Artificial Intelligence | For human mathematics

A human mathematics should reasonably also take the place of a male mathematics. Photo: Unsplash/Getty Images Technology companies are apparently finding more and more ways to “prove” the strength of their AI models. OpenAI currently claims to have solved over 700 mathematical problems. That sounds impressive, but not everyone is enthusiastic. Mathematicians, of all people, want nothing to do with AI solutions to mathematical problems. There is also a concern that this will shift the focus of science and at the same time its field of activity: instead of working on new findings themselves, researchers have to check and improve the results of machines. They continue to only function through human input.

At the beginning of September, the company OpenAI proudly announced that it had solved one of the so-called millennium problems in mathematics. On behalf of the group, a team using 10,000 AI agents and over 50 hours of computing power created a solution for the “Navier-Stokes existence and smoothness problem” – currently one of the trickiest mathematical problems.

The equations for describing physical forces in flows date back to the middle of the 19th century; one version is dated to 1843. But when formulating for certain cases, for example three-dimensional space, paradoxes arise that cannot be solved using previous mathematical means. In order to change that, the Institute, founded in 1998 by the Boston couple Clay, offered prize money of one million US dollars each for advances in seven areas of mathematics. The task in the Navier-Stokes case: to prove the equation for particularly complex cases or to formulate exceptions.

OpenAI published its work on its own website and submitted it to the Clay Institute. There they were unusually pleased - compared to the cliché of reserved scientists: "The CMI today shares the enthusiasm of the global mathematical community in view of the news that the Navier-Stokes problem has apparently been solved." The excitement is understandable. After all, it's about knowing more about whether the well-known laws of gravity apply to fluids in three-dimensional space with infinite acceleration in a finite time.

Not only is the solution newly acquired knowledge, but also the tinkering and sharing of it.

But not all mathematicians popped their corks in response to the news. There are other reasons for this besides questioning an entire profession. For one thing, OpenAI's claim to have solved the Navier-Stokes case has not yet been proven. The approach first has to be tested by people - a big challenge because it involves very complex mathematical relationships that have not yet been completely solved. The institute's examination normally takes two years. In fact, there are already indications of inconsistencies: an independent scientific team found that in the OpenAI publication, the machine-readable representation in code does not match the explanation, formulas and natural language provided.

The mathematicians Fabian Circelli and Anders C. Hansen from the University of Cambridge and Alexander Bastounis from Kings College London criticized this deviation in a joint publication at the beginning of October. They attribute the inaccuracy to the immense use of AI, more precisely to the principle of auto-formalization. Natural language such as formulas and ideas are automatically translated into code. This has the advantage that the calculations are easy to check.

In their paper “Navier-Stokes lost in Translation,” the authors argue that this approach also has disadvantages. Because translation errors are not an exception, but rather inevitable. Because the code is consistent, errors in the theoretical approach can be covered up - to put it simply: the calculation may work out even though the calculation method is wrong or does not fit the task. But that's exactly what proofs in mathematics are all about. They are intended to show whether and how a train of thought can be implemented mathematically completely and without contradictions. That's why not only the solution counts as newly acquired knowledge, but also the tinkering and the exchange itself are part of science.

In addition, there are disputes over the intellectual property of the solution approach. Mathematicians Tristan Buckmaster, a professor at New York University, and Levent Alpöge, who works for OpenAI competitor Anthropic, accuse OpenAI of profiting from their work without disclosing this. At the beginning of September, Buckmaster published unfinished work on the flow equation by himself and Alpöge - just hours before OpenAI announced that it had solved the Millennium Problem. Buckmaster responded to the rumor that OpenAI was working on a solution.

He was all the more surprised when he learned that the company followed the same approach as he and Alpöge: both first expanded Euler's equation and developed counterproofs for several special cases instead of proving the general validity of the equation. While this approach is consistent with the Clay Institute's rules, Buckmaster says it's not the most obvious approach. Since he and Alpöge also used OpenAI models in their work, Buckmaster suspects that their chats were incorporated into the training of the application.

OpenAI counters: Neither human researchers nor the AI ​​agents have ever seen Buckmaster and Alpöge's work before. The company clarifies on the X platform that no specific user data for the solution was retrieved. However, a loophole remains: the entire AI business model is based on copyright infringement. According to the terms of use, anonymized usage data is used “to improve the models” and sources cannot be reconstructed. Copyright infringement is difficult to prove. If someone writes information into a chat that the Internet doesn't publicly provide, even that can become AI knowledge.

At the end of September, a network for human mathematics was founded: the Association for Human Mathematics (AHM). She is reacting to the flood of mathematical publications by AI companies - but there is no cheering from her either. On the contrary: in a statement at the beginning of October, the AHM contradicted OpenAI's confident claim that the company was advancing the field of mathematics. Mathematicians would not have asked for AI to do this work. Instead, a scientific advisory board expressly advised OpenAI not to test its models on problems in higher mathematics.

The network accuses the company of deliberately ignoring ethical standards of scientific research. There is also a clear political interpretation of OpenAI's practice in the AHM statement: "The publication of over 700 files at once is not proof of scientific seriousness, but a demonstration of power."

The controversial actions of Sam Altman's company are not an isolated case. It fits with a political stance that uncritically idealizes the capabilities of technology applications. Donald Trump, for example, officially instructed the US authorities at the end of September to refer to such applications as “superintelligence” (SI). In an interview with the US broadcaster NBC in the spring, the former showmaster explained that he had never used a chatbot himself - but he knew everything there was to know about it. Also a form of superintelligence.

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