Stephen Wolfram

"Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable."

"I think there is an infinite collection of these local pockets [of reducibility]. Well never run out..."

"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."

"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, wed never be able to... know whats going to happen."

"Could it be that some place out there in the computational universe, we might find our physical universe?"

"Problem 9. What is the correspondence between cellular automata and continuous systems? Cellular automatat are discrete in several respects. First, they consist of a discrete spatial lattice of sites. Second, they evolve in discrete steps. And finally, each site has only a finite discrete set of possible values. The first two forms of discreteness are addressed in the numerical analysis of approximate solutions to, say, differential equations. ... The third form of discreteness in cellular automata is not so familiar from numerical analysis. It is an extreme form of round-off, in which each "number" can have only a few possible values (rather than the usual 216 or 232)."

"[In] Ancient Babylon... they were trying to predict three kinds of things.... where the planets would be, what the weather would be like, and who would win or lose a certain battle; and they had no idea which of these things would be more predictable than the other."

"Thats... the big discovery of this principle of computational equivalence of mine. ...This is something which is kind of a follow-on to Gödels theorem, to Turings work on the ... that there is this fundamental limitation built into science, this idea of computational irreducibility that says that even though you may know the rules by which something operates, that does not mean that you can readily... be smarter that it and jump ahead and figure out what its going to do."

"I think Computation is destined to be the defining idea of our future."

"It was the spring of 1978 and I was 18 years old. I’d been publishing papers on particle physics for a few years, and had gotten quite known around the international particle physics community (and, yes, it took decades to live down my teenage-particle-physicist persona). I was in England, but planned to soon go to graduate school in the US, and was choosing between Caltech and Princeton. And one weekend afternoon when I was about to go out, the phone rang. In those days, it was obvious if it was an international call. “This is Murray Gell-Mann”, the caller said, then launched into a monologue about why Caltech was the center of the universe for particle physics at the time."

"Im committed to seeing this project done. To see if within this decade we can finally hold in our hands the rule for our universe, and know where our universe lies in the space of all possible universes."

"[W]e live... in the pockets of reducibility. ...I should have realized [that] very many years ago, but didnt... [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]hats 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 thats... 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, thats 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."