Where do ideas come from? For the mathematician Dan Rockmore, it’s a lot of hard work, and a little bit of magic.
These stories suggest that an initial period of concentration—conscious, directed attention—needs to be followed by some amount of unconscious processing. Mathematicians will often speak of the first phase of this process as “worrying” about a problem or idea. It’s a good word, because it evokes anxiety and upset while also conjuring an image of productivity: a dog worrying a bone, chewing at it to get to the marrow—the rich, meaty part of the problem that will lead to its solution.
In college, I was about twenty hours into a twenty-four-hour take-home algebra exam when I became convinced that I’d hit a permanent block. I went to the weight room, where, bench-pressing in the midst of noisy midday regulars, I finally figured out the key to proving the irreducibility of a certain group of symmetries . It happened again a few years later, in Somerville, Massachusetts, when I was in the final stages of my Ph.D.
Whereas Erdős sought hyper-focussed vigilance, other eminent mathematicians have found a hazy drowsiness to be the most fertile state of mind. Poincaré described lying in bed in a half-dream state as the ideal condition for coming up with new ideas. The philosopher and mathematician René Descartes famously loved to lounge in bed in the morning and think .
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