On Friday morning, I meet my friend Nick so we can start working through Robin Hartshorne’s Algebraic Geometry together. We are both hungover. It feels like my brain is switched off—my tongue is reaching for words and not finding them. I make mistake after mistake, I am baffled by even the simplest exercise. But it’s not from the hangover, I haven’t done any maths in the four months since I graduated, and I’ve already forgotten basic definitions. Over four hours, Nick and I get through a massive two and a half pages.

‘I hate maths’, I say.

‘I hate maths, too’, says Nick.

‘Why are all my proofs by contradiction? Like, how do you even do construction? *There exists*. Then what? OK, let $\frak{a}$ be the ideal generated by… generated by a set?’

‘I can’t do construction because I have no intuition. *There exists* what? I’m so bad at it. Wait, what does it mean to be generated by a set?’

‘Something to do with generators?’ says Nick. I search Wikipedia for *ideal generated by a set.*

‘Nope! It’s the intersection of all ideals containing the set. Yeah, I wonder if it has anything to do with generators?’

‘Why would you say *generated* otherwise? Oh my God. Also, why say *zero set* when you could just say *locus*? We already know what a locus is. Check this out, he said *clearly*!’ We both laugh. ‘*Clearly*. I hate when they say clearly.’

‘Oh my God. We’re only at paragraph two.’

After our two-person seminar, my brain physically hurts and I want to have a nap. I try to do other work but it feels like there is glue in my head. I can barely read. Later in the afternoon, I write to Nick, ‘Can we please do that again next week? That was so fun, I miss maths so much.’

*

My friend Ali says people who like maths are masochists. I think this is true in a lot of ways. For instance, we brag about how many pages it takes us to write a proof, how many whiteboards we cover to rewrite a lecturer’s three-sentence solution, or how long it takes us to explicitly write out a textbook’s perfunctory ‘this proof is left as an exercise to the reader’. We aren’t satisfied with one proof, so we try another and another. We brag about how many times we wrote to show or there exists, about whose thesis had the most words, whose had the least (it was mine, I was concise, but I also had the most equations). We get hung up on wording, awkward phrasing, or how boring (too explicit) one text is, how devilish (too dense) is another.

A lot of these hang-ups have to do with writing and language. That’s because doing mathematics means doing writing, using language.

As a poet, I’ve been asked a few times to comment on the intersections between mathematics and poetry. I tend to hesitate in my reply. I flip-flop. I say either ‘they are very alike!’ or ‘they are hardly alike!’ Mathematics and poetry push the limits of language a lot—in terms of what words mean, how new words and concepts rise up out of imagery, how meaning shifts to accommodate advances in understanding. And although the points of likeness and difference between the two disciplines are ripe, I don’t consider these things in a conscious way when I’m doing maths, or poetry. Because, for me, to do one means to decide, if temporarily, against the other.

I think, when I am asked this question, that people want me to talk about the mathematics of poetry, the mathematics *in* poetry. I tell my parents I am writing this essay and my dad says, ‘It will be so easy, Anupama!’ But it isn’t, because I carry a lot of insecurity about my two competing, distinct obsessions and my role as commentator. When teasing out relationships between these seemingly unrelated categories and disciplines, though, I find them to be personal and human. The more I try to describe mathematics, the more I realise my understanding of it is heavily influenced by my understanding of poetry, and how poetry works. I want to talk about the poetry of mathematics, the poetry *in* mathematics.

*

In writing poetry, I am concerned with ambiguities and layerings, condensing content to make unexpected connections seem possible. I don’t want to, and I don’t think it is possible to, write explicit poems where meaning is exact to the words. Mathematics—though requiring clarity, exactness, and explicitness of expression—often trades in a similar layering of meanings. Ambiguities, correspondences, and unexpected connections being made to seem possible are fruitful avenues for innovation—unification is a goal in so many sciences.

A year ago, for instance, I took a class on category theory. We started the class by proving a result about fibre-products of abelian groups—that a certain map exists. A map is a kind of machine that takes in something and spits out something. Then, using an almost identical method, we proved an identically-worded result about fibre-products of *topological spaces—*also that a certain map exists. The only difference was that instead of taking definitions with respect to abelian groups, we were taking definitions with respect to topological spaces. The thing is, you don’t even need to know what a fibre-product, an abelian group, or a topological space is—only that it felt extremely wasteful and dull to prove identical results in essentially identical ways. The correspondences between these proofs made it seem possible that one could unify these apparently distinct things: abelian groups and topological spaces. One can: they are both *categories with pullbacks*. In other words, mathematics was in want of a theory capable of proving certain related results in general, rather than proving analogous statements ad infinitum. That theory is category theory.

‘Metaphor is for most people a device of the poetic imagination’, though our ‘ordinary conceptual system… is fundamentally metaphorical in nature’.^{1} Pure mathematics is conceptual and extremely subject to the device of metaphor. For instance, we are accustomed to the concept of *distance*, and this familiarity allows us to formalise the mathematical definition of a *metric*, and subsequently helps us understand it. The metaphor: a *metric* is a *distance* map. It takes in two points and tells you how far apart they are. Because of the abstract nature of many concepts in mathematics, there is a ‘need to get a grasp on them by means of other concepts that we understand in clearer terms’.^{2} And this is precisely the narrative of that first lecture on category theory: mathematics wanted a theory that engaged a kind of ontological metaphor. Some areas of mathematics are older, understood ‘in clearer terms’, their techniques more familiar. In that class, we allowed the relatively more concrete study of abelian groups to guide us. Our approaches were ‘morally’ the same, if not literally so.^{3}

*

Over the past few years I have often been told, usually by people who don’t study it, that mathematics is a language. I don’t always know what is meant by this. What is a language? Mathematics is *like* a language—it has a kind of vocabulary, a kind of grammar. But mathematical ideas are conveyed *through* language. I can’t read French very well, and so I can’t really read maths papers written in French. I can’t because the kind-of-vocabulary of mathematics is highly contextual, and its kind-of-grammar is *also* the grammar of the language through which it is mediated. However, fluency with mathematical concepts is a consequence of practice—not just of an encyclopedic understanding of the language. It is very easy to know the definition of a word like *irreducible variety* (even if it has to be translated from the French), for example, and still have no real notion of how to prove that a given variety is irreducible. To prove as much, you might need synonyms (‘equivalent definitions’), or specific characteristics of irreducible varieties, or notions of what irreducibility is *not*, or more—all of which are not immediately available in the definition of an irreducible variety.

