Category theory in mathematics is a relatively new and provocative branch of mathematics that has found many faithful followers and some critics. By relatively new I mean that category theory notions were first introduced only as far back as 1945. Criticism of the theory is often related to the level of abstraction it requires. But abstraction is also critically important to its strength. I’ve chosen to highlight things about category theory in these earlier posts:
Category Theory and the extraordinary value of abstraction
More on category theory and the brain
Quantum Mechanical Words and Mathematical Organisms (for Scientific American)
But the inspiration for this post is something I heard from mathematician Eugenia Chang. It was in a talk she gave, at the School of the Art Institute of Chicago, on The Power of Abstraction. Early in her presentation, Chang uses a turn of phrase that I like very much. Mathematics is useful, she says,
…because of the general light that it sheds on all aspects of our thinking.
Notice she doesn’t say, “on all aspects of things,” but rather “on all aspects of our thinking.” I believe this is important. There is an old tradition among educators to tell reluctant students that, while learning mathematics seems to have nothing to do with their day-to-day lives, or the issues they hope to explore, it’s value lies in the fact that it teaches us how to think. But what Chang is saying is bigger and more important than that. Shedding light on ‘thinking’ is not the same as teaching us how to think. Shedding light on thinking means that mathematics is telling us something about ourselves.
To clarify the value of abstraction Chang uses illumination again:
It’s just like when you shine a light on something (and that’s what mathematics is always doing – trying to illuminate the situation)…if we shine the light very close up, then we will have a very bright light but only a very small area. But if we raise the light further up, then we get a dimmer light, but we illuminate a broader area, and we get a bit more context on the situation…Abstraction enables us to study more things, maybe in less detail, but with more context.
Category theory, as she discusses it, is about relationships among things, the notion of sameness, universal properties, and the efficacy of visual representation. About sameness Chang makes the observation that nothing is actually the same as anything else, and that the old notion of an equation is a lie. I haven’t heard anyone apply the term ‘lie’ to a mathematical thing since my first calculus teacher complained about a popular (thick and heavy) calculus text! But the value of an equation, she explains, is that, while it identifies the way two things are the same, equality also points to the way they are different. 2 + 2 = 4 tells us that, in some way, the left side of the equation is the same as the right side, but it other ways, it is not. Equivalences in category theory are understood as sameness within a context.
When first introduced to the notion of equivalence classes in topology, I thought of it as a powerful offspring of equality, not a correction. But, either way, the broad applicability of category theory (even within mathematics itself) is certainly fueling its development. The Stanford Encyclopedia of Philosophy says this about it:
Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. Roughly, it is a general mathematical theory of structures and of systems of structures. As category theory is still evolving, its functions are correspondingly developing, expanding and multiplying. At minimum, it is a powerful language, or conceptual framework, allowing us to see the universal components of a family of structures of a given kind, and how structures of different kinds are interrelated. Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth.
Chang also wrote the account of the concept category in Princeton’s Companion to Mathematics. There she says the following:
An object exists in and depends upon an ambient category…There is no such thing, for instance, as the natural numbers. However, it can be argued that there is such a thing as the concept of natural numbers. Indeed, the concept of natural numbers can be given unambiguously, via the Dedekind-Peano-Lawvere axioms, but what this concept refers to in specific cases depends on the context in which it is interpreted, e.g., the category of sets or a topos of sheaves over a topological space.
If you look back at the earlier posts to which I referred, you will see how the simplicity of the abstractions can serve situations where traditional mathematical approaches contain some ambiguity. I’ve chosen to return to it all today because Eugenia Chang’s language has encouraged me to see mathematics the way I do, as a reflection of thought itself, among other things. Contrary to expectations, she says:
Mathematics is not definitive. It says, here is a point of view.
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