Probably every research mathematician has been asked the question, “How do you do mathematical research?” Some lay people think we simply figure out ways of doing bigger and bigger long multiplications. Many more people think that all the mathematics must have been discovered by now, so what are we doing?
These misunderstandings are about what we do research on rather than how we do it, the question I want to focus on here.
In 1945, Jacques Hadamard published a remarkable book, The Psychology of Invention in the Mathematical Field. He was a great mathematician himself, one of whose achievements was the proof of the Prime Number Theorem fifty years earlier. He examined his own mental processes, searched the writings of his predecessors, and sent questionnaires to many leading mathematicians and scientists of his time, including Albert Einstein.
Hadamard’s conclusion was that the process of mathematical discovery can be broken into four parts: preparation, incubation, illumination, and verification. First, one immerses oneself in the problem, thinking about all aspects. Then one takes a break from it, while a mysterious process takes place in the brain, completely below the level of consciousness. Only after this happens is it possible to have a sudden inspiration or breakthrough, the details of which are verified by further work afterwards.
His book was overtaken by the spread of behaviourism in psychology; the questions he was asking were regarded as unscientific, and the methods as too subjective, for a long time. Now that this philosophy has had its day, Hadamard’s question can be re-opened. Indeed, we have techniques for imaging directly the processes occurring in the brain of a mathematician, and further progress should be possible. Hadamard’s book was re-published recently by Princeton University Press under the title The mind of the mathematician.
Here are a couple of great mathematicians quoted by Hadamard, describing how some of their own breakthroughs came about.
Henri Poincaré, in a lecture at Société de Psychologie, Paris, made the following statement, which has been often quoted: by Hadamard; by Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance; by William Byers, How Mathematicians Think; and no doubt by others.
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.
The “prince of mathematicians”, Carl Gauss, also quoted by Hadamard, was much more brief and mystical:
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.
Here is an instance of a mathematician deliberately cultivating the “grace of God”. This is Kathleen Ollerenshaw, who spent her career as a public servant and local politician but never lost her interest in mathematics. The passage is taken from her autobiography To talk of many things, published by Manchester University Press in 1994:
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.
I was interested to see in last week’s New Scientist a brief account of research on players of the computer game Doom. The Brazilian researchers persuaded Doom-players to spend two nights in their lab. Before going to bed on the second night they played the game for an hour. The researchers monitored their dreams by simply waking them during periods of REM sleep and asking them what they had been dreaming about. The next day they played again and the researchers noted their relative performance. Those who had improved most had found elements of the game entering their dreams, but not obsessively. The researchers thought that the subjects had been rehearsing game-playing strategies or solving puzzles in the game while asleep.
Maybe something similar happens in any research …
I remember that, when I was a doctoral student, quite a long time ago, I spent a long time on Hadamard’s first stage, immersing myself in my research problem. When you do this, the problem becomes concrete in various strange and dreamlike ways; and indeed, in my dreams, I found myself wandering round a mysterious landscape which was in some manner the expression of my research problem. I have no doubt that this was the second stage of Hadamard’s process. I can remember the third stage as well. I went into the graduate common room at my college for a cup of coffee. As I picked up the cup, I suddenly saw very clearly what had to be done. As in Poincaré’s account, there was absolutely no need to verify my inspiration then and there. I knew it would work, as indeed it did; it led to my first mathematical publication, in 1969.