the particular manifestation 

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 influence 

can also be recognized in another idea held by some 

physiologists, the distinction between vitalism as philosophical 

speculation and the problem of scientific methodology. For 

those physiologists, even if there was a "vital" force underlying 

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 requirement 

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 proceeding, 

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 variable 

or invariable; in the former case the above conviction 

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 therefore 

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 discover 

the ultimate and unchangeable causes of natural 

phenomena. tO 

With the principle of the conservation of energy, the idea of


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 included 

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 controllable, 

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 conversion. 

Indeed, only difference, such as a difference of temperature 

or of potential energy, can produce results that are 

also differences.11 Energy conversion is merely the destruction 

of a difference, together with the creation of another difference. 

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 transformations, 

life and death, and that goes beyond equivalences and 

conservation of energy.12 Here we reach the most original contribution 

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 nineteenthcentury 

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 thermodynamics, 

which would lead to the first quantitative expression 

of irreversibility, was made by Sadi Carnot in 1824, before 

the general formulation of the principle of ;onservation of energy 

by Mayer (1842) and Helmholtz (1847). Carnot analyzed


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 producing 

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 physics 

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 having 

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 temperature 

change results from direct heat flow between two bodies 

at different temperatures. Since such flows have no mechanical 

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

.. 

... 


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 volume 

continues to change until the system has passed from the 

temperature of one source to that of the other. 

p 

' 

' 

' 

... 

.. 

.. 

.. 

... 

',Q 

... 

.. 

... 

.. 

... ......... 

.. _ 

_ _ 

T 

H 

---- T 

c 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.


It is quite remarkable that this description of an ideal thermal 

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 concerned 

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 discovered 

that the need for two sources and the formula for theoretical 

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 conversion 

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 cannot 

use without restriction the seemingly inexhaustible energy 

reservoir that nature provides. Not all energy-conserving processes 

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 equivalence 

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.


However, since the eighteenth century, the status of idealizations 

had changed. Based as it was on the principle of the conservation 

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 engine? 

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 conservation 

and compensation. In addition, it provided an opportunity 

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 thermodynamics. 

It was Fourier's heat propagation that Carnot had identified 

as a possible cause for the power losses in a heat engine. Carnot's 

cycle, no longer the ideal cycle but the "real" cycle, thus 

became the point of convergence of the two universalities discovered 

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 mechanical 

energy. Note the word "universal," which has obvious 

cosmological connotations. 

The world of Laplace was eternal, an ideal perpetual-motion 

machine. Since Thomson's cosmology is not merely a reflection 

of the new ideal heat engine but also incorporates the consequences 

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


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 mechanical 

effect. 

Thomson thus made a dizzy leap from engine technology to 

cosmology. Hs formulation of the second law was couched in 

the scientific terminology of his time: the conservation of energy, 

engines, and Fourier's law. It is clear, moreover, that the 

part played by the cultural context was important. It is generally 

accepted that the problem of time took on a new importance 

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 homogeneity 

and death, reminds us more of ancient mythological and 

religious archetypes than of the progressive complexification 

and diversification described by biology and the social sciences. 

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" society 

or for a moratorium on scientific research to the 

announcement of "scientific truths" concerning our disintegrating 

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 simultaneous 

use of thermodynamics and relativity. Yet most texts 

in this field are unanimous in predicting final doom. The conclusion 

of a recent book reads: 

The unpalatable truth appears to be that the inexorable 

disintegration of the universe as we know it seems assured, 

the organization which sustains all ordered activity, 

frem men to galaxies, is slowly but inevitably 

running down, and may even be overtaken by total gravitational 

collapse into oblivion.t4


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 intelligence 

as scientists of the twentieth century have tended 

to suppose.15 

In spite of the important progress made by Hawking and others, 

our knowledge of large-scale transformations in our universe 

remains inadequate. 

The Birth of Entropy 

In 1865, it was Clausius' turn to make the leap from technology 

to cosmology. At the outset he merely reformulated his 

earlier conclusions, but in doing so he introduced a new concept, 

entropy. His first goal was to distinguish clearly between 

the concepts of conservation and of reversibility. Unlike mechanical 

transformations, where reversibility and conservation 

coincide, a physicochemical transformation may conserve energy 

even though it cannot be reversed. This is true, for instance, 

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


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 energy 

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 environment. 

We can set up experiments in which heat is given up by 

the system instead of flowing into the system. The corresponding 

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 exchanges, 

we have irreversible processes inside the system, 

such as heat losses, friction, and so on. These produce an entropy 

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 monotonous, 

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 outside 

world, while the second refers to the irreversible processes 

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 entropy 

variation into two parts, it is useful to apply our formulation 

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


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 vessel, 

this quantity may vary both as the result of adding hydrogen 

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 hydrogen 

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 occurrence 

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 conservation, 

this was his main concern. Regarding the irreversible 

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 between 

entropy flow and entropy production can still be made. 

For an isolated system .that has no exchanges with its environment, 

the entropy flow is, by definition, zero. Only the production 

term remains, and the system's entropy can only 

increase or remain constant. Here, then, it is no longer a question 

of irreversible transformations considered as approximations 

of reversible transformations; increasing entropy corresponds 

to the spontaneous evolution of the system. Entropy thus becomes 

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 formulation 

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-


120 

creases to a maximum goes far beyond the technological problem 

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 ultimately 

lead the system to thermodynamic "equilibrium" 

corresponding to the state of maximum entropy. 

In Chapter I we emphasized the element of surprise involved 

in the discovery of Newton's universal laws of dynamics. 

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 conditions 

are interdependent processes. In this context irreversibility 

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, unlike 

dynamic objects, thermodynamic objects can only be partially 

controlled. Occasionally they "break loose" into 

spontaneous change. 

All changes are not equivalent for a thermodynamic system. 

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 modification 

of the boundary conditions (for example, ambient 

temperature). For an isolated system, equilibrium appears as


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, Nature 

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 according 

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 nonequilibrium 

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 dynamic 

states that may occur. 

We are now confronted with two baically different descriptions: 

dynamics, which applies to the world of motion, and 

*Physical chemistry often employs Avogadro's number-that is, the number 

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.


thermodynamics, the science of complex systems with its intrinsic 

direction of evolution toward increasing entropy. This 

dichotomy immediately raises the question of how these descriptions 

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, entropy, 

which increases uniformly for isolated systems. Entropy 

behaves as an attractor for isolated systems. 

How could the formulations of thermodynamics be reconciled 

with dynamics? At the end of the nineteenth century, 

most scientists seemed to think this was impossible. The principles 

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 toward 

dissipation had to be accepted as new axioms. This was 

the argument of the "energeticists" as opposed to the "atomists," 

who refused to abandon what they considered to be the 

essential mission of physics-to reduce the complexity of natural 

phenomena to the simplicity of elementary behavior expressed 

by the laws of motion. 

Actually, the problems of the transition from the microscopic 

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 concepts 

had to be developed to extend the physics of trajectories 

to cover the situation described by thermodynamics. Following 

in Maxwell's footsteps, Boltzmann sought this conceptual 

innovation in the theory of probability. 

That probability could play a role in the description of complex 

phenomena was not surprising: Maxwell himself appears


to have been influenced by the work of Quetelet, the inventor 

of the "average" man in sociology. The innovation was to introduce 

probability in physics not as a means of approximation 

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 compartments. 

The problem is to find the probability of the 

various possible distributions of particles between the compartments-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 particles 

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 different ways of putting one particle in one half and 

seven in the other half, if we suppose the particles to be distinguishable, 

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 distribution 

N1,N2• Its expression is P=N!IN1!N2!. 

For any given population, the larger the number of complexions 

the smaller the difference between N1 and N2• It is maximum 

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 

order of 1Q 2 3 values found in macroscopic systems, the overwhelming 

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 improbable. 

Boltzmann was the .first to realize that irreversible increase


in entropy could be considered as the expression of a growing 

molecular disorder, of the gradual forgetting of any initial dissymmetry, 

since dissymmetry decreases the number of complexions 

when compared to the state corresponding 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 dissymmetry, 

of all special distributions (for example, the set of particles 

concentrated in a subregion of the system, or the 

distribution of velocities that is created when two gases of different 

temperatures are mixed). This forgetting is possible because, 

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 overwhelming 

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 = N 2 • 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).


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 

will be 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 interpretation 

thus makes it possible to understand the specificity 

of t h e 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 system. 

This is the subjectivistic interpretation of entropy. Such 

an interpretation was acceptable when irreversible processes 

were considered to be mere nuisances corresponding to friction 

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 importance: 

life would not be possible without them. The subjectivistic 

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 probabilistic 

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 conditions 

available to a dynamic system. The second law therefore 

marks a radical departure from the mechanistic world of classical 

or quantum dynamics. Let us now return to Boltzmann's 

work. 

So far we have discussed isolated systems in which the number 

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


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 energy 

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 competition 

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 compared 

with the potential energy that results from the interactions 

of each molecule with its neighbors. We can imagine 

each particle as imprisoned by its interactions with its neighbors. 

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 fixed temperature are examples of "thermodynamic 

potentials." The extremes of thermodynamic potentials 

such as S or F define the attractor states toward which systems 

whose boundary conditions correspond to the definition 

of these potentials tend spontaneously. 

Boltzmann's principle can also be used to study the coexistence 

of structures (such as the liquid phase and the solid 

phase) or the equilibrium between a crystallized product and 

the same product in solution. It is important to remember, 

however, that equilibrium structures are defined on the molecular 

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.


Carnot and Darwin 

Equilibrium thermodynamics provides a satisfactory explanation 

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 statistical 

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 indefinitely 

without further interaction with their environment. 

When we examine a biological cell or a city, however, the situation 

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 "differences" 

created by the initial conditions have been forgotten, 

the number of complexions will be maximum. But it is impossible 

to understand spontaneous convection from this point of 

view. The convection current calls for coherence, for the cooperation 

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 Boltzmann's 

terms, it is an "improbable" state. If convection must 

be considered a "miracle,·· what then is there to say about life,


with its highly specific features present in the simplest organisms? 

The question of the relevance of equilibrium models can be reversed. 

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 Fa ust, who addresses to the alchemist who created 

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 temperature 

of the sun), but a world in which nonequilibrium remains 

the rule, a "lukewarm" world where equilibrium and 

nonequilibrium coexist. 

For a long time, however, physicists thought they could define 

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 fundamental 

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 evolution-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 immense 

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 selforganization, 

ever-increasing complexity.


To sum up our argument so far, equilibrium thermodynamics 

was the first response of physics to the problem of nature's 

complexity. This response was expressed in terms of the dissipation 

of energy, the forgetting of initial conditions, and evolution 

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 thermodynamics, 

time implies degradation and death. As we have seen, 

Diderot had already asked the question: Where do we, organized 

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 phenomena 

inside the system. This term is always positive except 

at thermodynamic equilibrium, when it becomes zero. For isolated 

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. Together 

with heat conduction, they form the prototype of irreversible 

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 fundamental 

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


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 concentrations 

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 whenever 

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 Y will be produced. A collision 

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 concentration, 

and these rates are expressed as a function of the 

concentrations of the reactants. The rate of change of concentration 

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 taking 

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 molecules 

of Y and B can set off the opposite reaction. A second 

reaction Y + B-.X +A thus occurs within the system described, 

bringing about a supplementary variation in the concentration 

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 equilibrium 

is therefore a typical example of an "attractor" state. 

Whatever its initial chemical composition, the system spontaneously 

reaches this final stage, where the forward and reverse 

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 between 

concentrations determined by the law of mass action 

corresponds to chemical equilibrium in the same way that uniformity 

of temperature (in the case of an isolated system) corresponds 

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 molecules 

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. Substances 

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 "enzymes." 

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 possible 

mechanism for the catalytic effect of enzymes is to present


134 

different "reaction sites" to which the different molecules involwd 

in the reaction tend to attach themselves, thus increasing 

the likelihood of their coming into contact and reacting. 

One very important type of catalysis, particularly in biology, 

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 enzyme. 

By attaching itself to the enzyme it stabilizes that particular 

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 molecule 

A is converted into a molecule X. Therefore we need X to 

produce more X. This reaction may be symbolized by the reaction 

"loop": 

A 

One important feature of systems involving such "reaction 

loops" is that the kinetic equations describing the changes occurring 

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 biology 

is that of crosscatalysis-for example, 2X + Y3X, 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 inhibition, 

which can also be represented by suitable feedback 

loops. 

The peculiar mathematical properties of the nonlinear differential 

equations describing chemical processes with catalytic 

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


,.... -----£ a 

D 

or 

A -.x 

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 metabolic 

functions. For example, the relation between nucleic 

acids and proteins can be described in terms of a crosscatalytic 

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 depends 

on the molecular mechanism, as can be verified by the 

examples already indicated. The thermodynamics of irreversible 

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 conduction. 

Fourier's law tells us that the heat flux J is proportional 

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 corresponding 

to a chemical reaction. Recall the reaction A+ X


-+ Y + B. We have seen how, at equilibrium, the ratio between 

concentrations is given by the law of mass action. As Theophile 

De Donder has shown, a "chemical force" can be introduced, 

the "affinity" a that determines the direction of the 

chemical reaction rate just as the temperature gradient determines 

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 immediately 

apparent that the affinity 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 chemical 

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, corresponds 

to the resolution of this conflict. Seen from this point 

of view, the specificity of thermodynamic affinity thus rephrases 

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.


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 frictionless 

pendulum. To neglect irreversible processes in dynamics 

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 chemistrychemical 

transformations characterized by reaction ratesare 

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 appear 

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 linear 

functions of the forces. The third field is called the "nonlinear" 

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-toequilibrium 

region. These are the famous "reciprocity relations." 

In qualitative terms, they state that if a force-say, 

"one" (corresponding, for example, to a temperature gradient)-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


138 

induces a process of diffusion of matter, we find that a concentration 

gradient can set up a heat flux through the system. 

The general nature of Onsager's relations has to be emphasized. 

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 microscopic 

assumptions. 

Reciprocity relations have been the first results in the thermodynamics 

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 equilibrium 

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, nonequilibrium 

thermodynamics bears mention here. We have already 

spoken of thermodynamic potentials whose extrema correspond 

to the states of equilibrium toward which thermodynamic evolution 

tends irreversibly. Such are the entropy S for is


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 production 

diS. Therefore, dS=O implies that deS= -diS


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 minimum 

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 characterized 

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: 

What 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 fluctuation 

may shift the system away from this minimum. The second 

law of thermodynamics, however, imposes the return 

toward the attractor. The system is thus "immune" with respect 

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 established 

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 fluctuation 

produced by the system or its environment. In some 

cases, the analysis leads to the conclusion that a state is "unstable"-in 

such a state, certain fluctuations, instead of re-


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 corresponding 

to minimum entropy production. 

Thermodynamics leads to an initial general conclusion concerning 

systems that are liable to escape the type of order governing 

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 hydrodynamics 

and fluid flow. For instance, it has long been known 

that once a certain flow rate of flux has been reached, turbulence 

may occur in a fluid. Michel Serres has recently recalled4 

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. Sometimes, 

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 spontaneous, 

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 truethe 

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 macroscopic 

scale, it is, on the contrary, highly organized on the 

microscopic scale. The multiple space and time scales involved 

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-


organization. Part of the energy of the system, which in laminar 

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 vertical 

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 surface. As a result of 

these boundary conditions, a permanent heat flux is set up, 

moving from the bottom to the top. When the imposed gradient 

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 molecules 

is produced, increasing the rate of heat transfer. Therefore, 

for given values of the constraints (the gradient of 

temperature), the entropy production of the system is increased; 

this contrasts with the theorem of minimum entropy 

production. The Benard instability is a spectacular phenomenon. 

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 number 

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 attributing 

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 between 

two boxes. Here also the number of complexions is 

large when there is disorder-that is, a wide dispersion of velocities. 

In contrast, coherent motion means that many molecules 

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 particularly 

obvious as far as the genesis of the new behavior is 

concerned. In the case of the Benard instability it is a fluctuation, 

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 imposed 

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 probability 

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 general 

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 "equilibrium 

structures" such as crystals. Benard cells are structures 

too, but of a quite different nature. That is why we have 

introduced the notion of "dissipative structures," to emphasize 

the close association, at first paradoxical, in such situations 

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 embedding 

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. Although 

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


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 differentthey 

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 speeding 

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 recently 

become a subject of interest. There is also another difference. 

All flows become turbulent at a "sufficiently" large 

distance from equilibrium (the threshold is measured by dimensionless 

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 particular 

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 reactions. 

In contrast with close-to-equilibrium situations, the 

behavior of a far-from-equilibrium system becomes highly specific. 

There is no longer any universally valid law from which


the overall behavior of the system can be deduced. Each system 

is a separate case; each set of chemical reactions must be 

investigated and may well produce a qualitatively different behavior. 

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 fundamental 

achievements of modern molecular biology (see Figure 4). 

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 variable 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. 

X

ORDER OUT 0 CHAOS 146 

Beyond the Threshold of Chemical Instability 

Today the study of chemical instabilities is common. Both theoretical 

and experimental work are being pursued in a large 

number of institutions and laboratories. Indeed, as will become 

clear, these investigations are of interest to a wide range 

of scientists-not only to mathematicians, physicists, chemists, 

and biologists, but also to economists and sociologists. 

In far-from-equilibrium conditions various new phenomena 

appear beyond the threshold of chemical instability. To describe 

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 an "Oregonator," and most recently a "Paloaltonator" 

!). 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 behavior 

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 becomes 

an unstable "focus" and the system leaves this focus to 

reach a "limit cycle."


0 

2 3 4 y 

Figure 5. This scheme represents concentration of component X vs. concentration 

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 plotted), 

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 oscillation 

period depends both on the kinetic constants characterizing 

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. Whatever the initial conditions, it approaches the 

limit cycle, the periodic behavior of which is stable. We therefore 

have a periodic chemical process-a chemical clock. Let 

us pause a moment to emphasize how unexpected such a phenomenon 

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


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 col or 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 observed, no one would believe that such a process 

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, communicate, 

which is of obvious importance in so many fields, from 

chemistry to neurophysiology. Dissipative structures introduce 

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 differences 

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 astonishing 

range of behaviors available to this system. Indeed, 

whereas at equilibrium and near-equilibrium the system remains 

spatially homogeneous, the diffusion of the chemical 

throughout the system induces, in the far-from-equilibrium region, 

the possibility of new types of instability, including the 

amplification of fluctuations breaking the initial spatial symmetry. 

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 


X hO 

hO.S8 

3 

o 

X h 1.10 

o 

h 1.88 

I--_ror.-____ 1 _ _____ _..., 

o 

X h2.04 

o 

h 3.435 

3 

21....-- - - - 

-- - --'" 

1 0 

o 

Figure 6. Chemical waves simulated on computer: successive steps of 

evolution of spatial profile of concentration of constituent X in the "Brusselator" 

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 coefficients 

for X and Y: 810-3,410-3• 

In addition, especially when the values of the diffusion constants 

of X and Y are quite different from each other, the system 

may display a stationary, time-independent behavior, and 

stable spatial structures may appear.


150 

Here we must pause once again, this time to emphasize how 

much the spontaneous formation of spatial structures contradicts 

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 dimensions 

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 simulation. 

Concentration X is a function of geometrical coordinates p,a in the 

horizontal plane. The location of the perturbation applied to the uniform unstable 

solution (X Q 

, Y0) is indicated by an arrow. 

This corresponds to a new, extremely interesting symmetrybreaking 

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.


Up to now it has been assumed that the "control substances" 

(A, B, D, and E) are uniformly distributed throughout 

the reaction system. If this simplification is abandoned, additional 

phenomena can occur. For example, the system takes 

on a "natural size ," which is a function of the parameters describing 

it. In this way the system determines its own intrinsic 

size-that is, it determines the region that is spatially structured 

or crossed by periodic concentration waves. 

These results still give a very incomplete picture of the variety 

of phenomena that may occur far from equilibrium. Let us 

first mention the possibility of multiple states far from equilibrium. 

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 reaction 

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).


