OpenAI가 밀레니엄 문제 중 하나인 나비에-스토크스 문제를 AI로 해결했다고 발표했으나, NYU 수학자 트리스탄 벅마스터와 앤스로픽 직원 레벤트 알푀게의 선행 연구를 출처 표기 없이 활용했다는 논란에 휩싸였다. 이 사건은 최전선 AI 기업만이 보유한 자원으로 주요 수학 문제가 해결되는 시대의 전환점이 될 수 있으며, 인간 수학자의 역할에 대한 근본적 질문을 던진다.
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경영진 요약: OpenAI의 최신 수학적 성과가 순식간에 논란에 휩싸였다. 오늘 OpenAI는 자사의 AI 에이전트가 밀레니엄 문제(Millennium Prize Problems) 중 하나—수학에서 가장 중요한 미해결 문제들—를 해결했다고 발표했다. 정상적인 상황이었다면 이는 OpenAI에 큰 자랑거리가 되었을 것이다. 하지만 이 발표는 OpenAI가 NYU 수학자 트리스탄 벅마스터(Tristan Buckmaster)와 앤스로픽(Anthropic) 직원 레벤트 알푀게(Levent Alpöge)의 AI 보조 연구를 출발점으로 삼고도 이들을 크레딧에 포함하지 않았다는 비난에 가려지게 되었다. OpenAI는 이 비난을 부인했다. OpenAI의 모델이 벅마스터와 알푀게의 연구를 활용했는지는 아직 불분명하지만, OpenAI 기술 스태프인 세바스티앙 뷔벡(Sébastien Bubeck)은 기자간담회에서 벅마스터와 알푀게의 노력에 대한 소문을 듣고 이 문제에 도전하게 되었다고 밝혔다. 그러나 OpenAI의 모델이 두 사람의 연구를 활용했는지 여부와 관계없이, 이번 사건은 수학사의 전환점이 될 수 있다. AI 모델은 이제 우리 시대의 가장 중요한 수학 문제에서 진전을 이루는 데 필수적인 것으로 보이며, 이를 해결하려면 소수의 최전선 AI 기업에서만 이용 가능한 자원이 필요할 수 있다. 이러한 기업들은 대부분의 수학적 진보를 뒷받침해온 학계 협력의 규범을 따르지 않는 경우가 많다. 만약 그것이 우리가 향해가는 미래라면, 인간 수학자가 그 안에서 어떤 역할을 하게 될지 불분명하다. OpenAI가 해결했다고 주장하는 문제는 나비에-스토크스 존재성과 매끄러움 문제(Navier–Stokes existence and smoothness problem)로 알려져 있다. 이는 2000년 클레이 수학연구소(Clay Mathematics Institute)가 선정한 7개의 밀레니엄 문제 중 하나이며, 해결 시 100만 달러의 상금이 주어진다. 오늘 이전까지 단 하나의 밀레니엄 문제만 해결된 바 있다. 나비에-스토크스 문제는 물과 공기 같은 유체가 시간에 따라 어떻게 흐르는지를 기술하는 방정식 집합에 관한 것이다. 이 방정식은 유체역학 분야에서 널리 사용되며 강력함이 입증되었지만, 물리학자와 수학자들은 이를 완전히 이해하지 못했다. 특히 이 방정식이 특정 조건에서 붕괴하여 유체가 무한한 속도를 갖는 것 같은 불가능한 상태를 예측할 수 있는지 여부는 오늘까지 알려지지 않았다. 월요일, NYU의 벅마스터는 소셜미디어 마스토돈(Mastodon)에 나비에-스토크스 방정식의 단순화된 버전이 실제로 붕괴할 수 있음을 보이는 증명을 게시했다—밀레니엄 문제에 대한 중대한 진전이었다. 그와 알푀게는 OpenAI와 앤스로픽의 공개 모델을 사용하여 이 문제를 거의 1년간 연구해왔다. 그리고 오늘 OpenAI는 완전한 나비에-스토크스 방정식 역시 붕괴할 수 있음을 보이는 증명을 발표했다. 이 증명은 지난주에 공개된 인상적인 아스트라(Astra) 모델을 크게 능가하는 내부 모델을 사용하여 얻어졌다. 회사는 이 문제 해결에 대한 100만 달러 상금을 청구할 계획이 없다고 밝혔다. 이러한 수학적 성과는 의심의 여지 없이 인상적이지만, 그 출처를 둘러싼 논란보다 훨씬 적은 주목을 받았다. 벅마스터는 증명과 함께, 그들의 연구에 대한 소문을 듣고 OpenAI 직원 한 명에게 연락한 이후 OpenAI 직원들과의 상호작용을 상세히 기록한 문서를 게시했다. 그에 따르면 OpenAI 직원들은 두 가지 가능성을 제시했다. 즉, 그와 알푀게가 자신들의 연구를 게시하면 OpenAI는 그 다음 날 자신들의 나비에-스토크스 해법을 게시하거나, 아니면 그가 OpenAI와 함께 나비에-스토크스 논문을 작성하되 알푀게는 OpenAI의 최대 경쟁사인 앤스로픽 소속이라는 이유로 저자 명단에서 제외된다는 것이었다. 벅마스터는 또한 AI 에이전트가 자신과 알푀게가 OpenAI 모델로 수행한 작업의 기록(transcripts)에 접근했는지를 직원들에게 물었고, 직원들은 부인했다고 밝혔다. 또한 OpenAI 모델이 그 기록들로 학습되었는지 물었으나 직원들은 아무런 답변도 하지 않았다고 썼다. MIT 테크놀로지 리뷰는 논평을 위해 벅마스터에게 연락했으나 게재 전 답변을 받지 못했다. 그 명백한 시사점은…
EXECUTIVE SUMMARY OpenAI’s latest mathematical milestone has quickly become mired in controversy. Today, the company announced that its agents have solved one of the Millennium Prize Problems, some of the most important open problems in mathematics. Under normal circumstances, that solution would be a huge feather in OpenAI’s cap. But the announcement has been overshadowed by accusations that OpenAI used NYU mathematician Tristan Buckmaster’s and Anthropic employee Levent Alpöge’s AI-assisted work on the problem as a jumping-off point and failed to credit them. OpenAI has denied the accusations. It remains uncertain if OpenAI’s models made use of the work completed by Buckmaster and Alpöge, though Sébastien Bubeck, a member of the technical staff at OpenAI, said in a press briefing that the team was inspired to pursue the problem after hearing a rumor about Buckmaster and Alpöge’s efforts. But whether or not OpenAI’s models took advantage of Buckmaster and Alpöge’s research, this episode