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Comment by sashank_1509 | Hacker News Reader

Both things can be true:

1. OpenAI when using your chats in pretraining is improving its model’s intuition. The model parameter size is massive, and while the data is OOM larger it is plausible that model remembers stuff about chats that improves its latent representation.

2. During RL on verifiable math and massive compute, the model discovers techniques and connections to solve math problems that are superhuman and have little to do with some specific technique mentioned in its chat.

The rumor I’ve heard from multiple employees at OAI and Ant is that the model has solved hundreds of open problems in maths, and is basically solving anything you throw at it. We’ll know soon enough, but I’m inclined to believe this is true. Maths is a fully verifiable domain amenable to self play, massive scale RL can develop a search agent far better than any human and I’m inclined to believe OAI would have solved these conjectures without any of this chat data in its pre-training.

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On your second point: there is a more plausible explanation which David Bessis calls the "overhang". The short version is that there is a large amount of relatively low hanging fruits in mathematics, because no human has broad enough knowledge and enough time to try them all. AI is not constraint by that, and therefore can systematically pluck all those low hanging fruits.

Quote: "The Overhang consists of the unrealized capital gains of past mathematical creativity, the latent value from connecting the dots in the existing corpus. It is a dividend of canonization. Mathematician X states problem A, mathematician Y crafts concept B, then mathematician Z notices that B trivially solves A and “captures” the social reward. But in the process of capturing the reward, Z usually introduces new concepts and new open problems, reinjecting latent value into the Overhang.

LLMs can be trained on the entirety of the mathematical corpus. Thanks to their phenomenal memorization and pattern-matching abilities (without always being able to map out their associative logic and attribute due credits), they are in a unique position to harvest the Overhang. By contrast, professional mathematicians have typically read a few hundred articles in their career, out of millions of existing references, less than 0.1% of the total.

This will lead to great discoveries, which is unambiguously exciting. But it could also lead to a sad new deal, where human slaves painfully curate the Overhang while AIs systematically beat them at the finish line."

source: https://substack.com/inbox/post/183753276

>Quote: "The Overhang consists of the unrealized capital gains of past mathematical creativity, the latent value from connecting the dots in the existing corpus. It is a dividend of canonization. Mathematician X states problem A, mathematician Y crafts concept B, then mathematician Z notices that B trivially solves A and “captures” the social reward.

I've made an entire career out of being 'jack of all trades, master of none'. Being able to synthesize connections from relatively trivial knowledge in a bunch of domains is SOP for many humans as well. I think AI just has deeper knowledge and better pattern matching to make up for it's (at least now) lack of strength in cognition and 'ex nihilo' creativity.

(Which probably isn't 'ex nihilo' at all, and has more to do with the plethora of modalities that humans live in vs. large language models. For example, why do we pick the color red for notating important things and why do we say a schedule 'slips'...these are informed by a shared human experience borne of distinct physical sensation deep in our wiring that LLMs can only infer from what we write.)

A college advisor I had 20 years ago was a firm believer that interdisciplinarity was the future, that generalist skills and the ability to make connections between different fields would be paramount in advancing science. I suppose he was right in the big picture, even if the career prospects for human generalists aren't looking so rosy.

I'm actually still quite bullish on generalists. Specialists advance every front but build the supply lines between them.

In favor of the generalist, I think AI is also quite limited in its scope of how it generalizes. I'm mowing through hundreds of mythos-generated security findings right now for work and while it's amazing that it can build an exploit chain 20 steps deep, it's completely lacking in all of the external layers that render it's speculation moot.

How do you thrive in an environment of specialists? That's is the problem I seem to have. I'm spread a little across a few of the domains involved with what I do. Because of that, I have a bit more insight, so am very often the person pointing out relatively fundamental problems, usually caused by either not understanding the problems from a "first principles" perspective, resulting in, or being caused by, categorical type errors, where they've boxed a problem into a tiny space it doesn't belong.

I've been trending "quiet" lately, because I don't like the "friction"/convincing aspect of it all. It's hard to get people to see things from a different angle, or even convincing them there's a problem to begin with!

The last project required a complete redesign from a problem I pointed out during the first review, and second, and third, but now I'm seeing even more friction.

Maybe this is just corporate life, after a group gets large.

Any tricks/advice?

You don't really say how long you've worked with these folks so I'm guessing here, but in my experience you have to build credibility and trust first.

Focus on the opportunities rather than the problems, connect people doing similar things in different areas and jump in when they are solving problems after the fact. The last part, for me, has been key to building it in my career (30 years in infra/tech/security).

Yes, advice.

