From "Jokester" by Isaac Asimov 1956:

"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...

That’s not untrue. But it’s also a misstatement of mathematical history. Many leading mathematicians historically have been highly competitive — Gauss comes to mind. Woe betide the lesser intellect that sent Gauss some ideas. The Newton Leibniz controversy was very serious business at the time in the UK and the continent. It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pythagoras circle.

Mathematics has always been highly competitive.

Tao's central point seems to be:

"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. "

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

I'm with @nilesh on this one, and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge. If a problem is "solved" (say, symbolically verified) without any insights gained, it doesn't seem very interesting to the profession.

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.

There's nothing that AI won't be able to mine and accomplish (aside from being literally human), it's only a matter of hardware and scale at this point. Generalized problem solving is a factor of search efficiency over the problem space. The actual software part is all figured out, the only open questions are how to do things efficiently and what the trade-offs are from a hardware perspective, but if hardware paradigms are unlocked then efficiency becomes a secondary factor for the problems we care about. Why bother making an LLM twice as fast if you can make a chip that can process 100mil TPS, for example. You're already in a ballpark where it can do anything you want, with plenty left to spare.

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.

We're having to rediscover in real time the extremely hard way, why enabling mass theft is so incredibly damaging to society. This is literally why we need a functional copyright system

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

Doesn't this just suggest that the next frontier for powerful AI models is to ask challenging questions, not simply solve them?

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.

It is now clear to me why the AI labs are sponsoring these mathathons: https://mathathonchallenge.com/. They are basically crowdsourcing human researcher data to get access to promising directions possibly later to scoop others.

I didn't realize that open math problems were a finite resource.

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.

This is worsened by OAI/Ant's strategy of grabbing the glory and running rather than spending effort trying to to advance understanding of math. Tao, who is quite sophisticated in use of AI, has said a lot about this, including in meme form: https://mathstodon.xyz/@tao/117068266026618494

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.

No matter what happened this must be a wake up call for all of us. We’re basically sharing everything we have with these companies/AI systems. This is wildly different than a human wiretapping our private messages. Because it is systematic and automated in an astronomical scale. There is no real privacy in this new world. Law? I think “National Security” is a good enough excuse to screen anything constantly, including foreign researchers in case they are close to a breakthrough.

This series of posts by Terry Tao is a direct response to the Navier-Stokes results (multiple results!) from the last 24 hours. The question is what is left after the levelling of mathematics, in all its senses, occurs? How can you protect a field that's under this much pressure in the next 6 months?

> [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.

Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.

Seems like in current cultural and economic context, short term extraction is what we’re going to do

> 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.

Aren't we in a similar position to what chess went through in the 2000s when Deep Fritz came out, and a desktop PC was able to defeat a reigning World Chess Champion? Did chess players just give up and stop playing? No, they didn't. They used these new chess engines to become better players. Computer programmers and mathematicians will probably go through something analogous.

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.

In short, after after training AI on an extraordinarily amount of human cognitive output, we are now facing the possibility that our ability to train by working on hard problems will be slowly stripped away at least in some domains.

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.

>"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"...

Other mathematicians for the last ten years : Open math problems being non renewably mined by Terrence Tao.

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.

It seems relevant that Terence Tao is the author of the paper that just about convinced everyone that the Navier-Stokes equations blow up in finite time, 12 years ago: http://arxiv.org/abs/1402.0290

We're running out of math. Maybe the president needs to establish a Strategic Math Reserve.

I think this highlights one of the fundamental differences between humans and our current AI systems. They can still only try to solve problems in the given well defined parameters they are given (with some exceptions). The human is able in the effort to solve problems to intuit where there may be new interesting problems adjacent to the current problem.

I don’t see why it makes a meaningful difference if a human solves a math problem versus AI - it seems like the same amount of understanding will come out in the end. Either the understanding will come from humans arriving at the proof in the former case, or the understanding will come from humans understanding the proof that the AI came up with in the latter.

