It's great to know that John Derbyshire just published a new book following his huge success with "Prime Obsession" (2003). This book on the history of algebra should be very appealing to Kanex et al. who are very enthusiastic about Category Theory etc.

http://www.amazon.com/gp/product/030909657X/ref=nosim/002-2666194-7089657?n=283155

Unknown Quantity:
A Real and Imaginary History of Algebra
by John Derbyshire

Reviewed by Luiz Henrique de Figueiredo

Derbyshire's previous book, Prime Obsession, was a tour de force whose goal was to explain the Riemann Hypothesis to non-mathematicians. The book was widely acclaimed as a success, an opinion that I share: I read the book with pleasure and learned some mathematics along the way. I thought Derbyshire succeeded in telling the interesting history behind the mathematics in pleasant and engaging prose, and in explaining (at least some of) the mathematics to non-specialists, a feat not to be sneered at given that the mathematics of the Riemann Hypothesis is certainly non-trivial.

So, having read Prime Obsession, I was curious and motivated to read Derbyshire's newest book, Unknown Quantity, which he describes as "a history of algebra, written for the curious nonmathematician". In this sense, Unknown Quantity has the same goal as Prime Obsession. However, the scope of Unknown Quantity is much larger, because algebra is a vast subject with a long history, dating back to the Babylonians, whose coherence is hard to see, especially for nonmathematicians.

Like Prime Obsession, Unknown Quantity contains large sections that describe the mathematics. In Prime Obsession this was done in the odd-numbered chapters, with the even-numbered chapters focusing on the history. In Unknown Quantity, Derbyshire chose to sprinkle "Math Primers" along the way (there are six primers among 15 chapters). There is a fair amount of mathematics in the main text as well. I have a feeling that a nonmathematician will need a lot of motivation to go through all this material. But such a reader will be rewarded with a reasonable sense of how algebra evolved from concrete (and theoretical!) problems handled by the Babylonians to the solution of polynomial equations, and then will get at least an overview of how algebra become abstract and pervaded all areas of mathematics.

Some high points in the book are: an engaging account the romantic story of the solution of cubic and quartic equations by Tartaglia, Cardano, and Ferrari; the use of complex numbers by Bombelli; the early attempts at a theory of equations via invariants and symmetric functions; the role of Lagrange (or should we say Vandermonde?) resolvents for solving the quintic. The book even contains a convincing "proof" of the Fundamental Theorem of Algebra.

On other topics, Derbyshire has not been as successful. Galois theory, despite the romantic aura around the short life of Galois (which Derbyshire argues is not totally warranted), is not a light topic, despite (or perhaps because of) its great beauty. It is hard to get the point across to nonmathematicians. The same can be said, even more strongly, of Kummer's ideals and Noether's ring theory. In particular, I think Derbyshire has failed to give a good account of Emmy Noether's work on invariants.

The book goes as near as possible to contemporary mathematics, discussing Klein's Erlangen Program, algebraic topology, algebraic geometry, algebraic number theory, category theory, and even Grothendieck's work on the modern foundations of algebraic geometry (Derbyshire concentrates mostly on Grothendieck's "colorful" life rather than on his mathematics, which is just as well, given how hard the latter is, even for mathematicians).

The prose itself is quite pleasant, as we have come to expect from Derbyshire. The book contains over 170 informative and sometimes entertaining endnotes, a good index, and 32 pictures of the main characters in the history of algebra.

In summary, I think Derbyshire has done at good job at portraying algebra and its journey toward abstraction from its roots in early civilizations. All interested readers will learn something about mathematics and its history. Readers with the right background will then be able to enjoy more mathematical accounts such as The Beginnings and Evolution of Algebra by Bashmakova and Smirnova and van der Waerden's classic A History of Algebra.


-----------------------------------------------------------------------------Publication Data: Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire. Joseph Henry Press, 2006. Hardcover, 376 pages, $27.95. ISBN 030909657X.
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Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.


Posted to MAA Reviews May 11, 2006.
Posted to Read This! June 7, 2006.

