THE COLLAPSE OF CHAOS

Jack Cohen is an internationally known reproductive biologist, who was a university teacher for thirty years, and has published nearly a hundred research papers. His books include Living Embryos, Reproduction; Spermatozoa, Antibodies and Infertility; and The Privileged Ape, a rather different look at human evolution. He now works with the mathematician Ian Stewart with whom he has explored issues of complexity, chaos and simplicity, producing several joint papers. He acts as a consultant to top science-fiction writers, designing credible creatures and ecologies, and frequently appears on radio and television programmes. His hobbies include boomerang-throwing and keeping strange animals.

Ian Stewart was born in Folkestone in 1945. He graduated in Mathematics from Cambridge and obtained a Ph.D. from the University of Warwick, where he is now Professor of Mathematics. He is an active research mathematician with over 130 published papers, and he takes a particular interest in problems that lie in the gaps between pure and applied mathematics. Ian Stewart has written or co-authored over sixty books, including Nature’s Numbers, shortlisted for the 1996 Rhône-Poulenc Prize for Science Books; The Science of Discworld (with Jack Cohen); The Collapse of Chaos, Fearful Symmetry; the bestselling Does God Play Dice?; Figments of Reality (also with Jack Cohen); and Life's Other Secret (several of which are published in Penguin). He is mathematics consultant for New Scientist and writes the ‘Mathematical Recreations’ column in Scientific American. In 1995 the Royal Society awarded him the Michael Faraday Medal for the year's most significant contribution to the public understanding of science.

THE BEGINNING

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THE COLLAPSE OF CHAOS

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How exquisitely the individual mind to the external world

Is fitted—and how exquisitely, too

The external world is fitted to the mind

And the Creation (by no lower name

Can it be called) which they with blended might

Accomplish.

WILLIAM WORDSWORTH

The next great awakening of human intellect may well produce a method of understanding the qualitative content of equations. Today we cannot. Today we cannot see that the water-flow equations contain such things as the barber pole structure of turbulence that one sees between rotating cylinders. Today we cannot see whether Schrödinger's equation contains frogs, musical composers, or morality—or whether it does not.

RICHARD P. FEYNMAN

PREFACE

At the heart of this book lies a paradox. The more we learn about the universe, the more complicated it appears to be, but we have discovered that beneath those complexities lie deep simplicities, laws of nature. How can simple laws explain complex behavior? Where does the complexity “come from”? How can the enormous diversity of life on earth have arisen from simple chemicals? How can a structure as complicated as the brain evolve? Over the centuries science has developed an extensive system of answers to such questions, a working philosophy known as reductionism, which shows how complexities on one level of description can be traced back to the interaction of large numbers of simple elements on a lower level. Much of the complexity of living creatures, for example, can be traced to the presence within them of the substance DNA. This truly gigantic molecule encodes a huge amount of information that tells an organism what to do when it develops.

The universe does not always seem complex. In our daily lives, we experience the world as a simple place—in fact, we would be unable to function if we had to grapple with the complexities as such. So in order to comprehend our world and humanity's place within it, we must do more than just explain higher-level complexities in terms of lower-level simplicities. We must also explain why, on every level of existence, we can deal with the world as if it were simple.

Where do the simplicities of nature come from? The conventional answer is that deep down inside, nature is simple: It functions on the basis of simple laws. Any large-scale simplicities that we observe—such as the spiral form of galaxies, or the tendency of a flock of geese to string out in a V—are just the underlying simplicities becoming visible on a higher level. Unfortunately, this answer is no longer convincing. Chaos theory tells us that simple laws can have very complicated—indeed, unpredictable—consequences. Simple causes can produce complex effects. Complexity theory tells us the opposite: Complex causes can produce simple effects. And conventional reductionist science tells us that inside the great simplicities of the universe we find not simplicity but overwhelming complexity. A galaxy's spiral arms contain myriads of stars, dotted almost at random. One of the deepest simplicities of biology is that the genetic material of almost all life-forms is constructed using the same giant molecule; but its workings are based on an intricate chemical code, whose unraveling for humans alone will require an effort comparable to the entire American space program.

The Collapse of Chaos shows how simplicity in nature is generated from chaos and complexity. These twin themes lie at the frontiers of modern thinking and are commonly confused with each other. A spate of recent books has emphasized one or the other as a general principle for understanding the natural world. We ask how the great simplicities of nature can persist within a chaotic universe. Our story combines chaos and complexity, and derives simplicity from their interaction. We show that the same simple large-scale features occur in many different complex systems because patterns of that kind do not depend upon detailed substructure.