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 Brusselator, 

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 Bailin, 

gives an idea of such very complicated attractor lines calculated 

for a model generalizing the Brusselator through the 

addition of an external periodic supply of X. What is remarkable 

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, introduced 

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 catalyst, 

cerium, manganese, or ferroin. 

INFLOW 

MALONIC . ::: ...rP U ::- M :: P :


Various experimental conditions may be set up giving different 

forms of autoorganization within the same system-a chemical 

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 relevance 

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 oscillations 

or to spatial structures . We have seen that the basic condition 

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 reaction 

chains activate each other's synthesis) provide the classical 

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 the reaction mechanisms are complex-in the 

Belousov-Zhabotinsky reaction, about thirty compounds have 

been identified. On the contrary, in the many biological examples 

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


154 

inhibitions is an essential step in understanding the functional 

logic of biological systems. This includes the triggering of syntheses 

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 explains 

the transmission and exploitation of genetic information 

is itself a feedback loop, a "nonlinear" mechanism. Deoxyribonucleic 

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 translated 

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 replication, 

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 complementary 

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 thermodynamics. 

On the other hand, molecular biology succeeded in relating 

living structures to a relatively small number of basic 

biomolecules. Investigating the diversity of chemical mechanisms, 

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 factory: 

on the one hand, they are the site of multiple chemical 

transformations; on the other, they present a remarkable "spacetime" 

organization with highly nonuniform distribution of biochemical 

material. We can now link function and structure. 

Let us briefly consider two examples that have been studied 

extensively in the past few years.


First we shall consider glycolysis, the chain of metabolic 

reactions during which glucose is broken down and an energyrich 

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 glycolytic 

cycle. 7 It has been shown that these oscillations are determined 

by a key step in the reaction sequence, a step activated 

by ADP and inhibited by ATP. This is a typical nonlinear phenomenon 

well suited to regulate metabolic functioning. Indeed, 

each time the cell draws on its energy reserves, it is 

exploiting the phosphate bonds, and AT P 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 phenomenon, 

a chemical clock. The theoretically calculated values 

of the chemical concentrations necessary to produce 

oscillation and the period of the cycle agree with the experimental 

data. Glycolytic oscillation produces a modulation of 

all the cell's energy processes which are dependent on ATP 

concentration and therefore indirectly on numerous other metabolic 

chains. 

We may go farther and show that in the glycolytic pathway 

the reactions controlled by some of the key enzymes are in farfrom-equilibrium 

conditions. Such calculations have been reported 

by Benno Hesss and have since been extended to other 

systems. Under usual conditions the glycolytic cycle corresponds 

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 thermodynamic 

point of view. Certain reactions are close to equi-


librium, and others are not. Not everything in a living system 

is "alive." The energy flow that crosses it somewhat resembles 

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 aggregation 

of slime molds, the Acrasiales amoebas (Dictyostelium 

discoideum). This process9A is an interesting case on the borderline 

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 Figure A. 

fruiting ? 

body 

spores 0 

00 

I 

(Y:J growth 

Q) 

\ 

((t'': kx 

\ , .. , I 

, - 

.. - - ' 

;'; aggregation 

\ 


In Dictyostelium discoideum, the aggregation proceeds in a periodic manner. 

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 periodic 

fashion. The other cells respond by moving toward the centers and by 

relaying the signals to the periphery of the aggregation territory. The existence 

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 existence 

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.), fluctuations 

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 aggregation 

centers. 

To understand the origin of the periodicity in the aggregation of D. discoideum, 

it is necessary to study the mechanism of synthesis of the chemotactic 

signal. On the basis of experimental observations one can describe 

this mechanism by the scheme of Figure B. 

_----+:.-- cAM P 

Figure B 

ATP 

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 enzyme, 

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 enzyme 

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 observed 

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 

< 

A 

Phosphodiesterase , 

k 

Figure C 

Three regions can be distinguished for different values of k and s. Region 

A corresponds to a stable, nonexcitable stationary state; region B to a stationary 

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 observed 

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 periodically 

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.


the environment in which these amoebas live and multiply becomes 

poor in nutrients, they undergo a spectacular transformation. 

(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 abundant 

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 resources. 

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 convergence 

of the amoebaes 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 attractor 

center and later by other cells through a relay mechanism. 

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 corresponding 

to a normal nutritive environment has become unstable-that 

is, the nutritive environment has become 

exhausted. The fact that under such conditions of food shortage 

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.


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 threshold. 

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 equilibrium. 

At some point we reach the threshold of the stability 

of the "thermodynamic branch." Then we reach what is generally 

called a "bifurcation point." (These are the points whose 

role Maxwell emphasized in his thoughts on the relation between 

determinism and free choice [see Chapter II, section 

3].) 

X 

,. 

, 

, 

I 

I 

\ 

\ 

\ 

\ 

A 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.


Let us consider some typical bifurcation diagrams. At bifurcation 

point B, the thermodynamic branch becomes unstable 

in respect to fluctuations. For the value Ac of the control parameter 

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 depends 

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 control 

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 interpretation 

of biological and social phenomena, but that it may play 

an important role in simple chemical processes is quite unexpected. 

Consider the bifurcation diagram represented in Figure 11. 

This differs from the previous diagram in that at the bifurcation 

point two new stable solutions emerge. Thus a new question: 

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 AAc there 

are two stable stationary states for each value of A (the formerly stable state 

becomes unstable).


they may represent either of the two nonuniform distributions 

of chemical X in space, as represented in Figures 12 and 13. 

X 

----r 

X 

---- r 

Figures 12 and 13. Two possible spatial distributions of the chemical component 

X, corresponding to each of the two branches in 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 macroscopic 

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 


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, half into the 

right. Here another interesting question arises: In the world 

around us, some basic simple symmetries seem to be broken.to 

Everybody has obser ved that shells often have a preferential 

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 "systematic" 

explanation. 

We have recently discovered a striking example of the fundamental 

new properties that matter acquires in far-fromequilibrium 

conditions: external fields, such as the gravitational 

field, can be "perceived" by the system, creating the possibility 

of pattern selection. 

How would an external field-a gravitational field-change 

an equilibrium situation? The answer is provided by Boltzmann'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 mountain 

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

. . .. .. . . . 


164 

Gravitation obviously will modify the diffusion flow in a reaction 

diffusion equation. Detailed calculations show that this 

can be quite dramatic near a bifurcation point of an unperturbed 

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 11. Suppose that for no gravitation, 

g=O, we have, as in Figures 12 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 external 

field. X is plotted as a function of parameter ". The symmetrical bifurcation 

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).


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 respect 

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 preferred 

and will be selected. 

The important point is that, depending on the chemical process 

responsible for the bifurcation, this mechanism expresses 

an extraordinary sensitivity. Matter, as we mentioned earlier 

in this chapter, perceives differences that would be insignificant 

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 fluctuations 

is another example of a system's spontaneous "adaptative 

organization" to its environment. Let us give an example12 

of self-organization as a function of fluctuating external conditions. 

The simplest conceivable chemical reaction is the isomerization 

reaction where AB. 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*. Consider 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 transformation 

from B to A absorbs heat: the higher the temperature, 

the faster the formation of A. But also the higher the concentration 

of A, the higher the absorption of light by A and its transformation 

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 appears 

one of the standard far-from-equilibrium phenomena: 

the coexistence of multiple stationary states. For the same val-


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 threshold 

between the first two. Such a coexistence of stationary 

states gives birth to the well-known phenomenon of hysteresis. 

But this is not the whole story. If the light intensity, instead 

of being constant, is taken as randomly fluctuating, the 

situation is altered profoundly. The zone of coexistence between 

the two stationary states increases, and for certain values 

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 

 

I ' ' I 

J ' 'I 

v 

• 

p': 

Q 

b1 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 diminishing. 

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 discontinuity: 

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.


ered as unavoidable in any "natural system." For example, in 

biological or ecological systems the parameters defining interaction 

with the environment cannot generally be considered as 

constants. Both the cell and the ecological niche draw their 

sustenance from their environment ; and humidity, pH, salt 

concentration, light, and nutrients form a permanently fluctuating 

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 stability. 

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 observed 

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 hydrodynamical 

or chemical systems. Let us consider Benard structures, 

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 complete.13 

The interplays among the frequencies produce possibilities 

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 nonequilibrium 

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 bifurcation) 

2: 

o 

0: 

u.. 

&£J 

U 

Z 

 

!:!! 

c 

Chaos 

Mixed - Mode Oscillations 

Chaotic 

and 

Periodic 

Relaxation Oscillations 

TIME 

Figure 16. Temporal oscillations of the ion Br- in the Belousov 

Zhabotinski reaction. The figure represents a succession of regions corresponding 

to qualitative differences. This is a schematic representation. The 

experimental data indicate the existence of much more complicated sequences. 

pect of the convection is largely destroyed. However, we 

should not confuse "equilibrium thermal chaos" and "nonequilibrium 

turbulent chaos." In thermal chaos as realized in 

equilibrium, all characteristic space and time scales are of molecular 

range, while in turbulent chaos we have such an abundance 

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 tropical 

forest; as represented, for instance, by the diversity of species, 

corresponds to the very archetype of order. Whatever 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 characteristic 

frequencies produces an increasing randomness stemming 

from the multiplicity of those frequencies. 

A remarkably simple road to "chaos" that has already attracted 

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 parameter 

values the system's behavior is periodic, with a period T; 

beyond this range, the period becomes 2T, and beyond yet another 

critical threshold, the system needs 4 Tin order to repeat 

itself. The system is thus characterized by a succession of bifurcations, 

with successive period doubling. This constitutes a 

typical route going from simple periodic behavior to the complex 

aperiodic behavior occurring when the period has doubled 

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 measurable 

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 parameter 

of the order of X 6 , we see that the system already has a 

wealth of possible stable and unstable behaviors. The "historical" 

path along which the system evolves as the control parameter 

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-


tween or among more than one possible future. Both the deterministic 

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 connected. 

This mixture of necessity and chance constitutes the 

history of the system. 

Solutions 

\ (c'),, 

' ,, 

' , 

' 

I 

I 

, 

(/ .. 

, - 

- 

- 

t---


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 interactions 

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 itthat 

modern science was born when Aristotelian space, for 

which one source of inspiration was the organization and solidarity 

of biological functions, was replaced by the homogeneous 

and isotropic space of Euclid. However, the theory of 

dissipative structures moves us closer to Aristotle's conception. 

Whether we are dealing with a chemical clock, concentration 

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 phenomena 

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 embryologist Hans 

Driesch believed that some immaterial "entelechy" was responsible 

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 observe embryological development on film, we "see" 

jumps corresponding to radical reorganizations followed by 

periods of more "pacific" quantitative growth. There are, fortunately, 

few mistakes. The jumps are performed in a reproducible 

fashion. We might speculate that the basic 

mechanism of evolution is based on the play between bifurca-


tions as mechanisms of exploration and the selection of chemical 

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 imperative 

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 substance, 

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 chemical 

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 particular, 

we may refer to Kauffman's work16 on drosophila. A 

reaction-diffusion system is taken as responsible for the commitment 

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 embryo 

as it results from successive binary choices. See text for more detail.


compartment would be specified by a unique combination of 

binary choices, each of these choices being the result of a spatial 

symmetry-breaking bifurcation. The model leads to successful 

predictions about the result of transplantations as a 

function of the "distance" between the original and final regions-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 biological 

systems where the embryo begins to develop in an apparently 

symmetrical state (for example, Fucus, Acetabularia). 

We may ask if the embryo is really homogeneous at the beginning. 

And even if small inhomogeneities are present in the initial 

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 transport 

appears as the only general mechanism capable of breaking 

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 antireductionists. 

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 required 

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 Nature, 

and Bergson with his simple, irrepressible, organizationcreating 

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::lc::mc::nts of a different order, with which we: 

try to imitate it artificially, but with which it remains in-


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 numerous 

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 antireductionists 

has often appeared as a conflict between the assertion 

of an external and an internal purpose. The idea of an immanent 

organizing intelligence is thus often opposed by an organizational 

model borrowed from the technology of the time 

(mechanical, heat, cybernetic machines), which immediately 

elicits the retort: "Who" 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 internal 

organizing power are expressions of an inability to conceive 

evolutive organization without immediately referring it 

to some preexisting goal. Today, in spite of the spectacular success 

of molecular biology, the conceptual situation remains 

about the same: Bergson's argument could be applied to contemporary 

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 biology 

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 functioning 

of a circuit may be deduced from the nature and posi-


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 processing 

systems, and motor response. We know of approximately 

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 generates 

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, emphasizing 

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 opportunity. 

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 differences 

in the system in the same way as the demon maintains 

temperature and pressure differences. If we accept this, biology 

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 different 

implications. We know today that both the biosphere as 

a whole as well as its components, living or dead, exist in farfrom-equilibrium 

conditions. In this context life, far from 

being outside the natural order, appears as the supreme expression 

of the self-organizing processes that occur.


176 

We are tempted to go so far as to say that once the condi· 

tions for self-organization are satisfied, life becomes as predictable 

as the Benard instability or a falling stone. It 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 behavior 

can be decisive. More generally, the "overall" behavior 

cannot in general be taken as dominating in any way the elementary 

processes constituting it. Self-organization processes 

in far-from-equilibrium conditions correspond to a delicate interplay 

between chance and necessity, between fluctuations 

and deterministic laws. We expect that near a bifurcation, fluctuations 

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 irreversible 

ones. This change of perspective affects chemistry in a 

striking way. As we have seen in Chapter V, chemical processes, 

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 concentrations 

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 deterministic 

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. Whenever 

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 trajectory. 

We cannot predict the details of temporal evolution. 

177


Once again, only a statistical description is possible. The existence 

of an instability may be viewed as the result of a fluctuation 

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 between 

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 fluctuations 

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. 

What 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 numbers, 

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 also 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 outcome. 

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 aspects 

of quantum theory. Since we can no longer simultaneously 

measure position and coordinates in quantum theory, 

classical determinism is breaking down. This was believed to 

be of no importance for the description of macroscopic objects

179 ORDER THROUGH FLUCTUATIONS 

such as living systems. But the role of fluctuations in nonequilibrium 

systems shows that this is not the case. Randomness 

remains essential on the macroscopic level as well. It is interesting 

to note another analogy with quantum theory, which 

assigns a wave behavior to all elementary 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 directions 

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 instead of X. We can then ask what is the 

probability at a given time of finding a number X for the concentration 

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 describes 

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


particles in a gas or a liquid. The mathematical form of this 

distribution is of no importance here. We merely want to emphasize 

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. Moreover, 

this law enables us to calculate the correlation between 

the number of particles X at two different points in space separated 

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 rand X' at r' (we cons.ider distances 

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 nonequilibrium 

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. Fluctuations 

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 nonlinear 

type of chemical reaction discussed in Chapter V, longrange 

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 become 

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 communication 

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 order. 

Here the situation is especially clear. At equilibrium molecules 

behave as essentially independent entities; they· ignore 

one another. We would like to call them "hypnons," "sleep-


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 equilibrium 

(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 fluctuation 

preceding the formation of a new structure can be followed 

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 Chapter 

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 distributed 

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 autocatalytic 

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 determines 

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


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 appears. 

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.

.. 


If this number is small compared with the total number of larvae, the cluster 

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 develop: 

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 homogeneous 

initial conditions. It would seem they correspond, in a bifurcation 

4) 

"' 

\1 

1 

"- 

.,.. 

' 

I .,. 

,. 

"' 

 

" 

\ 

.. 

A 

... 

- 

(,l 

l 

• 

Figure B. Self-aggregation at low density. The times are 0 minutes and 22 

minutes.

ORDER OUT OF CH AOS 

184 

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 . 

• 

Q 

100 

• 

80 

60 

40 

20 

10 

--f--- -- - 

,--..--- - 

- 

.. .... .. 

. . . 

- 

...... .. .... 

. ..... .. 

f 

--- r 

····] 

20 30 40 

. -- .. 

,- 

,-f 

.. · ·······t 

Figure C. Percent of the total number of larvae in the central cluster in 

function of time at three different densities. 

N 

CRITICAl NUCLEUS dKnc of e10 lot- nt ... nud

185 ORDER T;-iROUGH FWCTUATIONS 

N 

CR.tTiCAL NUCLEUS !l'owlh of• :or.-- inilinlnutk .. 1•··•1 

growth of a 30 la


186 

• > 

, , .. 

" 

., 

') 

., ' 

,. , 

., .I I 

I 

.. 

" 

... ... 

., " 

., \ 

I 

... 

I 

"' 

., 

' 

" 

, • 

., 

;) 

") 

1 

\. 

'\ 

., _, 

) 

... 

... 

'\ 

I 

... 

 

I 

 

,' 

'' I • 

,J 

' 

... 

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 appears 

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 construction 

of the base, has been shown by Grasse to be the 

result of what appears to be disordered behavior among termites. 

At this stage, they transport and drop lumps of earth in 

a random fashion, but in doing so they impregnate the lumps 

with 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


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 describe 

chemical or biological processes in which differences 

are leveled out and initial conditions forgotten, it cannot explain 

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 possibly 

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 nucleation 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 nucleation 

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 systems. 

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,·· the environment of the fluctuating region, 

always tends to damp fluctuations. These will be destroyed or


188 

(a) 

(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 dissipative 

structures. 

amplified according to the effectiveness of the communication 

between the fluctuating region and the outside world. The critical 

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 replication. 

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 replication 

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 result, 

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 numerous 

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


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 

When can we begin to speak about "evolution" in its proper 

sense? As we have seen, dissipative structures require far-fromequilibrium 

conditions. Yet the reaction diffusion equations contain 

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 occurring 

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 definite 

set of transformations of these units. The definition of the 

system is thus liable to be modified by its evolution. The simplest 

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, introduced 

in small quantities, lead to a new set of reactions among 

the system's components. This new set of reactions then enters 

into competition with the system·s previous mode of functioning. 

If the system is "structurally stable" as far as this


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 itself-if, 

for example, the kinetics whereby the "innovators" 

multiply is fast enough for the latter to invade the system instead 

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 system 

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 itself. 

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 Darwinian 

idea of the "survival of the fittest." 

Such ideas form the basis for the model of prebiotic evolution 

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 populations 

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


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 between 

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 continual 

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 believe 

that this is not the only aspect involved in pre biotic evolution. 

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 moving 

toward life were associated with the formation of mechanisms 

capable of absorbing and transforming chemical energy, 

so as to push the system into "far-from-equilibrium" conditions. 

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 important 

problems in evolutionary theory is the eventual feedback 

between macroscopic structures and microscopic events: macroscopic 

structures emerging from microscopic events would 

in turn lead to a modification of the microscopic mechanisms. 

Curiously, at present, the better understood cases concern social 

situations. When we build a road or a bridge, we can predict 

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 generate 

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.


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 defined 

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 available 

to the population. The logistic equation can be written 

dN/dt = rN(K- N) - mN, where r and m are characteristic 

birth and death constants and K the "carrying capacity" of the 

environment. Whatever 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. When this value is reached, the environ- 

K - ..!Il 

r 

N 

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. 

t


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 example. 

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 fluctuations 

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 Or2, we have, as in the 

continuous case, a uniform approach to equilibrium. For values 

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 parameters 

characterizing natural populations keep them from 

the chaotic region. Why is this so? Here we have one of the 

very interesting problems created by the confluence of evolutionary 

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-


ploiting exi.;ting resources or of discovering new ones (that is, 

K 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 temporary, 

and a logistically defined niche will be occupied successively 

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 Figure 

21.) Thus the logistic equation leads to the definition of a 

very simple situation where we can give a quantitative formulation 

of the Darwinian idea of the "survival of the fittest." 

The "fittest" is the species for which at a given time the quantity 

K-mlr is the largest. 

As restricted as the problem described by the logistic equation 

is, it nonetheless leads to some marvelous examples of 

nature's inventiveness . 

Take the example of caterpillars, who must remain undetected, 

since the slowness of their movement


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 undetected 

bring to mind the words of distinguished nineteenthcentury 

naturalist Louis Agassiz: "The possibilities of existence 

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 probability 

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 possessed 

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 problem. 

