OpenAI has spent the last few years planting flags across the increasingly difficult terrain in mathematics. This week, it claimed one of its biggest prizes yet: a solution to a legendary Millennium Prize problem. In normal circumstances, this would have been celebrated as a historic achievement.<\/p>\n<\/div>\n
Instead, many mathematicians have watched OpenAI\u2019s relentless advance with growing unease. To them, the company appears less like an enthusiastic newcomer than an impossibly well-resourced interloper, charging into problems they have dedicated their lives to studying with little apparent regard for long-standing norms or the consequences for those left in its wake. At the heart of that unease is a sense that OpenAI is doing mathematics for different reasons. Mathematicians want to advance the field. OpenAI wants to win.<\/p>\n<\/div>\n
Mathematics is not normally this dramatic, so how did things get this bad? <\/p>\n<\/div>\n<\/div>\n
This week, The Verge<\/em> spoke with more than a dozen mathematicians, including Tristan Buckmaster and Andreas Thom, who are at the center of recent controversies surrounding OpenAI\u2019s work in the field. Even those skeptical of the most serious allegations described a field shaken by the tech giant\u2019s conduct and fearful of what it might do next in its determination to trounce its rivals.<\/p>\n<\/div>\n Buckmaster has accused OpenAI of failing to adequately explain whether work he did through its tool Codex could have contributed to its recent successes. In a statement to The Verge<\/em>, OpenAI spokesperson Laurance Fauconnet strenuously denied that material from his prompts had played a role: \u201cWe can say categorically that it is impossible for Dr. Buckmaster\u2019s Codex prompts over the last two months to have influenced the system in any way, including training.\u201d<\/p>\n<\/div>\n Buckmaster remains unconvinced. \u201cGiven their behavior up until this point, one should take such statements with great skepticism,\u201d he said.<\/p>\n<\/div>\n Mathematics is not normally this dramatic, so how did things get this bad? A rumor was all it took for tensions to boil over.<\/p>\n<\/div>\n OpenAI says<\/a> it heard some researchers were making progress on Millennium Prize problems and decided to see whether one of its advanced, unreleased models could make headway too. It turns out it could. OpenAI says it took roughly 10,000 agents, tens of millions of dollars of compute, and just 88 hours to find a solution to the Navier-Stokes problem<\/a>, which concerns the flow of fluids.<\/p>\n<\/div>\n The company had also discovered who it was racing against: Buckmaster, an NYU professor, and Levent Alp\u00f6ge, a researcher at one of its fiercest rivals, Anthropic. Among several lines of research, the pair were pursuing Navier-Stokes, though had not yet completed a proof. Some details of what happened next are fiercely contested, but the two sides broadly agree on the basic sequence of events. One thing is particularly clear: Alp\u00f6ge\u2019s involvement was a problem for OpenAI, despite his saying<\/a> it was a \u201cpersonal collaboration\u201d independent of his work with Anthropic.<\/p>\n<\/div>\n Even those skeptical of the most serious allegations described a field shaken by the tech giant\u2019s conduct and fearful of what it might do next in its determination to trounce its rivals.<\/p>\n<\/div>\n<\/div>\n Buckmaster said<\/a> he contacted OpenAI after learning the company had become aware of their progress and was racing toward a solution of its own. He said discussions with OpenAI researcher S\u00e9bastien Bubeck grew contentious and, in his view, threatening, but the company offered a path forward for him \u2014 one that excluded Alp\u00f6ge. Buckmaster said he was offered practically \u201cunlimited compute\u201d to finish his own work, and the opportunity to be the sole author of OpenAI\u2019s paper announcing the breakthrough, which would of course credit its tools.<\/p>\n<\/div>\n \u201cAll I had to do was throw Levent under the bus,\u201d Buckmaster told The Verge<\/em> in a phone interview. He said he flatly rejected Bubeck\u2019s offer, which he viewed as a \u201cbribe,\u201d and also began questioning whether OpenAI may have benefited from his use of Codex, one of the company\u2019s AI tools he had been using to tackle the problem. OpenAI has denied that anyone \u2014 or any agent \u2014 accessed his specific user data, and until its more recent comments acknowledged it could not rule out the possibility data derived from his use of the products was used to improve the model.<\/p>\n<\/div>\n Buckmaster ultimately decided to go public with both his work and his account of OpenAI\u2019s conduct. His office, he said, had been transformed into something of a \u201cwar room,\u201d with colleagues helping scrutinize his mathematics, coordinate outreach, and even get in touch with lawyers.