This is what makes me think I am a mathematician
You begin with something, a problem that is horribly complicated and dense and opaque, and you don’t know how to even start dealing with it, or thinking about it. You push it, shove it, shake it, tear at it, turn it around constantly in your mind, look at it from different sides. You manage to peel away a few layers, but it remains complicated and dense and opaque. You put it aside, and think about something else for a few months …
You repeat this several times, sometimes for several years on end.
And then, one day, often when you’re not thinking about it at all, there is a … what? A hunch, a flash, a suspicion, a hint at an extremely simple idea that brings about an uneasy feeling, a vague sense of excitement. The excitement rises, and you decide to look at the problem again, because all of a sudden you have this crazy absolute certainty that this will work, even though you’d never be able to explain rationally why it should. It just seems to fit. It is simple and crazy and surprising enough to fit.
And yes, the problem starts to unravel, to fall apart into manageable pieces, and while you are working on it continuously for days and often weeks, your motivation and concentration are sustained by a strong feeling of satisfaction, because you are experiencing pure unmitigated beauty, simplicity, elegance, necessity.
You’ve solved the problem. You’re happy and satisfied.
You can’t face looking at it again for months and months. You feel you’re done with it.
But you have to tell other people about it, explain it, teach about it. And sometimes, the excitement comes back, because you see that the simple and crazy idea you had, manages to seduce other people as well.