Who Taught the Machines Mathematics?

In May, I wrote an essay called The Branch and the Wish about what might happen to mathematics when machines could produce the very things by which mathematicians have long measured their achievements. My concern then was not that machines would produce incorrect mathematics, but what might happen when they became very good at producing correct proofs. Over time, mathematics had come to identify its visible output—the theorem, the proof, the paper—with the activity itself, while the years of failed attempts, the development of intuition and taste, and the formation of a mathematician became harder to see. I borrowed from the stories of Kālidāsa and King Midas to describe the predicament. We had wished for proofs that could be made explicit, verified, and preserved independently of those who produced them. These were good wishes. But as machines began fulfilling them, we might discover what the wishes had left out.

When a mathematician announces a result, their reputation is tied to that claim. A mathematician may spend months, sometimes years, working on a problem and is expected to understand the argument and exercise judgment before announcing it. Mathematicians make mistakes (I have, too), and even the best have published incorrect proofs. But a serious error can affect how others view their work for years. OpenAI, on the other hand, can release hundreds of manuscripts, acknowledge that some may contain errors, and invite the community to examine them. The company receives attention for the number of results, while the consequences of individual errors are much more diffuse. I am not sure that the conventions through which mathematics has maintained trust were designed for this way of producing and announcing results.

Then there is the question of who will do the checking. A mathematician may need days or weeks to understand an unfamiliar proof, work through its details, determine whether the statement captures the original problem, and compare it with what is already known. Formal proof systems can help considerably, but deciding which results matter more is another matter. For that, we might need an additional machine trained to recognize mathematical taste. I do not doubt that something like this could be developed, but that was not what was presented here. If hundreds of results arrive at once, the attention required to make sense of them does not become available merely because they have been published. The company can generate them relatively cheaply, while mathematicians are left to decide which ones deserve their time. In my May essay, I worried that verification was a skill developed through long apprenticeship, and that this skill could deteriorate if mathematics were increasingly delegated to machines. Now there is another possibility: that the existing capacity for verification may simply be overwhelmed by the volume of material being produced.

I also found myself thinking about the countless hours I had spent training students in mathematics and theoretical computer science, and explaining the inner workings of my results to colleagues. We worked on the Unique Games Conjecture, sparsest cut, graph Laplacians, optimization, and related problems, including some that OpenAI now claims to have resolved. Much of this involved things that do not appear in textbooks or even in research papers. Why a particular reduction is useful, why a seemingly natural approach will not work, how to recognize when you are looking at the wrong problem formulation, and what to try when an argument gets stuck. Over years of doing mathematics, one learns umpteen small tricks and steps, often from one’s own teachers and collaborators. I passed on my own training and intuition, which in turn came from others. Some of it was written down, but much of it was not. It lived in conversations, on blackboards, in repeated attempts to understand why something worked or failed. I remember one occasion when a colleague convinced me to change my plans after STOC and accompany them to their university so that I could explain my latest result on metric embeddings in great detail. These exchanges took a great deal of time, and they were among the parts of academic life I cared about most.

Many mathematicians trained in universities, including some I taught and worked with, have moved into AI companies. I spent time in industrial research myself, and I never had a problem with this movement. Part of my understanding was shaped by earlier industrial research laboratories, where there was a culture of openness and researchers often participated freely in the broader mathematical community. I worried that the mathematical knowledge we passed on to one another might eventually be used to train machines to do mathematics, including the many unwritten tricks and intermediate steps we had learned to solve problems ourselves. I do not know exactly how this knowledge enters a machine, or how much of it can even be transferred this way. But I find myself wondering about the larger picture. Universities spend years training people, much of it with public support and through an intellectual culture in which knowledge is meant to be shared. Companies then recruit these people, gather expertise developed over generations, and use it to build systems whose purpose is increasingly to do the same intellectual work. The resulting capabilities become proprietary assets, built with resources and computing infrastructure the universities themselves cannot match.

I do not think this knowledge belonged to me or to any particular person. Nor do I think anyone owed me anything beyond what we owed one another as people working together. I learned mathematics because others gave their time and knowledge freely, and I wanted to do the same for the next generation. The possibility that a student would go on to do something I could not do was part of the point. But I had imagined this as a continuation of a human activity. I was helping someone learn to do mathematics, and perhaps that person would go on to teach others, discover things I had not discovered, and carry some of what we had learned into new directions. I could not imagine treating this knowledge as something to be guarded. Its value lay partly in the fact that it could be passed on. The same process of transmission can now become part of building systems intended to do more and more of this work without human mathematicians.

