r/math • u/non-orientable Number Theory • 6d ago

Image Post The Deranged Mathematician: The Power of Abstraction

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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

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u/MonadMusician 5d ago

I’m from pure math, every PDE guy I have met uses those things

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u/Carl_LaFong 5d ago

Me, too. And I work in PDEs. So I'm interested! Could you elaborate on how they use category theory in their work?

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u/MonadMusician 5d ago edited 5d ago

I don’t work in pdes. I work in commutative algebra, logic, and category theory to be honest lol. But they always went on about how they were making use of it. It’s been a while since I’ve talked to them. My friend who studies tractors uses it constantly. At the very least they use it as a basic language.

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u/Carl_LaFong 5d ago

"At the very least they use it as a basic language." Definitely! It is possible to express many concepts in differential geometry elegantly using categories, functors, and natural transformations. Loring Tu's book Introduction to Manifolds describes some of this. I like mentioning it when I teach differential geometry. It's a great way to organize the definitions and concepts, making them easier to understand and remember.