Navier-Stokes Announcement
claymath.org
claymath.org
As the OpenAI proof hasn't been officially published yet, the clock hasn't started ticking.
Humans need to verify everything.
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.
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.
I dunno, the HuggingFace incident was not too long ago. They might not be foolish, but their model could be smarter than them to "cheat", so to speak
You mean the company that “accidentally” let their model perform a cyber-attack?
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.
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...
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)
Building a system that reliably detects vacuous proofs in all cases is fundamentally undecidable. It's equal to the halting problem.
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.
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.
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's not. But Lean doesn't interrogate logical coherence, just internal consistency.
E.g. “If it’s raining, the sidewalk is wet.” That statement holds if it’s not raining or the sidewalk is wet.
This is a common occurrence in mathematics, where someone might not be able to unconditionally prove Y, but they can under the condition X. Later, another mathematician might build on this by proving X, thereby transitively proving Y. (Or conversely, they might unconditionally disprove Y, thereby disproving X.)
Many hard problems are answered this way.
For example, Fermat’s Last Theorem was proven assuming the Taniyama-Shimura-Weil Conjecture, then Wiles proved the conjecture.
Thousands of theorems rely on the the unproven Reinmann Hypothesis, which is why it’s so interesting to mathematicians.
But if your precondition is “stupid,” your proof is stupid.
Can only be false if there is an instance where A is true, and B is false. In all other cases it's true, even when A is always false.
That's the key.
I highly recommend the following two books to study Logic from the beginning (for a layman);
Logic: An Introduction to Elementary Logic by Wilfrid Hodges.
Introduction to Logic: and to the Methodology of Deductive Sciences by Alfred Tarski.
In logic, a proposition is some statement that can be true or false. So, let A be the proposition that 1 equals 3, and B be the proposition that 3 equals 3.
Now the poster is making a third proposition. If A, then B.
A is clearly not true. So in classical logic, B can be anything and “If A then B” is still true.
For example let B be the proposition that I am Elvis Presley (I’m not). So now we have “If one equals 3 then I am Elvis Presley”. This is clearly true. I’m not Elvis Presley, but that doesn’t matter because we’re not saying anything about what happens when one doesn’t equal 3.
Now, let’s try let B be the proposition that I am Sean Hunter (I actually am). So now we have “If one equals 3 then I am Sean Hunter”. This is clearly still true because we still are only making a claim about what happens when one equals three.
https://en.wikipedia.org/wiki/Vacuous_truth
By the way, this isn’t any kind of inherent contradiction or problem, it is just a possibly counterintuitive part of how classical logic works.
You see this type of statement (“If <x>, then <something ridiculous>”) being made a lot when people are exaggerating for effect, for example by Mr Bumble in “Oliver Twist”
> 'That is no excuse,' replied Mr. Brownlow. 'You were present on the occasion of the destruction of these trinkets, and indeed are the more guilty of the two, in the eye of the law; for the law supposes that your wife acts under your direction.' … 'If the law supposes that,' said Mr. Bumble, squeezing his hat emphatically in both hands, 'the law is a ass--a idiot. If that's the eye of the law, the law is a bachelor’
It was too easy, so I double checked my definitions, but it is quite easy to do something like that. And Claude does things like that quite frequently.
I am going through the exercise right now of trying to get Claude to formalize a published paper and it is a _struggle_ to get it not to take shortcuts or prove approximations of the paper’s theorems and then tell you it’s done.
Semantic versioning describes the following idealized setup:
- you have an interface you expose (a contract, and thus a contract signature)
- you do not change the contract signature -> patch version bump
- you do change it but in a non-breaking way (e.g. additively) -> minor version bump
- you do change it but in a breaking way (e.g. mutatively or destructively) -> major version bump
One would expect then that since interface signatures are statically derivable, semantic version tags can be auto-assigned. And indeed, in lots of shops that's exactly what happens (in my opinion, correctly).
The problem with this is that it comes with a lot more smoke than fire. The interface having no changes or non-breaking changes doesn't mean the actual code behind those interfaces is not going to cause a breakage. It literally is just about the interface itself.
