Comment by shubhamjain

The amount of money they're lately ploughing into proving math theorems is inconsistent with how societies and markets have priced pure mathematics. The entire US federal budget for math research is something like $100M annually. A single college football coach can already earn 10 percent of that.

Pretty much the only enterprise that historically pays some mathematicians handsomely is quant finance, but those people are actually compensated not for proving theorems but rather for statistical modeling and programming skills. And even that industry is so technologically driven these days that pure research mathematicians no longer hold a clear edge over strong programmers with undergrad level probability and statistics at their fingertips.

I don't think it will be the case, maths have real utility. I found something interesting at the intersection of combinatorics and information geometry. To be quite frank I don't understand what I'm doing. And yet, when I ask ChatGPT to use the framework we're developing to write an algorithm, it turns out it has quasi-parity with the state of the art. I have to measure absolute perfs to decide which one is better – theirs, not mine. Ok. Time to keep improving on what I have. And this implies dropping the code and going back to the blackboard doing more super abstract math that are way out of my league.

I think this is an interesting and good theory. They've probably eked out the large majority of the PR benefit at this point, so whether they continue in this vein will tell us a lot about their motivations for this work.

To take this to the next step, what happened after deep mind pretty much solved Go is that they started looking for the next set of things that hadn't been done yet. It does strike me as very likely that this will follow that same path.

Interesting path forwards, and probably partly true, but there are some important distinctions:

Go was a specialized application. All the math results come as a side effect of reading the whole internet, and it will keep reading the whole internet. It will keep practicing thinking questions. Actually, math might be one of the best ways to keep them contemplating and measure their contemplation abilities, so math will always stay in the loop.

Also, math might not be useful just for humanity, but also for AI, so the system might actively benefit from new math results itself. (Not sure if any of the recent proofs qualify, but future work might.)

> I wonder if top labs will soon abandon math progress like they did go and chess.

I definitely think that this is marketing, just "with good side effects". My doubt is when they will be able to move to "marketing with better side effects", that is, research with more concrete outcomes (health, materials etc.).

Problem is, that type of research is much harder. Some doubt that progress in such areas will be quick (https://www.noahpinion.blog/p/wheres-the-intelligence-explos...).

This misses the raw advantage of a good proof. It makes conceptualization simpler. In some ways math is like a hash list of of theorems. This list makes it simpler to prove other calculations, and will always be useful, to both humans and AI models. I can see two new directions 1 - the creation of specialist theorem models; that can answer questions efficiently about one topic and 2 - we probably need to incentivize and codify ownership of theorems; charging a proportion of the compute saved by using them. Ultimately enabling mathematicians to be paid our true market value!

I had the same idea recently. You've solved all the famous conjectures (all formulated by humans because humans found them interesting), what next? I doubt "AI formulated a math conjecture that nobody else cares about and immediately solved it" will produce that much hype. The actually interesting thing is indeed how mathematicians themselves will use these AI models going forward and how that will shape mathematics of the future.

AlphaGo and AlphaZero weren't generalized models. Math capability will presumably keep improving along with the other general capabilities, even if there wasn't a special RL focus for math itself.

Yeah chess is a good example. DeepMind came for publicity with AlphaZero. Arranged a match with Stockfish with rigged rules to make AlphaZero look better than it really was (it was amazing but the match wasn't fair) and then just published some games and went home.

I was bitter about that back in the day as I hoped for more answers, more matches, more "truth" about chess being shown. Soon after that community project Leela Chess Zero was started and not only surpassed original AlphaZero but added few hundred ELO points over it. Then the combination of NN and classical engines happened with NNUE and current Stockfish is again a few hundred ELO points stronger.

Today we pretty much know the truth in chess for all practical purposes. Human analysts/preparation experts focus on finding interesting path and opponent profiling (what is the most unpleasant for the opponent to face). They don't look for truth anymore. The game is doing great, it's more popular than it ever was.

They already proved the point

I disagree. Firstly, people in AI likely care about math on a personal level. Secondly math is useful. Playing go or chess is basically a party trick. Being useful gives it staying power.

But, I do think you are right that there will be some level of moving on. The spotlight is currently on maths and that won't last. It will move to some other area where there is more impact to be had. So while they might shift gears and put less focus on math, it will always be there as part of the portfolio.

