OpenAI announces solutions to 10 longstanding maths problems

OpenAI has revealed solutions to 10 longstanding mathematical problems that were found by its prototype AI model Astra. The announcement is the latest in a string of mathematical discoveries made by AI that are threatening to change the field beyond recognition. In May, an OpenAI model cracked a decades-old conjecture by Paul Erdős, causing a stir in mathematical circles. Last month, the Claude Fable 5 AI found a counterexample to the Jacobian conjecture, which had stood for nearly a century. Hundreds of other AI-led discoveries have been made in recent months. The newest solutions from OpenAI tackle problems ranging from how densely spheres can be packed into spaces with more than three dimensions to quantum game theory. But the one receiving the most attention is the discovery of a non-sofic group. Advertisement Soficity, first described in 1999, is the property of a group of operations that can be approximated by smaller, finite groups of permutations. You can think of it like a game played on an infinitely large chessboard being loosely approximated by games on a small one. Until now, all known groups had this property, so mathematicians proposed that all countable groups are sofic. Now OpenAI has discovered a single counterexample that disproves the claim. The news has sparked further hubbub among mathematicians who are having to adjust to a radical shake-up in their field. Elon Musk even said in a tweet that it was evidence we have reached the singularity – the point at which AI becomes self-improving and advances towards general intelligence at an accelerating pace. Francesco Fournier-Facio at the University of Cambridge has been studying the soficity problem since starting his PhD in 2020, and says he may well not have stayed in academia if AI had made this discovery back then. He is concerned that AI companies, particularly in this case, aren’t being transparent about how their solutions rely on prior human work. Fournier-Facio thinks OpenAI’s solution to the soficity problem relies heavily on papers by Andreas Thom and Gábor Kun that pushed the field ahead significantly, and that it is their work that should be celebrated more than this discovery of a counterexample. “In very broad strokes, the solution takes these two works from 2016 and 2019, pushes them forward and then does some clever tricks,” says Fournier-Facio. We will never know if humans could have arrived at the counterexample, but AI certainly couldn’t have if not for prior human work, he says. OpenAI’s initial announcement claimed that all 10 solutions “have seen no progress on the main result for at least a decade”, but Fournier-Facio complained this was incorrect and the company has since changed its statement. “Until AI shows that it can develop theory independently, it’s hard to believe that it could have come up with this [counterexample] independently,” says Fournier-Facio. He points out that many AI mathematical discoveries so far have focused on finding counterexamples, which can be easily and quickly checked, rather than developing new theory. “Developing theory, it’s very much less clear [if it is correct]: there’s no tick at the end. How do you know if you have developed the right theory, if you’re going a step in the right direction?” asks Fournier-Facio. Abhishek Saha at Queen Mary University of London says human mathematicians had also made significant progress with high-dimensional sphere-packing, and Astra has built on that work to arrive at a solution. Nevertheless, the release of these 10 solutions is still the most impressive display of AI mathematical prowess to date, he says. “Any one of them would be a significant and impressive achievement,” says Saha. “Some of them are not counterexamples; some of them are actually proofs, but they are all of the kind where you don’t have to build a huge new amount of theory. Instead, you [have to] very technically put together things that have been done, in unusual ways, and in very technical ways, and do something that no one had done before.” Saha is optimistic about the future of AI models in mathematics. “I would not be surprised if, in two or three years, they can actually build enough new theory to solve some of the deeper questions,” he says. OpenAI did not respond to an interview request for this article.