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 competition 

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 concern 

the organization of internally differentiated populations. 

Clear distinctions are absolutely necessary if we are to avoid 

confusion. In populations where individuals are not interchangeable 

and where each, with its own memory, character, 

and experience, is called upon to play a singular role, the relevance 

of the logistic equation and, more generally, of any simple 

Darwinian reasoning becomes quite relative. We shall 

return to this problem. 

It is interesting to note that the type of curve represented in


196 

Figure 21 showing the succession of growths and peaks defined 

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 discovery 

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 similar 

role in the situation described by the equation. 13 Thus, to 

choose but one example, not only did the spread of the steamship 

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 different 

way, a preexisting need that remains unchanged. However, 

in ecology as in human societies, many innovations are 

successful without such a preexisting "niche." Such innovations 

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 particular, 

the creation of a "demand," and even of a "need" for 

this demand to fulfill, often appears as correlated with the production 

of the goods or techniques that satisfy the demand. 

Evolutionary Feedback 

A first step toward accounting for this dimension of the evolutionary 

process can be achieved by making the "carrying capacity" 

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.


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 spatial 

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 variables 

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 population 

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 attained 

level of activity accelerate the increase of this activity. 

But each local level of activity is also determined by competition 

with similar centers of activity located elsewhere. The 

sale of produced goods or services depends on the cost of 

transporting them to consumers and on the size of the "enterprise." 

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 feedback 

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 corresponding 

to "superior" levels in Christaller's hierarchy-that 

is, implying exportation oo a greater range. Even if the initial


198 

65 

• 

66 

• 

69 

• 

64 

• 

62 

• 

67 

• 

65 

• 

65 

• 

67 

62 

• • 

66 

• 

63 

• 

62 

• 

66 

• 

66 

• 

66 

• 

65 

• 

63 

• 

63 

• 

62 

• 

67 67 

• • 

67 

• 

65 66 

• • 

61 

• 

60 

• 

69 

• 

&9 

• 

68 68 

• • 

Figure 22. A possible history of "urbanization." • have only function 1; 

• have functions 1 and 2; 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

199 

ORDER THROUGH FLUCTUATIONS 

61 

• 

66 

• 

67 

• 

64 

• 

62 

• 

64 

• 

62 

• 

67 

• 

66 

• 

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 min center.


200 

62 

• 

65 

• 

65 

• 

62 

• 

64 

• 

61 

• 

67 

• 

65 

• 

66 

• 

66 

• 

64 

• 

66 66 

• •

201 ORDER THROUGH FLUCTUATIONS


202 

i7 

• 

71 

•

203 

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 appearance 

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 domination 

structures. 

Whereas Christaller's symmetrical distribution ignores history, 

this scenario takes it into account, at least in a very minimal 

sense, as an interplay between .. laws," in this case of a 

purely economic nature, and the .. chance" governing the sequence 

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 particular, 

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 emphasized, 

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 .. counterintuitive" 

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 buildings 

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-


204 

tion, be it of animals, activities, or habits. What is presupposed 

is that each member of a given population can be taken 

as the equivalent of any of the others. But this general equivalence 

can itself be seen not as a simple general fact but as an 

approximation, the validity of which depends on the constraints 

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, particularly 

the prey-predator interaction. In this view, the typical 

evolution for a prey population will be the increase in the reproduction 

rate r. The predator will evolve toward more effective 

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 individuals 

becoming more and more able to learn from experience 

and to store memories-that is, individuals more 

complex with a longer period of maturation and apprenticeship. 

This in turn means individuals both more "valuable" 

representing a larger biological investment-and characterized 

by a longer period of vulnerability. The development of 

"social" and "family" ties thus appears as a logical counterpart 

of the K strategy. From that point on, other factors, besides 

the mere number of individuals in the population, 

become more and more relevant and the logistic equation measuring 

the success by the number of individuals becomes misleading. 

We have here a particular 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 separated 

from the question of the nature of the object investigated. 

We cannot ask the same questions about a population of 

flies that reproduce and die by millions without apparently


learning from or enlarging their experience and about a population 

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 phenomena. 

It is well known, for example, that structural anthropology 

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 models: 

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 organization 

of society. The anthropologist makes the logic of this behavior 

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 construction 

of the termites' nest, when interactions among individuals 

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 necessarily 

implies averages and reintroduces the question of stability 

and bifurcations. 

Which events will regress, and which arc likely to affect the 

whole system? What are the situations of choice, and what are


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? Questions 

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 "mechanical"? 

This question seems parallel to questions we have already 

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 translation 

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 stability 

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 circumstances, 

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 amplification 

of a small event are a legitimate matter for rational inquiry. 

Fluctuations do not cause the transformation of a systefll 's activity. 

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


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 particularly 

unfortunate use of the concepts and methods borrowed 

from physics18-unfortunate 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 optimization. 

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 consider 

optimization as the key to understanding how populations 

and individuals survive is to risk confusing causes with 

effects. 

Optimization models thus ignore both the possibility of radical 

transformations-that is, transformations that change the 

definition of a problem and thus the kind of solution soughtand 

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 rational 

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 nonequilibrium 

instabilities of great theoretical and practical interest.


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 incorporated 

into physics, while the ambitions to omniscience inherent 

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 progressive 

reinsertion of history into the natural and social sciences. 

Curiously, at every stage of the process, a decisive feature of 

this "historicization" has been the discovery of some temporal 

heterogeneity. Since the Renaissance, We stern society has 

come into contact with different populations that were seen as 

corresponding to different stages of development; nineteenthcentury 

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 complexity 

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 phenomena 

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


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 dynamics 

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 irreversible 

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 synthesis 

of centuries of experimentation as well as of converging 

lines of theoretical research. The same is true for thermodynamics. 

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 scientific 

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 becoming, 

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. 

213


We have reached the core of the problem. "What is time?'' 

Must we accept the opposition, traditional since Kant, between 

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 worried 

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 inevitable 

resignation. I remarked that all that occurs objectively 

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 towards 

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 Chapter 

III). Like Einstein, Bergson started with a subjective time 

and then moved to time in nature, time as objectified by physics. 

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 introduced 

the distinction between physical time and duration, a 

concept referring to existential time. 

But we cannot stop here. As J. T. Fraser says, "The resulting 

dichotomy between time felt and time understood is a hallmark 

of scientific-industrial civilization, a sort of collective 

schizophrenia. "3 As we have already emphasized, where classical 

science used to emphasize permanence, we now find 

change and evolution; we no longer see in the skies the trajectories 

that filled Kant's heart with the same admiration as the 

moral law residing in him. We now see strange objects: quasars,· 

pulsars, galaxies exploding and being torn apart, stars

215 REDISCOVERING TIME 

that, we are told, collapse into "black holes" irreversibly devouring 

everything they manage to ensnare. 

Time has penetrated not only biology, geology, and the social 

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 published 

by Einstein in 1917. 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 cosmological 

equations. We owe this discovery to the Russian astrophysicist 

A. Friedmann and the Belgian G. Lemaitre. At the 

same time Hubble and his coworkers were studying the motions 

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 Friedmann 

and Lemaitre was obvious. Yet for many years physicists 

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 Einstein 

the possibility of making the initial state of the universe 

more precise and perhaps finding there the explanation of cosmic 

rays, Einstein showed no interest. 

Today there is new evidence, the famous residual blackbody 

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; Einstein 

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 physicists 

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-


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 totalitarian 

claims of a conceptualization that, in fact, applies 

only to simple and stable objects? Why worry about the incompatibility 

between dynamics and thermodynamics? 

We must not forget the words of Whitehead, words constantly 

confirmed by the history of science: a clash of doctrines 

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 century, 

several physicists suggested abandoning determinism on 

the grounds that it was inaccessible in real experience. 4 Indeed, 

as we have already emphasized, we never know the exact 

positions and velocities of the molecules in a large system; 

thus an exact prediction of the system's future evolution is impossible. 

More recently, Brillouin hoped to destroy determinism 

by appealing to the commonsense truth that accurate 

prediction requires an accurate knowledge of the initial conditions 

and that this knowledge must be paid for; the exact prediction 

necessary to make determinism work requires that an 

"infinite" price be paid. 

These objections, while reasonable, do not affect the conceptual 

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 dynamics. 

In contrast, demonstrations of "impossibility" have a fundamental 

importance. They imply the discovery of an unexpected 

intrinsic structure of reality that dooms an intellectual 

enterprise to failure. Such discoveries will exclude the possibility 

of an operation that previously could have been imagined 

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 examples 

of statements of impossibility which have led to profound 

conceptual innovations.

217 .REDISCOVERING TIME 

Thermodynamics, relativity, and quantum mechanics are all 

rooted in the discovery of impossibilities, of limits to the ambitions 

of classical physics. Thus they marked the end of an exploration 

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 expresses 

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 requirements 

of relativity theory. In connection with the propagation 

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 propagation 

of all signals. Thus this limiting velocity plays a fundamental 

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 standard, 

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 according 

to the object's mass. The atom can no longer be regarded 

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 vari-


ous behaviors become qualitatively different, they also lead to 

a new conception of objectivity. No observer can transmit signals 

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 reference 

to its author. For "nonphysical" intelligent beings capable 

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 observers, 

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 subject 

to intrinsic constraints that identify us as part of the physical 

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. However, 

it left unchanged another fundamental characteristic of 

classical physics, namely, the ambition to achieve a "complete" 

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 mathematician 

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 nature. 

In this sense, relativity remains a continuation of classical 

physics. 

Quantum mechanics, on the other hand, is the first physical 

theory truly to have broken with the past. Quantum mechanics 

not only situates us in nature, it also labels us as "heavy"


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 conceptual 

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 cannot 

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 order 

to reach experimental results valid at thermal equilibrium, 

to assume that an exchange of energy between matter and radiation 

occurred only in discrete steps involving a new universal 

constant. This universal constant "h" measures the "size" 

of each step. 

In this case, as in many others, the challenge of irreversibility 

led to decisive progress in physics. 

This discovery remained isolated until Einstein presented 

the first general interpretation of Planck's constant. He understood 

that it had far-reaching implications for the nature of 

light. He introduced a revolutionary concept: the waveparticle 

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


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 mechanical 

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 characterizing 

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 consistent 

new theory. This theory incorporates Einstein's and de 

Broglie's wave-particle duality in the framework of a new generalized 

form of dynamics: quantum mechanics. For our purposes 

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 something 

new; the basic idea of quantum mechanics is that the 

Hamiltonian as well as the other quantities of classical mechanics, 

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 abstract. 

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 derivative 

represented by d/dx and suppose it acts on a function-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 number-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 operator 

are the "eigenvalues" of the operator. 

To each operator there thus corresponds an ensemble, a "reservoir" 

of numerical values; this ensemble forms its "spectrum." 

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 of quantum mechanics may thus be expressed 

as follows: to all physical quantities in classical mechanics 

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 essential 

point is that the concept of physical quantity (represented by 

an operator) is 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 energywill 

be identified with the eigenvalues corresponding to this 

operator. 

The introduction of operators opened up to physics a micro-


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 successfully 

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 discovering 

a single conceptual scheme common to all levels of description. 

A new mathematical language invented to deal with a certain 

situation may actually open up fields of inquiry that are 

full of surprises, going far beyond the expectations of its originators. 

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 demanded 

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 operators 

into quantum mechanics, their significance remains a subject 

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 mechanics. 

Operators now come into play as soon as, for one reason 

or another, the notion of a dynamic trajectory has to be discarded, 

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 eigenvalues 

correspond precisely to the numerical values the physical 

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-


pendent in the sense that we can ascribe to a coordinate a 

numerical value quite independent of the value we have ascribed 

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 expresses 

a relation between lengths (closely related to the concept 

of coordinates) and momenta. Therefore, positions and 

momenta can no longer be independent variables, as in classical 

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 textbooks 

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 factor 

of two. 

One fundamental property results from the relation between 

operators in quantum mechanics: the two operators q0 P and 

Pop do not commute-that is, the results of q0pP0p and of 

Pop%p applied to the same function are different. This has profound 

implications, since only commuting operators admit 

common eigenfunctions. Thus we cannot identify a function 

that would be an eigenfunction of both coordinate and momentum. 

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 momentum 

p, both have a well-defined value. This situation, unknown 

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 possible 

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 relations, 

and our discussion is admittedly oversimplified. But we 

wish to give our readers some understanding of the new problem 

that re:sult:s from the u:se of operators; Heisenberg's uncertainty 

relation necessarily leads to a revision of the concept of

ORDER OUT OF CHAOS 22 4 

causality. It is possible to determine the coordinate precisely. 

But the moment we do so, the momentum will acquire an arbitrary 

value, positive or negative. In other words, in an instant 

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 altered. 

These consequences of quantum mechanics were unacceptable 

to many physicists, including Einstein; and many experiments 

were devised to demonstrate their absurdity. An 

attempt was also made to minimize the conceptual change involved. 

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 parameters 

such as coordinates and momenta; but some of them 

would be made fuzzy by measurement, and Heisenberg's uncertainty 

relation would only express the perturbation created 

by the measurement process. Classical realism thus would remain 

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 designed 

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 ascribe 

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 access to a given reality. We have to decide which measurement 

we are going to perform and which question our ex-


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 objectivity, 

since in the classical view the only "objective" description 

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 complementarity,9 

which may be considered as an extension of Heisenberg's 

uncertainty relations. We can measure coordinates or 

momenta, but not both. No single theoretical language articulating 

the variables to which a well-defined value can be attributed 

can exhaust the physical conent 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 irreducible 

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 complementarity, 

a lesson that can perhaps be transferred to other 

fields of knowledge, consists in emphasizing the wealth of reality, 

which overflows any single language, any single logical 

structure. Each language can express only part of reality. Music, 

for example, has not been exhausted by any of its realizations, 

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-


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 mechanics 

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 determines 

the temporal evolution of the system. In quantum 

mechanics the role played by the canonical equation of classical 

mechanics is taken by the Schrodinger equation, which 

expresses the time evolution of the fu nction characterizing the 

quantum state as the result of the application of the operator 

Hop on the wave function \jJ (there are, of course, other formulations, 

which we cannot describe here). The term "wave 

function" has been chosen to emphasize once again the waveparticle 

duality so fu ndamental 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 equation, 

like the canonical equation of classical physics, expresses 

a reversible and deterministic evolution. The 

reversible change of wave function corresponds to a reversible 

motion along a trajectory. If the wave fu nction at a given instant 

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 mechanics 

do not include time. Time remains a number, not an operator, 

and only operators can appear in Heisenberg's uncertainty 

re lations. 

Quantum mechanics deals with only half of the variables of 

dassical mechanics. As a result, classical determinism becomes 

inapplicable, and in quantum physics statistical consid-


erations play a central role. It is through the wave intensity ttJ2 

(the square of the amplitude) that we make contact with statistical 

considerations. 

The standard statistical interpretation of quantum mechanics 

runs as follows: consider the eigenfunctions of some operator-say, 

the energy operator H0 P -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 various 

possible corresponding eigenvalues. 

Here again we notice a fundamental departure from classical 

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 nature. 

The first time was in Boltzmann's interpretation of entropy. 

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 achieving 

a complete description. (We shall see that today it is possible 

to overcome this attitude.) Here, as before, the use of probabilities 

was unacceptable to many physicists-including Einstein-who 

wished to achieve a "complete" deterministic 

description. Just as with irreversibility, an appeal to our ignorance 

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 evidence 

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 measurement 

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.


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 superposition 

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 accompanied 

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 measurement 

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 questions 

at the level of its formulation. 

The usual approach to this problem states that quantum mechanics 

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 irreversible 

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 definition 

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 measurement 

and even, for some, our mind-determine the evolution 

of the system that breaks the law of natural, "objective" 

reversibility. 11 Others have concluded that Schrodinger's equation 

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-


ibility and irreversibility shows that the classical idealization 

that describes the dynamic world as self-contained is impossible 

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 describe 

the functioning of our measurement instruments. Schrodinger'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 devoted-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 classical 

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 solution.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 evolution 

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 happen? 

This is the question of the physical mechanisms of irreversibility 

to which we shall turn in Chapter IX. But let us first 

summarize some other features of the renewal of contemporary 

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 mechanics. 

This is especially true regarding the role and meaning 

of time. In quantum mechanics, once the wave function at time


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 expressed 

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 produced 

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 precisely, 

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 antiparticles 

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 evidence 

that antimatter does not exist in our galaxy, but the possibility 

that it exists in distant galaxies cannot be excluded. We 

can imagine a mechanism in the universe that separates particles 

and antiparticles, hides antiparticles somewhere. However, 

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 Sakharov in 1966, and today much work is being 

done along these lines.14 One essential element of the model is


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 nonequilibrium 

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 particles 

of matter, including the proton, are unstable (however, the 

lifetime of the proton would be enormous, of the order of 1 Q 3 0 

years). Mechanics, the science of motion, instead of corresponding 

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 transformations. 

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 irreversibility, 

has therefore acquired a new urgency. From the field of 

engineering, of applied chemistry, where it was first formulated, 

it has spread to the whole of physics, from elementary 

particles to cosmology. 

From the perspective of this book, the importance of quantum 

mechanics lies in its introduction of probability into microscopic 

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-


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 stochastic 

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 it is necessary to move even farther away 

from deterministic descriptions of nature and adopt a statistical, 

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 antrlropomorphlc 

element alien to the ideals of physics. Nevertheless, 

in several important cases unidirectional time 

and unidirectional causality have been invoked, but always, 

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 

disagreement with Maxwell's equations-then so 

much the worse for Maxwells equations. If it is found to 

be contradicted by observation-well, these experimentalists 

do bungle things sometimes. But if your 

theory is found to be against the second law of thermodynamics 

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 physis that 

has been more often and more actively discussed than the rela· 

233


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 conflict, 

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 recoup 

the second law without giving up the formidable structure 

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 Boltzmann. 

Boltzmann's famous equation S = k log P relates entropy 

and probability: entropy grows because probability grows. 

Let us immediately emphasize that in this perspective the second 

law would have great practical importance but would be of 

no fundamental significance. In his excellent book The Ambidextrous 

Universe, Martin Gardner writes: "Certain events go 

only one way not because they can't go the other way but because 

it is extremely unlikely that they go backward. "3 By improving 

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 second 

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 processes. 

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 measuring 

beings exist on the earth or not, and whether or not, 

assuming they do exist, they are able to measure the details 

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

235 THE CLASH OF DOCTRINES 

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 approximations, 

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 probability 

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 

time n+1 1 

D EJ 

A 

A 

!-tottery 

B 

or N-k+1 N-k-1 

Figure 23. Ehrenfest's urn model. N balls are distributed between two containers 

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. 

B


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 probable 

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 process" 

(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 characteristic 

feature is the existence of well-defined transition probabilities 

independent of the previous history of the system. 

Markov chains have a remarkable property: they can be described 

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-{ quantity," 

which has the precise properties of entropy that we discussed 

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.


Figure 25. Time evolution of the J{ quantity (defined in the text) corresponding 

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 

t 

J{ = P(k,t) log J


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 proceeds 

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 decrease 

of .H has a very simple meaning: it measures the progressive 

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 averages. 

The :JiM quantity that decreases uniformly is expressed 

in terms of probability distributions and not in terms of individual 

events. It is the probability distribution that evolves irreversibly 

(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 onewaynss 

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 reversibly. 

It also illustrates the close relation between stochastic 

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 manner: 

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 consideration 

of the antecedent probability of the prior events. 7 

It is an important point, which has led to a great deal of discussion. 

8 Probability calculus is indeed time-oriented. The prediction 

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;"eversibility 

(further reinforced by the ambiguous analogy with information 

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 system, 

he cannot know the instantaneous state that simultaneously 

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 uniform 

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 irreversiblity 

something that links nature to the observer, the scientist 

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 universal 

as our ignorance. There should exist only irreversible 

processes. This is the stumbling block for all universal interpretations 

of entropy that concentrate on our ignorance of initial 

(or boundary) conditions. Irreversibility is not a universal 

propercy. In order to link dynamics and thermodynamics, a


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 1872, about 

thirty years before the discovery of Markov chains. His ambition 

was to derive a "mechanical" interpretation of entropy. In 

other words, while in Markov chains the transition probabilities 

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 attracted 

by the idea of evolution, and his ambition was to become 

the "Darwin" of the evolution of matter. 

