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perfect randomness. And the reason for this is that there are almost inevitably correlations even in the supposedly random environment.

In an ocean for example, the inertia of the water essentially forces there to be waves on the surface of certain sizes. And during the time that a boat is caught up in a particular one of these waves, its motion will always be quite regular; it is only when one watches the effect of a sequence of waves that one sees behavior that appears in any way random.

In a sense, though, this point just emphasizes the incomplete nature of the mechanism for randomness that we have been discussing in this section. For to know in any real way why the motion of the boat is random, we must inevitably ask more about the randomness of the ocean surface. And indeed, it is only at a fairly superficial level of description that it is useful to say that the randomness in the motion of the boat comes from interaction with an environment about which one will say nothing more than that it is random.

## Chaos Theory and Randomness from Initial Conditions

At the beginning of this chapter I outlined three basic mechanisms that can lead to apparent randomness. And in the previous section I discussed the first of these mechanisms—based on the idea that the evolution of a system is continually affected by randomness from its environment.

But to get randomness in a particular system it turns out that there is no need for continual interaction between the system and an external random environment. And in the second mechanism for randomness discussed at the beginning of this chapter, no explicit randomness is inserted during the evolution of a system. But there is still randomness in the initial conditions, and the point is that as the system evolves, it samples more and more of this randomness, and as a result produces behavior that is correspondingly random.

As a rather simple example one can think of a car driving along a bumpy road. Unlike waves on an ocean, all the bumps on the road are already present when the car starts driving, and as a result, one can consider these bumps to be part of the initial conditions for the system. But the point is that as time goes on, the car samples more and more of

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