On the Navier–Stokes Millennium Prize Problem

Mathematics has always been highly competitive.

Not sure I agree with this. AI generated proofs can still be analyzed and mined for useful insights. I suppose he's saying the process of banging our heads against the wall on a problem can itself yield useful insight? But what is stopping us from analyzing a proof after the fact. And if we can generate many different versions of a proof that should help us develop a much deeper understanding of the problem than we would have without being able to perceive the "proof landscape"...

What is going to happen is a complete revaluation of things like "finding a counter example to a famous problem". Even if someone finds a solution to a problem like this with pencil and paper, nobody will believe it, and they will assume that there was an AI involved.

Further, sitting and doing math with a pencil and paper will no longer be a reasonable strategy to build a reputation or career, beyond the benefit a mathematician gains to their own intuition and skill. People who work hard to build intuition and also use AI effectively will dominate the field.

In a world where everyone is using AI, the open problems that remain will be the ones that are AI resistant. This is no different that how things work now, mathematicians wait until they are fairly confident someone won't rapidly solve their problem before they start talking about it. They will do the same thing in the future, except in the future AI will be part of the toolset they use decide if they are ready to share yet or not.

Edit: Ok I believe I was generally right here, but I just read the details of what OpenAI did. They didn't solve a longstanding problem, they got tipped off to an approach a mathematician was using and would likely result in the solution very soon and they finished it first. If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.

I work at OpenAI, though not on the team that did this, and my understanding is:

- we decided to ask our model for Millenium problem solutions because of two reasons: (a) our new model was looking incredibly good and (b) we heard rumors that some Millenium problems had been solved and were curious if our models could solve them (the goal here was not to scoop any particular individuals and we were looking at many problems beyond these)

- we did not read any private chats (but of course the model was aware of prior research literature published to the internet)

- the proof generated by our model was very different from theirs and also goes far beyond the published literature

- we made an effort to jointly announce rather than immediately scoop (I understand Tristan was unhappy with the conversations; I know zero details here and I hope more is shared today)

Edit: Here's is Sebastian's take: https://x.com/SebastienBubeck/status/2097379411691516310?s=2...

>our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced)

[1] https://openai.com/index/research-acceleration-view-inside-o... [2] https://openai.com/index/an-alien-mind/

Is this buried under the drama or are the major OpenAI twitter accounts from the people involved in the drama desperately attempting to make this the story after everything else obviously got away from them?

Loops and parallel connections make transformer go brrr

This is sort of what OpenAI was supposed to be. I'll never understand how it was legal for them to turn it into a for profit corporation.

So I'm greatly excited what AI will bring about in physics, more so than in math, because in physics it's clear that our fundamental theories are missing a big piece of the picture, and given how easily AI crunches through Millenium prize problems I think it's possible that AI will come up with a viable grand unified theory uniting quantum mechanics and gravitation, or produce new predictions in other areas. There's enough contradictory or unexplained observational data available to make a ton of progress on the theory side I think. Exciting times ahead!

Math is like this too. The big problems they've been solving have been identified as interesting only through lots of prior effort.

"If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics."

BTW, there's also a problem of asking interesting questions that AIs aren't yet good at.

No one has found any principled walls of AI development yet. And empirical results are quite telling. So, I guess, those problems will not stand for long.

The natural sciences will soon start breaking too.

I will concede that AI seems likely to not invent a "research program" anytime soon.

It has no taste

This.

I do dislike the AI oligarchs as much as the next person, but I do find the thread full of complaining a bit depressing still.

If the result holds (and it looks it does), this may be one of the, if not the, biggest things to happen in computing to date. A lot bigger than e.g. Deep Blue beating Kasparov in chess or AlphaGo beating Sedol in Go.

- First off, to reiterate, WOW.

- Second of all, when does this end? Are we at the dawn of the singularity now?

- People are saying OpenAI "stole" this from the work of an OpenAI user. If so, that's pretty fucked - how can we trust them?