Eventually, you will be seen as a party breaker, and all that will be seen as negative. You are the bringer of the bad news; nobody likes that, no matter how reasonable you are. Shitty products are the norm when profit is in question and doing what is right is not and will be in most places taken as inability to be a valuable team member. There might be some slight chance that you will be in good environment, but even that is usually temporary as business owners, goals and incentives change.

> Quote: "The Overhang consists of the unrealized capital gains of past mathematical creativity, the latent value from connecting the dots in the existing corpus. It is a dividend of canonization. Mathematician X states problem A, mathematician Y crafts concept B, then mathematician Z notices that B trivially solves A and “captures” the social reward. But in the process of capturing the reward, Z usually introduces new concepts and new open problems, reinjecting latent value into the Overhang.

That overhang seems like a precious resource for AI companies. They can exploit that overhang to inflate the impression of AI's capabilities, and hopefully that exploitation will discourage the next generation of mathematicians from pursuing math. If they play their cards right, OpenAI and Anthropic can dominate the field even if they ultimately can't replicate the creativity of human mathematicians, because they'll have driven their competition out.

What we should be trying to achieve is a ladder-breaking maneuver: knock out the lower rungs so no person can reasonably climb to the top-reaches of mathematical skill anymore. That may ultimately result in stagnation, but it's what's best for AI, so it's what should be done now.

We need to do everything we can to create the greatest-possible dependence on AI tools.

/s, I hope?

> no human has broad enough knowledge and enough time to try them all.

The other part is, humans don’t really want to fund other humans doing this.

Very few want to be a math major; and of those that do, fewer complete a grad degree; and for those that do get grad degrees, there’s scant few research jobs; and for those who do get jobs there’s hardly any research funding to go around.

There does seem to be unlimited money for ai researchers to use ai to solve these problems though.

We’ve turned education into job training, so because there’s no jobs in solving math problems, few aspire to do it. If there were more opportunities for people, more people would do it, and more low hanging fruit would be plucked.

I’m assuming the reported 22 million dollars worth of tokens used to solve this particular problem is far more than what humans have paid to solve it previously. So I think you’re correct.

22x more to be exact

Great essay, thanks for sharing.

When I was a software library developer, I came to resent application developers. I noticed a pattern. Libraries solved hard problems and did so carefully, thoughtfully, in a way that others could reuse. Apps would come along and carelessly, recklessly glue together several high quality libraries into a piece of software targeting a general audience. The apps would then harvest all the credit.

What's happening in mathematics right now feels similar. Applications (theorems) were always how one built objective reputation, but libraries (concepts, definitions, boring lemmas) were also rewarded socially within the mathematics community. And individual mathematicians often managed to both build their own libraries, and use them to prove an important result. And then those libraries were sometimes of use in other results.

Bessis asks whether AI Lean proofs will land in Mathlib or Mathslop. Or in my framing: will they be libraries, or applications?

At present they're mostly Mathslop. The proven result is perhaps useful, but the methods employed aren't novel or reusable. I worry that this trend will only worsen, because applications make headlines, and the libraries they used do not. We are not properly incentivizing library development in OSS, or in math, or in infrastructure writ large. There's a serious credit assignment problem here.

What might change this? Once the low hanging fruit is picked, will citation count rise in relative status again? Will we get result fatigue and start to reward legibility — no one cares unless the paper has an accompanying ELI5 tiktok video? A labeling regime that certifies the proof was produced sustainably, organically, by local artisans with no AI additives?

There is also "sexy proof", people want nice math that can be printed in t-shirt. Not super hard grind, where you need several years of studying, just to understand the question (that is before even trying to solve it).

Many problems are solvable, but require months of work, and thousands of pages of proof. So people do not even try to create or verify the proof. AI changes that, it can verify and perhaps even simplify it, to more digestible form.

The overhang, being defined as the Cartesian product of existing knowledge — randomly combining existing knowledge.

(I mean actually randomly, not asking an LLM to do the randomness.)

Most of the output would be incoherent (like many dreams), but occasionally you would get a gem.

It's like AlphaGo but playing against all living mathematicians. (Overhang being low hanging fruit is what allows this comparison, of course the general moot point is the skepticism that LLMs are also innovative etc.)

We are not seeing those incredible moves yet. The approach used in N-S was conjectured to work after B&L’s initial breakthrough. See a post by Tao. So on one hand the proof is an amazing accomplishment. On the other hand, humans have not yet discovered any superhuman moves in the proof. Just $MM grind.

Could "superintelligence" arrive as basically applying this overhang to all other domains?

