>However, there were 34 noteworthy deviations from the published record that needed to be resolved, the most substantial of which related to the Unbalanced Group Theorem.
Doesn't this make it seem like Wang had the wrong intuition, if her meyhod does not generalize? She proved Kakeya only in dimension 3, and also not prove the Hausdorff 3D Kakeya. In retrospect it seems humans were asking the wrong question, if a couple dozen pages long manuscript can subsume so many other efforts.
AI has only been able to advance the math frontier for a couple of months, fully grasping the applications, connections, and possible generalizations takes longer.
They did have the right intuition; it's pretty uncontroversial to believe that local smoothing in Rd would have been resolved within the next decade without AI input. For example, Hong, Josh and coauthors were pretty far along the way to finishing the R4 case when they got scooped in the previous batch. The timeframes are too short, of course.
My original comment was perhaps badly written: my point was that the OpenAI proof does not subsume the previous efforts because they are proving something stronger than Kakeya using very geometric methods based on the discretized sum-product theory in the real line. This approach has basically revolutionized modern analysis. Kakeya is just one very publicly visible problem. All of the other byproducts are in total of much greater interest than the resolution of Kakeya itself.
It's like Fermat's last theorem: who cares about solutions to x^n + y^n = z^n? It's the underlying theory-building in algebraic geometry that is the actual interesting product in its own right. If someone finds an extraordinarily clever 1 page proof of Fermat's last theorem it does not mean that the progress on algebraic geometry was useless.
I want to point out as well that it's quite common for humans to find very short clever proofs of difficult old results. Those proofs have value of course but rarely do they generalize. One of the goals of mathematics is to understand the fundamental mechanism underlying why things are true; proofs (of any origin) that are too slick is antithetical to this.
The idea that OpenAI's proofs are fake is contradicted by the response of mathematicians. Were they fake, mathematicians would not consider this release important enough to comment upon[1,2], as vixra always provided a steady supply of such writings. Nor would they be able to write expositions and simplifications, of which several are published each day on arxiv (just search OpenAI on arxiv to find them).
But perhaps mathematicians are no longer able to be the best judge if a proof is valid. Yet nobody but them is able to even understand what is claimed to have been resolved.
> But perhaps mathematicians are no longer able to be the best judge if a proof is valid
It seems they are still the best judge of whether a proof is useful or insightful though. AI is going to reward hack and try to eke out a win whatever it takes - which is fine if you regard these problems as trophies to be won, but not so fine if you take them in the spirit intended.
The famous problems that OpenAI (primarily) are trying to trophy-shoot with AI are famous because some human mathematician such as Erdos had the taste to judge them as fruitful grounds for new math to be developed in the pursuit.
Until we're at the point where AI is capable of things like creativity, curiosity and continual learning, AI will remain reliant on humans to fundamentally advance math and give it new tools to apply. For humans to do this it is helpful if they are inspired by the challenges set by their peers, which is why AI solving these human challenges is so counter productive - the snake is eating it's own tail.
> It seems they are still the best judge of whether a proof is useful or insightful though.
How can you be so sure of that? Like we know that human mathematicians can judge usefulness or insightfullnes. But we don't really know whether AI can do it even better. Like I don't think anyone has set up an evaluation of that.
> Were they fake, mathematicians would not consider this release important enough to comment upon[1,2], as vixra always provided a steady supply of such writings.
Few articles on vixra are made by companies with ~$1T valuations.
What do you believe doesn’t reflect positively on the authors of that open letter? It raises valid concerns, and the statement you quoted is a factual observation.
When Thom, the mathematician who now alleges plagiarism, posted his digestion [1] of OpenAI's construction of a non-sofic group, he does not mention the proof being familiar. He even calls the crucial argument clever, without noting he thought of it first.
[1]https://mathoverflow.net/a/513885
That link is a helpful contribution to this discussion.
I'm not at all familiar with this area, but my reading is that he appears to call it out as a relatively obvious extension of his own work:
> It is a creative and at the same time elementary construction that uses not just property (T) for an application of my result with Kun, but also for the ambient group
G in order to overcome the problem, that the Γ-components might be of different size. Once this is achieved, the rest of the argument is straightforward.
Creative and at the same time elementary is where LLMs excel, generally speaking. It's why they are so good at writing code.
> On the other side, I was looking myself for such a mechanism ever since we wrote the paper in 2019 and admire the efficiency of this construction.
He seems to admit very clearly he does not see this as his own work. 'I was looking...' well why did he stop? Because the AI figured it out first.
It seems quite odd to me to 'admire the construction' of something, only for your opinion to sour once that something figures it out first.
I think a lot of the emotional reaction here is familiar to us non mathematicians: you spent years developing expertise, and then LLMs began producing competent work in areas that had previously required that expertise. That's understandably uncomfortable, but discomfort by itself isn't evidence of misappropriation.
>However, there were 34 noteworthy deviations from the published record that needed to be resolved, the most substantial of which related to the Unbalanced Group Theorem.
[1]https://axiommath.ai/research/the-enormous-theorem-formalize...