The book is in two parts. The first half is about what science knows; the second half is about how to think about what science knows—and what it doesn't. The first half is a guided tour of the Islands of Truth that have been mapped by conventional science; the second half is an adventurous and unorthodox dive into the Oceans of Ignorance that surround them. In the first half we can provide plenty of road maps to tell you in advance just where we plan to take you. We can't do that in the second half—because where we wish to travel, there are not only no road maps, but no roads. However, we do put up plenty of warning signs: Here Be Dragons.

In outline, the first half of The Collapse of Chaos explains how simple laws of physics lead, through the chemistry of DNA molecules, to the complexity of living creatures, evolution, intelligence, and human culture. This is the conventional reductionist view. Along the way we give a streamlined introduction to the central preoccupations of modern science, which we hope you will find accessible and well integrated. These areas include cosmology, quantum mechanics, the arrow of time, biological development, evolution, consciousness, intelligence, and culture. We present science not as a fixed body of established knowledge but as a developing body of ideas.

The pivotal chapter 7 argues that asking where complexity comes from is really the wrong question. A more important question is, Why is there any simplicity? The book's first half provides an accepted scientific database from which the second half can address this more subtle issue by viewing science in context. We argue that simplicities of form, function, or behavior emerge from complexities on lower levels because of the action of external constraints. The focus moves from things to rules that govern the behavior of things. We offer a novel and indeed sometimes heretical approach to the questions discussed in the first half. For example, we argue against the view of DNA as a genetic blueprint for organisms by demonstrating that in principle two totally distinct organisms might possess identical DNA. Does this happen in nature? Ah, that would be telling…. In place of the view that evolution is the direct result of chemical changes in DNA, we argue that most changes to the form of an organism occur without any genetic changes at all—and that even when the form stays constant, the genes may be changing substantially.

The final chapter combines content and context into two new concepts: simplexity and complicity. Simplexity is the tendency of simple rules to emerge from underlying disorder and complexity, in systems whose large-scale structure is independent of the fine details of their substructure. Complicity is the tendency of interacting systems to coevolve in a manner that changes both, leading to a growth of complexity from simple beginnings—complexity that is unpredictable in detail, but whose general course is comprehensible and foreseeable.

We don't want you to get the impression that because the second half exposes gaps in the orthodox theories of the first half we think the first half is wrong. On the contrary, we think it's right, as far as it goes. But it doesn't go far enough. The reductionist story is nowhere near as complete as it appears to be. However, we want to do more than just locate gaps; we want to explore how to fill them. We want to alert you to another, much more mysterious, world. We want to explain something that hasn't yet been formalized, so in place of formal definitions and descriptions we will offer illustrations, images, metaphors, examples…. Like thousands before us, we are trying to come to grips with “emergent phenomena”—collective behavior of a system that somehow transcends its components. Because it transcends them, it can't be “in” the components—so where is it? Tricky. Our collaboration on this book is, to us at least, an example of such a phenomenon. What has emerged from our collective deliberations includes many things that neither of us “knew” independently: We could only have written the book together.

If either of us were writing this on his own, he would be much surer he was right but (paradoxically) much more cautious in presenting his ideas. Instead, our joint voice knows that it is probably wrong all over the place but puts its ideas forward with immense confidence. We hope that even when we're wrong, our ideas will take your mind to new places—places well worth a visit. We believe that when we're wrong, we're constructively wrong—wrong in a more informative way than the orthodox story is right.

A preface is the traditional place to express thanks to friends and colleagues who were persuaded or bribed to read and criticize early drafts. They include Janet Brandon, Teri Bristol, Baron Mendes da Costa, Richard Craven, Dawn Ann Drzal, Alan Garfinkel, Brian Goodwin, Steve Gould (to whom special thanks for his sterling role as jokes consultant), Helen Haste, Kate Lyons, Robert Mash, Anne McCaffrey, Jacquie McGlade, Tim Poston, Terry Pratchett, Irving Rapaport, Lena Sarah, Rabbi Pete Tobias, and David and Helen Wake. We are also grateful to the Victorian computer scientist Augusta Ada Lovelace and the alien inhabitants of the planet Zarathustra, with whom we held many useful conversations, some of which have been recorded in the book. Although those conversations were entirely our own invention, we learned a lot by listening to what our characters were telling us. In particular we are grateful to Neeplphut for pointing out a serious misinterpretation of the image of an egg as a computer with a start-up disk.

This is the sort of book that has to have footnotes; but footnotes look so horribly academic. So we've put them all at the back, with enough information to indicate what they're about but with nothing at the front to show there's a footnote at all. The notes are a bonus, intended to be read once you've finished the main book. But if you want a precise reference to something, or think we ought to have qualified our statements a little more carefully, or think that we've said something stupid—take a look at the notes.

