Essay | How I used to study mathematics
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I probably began to study mathematics in earnest four or five years ago. Though “in earnest” may not be quite right: I never felt an inward longing for it, or the passion that other people described. To me it felt plain, like a glass of water, perhaps with a slightly sour taste. That was my impression of mathematics.
I really did not seem to have much talent for it. I had never heard of anyone in my family standing out among their peers in mathematics, and my marks in primary school in China were dreadful; before I left, I even failed once. Then I arrived here. As I used to mutter to the page, I was an uninteresting person. Others seemed to think so, and I thought so too. At first, age and cultural background and other distances kept me from getting along with my classmates.
But one has to do something to make a long school day pass. I had only just entered my twelfth year, and the stretch between sunrise and sunset still seemed long. So I imagined a strange world in my head and lived in it by myself. In simpler words, I daydreamed.
My grades were poor, with mathematics the one subject I could show for myself. Perhaps I heard praise from others and came to believe in mathematics; that is how the story should have gone. Yet, however hard I search my memory, I cannot find that praise. In any case, I began to study it.
At first I did not know what a proof or a definition was. Mathematics was no different to me from the other subjects. My method was rudimentary too. In 2019 the internet was already there: without a teacher, I found lectures online; without books, I copied the lecture notes from the screen onto a blank sheet. Exercises? I did not yet know that learning required practice and review. I had none, and simply kept going.
I stumbled along with fairly good results through primary and middle school. Yet those years left few traces in my memory. I know how I got through them, but cannot recover the details. It is as if an eraser rubbed hard across the past while I was not looking, leaving just enough faint marks to remind me that my present has a continuous history, that I did not appear from nowhere.
I was not well liked. I have no memory of recognizing that then; a few years ago I did not even know what being liked or disliked meant. Much of what I used to do seems strange to me now. I was once a fervent populist of the most typical kind, loud in speech and inactive in practice. And until the beginning of this year, I had a habit of breaking the few precious relationships I had every so often. After enough repetitions, it is hardly surprising that I was not popular.
But back to mathematics. My ambitions were always ahead of my abilities. Having barely swallowed the secondary-school material, I hurried on to university mathematics and imagined I was ready. I bought a Tongji University calculus textbook from China. I cannot remember why I chose it, but a little deduction suggests that the word “Advanced” on the cover attracted me. I wanted to do something different, to get ahead of others.
It was an ordinary textbook, and I was not putting much effort into mathematics. My daily study was more like checking off an online lecture. I came to believe that if I sat through a thirty- or forty-minute video at double speed, I had mastered its contents. The illusion did not survive meeting the exercises. If I could not solve a problem, I could not solve it. Every result has a cause; no subtle logic was needed to identify this one. I had learned the material neither deeply nor completely.
The old draft stopped here. Reading it now, I am not sure that last judgment explains everything about that time. I have kept it as a record of how I then saw myself.