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PaperclipPerfector  2d ago (text post)   7547 thread views

Culture War Roundup for the week of September 7, 2026

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The more important underlying issue is: frontier models are outcompeting aspiring academics in theoretical math, making open problems trivial, making them feel obsolete.

I think there's a workaround that lets academics (including Buckmaster et al) keep their pride and purpose.

Terrace Tao wrote a few days ago: AI largely autonomously solving open problems is bad, because the solution itself isn't very important, but the techniques (advances, insights, etc.) developed to find the solution and failed attempts are. These techniques assist future problems including ones with practical applications. Meanwhile LLM solutions are barely readable so we don't get any techniques from them, only the open problem solution.

So even if a model solves an open problem, it still makes sense for academics to do themselves, or at least break the model's solution into something comprehensible by humans, and receive credit for that. Buckmaster et al were already doing that:

I can say the first LLM generated proof Levent sent me was the most horrendous I have ever read; we verified it on Lean on August 22nd. Since this point, we have been working around the clock to understand this proof and turn it into something readable

So why don't they just finish their readable proof, publish, and get credit for that? Look, even if OpenAI hesitated to publish their allegedly stolen result, they still wouldn't be first, because another independent group has now also reached the solution. It's unfortunate that sometimes mathematicians reach the same discovery independently in parallel, but these ones still have a chance to be famous in a small group, just not in a (slightly larger?) small group.

AI largely autonomously solving open problems is bad, because the solution itself isn't very important, but the techniques (advances, insights, etc.) developed to find the solution and failed attempts are. These techniques assist future problems including ones with practical applications.

Well the technique in this case seems to be 'smash the problem with $10 million worth of compute.'

From Tao:

But there is now a scenario in which an autonomous AI harness, backed by an enormous amount of computational resources, performs this entire iteration internally, and ends up producing the final ansatz, and thence the solution to the Navier-Stokes regularity problem, while the AI company running the harness keeps the process to arrive at that ansatz almost completely out of public view. Technically, one of the most prominent open problems in mathematics would now be solved; but there would be almost no value added to mathematics as a consequence. It is theoretically possible that with some herculean (and heavily AI-assisted) additional effort by a third party, some portion of the process could be reverse-engineered to recover some actual insight and understanding from the solution; but this would be a far less efficient process than if the solution had been obtained via a diverse combination of both human mathematicians and machine assistance as mentioned above.

How would it be more efficient? We can read the trend on the chart. OpenAI's agent swarm apparently needed a human to put them on the right direction today, that seems possible. I don't understand bounded v unbounded or what exactly is going on here, the subtleties of different proofs. I don't trust OpenAI.

Say they gleaned a few insights off the other mathematician. What about in 6 months or 1 year? Unleash the swarm on both the practical problems and the theoretical problems, smash them all one after the last. More algorithmic improvement (the practical money-making kernels and AI algorithms, data techniques), more compute as Rubin starts being deployed, more money to spend as the ROI rises... An 'enormous amount of computational resources' will shift up 10x or 100x in absolute scale.

What does he think ASI is, what does he think OpenAI is trying to do?

Human mathematicians, even Terence Tao, need not worry about diminishing progress in mathematics in the medium term by AI eating the seed corn. Tao's actual insight and understanding will have no more value than Kasparov's in chess.

It hinges on whether the techniques humans learn to solve open problems can be applied to areas that LLMs can’t. If LLMs can apply their own techniques, or solve more open problems with more time and scaling that continues to be feasible, human-readable proofs won’t matter beyond niche curiosity. But even then, curiosity and learning for its own sake are intrinsically important to some people. Moreover, I think these techniques sometimes lead to insights (that, for example, lead to new inventions) which today’s LLMs lack the creativity to apply themselves.

It hinges on whether the techniques humans learn to solve open problems can be applied to areas that LLMs can’t

Fair enough but I think that window is closing. 'LLMs can't' is just going to shrink and shrink. It's already perilously small! If they can do Millennium problems, what can't they do in maths? They can't answer the other Millennium problems just yet? Judging by past trends, that's about six months to a year away!

OK, they can't deliver real gamechangers like quantum gravity or room temperature superconductors. But what are the chances that humans understanding whatever insane techniques the AIs used would help us get to quantum gravity before the AIs do that?

Isn't the accusation that all the frontier model did was train on the draft proof by Buckmaster, and then OpenAI published the model's work as original, in which case the frontier model is not really outcompeting mathematicians?

Or is my summary of the controversy incorrect? I agree with your post's conclusions about what it would mean for Buckmaster's accusations to be unfounded, but it is not clear to me either way.

I think that the draft proof was itself largely written by AI, and so the controversy is about which specific group of humans pressing the, “insert coins to solve mathematics”, button gets the credit.

EDIT: Also my understanding is that the draft proof was for a related-but-not-Millennium-Prize-worthy subproblem.

another independent group has now also reached the solution.

This isn't the solution, though, is it? The only difference between incompressible Euler and incompressible Navier-Stokes is that the former has no diffusive term and the latter does, so I'm certainly not going to suggest Tao was wrong that "There does not seem to be anything in principle preventing the methods from extending all the way to Navier-Stokes", but that diffusive term is pretty important in this specific case if they weren't able to extend their results to Navier-Stokes immediately ... and it's pretty important in this context in general. These are conservative equations where free energy in the system is bounded, so any blowup has to be a localized singularity that blows up larger and larger in tinier and tinier spaces. But, the tinier you get in space, the larger the derivatives with respect to space get, and the harder a diffusive term (which roughly speaking adds a force counteracting large second derivatives) fights against your developing singularity.

If you'd asked me to guess last month, I would have said that I'd expected blowups were possible in the Euler equations and not in the Navier-Stokes equations. In hindsight that would have been ignorance on my part, and probably embarrassing ignorance based on Tao's attitude above, but I still think Navier-Stokes was legitimately a harder problem to solve in this way than Euler was.

LLM solutions are barely readable

Depends on the problem. The Jacobian conjecture counterexample was something a good engineering Bachelors' could understand, and even its derivation seemed like math-grad-student level, at least for students focusing on differential geometry. Very nice.

Some of the other recent big results are so unreadable that I don't even tell anyone about them until I see people are confident in the formal verification. Erdös talked about proofs being "straight from The Book" of God's best proofs of every mathematical theorem, and claimed "You don't have to believe in God, but you should believe in The Book". Humans have also come up with some devilishly-convoluted proofs over the years too, but we at least have the good taste to hate it when that happens.

it still makes sense for academics to do themselves, or at least break the model's solution into something comprehensible by humans, and receive credit for that.

I strongly agree, if by "still" you mean "in Fall 2026". Beyond that? In early 2024 I couldn't get a frontier model to integrate a reaction-convection-diffusion equation by parts and give me a weak form without making a sign error, or get it to correct its sign error without making a different error instead. In mid-2026 they're coming up with proofs "straight from The Necronomicon", but they're proofs for problems that geniuses have failed to solve for decades. At this rate I'd be very surprised if humans are better than late-2028 LLMs at finding simpler proofs or at making the proofs we do have as easy to understand as possible.

This isn't the solution, though, is it?

The academics only had a solution to a subset of Navier-Stokes, like that group. OpenAI seems to be the only ones who completed the proof, although Tao suggested the academics' solution may have extended to it given a bit more time.

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