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But in a few percent of all cases, the behavior is much more complicated.
And it is my strong suspicion that in fact of all the current unsolved problems seriously studied in number theory a fair fraction will in the end turn out to be questions that cannot ever be answered using the normal axioms of arithmetic.
But the point is that mathematics has tended to ignore these, and instead to concentrate just on what are in effect limited patches of computational reducibility in the network of all possible theorems.
Yet all sorts of non-living systems—from crystals to flames—also do this.
So this means that there is in the end no difference between the level of computational sophistication that is achieved by humans and by all sorts of other systems in nature and elsewhere.
But with the new kinds of models based on simple programs that I explore in this book it becomes possible to capture all sorts of much more complex features that can only really be seen in explicit images of behavior.
No doubt there will quite quickly be all sorts of claims about applications of my ideas to the social sciences.
Bird songs
Essentially all birds produce calls of some kind, but complex songs are mainly produced by male songbirds, usually in breeding season.
Possible purposes [for systems]
As part of asking whether the rules for a system are somehow minimal for a given purpose, one can ask what properties the system has that could reasonably be considered a purpose at all.
Sometime later the Epicureans then suggested that a few types of elements might suffice, and an analogy was made (notably by Lucretius around 100 AD) to the fact that different configurations of letters can make up all the words in a language.