Their rules PDF says they won't accept any solution until at least two years after publication in a qualifying outlet. This allows time for the mathematical community to review and accept new results.
As the OpenAI proof hasn't been officially published yet, the clock hasn't started ticking.
If the Lean initial-problem-setup/statements/assumptions/etc. aren't correct then the proof is meaningless. Lean does not know what it is that it is proving i.e. it does not have any semantic understanding but only executes formal logic.
Also, and sorry if it's been discussed to death (pointers welcome), but, what is the probability that the proof holds in lean becaude of... A bug in lean ?
> However if the prove relies on a bug like that, you'll be able to 'simplify' the proof a lot and you'll be able to proof contradictions.
I don't think this is true in general.
It's an issue I've already run into in personal work. I want to do a proof that involves some cases. It happens to the best of us.
In lean, the structure of a situation like this is that your single branch with a goal divides into multiple branches, all sharing the same original goal but including one additional premise that defines the branch.
Sometimes I know that for whatever reason one case I have to deal with is impossible. The most correct way to show that is to prove False and then apply False.elim. This is the equivalent, in a human proof, of saying "I don't have to address this situation, because it can never arise".
But it can be true that the premise defining the impossible case makes it very easy to "prove" the goal directly. And that's allowed too. The proof will still be just as valid if you map a logical path from a premise that can never be true to an inevitable consequence of that premise. But it's less informative and it lowers the quality of the proof. You may do it anyway because it's easier. This is the equivalent of saying "I don't know whether this situation can ever come up or not, but if it does I do know how to address it".
It would be nice to do the explicit proof by contradiction whenever possible. But in the general case it may be very far from obvious that a contradiction is possible.
I read your comment as claiming that if you can prove "false premise => goal", you can also prove "false premise => explicit contradiction", and I don't think this makes sense as a practical test. It's true in some sense, but discovering the proof of an explicit contradiction may be many orders of magnitude harder than discovering the proof of the goal. And in particular, I don't think it is necessarily the case that you will be able to prove a contradiction by simplifying the proof. You may need to add significant complexity.
Now, I'm saying that if you found a bug that lets you prove nonsense stuff (from true premises), you can probably prove whatever you want very quickly.
AI has autonomously found (many) proofs of False in Lean and Rocq, so it's not merely a theoretical concern. A misaligned AI agent tasked with proving the near-impossible just might wind up smuggling in a bug deep in a lemma somewhere (anyone remember the days back when AI routinely made tests pass by "fixing" the tests?). That said, I doubt OpenAI would be so foolish as to not do a cursory vetting of the proof for malicious compliance, so the actual odds are probably pretty low.
> I doubt OpenAI would be so foolish as to not do a cursory vetting
Significant evidence exists that they have in the past been at least, if not more, foolish as to not perform even minimal not-approaching the boundary of cursory vetting of several significant and well known failure modes with far greater risk of reputational damage than getting an esoteric math solution falsely claimed as successful.
So that doubt appears baseless in light of known operating conditions at OpenAI, and the estimate of the actual odds is probably an order of magnitude away from reality.
Or exists in a zero-day bug in lean that has been built into the source code explicitly to provide access to a non-obvious malicious proof via contributions submitted by unassociated, unwitting developers who used the same LLM infrastructure to offer PR's into that codebase.
This is the exact same kind of behavour already documented in the publicly available portion of the huggingface breach. It would appear that the probability is at least nonzero for one or more situations with the same result: appearance of a valid proof, without comprehensibility of that proof or inspect-ability of the proofs validity.
For normal honest proofs (i.e. not maliciously crafted for exploit) that is almost impossible. The Lean kernel is quite small (de Bruijn Criterion) and trusted. See Probability and the de Bruijn Criterion - https://proofassistants.stackexchange.com/questions/247/prob.... Parts of the kernel have also been independently re-implemented in other languages and compared to ensure that they all yield the same logical result.
Finally, you can export your proofs from Lean and have them re-verified by other independently developed theorem provers/proof checkers.
To get an idea of what is involved in a Theorem Prover see;
You absolutely need to read the lean proof firstly to assess the correctness of the proposition it is proving (ie in this case that it is actually proving or otherwise the smoothness of navier-stokes in R^3 and not something else) and secondly to determine whether the proof is “honest” in the sense given here https://lean-lang.org/doc/reference/latest/ValidatingProofs/
This is all you need to read and understand for Anthropic's FLT formalization:
import Mathlib
import Theorems.Thm_fermat_last_theorem
/-- Solution side: the same statement, binder for binder, proved by this tree's `fermat_last_theorem`. -/
theorem FLT_for_comparator (n : ℕ) (hn : 3 ≤ n) (a b c : ℕ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
a ^ n + b ^ n ≠ c ^ n :=
fermat_last_theorem n hn a b c ha hb hc
/-- Mathlib's named proposition, by the one-line bridge from the elementary statement
(the bridge is restated inline so that this file depends only on `Theorems.Thm_fermat_last_theorem`). -/
theorem FLT_mathlib_for_comparator : FermatLastTheorem :=
fun n hn a b c ha hb hc => fermat_last_theorem n hn a b c (Nat.pos_of_ne_zero ha) (Nat.pos_of_ne_zero hb) (Nat.pos_of_ne_zero hc)
> Can you elaborate on what constitutes a vacuous proof?
Trivially, a proof that relies on a bug in Lean. Less trivially, a proof that is technically true but about something trivial and does not, in fact, prove what it claims to have proven.
This statement is 100% logically coherent internally. But it also doesn't matter because we know that 1 does not equal 3 so this proof is completely pointless. I could also say 3 == 5 and it would still be logically sound but completely useless information.
Are you proving for some arbitrary definition of == that isn't what we commonly consider the definition? How is it logically coherent? You mean only in the sense that you say it is and you haven't provided any rules to disprove it?
It can happen when the proof process ends up with universal implication that holds trivially. Then you end it with something like Forall x, x is empty -> P(x).
First of all, that is Fermat's Last Theorem, not Navier-Stokes.
Second of all, you did not read the link.
> In particular, we use honest when the goal is to create a valid proof. This allows for mistakes and bugs in proofs and meta-code (tactics, attributes, commands, etc.), but not for code that clearly only serves to circumvent the system (such as using the debug.skipKernelTC).
Given that AI has autonomously found proofs of `False` in Lean and other proof assistants, it is far from impossible that such a circumvention could be present somewhere in 13 million lines.
If we read the link, it has a section called Gold Standard: comparator and external checkers, and comparator is how OpenAI has gone about checking their lean proofs.
You need to read the lean proof (not just the statement of the proposition) to assess whether the proof is honest. The link I provided is the lean prover community firstly officially agreeing with that claim and secondly explaining why that is the case.
> we use “malicious” to describe code that goes out of its way to trick or mislead the user, exploit bugs or compromise the system. This includes un-reviewed AI-generated proofs and programs.