The first step toward the mechanistic interpretation of entropy 

was to reintroduce the concept of "collisions" of molecules 

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 frequency. 

We may also classify collisions-for example, distinguish 

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?


Maxwell demonstrated that this particular state, which is 

the thermodynamic equilibrium state, occurs when the velocity 

distribution becomes the well-known "bell-shaped curve," 

the "gaussian," which Quetelet, the founder of "social physics," 

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 associated 

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 example, 

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 evolution-natural 

selection-cannot be defined for one individual 

but only for a large population. It is therefore a statistical concept. 

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-


nection with chemical reactions) and to calculate the average 

number of collisions creating or destroying a molecule corresponding 

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 frequency 

of such events is evaluated as being proportional to the 

product of the number of molecules taking part in these processes. 

(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 velocity 

distribution becomes the equilibrium Maxwellian distribution. 

In recent years there have been numerous numerical verifications 

of the uniform decrease of J{ with time. All of them 

confirm Boltzmann's prediction. Even today, his kinetic equation 

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 experimental 

data. 

However, it is from the conceptual standpoint that Boltzmann's 

achievement is greatest: the distinction between reversible 

and irreversible phenomena, which, as we have seen, 

underlies the second law, is now transposed onto the microscopic 

level. The change of the velocity distribution due to 

free motion corresponds to the reversible part, while the contribution 

due to collisions corresponds to the irreversible part. 

For Boltzmann this was the key to the microscopic interpretation 

of entropy. A principle of molecular evolution had been 

produced! It is easy to understand the fascination this discovery 

exerted on the physicists who followed Boltzmann, including 

Planck, Einstein, and Schrodinger. t2 

Boltzmann's breakthrough was a decisive step in the direc-


tion of the physics of processes. What determines temporal 

evolution in Boltzmann's equation is no longer the Hamiltonian, 

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 conclude 

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 1872, 

objections were raised. Had Boltzmann really "deduced" irreversibility 

from dynamics? How could the reversible laws of 

trajectories lead to irreversible evolution? Is Boltzmann's kinetic 

equation in any way compatible with dynamics? It is 

easy to see that the symmetry present in Boltzmann's equation 

is in contradiction with the symmetry of classical mechanics. 

We have already seen that velocity inversion (v -v) produces 

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 describes 

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 velocity 

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 Boltzmann'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 dynamics. 

Boltzmann cannot, therefore, have "deduced" entropy 

from dynamics. He must have introduced something 

new, something foreign to dynamics. Thus his results ;an rep-


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 (1876) that this model can 

no longer be valid after a reversal of the velocities corresponding 

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 calculated 

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=100, (b) to N=484, (c) to N=1225.

.. 


with disks on lattice sites with an isotropic velocity distribution. 

The results follow Boltzmann's predictions. 

If, after fifty or a hundred collisions, corresponding to about 

I0-6 sec in a dilute gas, the velocities are inverted, a new ensemble 

is obtained.15 Now, after the velocity inversion, Boltzmann's 

J{ quantity increases instead of decreasing. 

I 

• 

• 

• 

- 

• 

. . • 

- 

. - 

• 

N:IOO 

• 

• 

• 

... 

- 

• u 

• 

• 

•• • 

• 0 

• . 

'\ 

on 

• 

• • 

• 

y 

 

0 

-. . . .... 

.. . - . 

. 

· 

·: : :" ... .. 

. ... . . .. . 2 '"· 

_,. • 

. 

••• t • 

, .. . 

... • 

. ... 

. 

.. 

- 

t 

' - 

. 

-'"' 

.... , ·· ... . 

.... ., 

·11-------::----------------- 

equit . 

... 

: 

0 20 TIME 60 

Figure 27. Evolution of :H when velocities are inverted after 50 or 100 

collisions. Simulation with 100 "hard spheres." 

A similar situation can be produced in spin echo experiments 

or plasma echo experiments. There also, over limited 

periods of time, an "antithermodynamic" behavior in Boltzmann's 

sense may be observed. 

But it is important to note that the velocity inversion experiment 

becomes increasingly more difficult when the time interval 

t0 after which the inversion occurs is increased. 

To be able to retrace its past, the gas must remember everything 

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


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 reasoning 

applies and those to which it does not. 

Once the problem was stated (1894), it was easy to identify 

the nature of this limitation.l6, 17 The validity of Boltzmann's 

statistical procedure depends on the assumption that before 

they collide, the molecules behave independently of one another. 

This constitutes an assumption about the initial conditions, 

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 predeterminable 

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 apparently 

purposeful behavior. 

But there is more. What does the transition from order to 

disorder signify? In the Ehrenfest urn experiment, it is clearthe 

system will evolve till uniformity is reached. But other situations 

are not so clear; we may do computer experiments 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 thermodynamic 

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 pioneering 

work remained incomplete.


Dynarnics and Thermodynamics: 

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. 18 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 specified 

by position (a vector with three components) and by momentum 

(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 dynamic 

states ompatible with the information we have concerning 

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 representative 

points in the phase space. We shall call p (q1 

• • • q 3 0, 

p1 • • • p30) this density in phase space where q1 ,q2 • • • q3n are 

the coordinates of the n points; similarly, p1 ,p2 • • • p30 are the 

momenta (each point has three coordinates and three mo-


menta). This density measures the probability of finding a dynamic 

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 trajectory 

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. Indeed, 

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 starting 

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 function 

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 possibility 

of consistently predicting a single well-defined trajectory. 

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. However, 

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

29 


meeting two particles a certain distance apart. In particular, 

correlations between particles, which we discussed in the preceding 

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 velocity 

distribution function. But this is not the case. The Hamiltonian 

equations discussed in Chapter 11 allow us to obtain 

an exact evolution equation for p without any further approximations. 

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 constant 

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 combination 

of the statistical point of view (the study of the "population" 

described by p) and the laws of dynamics. It also permits 

a more accurate representation of the thermodynamic equilibrium 

state. Thus, in the case of an isolated system, the ensemp 

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. 

q


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 distribution 

function p becomes uniform over the microcanonical surface. 

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 us any closer to the 

solution of the problem of irreversibility? Boltzmann's theory 

describes thermodynamic entropy in terms of the velocity distribution 

function f. He achieved this result through the introduction 

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 ensemble 

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 conclusion is immediate: :Jf0 is a constant 

and thus cannot represent entropy. With respect to Boltzmann, 

this appears as a step backward rather than forward! 

Though it is negative, Gibbs' conclusion remains very important. 

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 velocities, 

the result of collisions is randomization; therefore we 

can describe this process as a transition from order to disor-


der, but the appearance of correlations as the result of collision 

points in the opposite direction, toward a transition from disorder 

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 deduce 

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 results? 

Is any theory of irreversible processes in absolute conflict 

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 equations 

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 Newton'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. Irreversibility, 

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 important 

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.


Why have so many scientists accepted so readily the subjective 

interpretation of irreversibility? Perhaps part of its attraction 

lies in the fact that, as we have seen, the irreversible 

increase of entropy was at first associated with imperfect manipulation, 

with our lack of control over operations that are 

ideally reversible. 

But this interpretation becomes absurd as soon as the irrelevant 

associations with technological problems are set aside. We 

must remember the context that gave the second law its significance 

as nature's arrow of time. According to the subjective 

interpretation, chemical affinity, heat conduction, viscosity, 

all the properties connected with irreversible entropy production 

would depend on the observer. 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-living 

creatures capable of observing 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 thermodynamics 

to mere approximations due to our imperfect knowledge 

seem wrong headed when the constructive role of entropy 

is understood and the possibility of an amplification of fluctuations 

is discovered. Conversely, it is difficult to reject dynamics 

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 processes, 

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 materialism 

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 


standing it as being capable of producing man and his societies. 

Moreover, at the time Engels wrote his Dialectics of Nature, 

the physical sciences seemed to have rejected the mechanistic 

world view and drawn closer to the 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 view of 

these great discoveries, Engels came to the conclusion that the 

mechanistic world view was dead. 

But mechanicism remained a basic difficulty facing dialectical 

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 interpretation 

of irreversibility and to point out its weakness, it is 

not so easy to go beyond it and formulate an "objective" theory 

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 increase 

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 toward 

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 Boltzmann's 

attitude. He abandoned his attempt to prove that an objective 

arrow of time exists and introduced instead an idea that,


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 future. 

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 universe, 

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 thermal 

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 definition 

he will find that his own small region, isolated from 

the rest of the universe, is "initially" always in an improbable 

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 individual 

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 diagram 

proposed by Karl Popper (Figure 29). The arrow of time 

would be as arbitrary as the vertical direction determined by 

the gravitational field.


Arruw of tiM 1Jf c.hia 

•trett"h o£ tiM only 

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 cosmological 

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 illusion. 

Thus it makes our world an illusion, and with it all 

our attempts to find out more about our world. It is therefore 

self-defeating (like every idealism). Boltzmann's idealistic 

ad hoc hypothesis clashes with his own realistic 

and almost passionately maintained anti-idealistic philosophy, 

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 revolution 

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 microscopic 

theory of irreversible processes will require additional, 

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 following 

about destiny: 

. . . everything is governed by immutable laws • . . everything 

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 necessary 

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 publish 

it in spite of you.1 

257


However convincing they may sound, such a priori arguments 

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 astonished 

to discover the qualitative diversity presented by nature. 

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 irreversibility, 

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 manifestation. 

The microscopic theory has to account for two closely 

related elements. First of all, we must follow Boltzmann in 

attempting to construct a microscopic model for entropy 

(Boltzmann's .J{ fu nction) that changes uniformly in time. This 

change has to define our arrow of time. The increase of entropy 

for isolated systems has to express the aging of the system. 

Often we may have an arrow of time without being able to 

associate entropy with the type of processes considered. Popper 

gives a simple example of a system presenting a unidirectional 

process and therefore an arrow of time. 

Suppose a film is taken of a large surface of water initially 

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 number 

of distant coherent generators of waves the coordination 

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 reverse 

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 generate 

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 converge 

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 cosmological 

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 universe 

we find both reversible and irreversible processes. Similarly, 

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 

usual (Hamiltonian) formulation of dynamics. No entropy 

fu nction can be defined as a result of the T-violation. 

We are reminded of the celebrated discussion between Einstein 

and Ritz published in 1909.5 This is a quite unusual paper, 

a very short one, less than a printed page long. It simply is 

a statement of disagreement. Einstein argued that irreversibility 

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 advanced 

waves does not necessarily lead to a formulation of the 

second law. We need both types of arguments.


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 describing 

reaction-diffusion systems, left and right play the same 

role (the diffusion equations remain invariant when we perform 

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 symmetry-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 understanding 

temporal symmetry-breaking, in which we are primarily 

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 inversion 

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 solution 

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 realized 

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 propagated 

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 solutions 

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 us start with classical mechanics. As we have already 

mentioned, if trajectory is to be the basic irreducible element, 

the world would be as reversible as the trajectories out of 

which it is formed. In this description there would be no entropy 

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, introduced 

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 fa cto 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 rotate, 

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 reason 

for this is very simple: rotation requires more energy than 

oscillation (see Figure 30).


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 


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 unstable 

way. 6 Let us indicate one kind of trajectory (for example, 

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 region 

A to a smaller region within it, for the uncertainty remains. 

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.


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 conserved 

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 "imprisoned" 

in a very small "fraction" of the constant-energy (see 

Figure 32) surface formed by the intersection of all these invar 

-iant surfaces. 

p 

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. 

q


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 conserved 

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 "imprisoned" 

in a very small "fraction" of the constant-energy (see 

Figure 32) surface formed by the intersection of all these invariant 

. 

surfaces. 

p 

..--- ... 

- ... 

1),-/ 

. , , 

 

," , ---""'04\ 

" 

, I \ \ 

I , 

I , 

, \ 

I 

\ 

\ , 'I 

, rs-'" , 

I 

\ \ , " 

/ 

, / 

, 

,/ 

" 

'" 

..... _..-.,-. 

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. 

q


To avoid these difficulties, Maxwell and Boltzmann introduced 

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 chaotic 

(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 


Suppose that our knowledge of initial conditions permits us 

to localize a system in a small cell of the phase space. During 

its evolution, 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. Indeed, 

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 information 

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 network 

of filaments through phase space. 

p 

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. 

q


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 by 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 becomes 

p. The transformation is cyclic: after six times the initial cell is recovered. 

The square is deformed, but after six transformations we return 

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 systems-the 

first mathematical, the second of obvious physical

11 


relevance. The first system consists of a transformation that, 

for obvious reasons, mathematicians call the "baker transformation. 

"9, to We take a square and flatten it into a rectangle, 

then we fold half of the rectangle over the other half to form a 

square again. This set of operations is shown in Figure 36 and 

may be repeated as many times as one likes. 

q:1 

n 

 

1 

q=1 

2 p:1 

B 

q=1 

) 

q:11J 

p:1 

112 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 redistributed. 

The square corresponds here to the phase space. 

The baker transformation transforms each point into a welldefined 

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 system 

described by an initial condition such that a region A of 

the square is initially filled in a uniform way with representative 

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


p 

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. 

q 

always contains different trajectories diverging at each fragmentation. 

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 finding 

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). Arbitrarily 

small differences in initial conditions are amplified. As a 

result we can no longer perform the transition from ensembles


 

Q 

I 

I 

I 

I 

I 

'o 

I I 

I I 

I 1 

1 I 

1 I 

I 

' ----- 

- -o---- --:.. 

... 

.... .. ," 

... .. 

... .. 

... ... 

... " 

... 

" 

" 

, 

" 

/ 

" 

" 

• 

 

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. Statistical 

concepts are no longer merely an approximation with 

respect to some "objective truth." When faced with these unstable 

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 mathematician 

there is no place for probabilities. However, Poincare 

himself mapped the path leading to the answer to this problem.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 probability 

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,


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 knowledge, 

then he could get rid of all randomness. 

In conclusion, there is a close relationship between instability 

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 unshaded 

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 vertical 

partitions going into the past. These are the basic partitions. 

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 perform 

to go from the generating partition to the one under consideration.I2 

We therefore see that this type of system admits a 

kind of internal age.* 

The internal time Tis quite different from the usual mechanical 

time, since it depends on the global topology of the system. 

We may even speak of the "timing of space," thus coming 

close to ideas recently put forward by geographers, who have introduced 

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 contrary, 

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 welldefined 

time but only an "average" time corresponding to the superposition 

of the basic partitions out of which it is formed.


-m- 

- 

-=- 

past 

•1 0 z 

t 

genart1ting partition 

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 corresponding 

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 mathematical 

rigor, 14 but the result seems to us quite natural. As 

time passes, the distribution of ink reaches equilibrium, exactly 

like the distribution of balls in the urn experiment discussed 

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 shrinking 

vertical sections and, again, sufficiently far in the past we 

shall find a uniform distribution of ink. We may therefore conclude 

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 commen t it. Internal time provides us with a new 'nonlocal' 

description. 

When we know the 'age' of the system, (that is, the corresponding 

partition), we can still not associate to it a well-defined 

local trajectory.


We know only that the system is in a shaded region (Figure 

39). Similarly, if we know some precise initial conditions corresponding 

to a point in the system, we don't know the partition 

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 mechanics. 

It is because of the existence of this new alternative, nonlocal 

description, that we can make the transition from dynamics 

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 degenerate 

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 transition 

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 interpretations 

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 result 

of an alternative description of dynamics, a non-local description 

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 connection 

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 distribution 

concentrated on a line (instead of being distributed on a 

surface). This line may be vertical or horizontal. Let us look at


what will happen to this line when we apply the baker transformation 

going to the future. The result is represented in Figure 

40. The vertical 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 understand, 

the vertical 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 two realizations 

of dynamics, each involving symmetry-breaking and appearing 

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 random 

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-


pie physical interpretation. A contracting fiber corresponds to 

a collection of hard spheres whose velocities are randomly distributed 

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 collisions. 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 second 

law becomes a selection principle of initial conditions. 

Only initial conditions that go to equilibrium in the fu ture are 

retained. 

Obviously the validity of this selection principle is maintained 

by dynamics. It can easily be seen in the example of the 

baker transformation that the contracting fiber remains a contracting 

fiber for all times, and likewise for a dilating fiber. By 

suppressing one of the two Markov chains we go from an 

intrinsically random system to an intrinsically irreversible system. 

We find three basic elements in the description of irreversibility: 

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 increases; 

see Chapter VIII). There is, however , no contradiction; 

when we go from the dynamic description of the baker 

transformation to the thermodynamic description, we have to


modify our distribution function; the "objects" in terms of 

which entropy increases are different fro m the ones considered 

in dynamics. The new distribution function, p, corresponds 

to an intrinsically time-oriented description of the 

dynamic system. In this book we cannot dwell on the mathematical 

aspects of this transformation. Let us only emphasize 

that it has to be noncanonical (see Chapter II). We must abandon 

the usual formulations of dynamics to reach the thermodynamic 

description. 

It is quite remarkable that such a transformation exists and 

that as a result we can now unify dynamics and thermodynamics, 

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 emphasize 

only that at equilibrium, whenever entropy reaches its maximum, 

these objects must behave randomly. 

It also seems quite remarkable that irreversibility emerges, so 

to speak, from instability, which introduces irreducible statistical 

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 ar row 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 cannot 

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' describes 

the torment of a narrator who finds himself as unable 

to switch from one spatial direction to the other as we are to


"twirl time." In the fifth volume of Science and Civilization in 

China, Needham describes the dream of the Chinese alchemists: 

their supreme aim was not to achieve the transmutation 

of metals into gold but to manipulate time, to reach immortality 

through a radical slowdown of natural decaying processes. 

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 conditions 

fr om 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 progress 

can bring us closer to the velocity of light, but in the present 

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 information 

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 probability 

Peqm(k) at equilibrium as well as a non-equilibrium probability 

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 disappears. 

Therefore, to compare the Baker transformation with 

Markov chains, we have to make more precise the corresponding 

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 partitions 

between timet; and t=2. If we consider now Figure 39. 

we see that when ti is receding towards the past, the boxes will


become steadily thinner as we have to introduce more and 

more vertical subdivisions. This is expressed in Figure 41, sequence 

B, where, going from top to bottom, we have ti = 1, 0, 

- 1, and finally ti = - 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-equilibrium 

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 (sequence 

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 fo rmula 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.


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 excludes 

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 therefore 

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 associate 

an information content to each initial condition, and this 

information content itself depends on the dynamics of the system 

(as in the baker transformation we used the successive 

fragmentation of the cells to calculate the information content). 

Initial conditions and dynamics are no longer independent. 

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 evolution 

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 described 

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 Figure 

42. In the far distant past, there were no correlations between 

particles. Now, scattering has two effects, already 

mentioned in Chapter VIII. It disperses the particles (it makes 

the velocity distribution more symmetrical) and, in addition, it


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 follows: 

In the direct process, it makes the velocity distribution 

more symmetrical and creates correlations; in the inverse process, 

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 velocity. 

After the collision, the velocities are no more identical, and the scattered 

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, uncorrelated 

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.


particles enter, the correlations are destroyed through collisions, 

and uncorrelated particles re sult (see Figure 45). 

The two situations diffe r in the temporal order of collisions 

and correlations. In the first case , we have "postcollisional" 

correlations. With this distinction between pre- and postcollisional 

correlations in mind, let us re turn to the ve locity inversion 

experiment. We start at t = 0, with an initial state 

corresponding to no correlations between particles. During 

the time ot0 we have a "normal" evolution. Collisions bring 

the ve locity distribution closer to the Maxwellian equilibrium 

distribution. They also create postcollisional correlations be- 

0 0 

0 0 

before 

after 

Figure 44. Creation of postcollisional correlations represented by wavy 

lines; for details see text. 

tween the particles. At t0 after the velocity inversion, a completely 

new situation arises. Postcollisional correlations are 

now transformed into precollisional correlations. In the time 

interval between t0 and 2t0, these precollisional correlations 

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 stage s. During the 

0 0 

0 

 

0 

I before 

after 

Figure 45. Destruction of precollisional correlations (wavy lines) through 

collisions.

