alt Hacker NewsHow do these kinds of advancements in math happen? Is it a momentary spark of insight after thinking deeply about the problem for 20 years? Or is it more like brute forcing your way to a solution by trying everything?
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In this case, a ton of progress had already been made. The conjecture had been proved in some cases, reduced to a simpler problem in others. This couple went the last mile of solving the simpler problem in some particularly thorny cases.
You're really standing on the shoulders of giants when you rely on the classification of finite simple groups.
The structure of DNA was seen in a vision in James Watson's dream. Some say it's subconscious problem solving and I think most down to earth people agree with that, but some less down to earth people will absolutely attribute it to god (I'm in the latter). If we were to entertain a silly proposition, something in the universe could just move our story along, all of a sudden. These paradigm shifts just seem to appear.
My experience from proving a moderately complicated result in my PhD was that it's neither. There wasn't enough time to brute force by trying many complete solutions, but it also wasn't a single flash of insight. It was more a case of following a path towards the solution based on intuition and then trying a few different approaches when getting stuck to keep making progress. Sometimes that involves backtracking when you realize you took a wrong path.