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Find X By Michael Westmoreland In 1935, a prominent man eulogized one of his friends: “There is, fortunately, a minority composed of those who recognize early in their lives that the most beautiful and satisfying experiences open to humankind are not derived from the outside, but are bound up with the development of the individual’s own feeling, thinking, and acting. The genuine artists, investigators, and thinkers have always been persons of this kind. However inconspicuously the life of these individuals runs its course, nonetheless the fruits of their endeavors are the most valuable contributions which one generation can make to its successors.” If you have not yet read John Derbyshire’s delightful and challenging history of algebra, Unknown Quantity, you might well think that this passage was written by one artist, about another. In a sense it was: The author was Albert Einstein, and he was writing about Emmy Noether, who ranks among the most accomplished mathematicians. One of her greatest accomplishments was to discover a connection between certain mathematical structures and symmetries in physics. With this new book, Derbyshire establishes himself as one of our foremost expositors of mathematics. His previous effort in this area, Prime Obsession, was a gem; Unknown Quantity is a 100-carat necklace. While Prime Obsession focused on the Riemann Hypothesis, a single problem that has animated mathematicians for almost a century and a half, Unknown Quantity presents the entire two-millennia-long epic of algebra. Popular culture — in such films as Proof, Pi, and A Beautiful Mind — tends to portray creative mathematicians as exceedingly odd, even warped individuals. Derbyshire is closer to the truth in telling how ordinary are the lives of most mathematicians. He does note the occasional oddity: Galois’s fatally reckless romanticism, Grothendieck’s mystic environmentalism. But the drama of this history lies not in the struggles or eccentricities of the people but in the mathematics itself. Derbyshire both understands and respects the lay reader of mathematics. He knows that the typical reader will be unfamiliar with advanced mathematics, but he also respects that reader’s ability to learn something about it. Such an attitude requires courage on the part not only of the author but of the publisher as well. Most publishers will not touch a popular book with an equation or two in it; Joseph Henry Press is to be applauded. This does not mean that Unknown Quantity simply expects the reader to struggle on without assistance. Six mathematical “primers” are dispersed throughout the text. They cover such topics as the arithmetic of imaginary numbers, the basics of vector spaces, and the manipulation of polynomials. None of these primers requires the reader to know anything more than the basics of arithmetic. Any reader who carefully reads and attempts the problems will be rewarded with knowing some actual mathematics and not just knowing some new and recondite names. The book is challenging but highly rewarding. The reader will learn, among other things, how two plus two can equal one; how to take the square root of negative numbers; how to deal with more than four dimensions; and what the connections are between algebra and the other great branch of mathematics, geometry. What comes through in Derbyshire’s account is that mathematics is cumulative in a way other sciences are not. In astrophysics, the theory of planetary motion starts with Newton in the 17th century; studying Ptolemy’s epicycles from 1,500 years earlier would be quite beside the point. But new mathematics builds on the old. For example, to take the square root of -1 (an innovation from the 16th century) it helps to understand how to take square roots of positive numbers (a process understood by the ancient Babylonians). Derbyshire deals with this broad sweep in a masterful way, taking us from the work of Diophantus in the 3rd century a.d. up to the work of Grothendieck in the 20th century. One hopes that Unknown Quantity will catch the attention of at least two, overlapping audiences. The first consists of people who are interested in becoming mathematicians, from the curious high-school student to the recently graduated college math major. The younger group will benefit from a look at mathematics as it currently exists. (Most high-school students have, at best, dealt with mathematics that was state-of-the art in the 17th century.) As for the math majors, all too many wind up studying it because they have been told that they are “good at math” — while lacking a sense of what the overall intellectual landscape of math looks like. Unknown Quantity provides that necessary perspective, showing how algebra connects with other areas of mathematics and science. This is a wonderful thing to have; when you spend hours calculating the p-Sylow subgroups of groups of order ten — a common (if not popular) task for undergraduate math majors — it is easy to lose sight of the broader picture: for example, of the beautiful connections between group theory and quantum mechanics. The other audience for the book consists of anyone who wants to know what mathematicians do. In my case, I have been asked some amusing questions about my work. Do I sit around solving Ph.D.-level quadratic equations? Am I a sort of research CPA, doing arcane bookkeeping? Do I repair computers? While each of these tasks sprang from mathematics, the discipline moved on to explore new territory. A reader who wishes to understand the physical world and the life of the mind should have some understanding of mathematics. He should grasp that mathematics is done as much for its beauty as for its utility. States and corporations will spend treasure on mathematics because it is useful; men and women will spend their lives on it because it is beautiful. To understand that beauty, there is no better starting point than Derbyshire’s Unknown Quantity. Michael Westmoreland is a professor of mathematics and computer science at Denison University. |
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