I wanted to represent these [Fuchsian] functions by the quotient of two series; the idea was perfectly conscious and deliberate; the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and succeeded without difficulty in forming the series I have called thetafuchsian.
Just at this time, I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to 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’s sake I verified the result at my leisure.
Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, 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 the arithmetic transformations of indefinite ternary quadratic forms were identical with those of non-Euclidean geometry.
. . .
Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestible.
Henri Poincaré, lecture at Société de Psychologie, Paris; quoted by Jacques Hadamard, The Psychology of Invention in the Mathematical Field; Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance; William Byers, How Mathematicians Think; and no doubt others.
Two quotes from Simon Singh, Fermat’s Last Theorem:
Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into furniture, but gradually you learn where each piece of furniture is. Finally after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of — and couldn’t exist without — the many months of stumbling around in the dark that preceded them.
At first I supposed that I should be able to overcome the contradiction quite easily, and that probably there was some trivial error in the reasoning. Gradually, however, it became clear that this was not the case . . . Throughout the latter half of 1901 I supposed the solution would be easy, but by the end of that time I had concluded that it was a big job . . . I made a practice of wandering about the common room every night from eleven until one, by which time I came to know the three different noises made by nightjars. (Most people only know one.) I was trying hard to solve the contradiction. Every morning I would sit down before a blank sheet of paper. Throughout the day, with a brief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty.
Bertrand Russell (on his attempt to resolve his paradox)
I recently had an odd and vivid experience. I had been struggling for months to prove a result I was pretty sure was true. When I was walking up a Swiss mountain, fully occupied by the effort, a very odd device emerged — so odd that, though it worked, I could not grasp the resulting proof as a whole. But not only so; I had a sense that my subconscious was saying, “Are you never going to get it, confound you; try this.”
J. E. Littlewood, “The Mathematician’s Art of Work”, in Littlewood’s Miscellany (ed. B. Bollobás), Cambridge University Press, Cambridge, 1986.
Finally, two days ago, I succeeded, not on account of my painful efforts, but by the grace of God. Like a sudden flash of lightning, the riddle happened to be solved. I myself cannot say what was the conducting thread which connected what I previously knew with what made my success possible.
C. F. Gauss, quoted in Jacques Hadamard, The Psychology of Invention in the Mathematical Field, Princeton University Press, 1945.
It was at St Leonards, probably because it was a boarding school, that I discovered and developed as a positive habit the powers of what I call “subliminal learning”. We kept very regular hours and were never tired or stressed at school. Lights out at 9p.m., even when in the sixth form. Out of bed at 7a.m. The accepted wisdom is that one should relax before going to bed, emptying the mind of problems. As far as mathematics is concerned I could not agree less. We looked forward to “drawing-room” each evening, but I usually cheated and stole ten minutes back at my desk before bathtime. I would make certain to sort out in my head, as late as possible, what problems needed to be solved the next day and what might be usefully committed to memory. Before falling asleep, I “drew” with my finger any relevant geometrical figure or algebraic equation on the partitioning of the dormitory cubicle that formed a bedside wall. The result would be miraculous. Without fail, on waking in the morning, the details, the logical argument required or the facts that I needed to recall were clearly imprinted in my mind and, because of the clarity, any required solution would often be clearly “written” on the partition. For this to work, it is essential to make sure to wake at least five or ten minutes before the prescribed time for getting out of bed, giving oneself time to go over what has been resolved while asleep. This became honed to a fine art, without my ever telling anyone, and I have used the technique deliberately ever since.
Kathleen Ollerenshaw, To talk of many things: An autobiography, Manchester University Press, 2004.
The difference between a good mechanic and a bad one, like the difference between a good mathematician and a bad one, is precisely this ability to select the good facts from the bad ones on the basis of quality. He has to care!
Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance: An Inquiry into Values, Bodley Head, London, 1974.
I would like to briefly share with you a story about my father, which I believe is typical of him. One day when our family was having tea with some friends, he was enthusiastically talking about his work. He said: “I feel like I am somehow moving through outer space. A particular idea leads me to a nearby star on which I decide to land. Upon my arrival I realize that somebody already lives there. Am I disappointed? Of course not. The inhabitant and I are cordially welcoming each other, and we are happy about our common discovery.” This was typical of my father; he was never envious.
Hilda Abelin-Schur (daughter of Issai Schur), in A. Joseph, A. Melnikov and R. Rentschler (eds.), Studies in Memory of Issai Schur, Birkhäuser, Boston, 2002, p. xli.
(Thanks to Robin Whitty for this.)
I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with someting else I know aboout, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go.
Sir Michael Atiyah in Mathematical Intelligencer, reprinted in the Problem Corner of the European Mathematical Society Newsletter.