One of the most useful words in the English language is “despite.” Despite the success of conventional science, we think there should be more to the scientific endeavor than just the study of ever more refined internal bits and pieces. Despite not having read all the great philosophers, we're invading their space and reinventing their wheels. And despite its many flaws, we hope you'll find The Collapse of Chaos a stimulating contrast to rigid orthodoxy and emerge from it just a little different.

1 SIMPLICITY AND COMPLEXITY

A yeshiva boy—a young man studying in a rabbinical college—took instruction from three rabbis. A friend asked him his reactions.

“The first I found very difficult, disorganized, and poorly explained, but I understood what he was saying. The second was a lot clearer, and much more clever. I understood part of that.”

“And the third? They say he is very good.”

“Oh, he was brilliant! Such a magnificent, resonant voice—it flowed as if from the heart. I was transported to realms beyond my imagining! So articulate, so lucid—and I didn't understand a word.”

Do we live in a simple universe, or a complicated one? Common sense—the way we think when we go about our daily lives—treats the world as a simple collection of familiar objects bearing known relationships to one another. When we open the fridge, pour milk into a saucer, and put it down for the cat to drink, we don't need to grapple with the thermodynamics of refrigeration, the molecular structure of the chemicals that go together to make milk, the interatomic forces that hold ceramic materials together, or the patterns of electrical firing of cells in the cat's brain. Our own brains have found ways to structure our world and make it comprehensible without going down these ever more complex paths. If you look out of the window you will see about forty kinds of things—flowers, trees, birds, fences, cars, people, clouds—maybe two hundred if you look hard and make distinctions between daffodils and daisies. You transact most of your daily business using a basic vocabulary of some five hundred words.

Closer investigation, however, reveals complexities that seem too intricate for the human mind to comprehend. For example, the underlying complexity of the object that in commonsense mode we encapsulate by the term “flower” is staggering. First, think of all the funny bits and pieces that you find when you look in more detail, like stamens and pollen grains. Those are there so that the plant can reproduce by making seeds. To do this it often needs pollen grains from another plant of a similar kind. Bees are a common source of such grains, which they brush off when they visit one flower, and deposit on others. For their pains, bees are rewarded with nectar, which they make into honey. Flowers need bees and bees need flowers. How did this remarkable commerce between plants and insects come about?

Here's a second complexity of plants: They know a trick that animals don't. They can take carbon dioxide from the air around them, water from the soil beneath, and use light to pull these two chemicals to bits and recombine them to form more complicated chemicals such as sugars—a process known as photosynthesis. The trick is possible only because plants contain a rather complicated chemical, chlorophyll, which animals generally lack. How does chlorophyll work?

Yet another complexity, the most astonishing of all: Plants grow from tiny seeds. Is all the complexity of a mature plant somehow compressed inside the seed? Or does complexity arise spontaneously as the seed develops? Neither answer seems very satisfactory.

In order to answer this kind of question, we're going to have to dig deeper. What do we mean by “simplicity” and “complexity”? Can simple causes produce complicated effects, or must all of the complexity that we observe in the effect somehow be cryptically present in the cause? Is complexity a conserved quantity, or can you create complexity from nothing? The relation between simplicity and complexity in nature is one of the deepest and broadest questions that faces modern science.

COMPLEXITIES HAVE TEETH

The simplicities of our commonsense existence are the placid surface of a teeming ocean of complexity. This complexity is not just something outside us but hidden from view, like the ocean depths or the earth's geological strata. It is also, in a very real sense, inside us, just as chlorophyll is inside the plant. We have a comparably complicated chemical trick: Hemoglobin in our red blood cells transports oxygen around our bodies. And everything that we know about the world comes to us by courtesy of a major consumer of that oxygen, the most complex structure that we have yet encountered: the human brain. As the joke goes: “If our brains were simple enough for us to understand them, we'd be so simple that we couldn't“.

Faced with the incredible hidden complexities of the universe, it is not surprising that many people take refuge in the commonsense simplicities and prefer not to dig into what lies beneath them. “Ignorance is bliss”; “What the eye doesn't see the heart doesn't grieve over.” Unfortunately, the complexities of the real world have sharp teeth. Braking a car may seem a simple process: You just push the brake pedal and the car stops. However, if your mental picture of “brake” is no better than this—a kind of slogan, “Brakes are for stopping”—then you may get into terrible trouble on an icy road. If you understand a little bit more about how brakes work—they slow down the wheels' spin, so that friction between the tires and road can make the car slow down too—then you won't make that particular mistake. We are surrounded by technological gadgetry whose surface simplicity is becoming increasingly deceptive, as anyone who tries to program a video recorder or set a digital watch will discover. If you open up a Victorian steam engine you can get a fair idea of how it works. If you open the back of a television set nothing inside makes much immediate sense; indeed, most of the box is empty. Nature is like this too: Look at living creatures under a microscope and they make even less sense than a television. “No user-serviceable parts inside.”

