A wonderful little book that gave me insights into the mental processes underpinning thinking at the highest intellectual level was Jacques Hadamard’s seminal publication, The Psychology of Mathematical Invention. Hadamard, co-discoverer of a proof of the Prime Number Theorem, used a speech delivered by the great mathematician Henri Poincaré to describe a four-stage process that climaxes in a moment of discovery that mathematicians call “a eureka! moment.” Yet, this book is not about mathematics, but rather about the subconscious and how the brain solves deep problems.
Describing a eureka moment in his work on Fuchsian groups, Poincaré said:
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 indeterminate 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 incontestable.
Essentially, the process described by Poincaré and Hadamard that begins with hard work and ends with a discovery occurs in four stages:
Stage 1. Preparation: a period of intense conscious work that ends without a resolution of the problem.
Stage 2. Incubation: a period of relaxation when the conscious mind is at rest and the unconscious is exploring patterns and searching for connections and relationships in the underworld of the unknown knowns.
Stage 3. Illumination: a eureka! moment when an idea suddenly appears like a bolt out of the blue generating an immediate insight.
Stage 4. Consolidation: a conscious verification and precise articulation of the discovery that connects it to current knowledge.
This 4-stage process of illumination has been recognized by many great scientists and even those in the creative arts (including Robert Louis Stevenson and Paul McCartney.)
Computer scientist Gregory Chaitin, cofounder of Kolmogorov complexity described how his unconscious made connections:
I’m a great believer in the subconscious, in sleeping on it, in going to bed at 3 a.m. or 5 a.m. after working all night, and then getting up the next morning full of new ideas, ideas that come to you in waves while you’re taking a bath, or having coffee. Or swimming laps. So, mornings are very important to me, and I prefer to spend them at home. … I think of the subconscious as a new chemical soup that’s making new connections, and interesting combinations of ideas stick together, and percolate up to full consciousness.
This sudden appearance of solutions from the denizens of the hidden unconscious has also been reported in many fields other than mathematics. Jonas Salk, who developed the polio vaccine wrote:
It is always with excitement that I wake up in the morning wondering what my intuition will toss up to me, like gifts from the sea. I work with it and rely on it. It’s my partner.
Ray Kurzweil, referred to as “the ultimate thinking machine,” by Forbes magazine, wrote:
I’ve developed a method of solving problems while I sleep…I start out by assigning myself a problem when I get into bed…It could be any kind of problem, an issue with one of my inventions, a business strategy question, or even an interpersonal problem. I’ll think about the problem for a few minutes, but I try not to solve it. That would just cut off the creative problem solving to come…and then I go to sleep. Doing this primes my subconscious mind to work on the problem.
The Poincaré-Hadamard model for discovery suggests that during sleep, our brain attempts to organize all the fragments of perception gathered during consciousness and organize them into meaningful relationships. This period of incubation often climaxes in the illumination that is fast and frugal, and manifests as a eureka! moment. The final stage, consolidation, invokes what Kahneman refers to as the “System 2” type of slow arduous checking on this intuitive “aha!”. Throughout this entire discovery process, the System 1 and 2 modes of thinking conjoin in transforming the reflective habit of mind into a discovery.