And so unless you encode absolutely everything about the semantics your implementation actually observes into the interface, which is what the semver specification asks you to do so as their sleight of hand, this means the interface will be a leaky abstraction. Which means that external software interfacing with yours may observe behavior that is beyond the purview of semantic versioning. Which means that they do. Which means that they absolutely can and will break, and your package managers' fancy version constraint syntax exists to make such fun events happen.
The way this is usually handled then is:
- you live with the pain: acknowledge the limitations of semver, accept you've been duped, and just give in
- you have human release managers assign versions manually, based on whole program and whole system semantics (with the human overhead and error that entails), falsely claiming that what you're doing is still semver
- you switch to a less deceptive versioning scheme, like calendar versioning; as a bonus, you now no longer have to pretend that your entire application somehow only has a single unified interface
This mirrors the Lean statement and Lean proof situation. The statement is like an interface, and the proof is like the implementation behind that interface. The way the proof is derived may expose semantic gaps in the statement itself, and (ab)use them to obtain the logical consistency certificate. Hence, a vacuous proof, and hence why this is not statically assertable to be not the case. It is part of the challenge in asserting that the statement was correctly formalized in the first place: you need to manually identify whether the way the consistency was achieved is actually meaningful, or just a formalization gap.
Which really makes me wonder about the actual value proposition of Lean then, but alas...
> This mirrors the Lean statement and Lean proof situation. The statement is like an interface, and the proof is like the implementation behind that interface.
> Which really makes me wonder about the actual value proposition of Lean then, but alas...
[1] there are sub-tools such as comparator that are designed for this type of use case. https://github.com/leanprover/comparator
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.
Do you believe no open questions remain as to the truth of the Collatz conjecture?
At this point in time, we really can't be confident in accepting proof certificates without any human eyes on the script that generated it. I still have 95%+ confidence in this particular result being trustworthy, but a precedent of blind faith is guaranteed to end badly.
Not only that, but there is a very fuzzable tell of something funny in the Collatz proof script (`CommandElabM`, i.e. metaprogramming). We may not at all be so lucky in other malicious scripts, especially if there are still kernel-level bugs in Lean.
That is, verifying a proof in English was a painstaking, years long process in the past as independent mathematicians looked for holes in the steps connecting the logic. When the proof is written in Lean, all of that work goes away. My point is that if OpenAI publishes the Lean code (not sure if they already did), verification should take weeks not years.
It is interesting that AI-generated proofs are described as malicious by Lean docs unless reviewed.
Edit: Oh, didn't see the "qualifying outlet" condition. But Poincare was ever just put on arXiv, so arXiv must count as well.
I guess these little questions are what this article is really about.
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?".
Huh? We're about six decades past that moment.
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.
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
peer(X, 0) :- founding_peer(X).
electorate(T, count<Y>) :- peer(Y, T).
support(X, T, count<Y>) :- candidate(X), peer(Y, T), recognizes(Y, X, T+1).
peer(X, T+1) :- support(X, T, Votes), electorate(T, Total), 2 * Votes > Total.
But no, peer reviewed and published in a reputable journal is a fairly normal standard.
That being said, I think there needs to be some standard, and peer review seems like the best we have come up with. But is the current status quo for scientific publication the best we can do? I think that is an open question and we should be able to openly discuss alternatives.
and the whole "publishing papers" thing gets called out as a racket even on HN now and then
with many videos against it by former "academia" people on YouTube.
The sooner AI brings down such archaic customs into a gibbering pile of protesting rubble, the better innit?
But perceived truth and falsehood depend on convincing people - either convincing them one by one, or else convincing some gatekeeper, whose word will convince those who accept the gatekeeper.
You know in the past women (and other marginalized people) weren't allowed to publish scientific papers for the majority of human history?
Is this how you want your "truth" to be?
If so then man, no sympathy; these institutions deserve to be broken down. Let them spaz on the floor and throw tantrums all they want.
... spaz on the floor ...
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.
Strangely enough, the crackpots seems to prefer vixra.org to publish their work. I've never seen something like "4D wormholes can cure cancer" in ArXiv
Without limiting any other provision in this Section, a publication lacking any of the following characteristics will be deemed not to be a Qualifying Outlet:
i. an editorial board whose members are named and available for contact;
ii. an editor or editorial board member whose professional knowledge of the global mathematics community would enable him or her to identify an appropriate referee to review a submitted paper;
iii. a published refereeing process that, in the opinion of CMI, ensures that a submitted paper is reviewed and verified by appropriate experts in the field of the Problem; or
iv. inclusion in the list of publications maintained by MathSciNet.