Is this just a problem of adopting the wrong point of view? The ordinary city-dweller finds the jungle complex and incomprehensible but is entirely comfortable when surrounded by the ordered simplicities of New York, such as department stores, subways, taxicabs, drug-dealers, and muggers. The jungle-dweller is baffled by New York but is entirely at home with the snakes and the spiders in the nice, simple jungle. Crocodile Dundee rested entirely on this simple premise, but with the Australian outback in place of the jungle. An apparent complexity that may actually be simple is the fact that most holly bushes have spiky leaves only near the bottom. If you've never noticed this, take a look next time you pass one. It's very striking. Now, isn't it clever of the trees to save on the cost of making spikes and concentrate their efforts at the bottom, where animals might eat the leaves? How incredibly complex nature is! Except, of course, that we don't know whether it does cost the plant anything to make spikes. It may be harder to make all the leaves the same than it is to make them different, just as it's harder to make a flat piece of ground (all at the same level) than a sloping one (different heights). So the holly bush may be doing things the easy way, and it's just we who think it's complicated.

Even accepting that point, the universe still looks like a pretty complicated place if we remove our commonsense blinkers and look beneath our comfortable, illusory simplicities. If we don't want to be caught napping when that complexity decides to bite us, we must come to terms with it. There are two main approaches. Recondite professions (such as astrology or plumbing) claim to handle these hidden complexities in their own terms, but shy away from any attempt to explain their methods. The astrologer who casts your horoscope and predicts the approach of a tall dark stranger, and the plumber who produces an odd-shaped wrench to unscrew a nut that you didn't even know your sink possessed, are both keeping a lot up their sleeves. Science adopts a radically different approach. It claims to see beyond the apparent complexities to the underlying simplicities, which it calls laws of nature. By working with these simple laws, rather than trying to handle the complexities as complexities, science claims to render the world once more accessible to common sense. It is common sense on a more refined level, common sense with different intuitions; but when a physicist argues that perpetual motion machines are impossible because of the law of conservation of energy, the general line of thought is just as simple and transparent as the statement that the cat needs some milk because it's thirsty.

THIGHBONE EQUALS SPACESHIP?

Common sense, in short, has a great deal going for it—provided that it resembles the actual world sufficiently well for the purpose in hand. Common sense works when it is congruent to reality. When it is not, it can go horribly wrong—like, for example, throwing water onto a gasoline fire. The gasoline, still burning, floats on the water, and the fire just spreads. The slogan “Water puts out fires” sounds like common sense, but it goes wrong more often than it works. The trouble is, our brains mostly think in such slogans. The word “comprehend” originally meant “grasp.” To understand something is to grasp it with your mind, to make it into an object that you can hold as a unit. As the human race has evolved, it has developed this technique into a way of life.

By reducing complexities to underlying simplicities, science has allowed our brains to grasp the hitherto ungraspable. And once you have grasped something, you can use it, as a tool—provided that you know you have grasped it, so that you can play the same trick consciously again. You can then grasp at what we shall call a meta-level, a level of greater generality, by grasping the concept “tool.”

We will use the prefix “meta” frequently, to mean “a more general version of,” or sometimes “a higher-level way of thinking about.” For instance, meta-physics (a word that actually exists in dictionaries, in the form “metaphysics”) is a higher-level theory of how physics works; it's not a part of physics itself. “Meta-plumbing” isn't in the dictionary, but if it were, it would mean “a higher-level theory of plumbing.” Meta-plumbing would concentrate on such questions as how to look as though you really understand what you're doing when the living room is several feet deep in water—instead of how to make a good joint, which is ordinary plumbing.

