In a groundbreaking achievement, artificial intelligence (AI) systems have made significant progress in solving two major mathematical problems that have puzzled experts for decades. Recently, OpenAI announced that its AI agents had solved a substantial part of the Navier-Stokes problem, a complex mathematical equation that has been open for nearly 90 years. This achievement was accomplished in just 88 hours, with 10,000 AI agents working together and producing 130 billion words of reasoning.
The Navier-Stokes problem is a fundamental equation in mathematics that describes the motion of fluids and gases. The specific solution achieved by OpenAI's AI agents addresses the problem with an external force, similar to a fan blowing into water. While this is a significant breakthrough, the original problem without an external force remains unsolved. The solution was verified by a software called Lean, which checks mathematical reasoning step by step to ensure accuracy.
Another AI system, Claude, developed by Anthropic, has made a remarkable achievement in formalizing the proof of Fermat's Last Theorem. This theorem, proposed by Pierre de Fermat in 1637, states that there are no integer solutions to the equation a^n + b^n = c^n for n greater than 2. Andrew Wiles proved the theorem in 1994, but his proof was lengthy and complex. Claude's AI system translated Wiles' proof into a formal language that can be verified by a computer, producing 13 million lines of code and verifying over 30,000 steps of reasoning in just 11 days.
The achievement by Claude's AI system is significant, as it demonstrates the ability of AI to formalize complex mathematical proofs. Kevin Buzzard, a mathematician at Imperial College London, had been working on formalizing Wiles' proof since 2024 and confirmed the accuracy of Claude's work. Both achievements demonstrate the power of AI in exploring and verifying complex mathematical problems at a speed and scale that human teams cannot match.
While these achievements are impressive, it is essential to note that AI systems did not independently discover new mathematical concepts. Instead, they built upon existing ideas and research by human mathematicians. For example, the solution to the Navier-Stokes problem relied on ideas developed by mathematicians such as Diego Córdoba and Luis Martínez-Zoroa. Similarly, Claude's formalization of Fermat's Last Theorem was based on Wiles' existing proof.
The use of AI in mathematics has sparked discussions about the role of human mathematicians and the nature of mathematical discovery. Some researchers argue that AI systems can accelerate the discovery process, while others raise concerns about the ownership and credit for AI-generated results. Tristan Buckmaster, a mathematician at New York University, publicly accused OpenAI of profiting from his private research without proper credit.
Muhammad Sahimi, a chemist at the University of Southern California, proposes a framework for categorizing scientific problems in the context of AI. He identifies three types of problems: those where the formula is known but calculations are difficult, those where data is available but the formula is partially known, and those where neither the formula nor the data is available. This framework can help researchers better understand the potential and limitations of AI in solving complex scientific problems.
Key points
- AI systems have made significant progress in solving long-standing mathematical problems, including the Navier-Stokes equation and Fermat's Last Theorem.
- These achievements demonstrate the power of AI in exploring and verifying complex mathematical problems at a speed and scale that human teams cannot match.
- While AI systems have not independently discovered new mathematical concepts, they have accelerated the discovery process and raised important questions about the role of human mathematicians.
