Math and the incommunicable marvelous

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A multi-colored abacus used to calculate simple mathematical expressions. Photo by Skitterphoto from Pexels.

There are certain passages from books I’ve read that stay with me, and that later come to define certain moments or realizations. I imagine this is true for other people as well —  a beautifully layered series of words somehow winds itself around your thoughts and stays there. Sometimes it’s the result of an unexpected collision with something of beauty or peculiarity unarticulated but deeply known in your own mind. Other times it’s the words themselves — their collective motion and rhythm turns about and then settles in your mind, sometimes independent even of the passage’s meaning.  

For both reasons, there’s a passage from “A Portrait of the Artist as a Young Man” that I return to from time to time. It captures the aesthetic beauty and kinesis of math: “The equation on the page of his scribbler began to spread out a widening tail, eyed and starred like a peacock’s; and, when the eyes and stars of its indices had been eliminated, began slowly to fold itself together again. The indices appearing and disappearing were eyes opening and closing; the eyes opening and closing were stars being born and being quenched.” Even as an undergraduate math student with a crude mathematical skill set and limited mathematical experience, Joyce’s words affirm a characterization of an experience I have had: an encounter with something complete, whole and absolute, that with its beauty, compels you to draw nearer. 

Of course, that sort of encounter isn’t unique to math. It can happen in nature, or in a relationship, or in anything you’re studying — in whatever way it surprises you and draws you closer; in whatever way it saturates your mind and fills, but also opens, your mind to perceptions of beauty.  

However, and understandably, describing that sort of encounter in the context of math may seem exaggerated — even false. Part of the reason is that the way math is presented to us in school isn’t really what math is. Math in high school often involves learning how to do a certain type of problem in a specific way, and then repeating that procedure on a series of similar problems for homework. When I got to college I was surprised that my first math assignment, which is what I now consider to be a normal math assignment, was to write a couple proofs involving definitions and theorems we’d covered in class — no list of problems that I could grind out quickly while sitting in the back row of French class just by replicating a certain procedure (Désolée Monsieur Afantchao). Now, there was no procedure to follow, no single answer I could draw a neat box around at the end. Instead, I’d have to…well I’d have to think really hard about it. I’d have to be creative with using what I knew. I might have to collaborate with other students in the class to see what they came up with — and even when we might have completely different proofs written down, it was possible that each could be “correct”.  

Aside from how it’s traditionally taught, another reason why it can be challenging to express encounters of beauty in the context of math is that sometimes those encounters rely on surprise — and it’s impossible to be surprised by something if you have no initial expectations to confound. I expect the weather to be within a certain range of possibilities in October. That range is based on personal experience from having lived in the same place for most of my life. So when it snows in October, I’m surprised because it defies my expectations. In math, though, I don’t always have enough experience or intuition to have expectations. Sometimes, it’s not until after I’ve seen a theorem in multiple classes and also tutored it that I have a vocabulary and perspective to appreciate how beautifully counterintuitive it is. In a class sophomore year, I don’t think I had any expectations about how many prime numbers there should be. It’s not something I had really thought about before, so when I learned that there are infinitely many I wasn’t too shocked — there are, after all, lots of things in math that go on infinitely. Yet as I thought about it more, I realized how strange and fascinating the infinitude of the primes actually is. If a prime number can’t be divided by any other numbers, it really is quite remarkable that you can keep finding larger and larger numbers that can’t be divided by any of the millions and billions of numbers below. It’s even more remarkable that something so counterintuitive can be proved in just a few lines, and in many different ways.  

In a book of Borges short stories I was reading this week, there’s a beautiful sentence I copied down. It says, “[His] memory was a mirror of secret acts of cowardice. What story could he tell? Besides, the guests demanded marvels, while the marvelous was perhaps incommunicable …” Sometimes, the beauty of math is marvelous in incommunicable ways. I know, too, that there are parts of other people’s worlds and experiences that are also marvelous in incommunicable ways. It is the combination of that shared incommunicability; and our human inclinations to ponder, express, replicate, and then articulate beauty; that draws us nearer to one another — a marvelous and incommunicable thing itself.  

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