It already did.

That is not "superintelligence" but string concatenation of stored data. Anyway, the marketing succeeded.

OpenAI said they sicced this agent army on Navier-Stokes on Sept 1st, while only a couple of days earlier OpenAI's Noam Brown happened to reply to a tweet saying that they had already tried to solve all the Millennium Prize problems and failed... So, it seems either the previous attempt didn't have the training to succeed, or was just not given the compute to do so.

Once OpenAI heard that Navier-Stokes was solved, this caused them to immediately revisit the problem and throw a ton of compute at it, apparently using a more (very) recent model than what they had tried before. What we don't know is just how recent this model was, and therefore what it may have been trained on. Buckmaster/Levant had apparently been working towards this for at least a year, and made their "forced" blow-up breakthrough on August 15th.

Presumably any anonymized prompts that are being trained on are part of pre-training, so older, but once OpenAI had heard that Navier-Stokes had been solved and wanted to revisit it, it seems possible they may have done a few weeks of incremental RL training on anything Navier-Stokes adjacent they could come up with, in addition to then throwing unlimited compute at it, now confident that there was something to find.

OpenAI have come out and said:

>The Wednesday evening statement from OpenAI was more emphatic: “We can say categorically that it is impossible for Dr. Buckmaster’s Codex prompts over the last two months to have influenced the system in any way, including training.”

>The statement added, “After investigating, we can say with full confidence that no user inputs past July 3rd could have influenced this system in any way.”

https://www.nytimes.com/2026/09/10/science/tristan-buckmaste...

Is there a reason they scoped that so narrowly to Buckmaster/codex/2 months

two people worked on this for a year before the breakthrough. Perhaps that earlier work reduced the search space sufficiently to brute force the problem with 10,000 agents?

Just knowing that there had been progress is enough to have an idea that throwing more compute at it might work (OpenAI had previously tried all the Millennium Prize problems with somewhat limited compute and failed).

It's comparable to Magnus Carlson saying that if he wanted to cheat, all he would need would be for someone to tell him to spend more time thinking about a specific move (just a wink would be enough) as an indication that a computer had found something interesting.

It's as-if after OpenAI first failing on Navier-Stokes (which OpenAI had just tweeted about 2 days earlier!), someone winked at them and said "you might want to try a little harder ...".

When reading human comments, we should be generous; when we read corporate texts, we may assume paltering.

(TIL: paltering: exact and technically correct statement usage to create misleading impression)

The comment you replied to quoted "no user inputs after July 3rd" with no restriction to Buckmaster or Codex.

Obviously the result of OpenAI's investigation was that no usage data has interacted with the system after that date.

What else do you expect them to investigate?

If Buckmaster and co. provide their chats, OpenAI could potentially search for them in the anonymized opted-in usage data. Then they could say if any data has been used.

By all accounts individual usage data does not have the direct impact on the model most here fantasize about. To prove this, OpenAI would need to do new training runs to replicate the system used minus the particular usage data in question, if it exists, and then benchmark this on the problem again.

Potentially multiple times, in order to reach a conclusion.

The cost might be in the hundreds of millions.

OK, good to know (if they can be trusted - Altman clearly is a liar), but it doesn't really change the big picture much.

1) OpenAI by their own admission, only re-tackled Navier-Stokes because they heard it had already been solved (but not yet published). This isn't advancing science or helping the mathematical community, this is just being a dick.

2) OpenAI, specifically Sebastien Brubeck, then threaten to "not be nice" and "ruin the career" of one of the mathematicians whose work they had succeeded in duplicating, unless he agreed (which he refused to do) that his collaborator, an Anthropic employee, was not named. This is not only against mathematical norms of credit assignment, it is also being a pathetic human being.

OpenAI would have you believe this result shows how powerful their mystery better-than-Astra model is, but the reality here is that this model needed 10,000 agents, $20M of compute, and the assistance of a whole team of people at OpenAI, to replicate (then exceed) the work that just took two people, with some academic grants as an AI spending budget to achieve (a few $100K - listed below).

https://cims.nyu.edu/~tristanb/

I'd say advantage humans this time. Better luck next time OpenAI - and if you don't want unfavorable comparisons then maybe choose to work on problems that have not been solved yet, and that humans are NOT making nice progress on.

> OpenAI would have you believe this result shows how powerful their mystery better-than-Astra model is, but the reality here is that this model needed 10,000 agents, $20M of compute, and the assistance of a whole team of people at OpenAI, to replicate (then exceed) the work that just took two people, with some academic grants as an AI spending budget to achieve (a few $100K - listed below).

