Up to David L. Thompson's Homepage
Outline by Section:  


My thesis is that order is contingent. I will show this by an examination of chaos.

By "contingent" I mean that which is not necessary. By "necessary" I mean something for which a reason can be given to show why it had to happen, why it was predictable. The special technical meaning I give to "chaos" will become evident as the paper progresses.

My thesis is only about the order of the physical world, that is, the world studied by physics; logical, social, conceptual or other kinds of order are not my concern here.

I'll proceed by, first, examining what I call Rationalism (in a rather special sense), which I take to be the basis of Modern Science (that is, the science of Galileo, Descartes, etc.) and the notion of explanation it offers. Then, by means of a brief history and a few examples, I'll introduce the notion of chaos and use that notion to show that the Rationalist approach of keeping contingency in its place won't work. After arriving at my conclusion, I'll add an epilogue on the significance of the whole discussion.


Modern Science is dominated by Rationalism and the principle of sufficient reason. In opposition to a world run by wilful leprechauns, or arbitrary Spirits, Rationalism is the claim that there is a good reason for anything which happens; everything happens of necessity. In a perfectly rational world, there would be no contingency; nothing would "just happen." To make something intelligible scientifically is therefore to explain how it is necessary, or how it could not be otherwise.
In opposition to the Aristotelian reliance on forms, Modern Science gives a causal interpretation to rationality and claims that the physical world is governed by natural, causal laws. But why are there natural laws? One obvious answer, at least to some is, "Because God wills them." Note that the appeal to God's will may explain why there are laws, but does not tell us why each law is as it is. Indeed, since God is free and omnipotent, it makes each law ultimately wilful in the sense of arbitrary, and hence contingent and unintelligible.

For a rationalist, of course, such a conclusion is not very satisfactory. It makes order too contingent. Though it can never eliminate contingency entirely, modern philosophy and science attempts to reduce appeal to God's will in explaining why particular laws are as they are by restricting God's arbitrariness to the most foundational laws, thereby minimizing, as it were, the upsurge of contingency.

Consider, for instance, how to explain the position of Venus. Kepler's law offers an explanation. Kepler discovered that planets move in ellipses and, given the distance (67 million miles) of Venus from the Sun we can show how it must have an orbit of 225 days and so, knowing where it was 225 days ago, we can now understand why it necessarily has to be where it is today.

But why do the planets obey Kepler's law? Because God just happened to so will? For a Rationalist such a response is very unsatisfactory. Newton's theory of gravity is much better, for it makes Kepler's law, and many other laws, necessary and so intelligible, rather than contingent. But now the law of gravity is problematic, for it is, in its turn, arbitrary and wilful, i.e., contingent.

Under this Rationalist understanding of intelligibility, contingency cannot be ultimately eliminated. The best we can hope for is to push it further and further back, force it into a corner, and so postpone its appearance as long as possible. Rather than face the spectre of every law being contingent, the moderns aim at a "unified science" in which all superficial orders are to be explained by appeal to more foundational orders. Descartes' Rule #5, for example, directs us to "reduce involved and obscure propositions step by step to those that are simpler" and then reascend the steps to understand all the others.(1) Reductionism keeps contingency in its place by reducing the wilful intervention of God to an absolute minimum. Maybe we have to admit God as the creator of the most fundamental laws (perhaps the Law governing the Big Bang), but all other laws follow by necessity from them and so there is no further contingency to be faced.

Despite acknowledging that God has the power to create order where she wills, the moderns reject a natural world made up of many, fragmented and unrelated orders, each of them contingent, in favour of a hierarchy of orders all derived by necessity from the one fundamental order, the contingency of which cannot be eliminated.

Close to the bottom of this hierarchy is modern physics, i.e., mechanics, which assumes that the physical world is determined by causal laws which act like mathematical functions. Take, for instance, the Law of Falling Bodies, originally proposed by Galileo, which says that the velocity after t seconds, v(t), is given by

If a body is already falling at 100 metres per second - the "initial conditions" - and g = 10 m/sec/sec, then after 20 seconds its velocity will be 100 + (10 x 20) = 300 m/sec. We plug in the initial conditions and the formula predicts, or explains, the resultant situation.