283 


first, collisions are transformed into correlations; in the second, 

correlations turn back into collisions. Both types of processes 

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 decrease. 

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 "compensated" 

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 "thermodynamic 

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 Boltzmann's, 

the effect of correlations is retained in the new definition 

of :H. Therefore at the velocity inversion point t0 the J{ 

quantity will jump, since we abruptly create abnormal precollisional 

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 increases). 

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{ quantity 

has two different values, one for the system before velocity 

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.


284 

Suppose we wait a sufficient time before making the velocity 

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 persistent 

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 

t 0 2t 0 

Figure 46. Time variation of the .J l -function in the vlocity inversion experiment: 

at time t0, the velocities are inversed and J-l presents a discontinuity. 

At time 2t0 the system is in the same state as at ti_me 0, and .J l 

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 collisions 

and correlations. From the point of view of dynamics, collisions 

and correlations play equivalent roles. Collisions give 

rise to correlations, and correlations may destroy the effect of 

I


collisions. But there is an essential difference. We can control 

collisions and produce correlations, but we cannot control correlations 

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 thermodynamics. 

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 irreversible 

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 positive 

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 irreversibility 

only makes apparent the time-oriented polarized 

nature of the universe in which we live. 

As we have emphasized repeatedly, there exist in nature systems 

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 either 

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 generally 

accepted that the difference between particles and antiparticles 

could arise only in a nonequilibrium world. This may 

be extended to many other situations. It is likely that irreversibility 

also played a role in the appearance of chiral symmetry


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 microscopic 

irreversibility we have been concentrating on classical 

dynamics. However, the ideas of correlations and the distinction 

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 quantum 

mechanics. Even small classical systems, such as those 

formed by a few hard spheres, may present intrinsic irreversibility. 

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 entities, 

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 maximum 

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 correlations 

and coherence will appear. 

We come to one of our main conclusions; At all levels, be it


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 already 

mentioned, the concept of order (or disorder) is more 

complex than was thought. It is only in some limiting situations, 

such as with dilute gases, that it acquires a simple meaning 

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 interpretation 

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 be as complex as 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 min subjects in present-day physics is the problem 

of elementary particles. However, we know that elementary 

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 interaction. 

Even if we knew how to do that, it seems too radical a


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 hypothesis; 

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 "hypnons" 

could have been appropriate to the extreme conditions 

of temperature and density that characterized the early universe. 

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 thermodynamics 

opens new perspectives. 

In any case, it is striking how the situation has changed since 

the formulation of the second law of thermodynamics one hundred 

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, thermodynamic 

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 irreversibility) 

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 microscopic 

meaning of irreversibility. 

However, this requires basic changes in the way in which we


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 ndomness' 

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 as 

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 descriptions. 

Either we give ourselves a point in phase space, and 

then we don't know to which partition it belongs, and therefore 

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 description 

in terms of the distribution function p to a new one, p 

where the distribution p has an intrinsic arrow of time, satisfying 

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 betwen 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 de localization due to irreversibility. 

As emphasized in Chapter VII, the two great revolutions in


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 impossibility 

of measuring simultaneously coordinates and momentum. 

It is not astonishing that the second principle, which as well 

limits our ability to manipulate matter, also leads to deep 

changes in the 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 processes 

is at present well established. In contrast , the basic microscopic 

theory of irreversible processes is quite new. At the 

time of correcting the proofs of this 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 

In 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 questions 

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 becoming, 

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? Seventeenthcentury 

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


the core of nature's transformations and thus had exorcised 

time and becoming. This discovery of an order in nature produced 

the fe eling 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 phenomenon 

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 symmetry, 

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 symmetry 

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 Heisenberg 

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 

roof with its patina, the wood carvings in the church, constitute 

the whole castle. None of this should be changed by 

the fact that Hamlet lived here, and yet it is changed completely. 

Suddenly the walls and the ramparts speak a different 

language . ... Yet all we really know about Hamlet 

is that his name appears in a thirteenth-century chronicle 

. ... 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 Kronberg} 

The question of the meaning of reality was the central subject 

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 present 

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 participate. 

As it is concisely expressed by D. S. Kothari, "The simple 

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.


294 

Nearer to our own time, Einstein appears as the incarnation 

of this drive toward a formulation of physics in which no reference 

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 defend 

the cause of the multiplicity of coexisting "lived" times 

against Einstein. Einstein's reply was absolute: he categorically 

rejected "philosophers' time." Lived experience cannot 

save what has been denied by science. 

Einstein's reaction was somewhat justified. Bergson had obviously 

misunderstood Einstein's theory of relativity. However, 

there also was some prejudice on Einstein's part: duree, 

Bergson's "lived time," refers to the basic dimensions of becoming, 

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 outside 

the scope of physics. 

It is fascinating to follow the correspondence between Einstein 

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 philosophy, 

literature, and the problems that surround the core of 

human existence. Untiringly he kept asking the same questions: 

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 physicists 

the distinction between past, present and future is an illusion, 

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,

295 

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 science 

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 coordinate 

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 universe. 

What Einstein wanted to preserve above all was the objective 

meaning of this communication. However, we can 

perhaps state that Einstein stopped short of accepting that 

communication and irreversibility are closely related. Communication 

is at the base of what probably is the· most irreversible 

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 principle: 

to each initial condition there corresponds an "information." 

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 velocity 

of light as the maximum velocity of transmission of signals. 

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 causality. 

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 communication. 

We have already mentioned that irreversibility and 

communication are closely related. Norbert Wiener has argued 

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:


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 experience, 

and would have served to us as the natural explanation 

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 natural 

laws-by which that square wo_Id cease to exist. 

Our counterpart would have exactly similar ideas concerning 

us. Within any world with which we can communicate, 

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 relative 

simultaneity on the impossibility of transmitting information 

at a velocity greater than that of light. The whole theory of 

relativity is built around the exclusion of "unobservable" simultaneities. 

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 between 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 impossiblethat 

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 skepticism, 

when he grounded quantum mechanics on the ex.clusion

297 


of what the quantum uncertainty principle defines as unobservable. 

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 conclusion 

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 Boltzmann's 

interpretation of entropy, the basis of our interpretation 

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" expressed 

in terms of dynamical units remains constant. It is 

therefore of great importance that the existence of an evolutionary 

paradigm can now be established in physics-not only


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 irreversible 

processes shows that this requirement is satisfied for 

most systems of interest. Remarkably, the perception of oriented 

time increases as the level of biological organization increases 

and probably reaches its culminating point in human 

consciousness. 

How general is this evolutionary paradigm? It includes isolated 

systems that evolve to disorder and open systems that 

evolve to higher and higher forms of complexity. It is not surprising 

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 formulated 

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 participation 

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 appeared 

abruptly in a time-reversible world. The origin of irreversibility 

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 estranged science from philosophy. What had 

originally been a daring wager with the dominant Aristotelian

299 


tradition gradually became a dogmatic assertion directed 

against all those (chemists, biologists, physicians) for whom a 

qualitative diversity existed in nature. At the end of the nineteenth 

century this conflict had shifted from inside science to 

the relation between "science" and the rest of culture, especially 

philosophy. We have described in Chapter III this aspect 

of the history of Western thought, with its continual 

struggle to achieve a new unity of knowledge. The "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. Physicists have no privilege whatsoever to 

any kind of extraterritoriality. 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 recognized 

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 observer, 

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 participatory 

that we have explored through this book. 

In his Themesll Merleau-Ponty also asserted that the "philosophic" 

discoveries of science, its basic conceptual transformations, 

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-


300 

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 

dynamics 

irreversibility +- randomness +- 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 discussed 

them in Chapter IX. Once we have intrinsic irreversibility 

and entropy, we come in far-from-equilibrium systems 

to dissipative structures, and we can understand the timeoriented 

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 irreversibility 

that we can recognize reversible motion. Our diagram 

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 schemededucible 

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. 

I

301 FROM EARTH TO HEAVEN-THE REENCHANTMENT OF NATURE 

We have already emphasized that irreversibility is not a universal 

phenomenon. We can perform experiments corresponding 

to thermodynamic equilibrium in limited portions of space. 

Moreover, the importance 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 irreversibility 

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. Molecular 

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. Similarly, 

the arrow of time plays quite different roles in the universe 

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 science 

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 interrogation 

of nature, in the etymological sense that the poet is a 

"maker"-active, manipulating, and exploring. Moreover, science 

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 appreciate 

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


explicit inside physics the aspects of time he thought science 

was neglecting. 

Exploring the implications and the coherence of those fundamental 

concepts, which appear both scientific and philosophical, 

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 Lucretius. 

Leibniz introduced the strange concept of monads, noncommunicating 

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 noninteracting 

entities. These entities translate their own initial 

state throughout their motion, but at the same time, like monads, 

they coexist with all the others in a "preestablished" harmony: 

in this representation, the state of each entity, although 

perfectly self-determined, reflects the state of the whole system 

down to the smallest detail. 

All integrable systems thus can be viewed as "monadic" systems. 

Conversely, Leibnizian monadology can be translated into 

dynamic language: the universe is an integrable system.13 

Monadology thus becomes the most consequential formulation 

of a universe from which all becoming is eliminated. By 

considering Leibniz's efforts to understand the activity of matter, 

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 passage: 

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-

303 FROM EARTH TO HEAVEN- THE REENCHANTMENT OF NATURE 

perature. The consequence is that the energy of the universe 

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 concentrative. 

The dissipation of energy by the regular laws 

of nature is by these very laws accompanied by circumstances 

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 actual 

condition of the whole universe at the present time. J4 

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 spectrum. 

For him, being is inseparable from becoming. Whitehead 

wrote: "The elucidation of the meaning of the sentence 'everything 

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 conception 

of the natural genesis of the disturbance that causes 

things to be born. If the vertical fall were not disturbed "without 

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 repetitive 

connection between equivalent causes and effects governed 

by the laws of fate (foedera fati).


Denique si semper motus conectitur omnls 

et uetere exoritur [semper] novus ordine certo 

nee declinando faciunt 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 uniformly 

reversible trajectories existed, where would the irreversible 

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 beginning 

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 concatenation 

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 underlying 

a physical description and a philosophic, ethical, or religious 

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 Archime 

des to Clausius was opposed to the science of turbulence 

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 sciences 

are based on breaking this pact. Man is a stranger 

to the world, to the dawn, to the sky, to things. He hates

305 


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 instantaneous 

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 immut a ble heaven. In Aristotle's 

opinion, it was only the heavenly world to which we could 

hope to apply an exact mathematical description. In the Introduction, 

we echoed the complaint that science has "disenchanted" 

the world. But this disenchantment is paradoxically 

due to the glorification of the earthly 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, inferior 

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 boundaries 

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 hardness 

of jasper; let those who think the earth would be more 

beautiful after being changed into a crystal ball be transformed 

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 pendulum 

with its conservative motion, simple machines, planetary 

orbits, etc.-were found to correspond to a unique mathemati-


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 axis. 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 discovering 

the primacy of time and change, from the level of elementary 

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 themselves 

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 absolute 

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 description 

of the changes going on in the sciences today: 

Science has become involved in this adventure, our adventure, 

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, fixed for an instant in 

their ephemeral permanence. That is why the great dis-

307 


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 discovered 

without Einstein. Science is supposed to take a deterministic 

course, in contrast with the unpredictability involved 

in the history of the arts. When 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. Meteorites 

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 context 

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 scientific 

development, namely, the tendency to increasing specialization 

and compartmentalization; the identification of 

"normal" scientific behavior with.that of the "serious," "silent" 

researcher who wastes no time on "general" questions 

about the overall significance of his research but sticks to specialized 

problems; and the essential independence of scientific 

development from cultural, economic, and social problems.


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 research 

is based. It is in this way that they are given the criteria 

that define a problem as interesting and a solution as acceptable. 

The transition from student to researcher takes place 

gradually; the scientist continues to solve problems using similar 

techniques. 

Even in our time, for which Kuhn's description has the 

greatest relevance, it refers to only one specific aspect of scientific 

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 questioned. 

Instead of working in unison, the members of the community 

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, cultural 

exper.iences, and philosophic convictions are now expressed 

and often play a decisive role in the discovery of a new 

paradigm. The emergence of the new paradigm further increases 

the vehemence of the debate. The rival paradigms are 

put to the test until the academic world determines the victor. 

With 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. When 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 scientists 

have to deal with crises imposed upon them against 

their will. 

The questions we have investigated have led us to emphasize

309 


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 science 

is also characterized by a series of problems that are the 

consequences of deliberate and lucid questions asked by scientists 

who knew that the questions had both scientific and 

philosophical aspects. Thus scientists are not doomed to behave 

like "hypnons"! 

It is important to point out that the new scientific development 

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 concepts. 

In the preface of this book, we have emphasized that its 

French title (La nouvelle alliance) expresses the coming together 

of the "two cultures". Perhaps the confluence is nowhere 

as clear as in the problem of the microscopic 

foundations of irreversibility we have studied in Book Three. 

As mentioned repeatedly, both classical and quantum me-


chanics are based on arbitrary initial conditions and deterministic 

laws (for trajectories or wave functions). In a sense, 

laws made simply explicit what was already present in the initial 

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 century 

were precisely devoted to this problem. We have in mind 

Whitehead'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 western 

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 Whitehead 

and Heidegger. 

It would go beyond the aim of this book to develop this problem 

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 temporal 

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 belonging 

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.


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 phenomena 

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 fellowmen 

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 external 

universe appeared to be an automaton following deterministic 

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 physical 

universe. In each case you would be able to identify a 

cause and an effect. Whenever a stochastic description becomes 

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 question. 

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 impat of experimental


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 existence. 

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 in sights 

about the world around us and the world inside us is a satisfying 

feature of the recent evolution in science that we have tried 

to describe. 

It is hard to avoid the impression that the distinction between 

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 profound 

change in the scientific concept of nature and of the 

structure of human society as a result of the demographic explosion. 

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 ethical 

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


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 fluctuations 

may grow and change the overall structure. As a result, 

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 uncertain world that inspires no 

blind confidence, but perhaps only the same feeling of 

qualified hope that some Talmudic texts appear to have attributed 

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. 331. 

7. J. P. VERNANT et al., Divination et rationalite, esp. J. BOTTERO, 

"Symptomes, signes, ecritures" (Paris: Seuil, 1974). 

8. A. KoYRE , Galileo Studies (Hassocks, Eng.: The Harvester 

Press, 1978). 

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 Mathematicians 

to an Hostile Intellectual Environment," Historical Studies 

in Physical Sciences, Vol. 3 (1971), pp. 1-1 15. 

11. J. NEEDHAM and C. A. RoNAN, A Shorter Science and Civilization 

in China, Vol. I (Cambridge: Cambridge University Press, 

1978), p. 170. 

12. A. EDDINGTON, The Nature of the Physical World (Ann Arbor: 

University of Michigan Press, 1958), pp. 68-80. 

13. Ibid., p. 103. 

14. BERLIN, op. cit., p. 109. 

15. K. POPPER, Unended Quest (La Salle, Ill.: Open Court Publishing 

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, 1972). 

315


17. P. VA LRY, Cahiers, (2 vols.) ed. Mrs. Robinson-Valery, (Paris: 

Gallimard, 1973-74). 

18. E. ScHRODINGER, "A re there Quantum Jumps?" The British 

Journal fo r the Philosophy of Science, Vol. III (1952), pp. 109- 

10; this text has been quoted with indignation by P. W. Bridgmann 

in his contribution to Determinism and Freedom in the 

Age of Modern Science, ed. S. Hook (New York: New York 

University Press, 1958). 

19 .. A. EINSTEIN, "Prinzipien der Forschung, Rede zur 60. 

Geburstag van Max Planck" (1918) in Mein Weltbild, Ullstein 

Verlag 1977, pp. 107-10, trans. Ideas and Opinions (New York: 

Crown, 1954), pp. 224-27. 

20. R DDRRENMATT, The Physicists. (New York: Grove, 1964). 

21. S. MoscoviCI, Essai sur /'histoire humaine de Ia nature, Collection 

Champs (Paris: Flammarion, 1977). 

22. Quoted in Ronan, op. cit., p. 87. 

23. MoNon , op. cit., p. 180. 

Chapter 1 

1. J. T. DESAGULIERS, "The Newtonian System of the World, The 

Best Model of Government: an Allegorical Poem," 1728, quoted 

in H. N. FA IRCHILD, Religious Trends in English Poetry, Vol. I 

(New York: Columbia University Press, 1939), 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, 1961). 

4. La Science et Ia diversite des cultures, (Paris: UNESCO, PUF, 

1974), pp. 15-16. 

5. C. C. GILLISPIE, The, Edge of Objectivity (Princeton, N.J.: 

Princeton University Press, 1970), pp. 199-200. 

6. M. HEIDEGGER, The Question Concerning Te chnology (New 

York: Harper & Row, 1977), p. 20. 

7. Ibid., p. 21. 

8. Ibid., p. 16. 

9. "The Coming of the Golden Age," Paradoxes of Progress (San 

Francisco: Freeman & Company, 1978). 

10. See, for instance. P. DAVIES, Other Worlds (Toronto: J. M. Dent 

& Sons, 1980).

317 NOTES 

11. A. K oESTLER, The Roots of Coincidence (London: Hutchinson, 

1972), pp. 138-39. 

12. A. KoYRE, Newtonian Studies (Chicago: University of Chicago 

Press, 1968), pp. 23-24. 

13. In "Race and History" (Structural Anthropology II, New York: 

Basic Books, 1976), Claude Levi-Strauss discusses the conditions 

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 amplification 

phenomena) attests to an affinity between the problems of 

stability and fluctuation we discuss in Chapter VI as well as certain 

themes of the "structural approach" in anthropology. 

14. "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, 1971), 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 nature. 

See the efforts devoted by engineers till the twentieth century 

to the construction of perpetual-motion machines in A. Ord 

Hume, Perpetual Motion: The History of an Obsession (New 

York: St. Martin's Press, 1977). 

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. 

18. R. FEYNMAN, The Character of Physical Law (Cambridge, 

Mass.: M.I.T. Press, 1967), second chapter. 

19. 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, 1967), p. 12. 

21. 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


Rise of Modern Science (Edinburgh and London: Scottish Academic 

Press, 1972), esp. pp. 14-16. 

24. WHITEHEAD, Op. cit., p. 54. 

25. The famous text about nature being written in mathematical 

signs is to be fo und in 11 Saggiatore. See also The Dialogue Concerning 

the Two Chief World Systems, 2nd rev. ed. (Berkeley: 

University of California Press, 1967). 

26. 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, 1971), 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. " 

27. In his biography of dlembert-Jean d'Alembert, Science and 

Enlightenment (Oxford: Clarendon Press, 1970)-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 their relations with. the absolute sovereigns. 

28. EINSTEIN, Op. cit., pp. 225-26. 

29. E. MACH, "The Economical Nature of Physical Inquiry, " Popular 

Scientific Lectures (Chicago: Open Court Publishing Company, 

1895), pp. 197-98. 

30. 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 Internationales 

d'Histoire des Sciences, Vol. XX (1967), pp. 42- 

65, 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, No. 3 (1967), pp. 25-157. The four following paragraphs 

rest, for what concerns atomism and the conservation theories,

319 NOTES 

on W. Scarr, The Conflict Between Atomism and Conservation 

Theory (London: Macdonald, 1970). 

2. A. KovRE: , Galileo Studies (Hassocks, Eng.: The Harvester 

Press, 1978), pp. 89-94. 

3. In his history of mechanics-The Science of Mechanics: A Critical 

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. 

4. This at least is the conclusion of today's historians who began 

the study of the impressive mass of Newton's ·tchemical Papers," 

which till now were ignored or disdained as "nonscientific." 

