>Mathematics is probably the same way. There's a short-term rush when you pull the lever, but there's less desire to get invested in what comes out.
Perhaps it will take until the next generation to come along to really embrace the new AI-assisted way of doing mathematics. The current generation has too many reservations.
If you consider this event in isolation it is cause for celebration. But the controversy around this isn't so much about how the proof was obtained but what this means for the practice of mathematics going forward. It seems we can probably expect more and more results of this nature being dumped into the community. It's happened before that one person, Bill Thurston, was so successful in his field, proving theorem after theorem, that he inadvertently killed his field. People hesitated to enter his field, knowing that they could be scooped at any moment. It took years before his field recovered - and I think his famous essay was written in response to this.
It would be good if someone made a list of allresults obtained with AI so far just to see what kinds of problems AI excels at. Are there any that aren't of the existence-proof type?
Reductively, math can be said to be either problem solving or theory building - it seems the latter is a much harder thing to do right now.
This is also what I've been thinking. The result itself is amazing but it's not like this was completely unexpected. There has been a huge amount of progress on the problem in the last 10 years without which it seems unlikely today's full resolution would have been possible. It is not clear what strategy was taken but it sounds like it borrowed heavily from the two spanish mathematicians. Experts will scrutinize the proof and it will be interesting to see if anything truly original or unexpected was done, outside of known techniques, a move 37.
That doesn't really matter. Currently we're in the "this is a dog" (marks a muffin) stage of "AI as a scientist". It's totally reasonable to expect that considering how rapidly AI is advancing, the frontier of knowledge and research won't be universities anymore, but rather AI companies running their models in a loop.
Imagine that 20 years from now nobody really does science by hand because AI is just better at it, everyone just runs models, but these models require so much memory that only datacenters can realistically handle them, and it just so happens that the public gets access to nerfed models, while privately, companies actually break all asymmetrical encryption ciphers.
I find the framing a little strange, a sort of David vs Goliath (with his enormous computational resources at his disposal). Since Levent is at Anthropic whose internal models are presumably as capable as anything OpenAI has. So why wasn't Anthropic behind their effort? Why did Tristan use OpenAI's models when it should have been known was a potential outcome? I understand they wanted a normal math collaboration but presumably what Levent brought was his resources (as far as I can see Navier-Stokes is not his speciality). Normally these things are hashed out formally beforehand to avoid the sort of thing now happening.
They were working on it for almost a year, and Buckmaster has evidently been interested in Navier-Stokes for a while. This seems to be more of an innocent collaboration between two researchers than a strong company PR effort. Maybe Anthropic should have stepped in and made a large team to help them finish the proof (and maybe they tried and didn't succeed, who knows).
If what he wrote is accurate, it does suggest that OAI is effectively extremely hostile to cutting edge researchers (eg, if we hear rumors about your partial success on a problem that has huge PR benefits, then we'll assemble a strike team of researchers with unlimited compute to claim the win for ourselves, possibly by training on your data). It's also not a good look for them to request author removals based on company affiliations.
I think what you have in mind is more appropriate for more normal corporate projects and the like. But academic collaborations are not usually so political/'profit' driven, if that makes sense.
Presumably because this was something Levent did in his spare time and because it was not obvious that this work would eventually lead to a breakthrough.
> Why did Tristan use OpenAI's models when it should have been known was a potential outcome?
I'm sure in the past he had less cynical feelings about OpenAI and their penchant for academic fraud.
> I understand they wanted a normal math collaboration but presumably what Levent brought was his resources (as far as I can see Navier-Stokes is not his speciality)
I think you're not giving the guy enough credit in saying that his contribution came down to having an API key for Anthropic models.
> Normally these things are hashed out formally beforehand to avoid the sort of thing now happening.
How would that have helped? That agreement (which may well still exist) would not have involved OpenAI.
So what do you think his contribution was? His preprint record shows no research on fluids - and the statement says that the first LLM-generated proof Tristan received from Levent was 'the most horrendous I have ever read.' Levent is out for mathematical scalps whether it is in his field of expertise or not, and he has the resources to do it. And I am not saying he is not a very clever person, but the idea that you can bring yourself up to the forefront of research in PDEs, in particular NS, and contribute new ideas in less than a year is implausible.
They have messed things up, because Levent has a conflict of interest between his job at Anthropic and this independent work, and Tristan should have opted out of OpenAI training on their work (he probably didn't know about this). This doesn't justify OpenAI's despicable attempt to steal their work.
One of the cool facts about gravity is that the gravitational field of a body 'feels' spherical at a distance, even if those bodies are not spherical themselves. There are higher multipoles but they are increasingly suppressed the further you are from the body. So at least for gravity the spherical approximation turns out to be good and justified.
Beyond a "cool fact", this is the only reason that physics works, where I'm defining physics as the human study of physical laws, not the laws themselves. If we couldn't neglect higher moments systems from planetary motion to the large scale structure of the universe would be computationally intractable.
And it's not just gravity, it's also electromagnetism: charge arrangements (i.e. ions) look like point charges at a distance, meaning you can neglect the electron orbital structure. Light sources act like point sources at a distance too.
A lot of the equations you learn in introductory physics rely on the higher moments vanishing.
- The planetary motion equations (Kepler's laws) assume gravity as a point source (the three body problem can be approximated by replacing two of the bodies with 1)
- E = mgh -> assumes a homogeneous gravitational field, which is a special case of gravitational monopole near an equipotential surfaced. And of course parabolic motion and pendula only work because this holds.
This point of view is essentially codified in effective field theory, which tells us why so much of classical physics (EM, fluids, gravity) can be seen as low order approximations of some more complete theory that we found later on. One of the great insights from 20th century physics imo.
And if they were intractable, motions would be so irregular that pheneomena such as billion-year stable orbits would not be possible... leading to a lack of evolution and life generally. So it is a very good and necessary thing that cows remain mathematically spherical.
Scott Aaronson spoke about this in a colloquium where he said that this was mooted at OpenAI before the decision was made by Altman to not implement it for the reasons you describe.
Perhaps it will take until the next generation to come along to really embrace the new AI-assisted way of doing mathematics. The current generation has too many reservations.
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