“If a human did this, it would be an instant Fields Medal, no questions asked.”
That’s how Alex Kontorovich, chair of the department of mathematics at Rutgers, responded to an A.I.-generated proof that OpenAI, the artificial intelligence giant, released on Tuesday.
And that single proof — worthy, in Dr. Kontorovich’s assessment on social media, of arguably the highest honor in the field of mathematics — was just one among more than 350 findings that the firm released that day, all at once.
Not all of them were quite so remarkable. But Martin Bridson, a mathematician at Oxford and the president of the Clay Mathematics Institute, which established the coveted Millennium Prize Problems, called OpenAI’s release “breathtaking.”
“Until very recently, it would have been impossible to imagine that the frontiers of mathematics could move so far in one day,” Dr. Bridson said.
If there had been any doubt that the leaps achieved by artificial intelligence this year would transform the discipline of mathematics, the tens of thousands of pages let loose on Tuesday have put that doubt to rest. The puzzles apparently conquered by OpenAI include crucial open questions in nearly every subfield of math, including algebra, number theory, theoretical computer science, mathematical logic and topology.
None of the proofs fully resolve any of the five remaining Millennium Prize Problems, which were designated at the turn of the century as a way of celebrating mathematics and its vast frontier of mysteries. But the trove of findings seemed to include results that make tangential or related progress on all of them. Perhaps the most significant result — the one Dr. Kontorovich was responding to — was a proof of the “quasi-Riemann hypothesis,” a conjecture related to the best known of the Millennium problems, the Riemann hypothesis, and a possible steppingstone of sorts to its resolution.
Ken Ono, a professor the University of Virginia and the founding mathematician at Axiom Math, a math-focused artificial intelligence firm, wasn’t sure whether the quasi-Riemann solution would be a springboard for solving the original hypothesis. But he said that, as is expected with the original Riemann hypothesis, an encyclopedia of consequences would flow from this result, providing rich fodder for future breakthroughs.
“I imagine many mathematicians, like me, felt like they got punched in the stomach yesterday,” Dr. Ono said of Tuesday’s turmoil. “But that feeling is quickly giving way to excitement and determination. For those working to push mathematics forward, this is a challenge and an invitation to aim higher.”
(As of Thursday, OpenAI had made a number of updates to the public online repository; withdrawing three papers and making fixes to several others. A spokeswoman said: “Where errors are identified, we will work to correct them promptly and withdraw papers if no fixes can be found. As with other research manuscripts, this is an iterative process.” )
‘Exciting for Many, Frightening for Others, Devastating for Some’
Talking to mathematicians after the release, it is clear that it is a highly conflicting environment, motivating a wild range of reactions.
“The list of problems includes several that were guiding challenges in my own research,” Dr. Bridson said in an email, citing specific results in group theory, topology and algebra. “I have thought hard about all of these problems.”
The solutions announcement, he added, “will be exciting for many, frightening for others, and devastating for some.”
“In all cases,” he said, “the struggle to wrestle human understanding from the machines’ formalities will unleash greater ambition for what we achieved (working with A.I. agents) in this new era of mathematics. Some of our favorite mountains have been conquered, but behind them are bigger mountains that we can tackle with new equipment.”
He added: “We should embrace the power that the machines offer, but we should not be indentured to follow their lead.”
For Bryna Kra, a mathematician at Northwestern University, problem No. 145 on the list was of great interest — “the Rokhlin problem on mixing implying higher order mixing,” which answers a question from the 1940s. “The writing, however, makes it impossible to understand,” said Dr. Kra, former president of the American Mathematical Society, who is among a group of mathematicians who are proactively working to chart a course for the field through this period of “uncertainty and seismic change,” as she described it.
Overall, the results are “amazing,” Dr. Kra said, but now tens of thousands of pages of mathematical results need to be carefully checked. Typically checking and verification are done by peer review. “Peer review is gone in this process,” she said. “So, the question is, what kind of impact will these A.I.-generated results have? This is going to take a long time for the math community to digest.”
“Perhaps the most relevant issue is that by releasing this trove of material in this manner, the companies are in the process of destroying the ecosystem that made it possible for this work,” she said. “While this batch of papers has improved in the citations to the past literature, there are numerous indications that they have benefited from access to many arguments, only some of which are published. The mathematician who wants to make use of these amazing tools to advance their research instead risks giving their ideas away.”
The French mathematician Jean-Pierre Serre, an emeritus professor at the Collège de France in Paris — who has won two of math’s top prizes, the Fields Medal and the Abel Prize, and celebrated his 100th birthday last month — was naturally attracted to finding No. 46 in OpenAI’s list, which solves positively a conjecture he made almost 70 years ago.
But Dr. Serre is conflicted about doing math with A.I. He uses the technology for things like references, and, with the help of friends, for chasing down hunches about errors. “Mathematicians take pleasure in doing maths in two different ways: learning and finding new things,” Dr. Serre said. “Hence a conflict: A.I. increases the first pleasure and lowers the second one. The problem is it may lower the pleasure too much; that is especially serious for young mathematicians.”
‘A Kind of Vertigo’
Indeed, the cataclysmic upheaval will affect students the most, especially graduate students, those on the cusp of becoming a part of this intellectual community.
Thomas Carlson, a Ph.D. student at Montana State University, currently attending a semester-long program at the Simons Laufer Mathematical Sciences Institute (SLMath) in Berkeley, worried about some practical implications of such powerful A.I. systems doing math. “I think we have reached a point where we need to carefully consider the structure, requirements and value of a graduate degree in math,” Mr. Carlson said. “What can be done to ensure that a 2028 graduate has a degree with the same value of that of a 2023 graduate?”
Kai Shaikh, a Ph.D. student in mathematics at the University of Toronto, also currently at SLMath, observed that theorems are easy to use as benchmarks, but that they capture little of the wonderment at the heart of math. “Just like a photograph of a mile marker can’t provide much of the value of actually going on a hike,” he said, “it would seem to me that proofs without stories, or even stories without struggle or mystery, and first glimpses of strange beasts, are not very meaningful or valuable to us humans.”
Yuanning Zhang, a Ph.D. student in mathematics at Northwestern University, visiting at SLMath, said: “What I’m feeling is less a judgment than a kind of vertigo.” If these reported resolutions are verified, this would suggest that A.I.-assisted mathematical research is beginning to operate on an industrial scale.”
Initially, the rumors about these OpenAI results had it that there were solutions to 400 problems, and now the rumors are that further dumps are arriving soon. The number 400 had made Mr. Zhang think of “The 400 Blows,” a coming-of-age film by François Truffaut.
“The film is about a boy who is evaluated, judged and shut away from society, and no one really takes the time to understand him,” Mr. Zhang said. “Towards the end, he runs to the sea, and the film ends on a freeze frame that passes no judgment on whether this is an escape or a dead end. I don’t think that’s a bad picture of where mathematics stands with these results.”
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