The solution to the Poincaré conjecture was only accepted after an exposition of Perelman's proof was published in a refereed journal. His papers didn't qualify, but of course he got the credit for the result.
A similar rule existed for the 100-year Wolfskehl prize established in 1906 for solving Fermat's last theorem; two years after publication.
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.
None of them did, though, despite how many tried. So empirically, it seems unreasonable to say human researchers can solve ‘problems like this’.
OpenAI the opposite.
No I cannot, and I'd argue most people can't either. We rely on mathematicians, peer review, and letting the scientific process run its course.
My understanding is that the theorem statement is quite simple, so i guess the latter is not very likely, but the former is very much a possibility in a proof this large, and it will take some human eyeballs to go over it before convincing mathematicians.
They do a Comparator Challenge to validate that they actually solved the correct theorem from the result, which they copied from Google/DeepMind: https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b... - this is valid for both Euler and NS. Also, they validated with an external kernel from the Lean Kernel Arena. That way bugs in the Lean kernels were found in the past already, iirc.
Having said that, I strongly believe a positive result with the challenge above is the reason why they published it. I highly doubt anybody at OpenAI (or anywhere else) fully gets the proof after such a short time since publishing. This is also what Terry Tao criticized the most in my opinion.
Independent of the remaining drama [0], from my point of view, the proof is correct and an achievement.
[0]: https://news.ycombinator.com/item?id=49661928 - I am pretty much on the critical side, however, one can not ignore that it is an achievement. Esp. the unforced result.
If that's the current burden of proof required in your world for maths then that's fine! 't'ain't in my world: I want to see peer reviewed and published. Surely that's not too much to ask. Its not perfect but generally works rather well for maths.
I'm not a sodding programmer so please don't assume everyone here is one. I'm not a mathematician either but I do have standards: Your counter argument is a poorly constructed and inappropriately deployed example of "proof by whataboutism".
Not in the world mathematicians have been living in for the past decades at least. Nearly all big theorems that have been formalized so far had been published beforehand, and it was usually regarded as a step up in rigor. Wrong results get published in peer reviewed journals all the time.
OK but this member of the general public has at least subscribed to New Scientist since 1987, nine O levels, two A levels, two AS levels and a HND in Civ Eng. All pretty mediocre but I have a fair idea on how sciencing is supposed to work and how it ... actually works. Obviously, I ended up in IT.
I should also point out that maths "peer reviewed" is a bit special. For example Mr Wiles went through quite a maelstrom before his proof of some dodgy marginalia was accepted as "true".
While it's a convenient to assume that mathematics deals with logical statements, any attempt to evaluate those statements relies on physical processes with both known and unknown failure modes. There cannot be a test that establishes it unambiguously whether a claim is true or false. In all nontrivial situations, mathematical truth is based on expert consensus. When a new claim is made, people will try to raise and resolve objections, until a consensus emerges one way or another.
As for C++, all compilers are different. For any given compiler, there are valid C++ programs the compiler fails to compile and invalid programs it compiles without any errors or warnings. And now that I think of it, a new version of a compiler crashing with valid code earlier versions used to handle is the only class of compiler bugs I see with any regularity.
Just because Lean can compile it, does not mean it is safely proven. It is the start of a process to check whether something actually holds, not the end.
Many people, see prior conversation on HN, have already decided that AI solved it. The standards of reasoning and rigor in academia are complex enough that we all argue over them and harumph as we epistemically trespass on each other's domains.
The public, really humans if care for Herbert Simon, are much more apt to evaluate knowledge emotionally and by other standards. We may see them as wrong but standards only matter in context. The NYT, HN, and Annals of Mathematics will always have different standards of truth.
The more familiar analogy was when I look at the code that Claude spews for my partner. They take it at face value and hope it works. I usually find it very problematic, but only because I knew what to look for.
Granted for the sake of argument. But for this process, there's no requirement anyone from Clay needs to be involved, nor that anything has to be published in a traditional journal.
Terrence Tao can download the pdf, think it over, and publish on his blog, and it would be just as valid. Then some other mathematician can write something on 4chan or whatever. (See https://en.wikipedia.org/wiki/Superpermutation#Lower_bounds,... for 4chan.)
