팀 가우어스: LLM은 어떤 수학 문제를 잘 풀까?

Gowers's Weblog Mathematics related discussions « Thoughts about the Leiden Declaration What sort of maths are LLMs good at? For the sake of anyone who might read this blog post in the distant future (a month from now, say), let me mention that I am writing it a few days after OpenAI announced that it had solved ten major problems in mathematics and theoretical computer science, including the first construction of a non-sofic group, and a proof that the multicolour Ramsey number (where there are 3's) grows superexponentially in . The first was, to judge from various talks I have been to, one of the most important unsolved problems in group theory, and the second was a major open problem in Ramsey theory that I didn't necessarily expect to see solved in my lifetime, though of course such expectations now have to be revised. The reason I want to be clear about the timing is that I shall be discussing the current capabilities of LLMs in the full expectation that those will continue to change rapidly. So it is likely that in not too long from now, if there is anything interesting in what I write, it will be interesting mainly as a record of what the situation looked like in early August 2026. These results, and the other eight on the list, are extraordinarily impressive, but it still doesn't seem to be the case that LLMs are better than all humans at all aspects of mathematics. If they were, then their big speed advantage over us would mean that there would be much more of a flood of results. So it is natural to wonder about what kinds of problems LLMs are good at, and about where there is still room for improvement. I don't pretend to have a good answer to this question, where a good answer would be a crisp classification that would fit the current examples well, but it is an interesting exercise to try to rule out some bad answers, and to try to identify potential answers that aren't obviously contradicted by the evidence. Are LLMs particularly good at finding counterexamples? A first remark here is that LLMs are not just good at finding counterexamples: they can find proofs of difficult statements as well. However, it is notable that the most famous problems they have solved have almost all been with counterexamples rather than proofs. That is true of the two problems mentioned above, and also of the Jacobian conjecture and the unit distance conjecture. If one wants to theorize that LLMs are particularly good at finding counterexamples, then there are two things it would be good to do to make the theory more convincing. The first may sound unproblematic: it is to decide when solving a problem counts as finding a counterexample. Once that is sorted out, the second is to come up with a potential explanation of why LLMs would be particularly well suited to solving problems of that particular kind. What does it mean to find a counterexample? Why am I suggesting that it is not completely obvious what it means to find a counterexample? Surely, one might suggest, all it means is that you have a statement of the form "Every object of such and such a type has such and such a property," and you exhibit an object of the given type that does not have the given property. However, this doesn't always work. Consider a famous result of Vinogradov, which states that every sufficiently large positive integer is a sum of three primes. The negation of this statement is (or is equivalent to) the statement that for every positive integer there exists an integer such that is not a sum of three primes. In other words, it states that every positive integer has a certain property. Seen in this light, Vinogradov found an example of a positive integer that does not have the given property. Do we want to say that Vinogradov found a counterexample? Clearly not — the result should obviously be classified as a theorem and not a counterexample. Thus, we cannot just naively say that LLMs are particularly good at negating universally quantified statements: there has to be something about the nature of the universal quantification. With the three-primes example, it is clear that Vinogradov did not think, "How am I going to find with this property?" Rather, what he thought would have been more like, "I've got an integer that is very large. How am I going to show that it is a sum of three primes?" In other words, all his focus would have been on the universally quantified , with the existentially quantified being a sort of afterthought once the details of the proof have been worked out. In general, many interesting results, when they are stated formally, begin with an alternation of two or three (or more) quantifiers. The question then becomes to determine which is the first "interesting" quantified variable in some sense. Here's another example to illustrate the point, from the theory of finite-dimensional normed spaces. I'll give a few mathematical details for those curious, but if you don't care about those, then you can skip the next three paragraphs and should get the gist of what I am saying about this example. Let and be two -dimensional normed spaces and let be a linear map from to . We say that is a - isomorphism if there exists such that for every . By rescaling we can always take to be 1, in which case we have that for every . If , then this tells us that is an isometry. In general, the Banach-Mazur distance between and is defined to be the smallest such that there exists a -isomorphism from to . It is easy to see that the logarithm of the Banach-Mazur distance is a metric on the set of isometry classes of -dimensional normed spaces. A less easy fact, but still not too hard, is that the resulting metric space is compact: in fact, it is known as the Banach-Mazur compactum. It is natural to wonder what the diameter of the Banach-Mazur compactum is, and here things get interesting. A result of Fritz John states that every -dimensional space has distance at most from . (The idea of the proof is as follows: pick inside the unit ball of an -dimensional ellipsoid of maximal volume; that is the unit ball of a normed space that is isometric to ; it can be shown that the identity map is a -isomorphism between and .) From Fritz John's theorem and the (multiplicative) triangle inequality, it follows that for any two -dimensional normed spaces. That is, the diameter of the Banach-Mazur compactum is at most . But might it be substantially less than that? An indication that the answer is not obvious comes from looking at the spaces and . The identity map between these two spaces is an -isomorphism, but one can do much better by mapping the standard basis vectors not to themselves but to vertices of the unit cube, with the vertices chosen to be as orthogonal as possible. In particular, if there exists an Hadamard matrix, then the corresponding linear map is a -isomorphism. One can push this observation and deduce that for any the Banach-Mazur distance between and is . It is also easy to show that , so -spaces hardly improve on the easy lower bound, and do not improve on it at all in dimensions for which an Hadamard matrix exists. In 1981, Gluskin famously solved the problem by determining the correct asymptotics for the diameter of the Banach-Mazur compactum. Informally, what he showed was that the diameter is within a constant of the upper bound that follows immediately from Fritz John's theorem. If we make the quantification explicit, then the statement we end up with is , where I have written for the set of all -dimensional normed spaces. (If you want to argue that it is not a set, then let me specify in addition that the underlying vector space is .) In words, there is a positive constant such that for every positive integer there are -dimensional normed spaces and such that the Banach-Mazur distance between and is at least . I can't continue without very briefly describing the beautiful and highly influential idea Gluskin had for solving this problem. He took and to be