apply in the natural world. And indeed to assume that it does is effectively just to ignore the fundamental question of where randomness in nature comes from.

But beyond even such matters of principle, there are serious practical problems with the idea of getting randomness from initial conditions, at least in the case of the kneading process discussed above.

The issue is that the description of the kneading process that we have used ignores certain obvious physical realities. Most important among these is that any material one works with will presumably be made of atoms. And as a result, the notion of being able to make arbitrarily small changes in the position of a point is unrealistic.

One might think that atoms would always be so small that their size would in practice be irrelevant. But the whole point is that the kneading process continually amplifies distances. And indeed after just thirty steps, the description of the kneading process given above would imply that two points initially only one atom apart would end up nearly a meter apart.

Yet long before this would ever happen in practice other effects not accounted for in our simple description of the kneading process would inevitably also become important. And often such effects will tend to introduce new randomness from the environment. So the idea that randomness comes purely from initial conditions can be realistic only for a fairly small number of steps; randomness which is seen after that must therefore typically be attributed to other mechanisms.

One might think that the kneading process we have been discussing is just a bad example, and that in other cases, randomness from initial conditions would be more significant.

The picture on the facing page shows a system in which a beam of light repeatedly bounces off a sequence of mirrors. The system is set up so that every time the light goes around, its position is modified in exactly the same way as the position of a point in the kneading process. And just as in the kneading process, there is very sensitive dependence on the details of the initial conditions, and the behavior that is seen reflects the digit sequence of these initial conditions.

But once again, in any practical implementation, the light would go around only a few tens of times before being affected by microscopic