red string
how do you know if you love something?
27 April 2020
A theorem in mathematics is often a conditional statement, coming in the form of “if, then”.
Writing a proof to a theorem is like digging a tunnel from one end of the statement to the other. You start at one location (“if”), and you need to dig a tunnel to reach your destination (“then”). You know where it is you want to end up, and you have a feeling it should be possible to get there from where you are standing, but you’ve never gotten there from where you are. Anyhow, you’ll never get there if you don’t start digging. So you start digging.
Once you are underground, though, you don’t always know where the destination is relative to where you are. You kind of have an idea of the general direction you want to head in, but sometimes there’s bedrock you can’t get past, so you need to get around that, but after you’ve done so you might have no idea where your destination is, relative to your new position. Maybe your detour wasn’t so helpful after all, or maybe you should have gone around a different way. Or perhaps the original direction you had in mind was misleading. Anyhow, you’re lost.
Where were you going? The destination, right, what can we say about the destination? Sometimes, we can say quite a few things, and we know what sort of path leads up to the destination, so what we need to do is connect our path to that path. In this way, we can dig from both ends to form the connection.
Of course, not every conditional statement is true. Sometimes we can definitively show that two ideas are not connected, and other times we don’t know whether or not there is a connection. But once we’ve shown that the connection does exist, can we say that the two things were ever not connected? We may not have known the path, but it was there. Confirming a connection feels like discovering a truth, something undeniable, infallible, an idea that is true regardless of who is thinking it.
I thought I might love math.
It’s happened at least twice that I’ve fallen asleep thinking about a math proof that I couldn’t figure out. The day before a problem set was due, I’d turn a problem around again and again in my head, searching for a solution to no success. Defeated in the early hours of the morning, I resigned myself to sleep, figuring that I should rest enough to pay attention in math class the next morning. But I guess my subconscious or whatever is responsible for dreaming or what my brain does when I’m sleeping could not let it go. I woke up with a solution, and it wasn’t dream-gibberish—somehow the solution worked.
It felt like fate, to be able to come up with a path forward through the machinations of sleep. Of course, this is a process I had very little control over: sleeping through my problem set, or perhaps a midterm, would most likely not give me the answers I needed. The process is also completely unreliable. Sometimes I go to bed resigned, and I have to turn in a problem set with an incomplete or incorrect proof, to show that I did actually try to solve the problem. I’ve always had an amount of shame associated with turning in incomplete work, and so I’ve rarely ever handed anything in that wasn’t complete. But in writing a proof, I always feel that I either made a connection or I didn’t. There is no way to know if I’m on the right track if I don’t know that I am headed in the right direction. I can only show the thinking I have already done in an attempt to say “I tried, but fell short”, but this feels like a form of giving up on myself, disappointing in the way that telling yourself “you tried your best but didn’t succeed” is. I dreaded turning in incomplete proofs, so I can’t rule out that I was so anxious to complete the proof that my subconscious kept up the work I had started consciously in a desperate move to preserve my own self-worth. So maybe it wasn’t love, but self-preservation.
But did it mean something that my brain clung so vividly onto math in my sleep? Did it mean more that I was willing to cut my losses and try to get rest only to prepare to engage with more math the next day? Does that count as love?
At the very least, I can say that math captures my attention. Somewhere along the line I’d developed an unsavory ability to fall asleep in class, or rather, almost anywhere. I’d say it started when I overloaded my schedule in the second year of high school with water polo and swimming and too many classes. There was a certain period of time where I averaged maybe 3 hours of sleep a night: morning swim practice started at 6 AM sharp, ran until 7 AM, at which point my first class of the day would start. My last class would end at 2:45 PM, and by 3 PM I was back in the water, swimming until 5 PM. I would get home exhausted, napping until dinner around 7 PM, and it was only after dinner that I would start on my schoolwork for the day, maybe around 8 PM. Now, it wasn’t true that I had 7 hours of homework a night, but it was true that as the hours dragged on I got less and less productive. A task that should have taken half an hour would take twice that, easily. Finally I’d wrap up my work around 2 or 3 AM, and my alarm would go off at 5:30 AM the next morning. This schedule was only sustainable by planning to sleep through certain portions of my classes, as I could never manage to swim sets with my eyes closed.
