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And the Principle of Computational Equivalence now implies that this will normally be possible only for rather special systems with simple behavior. … So this implies that there is in a sense a fundamental limitation to theoretical science. … 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.
Despite the different underlying lattices the average of sufficiently many particles yields ultimately circular behavior in both cases—as implied by the Central Limit Theorem.
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. … Or does it somehow manage to do computations that are more sophisticated than the Principle of Computational Equivalence implies? … 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.
And indeed the Principle of Computational Equivalence implies that beyond some low threshold almost any axiom system should be expected to be universal. … And with the cellular automaton being a universal one such as rule 110 this implies that the axioms of arithmetic can support universality. Such universality then implies Gödel's Theorem and shows that there must exist statements about arithmetic that cannot ever be proved true or false from its normal axioms.
But the Principle of Computational Equivalence now implies that even given a model it may be irreducibly difficult to work out its consequences. … But computational irreducibility implies that in general they are not. Indeed it implies that even once the basic laws are known there are still an endless series of questions that are worth investigating in science.
But the Principle of Computational Equivalence also implies that the same is ultimately true of our whole universe. … For the principle implies that what goes on inside us can ultimately achieve just the same level of computational sophistication as our whole universe. … Yet now the Principle of Computational Equivalence implies that the computational sophistication of our intelligence should in a sense be shared by many parts of our universe—an idea that perhaps seems more familiar from religion than science.
For to say that something is meaningful normally implies that one somehow comes to a conclusion from it. And this typically implies that one can find some summary of some aspect of it—and thus some regularity.
And this in turn is in effect what allows history to be significant—and what implies that something irreducible can be achieved by the evolution of a system. … For it implies that all the wonders of our universe can in effect be captured by simple rules, yet it shows that there can be no way to know all the consequences of these rules, except in effect just to watch and see how they unfold.
But the Principle of Computational Equivalence implies that in fact there are all sorts of statements that simply cannot be decided by any computational process in our universe. … In some cases statements can in effect have default truth values—so that showing that they are unprovable immediately implies, say, that they must be true. … In computational systems, showing that it is unprovable that a given Turing machine halts with given input immediately implies that in fact it must not halt.
Its main significance is that it implies that if any detail of the initial conditions is uncertain, then it will eventually become impossible to predict the behavior of the system. But despite some claims to the contrary in popular accounts, this fact alone does not imply that the behavior will necessarily be complex.