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 chances are they never will publish it in any kind of useful format. Right now, the scientific community is outraged at OpenAI for going about their announcement in the least productive fashion they could have. It really does seem like they have no interest in progressing our understanding of maths outside of mining it for marketing material.

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.

> 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.

It’s easy to verify the lean statement, you don’t need to read the proof. That is part of the breakthrough

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

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

> The statement is so sterile they don't even mention who solved it.

Possibly because of the ongoing debate about who actually deserves credit.

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Unless CMI is playing a double game, hard to tell with academics, this is indeed a giant f-you to OpenAI.

> load-bearing

Apart from LLMs, who speaks this way?

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What a load od corporate drivel. "Interrogated"? Was their need for words so dire?

Per the rules in their link, the clock starts after publication in a qualifying peer-reviewed journal so, no, I don't think that's the reason.

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.

Based on recent history, we’re likely going to ignore this threat, continue doing this and perhaps call it vibe mathing.

>"You just let the machines get on with the adding up," warned Majikthise, "and we'll take care of the eternal verities thank you very much. You want to check your legal position you do mate. Under law the Quest for Ultimate Truth is quite clearly the inalienable prerogative of your working thinkers. Any bloody machine goes and actually finds it and we're straight out of a job aren't we?

No-one cares anymore, OpenAI does not care about the prize either or whatever the Clay institute has to say for that matter. It's about attention, capital and compute. If you can use the result to increase shareholder or company value, that is what matters.

Proving pharmaceutical drugs work on mental health issues has tremendous value even though we don't always (or often) understand the exact mechanisms in brain chemistry as to why it works

Outside of mathematics, I think it’s very rare for a single human to understand 100% of any complex undertaking, and this was true long before AI.

>without comprehending

Very on brand with using AI for everything.

Why is it a threat and not an opportunity? Imagine if in ancient times there was their Oracle that could produce mathematical proofs of any question you asked it, would you burn it or try to understand how it works and use it to ask questions you are stuck on?

No, it is not. Proving things without comprehending them built much of human civilization. Comprehension is a relatively new fad that came along in the 17th century with the scientific method.

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.

> 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.

I feel like this is answered in the very second paragraph?

> 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.

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.

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".

> 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.

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].

[1] https://arxiv.org/pdf/math/0008177

> 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.

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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.

> 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

> 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.

Who cares about the prize and the outdated methods?

OpenAI and Anthropic might have 3 millennium problems by December

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.

What did you expect? Degree of BS increases together with the population.

Gell-Mann Amnesia HN version?

My understanding is that it’s proof by counterexample, so the main thing to study would be the implications of the counterexample. The technique used to find it sounds like a lot of brute force. But I’m not a mathematician, and maybe the AI used some clever techniques to narrow in on it.

Clay will award the prize (or choose to not award it) when there is an expert consensus that the problem has been solved and the solution is correct. The rules merely state what that would mean in some typical cases. There is always an option that Clay changes the rules to match the reality better.

The result was never the point.

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.

They said they started working on millennium problems after hearing a rumor someone had found a solution to one of them. That's just bad taste. It indicates negative motivation rather than positive motivation from the start.

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.

I agree. I found it refreshing to get away from all of the AI drama and just enjoy the advancement in math.

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.