Saturday, July 3, 2010

Categorically Abstract Nonsense

My abstract algebra professor, Dr. Ernie Croot, used a phrase that first amused and then intrigued me: "This is the power of abstract nonsense." That was several months ago, and I've been tossing it around in my head ever since. I kept finding it a more and more packed phrase, even if it was only meant to be casual and comical. It turns out the phrase abstract nonsense was actually coined by mathematicians in the 1940s or so with the emergence of category theory.

My math coursework included only a teasingly brief introduction to category theory, really just enough to know that it's out there and that it adds even more levels of abstraction to what I saw in my abstract algebra courses. From what I understand, category theory generalizes notions including rings (which themselves generalize integers), groups (which generalize mappings and symmetries) and modules (which generalize vector spaces AND rings AND (abelian) groups). Even the least abstract thing just mentioned, the integers, is abstract. They came from what? Our need to count? But could we even conceptualize counting before having integers?

When theories aim to be applicable to everything, are they useful to nothing? The most generalized and abstract theorem I learned in Dr. Croot's class was the Fundamental Theorem of Finitely Generated Modules over Principal Ideal Domains. (Really I just like the name of it and wanted an excuse to mention it!) Now, there are tons of finitely generated modules, over tons of principal ideal domains. And this theorem tells you about any of them. But if you pick a special principal ideal domain, the integers, you get as a corollary the Fundamental Theorem of Finite Abelian Groups, which was pretty much the culmination of an entire previous semester course in group theory. Which is more important, the more general theorem or the corollary?

When I start my graduate studies in economics, instead of math, I don't expect the abstraction (or the nonsense!) to vanish. And while pure mathematicians make no qualms about valuing abstraction qua abstraction, I'm not sure there's such a consensus among economists.

It seems to be a pattern for me lately, that I read something that really fascinates me and then later find out it was written by a Marxist. Such is the case with Franco Moretti's Graphs, Maps, Trees: Abstract Models for Literary History. Moretti postulates:

Theories are nets, and we should evaluate them, not as ends in themselves, but for how they concretely change the way we work: for how they allow us to enlarge the ... field, and re-design it in a better way, replacing the old, useless distinctions ... with new temporal, spatial, and morphological distinctions.

Moretti's book comes from the field of literary history, and yet fits into a discussion that started with abstract algebra. (Only thanks to the hyperlinked Web did I ever come near it. I just can't get over the positive feedback between digital networks and knowledge networks.) Anyhow, I hope that in grad school I gain a better understanding of not only how to theorize, but why to theorize.

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