What Mathematics has Lost
Maryam Mirzakhani passed away recently. She was a first-rate mathematician, and the greatest living researcher in my former field of polygonal billiards. I fear that with her, we have lost our best chance of resolving the questions that once possessed me. But I have realized that we have actually lost much more; mathematics has not just lost her future work, but much of her past as well.
Mathematics is, at its heart, not a collection of theorems. It is about human understanding. It is often said among mathematicians that proofs are more important than theorems, and that definitions are more important than proofs. Mathematics is about understanding rather than knowledge, about finding frameworks and developing intuitions for logical structures. The first and still greatest insight into polygonal billiards was the unfolding: the observation that, instead of visualizing a billiard ball bouncing off the edge of a polygon, you can visualize the polygon reflecting across that edge while the ball continues in a straight line (I have built a demo of this here). From a coordinate geometry perspective, this does not change the calculations at all—a computer plotting the trajectory has to do just as much work as before. Yet all important theorems of polygonal billiards, including Mirzakhani's, flow from this observation.
While we treasure Mirzakhani's theorems, and moreso her proofs, the true value of her work lay in her unique intuitions that made them possible. Yet these are almost entirely absent from her papers. We can guess at them from the definitions she introduces, from the nature and structure of her proofs. But the originals were never truly laid out on paper, and only incompletely shared with her collaborators. To an extent, they died with her.
I do not mean to single out Mirzakhani as a bad writer. She was one of the better in our field, and we have only lost so much understanding with her because there was so much to lose. But it is not our custom to write down our intuitions. I have heard the process of writing a math paper described as "doing everything possible to disguise the fact that it was written by a human being". We have been trained to write mathematics in such a way that readers (within our specialized sub-field) may (if with difficulty) verify that our claims are correct, but never understand how or why we arrived at them. They are only truly comprehensible if you already share much of the underlying intuition of the author. To really read them, you must be able to build up the same intuitions the author did, using their writing only to check your work as you go. This has value, but not nearly as much as presenting the intuitions directly.
It does not have to be this way. These intuitions are developed through conference talks, seminars, and informal discussions. It is largely a collaborative process, and one carried out through words. Words like "squiggly-ness" and "kinda looks like", that embarrass us in formal writing, that we are loathe to put to paper lest they undermine our illusion that our mathematical process is as clean and logical as the product. But this is not the main way our pride gets in the way of explaining our underlying intuitions. In the words of the great geometer Mikhael Gromov:
This common and unfortunate fact of the lack of an adequate presentation of basic ideas and motivations of almost any mathematical theory is, probably, due to the binary nature of mathematical perception: either you have no inkling of an idea or, once you have understood it, this very idea appears so embarrassingly obvious that you feel reluctant to say it aloud; moreover, once your mind switches from the state of darkness to the light, all memory of the dark state is erased and it becomes impossible to conceive the existence of another mind for which the idea appears nonobvious.
I think he is too generous in saying that mathematicians lose the memory of their ignorance—rather I think we willfully pretend it never existed. As writers, we need only get over ourselves and admit in our writing that we are human. This burden falls on readers as well—we must not sneer at writers who deign to be understood.
There is one significant hurdle, which is that outside of collaboration, mathematics relies on the strange convergence of individual mathematician's intuitions after staring at a problem for long enough. This is what my well-meaning teachers meant when they told me that there was no substitute for hard work, that there was no book that could explain things to me, that even the best books could only show me what it was I had to learn. And while I have gained much intuition through collaboration, they were right that some of it I was only able to find through solitary thought. While I'm confident that other mathematicians converge on these intuitions—otherwise they could not write the proofs they do—I have no idea how to share them.
Why do we develop convergent intuitions, and how can they be shared? This is surely the most important question in mathematics, and yet it receives no formal attention from practicing mathematicians. Instead it is relegated to philosophers of mathematics. Is this really a satisfying state of mathematics? If there has been progress, it has not impacted the practice of mathematics. Today, the real substance of mathematics remains locked away in a few mortal minds.