\u201cStaggering.\u201d \u201cOverwhelming.\u201d \u201cUnprecedented.\u201d \u201cSurreal.\u201d \u201cPure insanity.\u201d<\/p>\n<\/div>\n
Those were among the descriptions more than three dozen mathematicians reached for in conversations with The Verge<\/em> as they tried to make sense of the flood of mathematical results<\/a> OpenAI abruptly dropped on the field this week. Amid the awe, excitement, and uncertainty over the sheer scale of the deluge was a deep-seated anxiety over what it all means \u2014 and what comes next. For all their different reactions, researchers agreed that simply understanding what OpenAI had released could take years, let alone figuring out where the mathematicians themselves fit in the field now changing around them. Many feared OpenAI would not wait that long before moving on \u2014 or releasing even more.<\/p>\n<\/div>\n In all, OpenAI released nearly 400 AI-generated results. These were spread across more than 700 manuscripts and covered a diverse array of mathematical disciplines, including combinatorics, several branches of geometry, number theory, theoretical computer science, algebra, topology, probability and statistical mechanics, and mathematical physics. The collection is so vast that OpenAI felt the need to publish guidance<\/a> on how to navigate the sprawling GitHub repository.<\/p>\n<\/div>\n The sheer volume of work makes even a preliminary assessment as to exactly what the company has released difficult. In the hours and days following the drop, most mathematicians The Verge<\/em> spoke with said they were still struggling to digest everything; several said that simply working through the roughly 40-page table of contents and abstracts took them the better part of an hour. \u201cJust going over the entire list of abstracts is overwhelming,\u201d said \u00c1lvaro Lozano-Robledo, a professor of mathematics at the University of Connecticut.<\/p>\n<\/div>\n Sprinkled among the hundreds of manuscripts are formalizations in Lean, a programming language and proof assistant that allows results to be verified computationally. These formalizations have proven instrumental in assessing some of OpenAI\u2019s previous mathematical claims, giving researchers confidence that a claim is logically correct even if they don\u2019t fully understand the argument behind it.<\/p>\n<\/div>\n \u201cIf the AIs would disappear now, as though there were aliens that came to Earth and then just left, we would be studying this for the next 10 years, trying to understand everything.\u201d<\/p>\n<\/div>\n<\/div>\n But the degree to which each result had been verified varied wildly. On GitHub, OpenAI acknowledged<\/a> that the results are \u201cat different stages of verification\u201d and that \u201cmany, but not all, of the manuscripts have been formalized.\u201d As of writing, fewer than half the manuscripts in the collection appear to have been described formally. OpenAI said only 300 top-line results out of 719 manuscripts had been formalized, around 42 percent, and that it \u201cwill update the repository with more formalizations as we obtain them.\u201d<\/p>\n<\/div>\n Several mathematicians complained to The Verge<\/em> about the lack of formalization, particularly given the sheer number of results, and stressed that even when Lean code accompanies a result, evaluation isn\u2019t instantaneous. Researchers must check that the formalization actually proves what the result claims, another time-consuming process, and several digging through the papers said that even where computer-verifiable proofs had been provided, the quality was inconsistent and the statements they verified did not always appear to map neatly onto the claims in accompanying manuscripts.<\/p>\n<\/div>\n Kevin Buzzard, a mathematics professor at Imperial College London, said he had identified numerous theorems in his area of work \u2014 algebraic number theory \u2014 of which only around six immediately \u201cstood out.\u201d Few, if any, of those appeared to be formally verified in Lean. \u201cHence, I either have to read possibly-not-correct slop, or wait for others to do the same, or wait for someone to formalise them before I can say for sure that the results are even correct.\u201d Buzzard\u2019s concerns were echoed by numerous other researchers.<\/p>\n<\/div>\n Buzzard was far from alone in worrying about AI \u201cslop.\u201d The term is a shorthand for low-quality, frequently erroneous AI-generated material that increasingly crops up online \u2014 and in the real world<\/a> \u2014 including academic papers<\/a>. As in other fields, mathematicians told The Verge<\/em> they have seen a huge uptick in such material produced with tools like ChatGPT and Claude in recent years. Much of it is confusing, hard to read, and demonstrates little understanding of the subject; it is especially shoddy when it comes to crediting other researchers.