Wednesday, December 18, 2013

Some mathematical principles

I was just thinking about the old days when I used to do math (although not particularly well), and I remember thinking that there are some principles in math. These are general "trueisms" in math that, unlike theorems, are neither provable nor even always true. Actually, I guess then they're "trueishisms". Anyway, here are a couple I know of:
  1. Conservation of proof energy. I think I heard of this from Robin Hartshorne when I was taking his graduate algebra course. The idea is that if you're trying to prove something hard, and there's a lemma that makes the result easy, then proving that lemma is going to be hard.
  2. In real analysis, if something seems intuitively true, there's probably a counterexample. For example, the existence of an unmeasurable set.  (2 and 3 courtesy of Fang-Hua Lin during his complex variables course, if I remember right.)
  3. In complex analysis, if something seems too amazing to be true, it probably is true. For example, everything about holomorphic functions.
  4. In numerical analysis, if you are estimating complexity or error bounds, log(n) for n large is 7 (courtesy of Mike Shelley, I think).
Any others that people know about?

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