Maths is a kind of battle, too—to know enough, to know it well enough and fast enough, to recombine and reuse ideas, to compute and calculate, always practising, writing, desperate to understand. Ultimately, to communicate. Good, clear writing is required. Beauty, though, is prized. Even as a student, there is a vision of beauty and an appreciation of it. And I don’t mean appreciation in the way of awe at a spectacle, I mean that the appreciation of beatify is institutionalised in the pursuit of mathematics.

My maths supervisor tells a story about beauty. As he puts it, the ‘definition of compact (in Walter Rudin’s *Principles of Mathematical Analysis)* was, perhaps, one of the most elegant possible seven-word combinations: ‘every open cover has a finite subcover”.^{4} One time, he was flicking through a book he’d lent to me (the second edition of Victor Kac’s *Infinite Dimensional Lie Algebras*, for those in the know), and paused just to say, ‘This is a beautiful book.’

Beauty, elegance, clarity, refinement—these are all highly desirable characteristics of maths, and they are qualities that are fostered in the classroom, discussed among peers, taught and passed down by mentors—who sometimes (that is, often) savagely edit your writing. I dream of having such mastery of my field that I can concern myself with style, with the aesthetics of my work. They are qualities that distinguish. Elegance in both writing and results is valued and respected, conjectures which are beautiful and refined are seen as better. As worthier, even, of pursuit.

So, maths is like a language because it *extends* a natural, spoken language. New words, symbols, expressions are developed in order to capture statements about a particular field of study. But, the language through which mathematical ideas are conveyed, are mediated, is extremely important. The exact meaning of words and the drifting of meaning in context are critical to understanding. Maths, to me, is about words-individually and in sentences. And mathematical practice is concerned with explication. Clarity of writing is paramount.

*

In the last few weeks of my degree, I would fall asleep dreaming of billowing squares and spheres, lines and arrows, infinite graphs. The boundaries between my personal and student life started to disintegrate. Math began to inflect every action I undertook. I thought about bundle morphisms while washing dishes; I imagined commutative diagrams describing the two actions of crossing to the opposite street corner—walking forward then crossing left is the same as crossing left then walking forward; I googled ‘groups’ to verify a definition, and was surprised by the first Wikipedia result: *a* *number of people or things that are located, gathered, or classed together*. Then I typed ‘crystal math’ to find a necessary resource for my thesis about $\widehat{\frak{sl}}_{2}$-crystals, and was utterly shocked to find recipes on how to cook methamphetamines. There was no longer an outside to my math-mind. Everyday shapes would undergo almost hallucinogenic homeomorphisms while I observed them—dinner chairs collapsing into genus-5 surfaces, glasses into discs.

Doing maths means doing writing, using language. But, maths is also an embodied practice. It is entirely bodied, because it involves the entire body. When it goes badly, the whole body suffers. When it goes well, the whole body is euphoric. For me, it is this full-bodied experience of mathematics that truly divides it from poetry. Poetry is hard, but it doesn’t give me headaches and make me need to nap. It doesn’t make me feel desperate and worthless and untalented and lazy. It doesn’t overpower my ability to recognise and label the boundaries of reality. Poetry is rewarding, but the stakes are lower. The reward is not like what happens in my body when I transition from the desperation of having no understanding or intuition on a concept *at all*, to the dazzling insight brought by unexpected perspectival shifts. Mathematics is fevered, consuming, total—it leaves no space behind, it is absolutely draining. It takes over my entire mind, until there is no room left to remember even how to open jars or touch on my Myki. Writing, for me, just is not. This might mean I am undedicated to poetry, or bad at maths. (Probably both, to some extent.) At the very least, it is not possible to do maths and be undedicated—there is no use for the dilettante mathematician.

*Anupama Pilbrow co-edits *The Suburban Review* and co-manages Poetry Donut: The Melbourne Poetry Reading Group. In 2015, she received the Dinny O’Hearn Fellowship, and her chapbook *Body Poems* will be released in 2018 with Vagabond Press. Her poems, reviews, and essays have been published in journals and anthologies including *Cordite Poetry Review*,* Rabbit Poetry Journal*,* JEASA*,* Southerly *and the* Hunter Anthology of Contemporary Australian Feminist Poetry*. Her work often deals with diaspora, dialogue, exchange, and gross stuff. She lives in Narrm/Melbourne.*

- Lakoff, George and Johnson, Mark.
*Metaphors We Live By*. [Print] Chicago, London: The University of Chicago Press, 1980. (3,3) - Lakoff, George and Johnson, Mark.
*Metaphors We Live By*. [Print] Chicago, London: The University of Chicago Press, 1980. (p. 115) - Murfet, Daniel. ‘Lecture 1: Categories.’ MAST90068 Lecture Notes. [PDF] Melbourne, 2016. Available at: http://therisingsea.org/notes/mast90068/lecture1.pdf [Accessed 3 Jan. 2018]
- Ram, Arun. The Little Book on Convergence. [Personal copy, manuscript] Melbourne, 2016.