Stephen Wolfram Quotes

[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...

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

[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.

[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.

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.

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.

That's... the big discovery of this principle of computational equivalence of mine. ...This is something which is kind of a follow-on to Gödel's theorem, to Turing's 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 it's going to do.

It's clear that we can go further than the quantum mechanics that I've known for the last fifty years.

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.

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.

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!

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... ...

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.