Ways to improve Mathematics (an introduction)

The goal of this continuing sequence of posts is to recognize the reality in how our subject is changing and how we should adapt.

Even in a hypothetical world where AI was infallible, essentially omniscient, benevolent, a wonderful expositor, and freely available, I believe that many people would still want humans to maintain and develop a deep understanding of mathematics. They would not want to leave mathematics entirely to the machines, any more than they would want to abandon other large swathes of human thought. For this post, I will take that desire as a starting assumption.

My core belief is that understanding mathematics is hard and AI is not going to fundamentally change that, even if it makes proving theorems in mathematics much easier. I remember driving to Wisconsin and listening to Jordan Ellenberg on Lex Fridman’s podcast. What struck me most from the podcast was the discussion of Fermat’s Last Theorem. Fridman was essentially arguing that since the statement of Fermat’s Last Theorem was so simple, there must inherently be a simple explanation of why it was true. This reflects a philosophical idea about science and mathematics that I think is fundamentally untrue: that any truth that is simple to state will ultimately be true for a simple reason. The easiest proof of Fermat may well not be the one found by Wiles (or maybe it will be). But consider instead something much older and established in mathematics, namely class field theory. If I hold any position in this post with conviction, it would be that, even with superhuman exposition, a human could not acquire a good understanding of the statements and proofs of class field theory without years of dedicated study^*. This is not an isolated example. Maintaining and developing human understanding across mathematics requires people who can devote substantial parts of their lives to it. If we value that understanding, there is a case for supporting those people and the communities in which they work, rather than leaving the whole enterprise to whoever wants to work on it in their spare time. That is a central part of the case for mathematics as a profession.

What is important to recognize, however, is that mathematics as a profession will surely be changing rapidly. Daniel Litt has told us that we need to be honest not only about the aspects of our field that will break with AI, but also about the aspects that are already broken. This is an amazing opportunity to fix some of these things! If we are going to defend mathematics, at least in part, as a means of increasing human understanding, then we ought to be rather more demanding about whether our own practices actually contribute to this understanding.

To me, there are (at least) four natural systemic issues in mathematics that we have to address. Addressing any one of them will require consensus building to achieve the massive shift necessary in our community norms. I plan to devote a blog post to each of the issues, so for now I will just introduce them. I am not attempting to be prescriptive, but rather simply to think aloud and hope for suggestions.

One thing we need to do in particular is ask whether our institutions and practices reward the production of mathematical understanding, or merely activities that we have come to treat as evidence of it. With AI, activities such as producing proofs will no longer be quite so closely coupled with understanding.

• Journal Articles: The journal system was, to put it politely, already struggling before AI. There are too many papers, and the refereeing system is bursting at the seams. At least in mathematics we have a large number of high-quality journals, which helps diffuse the power of editorial boards, given the career importance of publications. [Imagine a field where success could come only from publishing in a single journal; such fields exist.] As we go forward, we very much have an opportunity to reconsider how the publication system works in a radical way. If we don’t abandon it altogether, then we need to redefine what we hope to get out of it. I already have anecdotal evidence that submission rates to top journals are rising sharply, and I doubt that the current refereeing system can sustain such an increase. I don’t think it is as simple as demanding better exposition — it’s not unreasonable to expect AI to vastly improve in this dimension as well. If mathematics is a conversation, then what we would like to achieve are strands of interesting conversations that people are both invested in and listening to. This touches, in part, on the question of insularity raised below.
• Seminar Talks: I would say that the median mathematics seminar could (perhaps harshly) be described as a waste of time for both the participants and the speaker. If we are to claim that fostering mathematical understanding is one of our main goals, we certainly haven’t made much of an effort to reward good talks. One institutional obstruction has always been the expectation that people talk about their own work. What often gets lost when one does this is an explanation of why the broader question being addressed is interesting in the first place, the methods that have been used most successfully in the field in the past, and the most promising questions to consider in the future. (One can do this in a talk about one’s own work, but people frequently do not.)
• Insularity: The past few decades have seen an explosion in mathematics. But I feel this has come at the cost of mathematicians being less able to communicate; not only with people in other areas of mathematics, but sometimes even within their own field. Some have argued that this is an inevitable consequence of the growing difficulty of mathematics, but I suspect that once a mathematical subcommunity reaches a certain size, the impetus to reach out diminishes, to all our detriment.
• Ego: Perhaps the thorniest question of all: how do we shape the incentives in our field to produce better outcomes? For all that we emphasize understanding, it would be insane not to acknowledge the importance of ego, and the way that the desire to be the first person to prove something has motivated many of us. This moment is going to require a great deal of humility. If human understanding of deep mathematics is what we want to defend, it ought also to be what we reward.

^*: now I have the following in my head: