Fruitful Chaos

by Richard Hodges


Why can't we predict the future?

There is a feeling of contradiction between the certainty expressed in science about the "laws" behind phenomena and the fact, obvious to every practical person, that our attempts to predict the future fall far short of certainty.

As practical man, we might be content to let the matter rest there. As philosophical man, we have to inquire deeper: What does it mean that we can't predict the future?

Is it just that we fail to take enough into account? Or, is there something unpredictable in principle about nature? Is this unpredictability, this chaos, to be regarded merely as an impediment to our desire to know and control nature? Or, does it conceal a key to a new understanding which we have missed so far?

Science has only recently begun to ask these questions. Though the answers are not all yet in, a new vision is taking form. This vision indicates a previously unguessed relationship between what is and what can be known. There seems to be a truth here about the human situation-about the improbable and complex events of this world, about the mind and its wanderings, and about how these two, our outer and inner lives, play together as necessary partners.

Science appears to be about the external world. But all scientific ideas have correspondences-resonances-with facts of man's inner nature. Perhaps that is where scientific ideas originally come from. An approach to science is suggested which reverses this process: to listen for these resonances while studying scientific ideas.

The revolutionary idea of chaos may be placed in the context of two earlier revolutions which delineated the form which our scientific understanding of the physical world has today. The first scientific revolution was based on the idea that nature is governed by laws which are discoverable by observation and analysis. It began over ten thousand years ago with the study of the macrocosmos, the larger world which determines the conditions of our small existence according to the movements of the sun, moon, stars, planets, seasons, and tides. Requiring many human generations of organized observation, this revolution succeeded grandly and led to a vision of a clockwork system of precisely interlocking cycles of movements in the heavens. In such a view, the unknown is pushed back to the beginning: the initial condition, established by God, after which God might rest, since his further intervention had no place in a clockwork universe.

The second revolution began around the end of the 19th century. Science was facing the microcosmos, the constituents of matter, trying to understand how atoms and light work. Certain observations seemed to require us to admit an essential randomness into the very foundation of space and time. A radically new idea was developed: that there is a fundamental limitation to what can be observed. It is a matter of principle that not everything can be known.

To express the laws of observation required the development of a new mathematical model of matter. Echoing an idea found in ancient teachings, particles of matter are found to be packets or quanta of vibrations, much like musical notes. This is called the quantum principle. According to the mathematical laws of vibrations, a note has a certain degree of indefiniteness of pitch and a certain degree of indefiniteness of time (these two can be calculated and are reciprocally related). It is this indefiniteness which corresponds to the apparently paradoxical fact that we cannot simultaneously observe the position and the movement of a particle with perfect accuracy.

But the interaction of particles requires an act of observation (of one particle by another with which it is interacting). Thus, the limits of observation are realized as apparent randomness in the behavior of matter. The quantum God is an impartial observer, but he does not, can not, observe everything. And, he must work in partnership with a microscopic demon who randomly decides the outcome of every observation.

The principle that there are inherent limits in what can be known has resounded several times in the last century of thought. The quantum revolution was based on the idea that the limits of observation have a fundamental place in our understanding of nature. Relativity assumes that the structure of space-time is conditioned by the fact that there is no possible communication between any two events faster than the speed of light; we gain an appreciation of the scale of the universe known to science in understanding that the vast bulk of all time and space is hidden from us by this veil. The development of the philosophy of mathematics was greatly altered by a demonstration that there are things which are true which cannot be proven within any closed system.

This principle resurfaced again in the form of limitations on what can be predicted, the idea of chaos, which was the basis of the third revolution. The third revolution is happening now. This revolution studies a scale intermediate between the macrocosmos and the microcosmos. It examines the unpredictable phenomena of everyday experience, the mesocosmos: the weather, the shapes of landscapes and other natural things, the flow of water, the rise and fall of plagues and of stock markets.

