Psychological examples from science and mathematics

Jacques Hadamard

Jacques Hadamard, in his book Psychology of Invention in the Mathematical Field, uses introspection to describe mathematical thought processes. In contrast to authors who identify language and cognition, he describes his own mathematical thinking as largely wordless, often accompanied by mental images that represent the entire solution to a problem. He surveyed 100 of the leading physicists of his day (ca. 1900), asking them how they did their work. Many of the responses mirrored his own.

Hadamard described the experiences of the mathematicians/theoretical physicists Carl Friedrich Gauss, Hermann von Helmholtz, Henri Poincaré and others as viewing entire solutions with “sudden spontaneity.”

The same has been reported in literature by many others, such as Denis Brian, G. H. Hardy Walter Heitler, B. L. van der Waerden, and Harold Ruegg.

To elaborate on one example, Einstein, after years of fruitless calculations, suddenly had the solution to the general theory of relativity revealed in a dream “like a giant die making an indelible impress, a huge map of the universe outlined itself in one clear vision.”

Hadamard described the process as having steps (i) preparation, (ii) incubation, (iv) illumination, and (v) verification of the five-step Graham Wallas creative-process model, leaving out (iii) intimation, with the first three cited by Hadamard as also having been put forth by Helmholtz:

Marie-Louise von Franz

Marie-Louise von Franz, a colleague of the eminent psychiatrist Carl Jung, noted that in these unconscious scientific discoveries the “always recurring and important factor ... is the simultaneity with which the complete solution is intuitively perceived and which can be checked later by discursive reasoning.” She attributes the solution presented “as an archetypal pattern or image.” As cited by von Franz, according to Jung, “Archetypes ... manifest themselves only through their ability to organize images and ideas, and this is always an unconscious process which cannot be detected until afterwards.”