Bigsley the Oaf


Posted in Uncategorized by bigsleytheoaf on March 3, 2011

“What is Category Theory?”

I’m going to attack this from multiple angles. I hope that my explosion of this question will convey a sort of finite/non-existential fatalism which differs significantly from nihilism.

Descriptively: Category Theory is a formal system whose components are, fundamentally, objects and morphisms. The potential space of relationships between these components is sufficiently rich to describe most (all) mathematical systems which have classically been describes by sets and relationships between sets (e.g. Algebra, Topology, Geometry, Arithmetic, etc.) I want this to be clear – as far as I understand it (and please correct me if I am wrong) Category Theory is simply an encoding for other mathematical systems. This means that if you have some group or topological space you’re interested in then you can find a high-fidelity representation of this group or space in the category of groups or the category of spaces.

Intuitively: Category Theory is a systematic approach which focuses less on objects and more on the relationships between objects. More essentially, it is a system which exists because the intuition that relationships between objects define those objects (rather than the other way around) became sufficiently common within the mathematical community (1945 – we see an explosion of activity in the field following this – timeline here). Now, I feel a bit facile writing this since I am nowhere near sophisticated enough to understand what category theory is actually used for (sheaf what?), but it seems to me that what CT “really is” is a system which encodes a way of doing math or rather a community’s reflections on the way it does math.

Reflexively: Category theory includes, in its structure, a description of the way that we do math. How does it do this? Its structures generalize common ideas (limit, product, isomorphism, etc.) which appear all over math. Such a theory would not appeal to someone who did not see these ideas everywhere. Generalization only appeals to those who are burdened with redundancy and a description of redundancy with respect to a purely human act (like doing math) is a statement purely about humanity.

What I’m saying here is that if we do math honestly enough, long enough, with enough of an open mind, the components of category theory should pop out at us as a useful set of inter-related tools. An analogous situation can be found in computer programming w/r/t so-called “functional programming.” There is nothing fundamental about map/reduce (actually there is – hahaha!) but once one knows about these higher order functions one sees them everywhere.

How does this happen, though? Well, I have an idea, but it’s fairly conjectural. Whatever our minds do when they do math, it is systematic. This isn’t to say it’s necessarily deterministic or centralized – it could be a massively random, distributed system. It’s just to say that it’s systematic – it’s composed of objects interacting: transforming each other, destroying each other, creating each other, modulating each other, etc. But what if you could come up with a simple description of this system? What would be the utility of such a system?

I’m not sure I can answer this question. I’m not sure I can succinctly answer the question “what is the utility of Category Theory?”, either. Can you see what I’m getting at? Somehow, by being careful and thinking hard about what we’re doing when we do math, we’ve (I’m conjecturing) come up with a simple representation of the structures which represent the fields of math we’re familiar with.

But this brings us back to the matter of (finite) fatalism: now that we have CT and we think about CT, we need to think about what we’re doing and create a new formalism to represent that and so on. There really is no end. I mean, encoded in the question itself of “what is category theory?” is this attempt to rise above the system and understand it in terms of some more (primal?) system. We can ask “what are we doing when we do category theory?” and come to an answer sufficiently precise that it becomes a system X about which we can ask “what are we doing when we do X?” And so on.

But then WTF is the point of doing category theory? Are we just moving forward to ever-more-abstract representations of mathematics? To what end? Here’s the fatalistic point – it seems like the act of doing math requires some sort of delusional craze and that once you step out of it and look at what it really is it seems much more bleak and cold and meaningless than it did.

And so it seems with everything -> production to produce more, chemical methods for studying chemicals more effectively, bigger better models of physics to build bigger better tools to study physics, and so on.

This isn’t nihilism. I’m not saying that nothing has any meaning. I’m kind of saying that things have meaning, but we aren’t even barely scratching the surface.

It’s gorging yourself to bursting and finding you’ve only eaten a billionth of the cake.


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