Attending to the Subconscious
By Dr. Eugene MaierMost everyone has had the experience of trying unsuccessfully to recall a name or some other bit of information only later to have it unexpectedly come to mind. In a similar vein, there are those who have tackled a mathematical problem, found no solution, quit all conscious efforts to do so, and then later have a means of solution pop into their mind.
Several such experiences of the French mathematician Henri Poincaré are reported in Jacques Hadamard's little volume, The Psychology of Invention in the Mathematical Field. For a fortnight, Poincaré had been attempting to work out the properties of a certain collection of functions. He hadn't succeeded in doing so when he went on an excursion with a group of people and, he reports, "The incidents of travel made me forget my mathematical work. …we entered an omnibus to go some place or other. At the moment I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it," that the functions were simply another set of familiar functions in disguise. "I did not verify the idea; I should not have time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty."
"Then," Poincaré continues his report, "I turned my attention to the study of some arithmetical questions apparently without much success and. …I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty" that disposed of his questions. Poincaré reports a third, similar instance of "unexpectedly" and "unpreparedly" having a solution to a mathematical problem he had been investigating come to him while away on army duty.
Poincaré's experiences suggest that whereas one may consciously quit thinking about a problem, one's subconscious will continue the conscious efforts. Not only that, it can arrive at a solution and bring it to our conscious mind at an unexpected moment. No one knows exactly how it happens.
Andrew Wiles, the Princeton mathematician who provided a proof for Fermat's Theorem--something that had eluded mathematicians for 350 years--describes his problem-solving process in Fermat's Enigma, Simon Singh's story of Wiles achievement: "Basically it's just a matter of thinking. Often you write something down to clarify your thoughts, but not necessarily. In particular when you've reached a real impasse, when there's a real problem you want to overcome, then the routine kind of mathematical thinking is of no use to you. Leading up to that kind of new idea there has to be a long period of tremendous focus on the problem without any distraction. You have to really think about nothing but that problem--just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconcsious appears to take over, and it's during that time that some new insight comes." Where it comes from, Wiles says, is "a mystery."
It would be nice if it were otherwise; if there was a recipe we could give our students to get their subconscious to solve their problems. Lacking such a recipe, I tell my classes that solving a problem doesn't necessarily happen instantaneously and on demand; it can require simmering time. Then I give them the following instructions: "When you find yourself at a dead end--when you are no longer making progress towards the solution of a problem and are devoid of ideas, quit working at it. But as you quit, say to yourself something like, 'I'm going to quit working on this problem for the time being and when I come back to it, I will know more than I do now.' Then do something that diverts your mind. Above all, don't continue working on the problem until anger and frustration set in and you give yourself the message, 'To heck with it. I'm not going to waste any more time on this problem. I'll never get it anyway.'" My supposition is that the former message gives the subconscious permission to keep working on the problem while the latter message encourages it to drop the matter.
I've no hard evidence that heeding these instructions creates, on the whole, better problem solvers. I do know the procedure has worked for me on occasion, not just in mathematical problem solving but in other situations, ranging from figuring out a clue in a cryptic crossword to formulating funding proposals. It does lead students to an awareness of the role of the subconscious in creative thought and to attend to the circumstances of their moments of insight--the settings in which their "ahas" occur. And it helps students recognize that in problem solving, after a point, relaxing is more effective than working hard.