Sitting on the toilet one morning, it suddenly hit me: a Sudoku puzzle is a graph coloring problem in disguise. Such out-of-the-blue moments of mathemSitting on the toilet one morning, it suddenly hit me: a Sudoku puzzle is a graph coloring problem in disguise. Such out-of-the-blue moments of mathematical inspiration, which usually come after struggling with a hard problem for days and then engaging in a different activity, are among the topics that Jacques Hadamard explored in this interesting small book.
As P. N. Johnson-Laird notes in the preface of this edition, the book was prescient: when Hadamard set out to explore mathematical invention, he went against the dominant philosophy of psychology of the time, behaviorism, by using introspection and discussing mental processes. Henri Poincaré's famous lecture before the Société de Psychologie in Paris inspired Hadamard to undertake this study, so he quoted Poincaré extensively, but he also provided his own experiences and insights. More importantly, Hadamard surveyed some of the major mathematicians and scientists of the time, such as George Polya, Norbert Wiener, and Albert Einstein.
In Chapter VIII, Hadamard suggested that under certain circumstances, "even important links of the deduction may remain unknown to the thinker himself who has found them." He cited Pierre de Fermat, Bernhard Riemann, and Évariste Galois as examples. Each of these mathematicians made a mathematical statement and claimed he had a proof but did not enunciate it due to limitations of space or time. These statements were indeed proved (completely or partially) later using facts and theories that were unknown in the mathematician's time. These facts and theories represent, by themselves, significant discoveries, yet no conception of, or even allusion to them appear in any of the mathematician's writings. It may be speculative, but I find Hadamard's explanation of these "paradoxical cases of intuition" fascinating, and I wish he had elaborated on it more.
Despite its interesting content, I felt something was missing when I finished the book; a sense of closure, perhaps. But take this vague criticism with a grain of salt, because it could be a result of the discontinuity of my reading.
In short, this small book is worth reading, or at least skimming, as it provides a window into the creative processes of some of the great minds of the time. Who knows, perhaps that elusive answer that you were looking for will hit you while you are reading the book, especially if you are sitting on a toilet....more