Yearning for the Impossible: The Surprising Truths of Mathematics [Hardcover]

John C. Stillwell (Author)

Book Description

Publication Date: May 22, 2006 | ISBN-10: 156881254X | ISBN-13: 978-1568812540

This book explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress: - Irrational and Imaginary Numbers - The Fourth Dimension - Curved Space - Infinity and others The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. By imbedding mathematics into a broader cultural context and through his clever and enthusiastic explication of mathematical ideas the author broadens the horizon of students beyond the narrow confines of rote memorization and engages those who are curious about the place of mathematics in our intellectual landscape.

Editorial Reviews

Review

Stillwell weaves historical details into his writing seamlessly, helping to give the reader the true feeling that mathematics is more than just a bunch of people playing games with symbols, but rather a rich and rewarding intellectual endeavor important to the human enterprise. --MAA Reviews

Yearning For The Impossible is as much of a celebration of the greater understanding mathematics has brought to the world as it is a history and discussion of innovative concepts. and is highly recommended for library and personal reading shelves. --Wisconsin Bookwatch

Like the White Queen in Lewis Carroll s Through the Looking-Glass, mathematicians are called upon to believe in things that, at first glance, defy common sense and appear impossible.... . As Stillwell puts it, 'Mathematics is a story of close encounters with the impossible because all its great discoveries are close to the impossible.' --Science News

About the Author

John Stillwell, originally from Melbourne, Australia, is a Professor of Mathematics at the University of San Francisco. He earned his MSc in 1965 from the University of Melbourne and his PhD from MIT in 1970. His writing covers a wide spectrum of mathematics, including translations of classic texts by Dirichlet, Dedekind, Poincaré, and Dehn. He is also the author of Mathematics and Its History, The Four Pillars of Geometry, and Elements of Number Theory, among others. In 2005, he was awarded the Chauvenet Prize of the Mathematical Association of America for his expository writing

Product Details

• Hardcover: 244 pages
• Publisher: AK Peters, Ltd. (May 22, 2006)
• Language: English
• ISBN-10: 156881254X
• ISBN-13: 978-1568812540
• Product Dimensions: 9.1 x 6.1 x 0.9 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 4.8 out of 5 stars  See all reviews (9 customer reviews)
• Amazon Best Sellers Rank: #424,988 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

30 of 30 people found the following review helpful:

5.0 out of 5 stars Many of the mathematical ideas once considered impossible, October 15, 2007

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: Yearning for the Impossible: The Surprising Truths of Mathematics (Hardcover)

There are many great ideas in mathematics and what makes them unique is that many of them were considered impossible for a long period of time. In this book, Stillwell presents many of those ideas using an expository style that is both understandable and complete. The chapters are:

*) The Irrational - where the discovery of irrational numbers and how it shocked the Pythagoreans is explained. It forever destroyed the idea that everything could be completely expressed using only the integers. This discovery also made it clear that some things would forever remain unknown.
*) The Imaginary - this section describes the development of the "imaginary" numbers, where the impossible task of taking the square root of a negative number became routine.
*) The Horizon - where converging parallel lines allowed artists to perform what was considered impossible, give two-dimensional paintings a three-dimensional perspective.
*) The Infinitesimal - where splitting a figure into extremely small sections made it possible to easily solve an enormous number of complex problems.
*) Curved space - where the natural world of Euclid was suddenly overturned by the creation of curved worlds that are even more natural.
*) The Fourth Dimension - where the impossibility of structures having more than three dimensions is proven false. Along the way, imaginary numbers are made even more so by the development of the quaternions.
*) The Ideal - in this case, the impossibility of numbers having more than one fundamental factorization is overturned only to be partially restored.
*) Periodic Space - among others, M. C. Escher demonstrated that it is easy to place impossible objects on a canvas.
*) The Infinite - where it is demonstrated that not all infinities are alike, it is the case that some infinities have more elements than others.

Stillwell does an excellent job in pointing out that "impossible" is a difficult word to use in mathematics, as it is relative to the definitions of the object being examined. While there is absolute truth in mathematics, something lacking in many other areas of human endeavor, the truth is also often relative to how imaginative we are in our definitions.

Published in Journal of Recreational Mathematics, reprinted with permission

20 of 20 people found the following review helpful:

5.0 out of 5 stars Excellent, July 18, 2007

By 

Mark Shapiro (USA) - See all my reviews
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This review is from: Yearning for the Impossible: The Surprising Truths of Mathematics (Hardcover)

This book, which can be viewed as a prequel to Stillwell's "History of mathematics", is an excellent resource for someone who wishes to get a view of mathematics as a field of inquiry driven by the need to solve problems as much as by creative desire to uncover connections between seemingly unrelated ideas by people who made mathematics, such as Gauss, Hamilton, Abel, Euler, Riemann. There are lively short essays about these and other great mathematicians. When read along with regular (good) textbooks on, e.g., complex variables, geometry, the two Stillwell's books will lead to a much better understanding of mathematical ideas.

17 of 17 people found the following review helpful:

5.0 out of 5 stars Beyond Common Sense, May 29, 2007

By 

Lewis H. Robinson - See all my reviews
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This review is from: Yearning for the Impossible: The Surprising Truths of Mathematics (Hardcover)

I liked this book. I particularly liked Chapter 1, The Irrational, Chapter 5, Curved Space, and Chapter 6, The Fourth Dimension.

Chapter 7, The Ideal, is also excellent and alone worth the purchase price, albeit the reader needs to follow closely the notational details and diagrams. In fact Chapter 7 is the reason I bought the book in the first place. I had always struggled with this important concept and was pleasantely surprised upon finding a book--Stillwell's--that devoted a whole chapter to the subject at an introductory as well as historical level. The author follows the development of the notion of the ideal concept from Gauss, to Kummer, to Dedekind's final generalization, where the payoff comes in Section 7.8. "Ideals, or Unique Prime Factorization Regained".