But what if one just looks at possible sequences—as might be used for initial conditions—and asks whether any of them satisfy some constraint? … For if any sequence is going to satisfy the constraint one can show that there must already be a sequence of limited length that does so—and if necessary one can find this sequence by explicitly looking at all possibilities. … And indeed whenever the question one has can somehow involve looking at an infinite number of steps, or elements, or other things, it

suggest that if one is ever going to study many important phenomena that occur in nature one will also inevitably run into them. … So how then can one set up a reasonable idealization for mathematics as it is actually practiced? … To what extent does what one sees capture the features of mathematics?

One might imagine that one could find out whether a meaningful message had been communicated in a particular case by looking for correlations it induces between the actions of sender and receiver. But it is extremely common in all sorts of natural systems to see effects that propagate from one element to another. … But what if one were to notice some event happen to the sender?

But given some form of perhaps complex behavior, how can one find rules that will manage to generate it? … So what then can one do? … But while this is enough to see a tremendous range of behavior, there is no guarantee that one will in fact run across whatever specific features one is looking for.

One specifies the input to the computation by setting up an appropriate number of initial black cells. And then one determines the result of the computation by looking at how many black cells survive in the end. … But one can also get cellular automata to do more complicated computations.

particular string, then at each successive step one applies all possible transformations, so that in the end one builds up a whole network of connections between strings, as in the pictures below. In a sense such a network can then be thought of as representing the whole field of mathematics that can be derived from whatever set of axioms one is using—with every connection between strings corresponding to a theorem, and every possible path to a proof. But can networks like the ones below really reflect mathematics as it is actually practiced?

And indeed, if one takes the patterns from successive steps and stacks them on top of each other to form a three-dimensional object, as in the picture below, then this object has a very regular nested structure. … The facing page and the one that follows show patterns produced by two-dimensional cellular automata with a sequence of different rules. … Such pictures are the analogs for two-dimensional cellular automata of the two-dimensional pictures that I often generate for one-dimensional cellular automata.

Often one can identify features in common between the various mechanisms for any particular kind of behavior. … And so, for example, one might notice that most mechanisms for nesting can at some level be viewed as involving hierarchies in which higher components affect lower ones, but not the other way around. … So in the end one can indeed view most of the mechanisms that I have discussed in this section as being in some sense genuinely different.

simple arrangement of particles occurs in the middle of the evolution, then one can readily see that randomness increases in exactly the same way—whether one goes forwards or backwards from that point. … If one starts with this arrangement, then the randomness of the system will effectively increase whether one goes forwards or backwards in time from that point.

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. … 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.