AI does, and does not, change the way I do math

[This is a guest post by Rachel Webb. This blog post was initially written in a different file format and converted using AI. — T.]

The way I do math experienced a great upheaval once before, halfway through my PhD. I was in a seminar with a group of graduate students and faculty developing new discipline-specific writing courses. The facilitator asked the question,

What does it mean to write well in your discipline?

Then, as now, I use writing as a proxy for doing. As a mid-career grad student, I had a quick answer to this question: good mathematical writing, hence good mathematics, is clear and correct. I was surprised at the dissimilarity between my answer and the answer of the other seminarians (who were more experienced but also other-disciplined): good writing says something interesting.

This discussion unlocked for me the meaning of doing research. Doing research is searching for something interesting to say. (Of course, once you find it, you have to say it in a way that is clear and correct.) In mathematics, the search for something interesting is often driven by our search for understanding, as noted by Thurston and endorsed by many since. But I think the math research process has a creative aspect that is not completely captured by this search. By contrast, the task of making something interesting to say is like writing a novel whose characters are at once subject to constraints of human experience but also presented in a way that comments on human experience. The characters in mathematical novels are definitions, the story text is the lemmas and theorems; these are constrained by truth, but can also be chosen strategically or artistically to capture certain facets of truth.

This is, abstractly, how I currently approach math research: I seek to understand what is happening at a fundamental level, yes, but I don’t think there is a unique way to understand. I also seek to take the things I do understand—inevitably, these are just a subset of the phenomena I would like to understand—and craft a narrative from them that is beautiful and interesting to my fellow mathematicians. Concretely, I find that “interest” in a mathematical context often derives from applications, either to the real world, or to other math. It also derives from proximity to high-profile open problems that serve as centers of mathematical conversations.

I don’t see LLMs as changing that approach much, but I expect they will drastically affect how I execute it.

The execution changes because now I have access to a machine that has read all the books and knows how many standard lemmas go. This speeds up the research process immensely and turns some of my lands of mathematical fantasy into worlds I can realistically start exploring (dream bigger dreams, says Antieau). Of course, taking the interstate instead of the side roads has its tradeoffs, but for any given leg of the journey I can choose which route to take. I will return to this idea in a moment.

My approach to research does not change because critically, I’m not convinced that the advent of LLMs changes my metric for mathematical “interest.” If we have historically regarded a paper as interesting if it comments in some way on a Millennium (or similar) problem, need we lessen our interest if the problem has a solution? Certainly there are many solutions, even many shards of understanding that do not constitute full solutions. These are all very interesting. We might know how to get to the red city, but we can continue to map out the surrounding terrain, now all the more valuable for the economic and strategic opportunities arising from metropolitan proximity.

I have said something about how I will continue to do math research, but I have not discussed why: what reasons do humans have to do math, if LLMs are capable (hypothetically, say) of producing math that is even more interesting than the math we can create? I believe that there are economic reasons, but I will not discuss those (both Sahai and Strogatz-Townsend have some thoughts). Instead I present two humanistic reasons. Neither of these is unique to math, just as math is not the unique human practice whose execution is affected by the advent of AI.

The first reason for humans to do math is that math is interesting to us individually. I enjoy doing math, so I will keep doing it, even if machines are better at it. There is a threat that LLMs will take the fun out of doing math by tempting us towards knowing the answer over understanding the solution. The temptation must be resisted. I should use AI only in ways that make math research more enjoyable for me and allow me to get understanding along with my wisdom (Proverbs 4:7). This may require experimentation. In fact, I and others already make analogous choices to use technology only when it is helpful to us personally and not every time it is economically “correct.” For example, I persuade my kids every week to walk seven miles round trip to church, even though we could drive. The commute costs well over two hours, but the increase in understanding between family members from the shared time and suffering is worth it.

The second reason for humans to do math is that math holds shared interest for multiple people at once, and in this way it creates communities. In some sense, this is what the other-disciplined seminarians meant when they said good writing is interesting: they meant that good writing is interesting to other people and hence is a piece of a larger conversation. Math research is a tool for drawing people together, in student-teacher relationships, in collaborations, at conferences, and at department colloquia and tea-times. Again, AI could weaken these social bonds by making (fear of) scooping more common, or just by making it easier to ask a machine than a colleague. But could does not automatically imply will. To quote Wendell Berry, may the age of AI be the age of knowing our mathematical neighbor. “Friends, every day do something that won’t compute.”