Of All Their Possible Variations

“Mathematics, developing as it has from the middle of the nineteenth century up to the present day, has not ceased to create new spaces for the understanding.” 

— Fernando Zalamea

“The heart and soul of much mathematics consists of the fact that the ‘same’ object can be presented to us in different ways.”

— Barry Mazur

“Q: what can we do with quantum weirdness?

A: change the very language we use to discuss it!”

— Bob Coecke and Stephen Schanuel

At the beginning of 2020, I started a reading group for the book Seven Sketches in Compositionality, an introduction to the mathematical field of category theory by Brendan Fong and David Spivak. The book is explicitly geared toward non-mathematicians, framed around a series of practical examples and a central idea—compositionality—as a language for transformation and combination between different mathematical objects. Their thesis is that categorical thinking is not inherently reliant on formal mathematical training, but is instead—as in the diagram below, which they use to represent a monoidal category—relatable to an everyday context.

There were a few reasons for my wanting to start the group. I had been meaning to learn about category theory specifically for a while—it’s related to work that I do, and is something that quite a number of people I like are interested in, and I wanted to understand it (or at least, gain some idea of what it was exactly). Perhaps more markedly, it came from a nostalgia for doing maths with others, formed from somewhat rose-tinted memories of an undergraduate engineering degree, and sharpened by having tried to learn set theory by myself a few months before.

Since late summer of that year, the group has been on hiatus, due to a combination of the end of lockdown, good weather, and an increase in the complexity of the text. Despite being a relatively short period, however, there were a number of times where I found both the subject matter, and the context of learning with friends truly revelatory and transformative, in ways that I didn't expect.

It’s important to state here that I’m no expert in category theory, and neither will I make much of an attempt to explain its tenets here. One thing I noticed, however, when I was starting to invite people to come to the group, is how hard it is to find a satisfying definition of what category theory is, let alone articulate fully why I was interested in it. The reasons for this have become clearer with time—namely that a ‘category’ is not a definable thing, as such, but a lens for interpreting (mathematical) objects.

Category theory is often referred to (affectionately, and by its proponents) as ‘abstract nonsense,’ or, in the case of Fong and Spivak, ‘the primordial ooze’ where the definitions of important proofs feel abstract, non-sequitur, and sometimes circular. Pragmatic definitions tend to go something along the lines of describing it as ‘maths of maths:’ a tool for understanding and abstracting the representation of mathematical ideas, such that one can, by forgetting a few details, start to see the links between previously disparate examples of mathematical thought.

Emily Riehl describes categories as ‘contexts’ in which to think about a class of mathematical objects, with category theory itself providing a language to move between these vantage points. Continuing with the language metaphor, Barry Mazur describes objects and morphisms (two fundamental components of any category) as the ‘nouns and verbs’ of a mathematical sentence.  

F William Lawvere and Stephen Schanuel’s book Conceptual Mathematics uses diagrams as a primary representation from which to explore category theory. To demonstrate this, they frame a thought experiment of Galileo‘s—on understanding the flight path of a bird through space as a function of time—as an example of what category theorists call ‘composition,’ where simple representations (or ‘maps’) of the links between different mathematical spaces may be combined to reason about more complex objects. In this instance, they show how the flight path of a bird can be understood by composing the passage of time with the location of the bird’s shadow on the ground, and its corresponding level in the air. 

In Picturing Quantum Processes, Bob Coecke and Aleks Kissinger describe how finding a different language for quantum theory has allowed researchers to embrace the ‘quantum weirdness’ hitherto seen as a conceptual problem to be solved, rather than an exciting feature of quantum theory. They argue that the language of diagrams—itself informed by category theory and compositionality—is an elegant and exciting representation by which to reason about quantum processes. 

One of the more revelatory points in the group for me came with a passage in Seven Sketches that points out that diagrams are themselves categories—just another lens by which mathematical objects can be considered and manipulated. Not only that: the category of diagrams is a context essential to the understanding of so many mathematical objects! In this sense, categories act as portals to parallel universes, containing the same material things but representing ideas that from one perspective might be impossible to see, but in another are plainly obvious. 

In David Graeber's book The Utopia of Rules, in a passage on the affinity between artists and revolutionary movements, he remarks that “the ultimate, hidden truth of the world is that it is something that we make, and could just as easily make differently.” Mazur's eulogy for Alexander Grothendieck praises, amongst other things, his way of dealing with mathematical objects within the context of all their possible variations. These ideas seem to me the essence of what mathematicians describe as ‘categorical thinking,’ where everything exists in the nexus of its relationships with every other thing, and nothing is assumed but everything can be understood, if only it can be seen from the right vantage-point. There’s a lot to appreciate in how category theorists are so willing not just to admit but also celebrate the abstraction and the human-fabricated-ness of the systems that they contend with, and through doing so, to fundamentally reimagine structures that exist already. 

This is not to claim, of course, that there’s anything inherently liberatory about category theory. Far from it — part of the reason for category theory’s current vogue is to do with its application to relational database theory, specifically to knowledge graphs (also a reason for my interest in the topic). The power of these structures comes from representing information in context, itself an interesting and complex goal, but one that finds immediate uses powering the information infrastructure of Google, Facebook, Uber, and Linkedin. Part of the fairly consistent irony of category theory is that the abstract mathematics pioneered by pacifists and anarchists finds its applications in the most destructive technologies not long afterwards. But, perhaps, this should not be so surprising — there is nothing inherently liberatory about language either, but knowing that so many of our assumptions about it are constructed, and that there are ways to question these assumptions, can be incredibly powerful. 

For me, learning even a little bit about category theory has involved learning a very different attitude to the maths that I had spent a lot of my life trying to understand. One thing about the reading group that always surprised me was how something that felt impossible to grasp to any one of us alone could invariably be talked through and understood together. Mazur, in a really wonderful essay titled “When is one thing equal to some other thing?”(itself a dedication to category theorist Saunders Mac Lane), centers his argument around the diversity of ways that we might “come to terms” with the idea of the number 5. What a way to re-encounter the world you thought you knew intuitively, from an entirely different perspective! I think that this coming-to-terms felt (and feels) so revelatory to me because it was also shared.

Thank you to Owen, Rox, Giulia, James, Joel, SJ, and CATRG friends past and future.

This essay is from the Are.na Annual 2022, themed "portal."

References

Emily Riehl, Category Theory in Context

Barry Mazur, When is one thing equal to some other thing?

Barry Mazur, Thinking about Grothendieck

Bob Coecke and Aleks Kissenger, Picturing Quantum Processes

F William Lawvere and Stephen Schanuel, Conceptual Mathematics

Brendan Fong and David Spivak, Seven Sketches in Compositionality

David Graeber, The Utopia of Rules

Fernando Zalamea, Synthetic Philosophy of Contemporary Mathematics, trans. Zachary Luke Fraser