The Mathematician's Mind: The Psychology of Invention in the Mathematical Field

Well, the work is clearly outdated in many parts: that was inevitable. It is even a bit tedious, sometimes; and written in what sounded to me "a strange English"; after further reading I realized that it was not a translation, but it was Hadamard himself writing English... The most interesting parts of the book are, obviously, the "case studies" presented by Hadamard (including, of course, himself). I found this parade of minds very interesting because it helped me to understand a bit more my own way of thinking and my own (little) creativity. There is another most interesting point in this little book. Many mathematicians (perhaps all the true mathematicians) have a Platonic conception of an ideal world of numbers and mathematical concepts; in the course of their research they just "discover" these pre-existing, eternal truths. As a layman interested in many things, I have always agreed with this conception, but I have often wondered how and why such a Platonic view holds even today, against the recurrent blows of neopositivism and formalism, and so on. Reading Penrose's books I saw some clues, but this book suggests a clear answer: true mathematicians really see glimpses of this ideal world. One may argue that this could be just a way of our mind, so this ideal world is just an illusion cast by our mind (or I should say by the kind of mind which can develope into a mathematic mind). But there are many reasons to think that mathematics is a very real world which we can explore and discover. This is a very profound subject, involving the nature of our mind and that of the world around us, its being understandable, and the mysterious connections between the abstract world of pure mathematics and the physical world.

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.)

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."

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!