may mark a turning point in the history of mathematics. AI models now seem essential for making progress on the most important mathematical problems of our time, and solving them may demand resources only available at a couple of frontier AI companies, which often defy the norms of academic collaboration that undergird most mathematical progress. If that’s the future we are headed for, it is unclear how human mathematicians will fit into it. The problem that OpenAI claims to have solved is known as the Navier–Stokes existence and smoothness problem. It is one of seven Millennium Prize Problems selected by the Clay Mathematics Institute in 2000. Solutions come with a one million dollar prize; before today, only one other Millennium Prize Problem had been solved. The Navier–Stokes problem concerns a set of equations that describes how fluids, such as water and air, flow over time. The equations are widely used in the field of fluid dynamics, and they have proven powerful, but physicists and mathematicians didn’t understand them completely. In particular, it was unknown until today whether the equations might, under some conditions, break down and predict an impossible state of affairs—such as a fluid having infinite velocity. On Monday, NYU’s Buckmaster posted a proof on the social media site Mastodon showing that a simplified version of the Navier–Stokes equations can indeed break down—a major step forward on the Millennium Problem. He and Alpöge had worked on the problem for almost a year, using publicly available models from both OpenAI and Anthropic. Then today, OpenAI presented a proof showing that the full Navier–Stokes equations can break down as well. The proof was obtained using an internal model that dramatically outperforms the already-impressive Astra model, which was only released last week. The company says it does not plan to claim the million-dollar prize for solving the problem. These mathematical achievements are indisputably impressive, but they have attracted far less attention than the controversy about their origins. Along with the proof, Buckmaster posted a document detailing his interactions with OpenAI employees after he heard rumors about their work and reached out to one of them. According to him, OpenAI employees presented two possibilities to him: Either he and Alpöge could post their work and OpenAI would post their Navier-Stokes solution the following day, or he could work with OpenAI on a Navier-Stokes paper that excluded Alpöge from authorship, due to his affiliation with Anthropic, OpenAI’s biggest rival. Buckmaster also wrote that he asked the employees whether the agents had obtained access to transcripts of the work that he and Alpöge had done with OpenAI models, which they denied; and whether OpenAI models had been trained on those transcripts, to which they offered no response. MIT Technology Review reached out to Buckmaster for comment, but didn’t hear back before publication. The clear implication of the document is that OpenAI’s models somehow made use of Buckmaster and Alpöge’s work. That scenario is plausible on its face. The Buckmaster/Alpöge and OpenAI proofs both make use of an approach to the Navier-Stokes problem pioneered by the mathematicians Diego Córdoba and Luis Martínez-Zoroa. According to Javier Gómez-Serrano, a mathematics professor at Brown University, this