When OpenAI released hundreds of claimed solutions to some of the world’s hardest math problems this week, the frontier lab said that it had consulted an advisory group of elite mathematicians to avoid the controversy that came with the last time one of its models solved a long-standing problem in the field. But OpenAI fell short of those standards, particularly where the mathematicians emphasized the need for human understanding of a mathematical result. That’s especially concerning after a new paper highlighted gaps between the natural language and formally expressed solution to a million-dollar problem ostensibly solved by OpenAI’s models.
The Advisory Group on Mathematics and Artificial Intelligence, hosted by Princeton University’s Institute for Advanced Studies, is made up of nine prominent researchers at institutions around the world. The organization released guidelines for frontier labs solving math problems at the end of September. In a statement on the latest set of proofs, the AGMAI said that “it is ultimately up to the mathematical community to assess the extent to which our recommendations were followed successfully.” However, the organization’s first request was “to stop testing advanced mathematical problems on proprietary models.” OpenAI’s release explicitly says that it is evaluating its proprietary models using open research problems in mathematics.
The advisory group did not respond when asked by TechCrunch for a more thorough evaluation of OpenAI’s latest proof release. The lab clearly followed some of its principles, including releasing results as soon as possible and including information about how the models reached their conclusions. But not for all of them: Just ten of the 719 manuscripts included releases of the model’s chain of thought.
For papers that people don’t understand, the mathematicians suggested the proofs should be formalized — but just 42% of the proofs released by OpenAI had not undergone this process. Ultimately, it’s still not clear that OpenAI is taking “responsibility for ensuring that human understanding will follow” when releasing its proofs, in accordance to the AGMAI principles. AGMAI suggested that OpenAI should help fund the work of human mathematicians who will be required to make the lab’s solutions meaningful in any real way.
That problem is exemplified by a paper released this week by mathematicians at the University of Cambridge and King’s College in London that questions the way frontier labs are approaching these challenges. When AI models solve mathematical problems, they first create a “natural language” explanation, then try to express that result in Lean, a programming language that in theory confirms the accuracy of the proof by compiling it as code. However, there may be problems with the way the models translate their natural language proofs into code; this paper documents at least two discrepancies between the natural language proof and the Lean code behind the solution OpenAI has offered to a problem derived from the Navier-Stokes equations that describe the complex behavior of fluids.
These discrepancies don’t necessarily disprove either solution, but they do raise questions on whether we can simply rely on models to formalize their own solutions without human involvement. That’s one reason that AGMAI asked OpenAI to “include machine-readable metadata correlating the natural language and formal artifacts,” something that the frontier lab did not do with these releases. Mathematicians stress that when new results are discovered by humans, they take responsibility for them and engage with the broader community through papers, talks, and seminars.
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