One could imagine doing much as I did early in this book and successively looking at every possible rule for some type of system like a cellular automaton. … But as I discovered early in this book, many also do not, and instead exhibit behavior that is often vastly more complex.

In satisfiability what one does is to start with a collection of rows of black, white and gray squares. … The cellular automaton is set up to produce a vertical black stripe if the head of the Turing machine ever goes further to the right than it starts—as it does in cases 6 and 8.

But the remarkable fact that follows from Gödel's Theorem is that whatever one does there will always be cases where the approach must ultimately fail. … And in particular if an infinite process is computationally irreducible then there cannot in general be any useful finite summary of what it does—since the existence of such a summary would imply computational reducibility.

If one looks at recent work in number theory, most of it tends to be based on rather sophisticated methods that do not obviously depend only on the normal axioms of arithmetic. … But so long as one stays within, say, the standard axiom systems of mathematics on pages 773 and 774 , and does not in effect just end up implicitly adding as an axiom whatever result one is trying to prove, my strong suspicion is that one will ultimately never be able to go much further than one can purely with the normal axioms of arithmetic.

But just how complicated do the axiom systems for traditional areas of mathematics really need to be? … How complicated an axiom system does one need for this?

Four centuries ago we learned for example that our planet does not lie at a special position in the universe. … And nowadays the most common assumption is that it must have to do with the level of intelligence or complexity that we exhibit.

Emergence of reversibility Once on an attractor, any system—even if it does not have reversible underlying rules—must in some sense show approximate reversibility.

And although there are significant technical difficulties, one finds as the last few sections [ 8 , 9 ] have shown that the phenomenon of complexity can occur in continuous systems just as it does in discrete ones.

Does logic somehow stand out when one looks at these?

It does not help that most of the effects—at least individually—can be reproduced by mechanisms that seem to have little to do with the usual structure of quantum theory. … It is however notable that the vast majority of traditional applications of quantum theory do not seem to have anything to do with such effects. And in fact I do not consider it at all clear just what is really essential about them, and what is in the end just a consequence of the extreme limits that seem to need to be taken to get explicit versions of them.