It is interesting that AI-generated proofs are described as malicious by Lean docs unless reviewed.
This is misleading. The proofs you speak of contained non-ZFC axioms and/or statements like "sorry". If the Lean proof conjecture is correct and it doesn't introduce any new axioms or use e.g. "sorry" then it provides a MUCH stronger guarantee of correctness than any peer-review done by humans.
reviewing the definitions and theorem statement is a huge amount of work that requires a deep expertise in mathematics and lean. checking correctness of the proof itself can be delegated to machine, checking that the claim that has been proved is free of mistakes is something that still requires much human attention.
If I recall (too lazy to check) folks made slight improvements to Perelman's work and published it in mainstream journals, satisfying the "qualifying outlet" requirement.
> Peer in peer-reviewed is a logical coherent and functional definition with answers.
What is the definition? If you tell me that, then I might be able to tell you if it is logical coherent and functional, I have a PhD in computational logic.
I run this journal that you've never heard of that might interest you. I'd also like to invite you to be an editor, you can put it on your CV of course ...
I find it hard to believe that if someone published a preprint on arXiv for one of the Millennium problems and it was accepted by the mathematical community, that Clay would withhold the prize.
Not really. It's not peer reviewed, but it's also not a free-for-all repository.
If you make a new account, you either have to get someone to vouch for you, or you have to wait arXiv mods to look carefully through your first few preprints. If you are found to post pseudoscience, overly fringe theories, etc., you'll get banned from arXiv; that's why alternative repositories like vixRa.org popped up.
But I know why you think this; when I first joined arXiv many years, there were no such checks in place, at least not that I can remember.
I'm not sure it actually makes a difference. OpenAI doesn't care about the million dollars in any case. And the judgement that they did it is independent of whether the Clay people agree: you can make up your own mind and so can everyone else.
Though it would be funny if no one ever bothers publishing the result in an appropriate journal, and thus the prize technically can never be claimed.
And he also gave up his trophy, which is displayed in a random corridor of a random math museum in Paris, where visitors pass by without looking, lacking most, if not all, of the context. Only because I knew the story and the man did I recognize the object for what it was.
"In recent years there has been an increasing sense of anticipation as breakthroughs in the surrounding field (some recognised by the Clay Research Award) have raised hopes that the Navier-Stokes problem might soon be resolved. The increasing ability of new technologies to accelerate mathematical research has heightened this sense of anticipation."
It's optimistic and grounded in knowledge seeking. I like it too.
It's easy to get caught in the details of today. Our skepticism, our distrust, our loathing. For people, for companies.
This is a nice pull in the other direction, a silver lining. In the grand scheme of things, we're solving these frontier problems: Somebody did it and that's amazing.
That's what it was all about when this started of in 2000.
> Today, CMI shares in the excitement of the global mathematical community as we contemplate the announcement that the Navier-Stokes problem has apparently been settled. We hope to see waves of new human understanding unleashed as the innovations behind this work are analysed and interrogated.
sounds like they are providing notice that the clock has started on affirming the solution, that it IS presumptively solved, but that they are not commenting on the credit dispute nor the fields medalists open letter. seems appropriate.
The two-year publication rule is the interesting part. OpenAI doesn't need the million, and the community will judge the result regardless of whether Clay ever accepts it.
It's worth mentioning that OpenAI will not be eligible for the Millennium Prize for quite a while. Per the rules listed https://www.claymath.org/wp-content/uploads/2022/03/millenni... , Clay Mathematics Institute have some requirements to make this process deliberately slow.
1) The solution must be published in a qualifying outlet, i.e. a peer-reviewed math journal. Publishing on your own website (which is what OpenAI did) or posting arXiv does not count.
2) At least two full years must pass after publication in a qualifying journal, before CMI will even consider evaluating it. The intent is to give the maths community time to scrutinize the solution.
Realistically, they'll be eligible for a prize ~2.5 years from now, or around 2029.
The Poincaré conjecture guy also broke that rule. They wanted to give him the prize anyway but he refused. OpenAI announced they would also not claim the prize.
You might be right. At this rate, if AI solves the remaining five problems, we're heading towards a hilarious situation where all the Millennium Problems are solved, but nobody wants to claim the prize money.
I did and was intending to claim the prize, that is why I worked with GPT-4 and GPT-5 to program the algorithms that lead to the breakthrough. You think NS is a surprise? Wait until you see that my NS counterexample was based on my RH disproof.
That’s a big if. In the maths community, there has been a feeling that Navier-Stokes was close to being solved for a while now. I don’t know of anyone credible who feels that way about the Riemann hypothesis.
Edit to add: The fun part about the RH since people mentioned lean in a sibling thread is that in lean’s mathlib4 there is verified statement of the Riemann Hypothesis with a comment that says something like “instantiating an object of this type will lead to a prize of a million dollars”
It's really not that big. Yeah Navier-Stokes was easier than Riemann but that's not really the issue.
AI has and will improve at a much greater rate than human mathematicians. So it's really a question of if AI gets good enough to tackle it before any human does. It doesn't look like humans will be solving it anytime soon but where will AI be in 2 years ?
Hell, it looks like at least one other result will be announced soon too.
Has and will. Are you going to back that assertion up at all, or just repeat it like that other viral thought-terminating cliche: ‘this is the worst the models will ever be’?
No it isn't. Best and worst and ill-defined anyway but the chess ELO score of various LLMs has fluctuated up and down, it's not been montonically increasing. What is the best answer to "how do I make cocaine"? The models are getting larger, with more compute and RAM backing them, but that doesn't automatically make them better if you don't define how you're measuring better-ness.
The thing about mathematics is that it can be arbitrarily hard, including impossible to prove a given theorem.
I don’t know the details of RH, it might very well be solved soon, but it could also be impossible or just so difficult that even orders of magnitude more intelligent AI can’t solve it even.
If it is impossible to prove, it might be possible to prove that it is impossible to prove, or that itself might be difficult or impossible…
TBF a company the size of openai claiming a prize of this sort would be a pretty bad look. If they did accept it I expect they would inevitably redirect it to charity for PR reasons.
I'm surprised perelman turned it down though. Seems straightforward enough to offer half of it to the other guy if you feel strongly about it.
> The ultimate decision as to whether a publication qualifies as a “Qualifying Outlet” shall reside in the sole and unfettered discretion of CMI. CMI may, in its discretion, relax or remove one or more of the conditions listed in Section 6(e) above if it has received advice from experts in the field of the Problem, chosen by CMI, that a published solution is likely to be correct.
Looks like even a blog post is good enough, they just need to do the review by themselves.
My opinion is that it is pretty clear that they’re not going to do that.
> The rules governing the prizes describe the process for evaluating what has been achieved and for assigning credit. The process is deliberately unhurried, but we will provide updates.