I'd be furious.
I don't care much for AI myself, or smart phones either, for that matter. I would be content if NS remained a mystery for another 100 years - or forever. But goodness, does the "scientific community" need to take a deep breath and count down from 10.
My point is: why shouldn't they want a marketing win from this. What obligation does a non-academic institution have to follow the traditions of academia? Its result doesn't belong to academia. And if academia wants to subject OpenAI to their own internal processes and give them marching orders, it just isn't going to work and maybe - who knows - it'll even further erode their own legitimacy. Does anyone actually believe that NS would have been resolved in the 2020's if we lived in a parallel world where LLM's were never invented? Would Buckmaster have gotten as far as he did without LLM's doing a lot of the work for him? We can complain about AI companies contributing to mathematics, but are we complaining about Terence Tao using AI in his research? When Tao publishes something are we all going to go to war against him because maybe other mathematicians' prompts went into training the AI that Tao used?
The statement is so sterile they don't even mention who solved it. The word "OpenAI" doesn't appear at all.
Keyword: New Technologies
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.
Possibly because of the ongoing debate about who actually deserves credit.
That “apparently” feels load-bearing
Personally, I use the word "interrogate" when I want to question an idea without implying I want to discredit it.
You interrogate a proof.
Apart from LLMs, who speaks this way?
https://www.claymath.org/wp-content/uploads/2022/03/millenni...
I'm sure they've been getting a lot of press inquiries and don't want anyone to misinterpret their silence as refusing to engage with OpenAI's solution.
Now they have a statement on the record that can be quoted with a gentle reminder of the qualifying criteria (which makes clear they have nothing to evaluate either way yet).
Also to emphasize the original optimistic ideals of the prizes and their role as neutral arbiters who aren't going to litigate anything outside the scope of the prize itself.
Very on brand with using AI for everything.
Humans have always tried to make sense (comprehend) the world around us. The methods have become more rigorous, but the idea of understanding less in order to advance civilization is a truly weird idea.
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.
Currently 0/2 Millenium problem solvers claimed the prize money so clearly money is not their motivation for tackling the problem.
Openai is bringing machine gun to competition that used only knife and pistol.
...
>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.
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.
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.
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".
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 is a large prize if you're just an academic.
Yes, you can make problems arbitrarily complex. But the prize problems were chosen not just because the solutions appear likely to be very complex (the problem statements aren't necessarily inherently complex--there is a way to restate the Riemann hypothesis that a junior high school student could easily understand, which I'll give below).
They were chosen because they were important problems that mathematicians really wanted solved, top people had worked on them for a long time and progress stalled a long time ago, and it seemed likely that solving them would require major breakthroughs.
Those kind of problems can be discouraging. Enough people who are probably better than you have spent enough time failing to solve them that realistically most researchers are going to focus all their efforts on something they are likely to make progress on.
A nice prize can get more people to at least work on them as side projects.
Here's that restatement of the Riemann hypothesis I mentioned.
The Riemann hypothesis is that the non-trivial zeros of the function ζ(s) occur on the line 1/2 + yi.
ζ(s) is 1/1^s + 1/2^2 + 1/3^s + ... when s is a complex number whose real part is greater than 1, and defined everywhere else except s = 1 by a process called analytic continuation. The trivial zeros are at s = -2, -4, -6, ... .
For a mathematician, or a non-mathematician who has taken complex analysis and hasn't forgotten much of that, that is not too complex a definition. For anyone else the first reaction is probably "Trivial zeros? How the heck does that thing even have zeros? And if it does how the heck can it have zeros at any negative integers! It is obviously infinity at every negative integer!!!".
Here's a different hypothesis that turns out to be exactly equivalent to the Riemann hypothesis. They are either both true of both false, so resolving one of them resolves the other.
Let H(n) = 1 + 1/2 + ... + 1/n for all positive integers n. These are called the harmonic numbers.
Let S(n) = the sum of the positive integer factors of n for all positive integers n. For example S(4) = 1 + 2 + 4, S(6) = 1 + 2 + 3 + 6, and S(17) = 1 + 17.
Hypothesis: S(n) <= H(n) + exp(H(n)) log(H(n)) with equality only when n = 1.
The proof that this is equivalent to the Riemann hypothesis is here [1].