Instead of a few special brain slogans such as “Rocks are for throwing at rabbits” or “Fire is for burning things,” you can gear up a level to the meta-slogan “Tools are things that can enhance human abilities,” and start thinking about developing new tools out of what you have already grasped. This process of research and development leads to technology. Stanley Kubrick's now classic science-fiction film 2001: A Space Odyssey (with screenplay by Arthur C. Clarke and Kubrick) begins with a sequence of events whose star turn is the apeman Moon-Watcher. His investigations culminate in the discovery that you can beat a leopard's brains out with a bone. In a paroxysm of joy, Moon-Watcher tosses the thighbone tumbling end over end, high into the air—

The symbolism may be trite, but the message is deep. When protohumanity learned how to generalize about the structure of the natural world, to classify similar objects under identical labels—in short, to exploit the power of metaphor—it latched onto a wonderful trick for simplifying what would otherwise be complex beyond human understanding. You can't track a snake through the jungle if every leaf, every insect, every broken branch is seen as a unique individual. You have to distinguish things-that-sting from things-that-are-harmless, things-that-break from things-that-block-your-path, things-that-make-noises-when-stepped-on from things-that-don't. It's not easy to get the idea of making a hut from sticks and mud unless you have a good grasp of the stickiness of mud and the muddlability of sticks. And you have to achieve all of this with a considerable degree of rapidity: Label thy neighbor before it labeleth thee. The mental computations must be in real time, so something quick and dirty is the order of the day. We have to think in slogans, because a really high degree of congruence with reality takes too long. So a flash of black and orange is labeled “tiger,” when it might be just a funny-colored leaf—because tigers can bite. It's better to be safe than sorry.

WILD ELECTRONS

Once humanity grasped this trick—of attaching labels, generalizing, seeing the common simplicities in nature instead of the baffling complexities—then discovering the laws of nature, science, and technology was really just a lengthy exercise in research and development. The trick seems to work best when the underlying simplicities of the world are perceived on a mechanistic level: Physics is about electrons or atomic lattices rather than daffodils or prides of lions. Indeed, nature seems to simplify as we move toward subhuman or superhuman scales. On the ultramicroscopic scale, matter reveals itself as assemblages of huge quantities of particles of an extraordinarily limited number of types, everything being governed by the laws of quantum mechanics. On a pan-galactic level, there are just vast swaths of matter moving through empty space under the force of gravity, sliding gracefully and inevitably along the curved world-lines of general relativity. Modern physics claims to be on the verge of a final synthesis of the two theories, micro and macro, into a Theory of Everything.

On our human level, there seems to be a lot more going on: shoes and ships and sealing-wax, cabbages and kings. But science books tell us that this rich mixture, the stuff of which we build our lives, is no more than a logical, inevitable consequence of a few elegantly simple laws, those of quantum mechanics and relativity, or the presumably even more elegant laws of their long-sought synthesis. This claim is true in many respects, but the simplicity of physics is to some extent an illusion. We will explore both sides of the question as our story unfolds, but for the moment we fasten upon the possibly illusory nature of physics' simplicities. The point is not that physics deals in unrealities, but that its considerable successes are to a great extent dependent upon its choice of subject-matter.

The universe appears to simplify at nonhuman scales because we possess a very limited set of techniques for converting its behavior into human-scale effects, in both space and time. In order to observe nature on spatial scales markedly different from our own we must either amplify things (microscopes) or bring them closer (telescopes). We similarly alter time scales with high-speed or time-lapse photography, or by compiling records of observations for a very long time. We probably miss a lot of the fun by peering through glasses darkly. It takes a galaxy a million years to perform one stately revolution; how can we possibly appreciate the universe on the time scale of a galaxy? At the other extreme, subatomic particles go about their business at such a frenetic pace that we need the most expensive machines in the world just to catch a glimpse of them.

Physics takes a pragmatic and severely critical stance. It concentrates on simple, highly controlled systems; in return it expects impeccable agreement between experiment and theory. Thus quantum mechanics predicts, with spectacular success, the behavior of a single electron in the potential well of a proton; this is the hydrogen atom, whose theoretical analysis features prominently in the pages of any respectable physics text. The behavior of electrons “in the wild” is quite another matter, and while physicists generally believe that this is governed by the same laws that they have established experimentally for isolated electrons in a hydrogen atom, this belief is founded on extrapolation from the simple systems accessible to experiment (and—but only recently—on supercomputer calculations that simulate tiny quantities of bulk matter). By its nature, it cannot be tested directly, “in the wild.” Physicists would argue, quite properly, that in the absence of evidence that wild electrons behave differently from tame ones, the onus of proof is upon the skeptic. Physics deals with an invented, simplified world. This is how it derives its strength, this is why it works so well: Its raw material is of a type that can be placed in simple settings. Sciences like biology are less fortunate.

HOW DO WE DISCOVER LAWS?

