In case you haven't heard, this has been a bumper year for the number zero. All the world waited in anticipation to see the effect when computer calendars rolled over from "99" to "00", and legions of computer programmers spent their last night of the millennium (though we mathematicians know that really won't happen until next year) at the office or on call, waiting to be rushed in at the slightest sign that something had gone awry. Last year zero was even the subject of a "natural history" by Robert Kaplan (reviewed here). Now, five months later, Charles Seife has presented us with *Zero: The Biography of a Dangerous Idea*.

Of course, Seife's book is not a typical biography. There are no tell-all interviews with the number one or any of zero's other neighbors on the number line. In fact, the idea behind Seife's book is nearly identical to the idea behind Kaplan's book, and there is much overlap between the two books--right down to the stark white covers that adorn both of them. Both are books about mathematics aimed at amateurs, and their common goal is to tell the historical tale of mathematics through the eyes of the number zero.

Seife's book begins--of course--at Chapter Zero, with a story of how only recently a divide by zero error in its control software brought the guided missile cruiser USS Yorktown grinding to a halt. As Seife relates, "Though it was armored against weapons, nobody had thought to defend the *Yorktown* from zero. It was a grave mistake." Maybe it's not the pulse-pounding drama of a Tom Clancy novel, but it's enough foreshadowing to launch Seife on an essay which begins with notches on a 30,000-year-old wolf bone and ends with the role of zero in black holes and the big bang.

Chapters 1-5 cover much of the same material that Kaplan covers in his first eleven chapters--and that's to be expected. Historically, zero began as a necessity of place-value number systems. But before readers can grasp the value of a place-value number system like the Babylonians invented (and the necessity of place-holders in such a system), they must first see the other number systems that were employed and how the Babylonian system was superior. This is the subject of Chapter One. The next question is how the Greeks, in spite of being the master of almost every other branch of mathematics, missed the importance of zero. This leads Seife to a discussion of Pythagorean number philosophy, Zeno's paradoxes, and Aristotelian cosmology. This and a lengthy discussion of the history of the calendar we use today--and how it is fraught with error due to the problems bought on by the historical absence of zero--are the subject of Chapter two.

Chapters Three, Four, and Five cover the emergence of zero from the east and all of the problems and advances it entailed. Chapter Three begins with a discussion of the Indian adoption of the base-ten place-value number system and ends with Fibonacci's use of the Hindu-Arabic system in *Liber Abacci*. Chapter Four plays on the duality between zero and infinity--something Seife will do for the remainder of the book. It begins with Brunelleschi's introduction of perspective in 1425, in which the point at infinity is characterized as the "zero in the center of the painting [which] contains an infinity of space". He moves on to discuss Descartes' introduction of zero into geometry with his coordinate system, and the chapter ends with a discussion of Pascal's wager, where the algebraic properties of zero and infinity are used to compute the expected value of religious belief and atheism.

Chapter Five contains Seife's discussion of the origins of calculus--again with careful attention to the role of zero. It begins with the convergence of series. He explains that Suiseth was able to sum the series of terms *1/2, 2/4, 3/8, ... , n/2*^{n} (but he doesn't indicate how this was done). Next, he tackles Oresme's proof that the harmonic series diverges. The essential ideas behind integral calculus are then presented in a discussion of Kepler's *Volume-Measurement of Barrels*, but the most complicated example we're shown is an the approximation of the area of a triangle by eight rectangles. Newton's method for finding tangents is demonstrated by working through the example of finding the tangent to *y = x*^{2} + x + 1, and the controversies between Newton and Leibniz, L'Hospital and Bernoulli, and Berkeley and the entire English mathematical community are all presented briefly. The chapter finishes with a brief introduction to limits, nicely presented by showing how to solve Zeno's Achilles paradox (presented in a previous chapter) with a geometric series. (This is particularly refreshing to see. It's amazing how many philosophy students think that nobody has ever been able to answer Zeno. My only regret is that Seife didn't take the time to show his readers how easy it is to sum a general geometric series.) This entire discussion of calculus occupies only 25 pages.

