The AI takeover of mathematics has begun | The Verge
Overview
Tech Expand Amazon Apple Facebook Google Microsoft Samsung Business See all tech
Reviews Expand Smart Home Reviews Phone Reviews Tablet Reviews Headphone Reviews See all reviews
Details
Science Expand Space Energy Environment Health See all science
Entertainment Expand TV Shows Movies Audio See all entertainment
Policy Expand Antitrust Politics Law Security See all policy
Gadgets Expand Laptops Phones TVs Headphones Speakers Wearables See all gadgets
Verge Shopping Expand Buying Guides Deals Gift Guides See all shopping
Streaming Expand Disney HBONetflix You Tube Creators See all streaming
Transportation Expand Electric Cars Autonomous Cars Ride-sharing Scooters See all transportation
AIClose AIPosts from this topic will be added to your daily email digest and your homepage feed. Follow Follow See All AI
Posts from this topic will be added to your daily email digest and your homepage feed.
Report Close Report Posts from this topic will be added to your daily email digest and your homepage feed. Follow Follow See All Report
Posts from this topic will be added to your daily email digest and your homepage feed.
Tech Close Tech Posts from this topic will be added to your daily email digest and your homepage feed. Follow Follow See All Tech
Posts from this topic will be added to your daily email digest and your homepage feed.
Open AI showed that AI can tackle long-standing problems in mathematics. Experts are excited about the possibilities — and worried about what comes next for their field.
If you buy something from a Verge link, Vox Media may earn a commission. See our ethics statement.
Posts from this author will be added to your daily email digest and your homepage feed.
If you buy something from a Verge link, Vox Media may earn a commission. See our ethics statement.
Posts from this author will be added to your daily email digest and your homepage feed.
Mathematician James Maynard has spent a lot of time this past year “soul searching.” A professor at the University of Oxford and winner of the prestigious Fields Medal, Maynard told The Verge he’s been grappling with the future of his field as the traditionally slow-moving discipline hurries to adapt to AI.
Days before we spoke, Open AI revealed it had produced the solutions to 10 long-standing mathematics problems, some of which had confounded academics for decades. Like generative AI used to produce text and images or propose ideas in science and medicine, the technology learns patterns and connections from the vast amount of material it’s trained on and uses them to create something new. Applied to mathematics, that can mean combining known results, methods, and tools in new ways to attack a problem, sometimes drawing links between disparate fields or resurfacing concepts buried in academic literature.
For Maynard and other mathematicians The Verge spoke to, the announcement has added to a complex swirl of emotions about where their field is headed. There is palpable excitement at the prospect of accelerating mathematical discovery — but also apprehension, and in some cases despair, about what this could mean for the people who have dedicated their lives to the pursuit, and for the generations of future mathematicians who will follow them. Few doubt that a profound upheaval is already underway.
Few doubt that a profound upheaval is already underway.
The problems Open AI solved, using an advanced unreleased model known as Astra, spanned a wide range of mathematical fields, from the highly abstract to questions with practical implications. One breakthrough concerned how tightly spheres can be packed in more than three dimensions, a problem linked to how efficiently data can be encoded and transmitted. Another pushed the limits of error-correcting codes, which can help recover information from noisy signals. A third resolved two long-standing questions about how complex connected networks can become before structural patterns emerge. Other results on the list tackled problems in quantum game theory and the search for targets inside high-dimensional grids, with implications for techniques used in post-quantum cybersecurity.
One of the most attention-grabbing results concerned the existence of non-sofic groups, infinite mathematical structures that, roughly speaking, cannot be approximated by finite ones. Whether such structures existed at all had remained an open question for decades. It was attention-grabbing for another reason, too: a dispute over how much credit belonged to Open AI’s AI, and how much to the human mathematicians whose recent work it built upon.
Francesco Fournier-Facio, a mathematician at the University of Cambridge, told The Verge he and others working in that area believed Open AI’s original announcement minimized the contributions of researchers Andreas Thom and Gábor Kun, whose recent works helped lay the groundwork for the result.
When Open AI first published its announcement, it said it was sharing “results to problems that have been open and have seen no progress on the main result for at least a decade, and in most cases much longer.” It later changed this to say it was sharing “results, each of which resolves or makes substantial progress on a long-standing open problem.” The page contains no correction note or explanation for the change.
