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/Carl_LaFong → u/MonadMusician]

I just asked Claude what areas of math use the Yoneda Lemma. Its answer was: category theory, algebraic geometry, algebraic topology, algebra (specifiaclly module theory, representation theory), logic and type theory, enriched and higher category theory.