Yearning for the Impossible: The Surprising Truths of Mathematics

by John Stillwell

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.

Genres: Mathematics, Science, Nonfiction, Physics, Textbooks

230 pages, Paperback

First published May 22, 2006

About the author

John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University

Reviews

[WarpDrive · Rating 4 out of 5]

I quite enjoyed this book. It deals with subjects in mathematics which were once considered "impossible", but which were later addressed in a surprising and fascinating voyage of intellectual discovery.
- The first chapter, on the irrationals, is quite nice - the concepts are explained nicely and clearly.
- The second chapter, about the imaginary and the complex numbers, is quite nice too. But I must say that I was left a bit disappointed here:
a) yes there are very nice points about the geometry of complex number multiplication and corresponding rotations
b) yes, the mystical Euler's formula is there (and it could not have been there, ubiquitous as it is in mathematics and physics, probably the most beautiful formula so far discovered, formula which the physicist Richard Feynman called "our jewel" and "the most remarkable formula in mathematics").
c) yes, the author also correctly points out that terms like "complex" and "imaginary" are very misleading
BUT, on the negative side:
a) the whole subject is treated only superficially in a small 20-page chapter, and for example the fascinating subject of complex analysis is not even cursorily treated (at least, the Bezout's theorem is briefly mentioned)
b) and, while the author condemns terms such as "imaginary" and "complex" as misleading, he fails to mention that complex numbers are more "real" than real numbers, in the sense that without complex numbers it is simply not possible to fully describe reality. For example, without complex numbers modern quantum mechanics would simply not be possible: the wave function is a complex value function, and when you multiply operators or state vectors you actually have to multiply complex numbers (the matrix elements) according to the rules of complex multiplication. In quantum mechanics, the determination of measurement outcome probabilities always include the squaring of absolute values of complex probability amplitudes defined by the wave function. Complex numbers are simply fundamental for most predictions in modern science.
- Chapter 3 on projective geometry is quite interesting, although hardly mind-blowing
- Chapter 4 about the infinitesimal is quite nice - nothing new or revolutionary here, but item 4.8 where, with simple elegant trigonometric calculations and the application of geometric series, the beautiful result that "pi" is encoded by an alternating sequence of fractions with odd denominators (showing that the irrational, transcendent "pi" has much to do with natural numbers as with geometry) is really good
- Chapter 5 on curved space is really nice: in particular 5.4, where the concept of Gaussian curvature is beautifully explained, and where the beautiful Harriot's theorem (later extended by Gauss) is explained in a very nice manner.
- Chapter 6 on quaternions is done very nicely as well, but unfortunately it failed to really excite me, as quaternions are not so "cool" anymore - they have been at least partially superseded by other techniques in areas such as quantum mechanics (however they still play an important role in calculations involving three-dimensional rotations such as in three-dimensional computer graphics).
- Chapters 7 and 8 are pretty good
- Chapter 9 (on the "infinite") is really nice. It explains beautifully the concept of uncountable, of potential versus actual infinity, and the famous "diagonal argument" by Cantor is also explained. But I did appreciate that the less famous (but equally if no more beautiful) Harnack's theorem, is explained as well: it mind-blowingly demonstrated that "almost" all real numbers are irrational, and that "almost" all real numbers are transcendental (contrarily to what intuition might drive us to believe).

Overall, a nice and quite enjoyable book - please note that it requires at the very minimum high school maths, if not (preferably) at tertiary level.

[Charles · Rating 5 out of 5]

Many of the mathematical ideas once considered impossible

This is one of those books where I dislike the title, yet love the content. Mathematicians generally go where the necessity and reasoning takes them, so it is a misnomer to claim that they have a yearning for the impossible. Also, to argue that mathematical results are impossible at some point then are considered routine later is incorrect.
As mathematical knowledge has expanded, new discoveries sometimes contradict and in all cases extend previous knowledge. What was thought to be impossible earlier is demonstrated to be possible, so the earlier “fact” was simply a “significant misunderstanding.”
Stillwell discusses several of these “significant misunderstandings” in this book. They are:

*) The discovery of irrational numbers.
*) The discovery of imaginary (complex) numbers.
*) The discovery of perspective and points at infinity.
*) The development of calculus and the use of the infinitesimal.
*) The discovery of non-Euclidean geometry and curved space.
*) The discovery of the quaternions and the fourth dimension.
*) The discovery of the algebraic ideal and the loss of unique factorization.

When each of these discoveries was made, that discovery changed some aspects of mathematics forever. The previous ideas were not somehow just rendered “inoperable” but were shown to be restricted cases of a more general result. Irrational numbers were found to be the limits of sets of rational numbers, real numbers were found to be a subset of complex numbers and Euclidean geometry was found to be just one possibility among multiple options. Stillwell does an excellent job laying the historical foundations for these discoveries; he is to be commended for his historical accuracy.
While the word impossible certainly is a powerful word to catch your eye, it is inappropriately placed in the title of this book. However, that is really the only complaint that I will lodge about it. The rest is first-rate.

Published in Journal of Recreational Mathematics, reprinted with permission

[Chelsea M · Rating 4 out of 5]

It was a very interesting read, and it was around the right level for me.
One minor complaint is that the placement of diagrams weren't always ideal, and I ended up having to flip back and forth between pages to refer to them often. But otherwise, it was both entertaining and educational.

[Jilany]

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