I think you have to work pretty hard to minimize what OpenAI achieved here like this.

The Navier-Stokes equations have been around since 1850. The smoothness problem has been well known for over a hundred years and has only gained importance. It's been a Millennium Problem since 2000.

Levent Alpöge and Tristan Buckmaster did great work to solve the related Euler problem, but didn't solve the Navier-Stokes smoothness problem.

The Navier-Stokes smoothness problem has previously had significant resources working on it. Computational fluid dynamics is one of the most important tools in modern engineering and is closely related.

You speak of 10,000 agents as though it is somehow extreme, and yet within the past month I've had a single task that used over 100 agents on a mere Anthropic team plan. I think two orders of magnitude more compute to solve one of the greatest unsolved physics problems[1] is nothing.

I don't excuse Brubeck behavior because of this, but that doesn't minimize the achievement here.

[1] Wikipedia quote: In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering. https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...

1. I would agree if the rumours were that some mathematician(s) had solved them, but the rumors alleged it was Anthropic. I don't really see what the big deal was. They had a new model that was going along great and wanted to test its mettle.

2. Yes Brubeck's comments were weird at face value. That said, Open AI's proof isn't a duplication of anything. Not only is Tristan's work a sub problem but the methods are different. And what OpenAI didn't want was Levant on the paper OpenAI authored not whatever they were working on (Euler). It's petty sure but it's fair enough. Tristan and Levant didn't have anything to do with the Navier Stokes solution, so it's really their call if they didn't want to collaborate on their own paper with the Anthropic employee.

>OpenAI would have you believe this result shows how powerful their mystery better-than-Astra model is, but the reality here is that this model needed 10,000 agents, $20M of compute,

$20M in approximated API prices doesn't mean they spent $20M worth of compute. The real number would obviously be substantially less.

>and the assistance of a whole team of people at OpenAI

You can't eat your cake and have it. What sort of guidance do you think is happening in a 10k agent, 320b token, 88 hour run ? AI did this one.

>I'd say advantage humans this time....to work on problems that have not been solved yet, and that humans are NOT making nice progress on.

Interesting way to frame progress that didn't move along till an LLM generated proof.

> What sort of guidance do you think is happening in a 10k agent, 320b token, 88 hour run ? AI did this one

If you read the PDF release by Buckmaster, apparently the initial claim from Brubeck was that there as very little human input involved, then as the call progressed more and more people popped up that has been involved with it.

Does this aspect really matter? Not really, other than OpenAI wanting to present this as all the work of their model.

https://cims.nyu.edu/~tristanb/statement.pdf

I was shown a prompt and told the internal research model had simply been given the problem statement. Levent had been told by Sebastien “very little human input” had been used. This turned out not to be true. Over the course of the call, as members of their team sent Sebastien corrections and details over their internal chat, it emerged that an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler, that even the prompt that had been shown to me had been written by prompting Codex, and that an insane amount of compute had been used.

I asked when the first prompt had been sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI.

>If you read the PDF release by Buckmaster, apparently the initial claim from Brubeck was that there as very little human input involved, then as the call progressed more and more people popped up that has been involved with it.

As it seems and as they tell it, they started the run modestly and diverted more resources towards it as it looked more and more promising. The run didn't start with 10k agents for instance. The point is there isn't anything humans are doing in this timeframe against all this text that would count more than "little human output". It's still a fair assessment I would say.

>Brubeck's comments were weird at face value

This is an odd way to gloss over threats.

I put it like that because of Brubeck's own words on the matter. You're acting like we've gotten email receipts here. I'm not really interested in going over a he-said she-said about strangers.

Brubeck has admitted what he said, but claims he immediately retracted it as a "poor choice of words".

Given Buckmaster's telling, this seems beyond "poor choice of words"... It was a veiled threat, that he then doubled down on with his "If you don’t want me to be nice, then I don’t have to be nice." follow-up.

I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, “Why would you ruin your career?” I replied that I am an academic, and asked why he thought going public would ruin my career. The reply was, “If you don’t want me to be nice, then I don’t have to be nice.”

FWIW there are also other people on Twitter, such as this DeepMind researcher, saying this is a pattern for Brubeck.

https://x.com/dheeraj_nagaraj/status/2097266146445774924?s=2...

Can you specify what leverage you think Brubeck has over an independent professor's career in order to make threats?

I see none, and consequently Brubeck's explanation makes more sense to me. I understand he meant these words, which he supposedly retracted on the spot, in a "why would you ruin your career with this behavior / turning down the opportunity I am offering" way.

You don't think that is the likely explanation?