I would call this a form of extraction, though not a simple one. Industry has supported mathematics for a long time, and many important ideas and tools have come from industrial laboratories. Nor is there any reason why knowledge developed in universities should remain within universities. But there is a difference between industrial laboratories whose researchers participate openly in the development of a field and companies that gather publicly developed knowledge and expertise to create proprietary systems. The people who spent decades developing this knowledge have little say in what is now being built from it. The exchange I took for granted between universities and industry may no longer work quite the same way. And the institutions that once supported the free circulation of mathematical knowledge may find themselves increasingly dependent on companies that control the resulting technology.

Some of our most prominent mathematicians have encouraged the wider community to collaborate with AI systems. I understand the excitement. These systems can now do mathematics that would have been difficult or impossible to imagine only a few years ago. But what exactly are we encouraging mathematicians to do? When a mathematician spends hours working with a proprietary AI system, correcting its mistakes, explaining why an approach does not work, and suggesting better ones, they may think they are simply using a tool to advance their research. But they may also be sharing the mathematical judgment that took years, sometimes generations, to develop. Not every interaction is necessarily retained or used to train a model, and the arrangements differ across systems. Still, these interactions could become a source of further training and improvement. The tools may be useful, and that usefulness encourages people to work with them. But what is presented as human–AI collaboration may also become another way for companies to accumulate the mathematical community’s unwritten knowledge. In part, the encouragement comes from mathematicians whose judgment and standing carry weight within the community.

The release has also affected the relationship between AI companies and mathematicians. Even among those who have encouraged mathematicians to collaborate with AI, there is now dismay at how this release unfolded. An independent advisory group of mathematicians clarified that its involvement should not be understood as an endorsement of the process, and some mathematicians have called for an end to cooperation with OpenAI. I would not have expected a release intended to demonstrate mathematical capability to provoke such distrust. I also think of the students and postdocs who have spent years working on some of these problems. Whether the proofs turn out to be correct or incorrect, their work has already been disrupted. And if the proofs are correct, they cannot simply go back to exploring those problems as they did before. I can understand the anger, but I am not sure that withdrawing from AI is the answer. Younger mathematicians interested in these systems may find themselves caught between exploring them and remaining part of their mathematical communities. I would not want the choices to be either accepting the direction set by AI companies or refusing to engage with the technology altogether.

All those hours I spent explaining mathematics to students and colleagues, all the informal conversations, the failed approaches, the little tricks that never made it into papers—none of this counted for very much in the profession. What counted were the papers, theorems, citations, h-indices, prizes, Fields Medals, and other forms of recognition. I did not particularly mind that the conversations went uncounted. I was not doing them for recognition, and I would not want to put a price on them now. But much of what we did not know how to value in our own profession may now be among the most valuable things to companies building machines to do mathematics. We had already made the finished result the measure of mathematical achievement, while the activity through which mathematicians developed their understanding became secondary. Now that machines are beginning to produce those results themselves, perhaps we should also ask what we have been doing to mathematics.

If the response is to reorganize human mathematics around another set of awards, distinctions, and forms of professional recognition, this time meant to protect it from AI, I am not sure what will have changed. I do not particularly want to defend the mathematical profession as it exists. I have become disillusioned with much of it over the years. Mathematics existed before the institutions and incentives that now organize it, and it will presumably survive them. But when professional recognition becomes the organizing purpose of an intellectual life, something has already been lost. I would rather see mathematicians spend time on problems they find interesting, without worrying whether the work will lead to a publication, a prize, or a promotion. I would rather see more time spent teaching someone an idea in depth, as my teachers and colleagues did with me, without asking what professional advantage might come from the exchange. If machines can produce more theorems than we can, that need not mean there is less reason for a human being to do mathematics.

A student may spend years struggling with a problem without producing a celebrated theorem, yet acquire an understanding that is difficult to measure. A machine may solve the same problem in hours. If the result is all that matters, the machine is more efficient.

I had hoped that, when my children were older, I would spend an afternoon with them over a piece of paper, sharing some of the mathematical tricks that had once given me so much joy. Not because I expected them to become mathematicians, but because I wanted them to experience what others had patiently shown me: how a difficult problem can suddenly look simple when one sees it differently. I imagined us getting stuck, trying something else, and eventually finding our way through.

By the time they are ready, the answer may already be there before we have even begun.

I am not sure what place there will be for the afternoon I had imagined.