AI's Takeover of Math Is Underway

OpenAI has demonstrated that artificial intelligence is capable of resolving long-standing mathematical challenges, sparking both excitement among experts regarding future possibilities and concern about the trajectory of their field.

James Maynard, a distinguished professor at the University of Oxford and recipient of the prestigious Fields Medal, has dedicated a significant portion of the past year to "soul searching." He conveyed to The Verge his profound contemplation of the future of mathematics, a discipline traditionally characterized by its deliberate pace, as it now grapples with the rapid advancements of AI.

Just days prior to this discussion, OpenAI announced it had successfully generated solutions to ten complex mathematical problems, some of which had eluded academic resolution for decades. Similar to generative AI applications in text, image creation, or scientific and medical research, this technology learns intricate patterns and connections from vast datasets. It then leverages this acquired knowledge to synthesize novel outputs. In the context of mathematics, this translates to innovatively combining existing results, methodologies, and tools to address problems, occasionally forging links between disparate fields or resurrecting concepts embedded deeply within academic literature.

For Maynard and other mathematicians interviewed by The Verge, this announcement has intensified a complex array of emotions concerning the future direction of their profession. There is palpable enthusiasm for the potential acceleration of mathematical discovery, yet concurrently, apprehension, and in some instances, despair, about the implications for those who have dedicated their lives to this pursuit and for future generations of mathematicians. Few observers dispute that a profound transformation is already in progress.

Indeed, few doubt that a profound upheaval is already underway.

The problems tackled by OpenAI, utilizing an advanced, unreleased model named Astra, spanned a broad spectrum of mathematical domains, from highly abstract concepts to questions with practical applications. One notable achievement involved determining the optimal packing density of spheres in dimensions higher than three, a problem with direct relevance to efficient data encoding and transmission. Another pushed the boundaries of error-correcting codes, vital for recovering information from noisy signals. A third breakthrough resolved two enduring questions concerning the complexity thresholds at which structural patterns emerge within connected networks. Further results addressed challenges in quantum game theory and target identification within high-dimensional grids, with potential implications for post-quantum cybersecurity techniques.

Among the most striking results was the confirmation of the existence of non-sofic groups—infinite mathematical structures that, in simpler terms, cannot be approximated by finite ones. The question of their very existence had been an open problem for decades. This particular result also garnered attention for another reason: a controversy arose regarding the allocation of credit between OpenAI's AI and the human mathematicians whose recent foundational work was integral to the solution.

Francesco Fournier-Facio, a mathematician at the University of Cambridge, informed The Verge that he and his colleagues in the field believed OpenAI's initial public statement understated the crucial contributions of researchers Andreas Thom and Gábor Kun, whose recent studies had established the essential groundwork for this specific outcome.

OpenAI's initial announcement claimed 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." However, this statement was subsequently altered to declare that it was sharing "results, each of which resolves or makes substantial progress on a long-standing open problem." The webpage provided no explicit correction note or explanation for this revision.

Gábor Kun, a researcher at the Alfréd Rényi Institute of Mathematics in Hungary, recounted to The Verge that OpenAI had contacted him via email shortly before publishing its findings. He described the sweeping language of the original announcement as "rather comical," especially since the accompanying, more detailed research paper "clearly said that it builds on my results from 2016 and 2019," the latter coauthored with Thom. Kun characterized the original wording as "rather sloppy."

"It’s rather sloppy."

Following publication, OpenAI re-established contact with Kun. While he chose not to disclose the full email exchange, he did share an excerpt with The Verge in which an OpenAI mathematician explained that the initial wording was intended to refer to other results within the collection. "It was not intended to suggest that there had been no progress on this problem," the email reportedly stated. "We certainly agree that the argument relies crucially on your work." The mathematician also indicated that they would request a revision of the announcement's wording.

OpenAI spokesperson Laurance Fauconnet confirmed to The Verge that the online post was indeed 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 expressed wonder whether similar oversights might have occurred in the other results presented, given that those fell outside his specific areas of expertise.

A comprehensive evaluation of OpenAI’s results is inherently complex. Mathematics has become so specialized that few individual researchers possess the breadth of expertise required to thoroughly scrutinize all the diverse fields that Astra's solutions touched upon. OpenAI released over 250 pages of papers detailing the solutions, complemented by an additional 60 pages explaining "how the ideas came together," and each result was formally verified using Lean, a software for proof checking. While many mathematicians interviewed by The Verge admitted they could not personally assess some, or even any, of the solutions, there was a widespread consensus that OpenAI's achievement carries substantial scientific weight.

The problems, which OpenAI stated were solved by an internal iteration of its "next major model," Astra, were by no means trivial. Maynard noted that these were the types of questions that mathematicians and computer scientists had invested significant time in, repeatedly failing to find resolutions.

"There’s a general feeling that [solving] one of these 10 problems would get you a job in academia," remarked Yang-Hui He, a fellow at the London Institute for Mathematical Sciences. He recently returned from a four-week AI and mathematics research conference in South Korea, where he observed a widespread sentiment that the past six months had witnessed a "phase transition," with AI now delivering genuinely 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, OpenAI astonished mathematicians by announcing that an unnamed internal model had successfully cracked a conjecture by Paul Erdős, which had eluded mathematicians for nearly a century. In July, Harvard mathematician Levent Alpögetweeted that Anthropic’s Claude Fable 5 had disproved the notoriously difficult Jacobian conjecture with a small counterexample, overturning decades of efforts to prove its validity. These are among the latest instances of mathematically significant problems succumbing to AI.