I think there’s a venue where they start focusing on introducing hypotheses where the model currently can’t solve it, or maybe this is already happening?

Being able to present useful novel ideas would likely generate a lot of press, for a while. I don’t know how this would look since I’m useless at math, but Im sure there are plenty of unknown problems with massive implications, that once formulated can be solved.

Putting hallucination aside, LLM "theory of mind" has gotten worse over time. I feel it peaked in Opus 3 and Sonnet 3.5, GPT 4 and then GPT 4.5 for OpenAI. Since then, even with Opus 5.5, phrasing has needed careful crafting, in order that it not be taken too literally. OpenAI models suffer from this much more than Anthropic models but Claudes have backslid over time too.

This means when writing documentation, tutorials or commit messages, their output is often a garbled jumble. Assuming shared context, using invented terminology without explaining, leaking conversational states due to improper epistemic boundaries and failing to model the reader. This all usually leads to their freely generated explanations being terrible. Getting good explanations requires chaining questions that force them to line things up properly, which is not easy the less you know. These failures as something LLMs naturally struggle with make sense, given the nature of attention and RL with weak signals from human data.

Math is not merely a collection of proofs, it's a way of understanding. A proof presented in a manner that cannot be incorporated remains useless. It does not make it's way to physics like Riemannian geometry and matrix math did. This is no less true when done by humans too.

Your hallucination conclusion, checking if a proof is one, is exactly the counterproductive cost.

Most of us cannot verify that the claims in the OpenAI lore dump are in fact all correct. It will take tons of work from experts to do this. It took subject expert mathematicians to identify the discrepancy and disconnect in the Navier Stokes proofs, for example. LLMs will struggle to make use of their own proofs or turn them into knowledge that accumulates over time.

The act of proving is often more valuable than the proof itself. Human constraints and limitations force us to invent tools and abstractions that a 100,000 x 1M context swarm can bypass. The tradeoff from that AI swarm advantage is work that doesn't usually lend itself to being built upon. It's like doing all the side quests and reading all the books of an RPG versus min maxing a straight path with a guide. We might try to identify new abstractions, but the fact that we don't get access to CoT and that much of it will be illegible means mining LLM traces for what human mathematicians produce naturally will be a tedious chore.

> One is that AI will continue hallucinating in a manner that is not easy to verify

It is an old saw at this point, but what an LLM does still cannot be divided into hallucination and non-hallucination. This is literally an anthropomorphism trap.

Layers and layers of application-specific verification can reduce the risks inherent to LLMs, to a really remarkable degree, but nothing about what these tools are suggests that this problem will go away; it will just bubble up again somewhere else.

> assumptions which will probably not hold in the very near future [...] One is that AI will continue hallucinating in a manner that is not easy to verify

Hold up, that's an even bigger assumption in the opposite direction, and I don't see anything to support it.

At least in terms LLMs getting all the "AI" hype these days, there is no structural/mathematical reason to believe they won't continue to have the same problem they've always had of generating plausible text over rational text, and I don't think anybody even has a clear idea how it could eventually be accomplished.

I've seen "then the magic singularity occurs and somehow it solves the problem for itself", but I would classify that more as mysticism than engineering.

> but it is very productive in terms of hundreds of proofs being produced that had previously consumed uncountable hours of the brightest minds

You’re making the following assumptions:

1. the exercise of struggling to find proofs was not productive, but this is precisely how new techniques in math were produced. Brute forcing solutions doesn’t lend itself to the creation of much new mathematics (except maybe the exercise of developing verifiable proofs)

2. the point of doing mathematics is to be “productive” in the first place. This is silly. Many people get into mathematics because of the beauty of understanding, for example.

Even if you somehow have a 100% correct AI, it's not useful unless we can understand and internalise (and communicate) its results.

> One is that AI will continue hallucinating in a manner that is not easy to verify, second is that AI will not be enhanced to produced more simplified amd robust outputs, and third that a human will be required to do that.

There is literally not a single shred of evidence to indicate either of your supposed eventualities. The core technology of an LLM is sampling from a distribution so there is literally no way to make it deterministically robust (only probabilistically).

AI cannot explain chess moves it comes up with in an elegant way. What makes you think it will be able to do so for math?

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Expression of tech-faith is not intellectually honest argument.

