Navier-Stokes
11.09.2026An incredibly crazy week is coming to an end. One of the Millennium Problems, see also the official website of the Clay Mathematics Institute, the Navier-Stokes existence and smoothness problem, has apparently been solved by OpenAI. “ ‘Apparently,’ because the proof—which is more than 150 pages long—has not yet been fully verified by anyone, and formalization using Lean Code is not without its problems (agents do not always write ‘correct’ Lean Code, the kernel itself contains errors, etc.). Above all, however, I use the term “apparently” because the prevailing rules of scientific publication require careful peer review, and nothing of the sort has taken place. This is not the only controversy surrounding this achievement. Mathematician Tristan Buckmaster (Courant Institute of Mathematical Sciences, New York University) accuses OpenAI of plagiarism, fraud, and bullying, claiming that the company demanded that his co-author Levent Alpöge—a mathematician working at Anthropic—be removed from a publication in which the two had solved the Millennium Problem almost exclusively using an unusual technique.
In its response OpenAI stated that “it was “categorically” impossible for its A.I. system to have been influenced by anything Dr. Buckmaster had done in the past two months” (Source: NYT). The NYT article is by the way a god write up of what is known so far - another summary (in German) can be found with Heise and a more technical explanation is written by Terrence Tao, fields medallist.
These events have caused a stir in the mathematical community. Recently, the “Association for Human Mathematics” (AHM) was founded; its signatories include prominent mathematicians such as Peter Scholze, and the group is effectively calling for a ban on AI in mathematics.
Terrence Tao writes about the delicate role “child” questions curiosity and goals play for motivation of mathematics. He says “But, perhaps in order to impress the other adults observing (and, through public funding agencies and the like, financially supporting), we often downplay the “childish” aspects of the craft, in favor of the more tangible “serious” outcomes – including the hard, objective target of solving some designated open problem. Until recently, this polite fiction has worked, because human mathematicians knew, possibly subconsciously, to use both childlike and adult skills when solving these problems, even if they might only publicly admit to the latter. But modern Al tools, when directed without such expert supervision, can now be aimed all too easily at the ostensible goals of the field, without any incentive to take the slow, whimsical path. The flag is captured, the goal scored, and the problem is solved; but at the cost of lessons learned, insights gained, collaborations formed, and new targets located. In this modern era of heavy Al use, it is important to remember what it is like to be a child.” (Source: Tao’s Blog).
This finally brings me to my main point. Regardless of the extent to which OpenAI has exploited the contributions of Buckmaster and Alpöge—and given the reckless, albeit brief, history of all “Frontier Labs,” which flout the laws and social standards of their respective fields— the fundamental problem seems to be that “Frontier Labs” exploit the entirety of human knowledge without paying for it and without promoting the fields they exploit in any way. Mathematics, the most formalized of all sciences, is an easy target, but other human disciplines have also suffered as a result. Consider the visual arts, music, and so on. Already, 50% of all pop music appears to be AI-generated. The artists and musicians whose work was used to train the AI receive no compensation whatsoever—no royalties are the norm. In the long short term, this will erode the economic foundation for musicians and carries the risk of destroying centuries-old, vibrant cultural practices. Let’s hope it doesn’t come to that and that we can prevent this from happening in the sciences. (I will comment on the huge and unclear impact of AI on software engineering as a discipline in a different post, later.)
While scientists / employees are resigning from Antropic because AI will potentially kill humanity as it gets completely out of control, maybe the adversarial impact on sophisticated social systems we as humans have developed deserves more pressing attention.
But Tao is (still) concerned about another aspect: “The dual use capability of these most recent tools is indeed extremely concerning. There is a significant correlation between the capability of AI tools to solve pure mathematical problems, and their ability to potentially perform malicious acts such as exploiting security weaknesses in an online system, and the danger is very real that some major AI-triggered cybersecurity incident could occur in the near future which would be far more significant event in world history than, say, the production of a solution to all remaining Millennium prize problems. I am certainly supportive of efforts to slow down the development and release of even more capable models, and to impose genuine regulation on the industry, although the political environment currently is not exactly conducive to such policies, to put it mildly.” - let us keep any eye on it.
Update: On 9/11 25 Fields Medallists have expressed openly their concern about “A Severe Misalignment of AI in Mathematics”. As they write in the manifest which is worth reading in its entirety: “The mathematical community functions, in many ways, as a miniature version of humanity”.