If the Second Law was always obeyed, then one might expect that by now every part of our universe would have evolved to completely random equilibrium. … The picture on page 456 shows what happens if one starts rule 37R with a single small region of randomness. And for a while what one sees is that the randomness that has been inserted persists.

So this implies that one can in fact meaningfully associate a definite structure with non-planarity. … In general one can imagine having several pieces of non-planarity in a network—perhaps each pictured like a carrying handle. … For independent of any specific arguments about networks and their evolution, traditional intuition would tend to make one think that the elaborate properties of The K 5 and K 3,3 forms that lead to non-planarity in networks.

Earlier in this chapter we saw that if a network is going to correspond to ordinary space in some number of dimensions d , then this means that by going r connections from any given node one must reach about r d - 1 nodes. … If one starts, say, from an ordinary continuous surface, then it is straightforward to approximate it as in the picture below by a collection of flat faces. And one might think that the edges of these faces would define a network of the kind I have been discussing.

But I suspect that this fact will be very much easier to establish for some systems than for others—with rule 110 being one of the easiest cases. In general what one needs to do in order to prove universality is to find a procedure for setting up initial conditions in one system so as to make it emulate some general class of other systems. … The most obvious possibility is that one might be able to find special classes of initial conditions in which transmission of information could be controlled.

One can view asking about possible outcomes in a multiway system as like asking about possible ways to satisfy a constraint. … So as an example the picture on the facing page shows how one type of problem about a so-called non-deterministic Turing machine can be translated to a different type of problem about a cellular automaton. … And in the example shown, one sees that for two of these paths the head goes to the right, so that the overall constraint is satisfied.

But if one just observes the normal activities of the animal it can be remarkably difficult to tell whether they involve intelligence. … And one might think that if all such pathways could be found then this would immediately show that no intelligence was involved. … And in fact if one looks inside a human brain—say in the process of generating speech—one will no doubt also see definite pathways and definite rules in use.

But what is special about rules like those on the previous page is that they are the minimal ones that exhibit the particular feature of doubling their input. And in general if one sees some feature in the behavior of a system then finding out that the rule for the system is the minimal or optimal one for producing that feature may make it seem more likely that at least with sufficiently advanced technology the system might have specifically been created for the purpose of exhibiting that feature. … And in fact one might consider this not all that unlikely for the kind of fairly straightforward exhaustive searches that I ended up using to find the cellular automaton rules in the pictures on the previous page .

But when one proves a lemma one is in effect on a separate branch, which only merges with the main proof when one uses the lemma. And if one has nested lemmas one can end up with a proof that is in effect like a tree. … In the way I have set things up one always gets from one step in a proof to the next by taking an expression and applying some transformation rule to it.

More powerful axioms [for mathematics] If one looks for example at progressively more complicated Diophantine equations then one can expect that one will find examples where more and more powerful axiom systems are needed to prove statements about them. But my guess is that almost as soon as one reaches cases that cannot be handled by Peano arithmetic one will also reach cases that cannot be handled by set theory or even by still more powerful axiom systems. Any statement that one can show is independent of the Peano axioms and at least not inconsistent with them one can potentially consider adding as a new axiom.

And indeed the picture on the next page shows one of many examples in which starting from random initial conditions there continues to be very complicated behavior forever. … But dotted around the picture one sees many definite white triangles and other small structures that indicate at least a certain degree of organization.