<\/p>\n<\/div>\n Bubeck has rejected Buckmaster\u2019s characterization of the conversations on social media and in an interview<\/a> with The New York Times<\/em>. He acknowledged offering OpenAI\u2019s resources to help Buckmaster complete his own proof or to have him take over the writing of the company\u2019s. Strikingly, Bubeck said OpenAI had made similar arrangements with other mathematicians, though did not identify them.<\/p>\n<\/div>\n But his account nevertheless makes clear that Alp\u00f6ge\u2019s affiliation with Anthropic was a sticking point. \u201cFrom our perspective, how can we have an internal OpenAI project with an Anthropic employee?\u201d he told the Times<\/em>.<\/p>\n<\/div>\n If the goal is to compete in mathematics, there are few bigger trophies than solving a Millennium Prize problem. The seven problems, set out<\/a> by the Clay Mathematics Institute in 2000, are widely considered among the most formidable challenges in the field. Each carries a $1 million bounty for whoever solves it. Many had already endured decades of intense scrutiny by the time the prizes were established. In the quarter-century since, only one \u2014 the Poincar\u00e9 conjecture<\/a>, a topological problem concerning three-dimensional spheres \u2014 has fallen.<\/p>\n<\/div>\n For an AI company looking to prove that its models are the best at mathematics, then, they are irresistible targets. To Buckmaster and many other mathematicians The Verge<\/em> spoke to, that helps explain why OpenAI moved so ferociously when it heard others were closing in \u2014 particularly once a rival AI company appeared to be involved.<\/p>\n<\/div>\n For Buckmaster, the episode reinforced something he already believed strongly from a previous spell collaborating<\/a> with Google DeepMind: \u201cAll these tech people are obsessed\u201d with solving big famous problems and are \u201cobsessed with scooping,\u201d he said. \u201cIts all about competition.\u201d<\/p>\n<\/div>\n In that world, Tristan Buckmaster said there is an intense fixation on prestige, fame, being first, and being seen to be first. \u201cThat\u2019s the only currency,\u201d he said.<\/p>\n<\/div>\n<\/div>\n He said what often gets \u201clost\u201d when companies race to solve famous problems are the mathematicians themselves \u2014 not just the people whose accumulated work makes these breakthroughs possible, but the reasons they do mathematics to begin with. Yes, some may pursue prestige, but most are simply not trophy hunters. Andras Juhasz, a professor of mathematics at the University of Oxford, described mathematics as an elegant discipline that is part science, part art, with many different motivations driving those working there. \u201cOften there is no immediate practical application,\u201d he said. \u201cThey do it because it\u2019s beautiful. They enjoy it. It\u2019s the sense of discovery. It\u2019s natural.\u201d<\/p>\n<\/div>\n Unlike classroom-level mathematical exercises, frontier mathematics rarely has a prescribed route to an answer. Researchers can attack problems from any number of angles, some radically different, which makes the ideas that lead to a solution \u2014 and who developed them \u2014 especially important, perhaps more so than solving a problem itself. Mathematicians care deeply about this lineage because it is how the field expands, with new techniques and methods often proving more consequential than the problem they were designed to solve.<\/p>\n<\/div>\n To Buckmaster, his exchanges with Bubeck typify the chasm that separates the worlds of research mathematics and Big Tech, and highlight the differences between what is considered valuable in research. Reading from notes he took while chatting with Bubeck, he said the OpenAI researcher was visibly taken aback when he rejected the company\u2019s offer to take credit. \u201cI could see S\u00e9bastien\u2019s face. He was shocked when I said I don\u2019t care about the Millennium Prize,\u201d he recalled.<\/p>\n<\/div>\n Buckmaster said he had encountered a similar mentality among tech researchers before. In that world, he said there is an intense fixation on prestige, fame, being first, and being seen to be first. \u201cThat\u2019s the only currency,\u201d he said.<\/p>\n<\/div>\n Buckmaster isn\u2019t the only mathematician to come away from an encounter with OpenAI concerned about the company\u2019s motivations. Andreas Thom, a professor at the Technical University of Dresden in Germany, found himself at the center of a controversy<\/a> last month after OpenAI announced an impressive mathematical result that built heavily on work by him and fellow researcher G\u00e1bor Kun. The company quietly amended its announcement to acknowledge the pair\u2019s contribution without announcing or publicly disclosing the change.