The Psychology of Invention in the Mathematical Field

Rating:
Summary: An entry into the creative mathematical mind
Review: All humans are subject to the "aha moment" phenomenon, when a solution to a problem suddenly appears in your mind. In most instances, it is the sudden remembering of a name or a phone number, the rediscovery of something previously encountered and with nothing added. What is of most interest is when such a moment takes place and there is more there than there was before. This is what creativity is all about and it is one of the most difficult human activities to understand and predict.
Mathematics is very different from other fields, in that while new abstract ideas are built from old ones, the new idea is often quite different. Furthermore, it is sometimes not possible to use concrete representations such as diagrams to explain the idea. Hadamard refers to several mathematicians describing how their ideas came to them. From his evidence, mathematical inspiration is largely due to having a prepared mind. While the idea seems to appear fully formed in an instant, generally there was an extensive period of preparation to understand the background material. Once this was done, there was some significant mental effort that appeared to yield nothing, a time of relaxation and then suddenly the solution appeared. This is followed by the verification of the idea and the transliteration into a formal form for communication to others.
The three middle chapters, "The Unconscious and Discovery", "The Preparation Stage", "The Later Conscious Work" and "Discovery as a Synthesis: The Help of Signs" outline this sequence of events. In reading this book, it is clear that creativity is a skill that can be acquired and honed, at least to some extent. It was interesting to read the book and get the views of creative mathematicians on how they managed to invent new ideas.

Published in the recreational mathematics e-mail newsletter, reprinted with permission.

Rating:
Summary: Creativity by the creative
Review: Hadamard was one of the productive mathematicians of the late 19th and early/mid 20th century. Maybe his name isn't as broadly known as Gauss or many others. Still, the Hadamard Hadamard matrices appear in modern error correcting codes among other places, and his name is known in other contexts. This is someone whose creative work is still an active field of study, over a century after some of it was originated.

When someone of his stature decides to write about creativity, I am interested. He centers his inquiry on the thought processes of invention. That includes the place of sub-conscious thought, in several forms, and of abstract or verbal thought vs. imagery.

This study is wholly descriptive, not prescriptive. It simply relates the introspective sensation of Hadamard's thought process, and that of many others including Einstein and Mozart. There is little here about the the process of cultivating a creative talent, just description of how creators perceive their own talent in operation.

Although interesting, I found it somewhat dry. He cites many other studies of creativity, some of which are now hard to find. Many, however, come from an angle that I find un-helpful: the view a psychologist might take, rather than that of a mathematician or artist.

This brief book may be of historical interest, but probably won't benefit the working creative professional.

Rating:
Summary: Creativity by the creative
Review: Hadamard was one of the productive mathematicians of the late 19th and early/mid 20th century. Maybe his name isn't as broadly known as Gauss or many others. Still, the Hadamard matrices appear in modern error correcting codes among other places, and his name is known in other contexts. This is someone whose creative work is still an active field of study, over a century after some of it was originated.

When someone of his stature decides to write about creativity, I am interested. He centers his inquiry on the thought processes of invention. That includes the place of sub-conscious thought, in several forms, and of abstract or verbal thought vs. imagery.

This study is wholly descriptive, not prescriptive. It simply relates the introspective sensation of Hadamard's thought process, and that of many others including Einstein and Mozart. There is little here about the the process of cultivating a creative talent, just description of how creators perceive their own talent in operation.

Although interesting, I found it somewhat dry. He cites many other studies of creativity, some of which are now hard to find. Many, however, come from an angle that I find un-helpful: the view a psychologist might take, rather than that of a mathematician or artist.

This brief book may be of historical interest, but probably won't benefit the working creative professional.

//wiredweird

Rating:
Summary: First rate advice for science and math students
Review: 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!

Rating:
Summary: The Psychology of Math
Review: 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."

Rating:
Summary: A study of the mental workings of some great mathematicians
Review: 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. 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 retitled "The Mathematician's Mind" and republished in the Princeton Science Library series; it's not immediately clear from the Amazon page that this is so. The Dover edition is substantially cheaper.

A fascinating and informative book.

Rating:
Summary: A study of the mental workings of some great mathematicians
Review: 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. 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 retitled "The Mathematician's Mind" and republished in the Princeton Science Library series; it's not immediately clear from the Amazon page that this is so. The Dover edition is substantially cheaper.

A fascinating and informative book.