DH claims that chess masters beat computer chess programs not because they can look more moves ahead, but instead because they know how to recognize strategically good arrangements, and only consider moving to these (instead of all the possible moves).
He goes on to say that mathematicians do the same thing when proving theorems. Instead of considering all possible techniques, a good mathematician "smells the paths" before taking them, and only considers the good ones. He was close to saying that a good mathematician knows the proof that will work before attempting the problem.
I kind of agree with this last thought. Moreover, I find that I pick my research problems this way - if I think I know how to solve it, then I decide to spend time on it. I am not always right about the solution method, and sometimes this leads to interesting stuff. On the other hand I find this way of choosing problems somewhat cowardly - essentially letting publication pressure be a factor in choosing research problems.
What do you think? Certainly we tell students that there is an art to integration, but really the art is knowing how to solve the problem before you start. Having a sufficient bag of tricks (and having it well organized). Is research this way too? Are we really only looking at problems which we basically already know how to solve? My PhD thesis wasn't like this for me (although I believe it was for my advisor). To what extent are we really being courageous in our problem selection? Ever try to integrate x-x over R+. A simple looking integrable function*. Feeling brave?
*This challenge was given to me once at an airport bar by a smart-alec undergrad physics student. Clever really.
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