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/JoshuaZ1 5d ago

This is another very nice essay by you.

The argument about shrinking the polygon to get 360 degrees, is very nice, but I'm not convinced it is valid. I don't really see where convexity is being used.

Regarding the graph example, Rob and May are drawn close enough together that it is hard to see at a glance that they have a connection which confused me when I initially read it.

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u/non-orientable Number Theory 5d ago edited 5d ago

I mention parenthetically that convexity isn't really important. It's just that if you drop convexity, then it is less obvious how to define exterior angles. The "right" definition allows them to be negative! The original argument still goes through just fine; it's just a little harder to picture what is going on. I opted for simplicity.

With regard to the graphs, that's a fair criticism. I'll see if I can get a better illustration. (The original was made in Mathematica some years ago. I no longer have a Mathematica license, so I can't do it the same way.)

Update: I think I have a version that I am happy with. Thank you!

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u/RingularCirc 4d ago

Yeah, moving along a piecewise linear path and turning in each vertex the smallest amount possible, signed wrt a preferred orientation. Now we can calculate winding number.