This post on how the best thinkers in physics are those who can most ably explain technical concepts in non-technical language encapsulates why I think the current mathematical physics course is the best math and physics course I’ve taken in my life. The best math course I’ve taken, other than this, was similar, in that the teacher excelled at instilling understanding of abstract mathematical concepts by using highly accessible, intuitive language. At the same time, rigour isn’t sacrificed — the idea is to feel one’s way towards the solution while working entirely in the land of intuition, but to write the solution entirely in non-intuitive technical language.
Although there are no course notes and no official textbook for Geroch’s course, the solutions and comments on the problem sets constitute excellent resources. They represent both ends of the spectrum that one needs to be a successful mathematician or theoretical physicist. The solutions stick to the formalism alone, providing the standard proofs one sees in math textbooks. The comments represent the other end of the spectrum — they explain what the proof really means, and how one feels one’s way to it intuitively. An example of the kind of language the comments use:
Non-measurable sets are so frothy that they have excessive measure. Here, we know that X is virtually froth-free: Since
, where’s the froth? We want to show that X must be measurable, i.e., that frothlessness is a sufficient condition for measurability.
None of the textbooks on Lebesgue measure I’ve skimmed through explain the problem of excess outer measure in ‘frothy’ sets this way. This way, though, is infinitely more enlightening than the formal, axiom by axiom, inference by inference proofs that populate math textbooks. Eventually, one wants to express the idea of frothiness formally. But what comes first is an intuitive idea of what frothiness is like.
(Disclosure: Part of the reason I wrote this entry was to test the new implementation of LaTeX in WordPress.com posts. Now to avoid using equations as a crutch in areas where I lack sufficient intuition…)
