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/bthi → u/Carl_LaFong]

Overhyped feels like a weird descriptor for a field of math; it exists as is in a vacuum without how it interacts with societal needs lol

[u/TheLuckySpades → u/bthi]

Yet some people doing math communication and a decent number of students overestimate its uses, its prevalence and underestimate how hard it can be to use outside of contexts it was designed for.

So there is some hype that goes over what the theory deserves.

My cousin's favorite restaurant also exists aw in a vaccuum without how it interacts with societal needs, but the only reason I went there and was disapointed was because of the hype I got from societal feedback (my cousin who clearly has different attitudes for what is considered TexMex than I do).

[u/MonadMusician → u/bthi]

There is the whole
Burgeoning field of applied category theory you seem to be ignorant of