Kun, a researcher at the Alfréd Rényi Institute of Mathematics in Hungary, told The Verge Open AI reached out to him by email shortly before publishing to share its findings. He found the sweeping language in the original announcement “rather comical,” particularly as the more detailed research paper attached “clearly said that it builds on my results from 2016 and 2019,” the latter coauthored with Thom. “It’s rather sloppy,” Kun said.
After publication, Open AI contacted Kun again. He declined to share the email exchange in full but read The Verge what he said was an excerpt in which an Open AI mathematician wrote that the language had been intended to refer to other results in the collection. “It was not intended to suggest that there had been no progress on this problem,” the email read, he said. “We certainly agree that the argument relies crucially on your work.” The mathematician added that they would ask for the wording of the announcement to be revised.
Open AI spokesperson Laurance Fauconnet confirmed to The Verge that the post was subsequently updated. “We updated the language to better reflect the prior research these results build upon. Although the question of whether non-sofic groups exist had remained open for decades, our sofic group proof relies on important mathematical work published more recently, and we wanted to ensure those contributions were properly acknowledged.”
Kun said he wondered whether similar oversights might have been made in the other results, which were outside his areas of expertise.
Assessing Open AI’s results more broadly is complicated. Mathematics has become so specialized that few researchers are fully equipped to scrutinize all of the fields Astra touched upon. The company released more than 250 pages of papers laying out the solutions, along with 60 more pages describing “how the ideas came together,” and certified each result with Lean, software for verifying mathematical proofs. While many of the mathematicians The Verge spoke to said they couldn’t personally evaluate some or even any of the solutions themselves, all said there is a broad consensus that there seems to be real weight behind Open AI’s achievement.
The problems, which Open AI claimed were solved by an internal version of its “next major model,” Astra, were not trivial. Maynard said they were the kind of questions mathematicians and computer scientists had spent serious time thinking about, and repeatedly failed to resolve.
“There’s a general feeling that [solving] one of these 10 problems would get you a job in academia,” said Yang-Hui He, a fellow at the London Institute for Mathematical Sciences. He had just returned from a four-week AI and mathematics research conference in South Korea, where he said many felt there had been something of a “phase transition” over the past six months, with AI producing genuine and meaningful advances.
Many felt there had been something of a “phase transition” over the past six months, with AI producing genuine and meaningful advances.
In May, Open AI stunned mathematicians when it announced that an unnamed internal model had cracked a conjecture by Paul Erdős that had eluded mathematicians for the better part of a century. In July, Harvard mathematician Levent Alpöge tweeted that Anthropic’s Claude Fable 5 had disproved the fiendish Jacobian conjecture with a tiny counterexample, overturning decades of efforts to prove it was true. They are among the latest examples of problems mathematicians seriously care about falling to AI.
Maynard recalled that, until recently, this was not always the case. With some notable exceptions, he said AI breakthroughs in mathematics generated a lot of publicity while involving problems that had often attracted little serious attention from researchers. The speed at which this is changing has caught many researchers off guard, leaving the field scrambling to work out how to respond, and some wondering whether it can continue to survive in anything like its current form.
Many of the mathematicians The Verge spoke to seemed to still be working out what they thought about it all, while expressing surprise, even shock, at the speed of change. There was plenty of excitement, but He said his impression from the conference in South Korea, and from the field more broadly, is that many in the field are downplaying the significance of recent advances in a bid to “keep calm” about how quickly things are moving.
Money is at the heart of many of these concerns. “It’s not quite clear whether our universities are going to be willing to pay that much for our theorems,” said Colva Roney-Dougal, a professor at the University of St Andrews in Scotland. “Maths is a cheap discipline typically,” she said, gesturing to a whiteboard covered in her work. “Most of the time I don’t even bother getting a research grant. I don’t need one. I just get on with my job.”
That system could break down even if costs are relatively modest by AI standards. Open AI estimates that generating Astra’s 10 solutions would have cost around $2,000 in tokens at the current API prices for its Sol model. Researchers The Verge spoke to said the true cost was likely considerably higher, depending on how many problems and attempts preceded the successful ones. Open AI did not elaborate when asked about how the final list was assembled. But for a field used to operating on a shoestring budget, even the advertised price could be too much. Roney-Dougal fears researchers at smaller and less wealthy institutions could be locked out of some research entirely.
