For centuries people have asked, “What thinking processes do mathematicians and research scientists use in making discoveries that advance the frontiers of human knowledge?” Unfortunately, we know little of the mechanisms that lie “under the hood” of our mental awareness. In my post last Monday, I outlined Henri Poincaré’s theory of the subconscious and how our brain does major problem solving, unbeknownst to our conscious mind.
In his classic book, the Psychology of Invention in the Mathematical Field, Jacques Hadamard, explained that many mathematicians visualize relationships and make connections through a kind of intuition guided by imagery. For example, the Venn Diagram above, helps us understand the connections among intuition, wisdom and mental acuity as components of intelligence.
The greatest mathematical minds engaged in deep problem solving by prolonged and unrelenting reflection. Poincaré, Coxeter, and Perelman lived in spaces of more than 3 dimensions and “saw” relationships that were inaccessible by direct observation. They walked, strolled, and sat transfixed as they explored realms outside their present location, exuding all the characteristics of the preoccupied, “absent-minded professor.”
Poincaré’s description of his eureka! moment has been echoed by other mathematicians. H. S. M. Coxeter, who has been called the “greatest classical geometer of the 20th century”, described how he discovered the snub cube, a 4-dimensional figure having 96 vertices, 432 edges, and 480 triangular faces:
I had long been trying to extend to four dimensions the familiar construction for the snub cube…The snag was that, since the number of “nearest other white tetrahedra” was nine, the equality of their distances would impose eight conditions on the point to be selected: five more conditions than such a point could generally be expected to satisfy.
So, I went to bed and soon slept soundly. About 3 a.m. I awoke with the idea of using a symmetrical “isosceles” tetrahedron; … I could thus choose a point on the axis of symmetry and adjust its height so as to equate the distances of the two types of neighbouring point. I switched on the light and went into my living room to write it down, lest I might find the next morning that it had passed away like any ordinary dream. When morning came, there it was, ready for all the details to be filled in.
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.
It seems that visual imagery enables us to connect and extend our capacity for abstract thought. The statement, “A picture is worth a thousand words,” expresses the idea that Images and equations are much more concise than verbal descriptions in representing information. Imagine attempting to express Einstein’s field equation in words! Many of the greatest mathematicians and scientists have described the imagery they used in discovering relationships and connecting disparate entities. A precise knowledge of how this happens is not yet understood, but some discoveries may emerge as research in artificial intelligence attempts to simulate our mental processes.
