AI, Mathematics, and the Future of Science

Scott Douglas Jacobsen and Rick Rosner explore how artificial intelligence could transform mathematics and scientific discovery. They discuss automated proofs, simulation, brute force methods, and the gap between solving problems and understanding results. Rosner considers whether accelerating AI research could eventually surpass human contributions to mathematics, physics, and other sciences.

Scott Douglas Jacobsen: Let's pick a path and see how far we can take this one. Okay. The assumption for this section, with a future path for math development, humans are in the loop in some areas, and in other areas of math they are not in the loop. You have this new mathematical result on Navier-Stokes. OpenAI has announced a solution to the existence and smoothness problem.

So an internal AI system has produced a proof, formalized in Lean, showing that forced Navier-Stokes flows can develop singularities. The discussion on Terry Tao's blog also argues that we haven't really gained much conceptual understanding from the proof. So we did learn something quite big along the way? No.

So now, okay, let's say we have the full spectrum of mathematics and some new ecosystems, some new mathematical taxonomy of math that AI gives us insight into, in the same way that a multilingual neural translation system developed a shared internal representation when it was working with all these different languages. So it had this interlingua that was unusual to our way of thinking evolutionarily.

So it does that, and it solves a series of problems at approximately Millennium Prize Problem difficulty or more. Then the next iteration advances the mathematical discoveries on that new stratum it's done that is beyond normal human abilities. And then even further, the next iteration does that to the nth iteration.

What does that do to this new type of science, which is essentially what I was talking about years ago, but this principle of simulatability? Because that's what they're doing. They're writing, doing the pure math, but you can do this in other areas too, if the variables are tight and well-defined enough. And so you could run a simulatability matrix to actual new discovery that would not necessarily be within immediate human knowledge, but it would be within AI-styled informational networks. So let's go down this path.

Rick Rosner: To start, you can go back to Feynman. We've talked about this a million times. Around 1964, Feynman gives us a talk where he talks about possible paths of science and scientific progress. One is that the things that can be known, the amount of knowledge that it takes to know how the universe works in its entirety, is finite and fairly small. And so science can reach an end by knowing everything that can be known. That's possible path one.

Possible path two is the universe has levels. The more you look into it, the more there is to discover. So science moves forward, discovering more and more things, but never gets to the end because what constitutes the universe and the principles of the universe goes as layers and layers without end.

And possible path three is science reaches an impasse because the universe beyond a certain point is unknowable, and our ability to fucking know it reaches an end. So you can apply that kind of analysis to what's happening with math.

And my guess is that in drug research over the past, I don't know, 20, 30 years, it's gone, via automation, from getting and having a theory, “Hey, I think this stuff might work against this disease,” to just testing fucking everything. Let's test every substance we know via robotics. You've seen the pictures. They've got a thousand test tubes, and they've got a robot arm that's going drip into the thousand test tubes. And it's testing every possible substance against everything and making discoveries just through brute force, just by testing, testing. And you discover a lot of shit like that.

Jacobsen: I'm arguing you can do that. But at some point, even those chemical structures and interactions, those chains could be so well-defined, you could run it.

Rosner: Yeah, you can simulate. All right. So when I think of AI and mathematics, and maybe I'm wrong because maybe I don't understand what's going on, but I feel like when you have a problem in math, you call up, AI shows up in a truck that has every tool imaginable.

We've been doing math for a long time, and there's an incredible range of mathematical systems and tools, arcane little fricking variants and, I don't know, abelian rings and matrices and shit that not even most mathematicians have heard of. But the truck pulls up, and it's got 3,500 tools on the truck. Because when you're trying to fix stuff around the house, a lot of times you're doing a shitty job just because you don't have the right tool for it. And you don't want to go to Home Depot and spend 95 bucks on a specialized tool, if you even know what the tool you need is, for a tool that you'll need once.

But AI, it has all the fucking tools. And the driver of the truck, the AI, he's not brilliant, but he's got all the tools. And he pulls up, and he's like, “Well, let's just try every single fucking tool in the truck against this. And let's combine a couple of tools. Let's try every combination of two, three, five, eight tools in the truck and see if any of these things loosen this bolt.”

And AI, with all its tools and all its brute force and trying everything, plus AI has learned a few things. So AI has the equivalent of some intuition about which shit might work, but it's brute force plus some rudimentary pseudo-intuition. You loosen the bolt. You fucking prove a result about Navier-Stokes.

And all right, so you haven't... You just said that now that AI has proved this, what does that lead to? And you said, not much, except now we know a mathematical result with certainty. Now we know it's true.

The twin prime conjecture. Here's a simple one. We can prove that there's an infinite number of prime numbers. It's an easy proof. I won't do it, but there's a hypothesis that there's an infinite number of prime numbers that are two away from each other: five and seven, 11 and 13, 17 and 19, 29 and 31. That as you go down the number line, you should keep running into those. They become less frequent on average, but you should never run out according to that hypothesis. Nobody's been able to prove that.

Though on some TV show I was watching about AI, they presented it just offhand as something that AI had proven. But say AI proves that there's an infinity of these things. Does that get you anything more? Maybe not. In the case of most of these proofs, probably not. But that's the case of everything. Even when humans are doing the proofs, these proofs, they're nice to have, but nine out of 10 don't lead to further insight.

So I don't know, fucking right now, AI is a fucking tool truck, and it's pulling up to your mathematical problems. And the AI goes in and drags its toolbox and does a lot of fucking around and says, “Here, I fixed your problem.” And you don't have any idea how AI arrived at that. You may or may not be able to comprehend the chain of bullshit that went into the proof.

So what's the implication of all this? There's going to be a lot more math done. Depressingly, it won't be done by people. People are going to have to wade through. Eventually this truck is just idling by the curb, and it's just shitting out endless piles of math. And then the humans, and then the pile of math shit is spitting out more math shit. It's just one of those exponential fucking things.

Humans can go pick through the shit and maybe find nuggets of insight, find where AI may have missed the larger implications of some of this work it's done. But even that era of human discovery will be short-lived as AI learns to do that too. It's infected physics already. It'll infect everything.

I've read a science fiction story. It's probably 15 years old now. It takes the form of the Journal of Human Physics from the middle of the 21st century. And it's sad satire. It's that here's the shit we were, as humans, able to do. And it's basically nothing significant, nothing important, because AI has taken over everything. Go ahead.

Jacobsen: It's time to go.

Rosner: Okay. All right. Enjoy your sleep.

Scott Douglas Jacobsen is the Founder and Publisher of In-Sight Publishing (ISBNs: 978–1–0692343, 978–1–0673505) and Editor-in-Chief of In-Sight: Interviews (ISSN: 2369–6885). An accomplished journalist, his portfolio includes over 10,000 publications across 200+ outlets, including International Policy Digest, The Humanist, Basic Income Earth Network, Humanist Perspectives, and rabble.ca. He has served in leadership and board roles across global media and humanist organizations and holds memberships with the Canadian Association of Journalists, PEN Canada, and Reporters Without Borders.