ALGEBRA THEN AND NOW, May 16, 2006
Reviewer: Mead C. Whorton Jr. (Bastrop, LA United States)

John Derbyshire's Prime Obsession, the story of the Riemann Hypothesis, was a mathematical tour de force but Mr. Derbyshire has done it again. He has written an extraordinary book which traces the history of algebra from its beginnings in the Fertile Crescent nearly four thousand years ago to such modern day abstractions as Category Theory. To assist the reader who has never encountered higher undergradate mathematics or who has forgotten the content of courses taken long ago, Mr. Derbyshire has provided well written, concise MATH PRIMERS on such diverse topics as Cubic and Quartic Equations, Roots of Unity, Vector Spaces and Algebras, Field Theory, and Algebraic Geometry. These Primers are scattered through the text and serve as guide-posts for the reader as she/he treks through the historical development of Algebra. If you have ever wondered how Algebra began and what groups, rings, fields, vector spaces, and algebras are then purchase this book. The author has also done a wonderful job of bringing alive the many men and women who, through the centuries, created modern day abstract algebra. This is not a light read but the prose and logic are superb. The reader who is willing to invest the time to complete this book will emerge all the richer for completing a thrilling intellectual adventure of the highest order.


Abstraction brought down to Earth, June 1, 2006
Reviewer: Mike Birman (Brooklyn, New York USA)

Those of us who read and enjoyed Prime Obsession (even the title has a delicious tabloid flavor, reminiscent of Basic Instinct or Fatal Attraction) may have been most amazed at the very idea of popularizing something as arcane and difficult as the Riemann Hypothesis. What made that book work so well was Derbyshire's brilliant alternation between historical narrative and description with chapters that served as a mathematical primer on number theory and other background material. The mathematically challenged reader could peruse these more technical chapters or leave them be by choice: there was still much knowledge to be gained in either case. For the more mathematically sophisticated, a complete reading of the book served as a reasonably deep (if popularized) analysis of the famous Riemann Hypothesis. Short of tackling H. M. Edward's Riemann's Zeta Function, the classic discussion and much more difficult, Derbyshire provided the most cogent introduction to the RH.

Unknown Quantity is similarly constructed, with historical and biographical material alternating with chapters Derbyshire once again describes as mathematical primers. Although trained as a molecular biologist, I have a fairly strong background in mathematics. I still found much to learn. Especially interesting is the material on Vector Spaces and Algebras, the introduction to Hamiltonian Quaternions, Rings and Fields (with the vista of Abstract Algebra just over the hill) and a short introduction to Algebraic Geometry. I found even more to enjoy. The historical and biographical threads make the unfolding mathematics that much clearer and easier to visualize, hence more enjoyable. Derbyshire has produced another superb book that makes mathematics live and breath. To breath life into abstraction is a great gift. I reread Prime Obsession and will do the same for this newest work. If you find mathematics at all amenable to your taste, I urge you to sample this book. I look forward to being pleasantly surprised by the topic of his next work.


A Chatty History of Algebra, July 28, 2006
Reviewer: Walter Fekula (New York, NY USA)

"Unknown Quantity" is a history of algebra presented in Derbyshire's chatty, sometimes quirky style. It is compulsively readable. I kept stopping to remind myself I was reading a HISTORY OF ALGEBRA--and enjoying it! The math is kept at a decently low level. If you can get the idea of a polynomial (Derbyshire explains it up front), that will get you through most of the book. The explanation of groups is painstaking and pretty good, and I am fairly sure I got the main idea about rings and fields, too--though I'm afraid he lost me with "p-adic numbers." There is a strongly geometric angle in the later part of the book. This gives you some attractive diagrams--look at 13-2, the "ampersand curve," or 13-3, which beautifully illustrates the meaning of "variety"--but I found myself wondering whether the author was falling back on geometry because the algebra was just getting too hard to present to his chosen audience ("the curious nonmathematician"). When Derbyshire gets to the 20th century he pretty much gives up on trying to explain what the algebraists are doing, and concentrates on the personalities and the historical background. From the little he does tell us about 20th-century algebra, it's hard to see that he had much choice. I still have no idea what Alexander Grothendieck did in his day job, but he sure sounds like a fascinating character. Best chapter: "The Leap into the Fourth Dimension." I now understand what multi-dimensional spaces are all about and why people started thinking about them. Best chapter title: "Lady of the Rings" (that's Emmy Noether, another fascinating character). Best photograph: Hypatia--is this the first algebra book to include a picture of a naked woman? Best footnote: No. 142. (As with "Prime Obsession," the footnotes are wonderfully readable.) Still, "Unknown Quantity" shows even more clearly than "Prime Obsession" did that Derbyshire is at heart a novelist. I wish he'd give us another novel. Anyone who can make the history of algebra interesting is a born story-teller!