It is important to understand that what scientists now call chaos is not mere randomness. It is not due to a randomizing demon inherent in matter, as in the microcosmos. Rather, it is a consequence of the iterated action of ordinary mechanical laws of motion, which leads to results of previously unsuspected complexity and beauty. Also, it is not that we do not know the initial conditions: no matter how carefully we pour water over a perfectly smooth edge, it will break up in a chaotic pattern that we cannot predict. In the mesocosmos, such behavior is the rule, not the exception.

Let us look at an example of chaos. Start with a shallow pan of water, slowly rotating. The surface will settle down until it is perfectly smooth. If we want to follow the motion of a particle of water and predict where it will be in the future, it is easy to do: each particle rotates smoothly and regularly, like the hands of a clock. Even more important, if the flow is unexpectedly disturbed (a butterfly brushes the surface), the disturbance will soon settle down. The future motions of the particle of water will be only a little different from our predictions.

If we now gently heat the bottom of the pan, the heat will affect the movement of the water. Hot water is lighter than cold, and succeeds in setting up a vertical circulation so that it can rise to dissipate its heat to the air above.

A certain amount of heat can be accommodated without making the flow unsmooth. But if we increase the heat beyond a certain point, an irregular form of movement begins to develop, with cells of turbulent convection.

What is most important about this movement is that it is unpredictable. Suppose we could observe the exact movement of every particle of water in the pan at an instant. By using a powerful computer, we should be able to predict the movement at any future instant, using well-known physical laws. But now suppose a butterfly touches the water. Though its immediate effect is minuscule, after some time elapses we find that our predictions no longer apply.

This is an important principle about chaotic systems: any disturbance, no matter how small, eventually affects everything in the system. All predictions get scrambled.

The atmosphere and its weather is a chaotic system (in fact the heated rotating pan is a kind of simplified model of the atmosphere). If a butterfly flaps its wings in China, it may cause a hurricane to hit Texas a week later.

The butterfly effect means that it is useless to try to predict the weather more than a few days in advance. The unavoidable errors in the data which we feed the computer act like butterfly wings, and the computer's predictions diverge completely from reality in a week or so. Even if we could have absolutely exact initial data, and eliminate all the butterflies in the world, quantum randomness causes unpredictable movements of atoms. Finally, there is a kind of inner butterfly in the computer: computers cannot make perfectly exact computations; the results have to be rounded off at each stage of calculation. Since the computer's simulation of a chaotic system is also chaotic, these round-off errors soon cause the simulation to diverge from reality.

We are now forced to rethink the meaning of prediction. In the clockwork universe, there was the attitude that if you could measure everything accurately enough, you could use a computer to predict the future. You could represent large-scale processes by approximations without committing any essential errors in the computation. For example, the whole earth could be represented as a massive spinning sphere as far as its interactions with the sun and planets were concerned. To predict the weather in a non-chaotic atmosphere, you would just have to know the large-scale patterns of fronts, vortices, clouds, etc. The butterflies could be ignored.

Thus, the universe could be adequately simulated using a computer much smaller than the universe itself. Such a universe would be very wasteful, since we could replace it with a computer simulation without losing anything essential. We could even transubstantiate the universe into a series of computers, each simulating a whole universe as interesting as the "real" one. In fact, which universe is the "real" one? Maybe the universe we know, including ourselves, our thoughts, hopes, suffering, everything, is "just" a simulation running in a vast computer!

Obviously, there is something suspect about this thinking. For one thing, it leads to an awkward form of infinite regress (simulations within simulations within simulations). For another, there could be no life in such a universe, as we shall see later. But it is a realistic caricature of the way in which prediction was understood in thought inspired by the scientific vision of the clockwork universe.

In a chaotic world, this attitude is not tenable. We cannot ignore the butterflies. We cannot make valid predictions with only large-scale data. A computer capable of simulating the universe would have to take into account every detail, with absolute precision. It would have to be as large as the universe itself. The only simulation of the universe is the universe itself. It has been said, half-jokingly as scientists like to do when talking about God, that the reason God created the universe was that that was the only way he could find out what was going to happen. For the God of chaos, the universe itself is a co-participant in creation.

