Saturday, August 30, 2008

the minor dilemmas and decisions which arise from writing.

i intended to work for most of my free hours today, but all i seem to do is blog. so in the interest of doing work-related things, let me express a few thoughts:

ONE. i seem incapable of writing preambles to theorems anymore. often i sit and struggle and think of the right words (or the right point to make) to add before writing a precise mathematical statement in the form of a lemma, theorem, corollary, or proposition.

most of the time i give up, flip open my thesis, see how i wrote it there, and i think either:

* good enough; glad i thought of it before
    and then i paraphrase myself;

* god, that's crap. why didn't i just write
    <add mildly insightful comment here>?

    and so i do.

in other words, i cheat and i plagiarise myself. it makes me wonder how i ever managed to write this tome of a thesis.

TWO. i've often heard people say that "i'll probably write two or three papers out of my thesis," which leads me to think: wow. it must be nice to have that many good ideas.

my writeup is at 27 pages and counting. there's no introduction yet, and one application hasn't been added yet. if i never add it, then i might write a separate little note about it -- say, 10 pages -- but it would be a really weak note, as far as notes go. that theorem relies on two things:

* a main theorem, already written up and added to this first draft;

* another theorem which has essentially been proven,
    and if you know the subject well enough,
    you could probably reproduce it rather easily.

so as you can see, it wouldn't stand very well on its own. besides, once you combine two papers from the literature, then the same theorem would already be known. then again, one of those two papers doesn't exist yet. you see my point, though, right?

on the other hand, if i add the application into this paper, then it's headed into the unwieldy realm of 50+ pages. who in their right mind is going to read that, from a recently minted ph.d.?!?

THREE. there is also another theorem missing from the writeup and i'm not sure if it's worth the extra pages; i suspect it will need at least 7. the argument is long and technical and the method of proof is one that never sat well with me philosophically. [1]

also, i don't actually need the theorem to prove anything else, but it does round out the whole picture nicely. admittedly, in light of what i proved and wrote up thus far, i would ask a "natural" question that this theorem would answer.

again, the theorem wouldn't be able to stand on its own, not even in a note.

so what do you do when you do have ideas, but they cannot make papers on their own?

[1] the proof is correct, and i learned a lot by thinking it through, but the ideas don't feel like ones that i would have thought up. i had a lot of help with my second advisor and really, i think of it as his theorem, not mine.

he'd deny it, of course.

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