Mathematicians have been stunned by the accelerating capabilities of artificial intelligence, which in recent months have unraveled some of the toughest open math problems that had eluded humans.
On Tuesday, OpenAI deluged mathematicians with hundreds of new findings that span a wide swath of topics including algebra, number theory, theoretical computer science, mathematical logic and topology.
The 377 results follow OpenAI’s announcement last month that it had succeeded in cracking the Navier-Stokes equation — one of the so-called Millennium Problems, which were considered so challenging that a $1 million reward was offered for each solution.
The achievement showcased what the most advanced artificial intelligence models are capable of, but it also raised questions over whether OpenAI’s artificial intelligence systems were employing creative thinking or simply completing the final steps of a proof after borrowing ideas from human mathematicians’ work.
Like the Navier-Stokes result, the new mathematical solutions used a more advanced A.I. model that has not been released publicly.
A couple of weeks ago, OpenAI said that it is working with an advisory board of mathematicians that would offer advice on how artificial intelligence companies should communicate A.I.-produced results to the mathematicians. The group is hosted by the Institute for Advanced Study in Princeton, N.J., and is independent from any A.I. company.
Among other suggestions, the advisory board said that all of the prompts to the A.I. agents and the agents’ chains of thought should be released. On Tuesday night, the advisory board released a statement that called the public release “the beginning, not the completion, of the process of human understanding and the incorporation of the work into mathematical knowledge.”
“We want to create standards and practices so that results released from A.I. labs can be understood by mathematicians and can advance the field,” Melanie Wood, a mathematician at Harvard who is a member of the advisory board, said in an email.
Many of the proofs had been checked using Lean, a computer language that verifies that underlying logic.
For 10 of the solutions, the company also provided summaries of how the A.I. model came up with its solution. The average result took about three hours of computing, the company said.
“We have drawn on their advice and public recommendations to inform how we release these results,” OpenAI wrote in a posting on its website.
OpenAI, however, seems to be less interested in a key recommendation of the advisory board, that the A.I. companies should stop testing their proprietary models on advanced mathematical problems. “We want to state clearly from the start: we do not endorse this practice,” the advisory board wrote on Sept. 29.
The A.I. companies have given their models math problems — like those put to high school students during the International Mathematical Olympiad — that had been set up as benchmarks of performance.
But when the models were able to figure those out, the OpenAI team created new benchmarks of unsolved math problems at the cutting edge of mathematical research, and those are among what was released on Tuesday.
Dan Roberts, the research lead at OpenAI, said it was important to test the internal models to produce better tools, and the proofs were a byproduct of that testing.
Tristan Buckmaster, a mathematician at New York University who was working on the Navier-Stokes problem that OpenAI solved, said it remained unclear whether mathematicians using A.I. models like the ones from OpenAI were inadvertently providing the information that allowed A.I. to beat them to the final answer.
“There’s likely to be a bunch of results where they take someone’s work and then take it to completion,” Dr. Buckmaster said in an interview on Monday evening.
With so many results released at once, “I don’t think they’ve done their sort of due diligence at all,” he said.
Mathematicians are struggling with whether the accelerated pace of solving math problems helps or hurts their field if understanding the solutions lags behind.
“I think at the heart of this issue is that humans have two competing natures: a tendency to compete and a capacity to appreciate beauty,” said Kai Shaikh, a graduate student in mathematics at the University of Toronto. “To me this seems to be a case of the former attempting to strangle the latter.”
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