Where does this "will probably not hold in the very near future" come from? People correctly warn about extrapolating current things onto the future, but then just throw some vague "probabilities" without providing any argument why their "probably" is somehow more grounded than others.

Did you even RTFA? His argument absolutely doesn't make any assumptions about hallucinations, implicit or not. It's you who assumes Tao must have surely been complaining about hallucinations or some such. You've not addressed any of his arguments and moreover ask questions his post answers.

Here's a longer article which goes into a bit more of the details: https://terrytao.wordpress.com/2026/10/05/the-future-of-math...

In TFA I read (scroll up from the link anchor) this was explained: OpenAI are not giving talks - because they can't answer any questions about the model's work. There is little follow-up activity - the actual elaboration of human understanding of the new ideas is stifled since the problem is solved.

But you are right, this is not OpenAI's "fault". The problem is - as others have said recently - that many people in mathematics want recognition for solving open questions more than they want the answers to the open questions. Everything about the economics and social environment of Mathematics will have to change.

I think that this is exactly the same split we see in software: there are those who mainly enjoy the craft aspect of building software, and are uninterested in the product or business they are supporting. Others are primarily interested in the production of useful software or building a platform or company.

I've always been in both camps myself. When it became obvious that AI was going to destroy the craft aspect - at least two years before it actually could do so - I became very discouraged, even depressed. But once it was actually good at building software, I became very excited about all the stuff I could now build. Sadly, I think a lot of people in our field have never had something they really wanted to build.

I'm in the same boat, I guess I had this naive idea that unsolved math problems would mean something if they were solved. But it seems like a lot of them at least were more thought experiments than anything else.

Perhaps mathematics was never about proving things? I know it sounds like moving the goal post, and it certainly was what motivated mathematicians on a day-to-day basis, but bear with me for a moment. I think that beyond being a creative activity that humans enjoy, math was about building new tools and systems of thought. Axiomizing things we take for granted, logic, linear algebra and calculus (on which modern LLMs rely so heavily) are such examples. People chose to participate in this field because they found it enjoyable and satisfying in some way. As a side effect, society reaped the benefits every couple of centuries. Will harvesting open problems with AI ever give us these things, or will we just be left laundry pile of Lean formalizations?

Mathematics is less about the end product and having a healthy community of people to actually understand the proofs and eventually apply them.

That community won't exist as many people simply won't even enter the field because it's reprehensible and contemptible, not to mention boring, just to read machine-generated proofs and verify them.

Collaborating on, or at least working on unsolved problems is what motivates most people.

AI is like a cheat code in a video game. You get to the end faster but fewer people want to play if the cheat code is always on. You can't turn it off either because the very challenge is to do something unique.

What does it mean to “genuinely” contribute to mathematics? Because the definition you use is load-bearing ;) and it might be different from that of others.

Because they published a ton of slop papers with terrible English, impossible for humans (even experts) to understand, full of non-standard terms, invented jargon etc.

So effectively, stuff got proved, but people don't really understand how, so it's mostly fucking useless and done for OpenAI's marketing team, while also pissing off the maths world at large.

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I’m not deep in to math but the op tweets make sense to me. In that it’s not just the final proof that mattered, but the mind and understanding of the person who arrived at the answer. An LLM dumping the answer can’t elaborate on it, can’t tell the story of how they got there, etc. But it also deprives someone else of that achievement and learning.

You know it's valid. You're not working on incorrect assumptions. Surely there's value in that?

I tend to disagree with OP, for the same reason.

It's unclear what more could be expected than releasing the presumably already verified results and write-ups for each problem. Should they run a mathematics school too?

Then the comment goes on to argue AI labs were not interested in actually advancing mathematics, and that investments into AI were manically excessive.

IMO none of this follows and demand is there to justify the investments.

The comment then goes further to argue that AI labs were putting too much effort into pretending there was exponential progress rather than actually making progress.

The factual basis for this claim seems to be that OpenAI released math results and write-ups, and it's not even clear what more they could do on that topic.

That's a very negative opinion.

Look at how actual researchers are currently using the tools: they'll generally use the LLMs to create slop papers, sometimes supported by auto-formalizations, just like OpenAI does. Then they will go through the lengthy process of digesting the results, turning the often incomprehensible and poorly organised outputs into something that humans can understand and build upon, they will then give seminars on the results, further helping with dissemination. Doing so still requires expertise, and probably will for a good while.