Ever since that time, exclusively when I’m in class, I’ve conditioned myself to be able to sleep as soon as the thought of it enters my head. This is an extremely inconvenient habit. It’s resulted in the fact that there are very few classes I’ve taken in which I have never taken a nap. But those classes do exist, and are all the more notable for that fact. I did not sleep in my literature class my senior year of high school, and I have never slept in any of the math classes I’ve taken in college.
Perhaps it can be argued that I decided not to sleep in those classes, and that was why I didn’t. In either case, I had come to view a course that could capture my attention for the entire duration of the class as a luxury. Somehow, I had decided that there was not one minute of these classes that I could miss. So if this is love, is love a decision?
When I decided to go to Olin, I felt that I had committed myself to engineering. I wanted to become an engineer; I wanted so badly to love engineering. During the final day of my senior literature class in high school, after the decision to attend Olin had been made, my contribution to the class reflection devolved into a horrendously embarrassing fit of crying in front of my teacher and classmates, out of the realization that I’d likely never take another literature class again. To rationalize this episode, however misguidedly, I told myself that giving up literature classes was the blood sacrifice I needed to make this thing with engineering work. I was giving up something I loved for an engineering opportunity of a lifetime. I had to make engineering work. And despite all this, once in college, I would unfailingly find myself waking up halfway through another engineering lecture. Perhaps this is a normal phenomenon, but to me it felt like a sign. If becoming an engineer was a dream of mine, it was sleeping in part that had woken me up from it. I didn’t want to pursue something that I regularly zoned out of. Falling asleep in a subject I was supposed to love felt like a betrayal, so maybe I decided that I couldn’t love engineering. I looked elsewhere. After taking one math course I didn’t fall asleep in, perhaps I told myself that was love, and decided I had to stay awake in all the math courses that followed.
Is telling yourself that you love something the same as actually loving it? Maybe the reason I didn’t fall asleep was because I was afraid of getting completely lost in a class where one missed definition or one missed statement would leave me stranded in an increasing state of confusion. Maybe I didn’t want to sleep because to sleep was to betray this new decision I had made. Perhaps love is just the name I gave to this fear; perhaps when I feel intensely about something, love and fear are so intertwined they become indistinguishable.
I could theorize endlessly about this: perhaps it was just a matter of the professor. Maybe I had a really excellent literature teacher (I did). Most of my math professors have been similarly excellent: Professor Schultz (better known to students as Andy), my Number Theory professor (and Intro to Real Analysis and Galois Theory professor– can you tell I enjoy his classes?) who tells stories of his kids that make the whole class laugh; Professor Diesl, who made tea for us during Abstract Algebra office hours; Professor Hoffman (better known to students as Aaron), whose excitement inspired me to learn more mathematics after Linearity I & II; Professor Ambrus, who handed out chocolates to the class during his Discrete & Convex Geometry exams; and Professor Simonyi, who reassured me that I was welcomed to stay in Graph Theory, even after (completely!) bombing his first midterm; have all inspired me greatly, within mathematics and outside of it.
Anyhow, I had decided I was all-in for math after my sophomore year at Olin. I chose to spend my semester abroad in a mathematics-focused program in Budapest. During my time there, I experienced for the first time getting through a math class out of fear, and a belief that I had no choice but to understand the material, or else I would just straight up fail (something else that had never happened before). In those classes it was desperation that kept me awake, furiously copying down the increasingly alien symbols on the board. In one of these classes, at least once a lecture, the professor would pause, examine his handiwork, and mutter, “What letter [out of both the Greek and English alphabets] haven’t I used yet?” This would generally happen after he had exhausted some notation with subscripts. At first it was amusing, a sort of joke shared between the class and himself. It was self-congratulatory in a way: since each letter signified a distinct object, the work on the board was a unique code, and as a class we were conspirators sharing it. But as the ideas got increasingly complex, the items on the board became an alphabet soup to me, the code increasingly difficult for me to decipher. Instead of being amusing, the professor’s quip contributed to a feeling of doom. In one class, as I copied notes, tears started to fall, and I desperately tried to hide them (an impossible task in the front row) as I continued copying down what the professor was writing, two boards behind his current one.