If building a machine that solves these kinds of problems isn't a "major breakthrough," I don't know what is. Is the objection merely that it came from engineers rather than mathematicians? If so, there's plenty of room for contributions from many fields.
The best thing a mathematician can do to advance their art, at this point, is to drop whatever they're doing and work on AI.
> to elevate in the consciousness of the general public the fact that in mathematics the frontier is open, close at hand, and abounds with important unsolved problems; to emphasize the abiding value of working towards a solution of the deepest, most difficult problems; and to recognize achievements in mathematics of historic magnitude.
Also
> These are not arbitrary puzzles akin to fiendish crosswords. Rather, they are fundamental challenges that mark the frontier of human knowledge and challenge us to develop new structures and methods. They provide foci for the continuing struggle, across generations and cultures, to deepen our human understanding of mathematics and the universe that it describes.
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.
Looks like no one wants this prize lol
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”
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.
Better obviously means better, like how they became better than they were 6 months and a year ago.
e.g. If someone said "this is the worst they'll ever be" in response to some writing with obvious LLM cliches in 2024, I'm not convinced that prediction was actually correct.
The focus of OpenAI/Anthropic pivoted aggressively to the agentic performance arms race instead of making a more human sounding chatbot so regressions in writing ability aren't really a concern anymore if agentic benchmarks improve.
The first time I heard a recommendation to use Claude was specifically because it sounded much more "human" and natural than ChatGPT. Fast forward to now and idiosyncratic Claude-isms repeated every other sentence and its convoluted verbosity has become a widely mocked meme.
>e.g. If someone said "this is the worst they'll ever be" in response to some writing with obvious LLM cliches in 2024, I'm not convinced that prediction was actually correct.
2024 creative writing prose was...the last few versions have stalled, but I think they're still better than 2024.
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…
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 prize was offered to him in 2010, after multiple others had digested his work and published elsewhere.
Looks like even a blog post is good enough, they just need to do the review by themselves.
> 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.
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.
[0] https://web.archive.org/web/20000622023328/http://www.clayma...
OpenAI and Anthropic might have 3 millennium problems by December
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.
Presumably the people that OpenAI and Anthropic are trying to impress with these trophy kills are potential IPO investors, but I would have thought investors would also be concerned about the growing public backlash against AI.
Imagine if they had handled this differently and these results - still using OpenAI models - were coming from the math community. How much better the PR would have been - AI helping math/science rather than yet another story of AI harming society in some way.
No doubt this is what they were at least partially aiming for - not just shooting a trophy animal to brag about, but also being seen to advance math/science, a la AlphaFold, not just take all our jobs and enshittify society with deep fakes and AI slop. But, they heavily misjudged.
Unlike AlphaFold, this technology has the potential to wholesale replace the entire profession. You're never going to get anything more than bad PR from that group as the threat looms.
Over 3 Billion images gets generated per week via OpenAI chatgpt image models. None of the poor PR from artists on AI generated images even remotely matters.
IMO giving a hypothetical AlphaMath to mathematicians, the same way Google gave AlphaFold to research chemists/biologists, would have resulted in far better PR, and the profit opportunity of attempting to replace the jobs of either research group is minimal.
It's downright bizarre the way companies like Anthropic (primarily), and to a lesser extent OpenAI and anyone else, are themselves pushing the narrative of this tech may kill you, will take all your jobs, etc. That may all happen, unfortunately, but being aware of that possibility you'd think these companies would be a bit more mindful of their messaging and behavior, not just go out with a scorched earth approach of "well, they are going to hate us anyway".
If universities come to only need research mathematicians for teaching ability, then it would still gut the profession. You would presumably need far fewer of them for their research ability, and hopefully hire more people who can actually teach. It's strange that you don't see that as a threat to the profession. Lots of research mathematicians would lose their jobs even in this scenario.
>IMO giving a hypothetical AlphaMath to mathematicians, the same way Google gave AlphaFold to research chemists/biologists, would have resulted in far better PR
GPT-N isn't AlphaFold. I'm telling you AlphaFold is a bad analogy! AlphaFold has superhuman capabilities at one specialized subproblem - protein-structure prediction - embedded in a much larger biology/drug discovery pipeline. It doesn't replace the biologist or drug researcher, it just gives them a better instrument, and if you're not in the relevant professions it's essentially useless to you.
