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And although cellular automata remain some of the very best examples, we will see that a vast range of utterly different systems all in the end turn out to exhibit extremely similar types of behavior.
And so if we study computation at an abstract level, we can expect that the results we get will apply to a very wide range of actual systems.
We saw at the end of Chapter 6 that class 4 rules always seem to yield a range of progressively more complicated localized structures.
So this suggests that one might be able to get evidence about universality just by trying different possible encodings, and then seeing what range of other systems they allow one to emulate.
And I believe that it is this kind of intrinsic process—that we now know occurs in a vast range of systems—that is primarily responsible for the apparent freedom in the operation of our brains.
A proof must ultimately be based on an axiom system, and one might have imagined that over the course of time mathematics would have sampled a wide range of possible axiom systems.
But in a more compact network there may be intermediate definitions and concepts that can be used in a whole range of different theorems.
Or the ability to adapt to a wide range of different and complex situations.
For while in the past it might have seemed that the only way to generate primes was by using intelligence, we now know that the rather straightforward computations required can actually be carried out by a vast range of different systems—with no apparent need for intelligence.
However, as we will see on page 806 , traditional logic is in fact in many ways very narrow compared to the whole range of rules based on simple programs that I actually consider in this book.