Why do I care if I love a math class anyway? I clearly care about math, and isn’t that enough? Isn’t that enough reason to pursue something? I guess there’s a fear of not being able to sustain myself through a decision I’ve made. My high school swimming story ended that sophomore year of high school. Good for me, right, choosing to get a proper amount of sleep over my commitment to a sport? It didn’t exactly happen that way. I was supposed to get out of the pool, grab a kickboard and swim paddles for the next set of drills, but on my way back to the swim lane, I found myself on my back, staring at the sky. I had passed out on the pool deck, hitting my head on the water polo goal post on the way down, and now the paramedics were here. They said I had a concussion. I was out of school for the remainder of my sophomore year.
For a long time I’d say it was the concussion that kept me out of school, but it wasn’t exactly like that, either. I had tried going back to school, only to break into a cold sweat in class, gripping the sides of my desk, trying to focus on my breathing without huffing and puffing in a way that would disturb my classmates. Sometimes tears would just start rolling down my face. Despite all intentions I had of remaining discreet, my teachers noticed and redirected me to a school counselor. These were panic attacks, they said, and maybe I should consider lightening my course load. When I refused, they told me that they’d seen this before, high school students overworking themselves, and for what? I was participating in a rat race, and should just focus on the things that made me happy, or at least reduce the amount of stress by cutting out the things that induced it. But what made me happy and what caused me stress felt like the same thing. My classes made me happy, but I couldn’t give up even one assignment for one class. Swimming and water polo made me happy, but I couldn’t give up one practice. My inability to prioritize one over the other meant that I didn’t sleep, but I had to sleep.
After the concussion, whatever it was that pushed me through my schedule had disappeared. All that was left was the stress, so I cut out everything. I stopped swimming; I stopped going to school. I worried that I just didn’t want it badly enough, that if I did, I would have found a way to make it all work. I told myself whatever I chose to do in the future, I would want it badly enough and I would never quit like that again. So I needed to love or have passion for– or perhaps what I’ve been referring to as love is actually passion—what I was doing to ensure that I did not give up on it.
So pivoting again, this time away from engineering towards math, I guess I decided that I loved math, or that I had to love math, which makes it sound kind of awful. Is it still love if I am holding myself hostage to it? But this characterization feels wrong, because I’ve found so much beauty and joy in mathematics. It’s the feeling of tossing a ball in the air, and without looking, or really paying attention, catching it again– or rather, having it fall back in your hand, because you know the ball will land there. At its best, it feels like everything is right in the world. And how can it not be love, if it is something that makes you feel like everything is right in the world?
Math comes with logical certainties. It feels good to say something with certitude, which is something I feel like I’ve been able to do less and less. I can’t even decide whether or not I love math, man. Since it isn’t obvious, does it mean that I don’t love math, or like it for that matter? Is differentiating between the meanings of love, passion, like, enjoyment, care, etc. just an exercise of drawing various lines in the sand, movable depending on context? The boundaries in math feel much more rigid.
This rigidity can feel impersonal, existing outside of me in the way that numbers can go on forever. Maybe a number doesn’t exist until we need it, or notice its existence, or maybe numbers come from a construct so generous that we will always have the number we need. Do numbers exist for themselves, or do ideas exist only once acknowledged by someone else? Is infinity a matter of perspective, or is the quality of going on forever also the quality of existing anywhere you care to check for it? Maybe my love for math is something akin to a love for a god, a god that I admire, awe, and fear, but one that is indifferent to me. I may be building my sense of self around math, or the idea that I love it, in the way that a vine climbs a tree. The tree is indifferent to the vine, but to the vine, the tree is a source of life, a path towards the sun.