Perhaps this echoes the humanistic revolution which is fermenting in contemporary culture-the understanding that we are neither the authors nor the victims of our lives. Like the God of Chaos, our role is to participate in creation through reciprocal exchange with life.

It is only recently that scientists have begun to study chaos and to understand the laws which it obeys. For a long time, they ignored the very idea of chaos. Science worked, it seemed, by separating out an aspect of the world that was simple enough to study. Many scientists have a kind of faith that if you explain enough simple aspects of the world, you can explain everything. They tend to believe that the complexity of the real world is a side effect of simple underlying laws, a kind of ontological noise which it isn't necessary to study. But the real world, the mesocosmos, is essentially complex. And the idea of chaos offers us the first real approach to appreciating this complexity.

Though science could "explain" life, when it was faced with phenomena of practical importance, such as turbulence, it was almost helpless. In the 1930's, Russian scientists developed a theory which became the standard textbook way of calculating turbulence. But the calculations turned out to be wrong, because the model behind the theory was wrong. Nobody had bothered to observe what turbulence was actually like.

In the 1970's, a few scientists began to look at turbulence in a different way. The key idea is to imagine the set of all possible states of motion of the fluid as a "space." This space is called phase-space. We are going to have to try to understand this idea in order to get a feeling for chaos.

Each point in phase space represents one possible state of motion of the fluid at one instant. It is like the metaphysical idea of higher eternity [Ref 1], the continuum of everything that ever did, will, or could happen. Time is represented as a curving line moving in phase space, each momentary configuration of the fluid being followed in subsequent moments by new configurations.

Phase space has long been used as a way of visualizing a physical process. By standing above the worm's nest of individual events in time, you get a bird's-eye view of the process as a whole. This often enables you to see patterns that can't be seen from worm level.

As a theoretical tool, phase space is an essential tool enabling science's understanding of quantum mechanics and statistical mechanics (the theory that explains how heat is the random microscopic movements and collisions of myriads of molecules). With the recent appearance of computer workstations with image screens, phase-space plots have become practical tools for studying real systems in engineering, science, and medicine: the graphical plot of the phase-space history of a system reveals the essential nature of the system directly to the eye.

Before chaos, the kinds of behavior people looked for in phase-space plots were simple-the system path either spirals into a single point, or spirals into a regular loop. The first possibility represents eventual stasis; the second possibility represents eternal repetition of the same sequence of events. A third possibility occurs when randomness (not chaos) operates in the system, giving rise to a uniform cloud of paths in phase space. These different possible phase space plots are called attractors, because all states of movement fall into them.

Technically, the attractor is the set of points in phase space representing states near which the system spends most of its time. Thus, when you plot observed states of a system with black dots, the attractor is the figure that stands out to the eye.

What nobody had appreciated was that in real systems such as turbulent fluids and many other physical systems, under certain conditions which are actually very common, a different kind of attractor exists. Because of their apparently contradictory properties, they were coined with the name "strange attractors."

What was strange about them? Viewed as a pattern, a strange attractor etches an intricate filigree of points in phase space. This filigree has unusual properties: If you zoom in on a section of it and magnify it to any degree, it still reveals new intricacies of structure. Strange attractors are often very beautiful.

This property implies another fact: a strange attractor has more "black" in it than any one-dimensional attractor (such as a simple loop in phase space). But it has less "black" than the kind of cloud that results from randomness. In fact, it has more than one dimension and less than two dimensions. It has fractional dimension. For this kind of pattern, the term "fractal" was coined.

People had seen this kind of pattern in their phase-space plots before, but they had ignored or dismissed them, thinking of them as caused by mistakes or by malfunctioning apparatus. Only in the late 1970's, with the development of the chaos principle, did people begin to look with new eyes.