There is also a broader unease about the growing intrusion of commercial interests into a field that has largely operated in the open. Even as mathematics has become more computational, many tools researchers rely on are open source and freely available. The most capable models from companies like Open AI and Anthropic are proprietary and access is tightly controlled by the companies. While both have programs offering free access to academic researchers, that access is far from universal, and few of the researchers The Verge spoke to had been able to use the most sophisticated systems. Both Maynard and Roney-Dougal expressed hope that open-weight models could eventually close that gap, giving mathematicians access to tools without having to rely on a handful of big AI companies.
Access isn’t the only concern. Several researchers The Verge spoke with questioned whether the values of AI companies align with those of the mathematical community. With products to sell and enormous valuations for impending IPOs to justify, companies have every incentive to hype and exaggerate their contributions, they said, while underselling the human scholarship those results rely on.
“That’s the bit I’m most unhappy about at the moment,” Roney-Dougal said. “They’re treating our discipline as an advertising playground.”
“They’re treating our discipline as an advertising playground.”
Those concerns extend beyond the researchers The Verge spoke to. In June, mathematicians published the Leiden Declaration, a set of principles for the responsible use of AI in mathematics that has been endorsed by the International Mathematical Union and signed by more than 3,400 people. It urges policymakers, governments, the media, and other groups to not buy into “the hype” created by companies who “overstate the capabilities of their products.” The fear is that exaggerated claims will have real consequences for the field, convincing funders and governments that human mathematicians are less necessary than they actually are.
Open AI has been accused of doing just that with its 10 Astra advances, not least through its glossing over the contributions of Kun and Thom. Kun said his “feelings are quite ambivalent.” It was gratifying to see his work prove useful in resolving a significant problem, even if he wished he had been the one to finish it, and it brought him attention he otherwise might never have received, along with plenty of congratulations. His joke to well-wishers: He will be a “very famous unemployed” person.
Fournier-Facio feels considerably less ambivalent. “Most people will just look at the Open AI announcement and take it at face value,” he said. Few people, he argued, will have the time, expertise, or inclination to dig through hundreds of pages of technical papers to understand the human work behind the headline, particularly journalists or policymakers working quickly. “They’re just choosing the narrative that benefits them most,” he said. “It’s a lot more impressive to say that an AI system came up independently with something that humans have done nothing on for 10 years. It’s a lot less sexy to say that this is a kind of building on ideas from the past 10 years from humans and combining them in a clever way.”
“It’s a lot more impressive to say that an AI system came up independently with something that humans have done nothing on for 10 years. It’s a lot less sexy to say that this is a kind of building on ideas from the past 10 years from humans and combining them in a clever way.”
There is already an element of unfairness in how mathematics assigns credit, Fournier-Facio acknowledged. “The person that does the last step gets most of the credit, right?” He likened mathematical research to building a pyramid, with generations of work accumulated beneath the person who finally places the last stone. Maybe there wouldn’t have been this much insistence on making sure Kun and Thom were credited if it were a human solving this problem, he said. But with AI taking that final step, he worries that everyone else will wrongly “be seen as useless.”
Worse still: “In the case of humans, the human that puts the last stone in is not necessarily going to steal the job of all the people that built the pyramid.”
That fear is especially troubling when mathematicians consider how AI will impact the next generation of researchers. Many of the problems LLMs are beginning to solve are precisely the kind that graduate students “typically work on,” said Oxford professor Andras Juhasz. They are not always particularly flashy, but working through them is crucial for developing the skills, intuition, and habits needed to become successful researchers. They now risk being scooped by someone swooping in and solving it with AI, said ETH Zurich researcher Johannes Schmitt.
The effects are already beginning to be felt. Project work has “become a problematic form of assessment” for undergraduates as AI systems become more capable of completing it, Juhasz said. The tools may help students get answers quicker, but risk harming the overall mathematical understanding that comes through struggle. Maynard, meanwhile, is already worrying about how to futureproof research projects for students. “If the standard for a publishable paper is something that an AI can’t do, particularly when a Ph D is typically four years, you’re not trying to come up with a problem that AI can’t do now. It’s AI in four years’ time.”
