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This is a short (136 page) study of how creative thought works. Hadamard, a world-class mathematician best known for his proof of the prime number theorem in 1896, wrote this in the 40's, basing it on correspondence with many of the great living mathematicians of his time. The actual questions he posed are preserved in an appendix. Most of his respondents were mathematicians (and he limited his correspondence to the best minds in the field), but he did get information from several other fields, and cites data about physicists (a letter from Einstein forms another appendix), chemists, physiologists, metaphysicians, and so on. What he is trying to examine is a slippery subject, perhaps best explained by a quote. Here is a discussion of Sidgwick, an economist: "His reasonings on economic questions were almost always accompanied by images, and the images were often curiously arbitrary and sometimes almost undecipherably symbolic. For example, it took him a long time to discover that an odd symbolic image which accompanied the word 'value' was a faint, partial image of a man putting something on a scale." Hadamard gives his own mental images that accompany his following through the steps of Euclid's famous proof of the infinitude of primes. I won't reproduce that here for space reasons, but the contrast with Sidgwick's--and with other reports of mental activity--is fascinating. Many other examples are given, from Mozart to Polya to Galton to Poincare. Hadamard makes it clear that language and thought are not the same thing, contrary to a commonly expressed view among linguists.Read more ›
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18 of 21 people found the following review helpful
Not only is this book fascinating, it's the only one of it's kind. The book has also proved very useful to me in life. As a graduate student I used Poincaré's implicit `advice' (described in the book) in the following way. In electrodynamics we had a long problem sheet to hand in every two weeks. I started by writing down answers to all problems that I knew. Then, I thought about the next-easiest problem each day walking twice to and from the University (about 1 1/2 hours altogether). When the answer came I wrote it down and iterated the process. Before the end of two weeks most of the problems (from Jackson) had been solved. Poincari's advice is very good about giving the unconscious a chance to work. Phooey and double phooey on the silly, uncreative skinner-box types and other behaviorists who don't recognize the unconscious as the source of creativity!
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The Psychology of Invention in the Mathematical Field is a study on how research mathematicians go about the business of advancing their field. Jacques Hadamard, a prominent mathematician, wrote this psychology text over 50 years ago, after having done his best work 50 years prior. Although in some ways dated, both in content and in writing style, the book provides an interesting examination of the role of the conscious and subconscious in solving a problem, particularly the process of incubation and (seemingly) sudden inspiration. He brings up the roles intuition and logic play in the way various mathematicians go about their business. Hadamard also examines the influence of aesthetics in not just choosing a problem, but in solving it. He studies the choice of research direction, with the interesting comment that Hadamard himself avoided areas of research where there was already a great deal of activity. The book is short enough that if the subject interests you, it is worth your time. The text is also published under the title "The Mathematician's Mind."
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I give it 5 stars because it is unique. I.e. although not perfect, there is to my knowledge no better book on the difficult and elusive topic of how to come up with new ideas and insights on unsolved problems.
Even though Hadamard restricted his survey to great minds of his and earlier time, I have found that the lessons described there apply also to the rest of us.
As described well above, the mora is to "prepare the mind", that one must think and study a problem as hard as one can for a long time, and then the mind will often reveal the solution.
I myself have tried hard to solve problems unsuccessfully, then gone to bed and awakened in the night with a sense of mental energy. I have learned it is prudent to arise and see what is forthcoming, and occasionally have obtained the solution of my problem.
After being told otherwise by my linguistics professors, it was also reassuring that my way of thinking about problems, in pictures and vague shapes, not words, was shared by famous scientists.
I was also piqued as a young student by Hadamard's quote from a mathematics professor asking a student wanting a thesis problem if, in all his studies, he had not noticed anything needing further investigation. This habit of attempting to "find problems" is essential to research, but seldom mentioned in class.
(I have been a research mathematician now for some 30 years since reading this book in 1970.)
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This item: An Essay on the Psychology of Invention in the Mathematical Field