- Time to think about retiring from any knowledge work or business? This could be winner-take-all where a leading lab can button press any economic function, business process, or scientific discovery. 24 months of lead on Open Source might turn into virtual centuries of lead.

- Do "normies" even know what's happening?

Anybody who thinks the improvements stop here isn't paying attention. It hasn't been showing any signs of slowing down since 2018. And the curve isn't even linear! My god, next year is going to be insane.

If you need privacy, then you are going to have to pay full price for those tokens (API). This has been true since day one. Everyone knows it, I guess though this is the first time that it has become "real".

All that was just kicked in the teeth by a group with a lot of compute that was like “bro I heard on twitter that Navier stokes could be solved. Let’s try it.” That’s an existential level of engagement that almost no mathematician in history would like.

> 2) I never ever asked for Levent to be removed from authorship of his own work (as indicated by my text). I was surprised to learn during the call with Tristan that they had only solved Euler and not Navier-Stokes; after learning this we brainstormed possible paths forward. One option we discussed was that Tristan could be the lead author on a rewrite of OpenAI’s Navier-Stokes proof. It is in that context that I said “it would be simpler if Levent was not an Anthropic employee” because I felt it would be inappropriate for an Anthropic employee to author OpenAI’s work. Importantly it was admitted that internal Anthropic models had been used in their proof of Euler blowup; I therefore felt I could not consider Levent to be an independent academic. Another option I wanted to propose (but got cut short) is to offer access to our internal model so that they could try to finish their proof and bridge the gap between Euler and NS. Again I did not know how to navigate giving access to internal OpenAI IP to an Anthropic employee.

> 3) To reiterate it plainly: as my text clearly indicates, and as I said during our call, OpenAI's intention was to do everything possible to celebrate their mathematical achievements and the heroic efforts that they made on Euler. In the call I was immediately met with a litany of slander, including direct threats that if we were to announce Navier-Stokes he would immediately go to the press with a barrage of unfounded accusations. I refuted all these accusations but he replied “there is nothing you can do, I simply do not trust you”. I was confused why one would turn an incredible source for celebration (of their achievements!) into such bickering, which is when I said that I did not understand why one would risk their career [over unfounded accusations]. Genuinely, at that moment, I was trying to care for him and do a last ditch attempt to get a chance to give them all the credits that they deserve. I deeply apologize for this extremely poor choice of words, it is the opposite of what I was trying to convey. (I should say that I retracted them on the spot by the way.)

https://xcancel.com/SebastienBubeck/status/20973794116915163...

> While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .

2. The dark forest is fun for scifi stories, but is mathematically bunk anyway https://www.noahpinion.blog/p/the-dark-forest-hypothesis-is-... https://www.reddit.com/r/IsaacArthur/comments/1l06cnk/cool_w... https://www.projectnash.com/aliens-the-fermi-paradox-and-the...

When doomposting please actually say something substantive. Negative news always gets clicks/updoots; fight that human tendency.

Not to mention, it's still very much up in the air whether the model derived the answer of its own accord or sniped the important details from the researchers it was spying on.

To be fair, I think it's still an open question about how far it might surpass human capabilities.

I think it's clear that its speed of development will be significantly faster, but it's technically not proven that the frontier and problems don't themselves become increasingly difficult faster than any acceleration in intelligence past the point of human training, data and existing knowledge.

Should this be the case, we would see a rapid broadening of development, and a slow advance in the frontier in such a way that might surpass the collective capabilities of people, but not by very far.

Sure. A proof without an unknown amount of human steering (and/or stolen research) would be an unquestionable achievement.

To this day there's zero (0) evidence of any result by an LLM alone (maybe I'm wrong). If I just prompt ChatGPT right now with "give me a proof of the Riemann Hypothesis" and this thing delivers, I'm sold. But anything close to "yeah ChatGPT proved X with 5 years of 24/7 work with 10x Terrence Tao level geniuses" it really doesn't cut it.

