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Navier-Stokes Millenium Problem Solved by GPT
OpenAI says they solved the Navier-Stokes Millenium Prize problem with an internal model that is more powerful than GPT6, so a GPT 7 candidate or something close. OpenAI has been on fire lately really casting a shadow on Anthropic. First the hugging face incident, now the first to crack the Millenium prize problems.
As for the solution to the problem, they are claiming that smoothness is disproved and that the equations do break down.
They say that the equation can develop a singularity, which apparently means it is not a perfect model.
This is very exciting to me as I believe it to be the first scientific result of an AI model. Previous results have been basically irrelevant to the sciences. However, this result is of course still completely symbolic in nature, which is not surprising because LLMs are not embodied enough to collect data and analyze it autonomously.
While they are generating amazing PR recently, I will say GPT-6 is somewhat disappointing for coding. It is not the same leap that 5.5 to 5.6 sol was. It would appear that the models are getting better at running very long context chains while efficiency improvements and refinements in lower level tasks are lacking. Still, it's great work from OpenAI and it's plausible to me that if there's no singularity in general by 2030, there will be for mathematicians. They will no longer be meaningful for producing math, rather they will only be humans who understand it.
Assuming this is all true (and it seems to be, Buckmaster’s claims notwithstanding) then I unfortunately don’t see how mathematics survives this.
Chess survived, people care more about the drama between players than the literal games themselves.
Commercial art might shrink but art itself will survive in some form, just because so many people will continue to do it for fun and at least some people will pay extra for human art.
But who cares about a human-authored proof of RH when a machine proof is faster? I don’t know what the argument would be for continued institutional support of mathematics at anywhere near its current scale.
I'm in a minority here, but to me it's mostly been a confidence trick / running on fumes for many decades. As in, what the mathematicians do mostly does not matter, not because they aren't smart, but because their corners of math literally don't matter. People still heap tons of status on those people, I don't see how AI would change this because to me it's already not about results. However, I suppose now it democratizes the ability to produce math papers, so the status of the field will fall to more normal levels, which will feel really bad from the inside.
Yes and no, to varying degrees. Certainly there is still mathematics being produced even today that leads directly to new scientific discoveries and technological advancements; but it’s a vanishingly small percentage of all mathematics that has ever been produced (or at least, a vanishingly small percentage of everything that’s been produced post-1800 or so, when math started to become increasingly self-reflective and divorced from physics and engineering). I’ve seen more than one number theorist brag about what a great racket cryptography has been for number theory; you just have to make vague promises to the funding committee that your work will have “applications to cryptography”, and then you can go work on the problems that actually interest you. Much like chess, modern mathematics is essentially an arcane sort of sport that only few can play and few can appreciate.
To summarize my own position as briefly as possible, I’ve always been fine with allocating resources (tax dollars) to this small priestly caste so that they can continue with their game, because of its intrinsic worth and beauty, and I’d mostly be fine with continuing to do that even in an AI future. But very very few other people would. Even I have to admit that the existence of superhuman machine generated proofs kind of takes the wind out of the sails of the whole enterprise, in a way that it doesn’t for chess or art. Mathematics is creative, yes, but it’s also very deeply impersonal; probably, the methods that get us to RH first are going to look the same no matter what entity actually devises those methods, because that’s just how the contours of the mathematical landscape are intrinsically shaped. There’s very limited wiggle room to put your personal stamp of creativity on things; you’re really just offering yourself as a conduit for whatever the gods of mathematics decide to transmit through you. Because of that, it seems particularly fruitless to struggle for a proof of an open problem when you know that a perfectly good machine generated proof exists or could exist.
It's funny number theory is not needed for cryptography, potentially not even good for it. AES does not use it, assymmetric uses it the most but non-number theory asymmetric cryptography has existed since the 70s and might have better properties than number theory cryptography. So, the fake and ugly field (sorry, so many say it is pretty but I don't like it, it's very boring and dry and pointless) actually hurt the practical application it was used for.
I would keep funding them as smart people but I would force them to change their focus to physics and similar science. In other words I would revoke their funding for mathematics but would fund them for applied fields. We are probably missing a decade or two of science because fake math traps these people.
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