Definitions in math are satisfying to me because they are unambiguous about what they are. They do require a specific context: a perfect graph is not the same as a perfect number is not the same as a perfect field (of course, the thread here is that mathematicians have found constructs in math they believe are worthy of being called “perfect”). Within those contexts, some definitions implicitly tell us what the object is and what it is not. If a set is infinite, it is not finite. Zero cannot be one, unless we allow for everything to be nothing. Other definitions do not require this duality: a set can be open or closed, or both, or neither. But the conditions for each are defined in a way that makes it clear what quality is being described. The number “five” for instance, can describe fingers on a hand, or value of a nickel, or minutes before a meeting, which are all dramatically different things, but share an essential quality.
On the other hand, I’ve given myself plenty of labels to try and understand my identity, but to be honest sometimes I wonder if I’ve lost a foothold on who I am in this attempt. None of those labels are ever as straightforward as they initially seem. I took engineering classes, and asked myself if I was an engineer. Do I share that essential quality? What is the essence of an engineer? Then there’s a classifier like “depressed” and I wonder if I am because people with credentials told me so, from their medical and professional perspective. What is the essential quality of depression?
Depression is a word that has made it easier for other people to accept my actions but not for me to accept myself. I no longer know what to do with the word or with the actual thing. The diagnosis feels like a crutch, an excuse for the thoughts and actions I have that are difficult for me to accept, but maybe that’s just the depression talking. And if it is the depression talking, where does that end and where do I begin? Maybe it is all just me and regardless of what it is, it is something that I guess I must come to accept (and perhaps embrace)? But then when people ask, “Who are you?” I feel so much shame in talking about these parts of me. At the same time, it feels dishonest to only highlight the parts of me that I think other people want to see, or to present a version of myself that I want other people to see. It feels much easier to put a label on the parts of me that I am not proud of, and then point to them and say, “Oh, well, I have depression, so that is why I feel the way I feel. It isn’t really me, it is the depression.”
I wish I found comfort in the clinical framing of science. My depression might be caused by a lack of a chemical in my brain called serotonin, which can be supplemented by medication. It is maybe this neurotransmitter that is failing to make the right connections in my brain, failing to inspire the electrical impulses that constitute my mood and direct my actions and construct my sense of self in the right way. But these failed connections are the thoughts that I have, the thoughts that do construct parts of my sense of self. That isn’t to say that I didn’t take the medication; I did, but then I stopped, because I was afraid that the parts of me that grieved so intensely, at everything and nothing, would be blinked away like a bad dream. If I could do that, how could I begin to understand myself at all? And if we were to really stick with this metaphor, if nothing else, math introduced the idea that my dreams can bleed into reality, like when I completed the proofs in my sleep. Not all of my dream-world constructions fall apart in my waking one.
In math, once we have the definitions, we know what qualities of objects we are talking about, and from those we can make truthful statements, or theorems, logical certainties about these qualities. If I love math, then will math sustain me? If I am depressed, is the logical certainty that I should want to become not-depressed, or at least have the duty of trying not to be? Or are our definitions and theorems more fluid and less rigid, our contexts in a moving or subjective frame? Perhaps in this motion, there is room for contradiction, space that is not filled, allowable gaps in our logic. In math, logical gaps and contradictions within its structure are not allowed; consistency is valued over completeness, so while the gaps are there, the connections are left incomplete, which is to say the connection is not there at all. But I know that the same things that bring me joy can make me feel like I am not enough. I can lose myself to my depression, but without it I would not have myself at all. It is here that math is not enough for me; it cannot untangle my contradictions and hypocrisies, cannot sustain me through them. So if it can’t sustain me, in the end, do I love math?
I think I do.