Scientists began to reexamine familiar processes. They found that it is often very useful to know that a process is chaotic. For one thing, you know that it it is unpredictable-or at any rate, it is only predictable for short intervals of time. for another, you can hope to model the internal dynamics of a chaotic system, which may be very simple even though its behavior appears to be complex. You can also determine when irregular behavior is intrinsic to the system, caused by its own internal dynamics, rather than by external noise. And you can sometimes make a system less chaotic, or more chaotic, by modifying its internal dynamics.

Many natural phenomena are now known or suspected to be chaotic: fluid flow, including the weather, is chaotic; chaos in the solar system produces the peculiar orbit of Pluto; the fine structure of the rings of Saturn show the trace of a strange attractor in the gravitational interaction of Saturn and its satellites; chaotic vibrations have destroyed bridges and buildings. The search for chaos is very fertile today.

Sometimes chaos is a good thing. It is part of the design. Science is beginning to investigate how a certain degree of chaos is built into many aspects of the world of living things. This reveals meaning in seemingly perverse phenomena.

The human heart has an irregular beat, due to chaos. A healthy heart is more chaotic than an ill one [Ref 2]. This is believed to make the heart more sensitive to the different influences which play on it, from the autonomic nervous system, from the glands, and from the blood system of the body. A very ill heart can go into destructive oscillations, called fibrillations. The little chaos of the beat of the well heart is believed to help prevent this from happening. Modern cardiology visualizes the beating of the heart in phase-space and uses the methods of chaos theory to determine when fibrillation is likely to happen and what to do to prevent it.

The fluctuations of populations of plants and animals are often chaotic. While the attendant suffering may seem like a stiff price to pay, it sometimes happens that by maintaining a scramble of medium-scale fluctuations, chaos prevents large-scale booms and crashes which would extinguish species. As an application, it was recently shown that measles infection has a chaotic interaction with a city population. Knowing this, a strategy for timing vaccination of children was devised which is less costly than the aggressive "total inoculation" strategy of the past; it also eliminates the occasional great outbreaks that occur when a large susceptible pool builds up in spite of attempts at "total inoculation."

It may be that human communal life depends on the regulating action of chaos. On the scale of world geography and world history, human communities show a fractal pattern of distribution in space and time. The fluctuating chaos of local confrontations between individuals, groups, and species may function to discharge tensions and ameliorate the tendency of these tensions to build up into destructive and inhuman manifestations, such as war. Perhaps the current crisis in the world, which has been gathering since the debut of civilization, represents a breakdown in the natural chaos of a paleolithic Eden.

There is now good evidence that populations of neurons in the brain have chaotic patterns of activity, and that this prevents the brain from getting locked into one way of thinking. As with the heart, this may make the brain more sensitive to influences. Brain diseases such as epilepsy, parkinsonism, and schizophrenia may be produced by a breakdown in the adjustment of the brain chaos mechanism.

It turns out that the things which appeal to our senses have a fractal structure-the shapes of landscapes, the movements of tones in music, the surprises of relationships, all the things that give richness to life. Both the laws of nature which give rise to sense experiences and the brain mechanism which senses them obey the laws of chaos: as above, so below.

The idea of chaos applies to our inner experience also. The excitations in our brains which we call thoughts, feelings, relationships, struggles unroll endlessly without our being able to foresee them. The pattern which these movements trace in eternity is a fractal pattern. When we look closer and closer inside, ever new dimensions of meaning and structure are revealed. This is the gesture which is sacred for man: the mind looking within and facing always the unknown.

1. Ouspensky, P. D., Fragments of an Unknown Teaching

2. Goldberger, A.L. and West, B.J, "Applications of Nonlinear Dynamics to Clinical Cardiology", Ann. N.Y. Acad. Sci. V.504 (1987), pp. 195-212, and Goldberger, A.L and West, B.J. "Fractals in Physiology and Medicine", Yale J. Biol. 

© Copyright 1992 by Richard Hodges
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