This is it: the book critic's nightmare. A creature of unquiet dreams, wrought of the most tenebrous dregs of the Morphic philter. A chimera of ... well, you get the idea. This is a book about nothing, filled with scary math problems. If you were to find yourself reviewing a book in your underwear, late for a final exam, with wolves chasing you around a pink marble obelisk -- this is that book. As Charles Seife explains: "Consider the expression x/(sin x) when x = 0; x = 0 as does sin x, so the expression is equal to 0/0. Using L'Hopital's rule, we see that the ... " And then you wake up in your chair with that copy of the new Judith Butler still fluttering in your lap -- we all know the drill.
Actually, "Zero: The Biography of a Dangerous Idea" isn't nearly that dry, and it's not the only book on the topic to have hit the shelves recently. Science and mathematics have long had an odd tendency to generate two independent solutions to a problem at a single historical moment -- in the way that Gottfried Leibniz and Isaac Newton developed calculus independently in the 1660s -- and last October brought us Robert Kaplan's "The Nothing That Is: A Natural History of Zero," a playful, lyrically written text on the evolution, from ancient times on, of the idea of naught and the familiar oblong symbol that we use to denote it. Kaplan's book falls into the same tradition of liberal-artsy books on math as Paul Hoffman's "The Man Who Loved Only Numbers" and Robert Kanigel's "The Man Who Knew Infinity." Seife's book, while it also aims at a popular readership, is a more expository, more math-intensive (and perforce somewhat harsher) one, but equally rewarding, if you're willing to follow it where it wants to take you.
Seife begins in prehistory, before the advent of numerals, and shows that zero is a far stranger, more counterintuitive idea than it might seem today. With number systems that are based on counting and measurement, there's no occasion to conceive of nothing except as an absence of something concrete -- as no pebbles rather than some, nothing for dinner, no room at the inn. Back then, anything less than one simply registered as "aren't any," which makes perfect sense when you imagine the realm of infinite specters that zero the number calls into being. The empty room around you contains zero herds of buffalo, zero dead relatives, zero Scythian horse troops. With zero, they are there; you can count them. They total one less than one.
It was only with the advent of numerical notation and arithmetic that zero as a discrete concept became necessary, first as a simple place holder in the Babylonian number system, and later, with the Greeks, as an astronomical tool. But the Greeks, Seife writes, "didn't like zero at all, and used it as infrequently as possible." The Greeks were, if anything, a people obsessed with proportion, and, as Seife explains, the twin concepts of the infinite and the void (both of which are, like an infinite troop of no Scythian horsemen, contained in zero) played hell with the architectonic principles of both Pythagorean geometry and Aristotelian philosophy. Once you let zero in, some joker somewhere is going to try dividing and multiplying by zero, which produces all sorts of paradoxical results. Does, say, five divided by zero equal nothing, infinity or both? Is there really a difference between nothing and infinity? Once you get accustomed to thinking about that sort of thing, soon enough you'll have to start dealing with irrational numbers, such as pi, which Pythagoras tried assiduously to sweep under the rug. And at that point, everything the ancients thought they knew about mathematics begins to fall into ruin.
It was India that first domesticated zero, through the Hindu familiarity with the concepts of infinity and the void. Neither pagan Rome nor the Christian Europe of the Middle Ages had any truck with it; during this period it was disseminated through much of the East via Islam, and to some extent through the Jewish mystical tradition. Then came the Renaissance -- a time when much, indeed, was ado about nothing.
But from here on in, we're getting into some serious math. Seife explains the use of fluencies in Newton's calculus -- imaginary infinitesimal quantities used to round zero into a positive number -- and then proceeds through a number of heavy-duty equations to show why they "need never be thought of again." But of course, lots of their little infinitesimal friends keep coming back throughout the story, and the key to enjoying the latter half of Seife's book is actually to make an attempt at the math. Zero lurks at the heart of the calculus like a tiny black hole, pulling the equations into shape as though through the invisible force of gravity.
But from there through Georg Reimann's projective geometry, past the cabalistic mathematics of Georg Cantor and on into Einstein and string theory, it gets even more ephemeral and elusive. The math begins to drop out of the text, closing the aperture through which zero can be made visible. If you've been slacking on the equations, Seife blazes on ahead, turning corners faster than you can catch up. But if you've made even fumbling attempts at Newton et al., you'll discover that Seife has a talent for making the most ball-busting of modern theories (string theory again; basic quantum mechanics) seem fairly lucid and common-sensical.
It's all, as the Hindus knew, a play between the void and the absolute. And in that regard, the barrier that keeps many of us from understanding serious science and mathematics (that keeps these disciplines, much like serious art and literature, out of the public realm) might simply be one of focus -- of learning to see the infinite and the void on their own terms, as presences unto themselves that can be tracked and studied but never quite observed or caught.