proofs, lost and found.

during the technical part of a seminar talk today, i withdrew into my own thoughts and came up with a wrong idea for a proof.  it was clearly wrong because it was too general in scope:

it didn't rely on the hypotheses i had in mind,
and it would have applied to a setting in which a counter-example already exists.

so yeah .. quite wrong! (-:

what i didn't know yet was why .. and that intrigued me;
that's the interesting part, you see.

so i thought and thought in circles, a bit lost .. and before i knew it, i was clapping with the rest of the audience.  exiting the seminar room, i strolled back to my office, still lost.

---

sometimes i love mathematics because of all the little twists and turns that can come up. a proof is like getting lost in an unfamiliar part of town, but then finding your way back:

sometimes you remember when you return,
sometimes you still get lost and deeply suspect that there must be a better way than this ..

on the other hand, a good, polished proof is like taking a shortcut or a scenic detour, the sort of path that you'd show your friends if they were tagging along.

many mathematicians give the analogy of a toolbox, and that every job requires the right tools.  whenever i'm working on a proof, though, i feel like i'm living out an analogy of an escape route of sorts:

there's got to be a way out of this.
what if we tried this door? no good. it's locked;
wait, is the the air vent wide enough ..?

it would probably explain why my proofs seem .. weird, to me.

they're probably not good ones .. not in the above sense, anyway.  i never think to go back and polish them up .. not unless i feel like i too easily get lost and need a better map ..

.. and i'm never surprised when, in the midst of explaining one, that my audience gives me strange looks.