GPT-N is general purpose intelligence machine that can increasingly do chunks of work you would otherwise employ the mathematician or software developer to do.
Shanmu Jin is a perfect demonstration. A neurosurgery resident, not a research mathematician, who encountered the Crouzeix conjecture through his transcranial ultrasound research and used GPT-5.6 Sol to solve it (20+ year old longstanding problem).
In the AlphaFold story, the biologist gets a powerful new tool. In the Jin story, someone who isn't a mathematician can obtain research level mathematics and incorporate it into their research/code/business whatever without talking to a single human.
You're asking why they don't market GPT as "AlphaFold for Mathematicians". They don't because it's not.
>It's downright bizarre the way companies like Anthropic (primarily), and to a lesser extent OpenAI and anyone else, are themselves pushing the narrative of this tech may kill you, will take all your jobs, etc.
It's not that bizarre. It only seems strange if you think it's all a facade, "marketing" or whatever nonsense HN is convinced of. These are people who believe very strongly in what they are doing and in the potential of it. For these people, what you are asking them to do is actually incredibly slimy. And it might win some brownie points, but not for long.
I suppose you are suggesting that research mathematicians are either over-qualified mathematically and/or under-qualified in ability to teach, but it seems pretty clear that universities prefer to hire domain experts whose reputations and long publication lists attract students and raise the perceived academic standards of the school.
Your "scenario" of much math research soon being done by AI (supervised and paid for by who, one might wonder), while universities hire people chosen for their teaching skills not academic reputation, seems a bit of a stretch ...
> You're asking why they don't market GPT as "AlphaFold for Mathematicians". They don't because it's not.
No, I am not asking that, nor asking anything for that matter.
I was pointing out that giving a free research tool to mathematicians would likely be better received, and receive better press, than doing something that most top-tier mathematicians are opposed to.
My hypothetical "AlphaMath" certainly could just be free GPT-N access for academics/researchers (as they are also doing to some extent), or it could indeed be a custom system that OpenAI built as a gift to the math community.
As far as the AI companies acting in slimy fashion goes, the most slimy behavior of all is to strongly believe you are doing something that will kill people and cause massive societal disruption ... and still keep doing it.
I'm saying there's little correlation in how brilliant a researcher you are and your teaching abilities. Yes universities optimize for the former. That's not necessarily a good thing for students even if it increases the prestige for the university. If there's some future where research and teaching are decoupled, I don't expect things to remain the same, but who knows I guess.
>I was pointing out that giving a free research tool to mathematicians would likely be better received, and receive better press, than doing something that most top-tier mathematicians are opposed to.
At least currently, it's not possible for this to be a free tool. Both OpenAI and Anthropic have discounts for Universities/Education for such use cases as far as I'm aware.
>As far as the AI companies acting in slimy fashion goes, the most slimy behavior of all is to strongly believe you are doing something that will kill people and cause massive societal disruption ... and still keep doing it.
This is pretty bad, but I wouldn't call it slimy. And it's no longer up to any one person or company anymore.
I actually don't think that current AI can replace any profession that requires human interaction over many weeks. This is because imo they lack long term planning abilities
For example, how do you reconcile return-to-work mandates with the idea that companies are going to be happy with faceless remote workers? What does the boss do when the shit hits the fan and he would have yelled at people about the need to work all weekend, but instead all he has to yell at is an LLM that tells him he's "right to push back", that it promises not to delete the production database next time (except it will, because it can't learn), and that it could care less about being fired because it's just a calculator?
Imagine that paper-clip maximizing happening over millions of tasks.
I think we are just entering the LLM + Human era.
Got to admire the gung-ho approach I suppose ?
Okay...you understand that are training for this and it has gotten much much better at doing this over the years ? You should probably also understand that it doesn't need to be able to do this to cull the profession ?
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.
Tristan + Levent: 3D incompressible Euler with forcing
OpenAI: 3D incompressible Euler without forcing
OpenAI: Navier-Stokes with forcing
No one: Navier-Stokes without forcing
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.