See B. J. DoBBS, The Foundations of Newton's Alchemy 

(Cambridge: Cambridge University Press, 1975); R. WESTFALL, 

"Newton and the Hermetic Tradition" in Science, Medicine and 

Society, ed. A. G. DEBUS (London: Heinemann, 1972); and R. 

WESTFALL, "The Role of Alchemy in Newton's Career," Reason, 

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, summarized 

(quoted in DoBBS, op. cit., p. 13), "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 10,000 years 

ago." 

5. 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. 

6. The story of Newton's "mistake" is told in HANKINs's, Jean 

d'Alembert, pp. 29-35. 

7. G. L. BuFFON, "Reflexions sur Ia loi d'attraction," appendix to 

Introduction a I' histoire des minhaux (1774), To me IX of 

Oeuvres Completes (Paris: Garnier Frere s), pp. 75, 77. 

8. G. L. BuFFON, Histoire naturelle. De Ia Nature, Seconde Vue 

(1765), quoted in H. METZGER, Newton, Stahl, Boerhaave et Ia 

doctrine chimique (Paris: Blanchard, 1974), pp. 57-58. 

9. A. THACKRAY describes the way French chemistry became Buffonian 

in Atom and Power: An Essay on Newtonian Matter Th e 

ory and the Development of Chemistry (Cambridge, Mass.: 

Harvard University Press, 1 970). pp. 199-233. Berthollet's 

Statique chimique accomplished Buffon's program and also


closed it, since his disciples gave up the attempt to understand 

chemical reactions in terms compatible with Newtonian con· 

cepts. 

10. 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). 

11. 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 Science 

in Western Europe, ed. M. CROSLAND (London: Mac· 

millan, 1975). 

12. 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 century, 

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 nonprofessionals. 

13. Quoted in J. ScHLANGER, Les metaphores de I' organisme 

(Paris: Vrin, 1971), p. 108. 

14. 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, 1882). 

15. This problem is one of the main themes of French philosopher 

Michel Serres. See, for instance, "Conditions" in his La naissance 

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 Department 

of Johns Hopkins University. See M. SERRES, 

Hermes: Literature, Science, Philosophy. (Baltimore: The 

Johns Hopkins University Press, 1982.) 

16. See, about the fate of Laplace's demon, E. CASSIRER, Determinism 

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, 1980), p. 4. 

2. D. DIDEROT, d'Alembert's Dream (Harmondsworth:, Eng.: Penguin 

Books, 1976), pp. 166-67. 

3. D. DIDEROT, "Conversation Between d'Alembert and Diderot," 

d'Alembert's Dream, pp. 158-59. 

4. D. DIDEROT, Pensees sur /'Interpretation de Ia Nature (1754), 

Oeuvres Completes, Tome II (Paris: Garnier Freres, 1875), p. II. 

5. Diderot ascribes this opinion to the physician Bordeu in the 

Dream. 

6. See, for instance, 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 official 

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 academy 

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 of Science, ed. 

M. CLAGETT (Madison, Wis.: University of Wisconsin Press, 

1959), pp. 255-89. 

8. G. E. STAHL, "Veritable Distinction a etablir entre le mixte et le 

vivant du corps humain," Oeuvres medicophilosophiques et 

pratiques, To me II (Montpellier: Pitrat et Fils, 1861), esp. 

pp. 279-82. 

9. See J. SCHLANGER, Les metaphores de /'organisme, for a description 

of the transformation of the meaning of "organization" 

between Stahl and the Romanticists. 

10. Philosophy of Nature, §261. 

11. This is Knight's conclusion in "The German Science in the Romantic 

Period," The Emergence of Science in Western Europe. 

12. H. BERGSON, La pensee et le mouvant in Oeuvres (Paris: E ditions 

du Centenaire, PUP, 1970), p. 1285 ; trans. The Creative 

Mind (Totowa, N.J.: Littlefield, Adams, 1975), p. 42. 

13. Ibid., p. 1287; trans., p. 44. 

14. Ibid., p. 1286; trans. , p. 44.


15. H. BERGSON, L'evolution creatrice in Oeuvres, p. 784; trans. 

Creative Evolution (London: Macmillan, 191 1), p. 361. 

16. Ibid., p. 538; trans., p. 54. 

17. Ibid., p. 784; trans., p. 361. 

18. BERGSON, La pensee et /e mouvant, p. 1273; trans., p. 32. 

19. Ibid:, p. 1274; trans., p. 33. 

20. A. N. WHITEHEAD, Science and the Modern World, p. 55. 

21. A. N. WHITEHEAD, Process and Reality: An Essay in Cosmology 

(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 endeavor 

to describe in a positive way the organism as a whole. 

24. H. HELMHOLTZ, Uber die Erhaltung der Kraft (1847), trans. in 

S. BRUSH, Kinetic Theory, Vol. I, The Nature of Gases and 

Heat (Oxford: Pergamon Press, 1965), p. 92. See also Y. 

ELKANA, Th e Discovery of the Conservation of Energy 

(London: Hutchinson Educational, 1974) and P. M. HEIMANN, 

"Helmholtz and Kant: The Metaphysical Foundations of Uber 

die Erhaltung der Kraft," Studies in the History and Philosophy 

of Sciences, Vol. 5 (1974), pp. 205-38. 

25. H. REICHENBACH, 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 Wa tt to Clausius (London, Heinemann, 

1971). Particularly interesting in this respect is the convergence 

between on one hand the need determined by industrial problems 

and on the other the positivist simplifications by operational 

definitions. 

2. J. HERIVEL, Joseph Fourier: The Man and the Physicist (Oxford: 

Clarendon Press, 1975). In this biography we learn the following 

curious information: Fourier would have brought back 

from his trip with Bonaparte to Egypt a sickness causing permanent 

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 Journal 

for the Philosophy of Science, Vol . 9 (1976), pp. 293-3 19 and 

M. CROSLAND and C. SMITH, "The Transmission of Physics 

from France to Britain, 1800-1840," Historical Studies in the 

Physical Sciences, Vol . 9 (1978), pp. 1-61. 

5. For what follows, see Y. ELKANA, The Discovery of the Conservation 

of Energy Principle, as well as the famous paper by 

Thomas Kuhn, "Energy conservation as an Example of Simultaneous 

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 energy; 

see his book and "Helmholtz's Kraft: An Illustration of 

Concepts in Flux," Historical Studies in the Physical Sciences, 

Vol . 2 (1970), pp. 263-98. 

7. J. JouLE, "Matter, Living Force and Heat," The Scientific Papers 

of James Prescott Joule, Vol. 1 (London: Taylor & Francis, 

1884), 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, "Vitalism in the Nineteenth Century Scientific 

Thought: A 'JYpology and Reassessment," Studies in History 

and Philosophy of Science, Vol. 5 (1974), pp. 17-48. 

10. H.H ELMHOLTZ, "Uber die Erhaltung der Kraft," op. cit., 

pp. 90-91. 

11. G. DELEUZE, Nietzsche et Ia phi/osophie (Paris: PUF, 1973), 

pp. 48-55. 

12. In this study of Zola's "Docteur Pascal ," Feux et signaux de 

brume Paris: Grassel (1975), 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 system, 

and was now filled with dismay at the relentless degradations 

of fire. Hence the fierce, positive dilemma: perfect cycle 

without residue, eternal and positively valued, i.e., the cosmology 

of the sun; or else a missed cycle, losing its difference, irreversible, 

historical and despised-a cosmology, a thermogony of 

fire which must either be extinguished or destroyed, without alternative. 

One dreams of Laplace, whilst Carnot and the others 

have forever smashed the cubby-hole, the niche, where one


could sleep in peace; one is dreaming, that is certain: then 

cultural archaisms having returned through another door, 

through another opening of the same door, are powerfully reawakened: 

immortal flame, purifying blaze or evil fire?" 

13. The continuity between Carnot father and son has been emphasized 

by Cardwell (From Wa tt to Clausius) and Scott (The Conflict 

Between Atomism and Conservation Theory). 

14. P. DAVIES, The Runaway Universe (New York: Penguin Books, 

1980), p. 197. 

15. R DYSON, "Energy in the Universe," Scientific American, Vol. 

225 (197 1), pp. 50-59. 

1 6. What was particularly important was to grasp that, unlike what 

happens in mechanics, it is not just any situation of a thermodynamic 

system that can be characterized as a "state"; quite 

the contrary. See E. DAUB, "Entropy and Dissipation," Historical 

Studies in the Physical Sciences, Vol. 2 (1970), pp. 321-54. 

17. In his autobiography, Scientific Autobiography (London: 

Williams & Norgate, 1950), 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 temperatures. 

Thus, 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 energy 

differences (required by all transformation since only a difference 

can produce another difference) as an abstract ideal, but 

the sole object for a rational science. 

18. 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 1945 

(Paris: Dunod, 1947). 

19. R. CLAUSIUS, Ann. Phys., Vol. 125 (1865), p. 353. 

20. 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. 

21. R. CAILLOIS, "La dissymetrie," Coherences aventureuses, Collection 

Idees (Paris: Gallimard, 1973), p. 198. 

Chapter 5 

1. For what concerns the content of this and the fol lowing chapter, 

see P. GLANSDORFF and I. PRIGOGINE, Thermodynamic Theory 

of Structure, Stability and Fluctuations (New York: John Wiley 

& Sons, 1971) and G. NICOLls and I. PRIGOGINE, Self-Organization 

in Non-Equilibrium Systems (New York: John Wiley & 

Sons, 1977), where further references may be found. 

2. R NIETZSCHE, Der Wille zur Macht, Siimtliche Werke (Stuttgart: 

Kroner, 1964), aphorism 630. 

3. Which precise content can be given to the general law of entropy 

growth? For a theoretician physicist such as de Donder, chemical 

activity, which appeared obscure and inaccessible to the rational 

approach of mechanics, was mysterious enough to 

become the synonym of the irreversible process. Thus chemistry, 

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 Th. De Donder, L' Affinite (Paris: 

Gauthier-Villars, 1962) and L. Onsag'er Phys. Rev. 37, 405 (193 1). 

4. M. SERRES, La naissance de Ia physique dans le texte de Lucrece, 

op. cit. 

5. For more details concerning chemical oscillations, see A. 

WINFREE, "Rotating Chemical Reactions," Scientific Amer· 

ican, Vol. 230 (1974), pp. 82-95. 

6. A. GoLDBETER and G. Nicous, ''An Allosteric Model with 

Positive Feedback Applied to Glycolytic Oscillations," Progress 

in Theoretical Biology, Vol. 4 (1976), pp. 65-160; A. GOLDBETER 

and S. R. CAPLAN, "Oscillatory Enzymes," Annual Review of 

Biophysics and Bioengineering, Vol. 5 (1976), pp. 449-73 .. 

7. B. HESS, A. BOITEUX, and J. KROG ER, "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 Functional Order in Regulated 

Biochemical Cellular Systems," Advances in Chemical 

Physics, Vol. XXXVIII (1978), pp. 363-413. 

8_ R HEss, Ciba Foundation Symposium. Vol. 31 (1975), p. 369. 

9A.G. GERESCH, "Cell Aggregation and Differentiation in Die-


tyostelium Discoideum," in Developmental Biology, Vol. 3 

(1968), pp. 157-197. 

98. A. GoLDBETER and L. A. SEGEL, "Unified Mechanism for Relay 

and Oscillation of Cyclic AMP in Dictyostelium Discoideum," 

Proceedings of the National Academy of Sciences, 

Vol. 74 (1977), pp. 1543-47. 

10. See M. GARDNER, The Ambidextrous Universe (New Yo rk: 

Charles Scribner's Sons, 1979). 

II. O. K. KONDEPUDI and I. PRIGOGINE, Physica, Vol. l07A (1981), 

pp. 1-24; D. K. KoNDEPUDI , Physica, Vol. 115A (1982), 

pp. 552-66. It could even be that chemistry may bring to the 

macroscopic scale the violation of parity in weak fo rces; D. K. 

KoNDEPUDI and G. W. NELSON, Physical Review Letters, Vol. 

50, No. 14 (1983), pp. 1023-26. 

12. R. LEFEVER and W. HoRSTHEMKE, "Multiple Transitions Induced 

by Light Intensity Fluctuations in Illuminated Chemical 

Systems," Proceedings of the National Academy of Sciences, 

Vol. 76 (1979), pp. 2490-94. See also W. HoRSTHEMKE and M. 

MALEK MANSOUR, "Influence of External Noise on Nonequilibrium 

Phase Transitions," Zeitschrift fu r Physik B, Vo l. 24 

(1976), pp. 307-1 3; L. ARNOLD, W. HORSTHEMKE, and R. 

LEFEVER, "White and Coloured External Noise and Transition 

Phenomena in Nonlinear Systems," Zeitschrift fii r Physik B, 

Vol. 29 (1978), pp. 367-73; W. HORSTHEMKE, "Nonequilibrium 

Transitions Induced by External White and Coloured Noise," 

Dynamics of Synergetic Systems, ed. H. HAKEN (Berlin: 

Springer Ve rlag, 1980); for an application to a biological problem, 

R. LEFEVER and W. HORSTHEMKE, "Bistability in Fluctuating 

Environments: Implication in Tu mor Immunology," 

Bulletin of Mathematic Biology, Vol. 41 (1979). 

13. H. L. SwiNNEY and J. P. GoLLUB, "The Transition to Thrbulence," 

Physics Today, Vol. 31 , No.8 (1978), pp. 41-49. 

14. M. J. FEIGENBAUM, "Universal Behavior in Nonlinear Systems," 

Los Alamos Science, No I (Summer 1980), pp. 4-27. 

15. The concept of chreod is part of the qualitative description of 

embryological development Waddington proposed more than 

twenty years ago. It is truly a bifurcating evolution: a progressive 

exploration along which the embryo grows in an "epigenetic 

landscape" where coexist stable segments and segments 

where a choice among several developmental paths is possible. 

See C. H. WA DDINGTON, The Strategy of the Genes (London: 

Allen & Unwin, 1957). C. H. Waddington's chreods are also a 

central reference in Rene Thorn's biological thought. They could 

thus become a meeting point for two approaches: the one we are 

presenting, starting from local mechanisms and exploring the

327 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. 

16. S. A. KAUFFMAN, R. M. SHYMKO, and K. TRABERT, "Control of 

Sequential Compartment Formation in Drosophila," Science, 

Vol. 199 (1978), pp. 259-69. 

17. H. BERGSON, Creative Evolution, pp. 94-95. 

18. See C. H. WADDINGTON, The Evolution of an Evolutionist 

(Edinburgh: Edinburgh University Press, 1975) and P. WEISS, 

"The Living System: Determinism Stratified," Beyond Reductionism, 

ed. A. KOESTLER and J. R. SMYTHIES (London: 

Hutchinson, 1969). 

19. D. E. KosHLAND, · Model Regulatory System: Bacterial 

Chemotaxis," Physiological Review, Vol. 59, No. 4, pp. 811-62. 

Chapter 6 

J. G. NICOLlS and I. PRIGOGINE, Self-Organization in Nonequilibrium 

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 (1983), pp. I. 

3. See, for example, M. MALEK MANSOUR, C. VAN DEN BROECK, 

G. Nicous, and J. W. TURNER, Annals of Physics, Vol. 131, No. 

2 (1981), 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 termites," Insectes Sociaux, Journal International pour 

/'etude des Arthropodes sociaux, To me 24, No. 2 (1977), 

pp. 117-30. This first model is presently being extended in connection 

with new experimental studies; 0. H. BRUINSMA, ·n 

Analysis of Building Behaviour of the Termite macrotermes subhyaiinus," 

Proceedings of the VIII Congress IUSSI (Waegeningen, 

1977). 

5. R. P. GARAY and R. LEFEVER, · Kinetic Approach to the Immunology 

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 Ecology, 

" Bulletin of Mathematical Biology, Vol. 37 (1975), 

PP- 389-405; "Evolution, Population and Stability, " Proceedings 

of the National Academy of Sciences, Vol. 73, No. 3 (1976), 

pp. 665-68. See also R. CzAPLEWSKI, ']\_ Methodology for Eval-


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. 186 (1974), pp. 645-47; see also R. MAY, 

"Simple Mathematical Models with very Complicated Dynamics," 

Nature, Vol. 261 (1976), pp. 459-67. 

9. M. P. HASSELL, The Dynamics in Arthropod Predator-Prey Systems 

(Princeton, N.J.: Princeton University Press, 1978). 

10. B. HEINRICH, ·rtful Diners," Natural History, Vol. 89, No. 6 

(1980), pp. 42-5 1, esp. quote, p. 42. 

11. M. LovE, "The Alien Strategy," Natural History, Vol. 89, No. 5 

(1980), pp. 30-32. 

12. J. L. DENEUBOURG and P. M. ALLEN, "Modeles theoriques de 

Ia division du travail des les societes d'insectes," Academie 

Royale de Belgique, Bulletin de Ia Classe des Sciences, Tome 

LXII (1976), pp. 416-29; P. M. ALLEN, "Evolution in an Ecosystem 

with Limited Resources," op. cit., pp. 408-15. 

13. E. W. MoNTROLL, "Social Dynamics and the Quantifying of Social 

Forces," Proceedings of the National Academy of Sciences, 

Vol. 75, No. 10 (1978), pp. 4633-37. 

14. P. M. ALLEN and M. SANGLIER, "Dynamic Model of Urban 

Growth," Journal for Social and Biological Structures, Vol. 1 

(1978), pp. 265-80, and "Urban Evolution, Self-Organization 

and Decision-making," Environment and Planning A, Vol. 13 

(1981), pp. 167-83. 

15. C. H. WADDINGTON, To ols 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. 311-17. 

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. 186. 

2. The Philosophy of Rudolf Carnap, ed. P.A. ScHILP.P (Cambridge: 

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 (Bellagio, 

July 1981). 

4. See on this point S. BRUSH, Statistical Physics and Irreversible 

Processes, esp. pp. 616-25. 

5. Feuer has rather convincingly shown how the cultural context of 

Bohr's youth could have helped his decision to look for a nonmechanistic 

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, 1971) 

and D. SERWER, "Unmechanischer Zwang: Pauli, Heisenberg 

and the Rejection of the Mechanical Atom 1923-1925," Historical 

Studies in the Physical Sciences, Vol. 8 (1977), pp. 189-256. 

6. In Black-Body Theory and the Quantum Discontinuity, 1894- 

1912 (Oxford: Clarendon Press and New York: Oxford University 

Press, 1978), Thomas Kuhn has beautifully shown how 

closely Planck followed Boltzmann's statistical treatment of irreversibility 

in his own work. 

7. J. MEHRA and H. RECHENBERG, The Historical Development of 

Quantum Theory, Vols. 1-4 (New York: Springer Verlag, 1982). 

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 Mechanics, 

2nd aug. ed. (Reading, Mass.: Benjamin, 1976). See also B. 

o'EsPAGNAT, "The Quantum Theory and Reality," Scientific 

American, Vol. 241 (1979), pp. 128-40. 

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, 1968). C. GEORGE and I. PRIGOGINE, "Coherence 

and Randomness in Quantum Theory, " Physica, Vol. 99A 

(1979), pp. 369-82. 

10. L. RosENFELD, "The Measuring Process in Quantum Mechanics," 

Supplement of the Progress of Theoretical Physics .(1965), 

p. 222. 

11. 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'Espagnat 

and Jammer. 

12. B. MISRA, I. PRIGOGINE, and M. COURBAGE, "Lyapounov Vari-


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 Dissipative 

Processes in Quantum Systems," to appear in Proceedings 

of the National Academy of Sciences. Vol 80 (1983) 

4590-94. 

13. H. MINKOWSKI, "Space and Time," The Principles of Relativity 

(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. 71 (1930), 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. PLANCK, Treatise on Thermodynamics (New York: Dover 

Publications, 1945), p. 106. 

5. Quote by K. DENBIGH, "How Subjective Is Entropy?" Chemistry 

in Britain, Vol. 17 (198 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 between 

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 the 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 deterministic 

symmetry" and "macroscopic probabilistic asymmetry." 

