Free Will and Chaos - an Elementary Discussion
The subject of Free Will has perplexed philosophers for millennia. Both those who believe in an omnipotent and omniscient God as well as atheists who accept a purely mechanical view of the world have trouble with it. The study of "chaos" in the last few decades seems to provide a way out of the old paradoxes. In his 1987 book  James Gleick quotes Doyne Farmer (p. 251) saying "... (chaos) struck me as an operational way to define free will, in a way that allowed you to reconcile free will with determinism. The system is deterministic, but you can't say what it's going to do next." Edward Wilson provides a solid analysis of connection between free will and chaos in at least one of his books . In addition, there has been a plethora of articles on the web by several authors, both in the context of science (Paul Davies, Linas Vepstas, etc) and of religion (Rabbi Winston, Marc Perkel, etc). Unfortunately there has been a lot of hype on the significance of chaos (even Gleick's book suffers from that) and the quality of the web postings varies widely, from solid scientific to hyped-up "New Age." Amongst recent writers Matt Ridley  looks for an alternate explanation for free will but I found it rather unconvincing and, in essence, based in special case of chaos.
This essay is written for those who do not have the time or patience to read the primary references. It uses an example (that could be understood by readers who are neither trained philosophers nor scientists) to "de-mystify" chaos and explain the argument about its connection with free will. It is not attempt to take a position - only to explain a particular theory.
There are three broad issues: (a) whether deterministic systems can have unpredictable behavior - surprising as it might be the answer is solidly yes, based on mathematic reasoning; (b) whether unpredictable behavior provides the illusion of free will; the authors cited above think so and I agree with them but I do not expect all my readers to agree; (c) whether unpredictable behavior is the same as free will; there the argument becomes trickier and I deal with it only briefly. My personal opinion is that from a practical viewpoint the illusion of free will is as good as free will but I am sure many people will disagree but I am not going to press that argument beyond providing some quotes from Wilson.
I start with the concept of the continental divide, the mountain ridge (in the Rocky Mountains) that divides the land of the United States in two parts. All rain falling to the west of the divide flows eventually into the Pacific and all falling to the east of the divide flows eventually in the Atlantic (including the Gulf of Mexico). Let us try now to predict the ocean where a particular raindrop will end given its location. If the raindrop is over Ohio the answer is clearly Atlantic or if it is over California the answer is clearly Pacific. If the raindrop is over Colorado we have to know its location with more accuracy in order to predict to which ocean will end up. The closer the raindrop is to the continental divide, the higher the accuracy needed. At some point, when the raindrop is above the ridge itself, we may not be able to make any prediction (even if we take into account minute changes in the wind, temperature, etc) because the required accuracy is beyond our means. In such a case, it may look as though the raindrop has free will, deciding on its own whether to end up in the Atlantic or the Pacific. If we use scientific terminology we say that raindrops in a narrow zone over the ridge of the continental divide behave in a chaotic way: minutes changes in their location result in much large changes in their eventual destination. This example illustrates why philosophers and scientists have made the connection between chaos and free will.
In this example, only a tiny fraction of the rain drops appear to have free will. Consider now Figure 1 where the thick wiggly line represents a continental divide in a (very) roughly drawn map. Not only the continental divide is much longer (so more raindrops have "free will") we need only much higher accuracy for the rest of them in order to predict which ocean will end into. Still, a person looking at the map can tell which parts are part of the Pacific watershed and which are part of the Atlantic watershed.
The example of Figure 2 presents an even more wiggly continental divide. Not only the accuracy needed for predicting the destination of a raindrop has increased but simple inspection is not enough to identify the watersheds. I have placed two colored dots in the figure indicating the destination of two spots. We can also visualize a situation that the continental divide becomes even more wiggly by replacing straight line segments by zigzags. In such a landscape almost all points behave in a chaotic way and we can justifiably call it chaotic. In such a case almost all raindrops appear to have free will.
You may point that Figure 2 presents a very weird landscape and that we never see such mountain ridges. This is true because the underlying system is quite simple: the position of each drop is described by only two numbers, its longitude and latitude. We say that the system has two degrees of freedom. While chaotic systems with only two degree of freedom are indeed rather contrived, systems with more degrees freedom are almost always chaotic. This is the mathematical discovery of 30 or so years ago that led to the interest in chaos (and a lot of hype as well). While chaotic systems had been known for much longer, they were thought to be peculiarities. Instead they turned out to be the norm rather than the exception for systems with more than two degrees of freedom.
If we think of the process by which a human being makes a decision we may realize that it involves many variables, genetic predisposition, cultural background, education, perception of the environment and the intentions of others, etc. Even when we view the process as purely mechanical it is likely to be chaotic and therefore to have an unpredictable outcome. Is unpredictability of action the same as free will? Not necessarily, but from a practical viewpoint it appears the same way to an outside observer. If such an observer cannot predict how you will act in a given situation, then he/she may assume that you have free will.
One might argue that chaos provides the illusion of free will rather than free will. Wilson  provides two counter-arguments. One is that "Free will as a side product of illusion would seem to be free will enough to drive human progress and offer happiness." The other is a bit more complex (and I promised to keep this essay simple) but the key phrase is "Because the individual mind cannot be fully known and predicted, the self can go on ... believing in its own free will. ... in every operational sense that applies to the knowable self, the mind does have free will." (Emphasis in the original.)
Acknowledgment: I am grateful to Rabbi Adam Fisher for comments that led me to revise the material as to make it clearer. This does not necessarily mean that he agrees with the position taken.
Posted: January 2, 2006 (revised Jan. 4, 2006). Links to slides and tutorial added on Jan. 30, 2010.