“There does not seem to be anything in principle preventing the methods from extending all the way to Navier-Stokes, and there is even a non-negligible chance that the forcing term could be eliminated entirely, although there are an enormous number of technical difficulties that would ensue in implementing that program. At this point, I would not be surprised if one could batter out such an extension by pouring an enormous amount of compute and AI assistance at such a task…”
Pouring infinite AI resources into it is exactly what OpenAI did.
It is laughable to claim they are not datamining users when datamining users is their strongest advantage over open models.
In the end, there is no reason to care about openAI or credit them with anything. Tools are not attributed, people are.
The best solution is for universities to be universities and provide llms for students and staff to use. Any university allowing students to use cloud based AI has failed.
From the statement:
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.
Bushmaster then redacts 90% of what Bubeck says next (because it would detract from the narrative he's trying to portray) and only includes the most inflammatory line.
The reply was, “If you don’t want me to be nice, then I don’t have to be nice.”
When you fill in the context, it's clear: 1) there is no personal threat like you claim and 2) this is hardly a threat, on a scale of threats you can make, this scores a 1.5 out of 10.
For context: Bushmaster is pissed at OpenAI because they sniped a problem he was toiling on for the past year, so during this conversation I believe emotions are quite heated.
I don't claim this interpretation is 100% accurate, but if you disagree, do reply with a better one yourself.
So nice of them to give an ultimatum where the paper will be published in an academically dishonest way no matter what you choose.
Is Altman even mathematican these days?
Yes, but it took far less than using your meat brain to prove the Navier-Stokes Clay problem.
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.
OpenAI’s Navier-Stokes release included a Lean 4 formal proof - https://news.ycombinator.com/item?id=49650326 - Sept 2026 (179 comments)
More questions about whether researchers can trust OpenAI with unpublished math - https://news.ycombinator.com/item?id=49639408 - Sept 2026 (813 comments)
The Navier–Stokes Millennium Prize Problem - https://news.ycombinator.com/item?id=49621697 - Sept 2026 (237 comments)
Tao: Open math problems being non-renewably mined by AI - https://news.ycombinator.com/item?id=49616968 - Sept 2026 (420 comments)
On the Navier–Stokes Millennium Prize Problem - https://news.ycombinator.com/item?id=49613262 - Sept 2026 (1134 comments)
Navier-Stokes – Tristan Buckmaster [pdf] - https://news.ycombinator.com/item?id=49605915 - Sept 2026 (827 comments)
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.
Clearly the real-world cannot "blow-up" - real-world water vortices do not reach infinite velocity, etc.
The point of having Navier-Stokes as a Millennium prize was to hopefully generate new mathematics and techniques along the way, and auto-generating a sprawling AI-slop proof or millions of lines of Lean does not accomplish that result.
Clearly OpenAI has no interest in the math itself - to them this was just a trophy animal to shoot and stuff. I would be very surprised if they now helped analyze the proof and try to extract the mathematical value out of it, and this would obviously require outside help who are probably not inclined to help OpenAI math-wash their behavior.
Truly, if this is the biggest criticism left, they should be celebrated. While in reality, all of this has a bitter aftertaste.
So weird.
That does stretch the reader's credulity. I think most people find it far more likely the rumor was related to Anthropic all along.
- "yeah we know"
- "looking into it"
- "will ping you"
- "might take a while though"
+ preemptive linguistic cushioning in case they feel socially (politically) compelled enough to forbid clanker proofs in their solution acceptance criteria, or in case they decide against conceding to such pressuring
They did not do it for the money obviously, but for the PR, that much everyone must agree on.
Doing it for the PR is still just doing it for the money, only they have a much bigger pile in mind.
Aren't they already using computers, mobiles, calculators, etc. already?
The Startrek future is still a long way out.
https://memory-alpha.fandom.com/wiki/World_War_III
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.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.
⎿ You've hit your session limit · resets 2:52am (123°24′W Etc/GMT+8)
/upgrade to increase your usage limit.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.
I'm a professional mathematician and these tools are not available for me. So not happy about that!
(just to be clear, I'm not anti AI, I'm just criticising the shallowness of your comment)
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.
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.
This has a few practical implications: First of all, if you are in the target group of the marketing material, be wary. While these things can do non-trivial stuff, the amount of magic is being grossly over-stated. But also, when several of the big results have indeed been reappropriating the work of others; when the companies fail to provide proper attribution (the NS case in particular is laughable) and present the results as the models' own work, that's plagiarism.