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analysis used to study them must be processes based on natural laws. But at least in the recent history of science it has normally been assumed that the evolution of typical systems in nature is somehow much less sophisticated a process than perception and analysis.

Yet what the Principle of Computational Equivalence now asserts is that this is not the case, and that once a rather low threshold has been reached, any real system must exhibit essentially the same level of computational sophistication. So this means that observers will tend to be computationally equivalent to the systems they observe—with the inevitable consequence that they will consider the behavior of such systems complex.

So in the end the fact that we see so much complexity can be attributed quite directly to the Principle of Computational Equivalence, and to the fact that so many of the systems we encounter in practice turn out to be computationally equivalent.

Computational Irreducibility

When viewed in computational terms most of the great historical triumphs of theoretical science turn out to be remarkably similar in their basic character. For at some level almost all of them are based on finding ways to reduce the amount of computational work that has to be done in order to predict how some particular system will behave.

Most of the time the idea is to derive a mathematical formula that allows one to determine what the outcome of the evolution of the system will be without explicitly having to trace its steps.

And thus, for example, an early triumph of theoretical science was the derivation of a formula for the position of a single idealized planet orbiting a star. For given this formula one can just plug in numbers to work out where the planet will be at any point in the future, without ever explicitly having to trace the steps in its motion.

But part of what started my whole effort to develop the new kind of science in this book was the realization that there are many common systems for which no traditional mathematical formulas have ever been found that readily describe their overall behavior.

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