The AI maths crisis

AI will change the way mathematics is done, and there will be a lot of bad blood before it happens

In the last two months, artificial intelligence (AI) has triggered a crisis in mathematics.  

In August, OpenAI’s large language model (LLM) Astra released solutions to 10 open mathematical problems. The results caused two controversies. One was that the proofs were in Lean, a computing language, rather than in the traditional format of mathematical proofs. So these proofs were very difficult for humans to understand.   

The second issue: Andreas Thom, a mathematics professor at the Dresden University of Technology, said he suspected OpenAI stole his results. Dr Thom claims that he and his collaborator, Gabor Kun, had used ChatGPT to discuss an unusual approach to solving one of those problems. Astra adopted that precise approach.  

In early September, OpenAI announced an earth-shattering result. Astra’s successor, a more powerful LLM, had found some answers to the Navier-Stokes problem. The Navier-Stokes Equations (NSE) are foundational to the study of fluid dynamics. These were independently derived in the early 19th century by Claude-Louis Navier and Sir George Stokes.  

The equations are used extensively by physicists and engineers. But they can give some apparently absurd results, such as fluids being accelerated to infinite velocity, and occupying zero volume. This “blowup” or “singularity”, as it is described, is considered one of the seven biggest open questions in maths. In 2000, the Clay Mathematics Institute offered a $1 million prize for definitive answers.  

OpenAI published a 166-page proof that claimed such a blowup could indeed happen under certain conditions. That proof has not been definitively reviewed and accepted by mathematicians yet. OpenAI says it burned 88 hours of computation involving 10,000 AI agents to generate and verify the result.  

The result generated an immediate charge of plagiarism from Tristan Buckmaster of New York University, and his collaborator, Levent Alpöge, a mathematician who works for OpenAI’s rival firm, Anthropic. Dr Buckmaster said the duo was using an unusual approach to the NSE. They had been sharing their work on OpenAI’s Codex. Buckmaster-Alpöge also published a partial result of a simplified version of the NSE (the Euler Equations) just before OpenAI released its paper.  

When Dr Buckmaster alleged plagiarism, he also said that OpenAI had told him it would be happier if he published his results without crediting the Anthropic employee (who was working on this in his private capacity). OpenAI was defensive in response, saying that “while unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models??”.   

Within a few days, the mathematics community responded with an open letter signed by 25 Fields Medalists. (The Fields is considered the equivalent of a mathematics Nobel Prize and is arguably more stringent than the Nobels in its award criteria). 

The letter said there was a “severe misalignment” between the goals of AI companies using brute force to generate results and human mathematicians.  It claimed the AI approach doesn’t advance “conceptual understanding and insight”. Fears of plagiarism may also retard the usage of AI by the academic community.  

This isn’t just an outpouring of sour grapes and existential dread from a bunch of admittedly high IQ people, though there may be some of that. The ways in which AI proves mathematical propositions don’t necessarily advance conceptual insight. Plagiarism apart, AI-derived maths proofs may be a black box, as are many AI results in areas like financial modelling.  

Indeed, this has been an issue since computers started being used for proving mathematical propositions. A simple example of a computer-derived proof that did not advance insight was the famous “Four-colour Problem” (FCP).  

In the 1850s, Francis Guthrie looked at a two-dimensional map of the United Kingdom and noticed that no more than four colours were required to colour it. It was definitively proved in the 1970s that indeed, any and every two-dimensional map can be drawn using only four colours. Kenneth Appel and Wolfgang Haken used a computer to colour every possible map, using only four colours. The proof was incontrovertible and accepted. But we still don’t know “why”, and without that understanding, it is hard for human beings to develop further insights into the FCP.  

 The 1976 computer was laughably primitive compared to today’s hardware and algorithms. But this is what mathematicians are talking about. A proof without the “why” doesn’t help advance human understanding. It is something chess players complain about when they try to understand the way computer engines play fantastically accurate chess. 

Moving on, AI is going to continue tackling maths and it will derive more fabulous results. The maths community and their step-relations, the computer scientists, will have to find ways to translate such black-box results into humanly-usable insights. That will change the way mathematics is done forever, and there will be a lot of bad blood before it happens. 

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 The writer is a New Delhi-based independent journalist