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This is a book in the tradition of Euclidean synthetic geometry written by one of the twentieth century's great mathematicians. The original audience was pre-college teachers, but it is useful as well to gifted high school students and college students, in particular, to mathematics majors interested in geometry from a more advanced standpoint. The text starts where Euclid starts, and covers all the basics of plane Euclidean geometry. But this text does much more. It is at once pleasingly classic and surprisingly modern. The problems (more than 450 of them) are well-suited to exploration using the modern tools of dynamic geometry software. For this reason, the present edition includes a CD of dynamic solutions to select problems, created using Texas Instruments' TI-NspireTM Learning Software. The TI-NspireTM documents demonstrate connections among problems and--through the free trial software included on the CD--will allow the reader to explore and interact with Hadamard's Geometry in new ways. The material also includes introductions to several advanced topics. The exposition is spare, giving only the minimal background needed for a student to explore these topics. Much of the value of the book lies in the problems, whose solutions open worlds to the engaged reader. And so this book is in the Socratic tradition, as well as the Euclidean, in that it demands of the reader both engagement and interaction. A forthcoming companion volume that includes solutions, extensions, and classroom activities related to the problems can only begin to open the treasures offered by this work. We are just fortunate that one of the greatest mathematical minds of recent times has made this effort to show to readers some of the opportunities that the intellectual tradition of Euclidean geometry has to offer. ... Read more

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Customer Reviews (1)

A fascinating biography of a great scientist
The biography is written brilliantly and is quite fascinating. The 98-year long life of Hadamard was extraordinary, filled with fantastic joys, like playing the violin with Albert Einstein, and terrible hardships, as losinghis three sons in the two World Wars. He was a man of amazing activity andgreat social commitment, being involved in the struggle for human rights.In his lifetime, the world was a scene of tremendous perturbations andchanges in all respects, and so also in the area of science, where Hadamardwas a great driving force especially in mathematics, but also in physics,mechanics, psycology of invention etc. Little did he realise during hiscareer that his matrices later would be used in the coding theory oftoday.

The chapters about Hadamard's work in mathematical physicscontains a lot of untraditional, interesting and almost unknown material.It is hard to find elsewhere such a complete and clearly written survey ofthe history ofthe disproval of several "obvious" hypotheses.

The textis tastefully mixed with illustrations and the occasional anecdote makes ita quite entertaining read. I can higly recommend it to anybody, fromundergraduate students to high-level professionals. ... Read more

Product Description
Walter Rudin's memoirs should prove to be a delightful read specifically to mathematicians, but also to historians who are interested in learning about his colorful history and ancestry. Characterized by his personal style of elegance, clarity, and brevity, Rudin presents in the first part of the book his early memories about his family history, his boyhood in Vienna throughout the 1920s and 1930s, and his experiences during World War II.Part II offers samples of his work, in which he relates where problems came from, what their solutions led to, and who else was involved. As those who are familiar with Rudin's writing will recognize, he brings to this book the same care, depth, and originality that is the hallmark of his work.Co-published with the London Mathematical Society beginning with Volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners. ... Read more

Customer Reviews (2)

Portrait of a brilliant, deeply humane man
This is an inspirational and often moving biography of one of the great mathematicians of our century. Hadamard was not only a brilliant mathematician, but a stimulating mentor, a man of wide-ranging knowledge, insatiable curiosity, humility, and most of all, great humanity. When Japaninvaded Manchuria, he proposed that the League of Nations sendpeace-keeping forces there, only to face ridicule from the French press.Hadamard, more than most of us, deeply felt the tragic wastefulness of war.He lost two beloved sons in World War I. Of his son, Etienne, he once said:"what I did in mathematics is nothing compared to what he could do ifhe were alive today." (Half of the brilliant graduates of the EcoleNormale were killed in that war). I finished this book with feelings ofgratitude and regret--gratitude to Hadamard for sharing his remarkablegifts with such generosity, and deep regret that "fate" was sogrievously unfair to him.

Inspirational and moving
This is an inspirational and often moving biography of one of the great mathematicians of our century. Hadamard was not only a brilliant mathematician, but a stimulating mentor, a man of wide-ranging knowledge,insatiable curiosity, humility, and most of all, great humanity. When Japaninvaded Manchuria, he proposed that the League of Nations send peacekeepingforces there, only to face ridicule from the French press. Hadamard, morethan most of us, deeply felt the tragic wastefulness of war. He lost twobeloved sons in World War I. Of his son Etienne, he once said: "what Idid in mathematics is nothing compared with what he could do if he werealive today."(As the book points out, half the brilliant graduates ofthe Ecole Normale were killed in that war). I finished this book withfeelings of gratitude and regret--gratitude to Hadamard for sharing hisremarkable gifts with such generosity, and deep regret that"fate" was so grievously unfair to him. ... Read more

Product Description
Thoughtful and articulate study of the origin of ideas. Role of the unconscious in invention; the medium of ideas—do they come to mind in words? in pictures? in mathematical terms? Much more. "It is essential for the mathematician, and the layman will find it good reading."-Library Journal.
... Read more

Customer Reviews (7)

Understand the mind's inner creative workings
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.

the only book on the topic of doing mathematical research
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.)

An entry into the creative mathematical mind
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.

Creativity by the creative
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

The Psychology of Math
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." ... Read more

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Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork. ... Read more

Product Description
This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This scarce antiquarian book is a selection from Kessinger Publishings Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the worlds literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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############################################################################################################################################################################################################################################################### ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Lévi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence of unconscious mental processes in mathematical invention and other forms of creativity. Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field, remains an important tool for exploring the increasingly complex problem of mental life.

The roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. His discussion of this process comprises a wide range of topics, including the use of mental images or symbols, visualized or auditory words, "meaningless" words, logic, and intuition. Among the important documents collected is a letter from Albert Einstein analyzing his own mechanism of thought. ... Read more

Customer Reviews (3)

The Psychology of Invention in the Mathematical Field
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!

The Psychology of Math
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."

A study of the mental workings of some great mathematicians
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. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more