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being able to see a sequence of explicit bumps in the road, the initial conditions for the position of a point in the kneading process are encoded in a more abstract form as a sequence of digits.

But the crucial point is that the behavior we see will only ever be as random as the sequence of digits in the initial conditions. And in the first case on the facing page, it so happens that the sequence of digits for each of the initial points shown is indeed quite random, so the behavior we see is correspondingly random. But in the second case, the sequence of digits is regular, and so the behavior is correspondingly regular.

Sensitive dependence on initial conditions thus does not in and of itself imply that a system will behave in a random way. Indeed, all it does is to cause digits which make an arbitrarily small contribution to the size of numbers in the initial conditions eventually to have a significant effect. But in order for the behavior of the system to be random, it is necessary in addition that the sequence of digits be random. And indeed, the whole idea of the mechanism for randomness in this section is precisely that any randomness we see must come from randomness in the initial conditions for the system we are looking at.

It is then a separate question why there should be randomness in these initial conditions. And ultimately this question can only be answered by going outside of the system one is looking at, and studying whatever it was that set up its initial conditions.

Accounts of chaos theory in recent years have, however, often introduced confusion about this point. For what has happened is that from an implicit assumption made in the mathematics of chaos theory, the conclusion has been drawn that random digit sequences should be almost inevitable among the numbers that occur in practice.

The basis for this is the traditional mathematical idealization that the only relevant attribute of any number is its size. And as discussed on page 152, what this idealization suggests is that all numbers which are sufficiently close in size should somehow be equally common. And indeed if this were true, then it would imply that typical initial conditions would inevitably involve random digit sequences.

But there is no particular reason to believe that an idealization which happens to be convenient for mathematical analysis should

From Stephen Wolfram: A New Kind of Science [citation]