While Spain made history winning his second soccer World Cup on Sunday, a 34-year-old Harvard mathematician named Levent Alpöge asked Fable 5 for something interesting. Not that he tried to prove the Jacobi conjecture, but that he did the opposite: to refute it.
Argentina kicked us while Spain played soccer and Ferrán of my life and my heart scored a historic goal. By the time it was all over, two things happened. The first, that Spain, we insist, was making history with its second star on the shirt. The second, that a conjecture that had not been proven for almost 90 years was definitively refuted thanks to the help of an AI.
The aforementioned Jacobi conjecture suggests that a certain type of mathematical function could also work in reverse. It was formulated by Ott-Heinrich Keller in 1939, and in 1998 the mathematician Stephen Smale included it in his famous list of unsolved mathematical problems that were a challenge for mathematicians of the 21st century.
A discovery that leaves as many questions as answers
Alpöge has not explained in detail how he managed to reach that result or what prompt he used, but in his tweet he indicated that part of the credit was due to his “close friend fable“, the Anthropic model.
Abhishek Sahafrom Queen Mary University of London, explained that the very way of facing the problem was unexpected. The conjecture seemed intuitively correct, but no one was trying to refute it, but rather prove it.
Chris Bowman-Scargillfrom the University of York, pointed out something that he considers very important: it is one thing to find counterexamples that overturn conjectures and quite another to construct entire branches of mathematicswhich still requires human creativity. He gave as an example Fermat’s last theorem, solved by Andrew Wiles in 1994, and which forced him to develop dozens of pages with new and fascinating mathematical theories.
Several experts have already validated this refutation, something that was relatively easy to do. For Saha, AI advances in mathematics had already surprised —they did it with one of the problems recently—, but this went further: “It is probably the most important conjecture in which AI has played a relevant role so far in mathematics,” he said.
There is however some open questions: This particular counterexample refutes the conjecture with three variables, but a version with only two variables could be true. One thing is certain: mathematicians are taking advantage of this tool to help them in their work. The most famous in the world does it, of course.
Ivan Fesenkofrom Westlake University in China, was talking about this when he stated that AI models are becoming more and more capable: “Right now, AI can already produce math graduates. In a year, it will produce math PhDs. And then the question arises: Do we really need that many mathematicians if AI can do these things so well?”
It is a disturbing question, of course.
Image | Vitaly Gariev
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