[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.
British-American scientist and businessman (born 1959)
Stephen Wolfram (born 29 August 1959) is a British scientist known for his work in theoretical particle physics, cellular automata, complexity theory, and computer algebra. He is the creator of the computer program Mathematica.
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The remarkable thing is, what we've been able to do, is to make from this very... structurally simple underlying set of ideas, we've been able to build this... very elaborate structure that's both very abstract and... mathematically rich, and... it touches many of the ideas that people have had. ...[T]hings like string theory... ...
Can we use programs instead of equations to make models of the world? ...[I]n the beginning of the 1980s ...I did a bunch of computer experiments. ...It took me a few years to really say, "Wow, there's a big important phenomenon here that lets... complex things arise from very simple programs." ...[A] bunch of other years go by [and] I start of doing ...more systematic computer experiments ...and find ...that ...this phenomenon ...is actually something incredibly general... [T]hat led me to this... principle of computational equivalence... [A]s part of that process I said, "OK... simple programs can make models of complicated things. What about the whole universe?" ...and so I got to thinking, "Could we use these ideas to study fundamental physics?" ...I happened to know a lot about traditional fundamental physics. ...I had a bunch of ideas about how to do this in the early 1990s. I made... technical progress. ...I wrote about them back in 2002.
If we describe... heat... the air... it's this temperature, this pressure. That's as much as we can say... People [from the future] will say, "I just can't believe they didn't realize that there was this detail and all these molecules that were bouncing around, and that they could make use of that." ...One of the scenarios for the very long term history ...is the where everything... becomes thermodynamically boring... equilibrium. People say that's a really bad outcome, but actually... it's an outcome where there's all this computation going on... molecules bouncing around in very complicated ways, doing this very elaborate computation. It just happens to be a computation that right now, we haven't found ways to understand... [O]ur brains... and our mathematics and our science... haven't found ways to tell an interesting story about that. It just looks boring to us.
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.
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.
This is what you... learn from this principle of computational equivalence. ...[I]t's both a message of ...hope, and ...[that] you're not as special as you think you are... We're just doing computations like things in nature do computations, like those gas molecules do computations, like the weather does computations. The only thing about the computations that we do that's very special is that we understand what they are... because they're connected to our purposes, our ways of thinking...