Or why's there's no new branch of mathematics invented by AI? That'd be indubitably _novel_ and _creative_. But to my knowledge (and I'm eager to be educated) there's nothing like that. What are the HARD examples of novelty, creativity and genius you claim? For how people like you talk about AI I'd expect idk, a unified theory on fundamental physics, or a novel engineering solution for material science and nuclear fusion, or at least improve itself to not need a bazillion GPUs to emulate a 20 watts wetware. Sure it would infinitely easier to make OpenAI literally print money with any of the thousand problems easier to solve with such amazing intelligence than the NSE problem right? Honest question

Just because something is legal and permitted by terms of service doesn't mean it's morally right.

Better than the walled gardens of most journals where you can't even read half the papers without shelling over thousands of $$$

Highly persistent agents + vibe-coded security seems like a problem.

This is no different than scooping them.

https://mastodon.social/@tristanbuckmaster/11723647135247030...

He was also using LLMs to do it, so either way most of the credit goes to the LLM here.

Maybe that happened. What we know for sure is that this is definitely how ChatGPT works to the point where the possibility of this happening exists at all.

Don't get distracted by what may have happened, focus on the facts that we know, ChatGPT trains on user conversations, if you use ChatGPT to create something of value, you are not using the one true ring.

>our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced)

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

Threatening a research mathematician and dangling and $1M payday to dissociate from his research collaborators and to adopt OpenAI's narrative is bad stuff.

> "I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer."

OpenAI (i.e. this OP):

> "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models ."

One of the interesting threads here that is certainly relevant to the OpenAI writeup is the human role in the process. Buckmaster clearly points out that (exceptional!) mathematicians at OpenAI were certainly involved in correcting and guiding the process - and that their path/strategy was no doubt influenced by Alpoge & Buckmaster's work. It is always in OpenAI's interest to de-emphasize the role of people in the process, as is clearly the case here. Indeed, given sufficient compute and resources, I suspect Buckmaster could have also extended their approach to N-S.

> “You are creating your cool streaming platform in your bedroom. Nobody is stopping you, but if you succeed, if you get the signal out, if you are being noticed, the large platform with loads of cash can incorporate your specific innovations simply by throwing compute and capital at the problem. They can generate a variation of your innovation every few days, eventually they will be able to absorb your uniqueness. It’s just cash, and they have more of it than you. So the safest bet again is to stay silent, or at least under the radar. Best bet is to not disrupt - succeed at all … ?”

https://ryelang.org/blog/posts/cognitive-dark-forest/

https://news.ycombinator.com/item?id=47566442

    Our effort began on September 1st after hearing a rumor which we later realized was related to Levent Alpöge, an Anthropic employee, and Tristan Buckmaster, a math professor at NYU. After the completion of our full project and Lean verification (on September 6th), believing from the rumor they also had a solution of Navier–Stokes, we reached out to them to offer a concurrent release of our result and to recognize their priority in a joint announcement. At that point we found out that they had a resolution of the forced Euler problem. In these discussions we offered them visibility into all of the prompts we used and later to see the proof. We recognize the priority of their work on forced Euler and congratulate them on their remarkable mathematical achievement.

The drama comes from where OpenAI got the idea to use that route to tackle NS, since the authors maintain that no one could have plucked it out of thin air like the OpenAI research claim to have done.

[1] - "...I was shown a prompt and told the internal research model had simply been given the problem statement. Levent had been told by Sebastien “very little human input” had been used. This turned out not to be true. Over the course of the call, as members of their team sent Sebastien corrections and details over their internal chat, it emerged that an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler, that even the prompt that had been shown to me had been written by prompting Codex, and that an insane amount of compute had been used. I asked when the first prompt had been sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI.

I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer.

Two proposals were offered to me. The first was that we post our Euler result, and that OpenAI post its Navier-Stokes result the next day. The second was that, after posting Euler, I alone write a paper presenting the Navier-Stokes result, acknowledging that an internal OpenAI model had resolved it. Sebastien twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic. It was also said that if OpenAI posted after us, they would say that we deserved the Clay Prize, and that we were the “closest humans to the problem”. I declined both offers.