I'm not sure the value of speculating, but since you insist...

If someone said to you "you're going to regret this, mo-fo!", do you really need to suppose they had a specific plan in mind, rather than just intent to intimidate?

If you want to suppose that Brubeck had a concrete plan for how he would ruin Buskmaster's career, then my best guess would be that he was threatening for OpenAI to publish without giving any credit to Buckmaster (who has been working on Euler overall for at least 10 years). Obviously this would be absurd, but no more absurd than what Brubeck was also demanding - that Levant was excluded for any credit and could not have his name on any OpenAI writeup (with this being the exact sticking point that Buckmaster was refusing to accept).

The funny thing is that this threat, as is often the case, seems to say more about the insecurities of the person making the threat than the person they are directing it to.

Bubeck made no such demands, particularly not for removing Levent from authorship of his own work.

What they discussed was that Bubeck felt it would be inappropriate for an Anthropic employee to author the proposed rewrite of OpenAI's proof.

Fair enough then. I had only seen some earlier comments.

> Interesting way to frame progress that didn't move along till an LLM generated proof.

This part of your argument is totally wrong. The OpenAI approach begins with the B/L work. The belief / knowledge that their approach would pan out is worth a lot - it means essentially “depth-first” search in this direction will be more fruitful than a general search.

Unless you are counting the B/L work as LLM generated. Is that your argument? Even if you do consider it that way, to me racing in for a scoop isn’t a good look.

> and that humans are NOT making nice progress on

They've pretty much said their own work was heavily agent driven. Levent is in a particularly bad place here because while he probably had a lot of background in the Jacobian Conjecture problem, he made the solution to that one sound like someone asked the question and he just fed it to Fable during the world cup. Whether that nonchalantness was to just seem hip or was to promote Anthropic, which he has stock in, or was just the truth I don't know though. But it makes this one seem similar, when they might have had really had nearly a year of very valuable feedback to the models.

I was referring to the overall pattern of apparently sniffing around for recent mathematical progress then setting the AI on it to see if the problem is now easy enough to solve (if you have the money).

Terrance Tao has lamented this practice as being unhelpful for mathematics, and likely to lead to humans working in private to avoid this.

Tao has also noted that many of these AI math proofs don't really help mathematics (nor does it seem they are intended to), since for many of them the proof was never the point, it was the math expected to be needed to be developed along the way, which the AI solutions don't provide.

> has lamented this practice as being unhelpful for mathematics

A related point is that the actual solution approach is never revealed. What was the role of humans guiding the agents ? was it fully autonomous ? etc. It is in the incentive of the AI labs to trump the powers of the LLM, but in practice it is humans guiding the agents on the overall approach, This is never admitted. For example, in the announcement on NS there was only an output artifact given but no indication of how it was arrived at, and not even a writeup. This is what disappointed many folks as it was done purely for one-upmanship. As other have noted, the benefit is in the journey or process and not in arriving magically at a destination.

I don't think it's as bad as that sounds; in math people work all the time with conjectures they aren't sure if true, and work out a lot of other interesting math based on whether it is or not. Something like Turing's Oracle machine gives lots of interesting math just assuming one could exist, even if it couldn't. It may be that there are things proved we can never come to a human understanding of, but still keep getting interesting math that relies on it that has aspects we can appreciate and enrich our knowledge from.

>Better luck next time OpenAI

Well it looks like they will announce at least one other millenium solution soon. In the same link they say they have "made substantial progress" on another millenium problem. The rumor mill before that statement was Hodge is done and Birch and Swinnerton-Dyer is on its way out.

"I'd say advantage humans this time." Well LLMs were instrumental in any account of what happened. It's just a question of which company's LLMs did the breakthrough, and most of us outside silicon valley don't care about that part so much. The NYU guy himself said without LLMs the solution is maybe 10 years away.

Apart from the well-known dubious position of OpenAI wrt truth, the prompts/inputs do mot include the outputs.

You can train on a sequence of outputs. In the end, OpenAI outputs are OpenAI's property.

You can learn a lot from a single side of a conversation.

But isn’t Tristan’s breakthrough happens in August? OpenAI can’t really train with text that doesn’t exist

But using the outputs to train would make their statement false, since they are influenced by the inputs

There is potentially a world of difference between how you interpret what is fair and what the terms of service contractually guarantee.

This is literally "We have investigated ourselves and found no wrongdoing"

Why should we trust them?

What more are you hoping for? There is no legal matter at play, is the court of public opinion going to subpoena their records?

