Why do branches of math vary in proof styles and what category are different branches in? - MathOverflow

Why do branches of math vary in proof styles and what category are different branches in?

2009-11-04T22:38:04Z

Some branches of math seem to have reasoning which is more global. There is a lot of efficiency in the proofs because the reasoning transfers easily between proofs. For other branches of math, a lot of truths seem to be more local. The proofs tend to have lots of sub-cases and exceptions. There are fewer general principles. Does anybody know why branches of math vary like this? Can you place different branches of math on this scale from being dominated by more ad hoc to being dominated by less ad hoc proofs?

Answer by HW for Why do branches of math vary in proof styles and what category are different branches in?

2009-11-05T00:38:15Z

The only thing this reminds me of is Tim Gowers's nice article on the two cultures of mathematics, in which he compares and contrasts "geometry" (very broadly defined) and combinatorics.

The categories in the article don't exactly match up with the categories in the question, but perhaps there's some point of contact.

Answer by Jason Dyer for Why do branches of math vary in proof styles and what category are different branches in?

2009-11-05T02:17:59Z

This reminds me of Terry Tao's essay contrasting hard analysis and soft analysis:

At first glance, the two types of analysis look very different; they deal with different types of objects, ask different types of questions, and seem to use different techniques in their proofs.

and

I therefore feel that it is often profitable for a practitioner of one type of analysis to learn about the other, as they both offer their own strengths, weaknesses, and intuition, and knowledge of one gives more insight into the workings of the other.

Answer by MLevi for Why do branches of math vary in proof styles and what category are different branches in?

2009-11-05T02:34:57Z

Could some of it have to do with the difference between say considering very general structures versus working in a specific structure? For example, if one is trying to prove existence of a solution (in some sense) to PDEs, one may need to work on finding a priori bounds for some specific PDEs. The techniques used to find these bounds may or may not be able to be generalized (for example, many PDEs of interest have non-linear terms of differing "flavors"). Then on the other hand, is searching for a priori bounds alone something that the existence theories of these different PDEs have in common, if you will?

As far as "global reasoning," this type of reasoning would seem to be the correct approach when dealing with general structures (examples: Banach Spaces, Algebras, &c). If one is interested in proving statements about structures without imposing too many assumptions, then it would seem that thinking "globally" might be beneficial.