But if one uses the kinds of traditional mathematical models that have in the past been common, things can seem rather different. … And if one thinks only about this idealization one almost inevitably concludes that the system has very little computational sophistication. … And if one assumes that actual systems somehow always manage to find ways to satisfy such constraints, one will be led to conclude that these systems must be computationally more sophisticated than any of

One approach is illustrated in the picture on the next page . … And from this result, one might imagine that unprovability would never be relevant in any practical situation in mathematics. But does one really need to have such a complicated statement in order for it to be unprovable from the axioms of arithmetic?

Just thinking about it, one might not be able to come up with anything better. But if one in effect explicitly searches all 8 trillion or so rules that involve less than four colors, it turns out that one can find 4277 three-color rules that work. … But now the question is if one were just to encounter such a rule, would one be able to guess that it was created for a purpose?

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. It remains much easier to be sure of what is going on in a discrete system than in a continuous one.

The pictures at the top of the facing page show what happens if one uses several different underlying rules for the motion of each particle. And what one sees is that despite differences at a microscopic level, the overall distribution obtained in each case has exactly the same continuous form. … The top left shows three individual examples of random walks, in which each particle randomly moves one position to the left or right.

So one may wonder how one could ever expect to represent different kinds of components. … From looking at pictures like these one can begin to imagine that it could be possible to arrange localized structures in rule 110 so as to be able to perform meaningful computations.

But despite fundamental differences like this in underlying rules, the overall behavior produced by systems based on numbers is still very similar to what one sees for example in cellular automata. … This picture shows what happens when one starts with the number 16. After 180 steps, it turns out that all that survives are a few objects that one can view as localized structures.

But the point is that if one starts from some particular piece of behavior there are in general no such simple rules that allow one to go backwards and find out how this behavior can be produced. … So insofar as the actual processes of perception and analysis that end up being used are fairly simple, it is inevitable that there will be situations where one cannot recognize the origins of behavior that one sees—even when this behavior is in fact produced by very simple rules.

One might imagine, however, that if such systems were just to try patterns at random, then even though incredibly few of these patterns would satisfy any given constraint exactly, a reasonable number might at least still come close. … And in a 10×10 array the chance of finding a pattern where the fraction of squares that violate the constraints is even less than 50% is only one in a thousand, while the chance of finding a pattern where the fraction is less than 25% is one in four trillion. … So how can one do better?

measurements, so that the amount of information needed to pick out a single arrangement is essentially the length in digits of one such number. The pictures below show the behavior of the entropy calculated in this way for systems like the one discussed above. … The top plot is exactly for case (b); the bottom one is for a system three times larger in size.