Comment by SilverSlash
7 hours ago
In 2025, Hong Wang won the Fields Medal for proving the Kakeya conjecture in three dimensions. Here, OpenAI is claiming to have solved a stronger problem that proves the Kakeya conjecture in all dimensions!
That's so insane to me. This top mathematician (and colleagues) at her pinnacle spent years proving it in three dims and now OAI proves it in every dimension...
To be honest this doesn't feel as such impressive as the others. It was recently proven in 3 dimensions, so it could be "just" an extension of that. I haven't read the paper anyways so this is pure spectulation.
The question for me is, would OpenAI have been able to solve this without Hong Wang's work?
The question for some is, is whether the original authors would have produced their work without building off from the work of other mathematicians.
(which is somewhat valid). Though I think a far more interesting thing would be whether AI given the corpus of 19th century mathematics and physics reproduce what Ramanujam or what Einstein did.
:/
IMO that there is any work at all to build from is the result of a long tradition of taking the next step. Professor to student and so on. The danger of AI here is we get to take a leap, say, 10 steps - but then there is no more tradition to have the fortitude and mentoring necessary to continue the progress.
It all gets derailed while betting on RSI and never having to do math research again (but if an advanced theorem is proven, and no one alive is competent enough to comprehend it, does it even matter that we get to call it solved?)
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I think the question is rather, what will happen to human beings when any possible personal intellectual accomplishment will no longer be seen as interesting or relevant by other human beings when AI can do most of it?
It's all well and good to say that we should just go on these journeys for our own enrichment, but even so they will just be doing it alone...or else not do it at all because they need a shared human effort surrounding them to do it, and therefore end up depressed and immersed in drugs or media all day.
Of course we're awed at what AI can do now, because all of us observing it have had the luxury of having such opportunities before AI. I got my PhD before AI and I found satisfaction in participating in a shared math community, which I wouldn't have done if I were starting my PhD now, because I would find the current academic environment unacceptable and useless.
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> Though I think a far more interesting thing would be whether AI given the corpus of 19th century mathematics and physics reproduce what Ramanujam or what Einstein did.
I don't think so, it was a big leap forward in terms of creativity, imagination, rigor and experimentation. Einstein was able to "run" visual thought experiments inside his mind
That would be a very exciting experiment. I doubt you could find enough vintage training data to train a LLM enough to do it without it being contaminated by anachronistic data in the weights, but I would be absolutely amazed if it could reproduce that.
whether the original authors would have produced their work without building off from the work of other mathematicians.
That's rhetorical, right? Of course mathematicians built on what older mathematicians built and OpenAI's machine built on that. The real question whether OpenAI's machine has contributed something others will in turn build on.
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The proof uses completely different methods as what harmonic analysts used for the Kakeya conjecture prior so probably yeah.
Would Hong Wang would have been able to do his work without previous physicists work? That's the real question!
*her work
Yes. The all-dimension result follows a completely different approach from that of Wang and Zahl.
OpenAI's proof goes down a completely different route than the route taken for Kakeya in three dimensions and also the existing proof of local smoothing in two dimensions.
Hong Wang and Josh Zahl proved something a bit stronger, namely that the Kakeya conjecture holds for families of tubes satisfying the "Convex Wolff Axioms". From the outside this looks like a weird technical detail but if you unpack this you get an exceptional example of 'the journey is worth more than the destination'.
The idea for resolving Kakeya in R3 originates in the work of Jean Bourgain and a connection to the discretized sum-product phenomenon (the idea, in R, that "sum" and "product" are geometrically incompatible). This approach is summarized in Terry Tao's blog post from 12 years ago now [1].
Basically, the efforts to make this approach work resulted in the development of an entirely new and extraordinarily active subfield that one might call "continuous incidence geometry". This led to ideas like restricted projection theory (with spectacular applications in homogeneous dynamics and number theory), decoupling (another one of Thomas Wolff's ideas, and was realized by Bourgain and Demeter), the resolution of the Furstenberg set theorem, etc. There are very far-ranging applications, some of which even appeared on HN, and which a priori look totally unrelated, like the Heilbronn Triangle problem [2]
I know this is out of the context of most here on HN but let me say it in short: the development of the sum-product approach to Kakeya has resulted in a huge improvement of our understanding of the fundamental additive and multiplicative structure of Euclidean space.
OpenAI's proof of local smoothing in Rd shortcuts this entire matter. Don't get me wrong, the proof is phenomenal, and it's amazing that Local Smoothing admits a short, essentially self-contained proof (only uses some facts about elliptic PDEs which I can't comment about unfortunately since it isn't my area).
Hong Wang didn't win the fields medal for proving Kakeya in three dimensions (only). She won because of her incredible contributions to the development of continuous incidence geometry. The proof of local smoothing certainly contains a lot of very interesting insights into the wave equation but, if this proof had appeared in 2014 just like this, I do not think we would have had the same amount of development.
To me, this is the heart of the matter. For mathematicians, only having these incredible new proofs with no "social consequences" would be extraordinary. However, if we are not careful, the idea that math is "solved" will degrade the mathematics tradition in its own right. Where will be in 10 years, in 50 years? Humans, by virtue of being much worse at computations than computers, must build the "correct" abstractions in order to discover new mathematics. If this process is somehow lost, independently of the computer's ability to prove an infinite tower of new results, the mathematics tradition will suffer for it.
[1]: https://news.ycombinator.com/item?id=37465852
I mean, the models (starting with Bel clearly and Astra probably--though Astra probably is worse than top humans in some areas) are superhuman at math.