But my guess is that the only way that really convincing evidence will be obtained is if actual technological systems are constructed that can successfully be seen to emulate human thinking.

Indeed, as we saw at the beginning of this chapter , some cellular automata can perform only very simple computations, always yielding

Indeed, not only for cellular automata but also for essentially all of the other kinds of systems that we studied, we found that highly complex behavior could be obtained even with rather simple rules, and that adding further complication to these rules did not in most cases noticeably affect the level of complexity that was produced.

Several decades ago chaos theory pointed out that to have enough information to make complete predictions one must in general know not only the rules for a system but also its complete initial conditions.

And indeed I suspect the only reason that their failure has not been more obvious in the past is that theoretical science has typically tended to define its domain specifically in order to avoid phenomena that do not happen to be simple enough to be computationally reducible.

And the crucial point that turns out to be the basis for much of the success of traditional theoretical science is that in fact most standard mathematical functions can be evaluated in a number of steps that is far smaller than the numerical value of their input, and that instead normally grows only slowly with the length of the digit sequence of their input.

For normally it has assumed that if one can only find the underlying rules for the components of a system then in a sense these tell one everything important about the system.

And in fact I strongly suspect that even with just three pairs there is already computational irreducibility, so that in effect the only way to answer the question of whether the constraints can be satisfied is explicitly to trace through some fraction of all arbitrarily long sequences—making this question in general undecidable.

But at least if one only has to worry about a limited number of steps, it is always possible to set things up so as to get a system that is both complete and consistent, as in the third picture on the facing page .

If an integer equation such as x 2  y 3 +12 has a definite solution such as x  47 , y  13 in terms of particular finite integers then this fact can certainly be proved using the axioms of arithmetic. For it takes only a finite calculation to check the solution, and this very calculation can always in effect be thought of as a proof.