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

Reading Sam Altman's and Sebastien's tweets reads like something written by people who know they did dirty shit and are willing to cross any lines to "win". https://x.com/sama/status/2097385167002415140

OpenAI's leadership just can't help to continue to disappoint everyone with their lack of ethics or integrity.

This. What is worth a human life's attention? As little as a month ago, mathematics was valuable in part because only a small number of people could possibly make progress on the frontier. We are confronting an existential moment for a 4000+ year-old human cultural endeavor. The assumption that "mathematical thinking is hard" has been built-in at a number of important points in how we support mathematics and mathematicians.

We need a different model, and fast. Already, the research community is feeling unable to digest proofs fast enough to keep up with the output of AI models. The paper is 165 pages, and the discovery was finalized two days ago. What this means is that nobody really understands it. Nobody would accept OpenAI's proof in this amount of time, except that they formalized it in lean. The formalization alone would normally be another years-long (or career-long!) effort if the world was the way it was one year ago.

So, again, what efforts are worth a life's attention today? It's a harrowing change.

https://mastodon.social/@tristanbuckmaster/11723647135247030...

Which seems to be very directly accusing OpenAI of plagiarism

https://x.com/SebastienBubeck/status/2097379411691516310

https://x.com/sama/status/2097385167002415140

I tend to believe OpenAI on this. Their stated desires seem rational, and Buckmaster's account makes them sound like cartoon villains. It sounds like there may have been some things lost in translation along with some bruised egos. Seems like the most plausible explanation for what Buckmaster is claiming.

woah, this gives some credit to the rumor that openai finetuned a model over the course of a few days for this task, and maybe trained on the Chatgpt/codex history of the authors, including drafts of this research.

Glad to see this is the top comment. There's also https://news.ycombinator.com/item?id=49605915 which links directly. Corporate talking points where they try to set the narrative are going to get the big press and most discussion elsewhere, which is gross. But inevitably the press will muddle the priority question, and even if they didn't.. as usual OpenAI will even benefit from the accusation of bad behavior. Sigh.

Edit:

OpenAI have now admitted they were training on prompts at the time they made their breakthrough:

https://mastodon.social/@tristanbuckmaster/11723647135247030...

https://news.ycombinator.com/item?id=38433655

> Let's talk when we've got LLMs proving the Riemann Hypothesis (or any mathematical hypothesis) without any proofs in the training data. I'm confident in my belief that an LLM can't do that, and will never be able to. LLMs can barely solve elementary school math problems reliably.

https://news.ycombinator.com/item?id=42331654

> An LLM is like a well read college student with a nearly photographic memory that sometimes mixes things up. It's great for bouncing ideas off of and getting feedback on them. And yeah, it might product "novel ideas" by mixing and matching existing ideas, but LLMs will never create truly novel ideas. Not in their current form.

The paper didn't really answer the question sadly: their conclusion was just that humans rate LLM answers as more novel than human ones, but less feasible.

https://news.ycombinator.com/item?id=41522605

> Solving Millennium problems is a whole different ballgame. It's not known if these problems are solvable within ZFC axioms. (In one case, the Yang-Mills prize, stating the problem mathematically is part of the challenge.) All of the obvious applications of known tricks have been tried and failed. To solve such problems, one probably has to invent new and surprising mathematical definitions, building a framework in which the problem becomes solvable. This is something that LLMs will be crap at; the process of invention is not represented in any training data we have access to.

https://news.ycombinator.com/item?id=38435909

> LLMs cannot reason or use mathematics - in a way, they don't know what they are talking about. Why would such technology lead to superhuman smarts?

https://news.ycombinator.com/item?id=35752293

> But still, the questions in that test are "solved" in the sense of "I can take a dictionary and answers these questions with full certainty". Beyond established knowledge LLMs are monkeys with typewriters, at best.

