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Mathematician: this is category theory. No, it didn’t have anything to do with categorization, it just helps us understand how spaces can map to each other. Yeah I guess it’s kinda like graph theory or algebra, but not really. We made a category of graphs, and you can use the category of graphs to represent endofunctors on the category of categories.
-- It's about nothing.
-- No sets?
-- No, forget the sets.
-- You've got to have sets.
-- Who says you've got to have sets? Remember when we were talking about functions of functions? That could be a theory all by itself.
So they are metagaming mathematics?
I don't know that much of history, but by my accounts category theory is metagaming the metagame of the original mathematics metagamers.
So we're at least 4 metas deep at this point?
Just wait until the infinity mathematicians shows up and starts assigning cardinalities to the level of meta
It's a strange feeling to think you understand what you are reading until you get to the end, but you have given me that feeling. I was like "yeah category that's a word I know. Let's math the hell out of some categories." Then I recognized other words you said, but by the time I was at the end of your post I wasn't sure if I understood anything.
I don't mind feeling dumb. Honestly it helps keep my narcissism in check. I like math because I don't understand all of it even though it should be logical.
If it helps, category theory is affectionately referred to by mathematicians as "generalized abstract nonsense".
It can be very confusing, but it's sort of a field of math that helps to relate ideas on one area to similar ones in another domain.
Oh. Neat. Like a sort of language interpreter trying to explain an analogy from one language to another? The words might not mean the same when it comes down to the word for word interpretation, but the idea behind them is what matters.
Ok but can I use a graphing calculator to graph those graphs?
Also no! The “graphing calculator” is an abomination that should be more rightly called a plotting calculator. But that’s what happens when you let engineers in Texas name something.
I've read a fair few unintuitive mathematical things, but category theory has so far been the worst. Some things are just plain unintuitive and don't catch your attention. Then there are things that are intuitive and really do reel you in. Finally there are things that seem intuitive but become so complex that your comprehension inverses: what you thought you knew feels wrong because of the new things you learned.
The latter has been my experience with category theory.
It may have nothing to do with categorization, but has everything to do with categorification which is much more interresting anyway.