Despite a couple of terrible math teachers I had early in middle school (the kind that cause most people to loathe math for the rest of their lives), by the end I still fell in love with the subject. I spent much of my subsequent school years chaotically pursuing anything and everything that caught my eye mathematically. I got an intoxicating taste of the awesome experience of forming a conjecture and then, after pondering over it for days or weeks, finally proving it wrong or right. I went on to major in math during college at MIT, engaged in a few attempted research projects (one of which eventually led to a joint publication), and graduated with a math degree in 2020. I then stopped doing math for many years.
My departure from the subject was far from graceful. By the time I graduated, I was, I guess you could say, rather “burnt out”. For sure, much of that exhaustion was self-inflicted. For instance, taking a graduate-level measure theory course my freshman spring was, in retrospect, not the wisest decision. My brash, impatient impulse to advance as fast as possible in the topics that interested me while skipping over everything else shot me in the foot over and over again. Each time, I managed not to learn the appropriate lesson.
This is perhaps not too surprising, given the fact that I had been using math as a kind of crutch. Like many in our messed-up capitalist society, I have had my share of struggles with anxiety, depression and the like. In the years before falling in love with math, these struggles were particularly acute and were combined with a sense of purposelessness. Discovering at age 13 or so that I really liked math and that, with sufficient tenacity, I could advance considerably beyond the standard school curriculum gave me a sense of purpose and self-worth, and a way to run away from problems in my life and in the world around me. This naturally is not very healthy. It begins to eat away at the joy of engaging in the difficult struggle of learning math and (later on) of carrying out research. Math is hard, and it is hard for everyone. “There is no royal road to geometry,” as Euclid supposedly said, nor is there one to science in general as Marx later added. But when “success” in math (or any other pursuit) becomes a substitute for a sense of self-worth, the challenges and failures one comes up against become supposed signs of irremediable inadequacy. Not to mention it becomes easy to get sucked into a frankly silly competitive dynamic. For some of us in undergrad, for instance, the number of grad classes we took, and the fancy theories whose jargon we could regurgitate, became the (totally incorrect) metric by which we judged our future prospects. The result was to render the tortuous path of learning and discovery into a torturous one that ceases to actually be about learning or discovery.
My guess is that many people engaged in the sciences face some variant of the above problem at some point in their trajectory. In my case, it was certainly a major contributing factor to my lengthy departure from math. It was far from the only reason, however. In parallel to my “burnout,” I was also increasingly drawn to struggles against various outrages in society. From cozying up to Mohammed bin Salman while he was the lead architect of the atrocious US-armed Saudi war against Yemen, to establishing ties with Jeffrey Epstein, and much more, MIT provided no shortage of examples to fight against.
So, when I stopped doing math, I didn’t take any time to rest, or to reflect terribly much on my time doing mathematics. Instead, I threw myself into a different set of questions — political, historical, philosophical. I spent the next several years organizing, studying, and thinking a lot about the material forces shaping society in a way I had never done before. It changed my worldview considerably.
Indeed, it also thoroughly changed how I see mathematics.
Before that period, my view of the subject was — I now realize — vaguely Kantian. I believed that mathematics was the deepest expression of the structures of the mind, and that the mind’s structures were the only reality we could form definite knowledge of. Whatever objective reality — “things in themselves” — might be, we couldn’t know anything about it, we could only come to know how our minds structure it. Mathematics was beautiful precisely because it was the most penetrating investigation into the only “truth” we could hope to directly grasp. This view was, I now think, fundamentally inverted. Holding it also made it genuinely difficult to reconcile my love for mathematics with the jarring concreteness of social and political life. If the structures of the mind are what matter most, then what do I make of the fact that people face material oppression and exploitation right now, in ways that have nothing to do with mental structures?
My time away from math forced me to abandon this view. I now take it as a starting point that there is a real, objective, material world; that we represent it mentally and symbolically; that our representations improve precisely through the process of acting in the world and confronting the consequences; and that this process results in direct, albeit partial, knowledge of reality. From this vantage point, mathematics looks quite different. The objects of mathematics — structures, relations, invariants, numbers — are not constructs of the mind projecting itself outward to generate the only domain we can form knowledge of. They are real features of material reality that we have learned, through a long and genuinely social process, to perceive, abstract, and reason about.
This shift has made mathematics feel more, not less, interesting to me. It illuminates questions that I had either not considered at all or only considered through a thoroughly inverted, idealist lens: how has mathematical practice changed across different historical periods, and why? What does it mean that certain kinds of mathematics flourished in certain societies and not others? What is the relationship between theoretical and applied mathematics? And between the math used by professionals in the mathematical sciences versus the “everyday” math used by everyone else in the course of their working and personal lives? These are not merely sociological questions tacked onto the outside of mathematics — they are, I think, fundamental questions about what mathematics is and how it works.
This blog is, first of all, an attempt to actually do mathematics again, and to do so in an adventurous spirit. This will include working through my own proofs for theorems I like, posing a ton of mathematical questions (including, potentially, some open ones), and documenting the messy process of trying to answer them. But I also intend to make it a place for the kind of meta-mathematical reflection I have just been gesturing at. So, expect to see some explorations into the philosophy of math, its history, its relationship to other sciences, and its relationship to our class-divided society at large.
The first series, Pathological Periodicity, is squarely in the first category1 — a deep dive into the surprisingly rich world of periodic functions that began with a single exercise in Chapter 1 of a linear algebra textbook. Later posts will venture further into the second.
Footnotes
Ironically, Pathological Periodicity will take us into some deep set-theoretic territory involving the Axiom of Choice, where the philosophical view of mathematics I sketched above is seemingly hardest to defend. Stay tuned for a philosophical follow-up to that series.↩︎