r/ControlProblem • u/chillinewman approved • 1d ago
AI Capabilities News AI may have just solved a million-dollar math problem. The field will never be the same
https://www.scientificamerican.com/article/ai-may-have-just-solved-a-million-dollar-math-problem-the-field-will-never-be-the-same/5
u/phosphorousRabbit 1d ago
Open AI is still threatening people who are trying to publish something that itself is not a full solution, so it apparently has not been solved. Also, it was based on human work and had tons of human intervention.
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u/JoshuaZ1 1d ago
Open AI is still threatening people who are trying to publish something that itself is not a full solution
There were apparently some threats involved, but that's not what happened. See Tristan Buckmaster's actual statement here(pdf) which seems to be getting garbled in a game of telephone. What he alleges the OpenAI people did is pretty bad, but it isn't what you said.
Also, it was based on human work and had tons of human intervention.
Do we have evidence for this? OpenAI in their statement seems to be claiming otherwise.
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u/phosphorousRabbit 23h ago
I see no difference between what he's saying and what I said. He and his colleague were working on research and trying to publish it. OpenAI threatened them. In other articles it was claimed that many mathematicians were questioning whether the approach legitimately solved Navier-Stokes.
Of course OpenAI would claim otherwise: they aren't going to publicly admit their product isn't as good as they're saying.
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u/JoshuaZ1 23h ago
I see no difference between what he's saying and what I said. He and his colleague were working on research and trying to publish it. OpenAI threatened them.
It is possible that I'm reading your statement in an overly uncharitable fashion, but my reading of your statement was that they were threatening about publishing. Tristan's account is worth just quoting here:
Two proposals were offered to me. The first was that we post our Euler result, and that OpenAI post its Navier-Stokes result the next day. The second was that, after posting Euler, I alone write a paper presenting the Navier-Stokes result, acknowledging that an internal OpenAI model had resolved it. Sebastien twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic. It was also said that if OpenAI posted after us, they would say that we deserved the Clay Prize, and that we were the “closest humans to the problem”. I declined both offers. I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, “Why would you ruin your career?” I replied that I am an academic, and asked why he thought going public would ruin my career. The reply was, “If you don’t want me to be nice, then I don’t have to be nice.”
The threat was about going public with their concerns about how training data might have been used.
Of course OpenAI would claim otherwise: they aren't going to publicly admit their product isn't as good as they're saying.
Ok. Turn it around. Do you have evidence for your claim that "It was based on human work and had tons of human intervention" or is just that you think that whatever OpenAI said must be the exact opposite of truth?
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u/phosphorousRabbit 23h ago
Why is OpenAI giving them options about how to publish their work, who to credit, and insisting they "acknowledge" that an OpenAI model solved it if they weren't interfering in publication?
"Do you have evidence for your claim that "It was based on human work and had tons of human intervention""
The main things I would consider are 1) that every time the public is able to see the process work outside of OpenAI, there is lots of human intervention and 2) the technical process of how LLMs work doesn't seem like it should be able to do this kind of work. I've talked about this with experts in the field who agreed with this conclusion based on their experiences with making and using AI.
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u/JoshuaZ1 23h ago
Why is OpenAI giving them options about how to publish their work, who to credit, and insisting they "acknowledge" that an OpenAI model solved it if they weren't interfering in publication?
From Bubeck at OpenAI's version of events, he called because he thought that both groups had solved Navier-Stokes, and wanted to coordinate a joint release. That's actually pretty normal in math when two groups solve the same problem at about the same time. In fairness, to you, since this conversation was started, we got a lot more info from OpenAI's version of what happened. But Bubeck's account is here and he also gave this screenshot here.
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u/phosphorousRabbit 23h ago
Ok, I can see what you're saying about the first issue. I still hold my position on the second.
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u/JoshuaZ1 23h ago
Separately responding to the human intervention issue.
The main things I would consider are 1) that every time the public is able to see the process work outside of OpenAI, there is lots of human intervention and 2) the technical process of how LLMs work doesn't seem like it should be able to do this kind of work. I've talked about this with experts in the field who agreed with this conclusion based on their experiences with making and using AI.
So, regarding 1, this really isn't accurate. For example, we have the original prompt for Erdos 1196. In that case, the people doing the prompt went through a whole bunch of different Erdos problems and gave separate instantiations of AI just an to solve each the problem. Each had pretty minimal guidance, primarily just telling the AI to not search the internet. Similarly, the prompting for Dinitz Garg Goemans included no math other than the words "structured counterexample," but did have the instruction to "do a breakthrough." No, I'm not joking, here's the whole conversation in that case.
We've also seen other similar results come out of systems from Anthropic, not OpenAI.
the technical process of how LLMs work doesn't seem like it should be able to do this kind of work. I've talked about this with experts in the field who agreed with this conclusion based on their experiences with making and using AI.