This is all very well in theory. In practice we never actually know the exact initial velocity, time, or other parameter, since our physical measurements are never infinitely precise but always have some margin of error. So to apply any mathematical law in practice we need to assume what some have called the Principle of Continuity: if the initial conditions are changed slightly, then the result will be changed only slightly.


Yet this Principle, though presupposed for centuries, does not always hold. For instance, in the case of a double pendulum, once a certain starting angle is exceeded, very slight differences in the initial conditions cause catastrophic differences in the resultant behaviour in a way which is totally unpredictable. A system for which the Principle of Continuity does not hold is called a chaotic system(2). Its behaviour is "chaos" in the technical sense in which I'll be using the term in this paper. Chaos is sensitivity to initial conditions. In a chaotic system, very tiny differences in the initial conditions, even those so small as to be unmeasurable or to exceed the precision of our best computers, make such a difference to the outcome that the result is in practice unpredictable, random, and contingent. And this despite the fact that the system is still utterly deterministic, that is, still fully governed by natural, mathematical, causal laws.

Chaotic systems were scrupulously avoided by scientists for centuries. The turning point came in 1961 when a meteorologist, Edward Lorenz, ran two computerized weather simulations with initial conditions which differed, by accident, only in the fourth decimal place. After a period of time the two weather systems, initially almost identical, lost all resemblance to each other. Lorenz refers to this sensitive dependence of weather on initial conditions as the "butterfly" effect: the flap of a butterfly's wings in Mexico could make the difference, six months later, between a fine day in Newfoundland or a blizzard.(3)

Lorenz, however, didn't stop there. He pursued his investigation by studying a simplified model of air convection, a model which is mathematically identical to a waterwheel with leaky buckets. Such a wheel may rotate in either direction at varying speeds and may, occasionally, switch direction, all in a quite unpredictable way. It never repeats the same behaviour twice. Nevertheless, although the precise quantitative behaviour is unpredictable, Lorenz discovered that the system remained for long periods in one of two relatively stable patterns (rotating left, rotating right), and even after erratic changes of direction it would gravitate back into one of these two, qualitatively different, kinds of behaviour. He called these two patterns of behaviour "attractors." "Lorenz attractors," as they are now called, have since been discovered in the behaviour of many different systems based on various causal processes other than convection, weather or waterwheels. (Taps, for instance.)

So we have here three kinds of "orders." First there is the predictable order, beloved of Kepler, Galileo, Descartes, Newton, and Laplace, of systems subject to the Principle of Continuity. Second we have chaos, the mathematically non-predictable disorder in systems with sensitive dependence on initial conditions. Thirdly, we have in some cases the appearance of a qualitative order out of this chaos. Note that in all three cases we are still dealing with systems which are deterministic in the sense that they are governed by strict and precise mathematical causal laws.

It is important to understand that chaos has nothing to do with indeterminism or freedom. As I'm using the term here, chaos can only occur in deterministic systems. It is only because we know that the pendulums, or the weather, are totally and completely governed by causal laws that we can know that they are chaotic. It is from the mathematical equations which describe the behaviour of such systems that we can deduce their sensitive dependence on initial conditions. That is, chaos is determined.

Someone might object that chaotic behaviour is caused by force we have yet to discover. When we're investigating the real weather, there is always the possibility that there are some factors or causes that we have not yet discovered. Maybe the blizzard is actually caused by some as yet undiscovered force, or is due to the direct intervention of God or some extra-galactic alien. If so, then our inability to explain it on the basis of the physics we know would hardly be surprising.

But this is not a valid objection for, in many of these cases we can set up a computer simulation in which we have full control over all the factors governing the result. A computer simulation of the waterwheel, for example, produces the Lorenz attractors as faithfully as - indeed more faithfully than - the physical model. We know that the only causes and forces in the system are the parameters and laws we placed in the program: the water flow, the bucket mass and size, the wheel radius, etc. The appearance of the attractors cannot be due to unknown causes.

Another objections comes from those who argue that chaos is not real; it is just a matter of the limitations of our knowledge. After all, since the systems involved are deterministic, the exact behaviours are always in principle predictable.

