Friday, March 06, 2009

(2.1. Determinism and predictability. Jean Bricmont)

""""A major scientific development in recent decades has been popularized under the name of “chaos”. It is widely believed that this implies a fundamental philosophical or conceptual revolution. In particular, it is thought that the classical world-view brilliantly expressed by Laplace in his “Philosophical Essay on Probabilities” has to be rejected <4>. Determinism is no longer defensible. I think this is based on a serious confusion between determinism and predictability. I will start by underlining the difference between the two concepts. Then, it will be clear that what goes under the name of “chaos” is a major scientific progress but does not have the radical philosophical implications that are sometimes attributed to it.

In a nutshell, determinism has to do with how Nature behaves, and predictability is related to what we, human beings, are able to observe, analyse and compute. It is easy to illustrate the necessity for such a distinction. Suppose we consider a perfectly regular, deterministic and predictable mechanism, like a clock, but put it on the top of a mountain, or in a locked drawer, so that its state (its initial conditions) become inaccessible to us. This renders the system trivially unpredictable, yet it seems difficult to claim that it becomes non-deterministic <5>. Or consider a pendulum: when there is no external force, it is deterministic and predictable. If one applies to it a periodic forcing, it may become unpredictable. Does it cease to be deterministic?

In other words, anybody who admits that some physical phenomena obey deterministic laws must also admit that some physical phenomena, although deterministic, are not predictable, possibly for “accidental” reasons. So, a distinction must be made <6>. But, once this is admitted, how does one show that any unpredictable system is truly non-deterministic, and that the lack of predictability is not merely due to some limitation of our abilities? We can never infer indeterminism from our ignorance alone.

Now, what does one mean exactly by determinism? Maybe the best way to explain it is to go back to Laplace : “ Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it- an intelligence sufficiently vast to submit these data to analysis- it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present before its eyes.” [67] The idea expressed by Laplace is that determinism depends on what the laws of nature are. Given the state of the system at some time, we have a formula (a differential equation, or a map) that gives in principle the state of the system at a later time. To obtain predictability, one has to be able to measure the present state of the system with enough precision, and to compute with the given formula (to solve the equations of motion). Note that there exist alternatives to determinism: there could be no law at all; or the laws could be stochastic: the state at a given time (even if it is known in every conceivable detail) would determine only a probability distribution for the state at a later time.

How do we know whether determinism is true, i.e. whether nature obeys deterministic laws? This is a very complicated issue. Any serious discussion of it must be based on an analysis of the fundamental laws, hence of quantum mechanics, and I do not want to enter this debate here <7>. Let me just say that it is conceivable that we shall obtain, some day, a complete set of fundamental physical laws (like the law of universal gravitation in the time of Laplace), and then, we shall see whether these laws are deterministic or not <8>. Any discussion of determinism outside of the framework of the fundamental laws is useless <9>. All I want to stress here is that the existence of chaotic dynamical systems does not affect in any way this discussion. What are chaotic systems? The simplest way to define them is through sensitivity to initial conditions. This means that, for any initial condition of the system, there is some other initial condition, arbitrarily close to the first one so that, if we wait long enough, the two systems will be markedly different <10>. In other words, an arbitrarily small error on the initial conditions makes itself felt after a long enough time. Chaotic dynamical systems are of course unpredictable in practice, at least for long enough times <11>, since there will always be some error in our measurement of the initial conditions. But this does not have any impact on our discussion of determinism, since we are assuming from the beginning that the system obeys some deterministic law. It is only by analysing this deterministic system that one shows that a small error in the initial conditions may lead to a large error after some time. If the system did not obey any law, or if it followed a stochastic law, then the situation would be very different. For a stochastic law, two systems with the same initial condition could be in two very different states after a short time <12>.

It is interesting to note that the notion that small causes can have big effects (in a perfectly deterministic universe) is not new at all. Maxwell wrote: “There is a maxim which is often quoted, that ‘The same causes will always produce the same effects’ ”. After discussing the meaning of this principle, he adds: “There is another maxim which must not be confounded with that quoted at the beginning of this article, which asserts ‘That like cause produce like effects.’ This is only true when small variations in the initial circumstances produce only small variations in the final state of the system” ([77], p.13) <13>. One should not conclude from these quotations <14> that there is nothing new under the sun. A lot more is known about dynamical systems than in the time of Poincaré. But, the general idea that not everything is predictable, even in a deterministic universe, has been known for centuries. Even Laplace emphasized this point: after formulating universal determinism, he stresses that we shall always remain “infinitely distant” from the intelligence that he just introduced. After all, why is this determinism stated in a book on probabilities? The reason is obvious: for Laplace, probabilities lead to rational inferences in situations of incomplete knowledge (I'll come back below to this view of probabilities). So he is assuming from the beginning that our knowledge is incomplete, and that we shall never be able to predict everything. It is a complete mistake to attribute to some “Laplacian dream” the idea of perfect predictability <15>. But Laplace does not commit what E. T. Jaynes calls the “Mind Projection Fallacy”: “We are all under an ego-driven temptation to project our private thoughts out onto the real world, by supposing that the creations of one's own imagination are real properties of Nature, or that one's own ignorance signifies some kind of indecision on the part of Nature” <16> ([57], p.7). As we shall see, this is a most common error. But, whether we like it or not, the concept of dog does not bark, and we have to carefully distinguish between our representation of the world and the world itself.

Let us now see why the existence of chaotic dynamical systems in fact supports universal determinism rather than contradicts it <17>. Suppose for a moment that no classical mechanical system can behave chaotically. That is, suppose we have a theorem saying that any such system must eventually behave in a periodic fashion <18>. It is not completely obvious what the conclusion would be, but certainly that would be an embarassment for the classical world-view. Indeed, so many physical systems seem to behave in a non-periodic fashion that one would be tempted to conclude that classical mechanics cannot adequately describe those systems. One might suggest that there must be an inherent indeterminism in the basic laws of nature. Of course, other replies would be possible: for example, the period of those classical motions might be enormously long. But it is useless to speculate on this fiction since we know that chaotic behaviour is compatible with a deterministic dynamics. The only point of this story is to stress that deterministic chaos increases the explanatory power of deterministic assumptions, and therefore, according to normal scientific practice, strengthens those assumptions. And, if we did not know about quantum mechanics, the recent discoveries about chaos would not force us to change a single word of what Laplace wrote <19>.