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Yearning for the Impossible: The Surprising Truths of Mathematics

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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

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.

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".

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.

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.

While browsing this book on the shelf of the Singapore National Library one sunday, I was shocked and delighted to spot on the chapter 7 of "Ideal". Being a Mathematics student of Abstract Math 30 years ago, I could not find a satisfactory answer from my French University Math professors or any Math books who could tell me what does this "Ideal" concept really mean beyond its arcane definition, where did it derive from, and why most textbooks insert "Ideal" in the chapter of Ring Theory?
Prof John Stillwell did a beautiful job by explaining in simple layman language the historical background of Kummer's work on FLT (Fermat Last Theorem), who encountered the controversy of Fundamental Law of Arithematics with Algebraic Number extended Field [a+ SQRT(-b)]. So instead of giving up, Kummer 'faked' the 'Ideal' number which he guessed could resolve the conflict. It was after his death that Dedekind discovered prime ideal, principal ideal's existence hidden in the compound of other numbers (Greatest Common Divisors of Rationals and Algebriac numbers, to be exact). Beautiful fake and discovery story in Mathematics!

Mathematics professor John Stillwell (University of San Francisco) presents Yearning For The Impossible: The Surprising Truth Of Mathematics, a fascinating tour through the history of mathematics and the resounding impact new conceptual discoveries have made upon human civilization. Mathematical achievements such as concepts of irrational numbers, imaginary numbers, the infinitesimal, the fourth dimension, periodic space, and more are explored in terms accessible to lay readers, and their intersection with art, literature, philosophy, and physics among other disciplines is laid bare. 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.

If you look at the other reviews you'll see they are all full of praise. I really expected very much from this book, but the more I read in it the more I got disappointed. The material presented is indeed interesting, but the author's way of explaining things is quite often less than ideal. If this book had been written by someone like Paul Nahin, William Dunham or Adrian Paenza it would have been much better. Knowing much is one thing, explaining it the best way is another and unfortunately Stillwell isn't particularly good in the last thing.

Very helpful for a person, such as myself, who wants a clear understanding of mathematics especially geometry and how it relates to modern Cosmology.