Stephen Wolfram Quotes

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... ...

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.

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.

[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.

[F]iguring out where those pockets [of reducibility] are... is an essential thing... in science. ...If you just pick an arbitrary thing and say, "What's the answer to this question?" That question may not be one that has a computationally reducible answer. ...If you ...walk along the series of questions... you can go down this chain of reducible, answerable things, but if you just... pick a question at random... most likely it will be irreducible. ...When we engineer things, we tend to ...keep in this zone of reducibility. When we're thrown things by the natural world... [we're] not at all certain that we will be kept in this... zone...

What's happened is, for 300 years people basically said, "If you want to make a model of things in the world, mathematical equations are the best place to go. In the last 15 years: it doesn't happen. New models... most often are made with programs, not with equations. ...Was that ...going to happen anyway? Was that a consequence of my particular work and my particular book? It's hard to know for sure. ...Was there a chain of academic references? Probably not.

[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.

[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.

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.

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).

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.

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

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!

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.

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.