"Early in the history of Multivac, it had become apparent that there was one big bottleneck: the questioning procedure. Multivac could answer the problems of humanity, all the problems, if -- if it were asked meaningful questions. But as knowledge accumulated at an ever-faster rate, it became ever more difficult to locate those meaningful questions."

[0] https://web.archive.org/web/20150118004835/http://www.sffaud...

Mathematics has always been highly competitive.

I am no mathematician, may have misunderstood his point and would be delighted to receive any corrections.

Navier-Stokes is a bit different (because there's a prize attached, so "scooping" matters), but almost all interesting problems don't have any prizes attached.

Terry even says this: "In fact, it is now the identification of a promising problem which is the scarce and precious resource."

The creativity and insight needed to ask a question that Terry gets excited about is the next step. Perhaps OpenAI should create a set of challenging questions and offer a prize to solve them.

If theft becomes more profitable than genuine creation, then nobody will create anything. Then there's nothing to steal, at which point all progress collapses

The awkward part about all of this is that we're about to enter an age of extreme enslavement at the hands of the major tech companies if we do not focus on distribution of hardware and research, so that everyone can participate in the abundance and automate their daily lives. If we're beholden to frontier labs because they have hoarded all of the cutting edge hardware and we're left with overpriced scraps, we're collectively screwed. They will ensure a false economy is maintained so they can clutch onto a permanent class hierarchy of haves and have-nots and remain the key global decision makers. Automating hardware manufacturing is irrelevant if the hardware is not being distributed fairly, and is weighted to real scarcity instead of artifical scarcity.

Take Louis Vuitton for example. They can mass-produce their products for pennies, but they're artificially scarce and incredibly expensive. Imagine if ALL clothing was the price of LV. Now imagine this applies to every single thing you can purchase (or rather, rent - if some of these "elite" get their way), because they've cooked the economy and swallowed all industry. That's where we are headed if distribution and decentralization is not a priority for the world and we let labs like Anthropic pull off their regulatory capture stunts.

I recall a story about some famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest.

Clearly Tao knows a hell of a lot more than I do about this, but I'm surprised that math that close to completion.

The AI labs' approach to math is immature in a way they can't get away with in coding. In coding, they realize that a pile of code that technically works is not enough: they need the code output to be a foundation to build on, and they need their agents to work well with humans which means explaining things in a way that makes sense.

In math, their goal seems just to be to exploit mathematics' reputation as full of hard problems with a general population that can't tell a pile of Lean from a good proof. OpenAI pretty much said this work is just to show off at the end of the post. Anthropic said their FLT formalization is a research artifact they do not intend to clean up or improve in any way.

Besides uniting mathematicians in irritation at the labs, the other flaw with this strategy is that it ignores that organizing knowledge is part of intelligence, much like not just producing a mess that runs is part of programming. You can write a proof that uses algebraic geometry because someone organized what could have been a bunch of disparate ideas (or fragments of a Lean repo no one will read) into a toolbox where an expert can find the tool they need.

I hope they change tack. Perhaps instead of making an explicit strategy of taking the credit from mathematicians but doing little for actual understanding, they could let some math departments at their swarms or best models, ask for a bit of acknowledgement, and hopefully they approach it by trying to write good papers, simplify, etc. rather than just rushing for headlines. (Tao's post about digesting an LLM-generated proof https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the... is an interesting read for a sense of what he means by 'digestion'.)

On that last note, it's also important (Tao's also noted) for the mathematical community to properly value digestion and organization of results, so that given the incentives of mathematics and availability of new tools you end up with good papers and textbooks and so on, not just mathematicians taking the labs' current role of pushing incomprehensible-even-to-specialists proof code to repos.

[I]t is now the identification of a promising problem which is the scarce and precious resource. We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential.

In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained.

Presumably it is only a matter of time until these frontier models are used to create new interesting conjectures. I don't get Tao's line of reasoning.

"While it may be technically infeasible to completely prohibit the use of automated tools to perform indiscriminate solution extraction, I believe that we can still designate many classes of problems as being desirous of a careful analysis that not only solves the problem, but identifies insights from the solution process, and learn more about the difficulty landscape for nearby problems, and for which raw solutions without such analysis would be of negligible or even negative value for these purposes."

Not sure I agree with this. AI generated proofs can still be analyzed and mined for useful insights. I suppose he's saying the process of banging our heads against the wall on a problem can itself yield useful insight? But what is stopping us from analyzing a proof after the fact. And if we can generate many different versions of a proof that should help us develop a much deeper understanding of the problem than we would have without being able to perceive the "proof landscape"...

1. I give you a proof, you tell me if it's correct

2. I give you a theorem, you give me a correct proof

3. I give you nothing, you give me a theorem

1. is largely solved by modern LLMs and they took a big step toward 2. today with the Navier-Stokes proof. But they're definitely not there yet. It's unclear what progress is being made toward 3. for the time being that remains the realm of humans.

Pure mathematics is dead. Long live mathematics. I think all of interesting mathematics is applied mathematics in the end. Powerful AI means that the level at which we can do applied mathematics will be so much higher, though, and many more people will be able to be "mathematicians". The importance of pure mathematics is often argued for by citing examples of important applications that used pure mathematics invented a long time before the application became apparent. We can reverse this argument: by properly developing the mathematics our applications need, we surely will obtain all of interesting pure mathematics.