The question is 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, 1871), Chap. XXII; see also E. DAUB,

331 

NOTES 

"Maxwell's Demon" and P. HEIMANN, "Molecular Forces, Statistical 

Representation and Maxwell's Demon," both in Studies 

in History and Philosophy of Science, Vol. I (1970); this volume 

is entirely devoted to Maxwell. 

10. L. BoLTZMANN, Populiire Schriften, new ed. (Braunschweig 

Wiesbaden: Vieweg, 1979). As Elkana emphasizes in "Boltzmann's 

Scientific Research Program and Its Alternatives," Interaction 

Between Science and Philosophy (Atlantic, 

Highlands, N.J.: Humanities Press, 1974), the Darwinian idea of 

evolution is explicitly expressed mostly in Boltzmann's view 

about scientific knowledge-that is, in his defense of mechanistic 

models against energeticists. See, for instance, his 1886 lecture 

"The Second Law of Thermodynamics," Theoretical 

Physics and Philosophical Problems, ed. B. McGuiNNESS (Dordrecht, 

Netherlands: D. Reidel , 1974). 

11. For a recent account see I. PRIGOGINE, From Being to Becoming-Time 

and Complexity in the Physical Sciences (San Francisco: 

W. H. Freeman & Company, 1980). 

12. In his Scientific Autobiography, Planck describes his changing 

relationship with Boltzmann (who was first hostile to the phenomenological 

distinction introduced by Planck between reversible 

and irreversible processes). See also on this point Y. 

ELKANA, op. cit., and S. BRUSH, Statistical Physics and Irreversible 

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). 

13. H. POINCARE, "La mecanique et I' experience," Revue de Metaphysique 

et de Morale, Vol. 1 (1893), pp. 534-37. H. POINCARE, 

Lefons de Thermodynamique, ed. J. Blondin (1892; Paris: Hermann 

1923). 

14. See for a study of the controversies around Boltzmann's entropy, 

see on this point S. BRUSH, The Kind of Motion We Call Heat, 

op. cit., and Planck's remarks in his biography (Loschmidt was 

Planck's student). 

15. I. PRIGOGINE, C. GEORGE, R HENIN, and L. ROSENFELD, · 

Unified Formulation of Dynamics and Thermodynamics," 

Chemica Scripta, Vol. 4 (1973), pp. 5-32. 

16. D. PARK, The Image of Eternity: Roots of Time in the Physical 

World (Amherst, Mass.: University of Massachusetts Press, 

1980). 

17. 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 Irre versible Processes (Amsterdam: North Holland 

Publishing Company, 1976), as well as his commented anthology,


332 

Kinetic Theory: Vol. I, The Nature of Gases and Heat: Vol. II. 

Irreversible Processes (Oxford: Pergamon Press, 1965 and 1966). 

18. J. W. GIBBS, Elementary Principles in Statistical Mechanics 

(New Yo rk: Dover Publications, 1%0), Chap XII. For an historical 

account, see J. MEHRA, "Einstein and the Foundation of 

Statistical Mechanics, Physica, Vol. 79A, No. 5 (1974), p. 17. 

19. 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. 167, "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." 

20. L. BoLTZMANN, Lectures on Gas Theory (Berkeley: University 

of California Press, 1964), p. 446f, quoted in K. POPPER, Unended 

Quest (La Salle, Ill.: Open Court Publishing Company, 

1976), p. 160. 

21. POPPER, op. cit., p. 160. 

Chapter 9 

1. VoLTAIRE, Dictionnaire Philosophique. (Paris: Garnier, 1954.) 

2. See note 2, Chapter VII. 

3. K. PoPPER, "The Arrow of Time," Nature, Vol. 177 (1956), 

p. 538. 

4. See M. GARDNER, The Ambidextrous Universe, pp. 271-72. 

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, 1957); E. T. WHITTAKER, A 

Treatise on the Analytical Dynamics of Pa rticles and Rigid 

Bodies (Cambridge: Cambridge University Press, 1965). 

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, "Modern 

Ergodic Theory," Physics To day (Feb. 1973), pp. 23-29. 

9. For a more detailed study, see R. BALESCU, Equilibrium and 

Non-Equilibrium Statistical Mechanics (New York: John Wiley 

& Sons, 1975). 

10. V. ARNOLD and A. Av Ez, Ergodic Problems of Classical Mechanics 

(New York: Benjamin, 1968).

333 NOTES 

11. H. PoiNCARE, "Le Hasard," Science et Methode (Paris: Flammarion, 

1914), p. 65. 

12. B. MISRA, I. PRIGOGINE and M. CouRBAGE, "From Deterministic 

Dynamics to Probabilistic Descriptions," Physica, Vol. 98A 

(1979), pp. 1-26. 

13. D. N. PA RKS 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," Proceedings 

of the National Academy of Sciences, 80 (April 1983). 

15. 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 (1983), pp. 4590-4594. 

16. V. NABOKOV, Look at the Harlequins! (McGraw-Hill 1974). 

17. J. NEEDHAM, "Science and Society in East and West," The 

Grand Titration (London: Allen & Unwin, 1%9). 

18. See for more details B. MISRA, I. PRIGOGINE and M. CouR 

BAGE, "From deterministic Dynamics to probabilistic Description", 

Physica 98A (1979) 1-26.; B. MISRA and I. PRIGOGINE 

"Time, Probability and Dynamics", in Long-time Prediction in 

Dynamics, eds. C. W. Horton, L. E. Recihl and A. G. 

Szebehely, (New York, Wiley 1983). 

19. I. PRIGOGINE, C. GEORGE, R HENIN, and L. ROSENFELD, · 

Unified Formulation of Dynamics and Thermodynamics," 

Chemica Scripta, Vol. 4 (1973), pp. 5-32. 

20. M. CouRBAGE "Intrinsic irreversibility of Kolmogorov dynamical 

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, 1948). 

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, 1931), pp. 42-43. 

5. D. S. KOTHARI, Some Thoughts on Truth (New Delhi: Anniversary 

Address, Indian National Science Academy, Bahadur Shah 

Zafar Marg, 1975), p. 5.


334 

6. E. MEYERSON, Identity and Reality (New York: Dover Publications, 

1962). 

7. Described in H. BERGSON, Melanges (Paris: PUF, 1972), 

pp. 1340-46. 

8. Correspondence, Albert Einstein-Michele Besso, 1903-1955 

(Paris: Herman, 1972). 

9. N. WIENER, Cybernetics (Cambridge, Mass.: M.I.T. Press and 

New Yo rk: John Wiley & Sons, 1961). 

10. M. MERLEAU-PONTY, "Le philosophe et Ia sociologie," Eloge 

de Ia Philosophie, Collection Idees (Paris: Gallimard, 1960), 

pp. 136-37. 

11. M. MERLEAu-PoNTY, Resumes de Cours /952-/960 (Paris: Gallimard, 

1968), p. 119. 

12. P. VA LRY, Cahiers, La Pleiade (Paris: Gallimard, 1973), p. 1303. 

13. For what follows see also I. PRIGOGINE, I. STENGERS, and S. 

PA HAUT, "La dynamique de Leibniz a Lucrece," Critique "Special 

Serres," Vol. 35 (Jan. 1979), pp. 34-55. Engl. trans.: "Dynamics 

from Leibniz to Lucretius," Afterword to M. SERRES, 

Hermes: Literature, Science, Philosophy (Baltimore: Johns 

Hopkins Univ. Pr. , 1982), pp. 137-55. 

14. C. S. PEIRCE, The Monist Vo l. 2 (1892), pp. 321-337. 

15. A. N. WHITEHEAD, Process and Reality, pp. 240-41. On this 

subject, see I. LECLERC, Whitehead's Metaphysics (Bloomington: 

Indiana University Press, 1975). 

16. La naissance de Ia physique dans le texte de Lucrece, p. 139. 

17. LuCRETIUS, De Natura Rerum, Book II. ·gain, if all movement 

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 everlasting 

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. SERRES, op. cit., p. 162; 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. 

21. 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, 1949). 

24. A. NEHER, "Vision du temps et de l'histoire dans Ia culture 

juive," Les cultures et le temps (Paris: Payot, 1975), p. 179.

INDEX 

Note: Page numbers given in boldface indicate location of definitions 

or discussions of terms or concepts indexed here. 

Acceleration, 57-59 

Affinity, 29, 136 

Against the Current (Berlin), 2 

Agassiz, Louis, 195 

Alchemy, 64; affinity in, 136; 

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, 156-60 

Ampere, Andre Marie, 67, 76 

Anaxagoras, 264 

Antireductionists, 173-74 

Archimedes, 39, 41, 304 

Aristotle, 39, 40, 71, 85, 173, 

305, 306; on change, 62; 

notion of space of, 171; 

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, 119, 257-59; and 

heat engines, 111-15; Layzer 

on, xxv; meaning of, 289; 

and probability, 238-39; roles 

played by, 30 I 

Atomists, 3, 36; conception of 

change of, 62, 63; on 

turbulence, 141 

Attractor, 121, 133, 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, 310 

Belousov-Zhabotinsky reaction, 

151-53, 168 

Benard instability, 142-44; 

transition to chaos in, 167-68 

Bergson, Henri, 10, 79, 80, 

90-94, 96, 129, 173-74, 

301-2; on dynamics, 60; on 

time, 214, 294 

Berlin, Isaiah, 2, 11, 13, 80 

Bernoulli, Daniel, 82 

Berry, B., 17 

Berthollet, Claude Louis, 

Comte, 319n 

Besso, Michele, 294 

Bifurcations, xv, 160-61, 176, 

275; cascading, 167-70; in 

evolution, 171-72; 

fluctuations and, 177, 180; in 

reaction-diffusion systems, 

260; role of chance in, xxvi, 

170, 176; social, 313; and 

335


Bifurcations ( cont' d) 

statistical model, 205-6; 

theory of, 14 

Big Bang, xxvii, 288; and arrow 

of time, xxv, 259 

Biology, 2, 10; catalysts in, 

133-34; chemical reactions 

in, 131-32; "communication" 

among molecules in, 13; 

concepts from physics 

applied to, 207; and 

conversion, 108; evolution 

and, 12, 128; logistic 

equation in, 193-96; 

molecular, see Molecular 

biology; reductionistantireductionist 

conflict in, 

174; technological analogies 

in, 174-75; time in, 116; 

Whitehead on, 96 

Birchoff, 266 

Blake, William, 30 

Boerhave, Hermann, 105 

Bohr, Niels, 2, 74, 220, 224-25, 

228, 229, 292-93 

Boltzmann, Ludwig, xvii, 15, 

16, 122-27, 219, 227, 234-36, 

258, 259, 274, 286-87, 297, 

329n, 33ln; 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, 124 

Boltzmann's order principle, 

122-28, 142, 143, 150, 163, 

187 

Bordeu, 321n 

Born, Max, 220, 235 

Boundary conditions, 106, 

120-2 1, 125, 126, 138-39, 

142, 147, 151 

Braude!, xviii, xix 

Bridgmann, P. W., 316n 

Brillouin, 216 

Broglie, Louis de, 220 

Bruno, Giordano, 15 

Bruns, 72, 265 

Brusselator, 146, 148, 151, 152, 

160 

"Brussels school," xv 

Buchdahl, Gerd, 316n 

Buffon, Georges Louis Leclerc 

de, Comte, 65-67, 319n 

Butts, Thomas, 30 

Caillois, Roger, 128 

Calvin, John, xxii 

Cancer tumors, onset of, 188 

Canonical equation, 226 

Canonical variables, 70, 71, 

107, 222 

Cardwell, D., 322n 

Carnap, Rudolf, 214, 294 

Carnot, Lazare, 112 

Carnot, Sadi, 111-15, 117, 120, 

128, 140, 323n, 324n ; Carnot 

cycle, 112- 114, 117 

"Carrying capacity" of 

systems, 192-97 

Catalysis, 133-35, 145, 153 

Caterpillars, strategies for 

repelling predators of, 194-95 

Catherine the Great, 52 

Cells: Benard, 143; chemical 

reactions within, 131-32 

Chance, concepts of, xxii-xxiii, 

14, 170, 176, 203 ; see 

Randomness 

Change: motion and, 62-68; 

nature of, 29 1; of state, 106; 

in thermodynamic system, 

120-2 1; Whitehead on, 95 

Chemical clock, xvi, 13, 

147-48, 179, 307; 

communication in, 180; in 

glycolysis, 155; in slime mold 

aggregation, 159 

Chemical reactions, 127; in 

biology, 131-32; diffusion in, 

148-49; fluctuations and 

correlations in, 179-8 1; 

kinetic description of, 

132-34; self-organization in,

337 

144-45; thermodynamic 

descriptions of, 133-37; see 

also specific reactions 

Chemistry, xi, I 0; Bergson on, 

91; and Buffo n, 65, 66; 

conceptual distinction 

between physics and, 137; 

and conversion, 108; Diderot 

on, 82, 83; fluctuations and, 

177-79; inorganic, 152, 153; 

irreversibility in, 209; 

Newtonian method in, 28; 

relation between order and 

chaos in, 168; and "science 

of fire," 103; temporal 

evolution in, 10-1 1 

China, 57; alchemy in, 278; 

social role of scientists in, 

45-46, 48 

Chiral symmetry, 285 

"Chreod," 172 

Chris taller model, 197, 203 

Christianity, 46, 47, 50, 76 

Chronogeography, 272 

Chuang Tsu, 22 

Clairaut, Alexis Claude, 62, 65 

Clausius, Rudolf Julius 

Emanuel, 114, 115, 233, 234, 

240, 304; entropy described 

by, 117- 19 

Clinamen, 141, 303 , 304 

Clocks: invention of, 46; as 

symbol of nature, Ill; see 

also Chemical clocks 

Closed systems, xv, 125 

Collective phenomena, xxiv; in 

amoebas, 156-160; in insects, 

181-86; in human geography, 

197-203 ; in social 

anthropology, 205, 317n 

Collisions, 63, 69, 132, 240-42, 

270-7 1 ' 280-85 

Combinatorial analysis, 123 

Communication: description as, 

300; in dissipative structures, 

13, 148; and entropy barrier, 

295-96; and fluctuations, 

187-88; and irreversibility, 

INDEX 

295; molecular basis to, xxv, 

180; stabilizing effects of, 189 

Compensation, 107; Clausiuson 

Carnot cycle, 114; statistical, 

124, 133, 240 

Complementarity, principle of, 

225 

Complexions, 123, 124, 127, 

150; in Benard instability, 

142-43 

Complexity: dynamics and 

science of, 208-9; limits of, 

188-89; modelizations of, 

203-7 

Comte, Auguste, 104-5 

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, 114, 115; 

and entropy, 117, 118; 

principle of, 69-7 1 

Convection, 127; in Benard 

instability, 142 

Copernicus, 307 

Correlations: dynamics of, 

280-85 ; fluctuations and, 

179-8 1 

Cosmology, xxviii, 1, 10; 

entropy and, 117; mysticism 

and, 34; and 

thermodynamics, 115-17; 

time and, 215, 259; 

Whitehead's, 94 

Counterintuitive responses, 203 

Critical threshold see Instability 

threshold 

Critique of Pure Reason (Karit), 

86 

Cybernetics (Weiner), 295-96 

Darwin, Charles, xiv, xx, 128, 

140, 215, 240, 24 1, 25 1


Darwinian selection, 190, 191, 

194, 195 

David, Ben, 318n 

Democritus, 3 

Deneubourg, J. L., 181 

Density function p, 247-50, 

261, 264; with arrow of time, 

277, 289; or distribution 

function, 289; in phase space, 

265-72, 274, 279 

Deoxyribonucleic acid (DNA), 

154; dissymetry of, 163 

Desaguliers, J. T. , 27 

Descartes, Re ne , 62, 63 , 81 

Destiny, 6, 257 

Determinism, xxv, 9, 60, 75 , 

169-70, 176, 177, 216, 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, 156, 

157 

Diderot, Denis, 79-85, 91, 136, 

309, 321n 

Differential calculus, 57, 222 

Diffusion, 148-49, 177 

Dirac, Paul, 34, 220, 230 

Disorder, xxvii, 18, 124, 126, 

142, 238, 246, 250, 286-87, 

293 

Dissipation (or loss), 63, 112, 

I 15, 117, 120, 125, 129, 

302-03 

Dissipative structures, xii, xv, 

xxiii, 12-14, 142-43, 189, 

300; coherence of, 170; 

communication in, 148; 

cultural , xxvi 

Dissymmetry, 124; 163; in time, 

125; see also Symmetrybreaking 

Distribution function se 

Density fu nction p 

Dobbs, B. J. , 319n 

Dander, Theophile de, 136, 325 

Donne, John, 55 

Driesch, Hans, 171 

Drosophila, 172 

du Bois Reymond, 77, 97 

Diierrenmatt, E, 21 

Duhem, Pierre Maurice Marie, 

97 

Duration, xxviii-xix; Bergson's 

concept of, 92, 294 

Dynamics, II, 14-15, 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, 216, 

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, 122; 

reversibility in, 120; and 

science of complexity, 208-9; 

static view of, xxix; 

symmetry-breaking in, 

260-6 1; and theories of 

irreversibility, 25 1; theory of 

ensembles in, 247-5 1; 

twentieth-century renewal of, 

264-72 

Dyson, Freeman, 1 17 

Ecology, logistic equation in, 

192-93, 196, 204 

Eddington, Arthur Stanley, xx, 

8, 49, 119, 233, 291 

Edge of Objectivity, Th 

(Gillespie), 31 

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, 

271, 301, 307, 310; 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, 

218-20, 224; on scientific 

asceticism, 20-2 1; on 

simultaneity, 218; special 

theory of relativity of, 17; 

thought experiments of, 43; 

on time, 214-15, 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, 179 

Eliade, Mircea, 39-40 

Elkana, Y. , 323n, 33 1 n 

Embryo: development of, 

81-82; formation of gradient 

system in morphogenesis of, 

150; internal purpose of, 

171-73 

Encyclopedie, 83 

Energeticists, 234 

INDEX 

Energy, 107; dissipation of, 

302-3; and elementary 

particles, 287; and entropy, 

118- 19; exhaustible, Ill, 114; 

as invariant, 265; for living 

cells, 155; in quantum 

mechanics, 220-2 1; of 

unstable particles, 74; of 

universe, 117; see also 

Conservation of energy 

Energy conversion, 12, 108, 

114 

Engels, Friedrich, 252-53, 332n 

Engines, 12, 103, 105-07, 

111-15 

Enlightenment, the, 67, 79, 80, 

86 

Ensemble theory (Einstein 

Gibbs), 247-5 1, 26 1; 

equilibrium and, 265 

Entelechy, 171 

Entropy, xix-xx, 12, 14-18, 

117-22, 227; and arrow of 

time, XXV, 253-54, 257-59; 

and atomism, 288; as barrier, 

277-80, 295-97; in evolution, 

131; flux and force and, 135, 

137; law of increase of, xxix; 

in linear thermodynamics, 

138-39; mechanistic 

interpretation of, 240-43 ; 

probability and, 124, 126, 

142, 234, 235, 237-38, 274; 

production of, 119, 131, 133, 

135, 137-39, 142; as 

progenitor of order, xxi-xxii; 

as selection principle, 

285-86; subjective 

interpretation of, 125, 235, 

25 1-52; thought experiment 

on, 244; universal 

interpretations of, 239 

Enzymes, 133-34; feedback 

action of, 154; in glycolysis, 

155; resembling Maxwell's 

demon, 175 

Epicurus, 3, 305


Equilibrium, xvi; and baker 

transformation, 273; 

chemical, 133; chemical 

reactions in, 179-80; and 

entropy, 120, 131; evolution 

toward, 24 1-43; flux and 

force at, 135-37; in future, 

275, 276; and matter-light 

interaction, 219; maximum 

probability at, 286; and 

theory of ensembles, 265; 

thermal, 105, 116; thermal 

chaos in, 168; in 

thermodynamics, 12, 13, 

125-29, 138; velocity 

distribution in state of, 241; 

see also Far-fromequilibrium; 

