Align AI and Mathematics–To Something Else

Twenty five Fields medalists were initial signatories of the letter “A Severe Misalignment of AI in Mathematics” in which they opine that “The goals of the AI companies and the goals of the mathematical community are severely misaligned”. They are right. As they say, the goals of AI companies do not seem to be aligned with “the primary goal of conceptual understanding and insight”, and indeed, the “solutions [by AI companies] are [being] announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others.” This is true (but ironically, the signatories themselves say they released their letter quickly because they “did not have the time to have a more consultative process.”) It is also true that the goals of the mathematics community do not seem to be aligned with “the primary goal of conceptual understanding and insight”. This is clear, because if that were the goal, then the mathematics community would hold the view that “the most precious resources of [its] profession are students and ideas”, and that they would be “nurture[d] with great care”. But sadly students and ideas in mathematics have not been nurtured with great care. Since the letter by the twenty five Fields medalists seems to have been precipitated by the OpenAI announcement a solution to the Navier Stokes existence and smoothness problem, let’s take a look at how the mathematics community has nurtured some of the students who worked in that area, and their foundational ideas. Juliusz Schauder (Leray–Schauder degree): Schauder earned his doctorate in 1923, but antisemitism (by mathematicians) resulted in denial of university positions. Instead he taught high school while producing serious math. A few years later after the Nazis rose to power, Schauder asked for help from mathematicians around the world, and yet numerous mathematicians declined to help. He tried to get an invitation from Princeton and was denied. This wasn’t just a matter of getting a salary. His life was in danger. Eventually he did not even have access to paper to write down his work and he was murdered by the Germans. His wife Emilia hid with their daughter, for a while living in sewers to survive. Emilia was eventually captured and then murdered in a concentration camp. Was this “nurture with great care”? Oh, you say, but this was a long time ago! Olga Ladyzhenskaya (Ladyzhenskaya inequality): By the late 1950s Ladyzhenskaya was a leading mathematician in PDEs and fluid mechanics. In 1958 she proved global existence and uniqueness for the two-dimensional Navier–Stokes equations using what is now known as the Ladyzhenskaya inequality. This work became foundational to Navier–Stokes theory. That same year, at age 36, she was shortlisted for the Fields Medal but was passed over for Klaus Roth and René Thom. Oh, you say, but the Fields medal is merit based! Records of the 1958 Fields committee document that its decisions were not solely merit based. Friedrich Hirzebruch, the favorite, was eliminated because he apparently was doing fine and did not need further encouragement, while the committee agreed that Alexander Grothendieck was the most talented after Hirzebruch but figured he could win later. It would take another fifty-six years until the first woman, Maryam Mirzakhani, received a Fields Medal (notably out of the 25 Fields medal signatories only one is a woman). Was this “nurture with great care”? Oh, you say, but things have changed! Karen Uhlenbeck (Geometric analysis and nonlinear PDE): In 2019, Uhlenbeck became the first woman ever awarded the Abel Prize. But after receiving a PhD in 1968 and holding temporary positions at MIT and Berkeley, Uhlenbeck applied for jobs and was told matter of fact that “people did not hire women” and that women were supposed to “go home and have babies.” MIT, Stanford, and Princeton were interested in hiring her husband but not her. She was told that “nepotism rules” prevented the universities from hiring her (she examined this years later and the supposed rules could not be found). She eventually obtained a position at the University of Illinois at Urbana–Champaign, which she described as follows: “I hated Champaign-Urbana—I felt out of place mathematically and socially”. Was this “nurture with great care”? Oh, you say, but that’s just a single example. Cathleen Morawetz (nonlinear PDE and fluid dynamics): Morawetz was one of the leading mathematicians in nonlinear PDE and fluid dynamics. That’s not quite Navier-Stokes but adjacent. She was the first woman to direct the Courant Institute, president of the AMS, and the first woman mathematician awarded the U.S. National Medal of Science. When she raised the issue of how few women there were in mathematics before an American Mathematical Society governing body, Saunders Mac Lane replied, “Well, mathematics is a very difficult subject.” Was this “nurture with great care”? Of course every mathematician knows that the mathematics community does not nurture with great care all of its students and ideas. This is not a secret and it’s not an open secret. It’s just common knowledge. In the mathematics department at UC Berkeley, where I worked for 18 years, the department went from barring Julia Robinson (Hilbert’s tenth problem) from teaching mathematics for 35 years (also using the nepotism rule as an excuse and agreeing to appoint her only after she was elected to the National Academy of Sciences in 1976) to hiring Yuval Peres (Berkeley math professor ~2001 – 2011). A particularly shocking detail about Robinson is that she was required to document to the personnel office exactly what she worked on every single day (!) In Peres’ case eventually mathematicians publicly described at least seven cases or reports involving junior women, some of whom reportedly avoided conferences or lectures to escape his repeated advances. It was not surprising to me that just three years after I arrived at the UC Berkeley math department, the department lost an important NSF grant specifically due to poor student mentorship. And it’s outrageous that there is still a prominent faculty member at UC Berkeley (now emeritus) who even today insists there is no, and never has been, sexism in mathematics. One of the defining moments for me in the department was when a graduate student who I knew only in passing, came to my office in tears to tell me “I’ve just been told I’m stupid” (she is now a very successful professor of mathematics). I’d heard this insult made many times in the department. Once at a faculty meeting a senior professor was asked what he was doing to help one of his graduate students who was in his 9th or 10th year. The reply was “I’m doing nothing. He’s stupid”. These are not isolated anecdotes. Many mathematicians have experienced rejection not only of their ideas, but of themselves. Perhaps one of the most egregious examples is Yitang Zhang who was told by his advisor “no Chinese student is good“. Was this “nurture with great care”? I found it interesting that the 25 primary signatories on the “A Severe Misalignment of AI in Mathematics” signed the letter with “Fields medalist” in parenthesis. Why not their affiliation, or email address? Did they highlight that they are Fields medal winners to imply that the honor bestows upon them some kind of authority on the mathematics community? Sure- they all did some incredibly important mathematics and were recognized based on extraordinary mathematical ability. But receipt of a Fields Medal also depends on the judgments of a tiny group about which problems, fields, and styles of mathematics deserve recognition. Erdős remarked that “Szemerédi should have gotten a Fields Medal. [But] the people who decide are not that interested in combinatorics.” Indeed, prizes encode judgments about which mathematics, and therefore which mathematicians, matter. But regardless of Fields medal politics, the point is that mathematical ability is not the same as mathematical responsibility. Perhaps instead of foregrounding Fields medalists in a letter about “alignment,” it would have been more illuminating to hear from mathematicians whose careers were derailed by lack of mentorship and nurturing. So yes, there is severe misalignment of AI in mathematics. But alignment of AI with the existing mathematics community should not be the goal. The history of mathematics gives us little reason to treat the profession’s existing incentives, hierarchies, and institutions as a model. AI companies and mathematicians should instead be asking what both ought to be aligned to: understanding, attribution, intellectual generosity, and the nurturing of students and ideas, and not merely the production or recognition of results. Share this: Email a link to a friend (Opens in new window) Email Print (Opens in new window) Print More Share on X (Opens in new window) X Share on Reddit (Opens in new window) Reddit Share on LinkedIn (Opens in new window) LinkedIn Share on Facebook (Opens in new window) Facebook Related