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

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.

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.

I thought... I had a pretty good idea for what the structure of this... theory that's underneath space and time and so on might be like. ...I thought, "Gosh, in my lifetime... we might be able to figure out what happens in the first 10^-100 seconds of the universe. ...It's pretty far from anything that we can see today and it would be ...hard to test for what's right ...To my huge surprise, although it should have been obvious, ...we managed to get unbelievably much further than that. ...It turns out that even though there's this ...bed of computational irreducibility that ...all these simple rules run into, ...there are ...certain pieces of computational reducibility that ...generically occur for large classes of these rules, and... the big pieces of computational reducibility are ...the pillars of 20th century physics. That's the amazing thing, that general relativity and quantum field theory... turn out to be precisely the stuff you can say. There's a lot you can't say... at this... irreducible level where you.. don't... know what's going to happen. You have to run it [and] you can't run it within our universe... The things you can say turn out to be, very beautifully, exactly the structure that was found in 20th century physics...