<\/p>\n<\/div>\n OpenAI\u2019s previous mathematical write-ups were<\/a> widely<\/a> criticized<\/a> by experts for their sloppy nature, particularly their poor or nonexistent attribution<\/a>. In conversations with The Verge<\/em> ahead of the release, several researchers had taken to calling the impending flood of papers the \u201cslopocalypse,\u201d or similar variations on the theme.<\/p>\n<\/div>\n Whether the feared \u201cslopocalypse\u201d actually materialized is difficult to say, largely due to the bewildering volume of material released. Early indications suggest OpenAI took more care with papers this time around, or at least with some of them. Several mathematicians told The Verge<\/em> that their first impressions were far better than they had expected, though by their own admission that was hardly a high bar given the company\u2019s previous shoddy publications.<\/p>\n<\/div>\n \u201cIt\u2019s a big mess. It can cause a huge collapse in the academic culture and simply kill most of the faculties. It\u2019s a social problem and it seems that the AI labs are completely ignoring this issue.\u201d<\/p>\n<\/div>\n<\/div>\n But better does not necessarily mean good, let alone up to the standards usually expected of academic work making claims of this magnitude. With formal verification absent for many of OpenAI\u2019s claims, the quality of the accompanying papers becomes particularly important; they are the primary means by which mathematicians can confirm, understand, scrutinize, and contextualize the results.<\/p>\n<\/div>\n Producing rigorous mathematical papers is difficult work under even the best of circumstances. Doing so at this kind of scale is a formidable undertaking. OpenAI\u2019s models are pumping out mathematics at a dizzying speed and across a broad range of specialities, far outstripping the capacity of its human staff. The company simply does not have the breadth of expertise or resources to properly scrutinize its findings at the cutting edge of mathematics. Researchers told The Verge<\/em> that it shows.<\/p>\n<\/div>\n Many described papers that were difficult, sometimes practically impossible, to follow. \u201cThe write-up of the problem I know best made little sense after a quick read,\u201d Brendan Hassett, a mathematics professor at Brown, told The Verge<\/em>. \u201cIf this had been written by a person, I wouldn\u2019t spend any more time trying to understand it. Of course, this leaves 721 other preprints!\u201d (Hassett said this before OpenAI retracted three papers).<\/p>\n<\/div>\n Some researchers told The Verge<\/em> several papers they or their colleagues had noticed appeared to cover ground already trod by other mathematicians, though were wary of saying so publicly before they had a chance to properly review the material. Others pointed to the unusual brevity of the work, with results they would ordinarily expect to be developed over hundreds of pages compressed into a few dozen or less.<\/p>\n<\/div>\n Nalini Joshi, a mathematics professor at the University of Sydney in Australia, said a quick search through the release revealed little that overlapped with her own work, but noted that some of the papers she examined \u201chave short bibliographies.\u201d Given previous criticism<\/a> of OpenAI\u2019s crediting practices, she said she is \u201cwary that attribution in the papers may be lacking the complete story.\u201d<\/p>\n<\/div>\n But for all the slop and uncertainty, the overarching consensus is that the release contains some genuinely impressive work.<\/p>\n<\/div>\n While stressing the difficulties assessing the volume of material \u2014 and the need to properly verify the results \u2014 numerous researchers The Verge<\/em> spoke to said the work appeared to be of a very high caliber, despite shortcomings in its presentation. In a pre-AI world, they said, many of OpenAI\u2019s results would clearly have warranted publication in top-tier journals and could have been enough to secure an academic career for their authors. A handful were described as being the kind of work that could make a mathematician a serious contender for a Fields Medal, one of the discipline\u2019s highest honors.<\/p>\n<\/div>\n \u201cI either have to read possibly-not-correct slop, or wait for others to do the same, or wait for someone to formalise them before I can say for sure that the results are even correct.