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argued that it is responsible for much of the complexity we see in nature and elsewhere.

Yet so far I have given no fundamental explanation for the phenomenon. But now, by making use of the Principle of Computational Equivalence, I am finally able to do this.

And the crucial point is to think of comparing the computational sophistication of systems that we study with the computational sophistication of the systems that we use to study them.

At first we might assume that our brains and mathematical methods would always be capable of vastly greater computational sophistication than systems based on simple rules—and that as a result the behavior of such systems would inevitably seem to us fairly simple.

But the Principle of Computational Equivalence implies that this is not the case. For it asserts that essentially any processes that are not obviously simple are equivalent in their computational sophistication. So this means that even though a system may have simple underlying rules its process of evolution can still computationally be just as sophisticated as any of the processes we use for perception and analysis.

And this is the fundamental reason that systems with simple rules are able to show behavior that seems to us complex.

At first, one might think that this explanation would depend on the particular methods of perception and analysis that we as humans happen to use. But one of the consequences of the Principle of Computational Equivalence is that it does not. For the principle asserts that the same computational equivalence exists for absolutely any method of perception and analysis that can actually be used.

In traditional science the idealization is usually made that perception and analysis are in a sense infinitely powerful, so that they need not be taken into account when one draws conclusions about a system. But as soon as one tries to deal with systems whose behavior is anything but fairly simple one finds that this idealization breaks down, and it becomes necessary to consider perception and analysis as explicit processes in their own right.

If one studies systems in nature it is inevitable that both the evolution of the systems themselves and the methods of perception and

From Stephen Wolfram: A New Kind of Science [citation]