What is a mathematician?

What are examples where a mathematician subconscious derives a result after a break from consciously trying to prove a theorem?

• What are some examples where a result is subconsciously obtained and there was certainty that the new result was correct.  One famous example is from the French mathematician Henri Poincare.  He wrote, “For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions." Towards the end of those 15 days with several hours of conscious effort each day, he finally convinced himself that the functions do, in fact, exist and he had obtained some results identifying those functions. Then, he says: "Just at this time I left Caen, where I was then living, to go on a geologic excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when 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 transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience' sake, I verified the result at my leisure."

• Answer:

  Actually, pretty much any idea, mathematical or not, largely originates nonconsciously, both when you're consciously working on the problem and when you're not. It's a bit like choosing what words to use when talking: Even though you're consciously coming up with thoughts, their formulation using words happens unconsciously, and then you can consciously modify the words if you so choose. Likewise, with creative ideas, you consciously set up a problem, unconsciously find the scaffolding of a solution, consciously fill in the details the same way you try to reinterpret a bizarre dream in the morning as a sequence of events, and then consciously check that the solution is reasonable. If you're looking for examples of ideas where unconscious thought was essential to creating them, the answer is pretty much all of them. However, it's a bit of an overstatement to say that the subconscious derives the result. It's a bit more accurate to say that nonconscious processes provide fragments of an idea which happens to be one that leads to the result after some conscious thought. It's certainly true that taking breaks, both short ones and long ones, helps, but it's for a different reason: "A problem can't be solved with the same mindset that created it." A break allows your mind to let go of unproductive ideas. (If they were productive, you would have solved the problem.) The unconsciously generated idea fragments no longer try to modify a bad method to make it work, but instead try to fit together to make something new. As for having certainty that the unconsciously obtained result is correct, that's largely irrelevant. Anybody with some introspection can remember several times in their lives where they felt certain about something but were proven wrong. Drawing conclusions from people's reports of immediate certainty (e.g. love at first sight and your example with Poincare) is falling prey to http://en.wikipedia.org/wiki/Survivorship_bias. People like to talk about times when they were sure they were right and turned out to be right a lot more than they like to talk about times when they were sure they were right and turned out to be wrong.