\u201d<\/p>\n<\/div>\n<\/div>\n Stanford mathematician Jared Duker Lichtman said there were \u201ctens\u201d of results he would put in this category, including progress toward the Riemann hypothesis, a special case of the Hodge conjecture, and a solution to the four-dimensional Kakeya conjecture. These are major problems in mathematics. Riemann \u2014 arguably the most notorious unsolved problem<\/a> in the entire discipline \u2014 and Hodge are both among the seven famed Millennium Prize problems<\/a>. Respectively, they are concerned with the distribution of prime numbers and, very roughly, how complex geometric shapes can be understood in terms of simpler building blocks. Kakeya, meanwhile, roughly asks how little space is needed to rotate a needle or pencil<\/a> in every direction. A proof to the three-dimensional version of Kakeya was among the achievements NYU mathematician Hong Wang was awarded<\/a> a Fields Medal for earlier this year.<\/p>\n<\/div>\n \u201cIt\u2019s not the case that these are just silly problems that no one\u2019s ever heard of,\u201d mathematician Scott Armstrong said. \u201cMany of them are like very well-known problems that many people have tried for decades.\u201d<\/p>\n<\/div>\n As the dust began to settle, mathematicians were left confronting a landscape that had abruptly changed around them. Many of them had just watched years of work and carefully laid research plans evaporate in an instant, or knew colleagues who had. Across the field, whether reactions were laced with excitement, dread, despair, or something in between, there was a profound sense of disorientation.<\/p>\n<\/div>\n By the following morning, that mood had not altered much. Speaking by phone as he walked through Paris, Armstrong imagined that, had the Sorbonne not been closed amid ongoing student protests and his colleagues were gathered around their usual coffee machine, the atmosphere would have been rather \u201csolemn.\u201d<\/p>\n<\/div>\n In Scotland, Colva Roney-Dougal, a professor of mathematics at St Andrews University, described a similarly bleak atmosphere among colleagues and students. The deluge felt somewhat \u201chorrific,\u201d she said, but its arrival brought some relief after weeks of uncertainty. \u201cI have a bunch of friends and colleagues whose grant proposals have just been wiped out,\u201d she said.<\/p>\n<\/div>\n \u201cI have a bunch of friends and colleagues whose grant proposals have just been wiped out.\u201d<\/p>\n<\/div>\n<\/div>\n Armstrong said he knew of one group whose entire research program was practically \u201cwiped out\u201d by the release. Tristan Buckmaster, an NYU mathematician who was at the center of OpenAI\u2019s earlier dispute over the Navier-Stokes problem<\/a>, meanwhile, said he had already heard of \u201cthree people who had entire research programs obliterated.\u201d<\/p>\n<\/div>\n Similar stories surfaced repeatedly in The Verge<\/em>\u2019s conversations with mathematicians, though the disruption was concentrated heavily in some areas of the field. Francesco Fournier-Facio, a professor of mathematics at Heriot-Watt University in Scotland, said researchers in probability, combinatorics, and theoretical computer science appeared \u201cparticularly in shock,\u201d adding that some regions of his field, group theory, had been \u201cbulldozed.\u201d<\/p>\n<\/div>\n Armstrong and other researchers said some areas appeared to have been aimed at with almost military precision. \u201cThere were definitely some targets,\u201d said Armstrong. He singled out work in the area Wang received the Fields Medal for this year, as well as several Millennium Prize problems. A cluster of results in mathematical physics, he said, makes it clear OpenAI is pursuing Yang-Mills theory<\/a> \u2014 the mathematical framework underpinning much of modern particle physics \u2014 and its unresolved \u201cmass gap\u201d problem, one of the seven Prize problems.<\/p>\n<\/div>\n Elsewhere, researchers expressed a kind of surprised relief at how little their own work appeared to have been touched. Roney-Dougal said her own corner of the field \u2014 largely centered on group theory \u2014 appeared to have escaped relatively unscathed. She joked that it is fortunate she works in a \u201cvery unfashionable area,\u201d though later messaged to say she was feeling \u201cunsure\u201d after finding her work cited in one of the papers.<\/p>\n<\/div>\n Joshi similarly found little overlap with her work, which is largely focused on integrable systems, complex systems that can be solved exactly. She suggested that may be because her field is less driven by long-standing, formally stated conjectures and definitions and more by questions that wax and wane with developments in physics.<\/p>\n<\/div>\n As of October 8th, that record already listed numerous corrections, including revisions on more than a dozen manuscripts and the removal of three papers due to a \u201csign error\u201d it says invalidates an argument.