So yes, that is exactly what they should do. Alternatively, if they are too lazy or incompetent to put in the effort themselves, do what AGMAI proposed and fund a third party to help out.

> This does not affect people building new theories.

Actually, the problem is, somehow the skill of building new theories in math is directly tied to slaving hard over a problem. It's the very experience of slaving away that actually somehow causes ideas to form. Pretty much all mathematicians understand this. Yes, senior mathematicians now can form some new theories, but what about junior ones who will have very little experience in working hard on a problem by hand?

Of course, they could work on the problem by hand anyway, but they won't because no one will pay them when a machine can do it.

My grunt work is different than doing grunt work for a company to fix their problems so that they can make more money.

Honestly, that's a weakness shared by a lot of human-generated science as well.

I think in a ideal world your right.

I think a problem is that math seems like a deeply toxic, ego driven domain.

I think he argued that e.g because the navier stokes millennium problem ist considered solved now, you won't get any recognition for being the first human to solve.(How would you even proof you solved it yourself and not just regurgitated the ai proof?)

And since recognition is the main objective, noone would spend time on dissecting the proof, and perhaps finding some unique approach to solving the problem, that could be transferred to other open issues.

And therefore the problem is now "poisoned". Since it's assumed to be solved noone will research it, and the potential revelations won't be found

I think it's quite clear the mathematics panel didn't read most of the papers with the scrutiny it would take to publish it, if only because the lean certificates and the informal proofs are not 100% the same.

And if it's hard to understand (which seems to be the most common reaction) it's not exactly devoid of noise either

There was an opportunity for people to work with the AI to produce a proof, now it almost feels they're working against it.

How to make sure it doesn't evolve into some sort of Library of Babel of science?

And what use of that?

Etic is a factor of productivity, and the larger the contextual window is the weightier it becomes. That’s even integrated within the paperclip parabola.

If the focus in placed on maximizing some easily measurable output on a narrow perspective, situation is unlikely going to match a sweet spot of holistic equilibrium which is maximizing harmony and happiness through humanity as a whole.

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The problem is that one a person writes a 60 page proof in theory that person has spent an inordinate amount of time on the proof and can answer questions, describe some insight, etc etc.

If a random person is given a 60 page proof to digest and not the author, those hidden insights that _aren't_ in the paper might be completely inaccessible. Maybe the AI will "just" be able to provide the insights. Maybe. But pedagogy is tricky work, and despite these AIs being able to do all this fancy math we can't get them to write good cover letters yet, so....

Ultimately we might be left with just a bunch of intellectually unsatisfying proofs. This means way less drive to simplify the proofs or rework them.

End result: we generate a layer of "less efficient" mathematics, that won't get built upon. We will not actually have any shoulders upon which to stand.

> I'm not sure there's any evidence to suggest exploring and refining results is outside their capabilities

OP didn’t suggest that.

The bar has been raised. Everyone has to meet it now. An inelegant solution squatted onto the internet doesn’t count as discovery per se, even if it’s impressive.

But what should they do? They got all these proofs, should they just have sat on them?

That's not the hard part. The hard part still requires humans (and/or _maybe_ a bunch of tokens) and a lot of work.

And then what?

Proofs are valuable but I believe the mathematical community values understanding more. Proofs were previously a great way to develop understanding. Now, less so.

Not an expert, but explaining the proof and it being independently verifiable as important as just putting a paper of it. Reminds me of 1000 page proof of Goldbach’s conjecture that a Mathematician reached sometime back. He was told plainly that no one is going to invest time in verifying the proof because there’s a good chance there’s an error somewhere in between.

Proofs of open problems are valuable because we are assuming that proving the problem requires some new method or infrastructure in math to prove it. Basically proving open problems isn't actually useful if it doesn't develop new tooling for mathematics, which can help us create new open problems, solve other ones etc.

I recommend RTFA, it explains exactly that: why proofs should not be considered the goalpost, and why dumping all these AI-generated proofs might be an overall negative for mathematics as a whole.

Pretty tenuous connection to blame mathematicians for everything bad created in the world, that happened to use math.

This looks indeed like a public relations campaign, it would be helpful to the conversation to contribute your opinion, in a polite way.