> I agree but I have tried many times to intersect two ideas with a LLM that would be novel and the LLM can not do this at all. We shouldn't expect the stochastic parrot to be able to do this though and it is unfair to the stochastic parrot.

> It is like expecting a real parrot to say words it has never heard before.

> No one asks that of a real parrot because we don't anthropomorphize a real parrot like we do the LLM

https://news.ycombinator.com/item?id=41525962

There's allegations right now that the model essentially read the work of a human mathematician using AI to work on the problem and OpenAI is presenting his work as that of their model

How would they have gotten that mathematician's progress though? Did that guy also use OpenAI?

If that's the case, it only strenghtens their claims lol. If mathematician decide to use OpenAI's model to do the work, that only reiterates how strong their models are.

It should be clear to everyone reading this now that those generous compute quotes with the flat rate plans aren't charity.

I have a young daughter and my goal now is to provide a very broad and varied upbringing, exposing her to as many different perspectives and experiences that will lay the foundation of a broader ability to understand and adapt as the world changes ever faster. You no longer need to be an expert in anything, you need the ability to perform within the landscape that the present opportunities exist.

It feels like the "tide is rising" where the minimum level of skill applied to every aspect of everything will inexorably rise to "whatever an LLM can do", which is already pushing past PhD level.

In math, the question being asked is the validity of a logical statement. That is, there is some rigorous, logical statement which may or may not be true (or even provable, etc.), and the question is whether or not it is actually true or false (or even provable, etc.). Having a proof, fundamentally, means you have a logical statement which only assumes the axioms of the system you're working with and which shows that the statement you're trying to prove is deduced through that statement.

Basically, they already have the "answer" in the sense that the statement they want to prove/disprove/etc. is already known. What everyone doesn't/didn't have is the argument which starts from axioms and leads to that statement which is logically valid. A Lean proof IS this argument. Since it is just logic, it can be checked computationally.

For example, if I assert "2 is an even number," then I haven't proven that 2 is actually an even number yet, but I know that a valid proof of my assertion will end with the statement "2 is an even number". So the question I'd be trying to answer is "what is the line of logic, starting with axioms, which leads to the statement '2 is an even number'"? If I have that line of logic (as a Lean proof), then I can check that it is logically consistent, and if it turns out to be valid, then I can now assert that "2 is an even number" knowing that there is a proof of that statement.

This problem is no different. There is a logical statement corresponding to "Navier–Stokes Millennium Prize Problem" that everyone knows, but which nobody had been able to provide a proof (or counterexample, etc.) for until now.

If you think it'll slow down, you can do some of the same stuff you're doing now for lower pay while supervising an AI, maybe?

I talked about this at some length here, including a diagnosis of the structural issue we're facing as well as a path forward: https://news.ycombinator.com/item?id=49461333

But yeah, Terry Tao considered this exact situation in advance and is on record that this exact outcome (rushing to priority before an explanation) would be the worst possible result. https://mathstodon.xyz/@tao/117207849921390904

We will have to see whether any other millennium problems fall. I guess that in a year the scope of AI math will be much clearer, for now it's still a bunch of incidents of unclear pattern.

There are other examples though. For example, NP hardness of n^{1/400}-approx CVP. Like any NP hardness proof, this shows you can faithfully encode a hard problem (3SAT here iirc) in terms of another candidate hard problem. Not really a counterexample at all.

Turns out actually living some terrible catastrophe is only fun in the movies.

(This is the alignment problem of course)

This is the lynchpin behind everything, and I would describe it as "likely". Since I am not employed by any party to this dispute, my 1 opinion is more trustworthy than OpenAI blog poster's 1 opinion.

> Oh god.

Yes, a very reasonable reaction.

How far we have come :.)

the value to the researcher might not be all that big, but the value to the economy at large is gigantic

[1] https://en.wikipedia.org/wiki/Kolmogorov_microscales

[2] https://en.wikipedia.org/wiki/Knudsen_number

Wouldn't be surprising if they did. The prize money isn't worth the almost certainly negative PR.

OpenAI doesn't need a million dollars.