The Biography of a Dangerous Idea

By Charles Seife

Viking. 224 pp. $24.95 (forthcoming)

Reviewed by Curt Suplee

When nothing finally arrived, it changed everything. That is the theme of Charles Seife's lively and lucid history of zero and its many cognitive embodiments as symbol, tool and concept. It's a story that richly deserves telling in this three-cipher year. And Seife, a Washington correspondent for New Scientist, makes it a fascinating and memorable tale.

Despite our current familiarity with El Zippo (especially in post-holiday bank balances), zero is a ridiculously new idea in human history. Indeed, until about 500 years ago, Western civilization waddled along quite nicely without it. Sure, the absence of nought gave us our silly calendar system that goes from 1 B.C. to 1 A.D., thus guaranteeing that the new millennium actually begins next year. And it delayed the advent of advanced mathematics.

But nobody needed nothing to do the arithmetic of daily life. A farmer who had to tally zero pounds of grain would be bankrupt long before he discovered a deficiency in his number system. Moreover, even if we'd had it, we might not have been ready. Zero has the power to derange even the most rational calculations. Divide by it, and it produces its equally egregious cousin, infinity. Stick it into an equation, and you get freaky logical effects. (To illustrate, Seife includes a watertight proof that Winston Churchill was a carrot.)

We escaped early exposure, however, because the grand arbiters of Western philosophy, Seife argues, found the void (that is, the ontological aspect of zero) a hideous, nearly unthinkable prospect. The Hellenic mind abhorred a vacuum, so nothingness had no place in the Pythagorean-Aristotelian-Ptolemaic cosmos that shaped European thought until the Copernican revolution.

If zero the concept was anathema, however, zero the number became too useful to ignore. The number system of 7th-century India, which used a base-10 system with a rudimentary zero as a place holder, was dramatically more efficient than the clunky Roman numerals that encumbered monastic bean-counters. So when the Arabs imported the new symbol-set from India, and eventually made the system known in the West in the 12th century, "Arabic numerals" took over in a hurry.

Seife does a splendid job of unfolding this history within the larger context of alternative numbering systems developed by various cultures, such as the base-60 system of the Babylonians (who brought us the 60-minute hour) and the magnificent zero invented by the Maya, who were, alas, unknown in Europe.

Occasionally, the argument branches off, putting the author out on plenty of speculative limbs. Seife relates number systems to metaphysics, suggesting, for example, that India was more prone to embrace the void -- and hence zero -- because its cosmogony involved creation ex nihilo. And, of course, because in many Hindu theologies the entire sensual world is, at bottom, an illusion. Conversely, "the whole Greek universe rested upon this pillar: There is no void"; therefore, "before they could accept zero, philosophers in the west would have to destroy their universe."

Yup, that's hyperbole. And there's plenty of such rhetorical organ music throughout the narrative. But it's pedagogically useful in a book for the lay reader, and it makes for some amusing asides. For example, Seife takes Christianity to task for ignoring its own Judaic creation story -- in which things are "without form and void" before the Big Guy starts making stuff -- in favor of an everlasting Aristotelian plenum.

Having gotten zero to Europe within the first 85 pages, Seife spends the remainder of the book explaining what it did when it showed up. Sometimes this is pretty metaphorical. For example, he credits the development of perspective in painting to the idea of zero. Brunelleschi, he says, "first demonstrated the power of an infinite zero: he created a realistic painting by using a vanishing point." Charming, but purely figurative.

Most of the exposition, however, is a good deal more hard-headed, and contains some of the best, most comprehensible and engaging descriptions of math that anybody is likely to read anywhere. Seife tracks zero through the coordinate system of Descartes, whose axes literally gave zero a local habitation, to Pascal and Kepler, who found surprisingly different uses for it.

The author shines in explaining the role of zero in the invention of calculus by Newton and Leibnitz. Yes, there's calculus in here and a few equations, too. But Seife's carefully paced explanations and chummy style make the description the equivalent of an intellectual detective story that can be read comfortably by folks who had their last math course in high school, and cheated even then.

Both Newton and Leibnitz wanted to be able to describe the way natural objects behave -- which is almost never linear. Toss a ball in the air, and its velocity is constantly changing. So how can you calculate it at a given instant? By shaving the time interval as thinly as possible, which also cuts the distance to nearly nothing. Since speed is distance divided by time, and since the tiniest increment of either is zero, you end up with, um, 0/0. Obviously, this is nonsense. But it worked wonderfully well: "Calculus was so powerful that no mathematician could reject it." And, as Seife explains, ultimately they didn't have to; in the 18th century, Jean d'Alembert kindly produced the notion of "limits," which keeps these values from going to zero.

In four neatly brief final chapters, readers see the role played by zero (and cousin infinity) in modern number theory, geometry and set theory, as well as zero's function in such exotics of physics as the Big Bang, black holes, "zero point" energy of the vacuum (the latest infinite-energy fad among New Age fruitcakes), the 11 dimensions of string theory and, finally, the heat death of the universe -- "zero's final victory."

From the first page to the last, Seife maintains a level of clarity and infectious enthusiasm that is rare in science writing, and practically unknown among those who dare to explain mathematics. Zero is really something.

Curt Suplee writes on science for The Post's national news staff.