<\/p>\n<\/div>\n Thom described the ordeal as \u201cnot a very pleasant experience,\u201d but told The Verge <\/em>he had largely put it behind him until Buckmaster went public. His allegations prompted Thom to revisit an unresolved question<\/a> about OpenAI\u2019s breakthrough: whether conversations he and his colleagues had with ChatGPT about the research could have been used to help improve the models that ultimately cracked the problem he\u2019d spent years working on.<\/p>\n<\/div>\n Only OpenAI has the information needed to answer that question, Thom said. \u201cTo be honest, I suspect that they don\u2019t even know.\u201d The people training the models and using them to produce mathematical results are \u201ca different kind of people,\u201d he said. To him, that\u2019s hardly an excuse for the uncertainty. \u201cBecause it effectively means that they don\u2019t really care, right?\u201d<\/p>\n<\/div>\n \u201cThe prospect of competing with powerful AI companies, whose resources far exceed those available to academic research groups, could make them even more reluctant to pursue ambitious questions.\u201d<\/p>\n<\/div>\n<\/div>\n The tension echoes fights already playing out elsewhere. Writers, musicians, artists, and media companies have all challenged AI companies over systems built from vast stores of human-created work, often without permission, recognition, or compensation. While mathematics may seem a world apart, the underlying question is the same: What do companies owe to the people whose accumulated work they ingested to build their systems?<\/p>\n<\/div>\n In Thom\u2019s case, the question remains unresolved. OpenAI did not respond to The Verge<\/em>\u2019s question on whether data from conversations Thom and his colleagues had with ChatGPT could have contributed to the company\u2019s solution that built on his work.<\/p>\n<\/div>\n More broadly, Thom said he resents what he sees as a failure to recognize the \u201cthe communal effort that this entire community has put into all the research results\u201d underpinning AI\u2019s recent mathematical advances. Companies, he said, \u201care just now using [it] as if it was kind of nothing.\u201d<\/p>\n<\/div>\n \u201cI think there is a certain attitude that I don\u2019t like in that,\u201d he said.<\/p>\n<\/div>\n It\u2019s not that mathematicians are strangers to competition \u2014 researchers care deeply about priority and bitter disputes over who came first litter mathematical history \u2014 but while competition does not preclude cooperation, these are no ordinary competitors. Scooping in mathematics has historically been relatively difficult for obvious reasons: Very few people have the specialized expertise to swoop in on a discovery at speed. AI companies operate on a different scale. Researchers worry they could turn scooping into something of an industrial process mathematicians would have little chance of fighting back against, rapidly spinning up thousands upon thousands of agents and enormous amounts of compute whenever word spreads that a breakthrough is close.<\/p>\n<\/div>\n To Buckmaster, OpenAI could have easily collaborated with researchers rather than race them to results. Indeed, the company seemed perfectly willing to work with him. The problem was Alp\u00f6ge, or, more specifically, his ties to Anthropic.<\/p>\n<\/div>\n \u201cThey were in such a rush to publish, to beat Anthropic,\u201d he said. They barely took note of the researchers caught in the middle.<\/p>\n<\/div>\n For all the rush, OpenAI won\u2019t know whether it has won the Millennium Prize for solving Navier-Stokes for years. The Clay Mathematics Institute requires<\/a> a period of two years to have passed since a result was published, during which it must have \u201creceived general acceptance in the global mathematics community.\u201d For now, Navier-Stokes occupies a peculiar limbo: The Institute has removed it from its list of unsolved problems, though hasn\u2019t yet declared it solved. \u201cThe process is deliberately unhurried,\u201d the Institute said<\/a> in a statement.<\/p>\n<\/div>\n OpenAI, meanwhile, has already moved on. In a statement to The Verge<\/em>, OpenAI\u2019s Fauconnet said that \u201csince the completion of Navier-Stokes we have made substantial progress on another Millennium Prize problem,\u201d adding that the company is \u201cworking through how to share these results thoughtfully.\u201d<\/p>\n<\/div>\n Which problem remains unclear. Unconfirmed speculation<\/a> on social media suggests this could be the Hodge conjecture, which concerns, very roughly speaking, how complex geometric shapes can be understood in terms of simpler building blocks. Rumors<\/a> are also circulating that Anthropic is closing in on a Millennium Prize problem of its own.