Take, for instance, the way I understand mathematics. Like many, I seem incapable of understanding a particular concept unless I write it down, and like many analysts, it often helps me substantially if I draw a nice, descriptive picture: that way I may accrue some geometric intuition of the situation at hand.
Unfortunately, I also seem incapable of being satisfied with my own work. Regularly I reread my old work notes, think of them as sheer crap, and feel the urge to rewrite them. As a result, I've become accustomed to judging my work based on how long (I think) it will last [1] will last until I dispose of it.
I blame this on my habit of keeping a written journal, and years of half-assed creative writing. Editing and re-editing seem to me second nature if I am working on a matter of some substance, and I can never leave well enough alone.
It's the same with mathematics, such as my idiosyncracies with completing problem sets and doing research. I work and work, trying this idea and that idea, replacing this idea with a better one, but having a poor memory and little mental discipline, after a day full of work, I might be confused at what I was actually proving and mean to continue for the next day. The notes might seem too ragged, too specific to what I had in mind yesterday and not today, and I must summarise from scratch before moving on.
If anything, this reminds me of an anecdote about Archimedes: In the afternoon he would go to the beach, prove theorems by drawing in the sand, and then leave at sunset to pursue other matters. But each day he would arrive to see his diagrams washed away by the sea, and he'd have to reprove them quickly before being able to reach any new results.
One can only conclude that Archimedes must have had an excellent memory of proof, and amazing self-control; I know I never get any work done when I go to the beach. q:
At any rate, today is a writing day: I'm tired of half-forgetting which details I've proven and which ones I haven't. I've jotted down an outline of main ideas, and now I'm working out the gory details to each main idea, page by page. Maybe through this, my advisor might understand my ramblings when I see him next.
Here's hoping.
[1] To me, there is a substantial difference between work I've done and the paper on which I've written it. Due to errors or a better idea or technique, I'll often rewrite something: it's similar, improved work, but on paper it is truly a different manifestation. If ever I throw out the actual mathematics -- the latest version of the work -- then I have disposed of it.
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