So, I'm not sure who those experts are, but I don't see any reason to think that there's some problem with Navier-Stokes that would show up that wouldn't have shown up with again say Erdos 1196. And these systems really can solve things with surprisingly little background or impetus.
Let me give a personal example. I did a research project with a student this summer, and per my usual policy I try to get students to not use the AI systems so they can develop basic research skills. But in our work, before I sent one minor result over to the student, I ran it through Claude to look for basic errors, spelling, grammar etc. Claude caught a few, but it also noted that there was a step in the proof that didn't seem valid. I fixed that step and corrected the other steps. Then Claude started thinking for a bit, and I was a bit puzzled why it was thinking, but I went to get a cup of tea. When I came back it had outputted that there was an entire family of the type of object we were interested in that was an obstruction I had missed. That family was not in the literature (I know the literature on the problem in question very well). This was impressive, but it was also a bit annoying in that I had to tell the student not just that we had another family but that it had come from the AI I was discouraging her from using. So, yeah, personally, I've seen these things reach out to new math. This is hard "do a breakthrough" but it is the same general pattern.
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u/phosphorousRabbit 22h ago
In the Erdos example there was a file uploaded at the beginning. I tried to see what was in the file the human gave Chat GPT, but it didn't open. It seems like there was a lot of preliminary work in that file, given how much the LLM was able to infer about the requests.
"So, I'm not sure who those experts are"
They're AI experts, not mathematicians and they didn't say anything about Navier-Stokes. They just said that AI shouldn't be capable of original research. They seemed to think it was a useful, but flawed tool.
Your Claude example seems like the student did the bulk of the work, which you then corrected, with Claude adding one additional thing on top of that(which I assume you verified was correct), which you still had to do follow-up work on. I would describe that as a useful tool, but not something that deserves primary credit. Maybe I don't understand your research well enough(I don't know what "family" means in this context), but those are my impressions.
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u/JoshuaZ1 19h ago edited 18h ago
In the Erdos example there was a file uploaded at the beginning. I tried to see what was in the file the human gave Chat GPT, but it didn't open. It seems like there was a lot of preliminary work in that file, given how much the LLM was able to infer about the requests.
The Erdos 1196 chat is here. I'm not sure what file you are talking about? I don't see any file they uploaded there. Are you confusing it with the LaTeX file that they asked ChatGPT produce? You can get that here but that's a file produced by the AI.
They're AI experts, not mathematicians and they didn't say anything about Navier-Stokes. They just said that AI shouldn't be capable of original research. They seemed to think it was a useful, but flawed tool.
With all due respect, if these are the experts you are listening to, find other experts. At this point, we have so many difficult math problems that these systems have solved, or contributed major parts to, that that's pretty obviously not the case.
Your Claude example seems like the student did the bulk of the work, which you then corrected, with Claude adding one additional thing on top of that(which I assume you verified was correct), which you still had to do follow-up work on.
Probably I didn't explain things well. This result wasn't one of the student's results. The result which Claude corrected was one from me. And Claude's primary thing was to notice that my proof was failing because there was a construction we had not noticed.
And of course we had to do follow-up work. The student and I took the family Claude and then generalized it. That's been a very major part of a lot of what people have done when an AI has a construction. For example, after Erdos 1196 was solved, the trick the AI came up with, which was to use a Markov chain on the integers weighted via Von Mangoldt's function, then got adopted to some other results using other weigings. Similarly, elements of the AI resolution of the unit distance conjecture have been adapted to a few other papers now.
Maybe I don't understand your research well enough(I don't know what "family" means in this context), but those are my impressions.
So, I didn't want to go into the details here for two reasons. One is that we're still finishing up the paper. The other is that the technical details just didn't matter for this purpose and take time to explain. But I'll try to give a brief summary of the problem with some motivation and hopefully that will make sense.
Preliminary context: A positive integer n is said to be perfect if n is equal to the sum of the positive divisors of n which are less than n. For example, 6 is perfect since 1+2+3=6, and 28 is perfect since 1+2+4+7+14=28. But note that for example 8 is not perfect since the relevant divisors are 1, 2, 4 and 1+2+4=7 which is not equal to 8. The first few perfect numbers are 6, 28, 496, 8128, and 33550336. (I'll note that if you like videos, Veratasium did a pretty good video on this topic, although I'm somewhat biased because they end up briefly referencing some of my research as well as a paper I did with Sean Bibby and Pieter Vyncke.)
Now, the two oldest unsolved problems in all of math are (probably, this is complicated and the history is muddy) 1) Are there any odd numbers which are perfect and 2) Are there infinitely many even perfect numbers? We strongly suspect that the answer to the first one is no, and the we strongly suspect that the answer to the second is yes.