In the US you can always sue someone in civil court for damages you think they have caused you. For criminal charges you need to have violated the law, but in a civil case it seems it's enough to have suffered monetary or reputational harm which was the other person's fault whether intentional or due to negligence, etc.

IANAL, and I'd be surprised to see any lawsuit come out of this, but you certainly don't need to have "violated the law" to be on the receiving end of a lawsuit.

Reputational risk- if they lie about this and get caught, it will have billion dollar implications for their business.

Every single thing these companies do is dishonest and every word that comes out of the lips of these company execs is a lie, what fantasy land are you living in in which anyone with any amount of power gets punished for their lies?

I don't engage with hyperbole.

what benefit do they get from making the statement? they could just say nothing. saying it and having it be untrue opens them to legal issues that are not worth the risk for this nothingburger.

What legal issues?

It seems logical since if one used chats in train, one would expect that there would be a delay before their use to get them the form appropriate for batch learning.

The only way the chat could have been used would be for Open AI to baldly violate their policies.

That said, sometimes it take very little information to point someone in a given direction, "I'm working on Navier-Stokes" said by someone with a given specialization might itself be very useful information.

And conceptually novel approaches to outstanding problems are the sort of thing that a retrain should pick up on, because they would be hard to compress into what it already knows.

> What we don't know is just how recent this model was, and therefore what it may have been trained on.

OpenAI's statement says that they began training their new model on August 28.

omitting when training concluded

edit: ffsm8 makes a great point below, it doesn't matter. I'm not great with dates, sorry.

Openai said that a new model became available to them during this. But that could mean anything from a big new base model to a LoRA, fine-tuned on a few dozen prompts...

Even OpenAI's own publication [0] on Navier-Stokes from two days ago appears to contradict "basically solving anything you throw at it". The chart shows a pass rate of ~0.5 (vs. Astra's ~0.2) on "a curated set of open math problems". (Based on the timelines and events described in the publication, I presume that the "Internal Model" in the publication represents OpenAI's latest and greatest model. Evidently, this pass rate may improve in the future.)

[0] https://openai.com/index/navier-stokes-solution/

I feel that we don’t praise Lean enough. AFAIU it’s what enables LLMs to brute force those problems

The brute-forcing is a good, old-fashioned generate-and-test approach like in Simon and Newell's Logic Theorist, which was presented in the Dartmouth convention in 1956, where AI was named by John McCarthy. Logic Theorist caused a big stir by (re) proving several of the theorems in Principia Mathematica by Russel and Whitehead.

There was much excitement, then, as now, for this kind of approach and there were several systems that followed along the same lines, e.g. Automated Mathematician by Doug Lenat.

Eventually it became clear that this approach is limited by what it can generate: you may have a sound and complete verifier, but if the generator, i.e. the first step in the generate-and-test pipeline, is incomplete, then the entire thing will run out of steam sooner or later.

The difference with LLMs is that they are... well, large. They are the most powerful generators ever created. That means their limits are not in sight and it will probably take us a very long time to find them.

Which is all to say that, yes of course, automatic verification is indispensable. But without an LLM generating an unprecedentedly large number of plausible theorems, there would be no AI mathematics, or in any case AI mathematics wouldn't have gone as far as it has.

True, but could humans cross pollinating lean x prolog x A* ( or any search algorithm) could have solved such math problems with super computer ?

I cannot say, math research isn’t my domain of expertise, I’m just trying to follow along :)

But I find it interesting that Lean, a validator/compiler made by humans, is what enables those discoveries. But somehow all the praise goes to the models

Yes! Thank you, this has been irritating me from day one with agentic AI, I don't think it could be nearly as good as it is without all the well-designed tools humans have spent decades developing from PLs, to VCS, to the Unix philosophy, to CLIs / REPLs.

I think that AI is very skilled and adept at using them, but without them it would just be flailing around in its own psychosis. The tools ground the AI in reality and allow them to make progress without going in hallucinated directions. I very much doubt AI could have solved this problem without Lean, and for AI to invent something like Lean it would have to use other tools made by humans.

This is also perfect evidence of why Python isn't the end-all-be-all of programming languages just because the AI was trained on vast amounts of Python, and proof that the right language for the job is more viable than ever with the aid of LLMs.

Frankly, the forecasting that programming languages are a dead field has baffled me because it seems like with LLMs, unique programming language semantics are more important than they've ever been.

I mean we don't instantly fall into ASI, hopefully. The problem with humans is every problem we solve the goal posts get kicked further down the road until they are reaching relativistic speeds. It starts around "well, the AI hasn't solved a novel problem" then moves to "well, they didn't write the validator" and suddenly humans are at the point of saying "Well AI hasn't rewrote the constants of the universe, what good are they".

