A single release of hundreds of results

OpenAI published 377 new solutions to open mathematical problems, according to The Atlantic and Der Standard, which place the release on Tuesday, while Frankfurter Allgemeine places it on Wednesday. Der Tagesspiegel counts more than 700 manuscripts covering about 370 open problems, and Frankfurter Allgemeine counts 722 manuscripts. The Atlantic reports that about a third of the proofs address conjectures famous across mathematics, another third are significant within their subfields, and the remainder are more minor results or variants. Hundreds of people had spent dozens of years on some of these questions, the magazine notes, and the results came out in a single day. The sequence of events is shown below.

The Navier-Stokes claim and the dispute

On 8 September, OpenAI announced that a swarm of 10,000 AI agents had solved one of the seven Millennium Prize Problems, the Navier-Stokes equations, which describe flows such as wind and water, in 88 hours. The US Clay Mathematics Institute offers one million dollars for each Millennium Prize Problem solved. Mathematician Tristan Buckmaster accused the company of having ChatGPT appropriate his solution path after he had worked with the AI, and OpenAI rejects that claim. Frankfurter Allgemeine reports that some of the new results represent progress toward the Riemann hypothesis, the Hodge conjecture and the Birch and Swinnerton-Dyer conjecture, three of the six Millennium problems still open. The same paper reports that a proof of the Unique Games Conjecture was published on GitHub.

Verification and withdrawn papers

Verification is the open question. The Atlantic reports that it is too early for anyone to have verified the OpenAI work in detail, but several mathematicians told the magazine that at least some papers appear not to prove what they claim to prove. OpenAI has withdrawn three of its papers. The company also said the model had attempted about 3,600 further problems for which it did not release successful proofs, and it promised to pay mathematicians to study AI-produced results, without specifying the funding. An OpenAI spokesperson told The Atlantic by email:

We expect mistakes to be quite rare.

Mathematicians' reaction

Yang-Hui He of the London Institute for Mathematical Sciences at the Royal Institution spoke shortly after the Navier-Stokes proof was published:

The past six months have been the most dramatic in the entire history of mathematics since the time of Euclid.

Scott Aaronson, a prominent computer scientist, titled his blog post the day after the release "The Mathocalypse", according to Frankfurter Allgemeine. Hugo Duminil-Copin, who received the 2022 Fields Medal, wrote in an essay published before OpenAI's September announcement that AI proofs "nuke the mathematical landscape, making it increasingly difficult to inhabit after each blast." Il Sole 24 ORE reports that 25 Fields Medal winners signed a protest letter after the Navier-Stokes announcement. The Atlantic describes the release as a mathematical carpet-bombing, with hundreds of people having spent decades on some of the results.

Transparency and what comes next

After the controversy, OpenAI created a public GitHub repository, according to Il Sole 24 ORE, which reports more than 700 papers already posted, Lean formalizations of many proofs, ten documents summarizing the model's reasoning and an estimate of computing resources. OpenAI cites three hours of ChatGPT Pro reasoning for each result. The step followed advice from an independent advisory group on mathematics and AI at the Institute for Advanced Study, which OpenAI set up to verify its AI work. The spokesperson wrote:

We don't believe the future of mathematics is set. We're working with the math community to navigate the future collaboratively.

Il Sole 24 ORE describes the repository as a public archive of OpenAI's AI-assisted mathematics output.