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 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.

As I said above, it is at some level not surprising that questions will be considered interesting in a particular field only if the methods of that field can say something useful about them. … But in fact in its historical development mathematics has normally stuck to only rather few such systems—each one corresponding essentially to some identifiable field of mathematics, and most given on pages 773 and 774 .

And for each of these only a limited number of forms are allowed—all of which ultimately turn out to be equivalent to just the single forms shown on the facing page . … And so in effect they are also far from complete—for they can prove only equivalence results that hold for every single one of the various operators they allow.

undecidability—this is no longer the case, and as I discussed above I suspect that it will actually be quite common for there to be all sorts of short theorems that have only extremely long proofs. … Certainly they would provide what is in essence new information, but my strong suspicion is that in mathematics as it is currently practiced they would only rarely be considered interesting.

And the result of this is that most systems created by engineering are forced in some sense to seem simple—in mechanical cases for example typically being based only on simple repetitive motion. … And so for example at the end of Chapter 5 we saw several systems in which a simple constraint of achieving a particular outcome could in effect only be satisfied with fairly complex behavior.

Primes have divisors 1 and n only.

Initial conditions [for the universe] To find the behavior of the universe one potentially needs to know not only its rule but also its initial conditions. … But ideas like those on page 1055 —as well as inflation—tend to suggest that we currently see only a tiny fraction of the whole universe, making it very difficult for example to recognize overall geometrical regularities.

The evolution of a multiway system, first with every sequence explicitly shown at each step, and then with every sequence only ever shown once.

Examples of cellular automata that do allow information to be transmitted over large distances, but only in very restricted ways.

But it turns out that after only about 100 steps, three out of four of these patterns have resolved into simple forms.