Tuesday, July 13, 2010

a proof includes its details.

last week i speculated about this one problem, trying to determine what the theorem "should be."
i'd make a guess, think about what the proof would look like, what lemmas would be necessary;

i'd ponder counterexamples, wondering if anything immediate would fail;

i'd consider what would happen if the guess were wrong: could the work still go towards an interesting direction?
after a while, i realise that this was a self-deception of the exhaustive kind. it was like traversing a maze by always turning left or reasoning by depth-first search [1]. i don't think the platοnic world of mathematιcal ideas is that .. orderly.

if you really want to prove something,
then don't just talk about it;
either prove it, or go home!

so i just decided to prove a first guess, just to do something .. and immediately ran into technical difficulties. i don't know why, but all this time i had in mind this one assumption [2] and my reveries proceeded from there.

there's a lesson in this .. for me, anyway: the details matter. a rigorous proof is a guarantee; heuristics are not.

at any rate, it's time for something else: either hatch some new ideas for the problem at hand, or drop the problem and work on something else. maybe i'll flip a coin, tomorrow.

[2] this will sound naive, and perhaps there is a simple answer that i just don't see:

there are several good notions of sοbolev spaces on a metrιc space, where one can make sense of a generalised (sub-lιnear) gradιent. given one of these sobοlev functions, however, is there a good notion of difference quοtient, where these quοtient functions converge in the sobοlev norm to a gradiεnt?

admittedly, the question isn't well-posed, but it came up when i was trying to work through the aforementioned problem.

[1] on a more ridiculous note, it's best not to employ DFS on a date. (-:

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