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Thursday, January 11, 2007

Reading List: Unknown Quantity

Derbyshire, John. Unknown Quantity. Washington: Joseph Henry Press, 2006. ISBN 0-309-09657-X.
After exploring a renowned mathematical conundrum (the Riemann Hypothesis) in all its profundity in Prime Obsession, in this book the author recounts the history of algebra—an intellectual quest sprawling over most of recorded human history and occupying some of the greatest minds our species has produced. Babylonian cuneiform tablets dating from the time of Hammurabi, about 3800 years ago, demonstrate solving quadratic equations, extracting square roots, and finding Pythagorean triples. (The methods in the Babylonian texts are recognisably algebraic but are expressed as “word problems” instead of algebraic notation.) Diophantus, about 2000 years later, was the first to write equations in a symbolic form, but this was promptly forgotten. In fact, twenty-six centuries after the Babylonians were solving quadratic equations expressed in word problems, al-Khwārizmī (the word “algebra” is derived from the title of his book,
الكتاب المختصر في حساب الجبر والمقابلة
al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala,
and “algorithm” from his name) was solving quadratic equations in word problems. It wasn't until around 1600 that anything resembling the literal symbolism of modern algebra came into use, and it took an intellect of the calibre of René Descartes to perfect it. Finally, equipped with an expressive notation, rules for symbolic manipulation, and the slowly dawning realisation that this, not numbers or geometric figures, is ultimately what mathematics is about, mathematicians embarked on a spiral of abstraction, discovery, and generalisation which has never ceased to accelerate in the centuries since. As more and more mathematics was discovered (or, if you're an anti-Platonist, invented), deep and unexpected connections were found among topics once considered unrelated, and this is a large part of the story told here, as algebra has “infiltrated” geometry, topology, number theory, and a host of other mathematical fields while, in the form of algebraic geometry and group theory, providing the foundation upon which the most fundamental theories of modern physics are built.

With all of these connections, there's a strong temptation for an author to wander off into fields not generally considered part of algebra (for example, analysis or set theory); Derbyshire is admirable in his ability to stay on topic, while not shortchanging the reader where important cross-overs occur. In a book of this kind, especially one covering such a long span of history and a topic so broad, it is difficult to strike the right balance between explaining the mathematics and sketching the lives of the people who did it, and between a historical narrative and one which follows the evolution of specific ideas over time. In the opinion of this reader, Derbyshire's judgement on these matters is impeccable. As implausible as it may seem to some that a book about algebra could aspire to such a distinction, I found this one of the more compelling page-turners I've read in recent months.

Six “math primers” interspersed in the text provide the fundamentals the reader needs to understand the chapters which follow. While excellent refreshers, readers who have never encountered these concepts before may find the primers difficult to comprehend (but then, they probably won't be reading a history of algebra in the first place). Thirty pages of end notes not only cite sources but expand, sometimes at substantial length, upon the main text; readers should not deprive themselves this valuable lagniappe.

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