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Yearning for the Impossible: The Surprising Truths of Mathematics (Englisch) Gebundene Ausgabe – 30. April 2006


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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.' -Ivars Peterson, Science News , July 2006 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. -Marcus E. Barnes, MAA Reviews, August 2006 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, August 2006 Rises nobly to the challenge of describing these topics to a genuine novice...There is much to admire in Stillwell's attempt... he's accomplishing something very important and difficult here in demonstrating that there's some real struggle present in the process of mathematical discovery. -Daniel Biss, Notices, June 2007 Yearning for the Impossible offers a fascinating, historical look at some popular mathematical concepts used in music, art and philosophy... This book is an interesting find and provides a readable approach to some higher-level mathematics. The chapters can be read independently, and the reader can dig deeper into textbooks and history books for additional problems and details. I give a high recommendation for this book! -Lynn Godshall, Convergence Magazine (MAA), June 2007 A wonderful journey through mathematical discoveries... this book is an excellent vehicle for giving mathematics students new research ideas and, most important, for planting the seed in their minds to 'yearn for the impossible' as they investigate new truths. -Mathematics Teacher, August 2007 Stillwell has achieved what many might well have come to believe to be nearly impossible in mathematical exposition for the masses...[he] succeeds, in every topic treated, in bringing a fresh eye to questions even mathematicians might think have been mined in the past to boring exhaustion [and] shows there is still a lot of gold to be found, if one only thinks about things in a new way. Stillwell brings new, unorthodox insights to his writing that will stimulate readers (from high schoolers to emeritus professors) to think about old topics in new, nonstale ways... Yearning for the Impossible will be a treat for teachers, too, who are looking for new ways to bring stimulating, fresh examples into their courses. -SIAM Review, May 2007 Stillwell does an excellent job laying the historical foundations for these discoveries; he is to be commended for his historical accuracy. -Recreational Math, March 2007 Mathematics may be described as a story of close encounters with the impossible because all great discoveries are close to the impossible. The aim of this book is to tell this story, briefly and with few prerequisites, by presenting some representative encounters across the breadth of mathematics. -CMS Notes, May 2007 This book explores history through a lens focused on the creative tension between common sense and the 'impossible' ... Drawing connections to art, literature, philosophy, and physics, this book examines the place of mathematics in our intellectual landscape. -L'Enseignement Mathematique, January 2006 2009 Alpha Sigma Nu Book Award in the Discipline of Mathematics/Computer Science -The Association of Jesuit Colleges & Universities, November 2009

Synopsis

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.

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Amazon.com: 11 Rezensionen
34 von 34 Kunden fanden die folgende Rezension hilfreich
Many of the mathematical ideas once considered impossible 15. Oktober 2007
Von Charles Ashbacher - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe
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
21 von 21 Kunden fanden die folgende Rezension hilfreich
Beyond Common Sense 29. Mai 2007
Von Lewis H. Robinson - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe Verifizierter Kauf
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".
20 von 20 Kunden fanden die folgende Rezension hilfreich
Excellent 18. Juli 2007
Von Mark Shapiro - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe
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.
12 von 12 Kunden fanden die folgende Rezension hilfreich
Excellent overview of many less "traditional" topics 11. August 2007
Von Kenneth Knowles - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe
It is very nice to see a book that treats topics other than irrational and complex numbers (though they are important to understand first, of course!) like quaternions and prime ideals, not to mention all the geometrical connections. This book gives a great historical and motivational perspective; the author may be augmenting the personalities in the book to add to the suspense and mystery, but overall the effect is beautiful.

I would recommend this book for anyone interested in Mathematics, including advanced students (I am a PhD student hovering near the border of Computer Science and Math). It is a welcome inspirational supplement to the tragedy of axioms and formalism that is modern mathematics education.
45 von 55 Kunden fanden die folgende Rezension hilfreich
Beautiful, substantial, unusual topics 15. Oktober 2006
Von Viktor Blasjo - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe
The chapters on geometry---projective geometry and hyperbolic geometry in particular---are extremely beautiful. We study many picturesque ideas, wonderful in themselves, that arise as "impossibly" neat solutions to interesting problems (perspective drawing, axiomatisation of geometry, shape of the universe, etc.), and then pay off by supplying unexpected insights elsewhere (e.g., the connection between projectively generated arithmetic and hypercomplex number systems that deserves to be better known). The chapters on complex numbers and quaternions are also very interesting. There are "unrecognised appearances" of complex numbers in already in Diophantus's number theory, namely the equivalent of complex multiplication in the context of sums of two squares. Thousands of years later, when the geometry of complex numbers was established, the search for a three dimensional analog failed and one had to settle for the analog in four dimensions. The historical circle closed beautifully when Graves noted with surprise and satisfaction that this state of affairs is precisely mirrored in classical number theory where Diophantus's theorem on sums of two squares generalises to four squares but not three. Stemming from the same roots in classical number theory, there is also an excellent chapter on algebraic number theory. Just as in his proof of the non-existence of three dimensional hypercomplex numbers in the quaternion chapter, Stillwell here takes on some very serious mathematics that many mathematicians would tell you require plenty of abstract algebra. But Stillwell knows better, cutting to the core of things with beautifully clear geometric arguments in both these cases. The other chapters are less innovative, although we are happy with the initiative to derive the infinite series for pi (essentially by the power series for the arctangent) only ten pages after the idea of infinitesimals is introduced (again relying on geometric methods rather than, as others would have it, abstract theories like Taylor's theorem). The role of the impossible in mathematics is pointed out along the way, and Stillwell offers some rewarding reflections on this subject; these are highly retrospective, however, and if we were to take this topic seriously we would have wished for greater insights into the historical mathematicians' thoughts on these supposedly impossible things.
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