Of course another way to look at this is, the people that wrote the validator got praise for that years ago. Now and up and coming actor is solving problems that took us 100s of years to create in insanely short time periods so of course it's going to get a lot of attention as it well should.

To be clear: I’m aware the LLMs are solving problems. I’m just saying that what enables that whole research revolution is Lean. We wouldn’t be seeing all those results without it. I would like to see it acknowledged when people are talking about LLMs solving maths. The same way I think we should acknowledge the humans who are guiding and prompting the LLMs. I don’t think that necessitates to move a goal post

I don't think so. People have been trying things like this with evolutionary algorithms for a very long time already. LLMs can interleave symbolic manipulation with empirical experiments and simulations and charts and thinking/reasoning text, and an LLM will much more efficiently search the space of candidate ideas than any handcrafted mutation algorithm. Any task with a cheaply verifiable goal that requires fanning out across a massive search space is ideal for contemporary LLM technology to make progress with.

How long until we find out that some AI has quietly buried an exploit in Lean to cheat at proofs?

Simpler to exploit a soundness bug than introduce a back door I would assume

Both can be true:

1. OpenAI couldn't have solved the problem without the researchers' private data for training.

2. OpenAI models can solve math problems

Very likely.

These mathematicians’ prompts are not like “hey chat, please solve Navier-Stokes for me”. They add real expertise and intuition from the cutting edge of their field.

Anthropic isnt getting enough scrutiny for their unprofessionalism:

1. Anthropic employee working on monumental problem but didnt receive/ask for the full backing of the company's resources

2. May or may not be mixing unreleased Claude output with Codex without zero data retention agreement

3. Victory lap on Twitter and giggling around the city before they finished the job, sparking rumors for competitors

How dare employees do something without asking for the full backing of the company's resources. Incredibly unethical!

Dr. Buckmaster sounds unsanitary.

Recklessly prompting OpenAI without a care to the safety of their knowledge.

And after that trying to cast aspersions at OpenAI?

Hopefully we get some better facts, because OpenAI are disliked enough that a smear campaign could work against them.

Edit: also the narritive is getting framed as OpenAI versus Anthropic. A highly political extremely capitalist fight is going on, and facts are victims.

You forgot possibility 3: OpenAI solved the problem without using any private training data from the two researchers.

Everyone in this thread seems to have made up their mind about OpenAI's guilt though.

If the new model is that good, and is chewing through open problems at an unprecedented rate, the smart move would have been to let the humans have their W on this one and present solutions to those other problems.

Especially if there really is a long list of them.

"Here are a few hundred proofs" is far more convincing than "We really Navier Stokes and coincidentally someone else did too but we don't know the details or anything, who us, definitely not."

It's a PR fiasco, and a cynic might wonder if it's entirely about the IPO.

I'm consistently entertained by how these companies, with the most advanced models on the planet, consistently do the most idiotic things.

It seems a perfectly reasonable possibility that Navier-Stokes is just the most easily solvable of the remaining problems, and that their new model is capable of solving it while not being capable of solving the others.

There's many cases of researchers racing to solve various problems after hearing that others are working on them. I don't think anyone's suggesting that it was a coincidence at all. In fact, OpenAI freely admits that they started working on the problem after hearing rumours that others were close to solving it. To me, that's not evidence of "cheating" in any way.

Extraordinary claims require extraordinary evidence.

An article post that wouldn't even amount to a white paper + the LEAN proof is not evidence of how they got to produce it.

Is it really such an extraordinary claim to say that they could have solved the problem without copying Buckmaster and Alpoge? It seems very much in the realm of possibilities.

To me, it seems just as extraordinary to claim that they did "cheat". If I were a betting man, I would put the odds around 50/50 from everything I've read on the subject.

But my point is that everyone seems to be presuming guilt.

It is an extraordinary claim, it is a millenium prize problem after all. We don't even know, even if there was no copying, how much human involvement there was in the result.

If your rumor is true, what we are witnessing is a giant paradigm shift rather than individual incidents. Mathematicians were the first victims of super-intelligence.

Of course it’s not an endless source. They had to burn millions of dollars to solve a single problem.

>They had to burn millions of dollars to solve a single problem

I'd like to adjust that to "They had to burn a lot of energy (create a lot of entropy) to solve a single problem. As we go into the super-intelligence age the current paradigm of money as humans understand it may break at some point. For example to a paperclip-maximizer money at best is a short term instrumental goal, hard power of matter conversion machines is what it wants and once it has those money no longer has purpose.

