For nearly a century, the Jacobian conjecture has stumped mathematicians, but recent advancements in artificial intelligence have led to its potential refutation. Levent Alpöge, a mathematician from Harvard University, announced a significant breakthrough in a brief tweet on July 19, stating that the conjecture, which many believed to be true, has been disproven with the help of AI, specifically Claude Fable 5 from Anthropic.
This conjecture, proposed by mathematician Ott-Heinrich Keller in 1939, posits that specific mathematical functions should be reversible. It had become one of the 18 major challenges for 21st-century mathematicians, as outlined by Stephen Smale in 1998. Alpöge's tweet provided a concise counterexample comprising just 216 characters, and experts have identified it as one of the most complex mathematical problems solved by AI to date.
Although Alpöge did not respond to media inquiries, he expressed gratitude to his "close friend fable" in his tweet, hinting at the AI's role in the discovery that was made during the World Cup final. Anthropic has yet to comment on the matter.
Abhishek Saha from Queen Mary University of London remarked on the remarkable capabilities of AI in mathematics, highlighting that this development indicates AI’s growing proficiency in solving challenging problems. Saha pointed out that while AI has previously tackled difficult conjectures, Alpöge's findings mark a significant evolution in the field, with the Jacobian conjecture being a major milestone.
The mathematical statement Alpöge shared was straightforward to verify, and many have done so quickly. However, the crux of the inquiry now lies in the methodology used to attain the solution. Saha noted that some problems can be notoriously difficult to solve, yet solutions can be easily confirmed once uncovered. He speculated that Alpöge's approach went beyond merely asking the AI to search through data, suggesting that there was significant insight involved that remains unpublished.
Interestingly, the discovery raises further questions. While the counterexample disproves the conjecture with three variables, a two-variable version might still hold validity. Mathematician Chris Bowman-Scargill at the University of York commented on the remarkable capabilities of AI, indicating that while it can find counterexamples, the development of new mathematical theories will still necessitate human ingenuity.
He contrasted this with Andrew Wiles’ proof of Fermat's Last Theorem, which demanded the creation of extensive mathematical theories. He noted that the real creativity in mathematics often emerges from the journey of solving these conjectures, emphasizing the importance of this process.
Ivan Fesenko from Westlake University in China predicts that as AI models continue to advance, they will handle increasingly sophisticated math problems, fundamentally transforming the landscape of mathematics. He pointed out that current AI capabilities have reached a level where they can generate master's level work in mathematics, hinting that within a year, they might achieve PhD-level results. This raises critical questions about the role of human mathematicians in a future where AI can perform so adeptly in the discipline.