We're going to spend a lot of time talking about “laws of nature,” and it will help if we can develop some appropriate mental imagery. The most suitable one for our purposes is Isaac Newton's law of gravity, which states that the gravitational force between any two bodies is inversely proportional to the square of the distance between them. (It also states that the force is directly proportional to their masses, but we won't worry about that bit.) Newton discovered his law by thinking about possible causes for the elliptical shape of the planetary orbits, previously discovered by Johannes Kepler. Because Newton was an unusually good mathematicianhe's generally ranked among the top three of all time—it didn't take him too much thought to prove that a point particle obeying inverse-square-law attraction will pursue an elliptical orbit. He wasn't entirely impressed by this argument, though, because planets aren't points. Eventually he discovered that, on the assumption of an inverse-square law of gravity, spherical bodies produce exactly the same forces that they would if all their mass was concentrated at the central point. He was a lot happier after that. But the real clincher—dramatized in the almost certainly fictional story of the apple—was that the same law, correctly interpreted, also governs the motion of falling objects on earth. Apples and the moon both fall because they are attracted to the center of the earth. We don't normally think of the moon falling, mind you. But really it's falling all the time; it moves sideways in its orbit so that the amount of falling is just enough to compensate for the earth's curvature. Although the moon perpetually falls, it remains much the same distance from the center of the earth. Newton invented a brilliant thought experiment: A cannon on top of a tower fires a projectile at ever higher speeds. As the projectile falls, it also moves forward. At sufficiently high speeds it travels all the way around the earth and hits the cannon. It's in orbit—and still falling.

On page 13 is a diagram of the kind of thought process that is involved in digging out the law of gravity (Fig. 1). At the top are two different real-world phenomena: the falling apple and the orbiting moon. Two funnels lead down to a deeper layer of explanation, where we find in each case the same principle: inverse-square-law attraction. Notice that in this image we look down the funnels, but the arrow of causality, which we assume must be present, runs upward. The direction of discovery is from top to bottom, peering into the funnel, but the direction of explanation is from bottom to top. In just the same way, a simple map explains complicated territory, but you explore the territory to draw the map. (See Fig. 2.)

Newton's discovery of the law of gravity required lots of funnels. At their tops were things like Mars, Jupiter's satellites, and the wobbling of the earth and moon on their axes. At the bottom of every funnel he found the same thing: inverse-square attraction. So he named his find the inverse-square law of universal gravitational attraction—“law” because he thought that it was always rigidly enforced, and “universal” because he was convinced that it applied to all pairs of bodies throughout the entire universe.

Newton discovered other laws of nature using similar methods. Starting from Galileo Galilei's observations on the motion of bodies, he dug out a number of “laws of motion,” of which the most important is the statement that the acceleration of a body is proportional to the force acting on it. Especially in combination, Newton's laws were a spectacular success: They explained the “System of the World” to a remarkable degree of accuracy.

THEORIES DESTROY FACTS

As we've seen, you have to look hard and deeply at the world before you realize that it is complex. You have to look even harder to see that much of its complexity has simple causes. From Newton's time on, every scientist's dream was to discover a new law of nature—a new underlying simplicity that could be used to explain innumerable disconnected observational facts. But “law” is a word with curious and rather inappropriate overtones. Laws are invented by human societies to control individual or collective behavior; they proscribe certain activities and prescribe punishments for disobedience. But you can't break a natural law. Nevertheless, nature “obeys” simple laws. To the intense irritation of generations of schoolchildren, these laws seem to be mathematical. We human beings observe underlying simplicities in the way the universe works; we distinguish them by their mathematical elegance, and we call them laws of nature. This mathematical approach to the universe works; it underpins virtually all of science. “God,” said Plato, “is a geometer.” Paul Dirac called God a mathematician; James Jeans went further, defining God's specialty as pure mathematics. The best mathematics has a purity and elegance that somehow seem to capture the true essence of phenomena.

“Theories destroy facts,” said Peter Medawar at a Mensa conference in the 1960s. He meant that a general theory can remove the need to record huge quantities of isolated facts. One way to specify where the planets in the solar system are (or will be) on different dates is to draw up huge tables of numbers. However, once you've got Newton's laws, all you need is a much shorter list saying where they are, and how fast they are moving, at one instant of time. All else is a matter of routine calculation. Some thirty years ago the mathematician René Thom deplored biology as “a cemetery of facts.” Many biologists protested vehemently, thinking that Thom was deploring the facts. He wasn't. He was deploring the cemetery. He wanted a theory to organize all of the facts, and bring the cemetery to life.