Isn't it safe to say that all famous unsolved math problems will get a "massive amount of AI-powered effort" pointed at them regardless?

My model of mathematical intelligence for a little while now has been 3 levels:

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.

@Practal's comment is interesting:

> 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.

Strong disagree. They are of course infinitely renewable. Just work harder, Tao ;)

Open math problems, yes; also open source code, art, literature, and everything else as well. AI is a machine for turning commons into tragedies.

The way to tame a profit-maximizer is to make the most profitable choice the one that creates the most societal good.

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.

"In a world where the cost of answers is dropping to zero, the value of the question becomes everything"

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

What is this man talking about. You can speak physics into existance now, yet it still has to be proven with math. Until we are walking through worm holes and driving around in spaceships that travel in a warp drive could he even begin to say there is non-renewable. But even then..

There is also a large incentive for OpenAI to fold user conversations into the training process. Proving this happened is unduly hard, and given that professionals are using frontier LLMs for daily work, such training would make it easier for labs to scoop said professionals.

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.

AIs are putting humans out of work.

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

I guess we are.

Well, we name conjectures after the conjecturer not the (dis)prover so there is some incentive to be the guy who comes up with a hard problems. It is curious that we haven’t had something like this improve OR etc. problems. Perhaps not glorious enough.

Timing is everything https://nonlineartransform.substack.com/p/ai-swarms-timing-i...

Article arguing math is the next "human calculator".

If proofs are tropes, explanations are stories. There won't be an end to stories.

Oh man am I totally going to determine the 10^10^10th digit of π and cement my name in the annals of history.

I think this is where people will have to let go of the ego of being the "sole author" of a solution for us to move into the next era of human flourishing.

Why not let AI proof or disproof this Tao - PI - Riemann zeta hypothesis

It's the same pipeline problem coders have been talking about; once AI does all the work, how are people going to get the experience necessary to take part productively?

See also his previous thread from before the result was published (and before he knew it was coming [1]) on how a to this problem seemed increasingly likely to be solved by AI in a way that caused us to miss the insights that would traditionally be associated with solving it: https://mathstodon.xyz/@tao/117207849921390904

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

There seems to be a sense wher mathematicians are gamifying math, but are frustrated that AI labs are better at gamifying math.

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.

Tao seems to be stuck in a pure-math-for-the-benefit-of-pure-mathematicians mindset. The rest of us care about applications of math, not math itself.

Plenty of new open problems will come from applications, and applications are what actually matters. Pure mathematicians are wrapped up in math for the sake of math which is a fun academic game but not something the rest of us should care about.

The lack of reasoning traces in frontier model output is hurting science and progress... thats my take away from reading that, and why open source models are so critical and so needed, because they actually do expose the chain-of-thought reasoning traces recently missing from the frontier models (openAI, anthropic, etc). By encrypting and purposely hiding this important information from public inspection, it makes for a world where people lack the true understanding of how a problem gets solved.

I think he's suffering from a sort of static-universe fallacy. People aren't going to keep doing what they are doing, but secretly.

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.

This to me is another level of dishonesty. Imagine if a company producing math software starts to hear "rumors" someone is using their software to prove an important result and start massive runs of that software to beat the team. That may not be illegal per se but it is incredibly deceitful. I wonder if the original mathematicians should start a lawsuit for theft of intelectual property.

This is just the age of slop mathematics, if it doesnt lead to our lives neing improved none of this matters. Math peeps are being nerd sniped by AI in the same way SWEs (the worst ones) got sniped by claude code. Building solutions to problems that dont matter for the sake of doing it just because you can.

You'll ultimately waste a ton of time and get lapped by people doing real world work that actually improves the lives of regular people.

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I mean, hasn't it always been this way? If something valuable is within reach and you disclose it, someone else might reach for it?

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?

It sounds like a fancy way of saying kids will forget how to do math if they use calculators. The flattening of the space makes it hard to find new problems deemed significant? Maybe that went over my head, I’ll concede that point.

> 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.