Nonequilibrium 

Ergodic systems, 266 

Espagnat, B. d', 329n 

Esprit de systeme, 83 

Euclid, 171 

Euler, Leonhard, 52, 65, 82 

Everett, 228 

Evolution, xx, 12, 128; and 

arrow of time, xxv; 

bifurcations in, 171-72; 

biological, 153; Boltzmann . 

on, 240; chemical, 177; 

concepts from physics 

applied to, 207-9; cosmic, 

215, 288; Darwinian, 128; 

from disorder to order, xxix; 

entropy in, 119, 131; toward 

equilibrium, 241-43; 

feedback in, 196-203 ; 

logistic, 192-96, 204; 

·paradigm of, 297-98; in 

quantum mechanics, 

226-228, 238; toward 

stationary state, 138-39; 

structural stability in, 189-91 

Existentialism, xxii 

Expanding universe, 2, 19, 215, 

259 

Experimentation, 5, 41-44; and 

global truth, 44-45 ; Kant 

and, 88; universality of 

language postulated by, 51; 

Whitehead on, 93, 95 ; see 

also Thought experiments 

Falling bodies, Galileo's laws 

for, 57, 64 

Far-from-equjlibrium, xxvi, 

xxvii, 13-14, 140-45, 300; 

chemical instability in, 

146-53; in chemistry, 177; 

dissipative structures in, 189; 

in molecular biology, 153-59; 

prebiotic evolution in, 191; 

self-organization in, 176 

Faraday, Michael, 108 

Faust (Goethe), 128 

Feedback, 153; in biological 

systems, 154; in evolution, 

191, 196-203 ; between 

science and society, xiii 

Feigenbaum sequence, 169 

Feuer, 329n 

Feynman, Richard, 44 

Fluctuations, xv, xxiv-xxv, 

xxvii, 124-25, 140-41, 143; 

amplification of, 141, 143 

181-89; and chemistry, 

177-79; and correlations, 

179-8 1; in Markov process, 

238; on microscopic scale, 

23 1-32 

Fluid flow, 141 

Fluxes, 135-37; random noise 

in, 166-67; in reciprocity 

relations, 137-38 

Forces: generalized, 135-37; in 

reciprocity relations, 137-38 

Fourier, Baron Jean-Joseph, 12, 

104, 105, 107, 115-17 

Fraser, J. T. , 214 

Frederick II, King of Prussia, 

52 

Free particles, 70-72 

Free will, xxii 

Freud, Sigmund, 17 

Friedmann, Alexander, 215 

Fundamental level of 

description, 252-53

341 

Galileo, 40, 41, 50, 51, 305; on 

cause and effect, 60; and 

global truths, 44; and 

mechanistic world view, 57; 

thought experiments of, 43 

Galvani, Luigi, 107 

Gardner, Martin, 234, 259 

Gassendi, Pierre, 62 

Generalized forces, 135-37 

Geographical time, xviii 

Geography, 197; internal time 

in, 272 

Geology, 121; time in, 116, 208 

Gibbs, J. W., 15, 238, 247-5 1, 

261 

Gillispie, C. C., 31, 321n 

Glycolysis, 155-56 

Goethe, Johann Wolfgang von, 

128 

Gould, Stephen J. , 204 

Grasse, 181, 186 

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, 12, 66 

Guldberg and Waage's law (also 

Mass action, law oO, 133 

Hamilton, William Rowan, 68, 

%; see Hamiltonian 

Hamiltonian: equation, 249; 

function, 68, 70-7 1, 74, 107, 

220-2 1; operator, 221, 

226-27; and T-violation, 259 

Hankins, Thomas, 318n 

Hao Bai-lin, 151, 152 

Hausheer, Roger, 2 

Hawking, 117 

Heat, 12, 79, 103; conduction 

of, 104, 135; electricity 

produced by, 108; and heat 

engines, 12, 103, 106-7; heat 

engines, arrow of time and, 

111-15; propagation of, 

INDEX 

104-5; repelling force of, 66; 

specific, 106; transformation 

of matter by, 105 

Hegel, G. W. R, 79, 89-90, 92, 

93, 173 

Heidegger, Martin, 32-33, 42, 

79, 310 

Heisenberg, Werner, xxii, 220, 

292, 2% 

Heisenberg uncertainty 

relations, 178, 222-26 

Helmholtz, Hermann Ludwig 

Ferdinand von, %, 109-11 

Herivel, J. , 322n 

Hess, Benno, 155 

Hirsch, J. , 151 

History: of ideas, 79; open 

character of, 207; reinsertion 

of, into natural and social 

sciences, 208; of science, 307 

(cosmology) 208, 215, 

(geography) 197, 272, 

(geology) 116, 121, 208 

Holbach, Paul Henri Thiry, 

baron d ' , 82 

Hooykaas, R., 317n 

Hopf, 266 

Hubble, Edwin Powell , 215 

Humanities, schism between 

science and, 11, 13 

Hume, A. Ord, 317n 

Huss, John, xxii 

Huyghens, Christiaan, 60 

Hydrodynamics, 127; far-fromequilibrium 

phenomena in, 

141 

Hypnons, 180, 287, 288 

Hysteresis, 166 

Idealization, 41-43, 69, 

112-1 14, 115, 120, 216, 248, 

252, 305-06 

Impossibility, demonstrations 

of, 17, 216- 17, 296, 299-300 

Individual time, xviii 

Industrial Age , 111; combustion 

and, 103


342 

Information, 17-18, 250, 

278-79, 283, 295, 297-98 

Initial conditions (or state), 61 , 

68, 75, 121, 124, 128, 129, 

139-40, 142, 147, 248, 

261 -67, 270-7 1, 276, 278-79, 

295, 310 

Innovation, psychological 

process of, xxiv 

Innovative becoming, 

philosophy of, 94 

Instability, dynamic, 73, 

268-72, 276, 300; chemical, 

144-53; thermodynamic, 

141-42; threshold, 146, 147, 

160 

Insects, self-aggregation of, 

181-86 

Integrable systems, 71-72, 74, 

264-65, 302 

Internal time, 272-73, 289 

Intrinsically irreversible 

systems, 275-77, 289 

Intrinsically random systems, 

274-76, 289 

Intuition, 80, 91, 92 

Irreversibility, xx, xxi, xxvii, 

xxviii, 7-9, 63, 115; 

acceptance by physics of, 

208-9; and biology, 128, 175; 

in chemistry, 131, 137, 177; 

controversy over, 15-16; 

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, 128, 

189; formulation of theory of, 

105, 107, 117-2l; and limits 

of classical concepts, 261-64; 

and matter-light interaction, 

219; measurement and, 228; 

microscopic theory of, 242, 

257-59, 285-86, 288-90, 310; 

probability and, 16, 124, 125, 

233-40; quantitative 

expression of, Ill; from 

randomness to, 272-77; rate 

of, 135; in reciprocity 

relations, 138; 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, 165 

Jammer, M., 329n 

Jordan, 220 

Joule, James Prescott, 108-9 

Kant, Immanuel, 79, 80, 85-89, 

93, 99, 214 

Kauffman, S. A., 172 

Kepler, Johannes, 49, 57, 67, 

307 

Keynes, Lord, 319n 

Kierkegaard, Soren, 79 

Kinetic energy, 69-7 1, 90, 107, 

261 

Kinetics, chemical , 132-34 

Kirchoff, Gustav, % 

Knight, 321 n 

Koestler, Arthur, 32, 34-35 

Kolmogoroff, 266 

Kothari, D. S., 293 

Koyre, Alexandre, 5, 32, 35-36, 

62, 317n, 319n 

Kuhn, Thomas, 307-9, 320n, 

329n 

Lagrange, Comte Joseph Louis, 

52, %, 104, 324n 

Laminar flow, 141-42, 303 

Laplace, Marquis Pierre Simon 

de, xiii, 28, 52, 54, 66, 67, 

115, 323n; Fourier criticized 

by, 104 

Laplace's demon, 75-77, 87, 

27 1 

Large numbers, law of, 14, 178, 

180 

Lavoisier, Antoine Laurent, 2&, 

109

343 INDEX 

Layzer, David, xxv 

Lebenswe/t, 299 

Leibniz, Gottfried von, 50, 54, 

302, 303; formulae for 

velocity and acceleration, 58; 

monads of, 74 

Lemaitre, Georges, 215 

Lenoble, R., 3 

Levi-Strauss, Claude, 205, 317n 

Levy-Bruhl, L., 292 

Lewis, G. N., 233 

Liebig, Baron Justus von, 109 

Life: Bergson on, 92; 

compatibility with far-fromequilibrium 

conditions, 143; 

as expression of selforganization, 

175-76; and 

order principle, 127-28; 

origin of, 14; Romantic 

concepts of, 85; Stahl's 

definition of, 84; symmetrybreaking 

as characteristic of, 

163; temporal dimensions of, 

208; see also Molecular 

biology 

Light: velocity of, 17, 55, 

217-19, 278, 295, 296; waveparticle 

duality of, 219-20 

Limit cycle, 146-47 

Linear thermodynamics, 

137-40 

Liouville equation, 249, 250, 

266 

Logistic evolution, 192-96, 

203-04 

Look at the Harlequins 

(Nabokov), 277 

Lorentz, Hendrik Antoon, 270 

Loschmidt, 244, 246 

Louis XIV, King of France, 52 

Love, Milton, 195 

Lucretius, 3, 141, 302-5, 315n, 

334n 

Luther, Martin, xxii 

Lyapounov, 151 

Mach. Ernst. 49. 53-54. 97. 

318n 

Machines: Archimedes's, 41; 

ideal, 63, 69-70; mathematics 

and, 46; using heat, 103 

Macroscopic system, 106-07 

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, 133, 136, 

23 1 

Materialistic naturalism, 83 

Mathematization, 46; in 

Hamiltonian function, 71; 

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, 105; 

in far-from-equilibrium 

conditions, 14; interaction of 

radiation and, 219; new view 

of, 9; nonequilibrium 

generated by, 181; perception 

of differences by, 163, 165; 

properties of, 2; Stahl on, 

84-85; transition to life from, 

84; wave-particle duality of, 

221 

Maxwell, James Clerk, 54, 73, 

122, 160, 240, 24 1, 266 

Maxwell's demon, 175, 239 

Mayer, Julius Robert von, 109, 

Ill 

Measurement, irreversible 

character of, 228-29 

Mechanics, II, 15; 

generalization of, Ill; Hegel 

on, 90; and probability, 125; 

see also Dynamics; Quantum 

mechanics 

Medicine: Bergson on, 91; 

Diderot on, 82, 83 

Merleau-Ponty, M., 299 

Metternich, Clemens We nzel 

Nepomuk Lothar, Fiirst von, 

xiii


Meyerson, Emile, 293 

Microcanonical ensemble, 265 

Minimum entropy production, 

theorem of, 138-4 1 

Minkowski, H., 230 

Mole, 121n 

Molecular biology, 4, 8; farfrom-equilibrium 

conditions 

in, 153-59; 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, 189 

Moscovici, Serge, 22, 306-7 

Motion: and change, 62-68; 

complexity of, 75 ; instability 

of, 73; in mechanical engine 

vs. heat engine, 112; 

positivist notion of, 96; 

productton of, in heat engine, 

107; universal laws of, 57-62, 

83; see also Dynamics 

Nabokov, Vladimir, 277-78 

Napoleon, 52, 67 

Natural laws: belief in 

universality of, 1-2; 

mathematical concepts of, 46; 

Newton on, 28; primary and 

secondary, 8; social structure 

and views of, 48-49; timeindependent, 

2, 7; trials of 

animals for infringements of, 

48 

Nature of the Physical World, 

The (Eddington), 8 

Needham, Joseph, 6-7, 45, 48, 

49, 278, 322n 

"Neolithic Revolution," 5-6, 37 

Neumann, von, 266 

New Science, The (Vico), 4 

Newton, Isaac, xv, xxviii, 12, 

27-29, 76, 98, 104, 120, 124, 

234, 305, 319n; alchemy and, 

64; on change, 62, 63 ; 

eighteenth-century opposition 

to, 65 ; laws of motion, 70; 

and mechanistic world view, 

57; objectivity defined by, 

218; objects chosen for study 

by, 216; presentation of 

Principia to Royal Society, 1; 

second law, 58; see also 

Newtonian science 

Newtonian science, xiii, xiv, 

xix, xxv, xxvi, 37-40, 213; 

absence of universal constant 

in, 217; 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, Ill, 136 

Nisbet, R., 79 

N onequilibrium: cosmological 

dimension of, 23 1; difference 

between particles and 

antiparticles in, 285 ; 

fluctuations in, 178-80; 

innovation and, xxiv; and 

origin of structures, xxix; as 

source of order, 287; see also 

Far-from-equilibrium 

Non-linearity, 14, 134, 153, 

154-55, 197 see Catalysis 

Non-linear thermodynamics, 

137, 140 

Noyes, 152 

Nucleation, 187, 188 

Oersted, Hans Christian, 108 

Old Te stament, xxii 

Onsager, Lars, 137, 138 

Operators, 22 1-22, 225; 

commuting, 223

345 

Opticks (Newton), 28 

Optimization, 197, 207 

Order, 12, 18, 126, 131, 143, 

171-75, 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: glycolytic, 155; 

time- and space-dependent, 

148 

Ostwald, Wilhelm, 324n 

Pascal, Blaise, 3, 36, 79 

Pasteur, Louis, 163 

Pattern selection, 163, 164 

Pearson, Karl, 49 

Peirce, Charles S., 17, 302-3 

Pendulum, 16, 73, 216, 261-62 

Phase changes, 187 

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, 91, 92; changing 

perspective in, 8-9; 

complementary developments 

in biology and, 154; and 

concepts of change, 63; 

conceptual distinction 

between chemistry and, 137; 

Diderot on, 80-83; 

evolutionary paradigm in, 

297-98; inspired discourse of, 

76; introduction of 

probability in, 123; and laws 

of motion, 57; Lucretian, 

141 ; macroscopic, xii; 

objectivity in, 55 ; positivist 

view of, 97; of processes, 

INDEX 

243 ; and theology, 49; time 

in, 116; vitalism and, 84; 

Whitehead on, 95, 96 

Planck, Max, 121, 219, 242, 

324n, 329n, 33 ln; on second 

law of thermodynamics, 

234-35 

Planck's constant, 217, 219, 

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, 151, 236, 243, 253, 265 

27 1 

Poisson distribution, 179-8 1 

Pope, Alexander, 27, 67 

Popper, Karl, 5, 15, 254-55, 

258-59, 276, 317n 

Populiire Schriften 

(Boltzmann), 240 

Positive-feedback loops, xvit 

Positivism, 80, 96-98; Comte 

and, 104-5; German 

philosophy and, 109 

Potential: dynamic, 69-70; 

thermodynamic, 126, 138-40 

Potential energy, 69-70, 73, 107 

Prebiotic evolution, 190-91 

Pre-Socratics, 38-39 

Principia (Newton), l, 28 

Probability, 122-24; Einstein 

on, 259; at equilibrium, 286; 

in far-from-equilibrium 

conditions, 143; entropy and 

142, 274, 297; and 

fluctuations, 178, 179; and 

irreversibility, 233-40; in 

quantum mechanics, 227; 

subjective vs. objective 

interpretations of, 274; in 

unstable systems, 271-72 

Process: physics of, 12, 105, 

107, 243; Whitehead's 

concept of, 258, 303 

Process and Reality 

(Whitehead), 93, 96, 310


Proust, Marcel , 17 

Pulley systems, 41-42 

Quantization, 220 

Quantum mechanics, 9, II, 15, 

34, 218-22; causality in, 31 1; 

correlations in, 286; cultural 

background to, 6; 

delocalization in, 289; 

demonstrations of 

impossibility in, 217, 296-97; 

Hamiltonian function in, 70; 

Heisenberg's uncertainty 

relations in, 222-26; and 

Newtonian synthesis, 67-68; 

and probability, 125, 178-79; 

and reversibility, 61; temporal 

evolution in, 226-29, 238; 

thought experiments in, 43 

Quetelet, Adolphe, 123, 241 

Radiation: black-body, 209, 215; 

interaction of matter and, 219 

Randomness, xx, 8, 9, 126, 

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, 137-38 

Reduction of the wave function, 

227-28 

Reductionism, 173-74 

Reichenbach, H., 97 

Relation, philosophy of, 95 

Relativity, 9, 34, 215, 307; in 

astrophysics, 116; Bergson's 

misunderstanding of, 294; 

demonstrations of 

impossibility in, 217, 296; 

Einstein's special theory of, 

17; and elementary particles, 

230; and Newtonian 

synthesis, 67, 68, 229; static 

geometric character of time 

and , 230; and thermal history 

of universe, 23 1; thought 

experiments in, 43 ; and 

universal constants, 217 -18; 

and velocity of light, 295 

Religion: ancient Greek, 38, 39; 

resonance between science 

and, 46-5 1 

Residual black-body radiation, 

209, 215 

Respiration, physiology of, 109 

Reversibility: of canonical 

equations, 71; of 

thermodynamic 

transformation, 12, 112-13, 

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. , 321n 

Schrodinger, Erwin, 18-19 220, 

242, 329n 

SchrOdinger equation, 226-29 

Science and Civilization in 

China (Needham), 278 

"Scientific revolution," 5, 6 

Scott, W, 322n 

Sein und Zeit (Heidegger), 310 

Self-organization, xv; in Benard 

instability, 142; and 

bifurcations, 160-67; in 

chemical clock, 148; in 

chemical reactions, 144-45; 

and dynamics, 208; as 

function of fluctuating 

external conditions, 165-67; 

life as expression of, 175-76; 

in slime-mold aggregation, 

156; in turbulence, 141-42 

Serres, Michel, 104, \41, 303, 

304, 320n, 323n 

Shakespeare, William, 293 

Signals, propagation of, 217, 

218

3


Time (cont'd) 

277-78, 295-%; in everyday 

life, 16-17; global judgments 

of, 17; in Hamiltonian 

function, 70, 71; Hegel on, 

90; and human symbolic 

activity, 312; internal, 

272-73, 289; involved in 

turbulence, 141; meaning of, 

in physics, 93 ; as measure of 

change, 62; in nineteenthcentury 

physics, 117; 

oscillations dependent on, 

148; positivist view of, 97; 

progressive rediscovery of, 

208; in quantum mechanics. 

229-30; revision of 

conception of, 96; roots of, in 

nature, 18; social context of 

rediscovery of, 19; in 

thermodynamics, 12, 129; 

varying importance of scales 

of, 301; see also Arrow of 

time; Irreversibility 

Time Machine, The (Wells), 277 

Toftler, Alvin, xi-xxvi, xxxi 

Trajectories, 59-60, 68, 121, 

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, 167-68 

Turing, Alan M., 152 

Unidirectional processes, 

258-59 

"Unified field theory," 2 

Universal constants, 217-19, 

229 

Universe: age of, 1; aging of, 

xix-xx; disintegration of, 

116; energy of, 117; entropy 

of, 118; expanding, 2, 19, 

215, 259; history of, 215; in 

Newtonian dynamics, 59; 

nonequilibrium, 22932; 

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, 16, 301 

Velocity, 57-59; distribution of, 

240-42; instantaneous 

inversion of, 61, 243-46, 

280-85; of light, 17, 55, 

217-19, 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, llO 

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, 219-20, 

226 

Wealth of Nations (Smith). 103 

Weiss, Paul, 174 

Wells, H. G., 277 

Wey l, Herman, 311

34Q 

Whitehead, Alfred North, 10, 

17, 47, 50, 79, 93-96, 212, 

216, 258, 302, 303, 310, 322n 

Wiener. Norbert, 295-96 

Wycliffe , John, xxii 

INDEX 

Zermelo, 15, 244, 253, 254 

"Zero growth" society, 116 

Zhang Shu-yu, 151 

Zola, Emile, 323n

ilya_prigogine_isabelle_stengers_alvin_tofflerbookfi-org

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?