#503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory

I think it depends on what you mean by threads.

For me, the reason you prove results is so that you better understand the world around you.

Mathematics, to me, is a vehicle for abstraction and understanding things.

And when you have a major result, what's important is not just the truth of the statement, but what you've learned.

en route to doing that.

Usually people's first question is, okay, you proved something fantastic, but how did you do it? And then people, once they digest that new idea, that new definition, that new way of looking at things, there are lots of other problems nearby that fall.

So somehow you're building structures that help you attack not just one problem, but a whole sea of problems.

And, you know, it's not that we need fully formalized results to do that.

Sometimes we reason in very sloppy metaphors that, you know, help us conceptualize the world around us.

And how much of this is something Alex mentioned offline? Attention is all you need.

How much of this is the human ability to shine a light on particular areas that quote unquote matter? So the human ability to ask the right questions, the important questions.

That's always been the case.

So within this, before AI, there's a, you can think of like, here's the axioms and just abstractly from the axioms, you can get to logical deductions, right? There are theorems that we could prove from the axioms.

And there's a much, much, much larger space of theorems that we could prove than the space that human beings ever actually care about.

And so, you know, so by almost by definition, it's like a tautology that the theorems that we'll get excited about are the theorems that we care about as humans.

So maybe AI can convince us that there's some interesting structure.

Maybe it can say, hey, look over here, let me give you this new definition or something.

It hasn't done that yet.

And sometimes the mechanism by which we select

the things that we care about, it could have an application.

So Navier-Stokes is connected to fluids and all natural systems, but it could also be a thing that some kind of mixture of beauty and difficulty... that just pulls at us.

Maybe there's an intuition.

You know, the prime numbers are full of these really complicated problems that maybe don't have an obvious application in the immediate sense, but it pulls at us somehow.

And maybe as a symptom of the pursuit of the hard thing, great applied mathematics and engineering and solutions to real-world problems comes out.

So just as an example, a lot of times when I'm working on a problem, I might have some bigger goal that I want to accomplish, but usually what I do is I try to find the simplest, most embarrassing open problem that I still don't know how to solve.