<\/p>\n<\/div>\n As the two giants of AI race to collect yet more mathematical trophies, they are discovering that astonishing results alone are not enough to earn the trust of the community they are transforming.<\/p>\n<\/div>\n Mathematicians are beginning to push back. Many The Verge<\/em> spoke to, even the most enthusiastic proponents of AI in the field, worried the companies were having a chilling effect on research<\/a>, pushing mathematicians to be more secretive about unfinished work for fear someone may swoop in and beat them to it. Several said colleagues who had previously compiled lists of important unsolved problems were reconsidering the practice, concerned that what was intended as a useful resource for the field could instead become a list of targets for AI companies.<\/p>\n<\/div>\n Resistance is becoming increasingly public. In June, mathematicians published the Leiden Declaration<\/a>, a set of principles for the responsible use of AI in mathematics that has been endorsed by the International Mathematical Union and signed by nearly 3,900 people, an increase of nearly 500 people since I last covered it in mid-August. It urges policymakers, governments, the media, and other groups to not buy into \u201cthe hype\u201d created by companies who \u201coverstate the capabilities of their products.\u201d As the Millennium Prize controversy raged, OpenAI withdrew<\/a> its sponsorship of an undergraduate mathematics hackathon at Caltech following fierce opposition<\/a> decrying the intrusion of corporate interests and worries the event would create a deluge of low-quality \u201cslop mathematics.\u201d<\/p>\n<\/div>\n \u201cThey don\u2019t care anything about us as a community. It\u2019s all about this petty drama between two trillion-dollar companies that are acting like children.\u201d<\/p>\n<\/div>\n<\/div>\n Shing-Tung Yau, a professor of mathematics at China\u2019s Tsinghua University, an emeritus professor at Harvard, and a recipient of the prestigious Fields Medal, told The Verge<\/em> he worries about the potential effect on young researchers. \u201cWorking on hard problems already carries considerable risk for Ph.D. students and junior faculty,\u201d he said. \u201cThe prospect of competing with powerful AI companies, whose resources far exceed those available to academic research groups, could make them even more reluctant to pursue ambitious questions.\u201d<\/p>\n<\/div>\n Yau declined to weigh in on allegations that researchers\u2019 work may have been used by OpenAI, but said he is in favor of an independent review to establish what happened. More broadly, he worries that a lack of transparency and rush to announce first could obscure the intellectual lineage behind a breakthrough. Mathematical credit, he said, should reflect intellectual contributions, not who had budget for the most compute or made the loudest announcement.<\/p>\n<\/div>\n Yau pointed to another problem, too. OpenAI and other AI companies occupy a peculiar position as both the providers of important research tools and, in a way, researchers. It \u201craises a serious conflict-of-interest concern,\u201d he said, particularly as they could benefit from privileged access to customers\u2019 unfinished and unpublished work.<\/p>\n<\/div>\n \u201cThat concern deserves a substantive response,\u201d he said. \u201cIt should not simply be dismissed as ordinary competition.\u201d<\/p>\n<\/div>\n A lot of this growing sense of unease comes down to trust. Mathematicians do not have to accept the most explosive allegations against OpenAI to worry about a company that both provides their research tools and, simultaneously, competes with them.<\/p>\n<\/div>\n \u201cYeah, quite honestly, I don\u2019t think that some data security announcement or whatever will really solve it,\u201d Thom said. \u201cI don\u2019t really trust them.\u201d Buckmaster felt similarly: \u201cWhy should we trust anything they said?\u201d<\/p>\n<\/div>\n Buckmaster said the reaction from colleagues to his going public had been overwhelmingly positive. But there was an undercurrent of something else too: fear. He told The Verge<\/em> he initially intended to thank those who supported him when he went public with his experiences. He elected not to after many expressed discomfort at the idea of having their names publicly attached. \u201cThe reality is that mathematicians are actually scared of them,\u201d Buckmaster said, referring to the AI companies.<\/p>\n<\/div>\n And fear is hardly a solid foundation on which to build a productive and healthy research community. Buckmaster isn\u2019t convinced it matters much to those companies involved. \u201cThey don\u2019t care anything about us as a community,\u201d he said. \u201cIt\u2019s all about this petty drama between two trillion-dollar companies that are acting like children.\u201d<\/p>\n<\/div>\n\n