While I was traveling for the holidays this past weekend, I listened to several episodes of “The Joy of Why?” podcast, which explores new and revolutionary ideas or advances in mathematics and the sciences. I could probably write at least one article about each of the episodes, but the topic of one of them — category theory — is what I will try to cover for this week’s column.
Category theory, while a mathematical topic, is typically not covered in undergraduate or even graduate math programs, at least not directly. Described sometimes as “the mathematics of mathematics,” its goal is to formalize how mathematical objects are related to each other. Such objects can be related in lots of different ways, and each relation will give rise to a given set of patterns or other interesting properties. One can think of it as a mathematical game of categorization – thinking about a group of objects and placing them into categories based on how we’re relating the objects to one another.
A first question may rightly be: “what are these objects?” Well, they can be anything—which is what makes category theory incredibly abstract. However, the more abstract something is, the more analogies you can make for it. For instance, a group of mathematical objects could be a set of computer programs that perform certain functions. We could categorize these programs by their length, runtime, what types of functions they perform, what types of computer architectures they can run on, and so on. This is one recent application of category theory to the everyday sciences.
Another analogy involves linguistics. We could think of a language, or groups of languages, and categorize the words or groups of words in lots of different ways—part of speech, meaning, how often they’re used, etc. Chemists apparently have also used it for categorizing the vast array of molecules they deal with in the lab.
What was interesting about the podcast was how the guest, category theorist Dr. Eugenia Cheng, discussed potential applications to phenomena that most people would reserve for the social sciences. She talked about how category theory is very related to the concept of intersectionality, which has been used to study relationships among people from different racial, cultural, or socioeconomic backgrounds. Her belief seems to be that the possibilities for category theory are endless.
This idea was interesting to me, as it got me thinking about how math could be used to describe certain societal phenomena or other large-scale things that are typically thought of as too difficult to describe quantitatively. Maybe the answer lies in increasing the abstraction of concepts while retaining enough structure — this would be known as a set of “axioms” in mathematics — that can make those abstract concepts useful.
This also brings up the debate between pure and applied mathematics (which Stevens, for one, recognizes in the name of its math department!). Certain mathematicians believe there is a fundamental separation between these fields based on the interests and goals of pure vs. applied topics. I find this division to be rather silly, and it would be nicer if there were a more symbiotic relationship between the two. It seems that Dr. Cheng agrees, as she connects her very pure-math field to real-world issues throughout the interview.
Category theory, if anything, shows that we can rethink how we differentiate certain objects from each other, sometimes leading to miraculous breakthroughs. Since applied mathematicians rely on the foundational proofs derived by their “purer” counterparts, and pure mathematicians can gain intuition by thinking of specific applications of their work, we should make like the category theorists and remove this division between types of math. This could also help further break down barriers between math and the social sciences, which I think would also be a great benefit to society.