But what is the force of "in principle" here? A chaotic system is "infinitely sensitive to initial conditions" in the following, strict mathematical sense: for any given capacity for calculation (e.g., a 386 CPU) there is some point in time when the calculator's lack of precision (it can't calculate beyond 12 decimal places, say) will mean that the result may be inaccurate by any amount you choose. That is, we know for certain that any physical calculating device, be it organic or inorganic, be it even composed of every particle in the universe (which is finite), would be unable to predict the outcome of a chaotic process if that process goes on for long enough. Clearly our difficulty is not just that we need a larger research grant so that we can upgrade to a Pentium 133! Indeed, we can even calculate precisely how far out a calculator with a given precision can be after a given period of time.(4)

So the notion of "predictability in principle" is a disguised appeal to an actually (as opposed to potentially) infinite knower or calculator, i.e., to a Laplacean God. A chaotic system then, even though it must be deterministic, is predictable only by God. For an unbeliever, and I suspect for most believers too, "predictability in principle" just means unpredictability. The assumption that the concepts of "determined" and "predictable" are synonymous, an assumption we have more or less taken for granted since Descartes, Newton and Laplace, is not valid. Chaotic systems are determined but unpredictable.


If chaotic systems are unpredictable, how then can they be "understood?"

To respond to this question, let me first return to the analysis of the modern-rationalist account of understanding. For classical mechanics, as we've seen, understanding is to be found in the application of a general, causal law to initial conditions. A mechanistic explanation always has two components: the general law and the conditions to which it is applied. It is the general law that does the work of rendering the event intelligible; the second component, the conditions to which the law is applied, is like a parasite. From the viewpoint of intelligibility, the conditions contribute nothing. They just "happen to be the case." The necessity that the law delivers is the necessary link between two events which, in their individuality, are unintelligible. This is a Platonic theme: Science brings us intelligibility through universal laws which raise us beyond the irredeemable contingency of the hic et nunc.

Consider the Venus example. To understand the current position of Venus we not only need to appeal to Kepler's law, but also to the position and velocity of Venus a year ago (or whenever). Given its previous position, Kepler's law makes intelligible the present position of Venus, but this application of the law doesn't make the initial position necessary; it is, and remains, contingent.

The new study of chaos, however, changes this balance between the universal and the particular. With the loss of the principle of continuity, small changes in initial conditions result in huge changes in the outcome. A thorough grasp of the causal laws contributes little to our understanding of a chaotic system, though it is, of course, a prerequisite. It is as if the contingency of the conditions swamps the intelligibility offered by the laws.

The Lorenz attractors, for example, are structures which appear under many entirely different sets of causal laws. It is the conditions under which these laws are operating which result in the existence of these attractors. It is as if the contingent rather than the necessary has become the main source of intelligibility.

Let me put it another way. Natural causal laws are non-historical. I don't just mean that they do not themselves change; I mean that the laws do not depend on the history of the conditions under which they are operating. It doesn't matter how Venus came to be a member of our solar system: as long as it was in a certain place with a certain velocity 225 days ago, the law will explain its present parameters. If the parameters were exactly the same at some other time, t, then the results would again be identical at t+225. The source of order is in the eternal laws, not in the historical and accidental conditions, according to classical mechanics.

Under chaotic conditions, however, our understanding comes from grasping the history of particular events, the details of the actual system.

Most of our understanding of the waterwheel, for instance, comes from the way the system is set up rather than from the physical laws involved. Indeed, the physical laws governing air convection in the weather case and the laws governing the water flow are totally different, yet the Lorenz attractors apply in both cases because of the set-ups.

Take a more complex example. We can think of biological life as a myriad of chaotic structures which, according to Darwinian theory, have evolved from inorganic processes. Every organism is made up of material parts subject to physical laws. Yet appeal to these physical laws gives us little understanding of biological functions. We understand why marsupials have pouches for their young when we have been told the story of how marsupials came to fit into their particular, contingent niche in their environment. In general, biological intelligibility comes from understanding the history of evolution which led to this organism or process surviving in this niche. Physical processes are constrained to operate within biological conditions, yet it is the grasp of these particular, accidental conditions rather than the universal physical laws which deliver understanding.