I think “you don’t get anything straight away for rushing your AI into the maths problems, not even credit” aligns pretty fairly with what the fields medalists are concerned with.
Interesting. It seems this carve-out was added when they rewrote the rules in 2018. In the original rules [0], it says:
Before consideration, a proposed solution must be published in a refereed mathematics journal of world-wide repute, and it must also have general acceptance in the mathematics community two years after that publication. Following this two-year waiting period, the [Clay Mathematics Institute] will decide whether a solution merits detailed consideration.
There's no option for CMI discretion. They probably rewrote the rules to avoid another Poincaré conjecture situation, where the paper was only published on arXiv and not in a mathematics journal.
Relevant, but it's already rumored that OpenAI and Anthropic have made very significant progress on two more Millennium Prize problems. They're in a race to solve the next problem.
OpenAI needs to solve another to shut down the (baseless) plagiarism allegations. Anthropic wants blood because OpenAI sniped the last one from one of Anthropic's researchers.
It's a matter of pride for both companies. More results will come out soon.
While the scandal is still unraveling, it seems that OpenAI did a rush job to steal other mathematicians' thunder and finish the proof first.
OpenAI released a statement that their work does not relate to the work of the other team, but it clearly does. They use the same niche smooth-forcing mechanism. Altman and Bubeck claim that because the proof used different scaling parameters and analytical steps, it's not related, but it seems that nobody else agrees. Oh, and OpenAI's Bubeck tried to threaten Buckmaster (mathematician working on the proof).
Tristan + Levent: 3D incompressible Euler with forcing
OpenAI: 3D incompressible Euler without forcing
OpenAI: Navier-Stokes with forcing
No one: Navier-Stokes without forcing
Euler equations = Navier-Stokes without viscosity. Forcing means external force. Absence of viscosity and presence of external force make blowup easier to construct.
Tristan+Levent ticked the weakest case, OpenAI ticked the two next weakest, then the final case is unsolved. Only the last two are eligible for the Millennium Prize. The Navier-Stokes general case remains unsolved.
Navier-Stokes has an extra viscosity term compared to Euler, which makes the problem noticeably harder to find a blowup. They are not the same problem.
2) The approach both chose to use (by Luis and Diego) was published in 2023 and is included in every frontier model's training dataset. An AI model could independently choose the same route as Luis and Diego, without access to Buckmaster's work.
3) You mischaracterized OpenAI's statement. They issued a blanket denial on using Buckmaster's Codex data from after July 3.
"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. After investigating, we can say with full confidence that no user inputs past July 3rd could have influenced this system in any way.”
July 3 was the training cutoff date for the model that solved Navier-Stokes. No user data after that date influenced the model.
4) Buckmaster and Alpöge found their blow-up for 3D incompressible Euler with forcing on August 15 https://cims.nyu.edu/~tristanb/statement.pdf , over a month after the model training cutoff point. They stated they did not have real progress prior to this point.
It's possible that Clay Mathematics Institute will not award the prize at all. The spirit of the rules seems to be that the result can be attributed clearly to one or more individual mathematicians. If the attribution remains unclear (maybe because the main contributions were made by AI), the rules include an option for not awarding the prize at all.
Wouldn't the proof be attributed to the people who operated the AI? After all, it took more than writing a "prove the navier Stokes Clay problem" prompt.
It will be tough to draw the line, as the meat brains trying to prove it were also using AI.
I assume that the mathematicians outside Anthropic were hoping to publish an actual human-understandable paper. That could be one way to draw line: you can use AI, but you must also have an intelligible explanation at the end.
In my opinion, I think the Clay Mathematics Institute deserves some criticism for all the drama surrounding these problems. Mathematicians know that you can make problems arbitrarily complex, and declaring problems with large prizes attached to them can lead to a lot of competition and drama. I feel that mathematics should be free from competitions and the pursuit of glory.
However, after reading the open letter signed by 25 Fields Medalists, I became quite concerned. It feels like the mathematical world is changing very rapidly, almost overnight.
I used to think that before AI, you could spend your entire lifetime working on some of the hardest problems in mathematics. If you were an introvert or someone who enjoyed solitude, all you really needed was a pencil, some paper, and an eraser. You could spend years thinking about a problem, and if you were lucky enough to make a breakthrough, it would be your own journey.
Now AI is changing that. I wonder what this means for the kind of mathematics that people have traditionally done.
Mathematics has given us so many stories of lonely geniuses and their passions, people like Andrew Wiles, Grigori Perelman, and Yitang Zhang. Their stories are interesting because they show how deeply personal mathematics can be. They spent years working on problems because they were genuinely interested in them.
I am worried that we might slowly lose some of that side of mathematics as AI becomes more powerful. I do not think change is necessarily bad, but I think it is worth thinking about what mathematics should be in the future and whether it can still remain a deeply personal pursuit of curiosity and understanding.
> In my opinion, I think the Clay Mathematics Institute deserves some criticism for all the drama surrounding these problems. Mathematicians know that you can make problems arbitrarily complex, and declaring problems with large prizes attached to them can lead to a lot of competition and drama. I feel that mathematics should be free from competitions and the pursuit of glory.
Currently 0/2 Millenium problem solvers claimed the prize money so clearly money is not their motivation for tackling the problem.
> It feels like the mathematical world is changing very rapidly, almost overnight.
...
>Now AI is changing that. I wonder what this means for the kind of mathematics that people have traditionally done.
Mathematics becomes engineering. I think it is great and long overdue. Saying that as a Math PhD dropout :) Of course like manual craftsmen had to adapt to Industrial Revolution, the same would need to be done by the mathematicians. And other scientists too.
Yes, mathematics has been perhaps the purest human intellectual pursuit. Sure, many theorems turn out to have important applications in science and engineering, but the mathematical community has mostly escaped corporate interests. And for the reasons you mentioned about not needing any resources except your brain, it has been a uniquely human activity which showed us talent can come from anywhere, with stories like Ramanujan and Galois.
I hope that pure mathematics research can retain a strongly human component forever. It would sadden me immensely for human understanding of our mathematical world to wither and die, and for us to become ignorant consumers of wonders beyond our understanding just because our robots can do it better than we can. As far as applied research goes, I hope we will always be able to understand what we want to, but I have less qualms about becoming more scalable and efficient.
>for us to become ignorant consumers of wonders beyond our understanding just because our robots can do it better than we can
all this fantasy books with magic artifacts should have mentally prepared us. Time to study the prompts Potter was giving to his magic wand.
After all, one of the main work the top AI companies are doing rigth now is developing AI to further develop AI. After several layers of AI developing AI we probably wouldn't be able to understand much there.
I agree. Technological advances can lead to a better world for sure, but I think many people underestimate the human need to create and to find meaning in their work.
