[ Maestro Wolfram :] So one feature of black holes is—at least in the simplest case—is that they’re a place where time stops. So that means that any path in space, you’re going through this path, and it always comes to an end if you run into a black hole. So in mathematics, those paths are things like proofs. From this theorem, we derive another theorem, we derive another theorem, and so on and so on and so on.

Gödel’s theorem showed that there are mathematical theories, and it’s in fact the generic case, where there are proofs of arbitrarily great length. There are essentially infinite paths. So, when you have an undecidable theory, you have that no upper bound on the length of proofs. They can just wander around through proof space for arbitrarily long.

But if you have a decidable theory, a theory where every statement that you might make can be decided by a proof of bounded length, then in effect, that isn’t true anymore. So the paths always have to be finite. So it’s as if time stops. So what you end up with is what in a black hole is this physical path stopping, in meta-mathematics is the proof stops.

[Brian Greene:] So QED is like a singularity?

[ Maestro Wolfram :] Yes.

[Brian Greene:] QED in the sense of at the end of the proof.

Right. And so then it gets even funkier because in general relativity, we know that there are singularity theorems that tell one, when there’s enough energy and momentum, it’s inevitable you’ll form a singularity.

Okay,so the analogous thing—not completely worked out—but the picture in meta-mathematics is when there’s a high enough proof density, you will inevitably form a decidable theory. So when there are enough things that you can prove, you’ll inevitably be able to prove everything in finite time.

And so then it gets even funkier. If you say, “What’s the future of the physical universe?” Well, there are lots of possibilities, but one thing is that you form a bunch of black holes. When you have quantum effects, you get more than that, but roughly you form a bunch of black holes.

And so then the picture is the future of physics is like the future of mathematics because; what would happen in mathematics is there inevitably would form these singularities that are decidable theories. So just as the future of the physical universe might be a bunch of black holes, the future of the meta-mathematical universe might be the mathematical analog of black holes, namely decidable theories.

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Room 641A is a telecommunication interception facility operated by AT&T for the U.S. National Security Agency, as part of an American mass surveillance program. The facility commenced operations in 2003, and its purpose was publicly revealed by AT&T technician Mark Klein in 2006.