Maynard recalled that this level of impact was not always the norm. He noted that, with a few exceptions, earlier AI breakthroughs in mathematics often garnered considerable publicity but typically involved problems that had not attracted serious attention from researchers. The accelerating pace of this change has taken many researchers by surprise, prompting the field to scramble for a response and leading some to question whether mathematics can persist in its current form.

Many mathematicians interviewed by The Verge appeared to still be processing their thoughts on these developments, expressing surprise, and even shock, at the speed of change. Despite considerable excitement, He's impression from the South Korean conference and the broader field is that many are downplaying the significance of recent advances in an effort to "keep calm" about the rapid evolution.

Financial implications lie at the core 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," stated Colva Roney-Dougal, a professor at the University of St Andrews in Scotland. She added, gesturing to a whiteboard filled with her work, "Maths is a cheap discipline typically. 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."

This traditional system could face disruption even if the costs are modest by AI industry standards. OpenAI estimates that generating Astra’s ten solutions would have incurred approximately $2,000 in tokens at the current API prices for its Sol model. However, researchers interviewed by The Verge suggested the actual cost was likely significantly higher, depending on the number of problems and attempts preceding the successful ones. OpenAI did not provide further details when asked about the process of compiling the final list. Nevertheless, for a field accustomed to operating on minimal budgets, even the advertised price could prove prohibitive. Roney-Dougal expresses concern that researchers at smaller, less affluent institutions might be entirely excluded from certain areas of research.

Furthermore, there is a pervasive unease regarding the increasing encroachment of commercial interests into a field that has largely operated with open access. Even as mathematics has become more computational, many essential tools relied upon by researchers are open source and freely available. In contrast, the most sophisticated models from companies like OpenAI and Anthropic are proprietary, with access tightly controlled. While both companies offer programs providing free access to academic researchers, this access is not universal, and few of the researchers interviewed by The Verge had been able to utilize the most advanced systems. Both Maynard and Roney-Dougal voiced hope that open-weight models could eventually bridge this gap, enabling mathematicians to access powerful tools without exclusive reliance on a select few large AI corporations.

Access is not the sole point of contention. Several researchers consulted by The Verge questioned whether the values of AI companies truly align with those of the mathematical community. They argued that with products to sell and immense valuations to justify for impending IPOs, companies are strongly incentivized to exaggerate and hype their contributions, while simultaneously downplaying the human scholarship that forms the bedrock of those results.

"That’s the bit I’m most unhappy about at the moment," Roney-Dougal asserted. "They’re treating our discipline as an advertising playground."

"They’re treating our discipline as an advertising playground."

These concerns extend beyond the researchers directly interviewed. In June, mathematicians collectively published the Leiden Declaration, a set of principles advocating for the responsible application of AI in mathematics. This declaration has received endorsement from the International Mathematical Union and garnered over 3,400 signatures. It implores policymakers, governments, the media, and other stakeholders to resist "the hype" generated by companies that "overstate the capabilities of their products." The underlying fear is that exaggerated claims could have tangible consequences for the field, potentially convincing funders and governments that human mathematicians are less essential than they truly are.

OpenAI has been accused of precisely this behavior with its ten Astra advances, particularly through its minimization of the contributions from Kun and Thom. Kun described his "feelings are quite ambivalent." While gratifying to witness his work contribute to the resolution of a significant problem, he also expressed a wish that he had been the one to complete it. The attention he received, along with numerous congratulations, was something he might not have otherwise experienced. His jovial remark to well-wishers was that he would become a "very famous unemployed" person.

Fournier-Facio's sentiments are considerably less ambivalent. "Most people will just look at the OpenAI announcement and take it at face value," he stated. He argued that few individuals, especially journalists or policymakers operating under time constraints, possess the time, expertise, or inclination to delve into hundreds of pages of technical papers to comprehend the human effort behind the headlines. "They’re just choosing the narrative that benefits them most," he contended. "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."

Fournier-Facio acknowledged that an element of unfairness already exists in how credit is apportioned within mathematics. "The person that does the last step gets most of the credit, right?" he mused. He drew an analogy between mathematical research and constructing a pyramid, where generations of work accumulate beneath the individual who ultimately places the final stone. He speculated that perhaps there would not have been such an insistence on crediting Kun and Thom if a human had solved the problem. However, with AI now taking that final step, he worries that everyone else involved will mistakenly "be seen as useless."

Even worse, he added, "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."

This particular fear becomes especially troubling when mathematicians consider the potential impact of AI on the next generation of researchers. Andras Juhasz, an Oxford professor, noted that many of the problems large language models (LLMs) are beginning to solve are precisely the kind that graduate students "typically work on." These problems, while not always glamorous, are crucial for developing the essential skills, intuition, and habits required to become successful researchers. Johannes Schmitt, a researcher at ETH Zurich, expressed concern that these students now risk being "scooped" by someone who swiftly solves these problems using AI.

The ramifications are already starting to be felt. Juhasz observed that project work has "become a problematic form of assessment" for undergraduates, as AI systems grow increasingly capable of completing such tasks. While these tools might enable students to find answers more quickly, they risk undermining the holistic mathematical understanding that is typically cultivated through diligent struggle. Maynard, meanwhile, is already contemplating how to future-proof research projects for students. He highlighted the challenge: "If the standard for a publishable paper is something that an AI can’t do, particularly when a PhD 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