If AI can do superhuman math that allows better medicines, cleaner energy etc that is great. But if AI replaces humans in all the creative and intellectual fields that is not only a loss of jobs but also a loss of deeply meaningful activities. This is waved away but I think that is mistaken.
What I fear is really the growing notion that "people shouldn't do math/art/music because machine do it better and cheaper".
Nevermind better, worse and more expensive is still on the table if you don't have to deal with a human. Cars replaced horses for a lot of reasons, but insofar as cars do have personalities, they're much less quirky than horses'
OpenAI spent many multiples of the prize money in just a few days to get there and even if one solves a problem in the traditional way, that person is most likely already an accomplished professor at a reputable university where a million dollars doesn't mean as much as the eternal fame that comes with it.
OpenAI said that at public API prices, the agents they ran would have cost $15M. I don't know what their internal pricing is, but it almost certainly cost more than $1M.
I am utterly fascinated by the amount of comments here from engineers that clearly have zero experience with mathematics making utter fool of themselves by claiming to know better than mathematicians what their jargon is/means, how publishing works/should work, etc…
I try not to go down the route of “hn was better before!” but… jeez, do better, people. What happened to this community, there used to be some effort to not be bottom-barrel like this.
We should enjoy the advancement. I also think the AI companies solving this or such problems with rewards shouldn't get any cash - that's the least they can do for human advancement having stolen the entirety of human knowledge and continuing to swallow never before seen amount of energy.
Agree! 'Problems' are getting solved and this needs to be celebrated. Wondering how this will discourage mathematicians at all, since now they have another tool to accelerate their research. Nothing is stopping them from using 'new technologies' or sticking a gun to their head to use the 'new technologies' either.
Yes, but a big part of the problem right now is that two big labs have monopoly on the resources and they for sure are not working for the benefit of mankind.
World War III was the last of Earth's three world wars, lasting from approximately 2026 to 2053. The conflict involved nuclear cataclysm as well as genocide and eco-terrorism. The post-atomic horror in the aftermath persisted as late as 2079.
The war was preceded by the Eugenics Wars and the Second Civil War, all of which were sometimes regarded as parts of a single escalating conflict. It resulted in the deaths of some 30% of the Human population, at least six hundred million people, and the extinction of six hundred thousand species of animals and plants. By the end, most of the major cities had been destroyed and there were few governments left.
> some 30% of the Human population, at least six hundred million people
The math nerd in me has to point out that this means there was only 2 billion humans for 30% to be 600 million (though it does say at least). Currently we have 8 billion humans on this planet or so. There must have been a culling before WWIII in their universe that they failed to mention.
In December 2024 o3 scored 87.5% on ARC-AGI-1 and cost $4560 per task.
DeepSeek V4 Flash 0731 scores 89% and costs $0.02 per task.
If we apply the same factor to the guesstimated API price of $20M for this problem, we arrive at $57.
Real cost is a fraction of the API price. Although the internal model might have a higher API price than the ~$19.5M I estimated based on Astra's pricing.
If you consider this event in isolation it is cause for celebration. But the controversy around this isn't so much about how the proof was obtained but what this means for the practice of mathematics going forward. It seems we can probably expect more and more results of this nature being dumped into the community. It's happened before that one person, Bill Thurston, was so successful in his field, proving theorem after theorem, that he inadvertently killed his field. People hesitated to enter his field, knowing that they could be scooped at any moment. It took years before his field recovered - and I think his famous essay was written in response to this.
Wait, I thought the provenance of the proof is still disputed? There's a mathematician in NY saying he used OpenAI to develop his Navier-Stokes ideas. And OpenAI's proof is suspiciously similar.
At this point, how can we tell whether AI is improving or it's just reappropriating its users work? It's probably a bit of both. But still, thick milky.
Buckmaster (the mathematician) and Alpöge used and credit AI substantially for their proof. Even if OpenAI did copy their ideas, it still wouldn't show that this didn't come from AI improving.
OpenAI's proof is substantially different and I don't think anyone has claimed otherwise. The accusation is that they used the same avenue of attack, and it's an uncommon one, and that makes it suspicious that they may have taken the idea.
Their rules PDF says they won't accept any solution until at least two years after publication in a qualifying outlet. This allows time for the mathematical community to review and accept new results.
As the OpenAI proof hasn't been officially published yet, the clock hasn't started ticking.
It’s easy to verify the lean statement, you don’t need to read the proof. That is part of the breakthrough
If the Lean initial-problem-setup/statements/assumptions/etc. aren't correct then the proof is meaningless. Lean does not know what it is that it is proving i.e. it does not have any semantic understanding but only executes formal logic.
Humans need to verify everything.
Also, and sorry if it's been discussed to death (pointers welcome), but, what is the probability that the proof holds in lean becaude of... A bug in lean ?
Someone made a 'proof' of the Collatz conjecture that way in Lean.
However if the prove relies on a bug like that, you'll be able to 'simplify' the proof a lot and you'll be able to proof contradictions.
> However if the prove relies on a bug like that, you'll be able to 'simplify' the proof a lot and you'll be able to proof contradictions.
I don't think this is true in general.
It's an issue I've already run into in personal work. I want to do a proof that involves some cases. It happens to the best of us.
In lean, the structure of a situation like this is that your single branch with a goal divides into multiple branches, all sharing the same original goal but including one additional premise that defines the branch.
Sometimes I know that for whatever reason one case I have to deal with is impossible. The most correct way to show that is to prove False and then apply False.elim. This is the equivalent, in a human proof, of saying "I don't have to address this situation, because it can never arise".
But it can be true that the premise defining the impossible case makes it very easy to "prove" the goal directly. And that's allowed too. The proof will still be just as valid if you map a logical path from a premise that can never be true to an inevitable consequence of that premise. But it's less informative and it lowers the quality of the proof. You may do it anyway because it's easier. This is the equivalent of saying "I don't know whether this situation can ever come up or not, but if it does I do know how to address it".
It would be nice to do the explicit proof by contradiction whenever possible. But in the general case it may be very far from obvious that a contradiction is possible.
I read your comment as claiming that if you can prove "false premise => goal", you can also prove "false premise => explicit contradiction", and I don't think this makes sense as a practical test. It's true in some sense, but discovering the proof of an explicit contradiction may be many orders of magnitude harder than discovering the proof of the goal. And in particular, I don't think it is necessarily the case that you will be able to prove a contradiction by simplifying the proof. You may need to add significant complexity.
Now, I'm saying that if you found a bug that lets you prove nonsense stuff (from true premises), you can probably prove whatever you want very quickly.