Perhaps the pace of applied mathematics would rise sharply, given cheap intelligence. And this* may end up being the forefront driving progress in mathematics.

*Or maybe a split between the human domain and the practical real world. Where the human domain might end up with a variation of a "No machine contributions" policy. Sorta like the recent gcc policy.

Then an AI bot draws a false Ramanujan analogy, is corrected several times but keeps insisting.

At the end a meek comment that Tao helped in opening Pandora's box, without engagement or likes.

Modern academics have been conditioned to feel powerless and obedient. The proper way would be to ignore all AI proofs and not grant them access to journals, because OpenAI admitted themselves that plagiarism cannot be ruled out.

It’s like someone offers to build mag lev gym weights. It’s very cool that I can now lift the 500 pound weight with a finger. But what will I do when there’s no power and 500 pounds to lift?

Of course, cognition isn’t a single outcome problem like weight lifting. But we build cognition not wholly unlike how we build muscle: one needs resistance. Otherwise I’m not at all confident we “learn” in any depth.

Now that problem solving proved to be easy, mathematics will become even more interesting.

I have seen private correspondence between one mathematician working on Navier-Stokes and OpenAI that makes it sound like OpenAI deliberately scooped this Navier-Stokes result. The alleged correspondence also contained veiled threats if said mathematician went public.

@tao curious to know what makes LLM fundamentally different compared to (other) automatic theorem provers?

It highlighted that "automatic" is a spectrum. And the norm is shifting toward "fully" (even if the process is chaotic).

I would like to see the Clay Institute give zero recognition for formalizations without human-readable proofs. That would incentivize OpenAI to scram or create something that's actually useful.

Article arguing math is the next "human calculator".

https://www.youtube.com/watch?v=dcolM6W5Odc

Yes. We already knew this. Are we actually surprised it's happening?

I guess we are.

That's how I view modern math. There's the occassional discovery, like Linear Algebra, or the Fourier Transform which has basically allowed math to improve civilization to such a degree, that they deserve to be counted among the most important works of civilization.

Yet the best these weird math theories usually provide is proving something is fundamentally hard to compute, or distributed in some peculiar way, or maybe knock down some of those aforementioned problems into easier tiers.

This all ends up practically in cryptography. Hashes, symmetric asymmetric crypto, zero knowledge proofs etc. etc.

These are well and useful, but not that useful and the new ones often end up functionally identical to the old, from an utilitarian point of view.

It seems like new age requires a new form of publication instead of sticking to 17th century concept of a "paper".

Jokes aside, this seems like a pretty weird take. What's stopping mathematicians to make this renewable?

Why not spend some time and effort (presumably using AI) to pose new open problems that are fundamental in nature?

As a field, math could start crediting the person who comes up with a great question, rather than the AI brute forcing a LEAN proof.

If an AI solves a problem in an unenlightening way, then there's no reason for mathematicians to stop studying it. Pythagoream Theorem has hundreds of different proofs!

If an AI solves a problem in an enlightening way, mathematicians should study it and propose extensions.

[1] https://mathstodon.xyz/@tao/117219101339291693

[Fig Tree] "By end of the story Gilgamesh is human; He is Us; We see the Deep; That's our gift as humans; We fight to understand something no god could ever could (emergence and richness from scarcity); The gods don't need to fear death so they don't have to live an impassionate life; It means nothing to them to run out of time; "Seeing the deep" is the apex of human potential, not a transgression into godhood; It's something that only humans can do."

What is going to happen is a complete revaluation of things like "finding a counter example to a famous problem". Even if someone finds a solution to a problem like this with pencil and paper, nobody will believe it, and they will assume that there was an AI involved.

Further, sitting and doing math with a pencil and paper will no longer be a reasonable strategy to build a reputation or career, beyond the benefit a mathematician gains to their own intuition and skill. People who work hard to build intuition and also use AI effectively will dominate the field.

In a world where everyone is using AI, the open problems that remain will be the ones that are AI resistant. This is no different that how things work now, mathematicians wait until they are fairly confident someone won't rapidly solve their problem before they start talking about it. They will do the same thing in the future, except in the future AI will be part of the toolset they use decide if they are ready to share yet or not.

Edit: Ok I believe I was generally right here, but I just read the details of what OpenAI did. They didn't solve a longstanding problem, they got tipped off to an approach a mathematician was using and would likely result in the solution very soon and they finished it first. If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.

Just because the length of the arm is longer with an AI org doesn't mean it's somehow fundamentally a different system.

The future is that if you don't use AI your work is a lot easer to reach against someone else who has it.

That dude hand writing code with punchcards can be lapped by a 20yo with python, what's different?

We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential.

In other words, the "right" people need to solve it: the mathematicians who made it their job and not the people working to push AI models forward?

Struggling to understand how a solution to a millennium problem like this isn't a net positive. Presumably Open AI employs mathematicians in these efforts anyway. And I can think of far worse uses of the AI compute resources.