thought of as computations. But though we might have imagined that such computations would always be vastly more sophisticated than those performed by simple programs, the Principle of Computational Equivalence implies that they are not. And it is this equivalence between us as observers and the systems that we observe that makes the behavior of such systems seem to us complex.

One can always in principle find out how a particular system will behave just by running an experiment and watching what happens. But the great historical successes of theoretical science have typically revolved around finding mathematical formulas that instead directly allow one to predict the outcome. Yet in effect this relies on being able to shortcut the computational work that the system itself performs.

And the Principle of Computational Equivalence now implies that this will normally be possible only for rather special systems with simple behavior. For other systems will tend to perform computations that are just as sophisticated as those we can do, even with all our mathematics and computers. And this means that such systems are computationally irreducible—so that in effect the only way to find their behavior is to trace each of their steps, spending about as much computational effort as the systems themselves.

So this implies that there is in a sense a fundamental limitation to theoretical science. But it also shows that there is something irreducible that can be achieved by the passage of time. And it leads to an explanation of how we as humans—even though we may follow definite underlying rules—can still in a meaningful way show free will.

One feature of many of the most important advances in science throughout history is that they show new ways in which we as humans are not special. And at some level the Principle of Computational Equivalence does this as well. For it implies that when it comes to computation—or intelligence—we are in the end no more sophisticated than all sorts of simple programs, and all sorts of systems in nature.

But from the Principle of Computational Equivalence there also emerges a new kind of unity: for across a vast range of systems, from

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