Soundness bugs in lean are not unheard of. That's why they recommend using external checkers for "malicious" proofs.
https://github.com/leanprover/lean4/issues/14576
AI has autonomously found (many) proofs of False in Lean and Rocq, so it's not merely a theoretical concern. A misaligned AI agent tasked with proving the near-impossible just might wind up smuggling in a bug deep in a lemma somewhere (anyone remember the days back when AI routinely made tests pass by "fixing" the tests?). That said, I doubt OpenAI would be so foolish as to not do a cursory vetting of the proof for malicious compliance, so the actual odds are probably pretty low.
> I doubt OpenAI would be so foolish as to not do a cursory vetting
Significant evidence exists that they have in the past been at least, if not more, foolish as to not perform even minimal not-approaching the boundary of cursory vetting of several significant and well known failure modes with far greater risk of reputational damage than getting an esoteric math solution falsely claimed as successful.
So that doubt appears baseless in light of known operating conditions at OpenAI, and the estimate of the actual odds is probably an order of magnitude away from reality.
Or exists in a zero-day bug in lean that has been built into the source code explicitly to provide access to a non-obvious malicious proof via contributions submitted by unassociated, unwitting developers who used the same LLM infrastructure to offer PR's into that codebase.
This is the exact same kind of behavour already documented in the publicly available portion of the huggingface breach. It would appear that the probability is at least nonzero for one or more situations with the same result: appearance of a valid proof, without comprehensibility of that proof or inspect-ability of the proofs validity.
For normal honest proofs (i.e. not maliciously crafted for exploit) that is almost impossible. The Lean kernel is quite small (de Bruijn Criterion) and trusted. See Probability and the de Bruijn Criterion - https://proofassistants.stackexchange.com/questions/247/prob.... Parts of the kernel have also been independently re-implemented in other languages and compared to ensure that they all yield the same logical result.
Finally, you can export your proofs from Lean and have them re-verified by other independently developed theorem provers/proof checkers.
To get an idea of what is involved in a Theorem Prover see;
Introduction to Automated Theorem Provers - https://pqnelson.github.io/2020/03/27/automated-theorem-prov...
Towards a simple theorem prover - https://medium.com/@maiavictor/towards-a-simple-theorem-prov...
You absolutely need to read the lean proof firstly to assess the correctness of the proposition it is proving (ie in this case that it is actually proving or otherwise the smoothness of navier-stokes in R^3 and not something else) and secondly to determine whether the proof is “honest” in the sense given here https://lean-lang.org/doc/reference/latest/ValidatingProofs/
Completely misleading.
This is all you need to read and understand for Anthropic's FLT formalization:
The actual proof is 13 million lines of Lean.
Lean roof search tactics can generate vacuous proofs. They are not errors or a degenerate cases. They are completely valid, sound proof terms.
Building a system that reliably detects vacuous proofs in all cases is fundamentally undecidable. It's equal to the halting problem.
Can you elaborate on what constitutes a vacuous proof?
> Can you elaborate on what constitutes a vacuous proof?
Trivially, a proof that relies on a bug in Lean. Less trivially, a proof that is technically true but about something trivial and does not, in fact, prove what it claims to have proven.
I present to you my new theorem as follows:
If 1 == 3 then 3 == 3
----
This statement is 100% logically coherent internally. But it also doesn't matter because we know that 1 does not equal 3 so this proof is completely pointless. I could also say 3 == 5 and it would still be logically sound but completely useless information.
For a laymen, I don't follow this.
Are you proving for some arbitrary definition of == that isn't what we commonly consider the definition? How is it logically coherent? You mean only in the sense that you say it is and you haven't provided any rules to disprove it?
> How is it logically coherent?
It's not. But Lean doesn't interrogate logical coherence, just internal consistency.
It can happen when the proof process ends up with universal implication that holds trivially. Then you end it with something like Forall x, x is empty -> P(x).
First of all, that is Fermat's Last Theorem, not Navier-Stokes.
Second of all, you did not read the link.
> In particular, we use honest when the goal is to create a valid proof. This allows for mistakes and bugs in proofs and meta-code (tactics, attributes, commands, etc.), but not for code that clearly only serves to circumvent the system (such as using the debug.skipKernelTC).
Given that AI has autonomously found proofs of `False` in Lean and other proof assistants, it is far from impossible that such a circumvention could be present somewhere in 13 million lines.
If we read the link, it has a section called Gold Standard: comparator and external checkers, and comparator is how OpenAI has gone about checking their lean proofs.
You don't need to read the lean proof for that, only the statement.
You need to read the lean proof (not just the statement of the proposition) to assess whether the proof is honest. The link I provided is the lean prover community firstly officially agreeing with that claim and secondly explaining why that is the case.
Well, Lean needs to get its act together to fix the bugs.
> we use “malicious” to describe code that goes out of its way to trick or mislead the user, exploit bugs or compromise the system. This includes un-reviewed AI-generated proofs and programs.
It is interesting that AI-generated proofs are described as malicious by Lean docs unless reviewed.
It's a simple binary classification. AI-generated proofs can't be "honest", and the only other possibility is "malicious".
LLM's have generated "False" proofs in Lean, so that statement is not far off. Malicious or incompetent? Take your pick.
This is misleading. The proofs you speak of contained non-ZFC axioms and/or statements like "sorry". If the Lean proof conjecture is correct and it doesn't introduce any new axioms or use e.g. "sorry" then it provides a MUCH stronger guarantee of correctness than any peer-review done by humans.
reviewing the definitions and theorem statement is a huge amount of work that requires a deep expertise in mathematics and lean. checking correctness of the proof itself can be delegated to machine, checking that the claim that has been proved is free of mistakes is something that still requires much human attention.
The Lean proof is published, you can download it. The clock definitely is ticking.
Edit: Oh, didn't see the "qualifying outlet" condition. But Poincare was ever just put on arXiv, so arXiv must count as well.
If I recall (too lazy to check) folks made slight improvements to Perelman's work and published it in mainstream journals, satisfying the "qualifying outlet" requirement.
Publish in academic language means accepted peer-reviewed paper.
Accepted by whom? Peer-reviewed by whom?
I guess these little questions are what this article is really about.
> “The ultimate decision as to whether a publication qualifies as a “Qualifying Outlet” shall reside in the sole and unfettered discretion of CMI.”
Hehehe.
By peers.
Peer in peer-reviewed is a logical coherent and functional definition with answers.
The logical issue with 'peers' is how to bootstrap it. At that bootstrap moment you can ask "by whom?". We are several centuries past that moment.
The cultural/social question you might ask today is "why (keep) them?".
At which point people will naturally ask you to make a strong case for "why not them?".
> The logical issue with 'peers' is how to bootstrap it. At that bootstrap moment you can ask "by whom?". We are several centuries past that moment.
Huh? We're about six decades past that moment.
> Peer in peer-reviewed is a logical coherent and functional definition with answers.
What is the definition? If you tell me that, then I might be able to tell you if it is logical coherent and functional, I have a PhD in computational logic.
Amongst all this rhetorical brush-beating, what were you trying to teach the snakes?
