Wednesday, May 24, 2006

Amazon book review: Unknown Quantity

A fine biographical history of algebra

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

Publisher: Joseph Henry Press (May 15, 2006)
Language: English
ISBN: 030909657X

I had high hopes for UQ. My hopes were not dashed, but I wasn't as uplifted and exalted as I was with Derbyshire's excellent "Prime Obsession". If one comes upon these two books for the first time, one should definitely read UQ first.

As in "Prime Obsession", Derbyshire writes very appealingly about the history of the times and about the mathematicians themselves. The biggest issue is that the book is too small for such a huge subject. It's only 320 pages long with 32 pages of notes.

Derbyshire's portraits of algebraists in his book are uniformly delicious. His bio of Alexander Grothendieck reminded me of the life of former world chess champion, Bobby Fischer. Grothendieck was as unworldly, uninformed, naively opinionated, anti-American, and brilliant as Fischer. We find him now holed up in a remote village in the Pyrenees, where "he is known to come up with ideas like living on dandelion soup and nothing else."

Or Solomon Lefschetz, the algebraic geometer, who lost both his hands in an industrial accident. He was "energetic, sarcastic, and opinionated", and something of a character. His most famous quote: "It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry."

I think that Derbyshire had to edit severely. His introduction to "Unknown Quantity" says that it was "written for the curious nonmathematician." Perhaps he should have said, "written for the college math major who decided not to pursue a career in mathematics." I studied math in college but I didn't get a degree. I mention this because I was disappointed by the blandness with which he writes of the "simple substitution" one can make if one only notices the "simple algebraic fact" that turns a general cubic equation into a depressed cubic equation. It's something I never encountered in high school or college. Granted, that was 35 years ago now. I tried for quite some time over a period of three days to derive the "simple algebraic fact" for myself before moving on with the rest of the Cubic and Quartic Equations chapter, but I couldn't. And this was only page 58!

Derbyshire's math primer interludes are designated with initials. So, instead of referring to a section in "Cubic and Quartic Equations" as section 4.7 as he does for all the non-primer chapters, he uses abbreviations: section CQ.7 for "section 7 of the Cubic and Quartic Equations chapter". A bit annoying, actually. If one wishes to brush up on a concept by re-reading, one has to refer first to the table of contents to find it.

However, there are plenty of interesting things to learn for the "curious nonmathematician". For example, the complex cube roots of the number 1. I found this fascinating. I didn't do much with complex numbers in school. Derbyshire whetted my appetite for them in "Prime Obsession" and UQ sated me!

I loved matrices and determinants in high school. They made complete sense to me. The section on the discovery and application of matrices was a comfortable interlude of re-discovery. So was the discussion of Boolean algebra. A few of the chapters weren't. I felt myself to be way outside the book's target audience even though I'm a "curious nonmathematician".

UQ is not as sprightly as "Prime Obsession". The jokes are there, just more widely spaced and drier. The best chuckle I had was in one of the notes in which the author wrote of a video "demonstrating one of the 20th century's most fascinating discoveries in topology: how to turn a sphere inside out." He recalls that he "used to bring it out and play it to dinner guests as a conversation piece, but this was not an unqualified social success."

Derbyshire continues his practice (begun in "Prime Obsession") of collecting all of the footnotes into a set of endnotes at the back of the book. As I said, the best laugh of the book is in one of those notes, but if you don't care to keep flipping back there to find the right note, you might miss it. At least the notes are numbered consecutively; the numbering doesn't start over again with each chapter. But I sure enjoy sidebars and footnotes as part of the text.

Part of what made "Prime Obsession" so vivid were the illustrations. Algebra uses symbols that don't lend themselves to illustration very readily, except where geometric figures can be plotted from algebraic equations. But even in his chapter on geometry Derbyshire's use of illustrations is stingy.

UQ did prove useful right away, though. In skimming through references to help my 15-year-old son find material for a paper on Archimedes, I found a mention of Archimedes' Cattle Problem in the 11th edition of the Encyclopaedia Britannica. The invaluable Mathworld web site contained a more detailed article. I worked on the problem for some time while reading UQ, but I didn't pursue it to a solution. I now realize, though, that both Newton's method and the use of determinants would have solved it...both of which are covered in detail in UQ!

"Unknown Quantity" is a fine historical and biographical treatment of algebra, with engaging writing and plentiful -- though brief -- explorations of a multitude of algebraic topics. There's plenty of meat here for the "curious nonmathematician".

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