Wednesday, July 02, 2008

Chaos: A Note To Philosophers

For some reasons (not too hard to guess) I was recently exposed to a number of texts (both oral and written) on the relation between science (or physics) and religion. In those, a recurring misconception is a misunderstanding of the meaning of "chaos":

In the humanities it seems, an allowed mode of argument (often used to make generalisations or find connections between different subjects) is to consider the etymology of the term used to describe the phenomenon. In the case of "chaos", wikipedia is of help here. But at least in math (and often in physics) terms are thought to be more like random labels and yield no further information. Well, sometimes they do because the people which coined the terms wanted them to imply certain connections, but in case of doubt, they don't really mean something.

A position which I consider not much less naive than a literal interpretation of a religious text when it comes to questions of science (the world was created in six days and that happened 6000 years ago) is to allow a deity to intervene (only) in the form of the fundamental randomness of the quantum world. For example, this is quite restricting and most of the time, this randomness just averages out for macroscopic bodies like us making the randomness irrelevant.

But for people with a liking for a line of argument like this, popular texts about chaos theory come to a rescue: There, you can read that butterflies cause hurricanes and that this fact fundamentally restricts predictability even on a macroscopic scale --- room for non-deterministic interference!

Well, let me tell you, this argument is (not really surprisingly) wrong! As far as the maths goes, the nice property of the subject is, that it is possible to formalise vague notions (unpredictable) and see how far they really carry. So, what is meant here is that the late time behavior is a dis-continuous function of the initial conditions at t=0. That is, if you can prepare the initial conditions only up to a possible error of epsilon, you cannot predict the outcome (up to an error delta that might be given to you by somebody else) even by making epsilon as close to 0 as you want.

The crucial thing here if of course what is meant by "late time behavior": For any late but finite time t (say in ten thousand years), the dependence on initial conditions is still continuous, for any given delta, you can find an epsilon such that you can predict the outcome within the margin given by delta. Of course the epsilon will be a function of t, that is if you want to predict the farther future you have to know the current state better. But this epsilon(t) will always (as long as the dynamics is not singular) be strictly greater than 0 allowing for an uncertainty in the initial conditions. It's only in the limit of infinite t that it might become 0 and thus any error in the observation/preparation of the current state, no matter how small leads to an outcome significantly different from the exact prediction.

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