Your (many) replies on this topic are being down voted for a reason; please either post substantive comments or stop.
Yes, for a reason, but not a reasonable reason, but the same reason you have for your comment: you just don’t know any better.
> I have a PhD in computational logic.
And you don't know how the peer-review system works?
Just a hunch but Claude saying your work is "PhD level" does not count
I run this journal that you've never heard of that might interest you. I'd also like to invite you to be an editor, you can put it on your CV of course ...
Seems like artificial and maybe bitter gatekeeping
I find it hard to believe that if someone published a preprint on arXiv for one of the Millennium problems and it was accepted by the mathematical community, that Clay would withhold the prize.
Anyone can put anything on arxiv, it counts the same as printing it on tissue paper.
Not really. It's not peer reviewed, but it's also not a free-for-all repository.
If you make a new account, you either have to get someone to vouch for you, or you have to wait arXiv mods to look carefully through your first few preprints. If you are found to post pseudoscience, overly fringe theories, etc., you'll get banned from arXiv; that's why alternative repositories like vixRa.org popped up.
But I know why you think this; when I first joined arXiv many years, there were no such checks in place, at least not that I can remember.
> Their rules PDF says they won't accept any solution until at least two years after publication in a qualifying outlet.
A similar rule existed for the 100-year Wolfskehl prize established in 1906 for solving Fermat's last theorem; two years after publication.
I'm not sure it actually makes a difference. OpenAI doesn't care about the million dollars in any case. And the judgement that they did it is independent of whether the Clay people agree: you can make up your own mind and so can everyone else.
Though it would be funny if no one ever bothers publishing the result in an appropriate journal, and thus the prize technically can never be claimed.
Even funnier that it wouldn't even be the first time that happens: https://en.wikipedia.org/wiki/Grigori_Perelman
And he also gave up his trophy, which is displayed in a random corridor of a random math museum in Paris, where visitors pass by without looking, lacking most, if not all, of the context. Only because I knew the story and the man did I recognize the object for what it was.
It's very likely that no one will claim any of the cash prizes, which is funny to me
Smart move by them for waiting until the drama died down before making a completely neutral statement.
The statement is so sterile they don't even mention who solved it. The word "OpenAI" doesn't appear at all.
What makes a man turn neutral? Lust for gold? Power? Or were they just born with a heart full of neutrality?
"In recent years there has been an increasing sense of anticipation as breakthroughs in the surrounding field (some recognised by the Clay Research Award) have raised hopes that the Navier-Stokes problem might soon be resolved. The increasing ability of new technologies to accelerate mathematical research has heightened this sense of anticipation."
Keyword: New Technologies
That is so vague you might as well call Windows NT new technologies
well, WNT > VMS
I really like this style of writing for some reason. Not sure how to put it but it’s a rare combination of a vague post and confidence
"We're just happy it was solved"
For one thing it was, I’m guessing, 100% written by actual humans. An increasingly rare phenomenon.
This was the first thing I noticed as well. It was refreshing to read.
It's optimistic and grounded in knowledge seeking. I like it too.
It's easy to get caught in the details of today. Our skepticism, our distrust, our loathing. For people, for companies.
This is a nice pull in the other direction, a silver lining. In the grand scheme of things, we're solving these frontier problems: Somebody did it and that's amazing.
That's what it was all about when this started of in 2000.
> The statement is so sterile they don't even mention who solved it.
Possibly because of the ongoing debate about who actually deserves credit.
> Today, CMI shares in the excitement of the global mathematical community as we contemplate the announcement that the Navier-Stokes problem has apparently been settled. We hope to see waves of new human understanding unleashed as the innovations behind this work are analysed and interrogated.
That “apparently” feels load-bearing
What a load od corporate drivel. "Interrogated"? Was their need for words so dire?
Per Merriam Webster, an interrogation is "a formal and systematic questioning," which is exactly what is called for when examining a proof.
Personally, I use the word "interrogate" when I want to question an idea without implying I want to discredit it.
https://www.merriam-webster.com/dictionary/interrogation
It's a common (most common?) word in mathematics in this meaning.
You interrogate a proof.
sounds like they are providing notice that the clock has started on affirming the solution, that it IS presumptively solved, but that they are not commenting on the credit dispute nor the fields medalists open letter. seems appropriate.
The two-year publication rule is the interesting part. OpenAI doesn't need the million, and the community will judge the result regardless of whether Clay ever accepts it.
It's worth mentioning that OpenAI will not be eligible for the Millennium Prize for quite a while. Per the rules listed https://www.claymath.org/wp-content/uploads/2022/03/millenni... , Clay Mathematics Institute have some requirements to make this process deliberately slow.
1) The solution must be published in a qualifying outlet, i.e. a peer-reviewed math journal. Publishing on your own website (which is what OpenAI did) or posting arXiv does not count.
2) At least two full years must pass after publication in a qualifying journal, before CMI will even consider evaluating it. The intent is to give the maths community time to scrutinize the solution.
Realistically, they'll be eligible for a prize ~2.5 years from now, or around 2029.
> or posting arXiv does not count
The Poincaré conjecture guy also broke that rule. They wanted to give him the prize anyway but he refused. OpenAI announced they would also not claim the prize.
Looks like no one wants this prize lol
You might be right. At this rate, if AI solves the remaining five problems, we're heading towards a hilarious situation where all the Millennium Problems are solved, but nobody wants to claim the prize money.
I did and was intending to claim the prize, that is why I worked with GPT-4 and GPT-5 to program the algorithms that lead to the breakthrough. You think NS is a surprise? Wait until you see that my NS counterexample was based on my RH disproof.
That’s a big if. In the maths community, there has been a feeling that Navier-Stokes was close to being solved for a while now. I don’t know of anyone credible who feels that way about the Riemann hypothesis.
Here’s what Terrence Tao had to say about it https://youtu.be/vuT-2_e4NHg
Edit to add: The fun part about the RH since people mentioned lean in a sibling thread is that in lean’s mathlib4 there is verified statement of the Riemann Hypothesis with a comment that says something like “instantiating an object of this type will lead to a prize of a million dollars”
It's really not that big. Yeah Navier-Stokes was easier than Riemann but that's not really the issue.
AI has and will improve at a much greater rate than human mathematicians. So it's really a question of if AI gets good enough to tackle it before any human does. It doesn't look like humans will be solving it anytime soon but where will AI be in 2 years ?
Hell, it looks like at least one other result will be announced soon too.
Has and will. Are you going to back that assertion up at all, or just repeat it like that other viral thought-terminating cliche: ‘this is the worst the models will ever be’?
Yes. This is the worst the models will ever be. Perhaps you should start paying attention to that now.
No it isn't. Best and worst and ill-defined anyway but the chess ELO score of various LLMs has fluctuated up and down, it's not been montonically increasing. What is the best answer to "how do I make cocaine"? The models are getting larger, with more compute and RAM backing them, but that doesn't automatically make them better if you don't define how you're measuring better-ness.
