Claude-Louis Navier was an engineering professor in Paris in the 1820s, where he was known for his deep analytical knowledge of bridge-building, but also blamed for relying too much on theory when a bridge he designed cracked and had to be dismantled.
George Gabriel Stokes was the child of a church rector from the small Irish village of Skreen, whose talent for doing sums eventually led him, in 1849, to become the Lucasian professor of mathematics at Cambridge University, a position once held by Isaac Newton.
Born 34 years apart in different countries, the two never met. Yet, their names are forever entwined thanks to a set of equations they separately formulated, spawning a notoriously pertinacious math problem.
This week, the Navier-Stokes existence and smoothness problem, as it is known, made headlines after it was declared cracked (like a puzzle, not a bridge) by artificial intelligence.
If verified, the breakthrough is both spectacular and controversial. Mathematicians witnessing first-hand the transformation of their profession by AI are finding it hard to keep up with the growing capabilities of the technology. For some, this week’s news is another sign that, instead of competing with AI, their skills may be better put toward managing its risks.
“I think a lot of us are constantly recalibrating, almost minute by minute, given the pace we’re going at here,” said Deanna Needell, a Vancover-based professor of mathematics at UCLA and the University of British Columbia.
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A diagram of a rapidly spinning vortex that grows more elongated toward the centre. This scenario lies at the heart of OpenAI’s demonstration of a way in which the Navier-Stokes equations, which describe fluid flow, can fail.Supplied
The Navier-Stokes problem concerns the motion of fluids in three dimensions, and whether or not there are circumstances where the equations used to describe them can fail.
Last Tuesday, OpenAI, the tech company behind ChatGPT, said that its AI agents, after 88 hours of computation, had found a situation where this occurs.
This immediately captured media attention, in part because Navier-Stokes is one of seven “millennium problems” that were listed in 2000 as outstanding challenges by the Colorado-based Clay Institute of Mathematics. Each problem is attached to a US$1-million prize for anyone who can provide a verified solution. Prior to this week, only one had been solved, in 2003.
No sooner had OpenAI made its announcement than Tristan Buckmaster, a mathematician at New York University, revealed that he had been working with a collaborator, Levent Alpöge, at Anthropic, an OpenAI rival, and they were closing in on their own AI-assisted solution to the problem.
The situation is complicated by the fact that Dr. Buckmaster used OpenAI’s tools for some his research, though OpenAI has said it’s not possible that its model somehow found and used this work to gain the lead in the discovery. Then there’s the question of whether such advancements should be made public in the traditional way, through publication in a peer-reviewed journal.
It’s too soon to know how credit and prize money (if any) will be apportioned, or how history will view the matter. What is not in doubt is that AI has ascended to the highest levels of mathematics with remarkable speed – and likely changed the field forever.
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The prospect that it could soon climb higher than the brightest human minds is both fascinating and scary.
“I expect most of our research projects ongoing in the world right now to be solved fairly quickly,” said Jacob Tsimerman, a University of Toronto professor. In July, he became the first Canada-based winner of the prestigious Fields Medal, known unofficially as Nobel Prize of mathematics. He added that machines may also “autonomously go on and do math that we haven’t even thought of yet.”
It’s fair to say that many people think of advanced mathematics as the hardest thing the human brain can do. At the professional level, it requires raw intellectual power and years of training to make a meaningful contribution.
One might therefore conclude that mathematics would be among the domains least susceptible to competition from AI. Sure, ChatGPT can write a greeting card poem, but solve a millennium problem?
In fact, Dr. Tsimerman said, what makes math hard for humans is also what makes it amenable to exploration by AI. Unlike the world of everyday experience, the structure of mathematics is formal and logical. Theorems can be proven true or false, with no grey area in between. And, most importantly, from the days of Pythagoras to the present, math is something we perform in an idealized, abstract environment that AI is well-equipped to operate in.
“Math is a closed loop. You can do it just by thinking,” Dr. Tsimerman said.
It was Dr. Tsimerman who instigated the second big math story of the week. A few months ago, after winning the Fields Medal, he revealed he would be working with OpenAI on safety in the use of artificial intelligence – a topic that has become his primary focus. On Tuesday, Dr. Tsimerman said he was launching a new centre dubbed the Mathematical AI Safety Institute in California’s Bay Area, where OpenAI is based.
“Specifically, we’re going to be developing theories with the intention of building safe superintelligence,” Dr. Tsimerman said. The point, he said, is not simply to program AI to avoid bad behaviour, but to make it think in ways that align with human well-being, and follow instructions “not just to the letter, but also in spirit.”
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This is a tricky and nuanced pursuit. Until now, Dr. Needell said, AI safety issues have often been regarded as a “masking” problem, which programmers try to address by building guardrails to prevent AI from revealing dangerous knowledge it may have acquired.
Such a stopgap is problematic, however, because it means the knowledge is still in the machine and discoverable in principle. What may be more effective is adjusting the algorithms so they are unable learn certain things in the first place.
Dr. Tsimerman said he hoped current approaches to AI safety will come to be regarded as clumsy early attempts as efforts progress.
The need for such progress was starkly underscored on Thursday, when Anthropic revealed that it had blocked unnamed actors who were trying to use its large language model Claude for research that could lead to a biological weapon.
“I definitely think the work we’ll be doing will be applicable to that problem,” Dr. Tsimerman said. But he added that while the new institute would be concerned with the mathematical side of AI safety, experts in other fields and society at large need to press policy-makers to make those safeguards a priority.
“This is very much an all-hands-on-deck situation,” he said.
This is why an AI victory over the Navier-Stokes problem may pose other challenges. If young researchers decide there is little point in trying to tackle problems that AI will soon master, there will be fewer experts left who can evaluate whether AI is doing something useful or safe.
Davide Gaiotto, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ont., said he can still recall the feeling he experienced at the age of 25, when he realized he had moved from “learning stuff to creating new knowledge.”
He said a key concern in the field now is whether today’s graduate students in math-driven fields will get to feel the same.
If there is a glimmer of optimism, Dr. Needell said, it may lie with what humans possess that machines still lack: real-life experiences and interactions that help to determine what we pay attention to.
“I think there is a human contribution in deciding what is important, what we should be thinking about,” she said. “And I think that is still true for mathematics.”