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So when computational irreducibility is present it is inevitable that the usual methods of traditional theoretical science will not work. And indeed I suspect the only reason that their failure has not been more obvious in the past is that theoretical science has typically tended to define its domain specifically in order to avoid phenomena that do not happen to be simple enough to be computationally reducible.

But one of the major features of the new kind of science that I have developed is that it does not have to make any such restriction. And indeed many of the systems that I study in this book are no doubt computationally irreducible. And that is why—unlike most traditional works of theoretical science—this book has very few mathematical formulas but a great many explicit pictures of the evolution of systems.

It has in the past couple of decades become increasingly common in practice to study systems by doing explicit computer simulations of their behavior. But normally it has been assumed that such simulations are ultimately just a convenient way to do what could otherwise be done with mathematical formulas.

But what my discoveries about computational irreducibility now imply is that this is not in fact the case, and that instead there are many common systems whose behavior cannot in the end be determined at all except by something like an explicit simulation.

Knowing that universal systems exist already tells one that this must be true at least in some situations. For consider trying to outrun the evolution of a universal system. Since such a system can emulate any system, it can in particular emulate any system that is trying to outrun it. And from this it follows that nothing can systematically outrun the universal system. For any system that could would in effect also have to be able to outrun itself.

But before the discoveries in this book one might have thought that this could be of little practical relevance. For it was believed that except among specially constructed systems universality was rare. And it was also assumed that even when universality was present, very special initial conditions would be needed if one was ever going to perform computations at anything like the level of sophistication involved in most methods of prediction.

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