In Newton's day, scientists (they called themselves natural philosophers then) held a rather strong view of laws of nature: They thought they were true. However, there has since been what Thomas Kuhn calls a paradigm shift—a change in worldview. Newton's law of gravitation has been replaced by something that predicts the observed motion more accurately: Albert Einstein's theory of general relativity. Einstein's theory, which views matter as the curvature of space-time, and gravity as the distortions in motion that result from this curvature, explains a number of oddities that do not fit the Newtonian picture. These include a slow drift (“precession”) of the orbit of Mercury, and the bending of rays of starlight by the sun's gravitational field. Nowadays we can observe the light from a distant quasar, bent by an intervening galaxy that acts as a “gravitational lens.” The result is that we see several images of the same quasar, a kind of cosmic mirage.

In a different paradigm shift, Newton’s laws of motion were replaced by quantum mechanics, in which mass and energy come in tiny, indivisible packages, and everything is as much a wave as a particle. Relativity and quantum mechanics are both systems of laws, in the same way that Newton's laws were. In particular, if you look down a funnel from either chemistry or cosmology, you see quantum mechanics at the bottom (Fig. 3).

Science's great generalizations are produced in just this manner—by finding similar laws, similar simplicities, inside different complexities. Systems that have the same deep similarities must obey the same simple rules.

As we'll describe at greater length in the next chapter, quantum mechanics is implicated in both the atomic structure of chemical elements and the Big Bang that gave rise to them. Cosmologists in particular are very impressed by this closing of the logical circle from the microscopic world to the macroscopic world. Like the scientists of Newton's time, many of them think that this proves that the law of nature expressed in the Big Bang is true. We don't even learn from history that we don't learn anything from history.

Even though relativity and quantum mechanics are better laws than Newton's—they fit more aspects of the real world more accurately—scientists still find Newton's laws extremely useful. The mathematics involved in Newton's laws is simpler; the mental picture of masses and forces is on a more human scale. Newton's laws break down badly only at the two extremes of the microscopically small or the astronomically large: When bodies move very fast, or have enormous densities, or are very small. For most human-scale phenomena, Newton's laws survive, essentially for reasons of convenience.

Biology too has “laws,” but these are less frequently expressed in mathematical language. An example is chlorophyll. Millions of different plant species build carbohydrates from sunlight by the process known as photosynthesis, and down the funnel from each species lives the same chemical: chlorophyll. You don't need a different method for each different species of plant. The word “law” is not so appropriate here: Chlorophyll is an underlying universal. But the role of a generality is exactly the same as that of a law, and for our purposes we'll consider statements such as “Chlorophyll is responsible for photosynthesis” to be just as much a law of nature as Newton's law of gravity is.

IS THE UNIVERSE A COMPUTER?

Either because the laws of nature are couched in mathematical symbolism, or because science cannot progress safely in the presence of ambiguity and imprecision, scientists tend to express natural laws as mathematical statements. It would be wrong, however, to read too much into this. Consider a stone tumbling down a hillside, bouncing off rocks and molehills, until it reaches its final resting place at the bottom of the slope. If the stone really is implementing mathematical laws, then in a few seconds it will have performed a series of calculations beyond the capabilities of the fastest supercomputer. But is that really what the rock is doing? Measuring its own position to the hundreds of decimal places that we know are needed to guarantee the “correct answer”? Computing its way from collision to collision in an orgy of dynamical equations? Some physicists and philosophers think so; in their view, information, rather than matter, is the basic material of the universe. The universe itself then becomes a supercomputer of unprecedented speed and power, busily pursuing the consequences of its “program,” its program being the laws of nature.

Alternatively, the simple laws that we consider fundamental may not be fundamental at all, but just approximations of how nature behaves, or consequences of that behavior. We now know that Newton's laws are not rigid rules that nature just obeys; they are excellent but sometimes inaccurate descriptions of what nature does. They are not nature's laws but human laws, and like all human laws they can be broken. Indeed, according to another human law, Murphy's, they always will be—an interesting case of self-reference. If nature breaks our laws, then our calculations will bear no relation to the way in which nature actually works. We may use the laws of dynamics to calculate where the stone will fall; but that's not how the stone does it. It certainly can't if we've got the wrong laws. The Newtonian stone has no choice; it is forced to fall wherever it does. In this view the universe is a machine rather than a computer; it is composed of matter, and it is in the nature of matter to behave in ways that happen, coincidentally, to mimic certain computations that appeal to humans.

That was a classical picture of a moving stone. The quantum picture is more subtle, and far stranger to human intuition. In a quantum view, the subatomic particles that make up the stone actually follow all possible paths, consistent with the laws of quantum mechanics. According to the quantum paradigm, what we see is the superposition of all of those potentialities. It just happens that the result of this strange process looks like a lump of rock moving under Newtonian laws. In this picture the Newtonian laws are viewed as mathematical consequences of the real quantum laws, valid for modest but bulky quantities of matter moving at moderate speeds.

