The Secret to Solving the Problem

There is often no recipe - chatGPT included - for cracking open an unsolved problem other than creative thinking. Where does human creativity come from? I have never carried out a formal study of this topic. But a general curiosity about the subject made me pick up a copy of The Psychology of Invention in the Mathematical Field by Jacques Hadamard.

Hadamard was a great mathematician. As a physicist I have encountered his work in the form of the Cauchy-Hadamard theorem, and the Hadamard matrices, which form the basis of the Hadamard gate in quantum computation. I have appreciated many times his famous saying about how useful complex analysis is when solving problems with real variables. The actual remark can be found in the book quoted above, on page 123.

Hadamard was interested in mathematics not only from a technical perspective but also from a pedagogical and psychological viewpoint. He had been motivated to investigate the question of mathematical creativity after hearing a talk by Poincaré on the subject. What I found interesting about Hadamard's book:

i) the nuanced comparisons between discovery and invention

ii) the suggestion that scientific truth is born from poetic emotion

iii) a 'creativity' questionnaire, assembled by the psychologists Claparede and Flournoy (Appendix I). Although aimed at mathematicians, I think it's fun and interesting for anyone to take.

iv) statements about their own creative process by Einstein (Appendix II), Gauss, Hermite, Helmholtz, Poincare, Mozart, Norbert Wiener, Pólya, Paul Valery (the poet), etc.

v) Hadamard's admission that he needs to walk in order to think - at last, a trait that I share with the great man!

v) extensive discussions about the role played by the unconscious in creative discovery

vi) the refrain that typically creative inspiration comes only after a period of fruitless labor.

Hadamard's work has apparently been confirmed and extended by subsequent researchers. His book was published in 1945 - it's an oldie, but a goldie.