To answer this question adequately would require much more space than is available on this post. Therefore, I will focus on only one example of an important approach to problem-solving that is widely employed by those of high mental function.
An old adage about dealing with an intractable problem states, “Sleep on it.” For centuries, people have discovered that when they went to sleep while ruminating over a problem, they often awoke in the morning with a solution. However, there was no science to explore whether this adage was an “old wives tale,” or whether it had substance, until Freud, in the 19th century, proposed the idea of a subconscious mind.
In a speech delivered to the Société de Psychologie in 1908, the great mathematician, Henri Poincaré described his unconscious mental activity that exploded in a flash of illumination, revealing a solution to the problem that had haunted him for months:
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 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.
…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.
In his seminal publication, The Psychology of Mathematical Invention. Jacques Hadamard, co-discoverer of a proof of the Prime Number Theorem, used the Poincaré narrative to describe a four-stage process that climaxes in a eureka! moment.
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–known knowns.
Poincaré’s description of his eureka! moment has been echoed by other mathematicians, including H. S. M. Coxeter, who has been called the “greatest classical geometer of the 20th century. High achievers in other fields, who also attributed their discoveries to unconscious brain activity during their sleep, include Dr. Jonas Salk, discoverer of the polio vaccine, author Steven King and musician Paul McCartney.The music for the famous hit song Yesterday came to Paul McCartney in a dream, so vivid that he thought he must have heard it while conscious. He wrote:
I was living in a little flat at the top of a house and I had a piano by my bed. I woke up one morning with a tune in my head and I thought, ‘Hey, I don’t know this tune – or do I?’ It was like a jazz melody. My dad used to know a lot of old jazz tunes; I thought maybe I’d just remembered it from the past. I went to the piano and found the chords to it, made sure I remembered it and then hawked it round to all my friends, asking what it was: ‘Do you know this? It’s a good little tune, but I couldn’t have written it because I dreamt it.’
It seems that while we are sleeping, our brains scan all the ideas and images that it processed during consciousness and matches them in some random way, occasionally finding a connection that was previously blocked by our conscious mind. People of high intellect take advantage of this process and often engage intensely on a problem before going to sleep. It doesn’t always yield results, but it does often enough to make it a viable strategy for problem-solving.
