Where do ideas come from?
I began to wonder about that particularly just over a week ago. I'd been set a complex analysis problem to do for my review course; the results of which are visible here. I looked at it and looked at it and looked at it periodically for several days, but never got anywhere at all. Then, while I was riding a bus back to Capitol Hill (the one in Seattle, not the one in DC), staring out the window and thinking mainly about how poor the suspension on a bus is, a thought suddenly popped into my head, like a tiny weasel into a bag of crack-coated eggs. 'Herglotz,' it said. 'Herglotz.' Which meant, of course, that the way to relate the convergence properties of the real part of an analytic function to the convergence of the whole thing is, locally, through the Herglotz integral formula. And so it went. After that, the problem was a straightforward matter of patching up all the details, finding the holes in my argument and filling them. A very sequential, logical, and above all conscious procedure, qualitatively very different from the flash of inspiration preceding. On reflection, this seems to be the way I usually do math. I'll think about a problem for a bit, maybe get bored and go do something else, have a bath, belittle a certain gay conservative blogger, juggle ferrets or whatever it is I do with my free time...And then, in the middle of something completely unrelated, I'll suddenly realise how to attack the math problem. The same thing happened again Sunday, while I was drinking beer (Boulevard Wheat, brewed in Kansas City, and very palatable for a lighter beer) at a pizza joint called the Airliner in Iowa City. (They include Muenster cheese on their pizzas.) Then there follows a tedious and very sequential, focussed process of turning an idea into a proof, which is absolutely vital, yet which does not usually yield a solution on its own. Really nice ideas seem to come in flashes, after a certain amount of initial puzzling and then distraction. Ideas wrought entirely out of sequential, step-by-step brute forcery are usually unsatisfying. They do not seem pretty. Pretty ideas flash.
It almost seems like cheating, in a way. I'm not conscious of doing any actual work to concoct these ideas; they just happen. I'm foisting most of the work off onto my unconscious mind, I suppose. Part of me is thinking about it, just not the part of me I'm aware of. I love the unconscious mind; whatever sins he may be guilty of, I do applaud Sigmund Freud for the unconscious.
Why is Unconscious Luke so much more mathematically insightful than Conscious Luke?
(Luke being, as it were, me.)
Is this more evidence for the bimodality of the mind's logic? Is there, as Ignacio Matte Blanco hypothesised, an asymmetrical, Aristotelian logic of the conscious mind, linear and, well, logical, and a symmetrical logic of the unconscious, processing information in a relentlessly and unreasonably even-handed way that does not distinguish between P and not-P, and thus able to leap tall theorems in a single bound? What is it about planning an attack on mathematics that makes symmetrical reasoning more fecund?
Can I get paid to let my unconscious do all the work?
Posted by aloysius at August 12, 2003 11:53 PM | TrackBack |