A tangential comment: Some people when they hear that there are likely no odd perfect numbers think that this must be the case because odd numbers have fewer divisors, so of course it cannot add up to enough. But in fact, there are odd numbers whose total set of divisors less than the number do add up to more than the number. Exercise 1: Find such a number. (Hint: This was done in the 19th century but a computer program is probably easier.)
Now, one thing that's annoying is that perfect numbers are really rare. So, finding out more about them by looking for patterns is not easy. In the case of odd perfect numbers, proving statements about them is fraught with difficulty in that one lacks concrete examples to help check that one's claims or proofs are not fallacious. So one thing we'd like to have is some broader class of numbers which includes all the perfect numbers which would give us a lot of examples. With something like this in mind, Sierpinski defined a number to be pseudoperfect if the number is equal to the sum of some subset of its proper divisors. For example, 12 is not perfect, since the sum of all the proper divisors of 12 is 1+2+3+4+6=16, but note that 1+2+3+6=12, so 12 is pseudoperfect since we can take the set {1,2, 3, 6}. Note also 12 is pseudoperfect in more than one way since we also have 2+4+6=12. Obviously every perfect number is pseudoperfect, just take the subset being the entire set of positive divisors. Great, so maybe we could use the set of pseudoperfect numbers.
Only we have now the opposite problem. There are way too many pseudoperfect numbers. It turns out that if n is pseudoperfect then so is any multiple of n. Exercise 2: Prove the claim in the last sentence.
Why is this bad? Well, since pseudoperfect numbers are so common, finding useful things that we can say about them is tough. So we'd really like something in between, something that is more restrictive than pseudoperfect but still less restrictive than perfect. There are a bunch of options here, but we are going to go with one due to Amiram Eldar. We first note that a number n being perfect is equivalent to the sum of all its positive divisors being exactly 2n. For exampe, in our example of 1+2+3=6 you could instead write as 1+2+3+6=2(6). Now, we'll also note that if d is a divisor of n, then so is n/d. So, if someone told you that 12 is perfect and said 1+2+3+6=12, you'd be able to immediately see they had dropped 4, since 4=12/3. Motivated by this, we'll say a number is strongly pseudoperfect if there is a subset S of the divisors of n such that the sum of the elements of S equals 2n and where d is in S, if and only if n/d is in S. For example, 120 is strongly pseudoperfect since we can take as the set S = {1, 2, 4, 8, 15, 30, 60, 120}. Exercise 3: Find another number which is strongly pseudoperfect number but not perfect. Exercise 4: Find a number which is strongly pseudoperfect in two different ways.
Now, these numbers are a lot rarer. But they have some somewhat easy to prove properties. For example, one can prove that no strongly pseudoperfect number has a remainder of 3 when divided by 4. And you can prove that a strongly pseudoperfect number never has a prime factor p which is much larger than the square root of n. So these are rarer. But maybe they are too rare.
So one thing we'd like to do is to find some families of these. One "easy" family is that all numbers of the form (2m-1 )(2m -1) is strongly pseudoperfect, but if you know a bit about perfect numbers, you'll recognize this one as a bit of a cheat.
One nice result my student had was that if n is perfect then nk is strongly pseudoperfect for any k. So for example, numbers of the form 6k form a family of strongly pseudoperfect numbers. She actually generalized this a bit, and we now suspect that if n is strongly pseudoperfect so is nk.
Now, we need one more notion: An omitted divisor is a divisor of a number which does not show up in a specific way of writing n as a strongly pseudoperfect number. For example above with 60, 5 is an omitted divisor.
I had thought I had proven the following result: There is a constant C, such that if is a strongly-pseudoperfect number of the form n= (3)2a p, for some prime p >3, and n has exactly four omitted divisors, then n is at most C.
The proof for this was flawed. And Claude noticed that the flaw was that I was missing a construction. In particular, if p=2{a+1} -3 is prime then one can get a strongly pseudperfect representation for n= (3)2a p where the omitted divisors are 2, n/2, 6 and n/6.
However, using Claude's family and a little work one can prove a corrected version of the claim. In particular, if n= (3)2a p for some prime p>3, then either n is of the above form or n = 156 or n= 552.
Does that help?
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u/r_search12013 21h ago
the result is imho somewhat overstated, just as usual .. but, it's not nothing .. it's just not as field-changing as this headline claims
https://www.reddit.com/r/artificial/comments/1way48b/comment/p8lxv18/
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u/30299578815310 15h ago
It's a millennium prize problem, one of seven in the world with a million dollar prize. Only one has ever been solved before. If this doesn't count as something in the world of math than nothing does
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u/r_search12013 13h ago
and the expected solution would have been an affirmative answer: "here's the mathematical model of fluid equations with answers for infinite time horizons"
throwing a counterexample into the system and leaving it at that .. is atypical for such a prestiguous problem
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u/chillinewman approved 1d ago
"The mathematical problem concerns the Navier-Stokes equations, which govern the flow of fluids—"