In other areas of science, especially those where really accurate measurements or repeatable experiments aren't possible, people nowadays tend to speak of “models” rather than “laws.” They look for underlying rules and regularities that explain a limited range of phenomena in simple, graspable terms. From that point of view, “laws” may be just spectacularly successful, very simple, models. The important thing is that, even though we can't be certain that what we think of as laws of nature are actually true, we do see a lot of patterns and regularities in the world, and we can use these patterns very effectively to bring certain aspects of the world under our control. For instance, the laws of aerodynamics work sufficiently well that airplanes designed using those laws stay up. The vast bulk of evidence, while not quite so conclusive, points to the flight of birds as a consequence of those same laws. However, we can't yet start with aerodynamics and end with a proof that a bird, too, will stay up; but despite such admitted uncertainties, there still seem to be simple laws at work. It's just that some operate further behind the scenes than others. Indeed, the further behind the scenes the laws are, the more we tend to think of them as being “fundamental.”

CONSERVATION OF COMPLEXITY

Many people seem to have an intuition that can be most conveniently described as “conservation of complexity.” They expect complicated effects to have complicated causes (and simple things to have simple causes). The eye is a very complicated organ; therefore, a simple explanation like Darwin's theory of evolution must be wrong. Economic inflation is a simple effect; therefore, there must be a simple cause, therefore an easy way to deal with it. Human consciousness is enormously complicated; therefore it cannot result from laws of nature: it must have some supernatural element. Early theories of biological development contended that inside every human sperm there is a homunculus, a tiny person perfect in every exquisite detail. The argument was that you can't make a person unless you've already got one, albeit in cryptic form.

We may tentatively define the complexity of a system as the quantity of information needed to describe it. One small whole number conveys very little information, and can describe only very simple things such as the position of a dot on a computer screen; but a sufficiently long list of numbers can describe far more complicated things. (A videotape is, at root, just such a list, and you can put a description of anything onto a long enough strip of videotape.) If you think that complexity—in this “bit-counting” sense—is conserved between cause and effect, then there is only one way in which simple laws can produce complex effects. Any complexity in nature, such as the intricate collisions of molecules in a gas, must arise because huge numbers of objects are interacting. When simple laws govern systems with a large number of variables, the underlying order is obscured by our inability to track every component, and it becomes inaccessible to our limited brainpower.

Within the last decade this view of the origin of complexity has been strongly challenged from a variety of directions. At the frontiers of today's mathematics are startling paradoxes about the way the world can change. In particular, we now know that rigid, predetermined, simple laws can lead to behavior so irregular that it is to all intents and purposes random. Simple rules can produce incredibly complex effects. An example is the infamous Mandelbrot set, which we describe in chapter 6; it is one of the most intricate geometric objects ever to have decorated a teenager's wall (Fig. 4). However, the computer program that generates it is only a few instructions long. This phenomenon—of vastly complex effects arising from simple causes—is known as chaos, and there is plenty of evidence that it is widespread. Stuart Kaufmann of the Santa Fe Institute has pioneered the study of a converse process, “antichaos,” in which complex causes produce simple effects; this is also widespread. Complexity can get lost as well as being created.

FIGURE 4
The Mandelbrot set (a). If any region of the set is magnified, new and intricate detail appears, as in the sequence of blow-ups, b-h. (Peitgen and Saupe, The Science of Fractal Images)

We can still save the conservation principle, but only if we refine our concepts of simplicity and complexity to include the processes that generate phenomena as well as the phenomena themselves. Suppose we say that a system is simple if it can be prescribed by simple rules, rather than described by simple lists of numbers. This transfers the problem of complexity from results to processes, from effects to causes. It turns out that simple processes can generate complex results, in the sense that a process defined by a small list of numbers can produce effects that can be described only by huge lists. One interesting side effect of adopting this proposal is that it then becomes very hard to tell whether or not something is “really” simple. We can no longer just look at it and count numbers; we must ask ourselves whether it might result from some short list of secret rules. Which rules? The sky's the limit.

THE BEGINNING

THE COLLAPSE OF CHAOS

CONTENTS

PREFACE

1 SIMPLICITY AND COMPLEXITY

COMPLEXITIES HAVE TEETH

THIGHBONE EQUALS SPACESHIP?

WILD ELECTRONS

HOW DO WE DISCOVER LAWS?

THEORIES DESTROY FACTS

IS THE UNIVERSE A COMPUTER?

CONSERVATION OF COMPLEXITY