Sometimes the proof isn't intuitive. Sometimes "it turns out" that if you just crank through this ugly mess of symbols, you get out this result. This is more common on the applied side of mathematics than on the pure side, but I've seen it in both.
Sure, rigorous proofs are often obtained by "proceeding formally", but there's usually (not always, granted) some more intuitive reason as to why something should be the case, otherwise nobody would have tried to prove them in the first place.
Although some proofs can accumulate cruft.
E.g. there was a small and nice intuitive prove of a reasonable theorem in the first place, but scores of people found ways to generalize it, and in the end you have a mess of symbols proving a hopelessly general lemma.
Sometimes even an “intuitively hand-waved” proof substitute, it turns out, is well beyond the scope of the course.
(Then again, I’ve had uncommonly thorough math courses, and would likely share your frustration.)
Re:“It turns out”
by roundsquare on Thursday March 04, 2010 @02:53AM
(#1166415)
otherwise nobody would have tried to prove them in the first place.
Not really. Sometimes someone sees a bunch of examples and wonders if there is a generalization/theorem. So they set out to prove it and 3 pages of symbol manipulation later you have a proof.
"It turns out" is often used in mathematics lectures, when the lecturer is trying to keep things relatively informal and is avoiding proving results. Blind assertions have no place in mathematics - it's much better to provide an intuitive explanation in the spirit of the proof.