15 of 15 people found the following review helpful:
4.0 out of 5 stars
A study of the mental workings of some great mathematicians, January 16, 2001
This review is from: The Mathematician's Mind (Paperback)
This is a short 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. He cites Max Muller's comments equating thought and language, and acknowledges that for Muller it may be so, but convincingly demonstrates, by quoting numerous other mathematicians, that it is not true for everyone. The further conclusion, that the process of creative thought, while following similar patterns in similar discipline, can vary dramatically, is as far as Hadamard can go with the data he has.
One other note: this book was originally titled "The Psychology of Invention in the Mathematical Field" and is available under that title from Amazon, published by Dover Books. It's not immediately clear from the Amazon page that this is so. The Dover edition is substantially cheaper.
A fascinating and informative book.
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8 of 9 people found the following review helpful:
4.0 out of 5 stars
The Psychology of Math, April 2, 2001
This review is from: The Mathematician's Mind (Paperback)
The Mathematician's Mind 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 Psychology of Invention in the Mathematical Field."
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6 of 7 people found the following review helpful:
5.0 out of 5 stars
The Psychology of Invention in the Mathematical Field, March 3, 2005
This review is from: The Mathematician's Mind (Paperback)
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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