A beginning for mathematics

From the article:

Here I want to lay out, instead, a positive vision of the future of mathematics, and the human practice of mathematics. I claim we can deepen human understanding even as the production of interesting mathematics becomes less dependent on it.

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In the course of this change, we will have to decide what to hold on to and what to throw away. Some things I would like to preserve: learning seminars; serendipitous conversations that spark an idea; students knocking on a professor’s door to chat about math. A robust community learning exciting new mathematics. Thousands of people that, together, slowly start to resolve their confusion.

I worry that much of what has been written on this topic, including some of my own past writing, focuses too much on trying to preserve the precise shape of the institutions of academic mathematics, rather than our values. How can we preserve the journal and peer review system?⁵ How can we protect the arXiv? How can we keep our role as gatekeepers? If you have internalized the fact that existing AI systems can produce relatively high quality results for the marginal cost of a few dollars, the idea that any semblance of the current equilibrium can survive what’s coming is absurd.

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Before I propose some relatively concrete steps we can take, let me remark on what we’re trying to protect mathematics from. There is a lot of anger at AI labs, and certain individuals at those labs. But whatever our judgment of the labs, we need a plan that does not depend on AI capabilities disappearing. The basic issue is not the labs’ behavior, ethical or not.⁶ It’s the technology itself. I think there is some belief that the labs will “move on” from math next year, be nationalized or broken up, or that a financial bubble will pop, somehow returning things to normal, or… But there is no way our institutions can survive unchanged when anyone with a laptop and a few hundred dollars can generate what would have been an Annals paper last year. AI does not care if you are anti-AI.

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In my view we should welcome interesting mathematical results regardless of provenance. But our institutions have historically relied on the same signal to indicate both mathematical progress and mathematical expertise. These now must be distinguished.

I propose the following reconceptualization of the goal of a mathematics PhD: to become a world expert on some interesting, deep topic, and to be able to convey that interest and understanding to others. Part of operationalizing this might be a thesis, but the degree would be awarded primarily on the basis of a rigorous defense, in which the student explains the topic to their examiners until they are satisfied. While we might require the topic to be original, its provenance—AI or not—is irrelevant.⁸

How different would this look from current PhDs? I think students would still meet with an advisor, who might suggest a topic. That topic could be explored with AI assistance, or not, but the student would be responsible for understanding it; it might be much more open-ended and larger than the typical PhD is currently. The student would be trained to ask interesting questions and try to resolve them, by whatever means. To keep students on track, there might be regular meetings in which the student is asked to independently work through an unfamiliar example, apply a technique in a new case, etc.

The allocative aspects of our job (hiring, graduate admissions, etc.) are in dire need of reform if we want to retain human mathematical expertise. Broadly speaking I think we should focus on rewarding skill in the parts of our jobs that cannot be automated: the internal (e.g. understanding mathematics) and social-relational parts, and operationalizations that hew as closely to those aspects of the profession as possible. For example, talks and sustained mathematical discussion now demonstrate understanding much better than papers. Once AI systems improve at exposition and “digestion,” this will be even more the case. We already interview faculty hires; we must now do the same for graduate admissions.

I think we should try to foster a robust seminar culture in which speakers are expected to explain their topic to the audience’s satisfaction. Much has been written recently (by myself among others) about the fact that we are primarily interested in understanding, not merely the truth value of mathematical statements. If that is the case, let us make sure we actually understand each other.

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A student will be confused. They will knock on their professor’s door. Maybe the two of them will ask a model for help, or maybe not, but first they might spend some time at the blackboard thinking through the question. And the model might give them a beautiful explanation, but we all know that’s not enough; no one can understand mathematics for us. We have got to do the work.