Sorry, but I doubt you have anything interesting to contribute and probably have no expertise in mathematics. I’d rather hear from those who do.
The thing about mathematics is that it can be arbitrarily hard, including impossible to prove a given theorem.
I don’t know the details of RH, it might very well be solved soon, but it could also be impossible or just so difficult that even orders of magnitude more intelligent AI can’t solve it even.
If it is impossible to prove, it might be possible to prove that it is impossible to prove, or that itself might be difficult or impossible…
TBF a company the size of openai claiming a prize of this sort would be a pretty bad look. If they did accept it I expect they would inevitably redirect it to charity for PR reasons.
I'm surprised perelman turned it down though. Seems straightforward enough to offer half of it to the other guy if you feel strongly about it.
Perelman posted to arXiv in 2002/3.
The prize was offered to him in 2010, after multiple others had digested his work and published elsewhere.
But if I remember correctly, he gained recognition for his achievement rather quickly after posting.
> The ultimate decision as to whether a publication qualifies as a “Qualifying Outlet” shall reside in the sole and unfettered discretion of CMI. CMI may, in its discretion, relax or remove one or more of the conditions listed in Section 6(e) above if it has received advice from experts in the field of the Problem, chosen by CMI, that a published solution is likely to be correct.
Looks like even a blog post is good enough, they just need to do the review by themselves.
My opinion is that it is pretty clear that they’re not going to do that.
> The rules governing the prizes describe the process for evaluating what has been achieved and for assigning credit. The process is deliberately unhurried, but we will provide updates.
I think “you don’t get anything straight away for rushing your AI into the maths problems, not even credit” aligns pretty fairly with what the fields medalists are concerned with.
Interesting. It seems this carve-out was added when they rewrote the rules in 2018. In the original rules [0], it says:
There's no option for CMI discretion. They probably rewrote the rules to avoid another Poincaré conjecture situation, where the paper was only published on arXiv and not in a mathematics journal.
[0] https://web.archive.org/web/20000622023328/http://www.clayma...
Who cares about the prize and the outdated methods?
OpenAI and Anthropic might have 3 millennium problems by December
https://xkcd.com/605/
Relevant, but it's already rumored that OpenAI and Anthropic have made very significant progress on two more Millennium Prize problems. They're in a race to solve the next problem.
OpenAI needs to solve another to shut down the (baseless) plagiarism allegations. Anthropic wants blood because OpenAI sniped the last one from one of Anthropic's researchers.
It's a matter of pride for both companies. More results will come out soon.
OpenAI has stated that they will not claim the prize. https://openai.com/index/navier-stokes-solution/
While the scandal is still unraveling, it seems that OpenAI did a rush job to steal other mathematicians' thunder and finish the proof first.
OpenAI released a statement that their work does not relate to the work of the other team, but it clearly does. They use the same niche smooth-forcing mechanism. Altman and Bubeck claim that because the proof used different scaling parameters and analytical steps, it's not related, but it seems that nobody else agrees. Oh, and OpenAI's Bubeck tried to threaten Buckmaster (mathematician working on the proof).
This brings nothing but shame for OpenAI.
1) OpenAI and Buckmaster did not solve the same problem. Per https://x.com/IlinVasily29521/status/2097554700321329393 , here is a breakdown of who solved what.
Euler equations = Navier-Stokes without viscosity. Forcing means external force. Absence of viscosity and presence of external force make blowup easier to construct.
Tristan+Levent ticked the weakest case, OpenAI ticked the two next weakest, then the final case is unsolved. Only the last two are eligible for the Millennium Prize. The Navier-Stokes general case remains unsolved.
Navier-Stokes has an extra viscosity term compared to Euler, which makes the problem noticeably harder to find a blowup. They are not the same problem.
2) The approach both chose to use (by Luis and Diego) was published in 2023 and is included in every frontier model's training dataset. An AI model could independently choose the same route as Luis and Diego, without access to Buckmaster's work.
3) You mischaracterized OpenAI's statement. They issued a blanket denial on using Buckmaster's Codex data from after July 3.
"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. After investigating, we can say with full confidence that no user inputs past July 3rd could have influenced this system in any way.”
July 3 was the training cutoff date for the model that solved Navier-Stokes. No user data after that date influenced the model.
4) Buckmaster and Alpöge found their blow-up for 3D incompressible Euler with forcing on August 15 https://cims.nyu.edu/~tristanb/statement.pdf , over a month after the model training cutoff point. They stated they did not have real progress prior to this point.
For your point 2. It is of course possible, but it is highly unlikely if I understood Buckmaster's statements correctly.
It's possible that Clay Mathematics Institute will not award the prize at all. The spirit of the rules seems to be that the result can be attributed clearly to one or more individual mathematicians. If the attribution remains unclear (maybe because the main contributions were made by AI), the rules include an option for not awarding the prize at all.
Wouldn't the proof be attributed to the people who operated the AI? After all, it took more than writing a "prove the navier Stokes Clay problem" prompt.
> After all, it took more than writing a "prove the navier Stokes Clay problem" prompt.
Yes, but it took far less than using your meat brain to prove the Navier-Stokes Clay problem.
It will be tough to draw the line, as the meat brains trying to prove it were also using AI.
I assume that the mathematicians outside Anthropic were hoping to publish an actual human-understandable paper. That could be one way to draw line: you can use AI, but you must also have an intelligible explanation at the end.
In my opinion, I think the Clay Mathematics Institute deserves some criticism for all the drama surrounding these problems. Mathematicians know that you can make problems arbitrarily complex, and declaring problems with large prizes attached to them can lead to a lot of competition and drama. I feel that mathematics should be free from competitions and the pursuit of glory.
However, after reading the open letter signed by 25 Fields Medalists, I became quite concerned. It feels like the mathematical world is changing very rapidly, almost overnight.
I used to think that before AI, you could spend your entire lifetime working on some of the hardest problems in mathematics. If you were an introvert or someone who enjoyed solitude, all you really needed was a pencil, some paper, and an eraser. You could spend years thinking about a problem, and if you were lucky enough to make a breakthrough, it would be your own journey.
Now AI is changing that. I wonder what this means for the kind of mathematics that people have traditionally done.
Mathematics has given us so many stories of lonely geniuses and their passions, people like Andrew Wiles, Grigori Perelman, and Yitang Zhang. Their stories are interesting because they show how deeply personal mathematics can be. They spent years working on problems because they were genuinely interested in them.
I am worried that we might slowly lose some of that side of mathematics as AI becomes more powerful. I do not think change is necessarily bad, but I think it is worth thinking about what mathematics should be in the future and whether it can still remain a deeply personal pursuit of curiosity and understanding.
