I’ve spent the last two days going over my problem, going over my approach, finding new gaps in my proof, fixing them… wash, rinse, repeat. It’s amazing that this vision of math as “getting to the right answer on your first try” even exists. I have to make, unmake, remake so many mistakes to get where I’m going. I think all mathematicians work that way.
Einstein famously said, “Do not worry about your problems with mathematics, I assure you mine are far greater.” I don’t think he said this to brag about how advanced he was: I think he genuinely had a lot of trouble with math. Somehow, a big part of the experience of math is trouble. Frustration is the status quo. But when you get something—the thrill!
Speaking of which, I’m feeling pretty good that I seem to have my argument at least mostly patched up at this point. I think I benefited quite a bit from certain mistakes I made along the way. To quote Simon Singh’s Fermat’s Enigma:
While Shimura was fastidious, Taniyama was sloppy to the point of laziness. Surpisingly this was a trait that Shimura admired: ‘He was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this and tried in vain to imitate him, but found it quite difficult to make good mistakes.’
Math doesn’t happen in a straight line. If I hadn’t made as many mistakes in my thinking about this problem, I don’t think I would have solved it.
Of course, I should sleep on it to see if I have a real solution. Can’t sound the fanfare too quickly.