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?
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1$\begingroup$ -1. I don't like this question. It's way too subjective, and seems to be (implicitly or maybe not-so-implicitly) based on the assumption that being "more ad hoc" is necessarily a bad thing. $\endgroup$– Kevin H. LinCommented Nov 5, 2009 at 0:18
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$\begingroup$ I edited it. I hope it sounds less biased now. $\endgroup$– Kim GreeneCommented Nov 5, 2009 at 0:32
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2$\begingroup$ I'm sorry, I still don't like the question, maybe for the same reason Kevin doesn't. It seems like a very good "discussion" question, but I don't think MathOverflow is well-suited for discussions. (And I'd need to be shown a fair amount of evidence before I believe that whole branches of mathematics are uniform and classified by their rates of "efficient", "global", or "ad hoc" proofs. I've seen plenty of results within the same area with different proofs that vary on these qualities.) $\endgroup$– Theo Johnson-FreydCommented Nov 5, 2009 at 4:18
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1$\begingroup$ @Theo: I do not think that it is any more subjective than a question about what is a 'good' book for learning analysis. I think this is a question with real consequences for people learning and some people find one style of math easier than the other. I also don't think there is evidence in the comments so far of big differences in how people are interpreting the question as one would expect if it was very subjective. $\endgroup$– Kim GreeneCommented Nov 5, 2009 at 4:41
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2$\begingroup$ The problem here is that the question is perhaps "subjective and argumentative", which we've been taking as grounds to closing. Mathoverflow isn't suitable for questions that require or deserve discussion. I won't close, because I actually like the question. Someone else may! $\endgroup$– Kim MorrisonCommented Nov 5, 2009 at 7:17
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3 Answers
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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.
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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.
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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.
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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.
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2$\begingroup$ The PCM article about PDEs seems to adopt this philosophy; in particular, the article argues at length that PDEs are not a subject where general questions, such as that of classification, are meaningful. Instead one concentrates attention on the most interesting PDEs, which seem invariably to be the ones associated to interesting physical phenomena. $\endgroup$ Commented Nov 5, 2009 at 2:54