Since I seem to like finding out about how other people work and think, I was delighted to come across Henri Poincare’s description of how discoveries are made in mathematics in a large number of cases. He says it is a three stage process, which he illustrates with some of his own discoveries. The first stage is to work on a problem, attempting proofs and playing with definitions. This might not result in anything concrete but it will get your mind working. The next step is to take a break and do something unrelated. Often, he says, the subconscious will continue working on the problem and you will get a flash of insight. This flash is, apparently, often times correct. The final step is to sit down and check the insight to make sure it works and draw out a few consequences while it is still fresh and active in your mind. Something like this was also described by George Polya in his How to Solve It. Although, Polya didn’t endorse Poincare’s crazy analogy for why this is. Poincare described the mind as being like a great chamber full of Lucretius’s hooked atoms. The atoms are ideas and are stuck to the wall of the chamber. The initial phase of work shakes some free to float around and crash into each other. When two ideas fit together (hook each other), they produce insight. Further work verifies this and pulls more atoms from the wall. Somewhere I read that Hilbert had a garden near his office. He set up a blackboard near the garden and would garden for a while, then walk over to write things on the board, then go back to gardening. He would mix this up by riding around on a bike occasionally. That sounds like he was engaging in a cycle like Poincare described. I wonder if there is any more than scattered anecdotal evidence that this is a way to produce insight. I think something like this works for me. Working hard on a problem for a class but not solving it before going to bed tends to result in figuring it out over breakfast or lunch. (Although occasionally it results in bad dreams. Raise your hand if you’ve ever dreamed about modal logic.) The question, I suppose, is: does insight come according to a pattern like the above in all disciplines or are there some where insight only really comes by slogging along at something for a long long time?

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