Showing Text From Page | View Full page with images

to state will also be easy to prove. But experience suggests that this is far from correct. And indeed there are all sorts of well-known examples—such as Fermat's Last Theorem and the Four-Color Theorem—in which a theorem that is easy to state seems to require a proof that is immensely long.

So is there an analog of this in multiway systems? It turns out that often there is, and it is that even though a string may be short it may nevertheless take a great many steps to reach.

If the rules for a multiway system always increase string length then it is inevitable that any given string that is ever going to be generated must appear after only a limited number of steps. But if the rules can both increase and decrease string length the story is quite different, as the picture on the facing page illustrates. And often one finds that even a short string can take a rather large number of steps to produce.

But are all these steps really necessary? Or is it just that the rule one has used is somehow inefficient, and there are other rules that generate the short strings much more quickly?

Certainly one can take the rules for any multiway system and add transformations that immediately generate particular short strings. But the crucial point is that like so many other systems I have discussed in this book there are many multiway systems that I suspect are computationally irreducible—so that there is no way to shortcut their evolution, and no general way to generate their short strings quickly.

And what I believe is that essentially the same phenomenon operates in almost every area of mathematics. Just like in multiway systems, one can always add axioms to make it easier to prove particular theorems. But I suspect that ultimately there is almost always computational irreducibility, and this makes it essentially inevitable that there will be short theorems that only allow long proofs.

In the previous section we saw that computational irreducibility tends to make infinite questions undecidable. So for example the question of whether a particular string will ever be generated in the evolution of a multiway system—regardless of how long one waits—is in general undecidable. And similarly it can be undecidable whether

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