Counterexample

In the summer of 1969, I was twenty-one years old and still missed England. Back then, I didn’t have an ambition to be anything in particular. I’d left college in Boston after two and a half years without a degree, to work on a political campaign in the city. There was a lot of anti-war sentiment around and I became politically active, probably swept up by the spirit. But after taking the job, I had a big come-down. The reality was very mundane. Sitting at a desk, answering the phone, making cold calls to lists of donors: it wasn’t what I’d signed up for. Mostly, I was impatient. When I quit, I remember the section chief wishing me well. And that was that. 

I planned on staying in America for graduate school, but couldn’t decide on a subject. Sitting in that office, I’d often read through physics textbooks, especially things on statistical mechanics. At that time, I remember a strong urge to really nail things down, and understand every line. Although my short career in politics had been a failure, there was one good thing to come of it, confirming in me a desire to pursue academic work. Of course, my ideas then were all wrong. For example, I thought that you could only move on to a new subject when you understood something thoroughly. In reality, things only start to make sense when you look at the big picture. This may well be true of life more generally.

I enrolled in 1970, at first in the physics programme. At that time, particle theory was going from strength to strength, but I didn’t understand any of it. After a few courses in quantum mechanics, I felt sure it wasn’t for me. I didn’t like the way the physicists worked, maybe because part of me felt like they were making it all up. Looking back, I was making far too much of a few bad experiences, since physicists have their own way of doing things. I also believed that all the other students were far ahead of me. It was only much later that I realised, you never really know how others are finding it. 

After two unhappy terms I switched to mathematics, but things didn’t get much better. Suddenly I was plunged into a world of abstract algebra and advanced geometry, totally lacking a formal background in any of it. I don’t think it’s an exaggeration to say I didn’t understand a single thing during the harder lectures. Nevertheless, some classes went better. I enjoyed functional analysis, perhaps because I lacked imagination. At some point during my fourth term, I remember feeling more at ease. But if things did improve, it was only by a fraction.

You once asked me if I’d always wanted to be a mathematician. Honestly, I can’t remember. I don’t think so. Growing up in England, I liked reading science fiction. I think my father had always harboured unrealised scientific ambitions, and it’s possible that some of that trickled down to me. He was a big fan of Richard Feynman. When I told him that I wanted to study in America, I remember him saying that it must have been a marvellous country to produce a man like Feynman, whilst England and China lagged far behind. I also remember my mother telling me that she came from a long line of successful Chinese doctors. But I have never looked into that carefully. 

I do know that during graduate school, most of my studying was just for pleasure. I had wonderful nights spent in the library, slowly working my way through stacks of textbooks. Nowadays, I feel like I never really thought about what would come of it, but most likely I’m romanticising things. Either way, as much as I studied, in those years I never distinguished myself. There were some extremely brilliant young mathematicians in my cohort. I just made my way as best I could. 

At that time, I became interested in a field known as Algebraic Topology, which involves the study of shapes. One day, a professor directed me towards a paper, written by an Italian mathematician, Di Giorgi, containing several open problems, in particular the Balloon conjecture. I really liked the paper, mostly because I was able to understand it. And when I say I understood it, I mean just barely. Perhaps in the same way I “understand” Chinese. I know I’ve explained the conjecture before, so I won’t bore you with the details. To refresh your memory: some shapes are “like” spheres, when you stretch them in the right way. For example, a cube is really just a sphere, if you imagine pumping air into it. On the other hand, a donut isn’t, because of the hole in it. Di Giorgi believed that a certain class of shapes are really just spheres, after stretching. But many equally brilliant mathematicians disagreed.

It was also around then that I met Eileen, who was working on her doctorate in anthropology. One day, I passed her in the library, and was struck by the fierce, determined way she had of reading. I myself never studied particularly intensely, and was always getting distracted. Eileen had a huge book in front of her, open to a map of South America. I remember her black hair, as well as the same intense stare she still has today. For me, it was love at first sight. But sometimes memory plays tricks on you, and I may have been more intimidated than anything else. Either way, I’m sure you don’t want to hear much more about this. 

Eileen and I would spend whole days at the museums, where she would explain the paintings and sculptures. To me, it was like discovering a whole new world. She talked about Renaissance artists with a passion I had never imagined possible, and I was fascinated by her way of seeing things. I also remember explaining some mathematics to her, and being immensely impressed by her ability to understand it. This must have been during the autumn of my fourth year, because I always associate that period with fallen leaves. I don’t know whether Eileen and I thought much about the future. Perhaps we did, but I can’t remember. I also think that the Balloon conjecture was taking up more and more of my time. 

In mathematics, one often seeks to disprove a certain statement by coming up with what’s called a “counterexample”. For example, if I said: “all prime numbers are odd”, a counterexample would be the number two, which is even and prime. As I approached the end of my doctorate, I became increasingly interested in finding a counterexample to the Balloon conjecture. Because many brilliant people believed Di Giorgi wrong, I was caught in the same way of thinking, and tried many ways of refuting him. But there was always some subtlety that eluded me, and I became frustrated. By then, I had tried many ways of stretching and gluing the shapes that I knew about, all to no avail. At one point, I even announced to colleagues that I’d found a counterexample, only to be humiliated when they found an error. Before mathematics, my whole life had been a sequence of failed starts. It seemed as though things were heading that way again. 

I don’t remember exactly when my thinking changed, but there was always a remarkably delicate way that my counterexamples failed. Years later, I learned of an Einstein quote which says that nature is clever, but not dishonest. Although I couldn’t have known it then, this was also true of the conjecture I was working on. If it was wrong, it didn’t make sense for all of my attempts to fail so precisely. So after some time, I decided that Di Giorgi must be correct. And then, within a few weeks, I managed to prove it. For several years, I’d been trying one thing, when all I had needed to do was try the opposite. That was how my mathematical career got started, more or less. Some colleagues judged the work important, and things got going.

I’m not sure why I decided to write this letter. Eileen thinks you might be having some trouble, but you don’t need to tell me about it. I’m sure it’s nothing serious. It would be dishonest of me to draw a moral out of these ramblings, mostly because I can’t really remember why I did the things I did. Nowadays, it feels as though anything I achieved, happened by accident. At the same time, if you’d known me then, I’d probably have told you a very different story. Old men have funny notions, which they like to pass off as wisdom. 

If the trouble you find yourself in feels rather more serious, remember that sometimes young men may also imagine problems where there aren’t any. Looking back, it seems like I was creating obstacles for myself, mostly through impatience. Things turned out fine, despite my efforts to the contrary. Either way, I doubt that any of these silly musings will be of much use. 

I can’t rely on my memory like I used to. But some things are still clear. In the fall of 1976, a week after I proved the Balloon conjecture, Eileen and I had our first child. The two greatest things that ever happened to me, occurred in the space of a few days. And although I’m sure of very few things, this I say with certainty: you were the beginning of everything that really mattered. The best thing that mathematics ever brought me was Eileen and you. The other day, a science magazine asked me to write a personal article about my proof. But I couldn’t recall the details, because it felt like things just fell together. I guess, to state the obvious, this letter is my first attempt at writing it. In that sense, it’s logical for me to send it here first. The reason being, son, that whenever I think about those times, all I can remember is your mother, and you.

By Oisin Kim for CRoB Digital, edited by Haley Zimmerman

Artwork by Madeline Elmitt