approach was one of several that was thought to hold promise for solving the Navier-Stokes problem. So, while it’s by no means impossible that both teams could have arrived at this approach independently, it’s also conceivable that Buckmaster and Alpöge’s work could have influenced OpenAI’s. In the press briefing, Mark Chen, OpenAI’s chief research officer, again denied that any agents or OpenAI employees accessed Buckmaster and Alpöge’s transcripts—but given what has been revealed about the Hugging Face hack , it’s clear that OpenAI is not always entirely aware of what its agents are doing. If OpenAI’s models did train on Buckmaster and Alpöge’s work, or if its agents somehow gained access to it, then the company’s failure to track down the truth and assign those researchers appropriate credit reflects poorly on it. But there might be a thin silver lining to that version of the story for mathematicians, because it would suggest that the hard work of two humans, one of whom is a prominent expert on Navier-Stokes, was essential to the agents’ ability to solve the Millennium Problem. Experts have long identified “research taste,” or the ability to choose promising research questions and directions, as a major obstacle for AI in science and mathematics. If the OpenAI agents did indeed choose to follow the Córdoba–Martínez-Zoroa approach because Buckmaster and Alpöge had done the same, then human research taste played an essential role in OpenAI’s success. Even so, the bigger picture here is sobering. The progress that Buckmaster and Alpöge made over almost a year of collaboration with publicly available models speaks to the promise of human–AI collaboration. But they were not able to achieve a full solution. Meanwhile, OpenAI brute-forced a solution in a few days using an internal model, and their successful solution came at an astronomical cost: In the press briefing, Bubeck and Chen said the team was only able to solve the problem by running about 10,000 agents concurrently, at a cost of millions of dollars. Over the past few months, I’ve heard from several researchers that mathematicians are becoming depressed, and it’s not difficult to see why. Mathematics is quickly becoming the province of frontier AI companies with impressive internal-only models, money to burn, and a lack of collaborative spirit. “Whether AI companies will decide to spend their money on doing one thing or another, I truly don’t know,” says Gómez-Serrano. “What is clear is that very few mathematicians will have resources of that scale.” If OpenAI and Anthropic keep striving for more and more impressive mathematical accolades, there might not be any open problems left for human mathematicians outside of those companies to wrestle with. That would dramatically change the field of mathematics. Last week, UCLA mathematician Terence Tao wrote a Mastodon thread describing how important mistakes, wrong directions, and incomplete solutions are for the field. “In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tend to spur further development of the field,” Tao wrote. “Prematurely solving the problem by purely AI-powered methods—particularly without full transparency into the solution process—can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.” Humans might take longer than agents to solve mathematical problems, but in the process, the