But while these procedures could easily be implemented as programs, they are in a sense not based on what are traditionally thought of as ordinary mathematical functions.

For while ordinary human language has little trouble describing repetitive and even nested patterns, it seems to be able to do very little with more complex patterns—which is in a sense why this book, for example, depends so heavily on visual presentation.

Starting around the middle of the region, however, the behavior becomes quite different from region (a): while region (a) yields an object that allows information to pass through, region (g) yields one that stops all information, as shown in regions (h) and (i).

And while the details are different in each case, the general features of the behavior are always rather similar.

For while in simple cases complicated molecules may for example arrange themselves in configurations that minimize energy, the evidence is that in more complicated cases they typically do not.

But while it is known that many systems in nature are made up of discrete elements, it is still almost universally believed that there are some things that are fundamentally continuous—notably positions in space and values of quantum mechanical probability amplitudes.

So while the behavior of the first two systems on the facing page is readily seen to be computationally reducible, the behavior of the third system appears instead to be computationally irreducible.

Indeed, as a practical matter what usually seems to happen is that we receive external input that leads to some train of thought which continues for a while, but then dies out until we get

At some rather abstract level one can immediately recognize a basic similarity between nature and mathematics: for in nature one knows that fairly simple underlying laws somehow lead to the rich and complex behavior we see, while in mathematics the whole field is in a sense based on the notion that fairly simple axioms like those on the facing page can lead to all sorts of rich and complex results.

And so, for example, what are usually viewed as more successful areas of pure mathematics may have more compact networks, while areas that seem to involve all sorts of isolated facts—like elementary number theory or theory of specific cellular automata—may have sparser networks with more tendrils.