Stephen Wolfram Quotes

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.

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.

It's a lot easier for one person to have a crisp new idea than it is for a big committee... It can happen that you have a great idea but the world isn't ready... for it. ...This has happened to me plenty. ...It's actually a pretty good idea, but... either you're not really ready for it, or the ambient world isn't... and it's hard for the thing... to get traction.

I think there is an infinite collection of these local pockets [of reducibility]. We'll never run out...

What we realized is that... these theories are generic to a huge class of systems that have these particular very unstructured, underlying rules. ...[P]eople have been struggling for a long time... How does general relativity, the theory of gravity, relate to quantum mechanics? They seem to have all kinds of incompatibilities. ...What we realized is at some level they are the same theory!

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

It's not... something where you say... you've got the fundamental theory of everything, then... [you can] tell me whether... lions are going to eat tigers or something. ...No, you have to run this thing for ...10⁵⁰⁰ steps ...to know ...You say ...run this rule enough times and you will get the whole universe. ...That's what it means to ...have a fundamental theory of physics ...You've got this rule, it's potentially simple... You've kind of reduced the problem of physics to a problem of mathematics... as if you generate the digits of pi.

It's 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. It's just sampling what's out there in the computational universe.

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...

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... ...

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).

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.

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.

I had a very selfish reason for building Mathematica. I wanted to use it myself, a bit like Galileo got to use his telescope four hundred years ago. But I wanted to look, not at the astronomical universe, but at the computational universe.