<\/p>\n<\/div>\n<\/div>\n Whether their own work had been directly affected or not, most mathematicians The Verge<\/em> spoke to shared a sense that the field had crossed some kind of threshold and was no longer the same as it had been a day earlier. Constantin Kogler, a researcher at the Institute for Advanced Study, called it \u201cthe most important single moment in the history of mathematics,\u201d while Yang-Hui He, a fellow at the London Institute for Mathematical Sciences, reached back millennia, comparing AI-driven developments this year to the publication of Euclid\u2019s Elements<\/em>, one of the discipline\u2019s most influential works.<\/p>\n<\/div>\n Few researchers were quite so grand in their assessments, but there was a general consensus that mathematics was changing fast \u2014 and that mathematicians would have to change with it.<\/p>\n<\/div>\n OpenAI knew the release would be disruptive and had made some effort to soften the blow and engage with the mathematical community. After several bruising encounters with researchers earlier this year, the company turned to mathematicians themselves for help, working with<\/a> the newly formed Advisory Group on Mathematics and Artificial Intelligence (AGMAI) on how to release the results responsibly.<\/p>\n<\/div>\n AGMAI had already laid out<\/a> what it believed responsible engagement would be. AI labs should ideally release papers \u201cthat a human understands\u201d or otherwise provide the necessary \u201csupport for the additional mathematical activities that are needed for humans to be able to understand and assimilate their AI-generated mathematical output and identify possible applications of it.\u201d They said that, where possible, proofs should be formalized, and that companies should make public what models and prompts they used to create them. The group also urged AI labs to stop treating mathematical releases as \u201cmarketing vehicles to promote their models\u201d and should \u201cstop testing advanced mathematical problems on proprietary models\u201d that are inaccessible to the broader scientific community.<\/p>\n<\/div>\n \u201cIt\u2019s not the case that these are just silly problems that no one\u2019s ever heard of. Many of them are like very well-known problems that many people have tried for decades.\u201d<\/p>\n<\/div>\n<\/div>\n OpenAI followed some of the group\u2019s advice. It said it would be funding a series of workshops and conferences around its work in mathematics to help the community process and understand its work, though it provided no details as to what these might look like or when they may occur. The company also disclosed significantly more information than it had in previous releases \u2014 again, researchers said this was a low bar \u2014 including that its model attempted more than 4,000 problems and that a typical result used around three hours of ChatGPT Pro thinking compute. It has also implemented \u201cprotocols for paper revisions and citations\u201d and on GitHub said it \u201cwill preserve the public release history\u201d of the collection. As of October 8th, that record already listed numerous corrections, including revisions on more than a dozen manuscripts and the removal of three papers due to a \u201csign error\u201d it says invalidates an argument. OpenAI did not respond on the record to The Verge<\/em>\u2019s request for more details.<\/p>\n<\/div>\n That change addresses a key source of friction with mathematicians, some of whom speak of a kind of \u201cparanoia\u201d around the company\u2019s published claims. The company has shown a habit<\/a> of surreptitiously altering press releases and papers in response to criticism without clearly disclosing those changes.<\/p>\n<\/div>\n But OpenAI did not follow all of the group\u2019s recommendations \u2014 including some of its most consequential. It did not identify the model behind the release nor did it disclose the prompts used or the full set of problems the model attempted. OpenAI also appeared to acknowledge that its formalizations were lacking \u2014 it said it will add more as it obtains them \u2014 and that papers were subpar, saying that \u201cfor future releases, we are committed to further improving the quality of the papers via the citations, mathematical exposition, and presentation of the results for better understanding.\u201d<\/p>\n<\/div>\n \u201cThe write-up of the problem I know best made little sense after a quick read.\u201d<\/p>\n<\/div>\n<\/div>\n Researchers The Verge<\/em> spoke to questioned why OpenAI couldn\u2019t have improved the quality of these papers and expressed disappointment that it seemingly couldn\u2019t be bothered to formalize \u2014 or even check, in the case of the retracted papers \u2014 many of the results ahead of time. Information like the prompts used would have also been incredibly useful in lessening what many felt was the burden of assessing the flood of material the company suddenly dropped on them.<\/p>\n<\/div>\n