But the time periods covered in Chapters 3-5 were also times of tremendous philosophical and cultural change in the West. While Hindu philosophy had embraced the void, Aristotle--and consequently the church--had rejected it because "Nature abhors a vacuum". Seife does an excellent job of relating how the dual ideas of emptiness and infinity were shaping the cultural changes taking place in the Renaissance. In fact, the bulk of Chapter Four, entitled *The Infinite God of Nothingness*, centers around the struggles between the church and Renaissance scientists over the nature of the universe. Copernicus' heliocentric model of the solar system had banished Aristotle's (finite) universe and the centrality of the church. As Seife relates:

Nicholas of Cusa and Nicolaus Copernicus cracked open the nutshell universe of Aristotle and Ptolemy. No longer was the earth comfortably ensconced in the center of the universe; there was no shell containing the cosmos. The universe went on into infinity, dotted with innumerable worlds, each inhabited by mysterious creatures. But how could Rome claim to be the seat of the one true Church if its authority could not extend to other solar systems? Were there other popes on other planets?

Descartes' attempt to rebuild a rational belief in God on the understanding of the infinite is also presented well by Seife: "Since we have a concept of an infinite perfect being in our minds, . . . this infinite and perfect being--God--must exist". But Descartes too was tripped up eventually because he couldn't bring himself to accept "infinity's twin", the void. Pascal's experiments with atmospheric pressure, also described by Seife, would lead to that.

The theme of the duality between infinity and zero is continued in Chapter Six. The chapter begins with projective geometry, but the heart of the Chapter is the complex number system and how it leads to the Riemann sphere, where the antipodal nature of zero and infinity are presented as the ultimate intuitive expression for the duality between zero and infinity. The chapter finishes with a discussion of Cantor's cardinal numbers, and includes both a proof of the uncountability of the real numbers and an excellent intuitive justification for why the rational numbers are a set of measure zero.

Chapters Seven and Eight follow the history of modern chemistry and physics by tracing the role of zero and infinity in the formulation of various modern scientific laws. The first stop in Chapter Seven (entitled *Absolute Zeros*) is Charles' law and absolute zero. Then, by way of a description of thermodynamics and statistical mechanics, the discussion shifts to the historical question of what constituted light. From here, Seife describes the Raleigh-James law and the problems that lead to the formulation of quantum mechanics. By the end of the chapter, Seife has managed to fit in Heisenberg's uncertainty principle, general relativity, black holes, wormholes, and their application to interstellar space travel. In Chapter Eight, Seife moves on to describe the more recent topics of string theory, the big bang, and the question of how the universe will ultimately end.

Like Kaplan, Seife doesn't present anything that experts and avid amateurs in the history of mathematics haven't heard before (though his descriptions of modern advances in physics may be enlightening for some). In fact, broad brush strokes such as "[t]he Egyptians, who had invented geometry, thought little about mathematics" are bound to ruffle a few feathers. But newcomers to mathematics will be enlightened and excited by Seife's enlivening account. His prose is clear and uncluttered and his vignettes in the history of mathematics and science are informative, entertaining, and show the novice the cultural, philosophical, and scientific significance of mathematical ideas. In fact, though it's impossible to do much justice to anything when covering all of mathematical history in 215 pages, if you've been looking for a book that will give your teenage protégée an introduction to the history of mathematics (and the western world!) and can be consumed in a weekend, this is probably it. Be warned, though: Seife assumes less mathematical maturity--and far less cultural maturity--than Kaplan does in his story of zero, and even something as elementary as the modern definition of the derivative is relegated to an appendix. (Other important results, such as a proof that Winston Churchill is a carrot and instructions for constructing a time machine out of wormholes, are also given in the appendices.)

Of course, the real question is whether either Seife or Kaplan has written the definitive story of mathematical zero. In the end, Seife follows the story of zero into physics while Kaplan follows it into philosophy. But neither one seriously addresses what mathematicians of the twentieth century have done with zero. The fact is that mathematicians have looked around and found zeros everywhere. For instance, the simple axiomatic requirement which says that, for any *a*, we have "*a + 0 = a*" is one of the most powerful and prevalent ideas in mathematics today. This basic algebraic statement about the nature of zero certainly lacks the grandeur of galactic wormholes and the emotional gravity of a poem by Sylvia Plath, but the fact that mathematicians--true to the example of Euclid thousands of years ago--are still sifting through the ideas of mathematics and trying to discover what is most essential about them certainly deserves more than a passing mention.

Andrew Leahy (

aleahy@knox.edu) is Assistant Professor of Mathematics at Knox College.