“Many of the people that I talked with are really, really scared.”
“I’m not that depressed about it,” she said. “But plenty of people are.”
The issue goes beyond how many problems AI can solve. Mathematics does not progress by simply checking problems off a list. A solution can matter most for what comes next. The best results can open up entirely new questions, techniques, or even fields of research. It remains to be seen whether AI can do that. Conversely, it is too early to dismiss the problems falling to AI as simply low-hanging fruit. Fully understanding what these results contribute, and how important they ultimately prove to be, will take time.
So far, Schmitt is unconvinced. He said he has seen little evidence of major AI contributions opening up new areas of thought and inquiry like this, raising the unsettling prospect of an increasingly lopsided endeavor. “We might be heading for a somewhat imbalanced situation, where lots of good and interesting problems get mowed down by AI agents,” he said, without those solutions feeding back into the creation of fruitful new directions for researchers to pursue.
“We might be heading for a somewhat imbalanced situation, where lots of good and interesting problems get mowed down by AI agents.”
Not everyone is so unsettled. The London Institute’s He was strikingly optimistic about what comes next. He said some contraction in the number of people pursuing traditional academic mathematics could even be a “healthy direction,” pointing to an already brutal job market for mathematics graduates. Others saw the potential for AI to broaden who gets to participate in research, allowing undergraduates and researchers without access to expertise concentrated at elite institutions to develop their ideas and potentially produce publishable work that might previously have been beyond their reach. For established researchers, meanwhile, handing off more routine work to AI could free up time to focus on more creative and conceptual parts of mathematics.
For all the fears and hopes, nobody knows where this is going. AI is moving too fast, and the mathematics it is producing is still too fresh to judge what its impact may be. Several researchers worried that the field could be reshaped for the worse by claims about what AI could become before anyone has had time to understand what it actually means.
When asked whether he thought any of Astra’s results were worthy of a Fields Medal, one of mathematics’ highest honors and an award he himself received four years ago, Maynard said his initial impression was that they fell short.
“The ones that I’ve looked at, they have the flavor of being very impressive results,” he said, but not necessarily ones that elevate mathematics at a conceptual level. For now, he said he sees more evidence of existing techniques and methods being pushed further and connected in clever ways than of some profound new mathematical ideas.
“But I think it’s also too early to say,” he said. “Sometimes it takes time and perspective to realize, ‘Oh, here is really a fundamental new idea.’”
At the speed it is going, AI may not give mathematicians much time to figure it out.
Robert Hart Close Robert Hart AI Reporter Posts from this author will be added to your daily email digest and your homepage feed. Follow Follow See All by Robert Hart
Posts from this author will be added to your daily email digest and your homepage feed.
AIClose AIPosts from this topic will be added to your daily email digest and your homepage feed. Follow Follow See All AI
Posts from this topic will be added to your daily email digest and your homepage feed.
Open AIClose Open AIPosts from this topic will be added to your daily email digest and your homepage feed. Follow Follow See All Open AI
Posts from this topic will be added to your daily email digest and your homepage feed.
Report Close Report Posts from this topic will be added to your daily email digest and your homepage feed. Follow Follow See All Report
Posts from this topic will be added to your daily email digest and your homepage feed.
Tech Close Tech Posts from this topic will be added to your daily email digest and your homepage feed. Follow Follow See All Tech
Posts from this topic will be added to your daily email digest and your homepage feed.
Watching Roku’s AI channel is like eating from a trough
Zuckerberg’s yacht was closer, but someone else saved a stranded boat
Anker’s travel charger works in almost any country and is down to $20
Costco’s great Switch 2 console bundle includes over $100 in free stuff
Key Takeaways
- Tech Expand Amazon Apple Facebook Google Microsoft Samsung Business See all tech
- Reviews Expand Smart Home Reviews Phone Reviews Tablet Reviews Headphone Reviews See all reviews
- Science Expand Space Energy Environment Health See all science
- Entertainment Expand TV Shows Movies Audio See all entertainment
- Policy Expand Antitrust Politics Law Security See all policy
