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So if such processes can correspond to the evolution of systems like cellular automata, then it follows at least formally that differential equations should be able to do in finite time computations that would take a discrete system like a cellular automaton an infinite time to do.

But just as it is difficult to make an analog computer faithfully reproduce many steps in a discrete computation, so also it seems likely that it will be difficult to set up differential equations that for arbitrarily long times successfully manage to emulate the precise behavior of systems like cellular automata. And in fact my suspicion is that to make this work will require taking limits that are rather similar to following the evolution of the differential equations for an infinite time.

So my guess is that even within the formalism of traditional continuous mathematics realistic idealizations of actual processes will never ultimately be able to perform computations that are more sophisticated than the Principle of Computational Equivalence implies.

But what about the process of human thinking? Does it also follow the Principle of Computational Equivalence? Or does it somehow manage to do computations that are more sophisticated than the Principle of Computational Equivalence implies?

There is a great tendency for us to assume that there must be something extremely sophisticated about human thinking. And certainly the fact that present-day computer systems do not emulate even some of its most obvious features might seem to support this view. But as I discussed in Chapter 10, particularly following the discoveries in this book, it is my strong belief that the basic mechanisms of human thinking will in the end turn out to correspond to rather simple computational processes.

So what all of this suggests is that systems in nature do not perform computations that are more sophisticated than the Principle of Computational Equivalence allows. But on its own this is not enough to establish the complete Principle of Computational Equivalence. For the principle also implies a lower limit on computational sophistication—making the assertion that almost any process that is not obviously simple will tend to be equivalent in its computational sophistication.

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