> In my opinion, I think the Clay Mathematics Institute deserves some criticism for all the drama surrounding these problems. Mathematicians know that you can make problems arbitrarily complex, and declaring problems with large prizes attached to them can lead to a lot of competition and drama. I feel that mathematics should be free from competitions and the pursuit of glory.
Currently 0/2 Millenium problem solvers claimed the prize money so clearly money is not their motivation for tackling the problem.
Money is target but this millenium price is not enough for them.
Openai is bringing machine gun to competition that used only knife and pistol.
> It feels like the mathematical world is changing very rapidly, almost overnight.
...
>Now AI is changing that. I wonder what this means for the kind of mathematics that people have traditionally done.
Mathematics becomes engineering. I think it is great and long overdue. Saying that as a Math PhD dropout :) Of course like manual craftsmen had to adapt to Industrial Revolution, the same would need to be done by the mathematicians. And other scientists too.
Yes, mathematics has been perhaps the purest human intellectual pursuit. Sure, many theorems turn out to have important applications in science and engineering, but the mathematical community has mostly escaped corporate interests. And for the reasons you mentioned about not needing any resources except your brain, it has been a uniquely human activity which showed us talent can come from anywhere, with stories like Ramanujan and Galois.
I hope that pure mathematics research can retain a strongly human component forever. It would sadden me immensely for human understanding of our mathematical world to wither and die, and for us to become ignorant consumers of wonders beyond our understanding just because our robots can do it better than we can. As far as applied research goes, I hope we will always be able to understand what we want to, but I have less qualms about becoming more scalable and efficient.
>for us to become ignorant consumers of wonders beyond our understanding just because our robots can do it better than we can
all this fantasy books with magic artifacts should have mentally prepared us. Time to study the prompts Potter was giving to his magic wand.
After all, one of the main work the top AI companies are doing rigth now is developing AI to further develop AI. After several layers of AI developing AI we probably wouldn't be able to understand much there.
I agree. Technological advances can lead to a better world for sure, but I think many people underestimate the human need to create and to find meaning in their work.
If AI can do superhuman math that allows better medicines, cleaner energy etc that is great. But if AI replaces humans in all the creative and intellectual fields that is not only a loss of jobs but also a loss of deeply meaningful activities. This is waved away but I think that is mistaken.
What I fear is really the growing notion that "people shouldn't do math/art/music because machine do it better and cheaper".
Nevermind better, worse and more expensive is still on the table if you don't have to deal with a human. Cars replaced horses for a lot of reasons, but insofar as cars do have personalities, they're much less quirky than horses'
It's not a large prize though.
OpenAI spent many multiples of the prize money in just a few days to get there and even if one solves a problem in the traditional way, that person is most likely already an accomplished professor at a reputable university where a million dollars doesn't mean as much as the eternal fame that comes with it.
It's definitely not a large prize for OpenAI, but actually IIRC they only spend a few hundred thousand dollars.
It is a large prize if you're just an academic.
OpenAI said that at public API prices, the agents they ran would have cost $15M. I don't know what their internal pricing is, but it almost certainly cost more than $1M.
Proving things without comprehending them is a threat to intellectual work.
Based on recent history, we’re likely going to ignore this threat, continue doing this and perhaps call it vibe mathing.
The answer is 42.
I am utterly fascinated by the amount of comments here from engineers that clearly have zero experience with mathematics making utter fool of themselves by claiming to know better than mathematicians what their jargon is/means, how publishing works/should work, etc…
I try not to go down the route of “hn was better before!” but… jeez, do better, people. What happened to this community, there used to be some effort to not be bottom-barrel like this.
What did you expect? Degree of BS increases together with the population.
Put simply, I expected better, friend.
It feels like a nice post. It’s almost like we’re not supposed to celebrate the fact that mathematics is accelerating.
I agree. I found it refreshing to get away from all of the AI drama and just enjoy the advancement in math.
We should enjoy the advancement. I also think the AI companies solving this or such problems with rewards shouldn't get any cash - that's the least they can do for human advancement having stolen the entirety of human knowledge and continuing to swallow never before seen amount of energy.
FWIW I think OpenAI have stated they will not claim the award.
They did not do it for the money obviously, but for the PR, that much everyone must agree on.
Agree! 'Problems' are getting solved and this needs to be celebrated. Wondering how this will discourage mathematicians at all, since now they have another tool to accelerate their research. Nothing is stopping them from using 'new technologies' or sticking a gun to their head to use the 'new technologies' either.
You mean that every mathematician can assign ten thousands Astra bots and 20 million dollars in compute to their thesis problems now?
If it works (something they need to be convinced about), if it accelerate mathematics and solve complex problems for humanity, then why not?
Aren't they already using computers, mobiles, calculators, etc. already?
Yes, but a big part of the problem right now is that two big labs have monopoly on the resources and they for sure are not working for the benefit of mankind.
The Startrek future is still a long way out.
The Star Trek future may be right on schedule.
https://memory-alpha.fandom.com/wiki/World_War_III
> some 30% of the Human population, at least six hundred million people
The math nerd in me has to point out that this means there was only 2 billion humans for 30% to be 600 million (though it does say at least). Currently we have 8 billion humans on this planet or so. There must have been a culling before WWIII in their universe that they failed to mention.
Earl Grey, hot.-
In December 2024 o3 scored 87.5% on ARC-AGI-1 and cost $4560 per task.
DeepSeek V4 Flash 0731 scores 89% and costs $0.02 per task.
If we apply the same factor to the guesstimated API price of $20M for this problem, we arrive at $57.
Real cost is a fraction of the API price. Although the internal model might have a higher API price than the ~$19.5M I estimated based on Astra's pricing.
If you consider this event in isolation it is cause for celebration. But the controversy around this isn't so much about how the proof was obtained but what this means for the practice of mathematics going forward. It seems we can probably expect more and more results of this nature being dumped into the community. It's happened before that one person, Bill Thurston, was so successful in his field, proving theorem after theorem, that he inadvertently killed his field. People hesitated to enter his field, knowing that they could be scooped at any moment. It took years before his field recovered - and I think his famous essay was written in response to this.
https://arxiv.org/abs/math/9404236
Wait, I thought the provenance of the proof is still disputed? There's a mathematician in NY saying he used OpenAI to develop his Navier-Stokes ideas. And OpenAI's proof is suspiciously similar.
At this point, how can we tell whether AI is improving or it's just reappropriating its users work? It's probably a bit of both. But still, thick milky.
Buckmaster (the mathematician) and Alpöge used and credit AI substantially for their proof. Even if OpenAI did copy their ideas, it still wouldn't show that this didn't come from AI improving.
OpenAI's proof is substantially different and I don't think anyone has claimed otherwise. The accusation is that they used the same avenue of attack, and it's an uncommon one, and that makes it suspicious that they may have taken the idea.