These examples lead me to reject the rationalist and reductive approach and hold instead that understanding comes more from recognizing a pattern or attractor on its contingent home turf rather than analyzing a universal, causal law.


But if we can't appeal to universal causal laws, how do we justify the existence of order?

The question could mean a number of things. It could be asking for evidence that there actually is order. Such a question is self-answering. By something like the anthropocentric principle, some order, indeed a huge amount of order, is necessary for the question to be posed in the first place. Only highly ordered beings could ask any question.

But the question could be more like the infamous metaphysical question, "Why is there something rather than nothing?" It could be asking for an "explanation" of order, in the rationalist sense of an account showing why the existence of order is necessary. If taken in an absolute sense - why order as such is necessary - such a question must be nonsense, for how could there be any account which did not itself appeal to a prior order for its intelligibility?

A more sensible interpretation of the question would be: Can the order of specific systems be explained by some more fundamental order? This is the rationalist-reductionist approach taken by much of modern philosophy, the hope of reducing contingency to a minimum. It is this hope that the analysis of chaos forces us to abandon. Chaotic order is not a rare exception in a universe normally guided by the Principle of Continuity. Most systems in nature are chaotic; it is Continuity which is the exception. Attractors and other forms of chaotic order cannot be reduced to one fundamental order where contingency can be cornered. Order appears miraculously, unexplainably, at many levels and in various areas of the physical world. The ideal of a unified science is unattainable. The world is ordered by a multitude of forms and cannot be reduced to one uniform, causal mechanism. This, of course, is what Aristotle tried, unsuccessfully, to tell Descartes at the beginning of the modern era.

The appearance of schemes of order is unpredictable, unexplainable and contingent. Perhaps this should not be surprising. After all, in the rationalist-modern approach, order was ultimately contingent too, though reductionism may have postponed this realization by hiding this contingency from view. Now, as then, order is contingent; it just happens.


Apart from its intrinsic interest for our understanding of science, order, and contingency, chaos holds promise as a model for the relationship of mind to brain. Neural processes are determined by biochemical, causal laws. Cognitive unities such as beliefs, desires, or decisions, can be thought of as analogous to attractors, that is, as relatively stable structures which groups of neurons fall into under the impetus of perceptual causes or other internal, prior structures. The order to be found in mental life would then not violate the underlying deterministic physical causes but still not be reducible to, nor predictable by, these causes.

Edelman has recently combined the evolutionary and the neural with his concept of neural Darwinism. Groups of neurons give rise to attractor-like structures which then compete with other groups within the brain. The groups which are fittest within the brain environment survive while other neural groupings dissipate or die out. Again, most of the explanatory force comes from the history of the system. We understand why a certain cognitive structure persists when we grasp the events that led up to it. Appeal to physical laws gives us no more insight in these cases than it does in the case of Lorenz attractors.

Applying such an analogous scheme to the philosophy of mind may allow us to preserve neurally unpredictable mental life while still integrating it into a deterministic natural and neural world.


1. . Descartes, Rules for the Direction of our Native Intelligence, #5.

2. . Kellert offers a more precise definition: "Chaos theory is the qualitative study of unstable aperiodic behaviour in deterministic nonlinear dynamical systems." Stephen H. Kellert, In the Wake of Chaos. Chicago: University of Chicago Press, 1993.

3. . James Gleick, Chaos: Making a New Science. Penguin, 1988, gives us Lorenz's story on pp. 9-33.

4. . The Lyapunov exponent can be used to give us a measure of the strength of the chaos in a system, that is, how soon, given our levels of precision, we will run into unpredictability. (See Kellert, p. 19.)


Rene Descartes, Rule for the Direction of our Native Intelligence, in Descartes: Selected Philosophical Writings, Trans: John Cottingham et al. Cambridge: Cambridge University Press, 1988.

Gerald M. Edelman, The Remembered Present. New York: Basic Books, 1989.

James Gleick, Chaos: Making a New Science. Markham, Ontario: Penguin, 1988.

Stephen H. Kellert, In the Wake of Chaos. Chicago: University of Chicago Press, 1993.