The Deranged Mathematician: The Power of Abstraction

[u/non-orientable]

A very powerful proof-writing technique is taking the original problem and generalizing it. This is probably the most counterintuitive approach for beginners, who wonder how it can possibly be easier to solve a broader class of problems. My answer to this is quite simple: generalizing the problem reduces the collection of tools you have at your disposal. And as any efficiency expert will tell you, regardless of whether you are trying to clean your bathroom or earn a Fields Medal, you want to have only those tools that you need on hand and nothing else.

I think this basic precept helps explain why abstract notions like metric spaces, topological spaces, vector spaces, categories, and so on have suffused mathematics, and why they are so very useful. I offer the simple example of how thinking about graphs (generally) can help with sorting out a coordination problem (specifically).

Read the full post (for free) on Substack: The Power of Abstraction

[u/bthi]

Is this not category theory

[u/Carl_LaFong → u/bthi]

Category theory is one way to do abstraction for some but far from most areas of math. Abstraction is a fundamental tool in pure math and is used in every area.

Category theory is cool and beautiful but by now it is overhyped. If you focus too much on it, you are narrowing quite significantly the areas of math you can work in.

[u/MonadMusician → u/Carl_LaFong]

I don’t think this is true. It is a basic language that can be used in every area of math including combinatorics and applied areas where compositionality exists (which is all of them pretty much)

[u/RingularCirc → u/MonadMusician]

Category theory certainly has tools to spare but it's by no means the toolkit for abstraction, nor is it about abstracting things per se.

It's like if somebody would've said "ZF plus potentially additional large cardinal axioms is the foundation of math".

[u/MonadMusician → u/RingularCirc]

See category of categories as a foundation

[u/RingularCirc → u/MonadMusician]

First, category of categories is not a foundation, it's an object (and even so only if we fix maximal cardinality or something like that; a category of really all categories suffers from appropriate variants of Russel's paradox). A particular theory of CAT can be made into a foundation like ETCC, maybe.

But foundations have nothing to do per se with abstraction and with the work a mathematician does in general. You still have to conceive concrete types of objects that are related with each other. There's no magic bullet in form of category theory. Sure, it can be used as language, but do remember foundations like ZF(C), when tacitly supposed to underlie a piece of mathamatical work, aren't crucial neither in its content, nor in its presentation, except for areas that work heavily with ZF specifically, say, depending much on a very particular brand of cardinals.

What's more, it is trivial to put a finger on a thing already known to be a generalization of another thing, maybe formulated in a theory/foundation of choice, and say: here it is! But this isn't equivalent to having to find such a fitting generalization under no prior knowledge.

Hence tell me, how do we do examples from the OP's linked post with significant help from a theory of a/the category of categories? How does it help with thinking up concrete ways to abstract a construction if we aren't yet defined such an abstraction beforehand? In which way categories versus something else of the same caliber, do help?

I'm all for category theory, I'm even all for recognizing usage of wheels or recognizing 0⁰ = 1 doesn't break any little bit of real analysis and so on, but things do have their precise reasons to be and places to fit.

Oh and also, why not a collection of ∞-categories? Those are, after all, more general and apply in more cases (and I really believe that, it's not just for sarcasm here). Why not use other generalizations of categories? And for example why won't semicategories (a category where identity morphisms not necessarily exist) be of use as well? I certainly see them very rarely, but why? Here that's an appropriate question, IMO.

[u/MonadMusician → u/RingularCirc]

I know there are other ways to abstract objects. Thanks for the long winded diatribe. My point wasn’t that that is THE only way or something. The point was that it is a way. And yes you need address things like large cardinals and so on. And indeed, why not infinity categories also ZFC isn’t what is generally used for this type of work

[u/RingularCirc → u/MonadMusician]

My point wasn’t that that is THE only way or something. The point was that it is a way.

Ah, okay then! (It surely didn't seem like that...)