> How many human-years were invested in solving the same problem, before they were overtaken in the last few days by an AI? 90? Not even: that's just the time since Jeal Leray's statement of the problem in 1934. 26, if you want to count the time since 2000 when the Clay Institute named it as one of its Millennium Prize problems.

I hope this isn't actually news to you, but: There is more than one human. There is even more than one mathematician.

If there happen to have been as many as four humans working on Navier-Stokes at any given time since the year 2000, then that's more human-years applied to the problem than agent-years.

> How many mathematicians have worked on the problem? 10k? Not likely.

You don't get to count the factor of 10k once when working out how many agent-years OpenAI gave to the problem and again when demanding that for parity there would need to have been 10k mathematicians on it.

> And all that's without even considering whether the AI based its proof on carelessly shared work by the humans. Or rather, yes, let's consider that: it totally did.

Let's suppose that indeed what Buckmaster and Alpöge had done was in the model's training data. Well, it didn't enable Buckmaster and Alpöge to solve the problem for Navier-Stokes (they could only do Euler), and it did enable OpenAI's model to do that.

Also: we don't actually know that what they'd done was in the training data; the latest bits of what they'd done that could plausibly have been in the training data were from before when Buckmaster said they progressed from preliminaries ("We worked through the literature and upgraded various preliminary results") to actually making substantial progress on the problem ("This was until about a month ago, when we had real progress"); and from what Buckmaster wrote it sure seems like a lot of the Buckmaster/Alpöge progress was in fact done by LLMs. (E.g., Buckmaster says that he and Alpöge have been working frantically to try to understand the proof for their Euler solution. That sounds to me much more like "an LLM did this thing" than "we figured out all the hard bits and the LLM did nothing more than filling in a few details".)

Buckmaster's own account of things is that all the really clever ideas were those of Córdoba and Martínez-Zoroa. (Which are already out there in the open literature, and there is nothing remotely improper about making use of them.) And my understanding (but, note, I am not an expert on fluid dynamics or PDEs and I could be wrong) is that actually the OpenAI model's construction is quite different from that of C&MZ. On what basis are you confident that "the AI based its proof on" what B&A did?

(For the avoidance of doubt: I am not arguing that what OpenAI did was OK. Even if they actually didn't train at all on any of the Buckmaster/Alpöge chats, it's very much not good professional ethics to hear that someone else is working on something and rush to try to scoop them, and there is absolutely no question that they did that. The question here is how impressed we should be by the model's mathematical prowess.)

> A success rate of 17%?

A success rate of 17% on problems of this difficulty and significance is something that for any human being would be a career-defining triumph.

> We have thousands of problems solved by humans in the last 2k years and how many by AI? A couple dozen? Oooh scary!

That would be a more convincing argument if the AIs, like the humans, had been around and trying to solve those problems for the last 2k years. However, as you might have noticed, the state of the art in AI was rather primitive 2000 years ago.