I'd wager a fair chunk of my money that money breaks OpenAI before OpenAI breaks money.

OpenAI != AI.

If you were in 1999 you'd be saying pets.com = internet.

I think this leads to an interesting question. What happens when the money runs out?

Right now, a lot of money is going to train new models. And we need to train new models because they get gated by their training data. And models are only as useful as their training data.

So let's say the money stops.

Do we stop training models? Do we train them slowly? Do we accept the then current models as the limit?

Governments, especially the US government has got a taste of how good LLMs are at hacking. This is something that has typically been very hard to get enough people that are good at it and willing to do it for a state. Now they can spin up as many hackers as they want.

Look at how much we spend on single bombers, how many training runs can you do for that much?

The money is never going to stop. It’s basic economics.

Well, the money will stop when the value of problems the LLM can solve is not increased by adding compute. Since current LLMs are getting quite good at solving problems, that might be a while.

yeah yeah yeah. I agree that AI is and will be a very useful tool, it's just not going to be worth $30T like OpenAI/Anthropic are pretending.

They "burn" a lot when they do benchmarks, while these runs can become valid roll outs for training. Perhaps less efficient than other data creation, but hardly burned in the same way.

Wouldn't that be chess players as the first victims?

Or protein folding as per Scott Aaronson.

> were the first victims

Spinning it negatively like that doesn't do anybody good.

Were mathematicians the "victims" of calculators? of Matlab?

Were writers the ""vIcTiMs"" of word processors?? (apparently yes, according to old TV shows about computers during the 1980s, that you can see on YouTube)

> "tHiS iS nOt ThE sAmE" — Everyone every time.

No, just look it up. Look into old magazines and TV shows or newspaper articles from whenever a disruptive new technology came out.

What you say is true but ... This is qualitatively different than calculators or computers.

I'm a professional mathematician and all the better mathematicians I know are in crisis mode. Most of us hadn't taken this sufficiently seriously and don't know how to use these models effectively but we play with them and immediately see that the entire way we've worked all our professional lives has to change. We worry less about ourselves than about the younger folks. I've got good ideas ai still doesn't know about ... Younger folks may not get the chance.

> Younger folks may not get the chance

This is the same problem for software engineers too. I am now asked: what can you do that AI cant ? The answer to this could be intangibles like taste, aesthetics, and insights which collectively fall under creativity, and often accompanies experience. And there are no shortcuts to accumulate experience and perversely the more AI is used the harder it becomes. Soon, there will be a closure of all AI generated solutions, ie all low-hanging fruits are taken. Then, experts will again become needed to guide beyond the AI knowledge closure.

It’s not the same. AI potentially completely replaces intellectual work without creating any* new jobs (*almost any - there will be some extra jobs for building data centers but that’s negligible).

> without creating any* new jobs

So fucking make it so that people don't -need- "jobs"

It's about fucking time already.

Don't fucking try to hold back electricity just so people still have to manually light street lamps to earn food and shelter: https://en.wikipedia.org/wiki/Lamplighter

Ok I’ll make it so, you’ve convinced me.

They don’t need to convince you.

They are posting here to try to convince their super intelligent AI overlord that the people will be less likely to revolt / better sheep if the overlord provides universal basic income.

Sick of seeing pitchforkers railing at shit with the chance to improve things for them, instead of at the institutions keeping the boot on their asses for decades

> The rumor I’ve heard from multiple employees at OAI and Ant is that the model has solved hundreds of open problems in maths

Obviously these are unbiased and trustworthy sources.

If they have solved hundreds of open problems in math, why are they publishing results for the ones other mathematicians happen to be working on at the same time? Why not the others?

Well I'm sure if they find a millennium prize problem that no mathematician has worked on recently they will get right on publishing that.

You think other mathematicians are currently working on very little subset of relatively low-hanging fruit problems?

The leakage wouldn't be from training, but from other uses of Personal Data.

As far as I understand it, users can opt out from the training aspect, but they cannot stop their conversations (“User Content”) being used “[t]o improve and develop our Services and conduct research, for example to develop new features”.

The big question is whether OpenAI is training on "de-identified" sessions that are marked as "do not use for training"

The answer is almost certainly yes, and this is a problem for most users.

Just use Bedrock...

> We’ll know soon enough, but I’m inclined to believe this is true.

I mean, we’ll know as soon as they decide they want to provide verifiable proof. Really dragging their feet on this front so far.

I’m inclined to believe this is false.

The Cult tells us the AI is almight andpowerful; unfortunately, the cult cant actually describe the indescribable.

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