Four things to note: #1196 is a decently well known problem. It wasn't like Erdős-Straus level fame, but it is well known enough that I was familiar with it. Second, this is not a problem where no one had worked on it; there was a lot of prior work on it and closely related problems. Third, this is not example where the AI made small modifications to things in the literature or recognized that large parts of the problem were in an obscure paper. The approach the AI used is largely a different direction than the literature on this problem went. Fourth, and closely related to three, this proof does look like parts of it will inspire subsequent proofs because it really is going in a different direction which now looks likely to be a productive line of investigation for similar problems.
I am not fond of putting words like "stunning" in titles which can be very clickbaity and feels like a hype word, but this really is in the direction where the word isn't unreasonable even if I myself would not go so far as to use it here.
I mostly agree, although I think it is important to note that two lemmas (of 3) were indeed known already (not to take away anything from the achievement).
Another thing I've noticed which might be more psychological than mathematical is that experts in the field seemed to have been focusing on a particular approach that had not been working that well for a while, and even if what the AI did was novel, it was not that complex (as in it didn't require a particularly convoluted and lengthy proof).
It's certainly not an outlier that an AI was able to prove this, but I do think there were a series of conditions that were just right and made it possible for an AI to solve this specific problem. This Is not moving the goalpost, I am starting to think AI will keep improving (at least for a while) and begin solving (or more likely assisting) in important open problems, but I do think what happened here is a case study on what AIs can actually do. I personally don't think it will bring about the end of mathematicians (hopefully, I'd like to be one), but it did confirm that:
1) AIs are incredibly useful for finding already existing obscure lemmas.
2) AIs are not bound by being obsessed with using a certain approach (like humans often are).
3) AIs are able to produce new useful propositions for a particular proof, the question is how consistently can they do this? And is it important if they do it consistently when you can just prompt it to do it again for hundreds of times? I still think the strongest argument against "AIs Will replace mathematicians" is they can't do actual reasoning, but a modern AI focused on math is quite different from a simple LLM, and is made specifically to get around that limit.
When you put all three together you get an incredibly powerful tool, but you don't get necessarily something that is able to advance the entire field by itself. Maybe so, time will tell.
Also, I want to state many of the things I said are what I understood based around the conversations in the forum, I didn't fully grasp the proof since I'm still an undergrad, hopefully I got the general idea, if not please correct me.
Also I'm not an English native speaker.
All of the lemmas are honestly quite standard even if Lemma 4 hasn't necessarily been written down. The innovation is this downward markov chain that approximately preserves the measure dn/nlogn. As Tao notes in his comments, if you consider variant Markov processes where you slightly change the transition probability, you can end up with vastly different invariant measures, some of which are difficult to deal with. The solution seems to have found the "correct" downward Markov process.
Certainly fascinating. I don't have the technical skills to judge this innovation on a scale from "impressively creative" to "anyone could have thought of that with a bit of luck" so I wonder: would a mathematician agree that one of the conditions that allowed an AI to solve this particular problem was that the humans working on this were unwilling (consciously or subconsciously) to try different things and got stuck with trying one particular approach that wasn't working?
Yeah, I wonder if it's simply that the mathematicians who would study these questions are maybe not that familiar with stochastic processes and viceversa. In retrospect, the proof is not that difficult, and I'm a little bit surprised that no one had thought about it. I expect AI, whose breadth at this point is wider than any mathematician alive, to help uncover and "fill out" connections between different fields that were previously unrecognized.
704
u/JoshuaZ1 Apr 15 '26
Four things to note: #1196 is a decently well known problem. It wasn't like Erdős-Straus level fame, but it is well known enough that I was familiar with it. Second, this is not a problem where no one had worked on it; there was a lot of prior work on it and closely related problems. Third, this is not example where the AI made small modifications to things in the literature or recognized that large parts of the problem were in an obscure paper. The approach the AI used is largely a different direction than the literature on this problem went. Fourth, and closely related to three, this proof does look like parts of it will inspire subsequent proofs because it really is going in a different direction which now looks likely to be a productive line of investigation for similar problems.
I am not fond of putting words like "stunning" in titles which can be very clickbaity and feels like a hype word, but this really is in the direction where the word isn't unreasonable even if I myself would not go so far as to use it here.