Computational reducibility may well be the exception rather than the r — Stephen Wolfram

"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."

"Its always seemed like a big mystery how nature, seemingly so effortlessly, manages to produce so much that seems to us so complex. Well, I think we found its secret. Its just sampling whats out there in the computational universe."

"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."

"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."

"[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."

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

"In the life of the mass-order, the culture of the generality tends to conform to the demands of the average human being. Spirituality decays through being diffused among the masses when knowledge is impoverished in every possible way by rationalisation until it becomes accessible to the crude understanding of all."

"I say this to you because we Spaniards are a forgetful people, because we are used to living for the moment, because we do not look back, because we do not know how to see the chain of heroes, because we do not contemplate the sum of sacrifices."

"Sharon Tate was my best friend. Once, we were roommates. She introduced me to my husband. She was the godmother to my baby daughter who is named for her. In the six years time that I knew her, she never said an unkind word about anyone."

"Long time to see. (VS: Tapion)"

"Most mathematicians prove what they can, von Neumann proves what he wants." Once in a discussion about the rapid growth of mathematics in modern times, von Neumann was heard to remark that whereas thirty years ago a mathematician could grasp all of mathematics, that is impossible today. Someone asked him: "What percentage of all mathematics might a person aspire to understand today?" Von Neumann went into one of his five-second thinking trances, and said: "About 28 percent."

"Children must be free to think in all directions irrespective of the peculiar ideas of parents who often seal their childrens minds with preconceived prejudices and false concepts of past generations. Unless we are very careful, very careful indeed, and very conscientious, there is still great danger that our children may turn out to be the same kind of people we are."