“[S]cience has become used to... using the little... pockets of computational reducibility ([A]n inevitable consequence of computational irreducibility... There have to be these pockets ...scattered around.) to be able to find those cases where you can jump ahead.”

—  Stephen Wolfram

Stephen Wolfram: Fundamental Theory of Physics, Life, and the Universe (Sep 15, 2020)

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Stephen Wolfram 16
British-American computer scientist, mathematician, physici… 1959

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“[W]e live... in the pockets of reducibility. ...I should have realized [that] very many years ago, but didn't... [I]t could very well be that everything about the world is computationally irreducible and completely unpredictable, but... in our experience of the world there is at least some amount of prediction we can make. ...[T]hat's because we have ...chosen a slice of ...how to think about the universe, in which we can... sample a certain amount of computational reducibility, and that's... where we exist. ...It may not be the whole story about how the universe is, but it is that part of the universe that we care about and ...operate in. ...In science, that's been ...a very special case ...science has chosen to talk a lot about places where there is this computational reducibility... The motion of the planets can be ...predicted. The... weather is much harder to predict. ...[S]cience has tended to concentrate itself on places where its methods have allowed successful prediction.”

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“If we want to have a predictable life... then we have to build in these... pockets of reducibility. If we were... existing in this irreducible world, we'd never be able to... know what's going to happen.”

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“Computational reducibility may well be the exception rather than the rule: Most physical questions may be answerable only through irreducible amounts of computation. Those that concern idealized limits of infinite time, volume, or numerical precision can require arbitrarily long computations, and so be formally undecidable.”

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[Undecidability and intractability in theoretical physics, Physical Review Letters, 54, 8, 1985, 735–738, 10.1103/PhysRevLett.54.735, https://www.stephenwolfram.com/publications/academic/undecidability-intractability-theoretical-physics.pdf]

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“If you think about things that happen, as being computations... a computation in the sense that it has definite rules... You follow them many steps and you get some result. ...If you look at all these different computations that can happen, whether... in the natural world... in our brains... in our mathematics, whatever else, the big question is how do these computations compare. ...Are there dumb ...and smart computations, or are they somehow all equivalent? ...[T]he thing that I ...was ...surprised to realize from ...experiments ...in the early 90s, and now we have tons more evidence for ...[is] this ...principle of computational equivalence, which basically says that when one of these computations ...doesn't seem like it's doing something obviously simple, then it has reached this ...equivalent layer of computational sophistication of everything. So what does that mean? ...You might say that ...I'm studying this tiny little program ...and my brain is surely much smarter ...I'm going to be able to systematically outrun [it] because I have a more sophisticated computation ...but ...the principle ...says ...that doesn't work. Our brains are doing computations that are exactly equivalent to the kinds of computations that are being done in all these other sorts of systems. ...It means that we can't systematically outrun these systems. These systems are computationally irreducible in the sense that there's no ...shortcut ...that jumps to the answer.”

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