A Natural History of Zero

By Robert Kaplan

Oxford Univ. 225 pp. $22

Reviewed by Rudy Rucker

The Nothing That Is treats the evolution of the mathematical notion of zero. The history of this particular idea doesn't quite fill out even a short volume, so there's additional material about the scientific and philosophical notions of nothingness. The book is loaded with quotes and illustrated with pleasant line drawings by the author's wife.

The original role of zero seems to have been as a placeholder, which is still one of its main functions. It's thanks to the zeroes that the digit 3 has a different meaning in 3, 30, and 3000. There was a centuries-long resistance to thinking of zero as a proper number. As a French writer in the 15th century put it, "Just as the rag doll wanted to be an eagle

. . . the zero put on airs and pretended to be a digit." Kaplan speaks of "the uncomfortable gap between numbers, which stood for things, and zero which didn't." In an interesting turn of thought, Kaplan notes that the introduction of a variable like x for the unknown quantity in a mathematics problem is akin to the use of zero as something that affects the digits around it rather than standing for a simple absence.

There's a popular tradition that the Indians invented zero, and the book has a good discussion of whether they may have gotten the notion from the Greeks or the Babylonians. Kaplan eloquently describes the difficulty of figuring this out: "Who next can we make out in the haze, and what is that haze itself? Its particles are words, colliding with each other and diffracting the light of ideas." In any case, Hindu philosophy seems particularly congenial to the notion of a quantity to represent nothing. "For the Hindus there is no unqualified nothingness . . . substance for them cannot disappear but can only change its form or nature: this fullness -- brahman -- pervaded the universe."

Kaplan makes the novel point that the invention of double-entry bookkeeping in the Renaissance had profound philosophical consequences. Bookkeeping legitimizes zero, but, even more, it paves the way for the conservation laws of physics. It's always fascinating to think of a new tool or medium as completely changing the way that people think. What unexpected philosophical effects, for instance, will the advent of Web pages lead to?

One might expect The Nothing That Is to be an easy read, but I found it tough going. One of my problems with the book is its clubby, discursive tone. Kaplan frequently goes off on cryptic tangents. It's a bit like having a professor who makes too many private jokes. A sample passage begins: "An increased likelihood, then, that the Greek hollow circle for zero came from the impression of stones removed from a sand-covered counting board. If there were words for the little pleats between the possible and the probable we could choose one of them here -- some junior member of the family that gave us would, should and could. Without them, let us fold up the corner of this conjecture lest we find ourselves putting words in ancient mouths as some anonymous scholar did in the eleventh century to Boethius."

A second problem is that Kaplan doesn't always explain the mathematics behind the ideas he discusses. For me, the most intriguing figure in the book was a spindle of wrapped curves relating to a graph gotten by raising a variable x to the power x. But there's no explanation of exactly how the graph was generated. Dutifully I went to the 78 pages of Notes and Bibliography which are online at the Oxford University Press web site but not printed in the book. And here I found this gem: "For details see Mey passim (on whose article the two spindle diagrams are based)." I guess I'm crabby because, after all my effort, I still don't know how the spindle graph was created.

But enough carping. There are many wonderful things in the book, and if it sometimes takes a little time to figure things out -- well, what's the rush? The Nothing That Is includes an excellent discussion of the Mayan calendar: Apparently the Mayans were afraid that time would eventually come to a stop. Lacking our notion of a linear infinity of years, they invented a hierarchy of calendars so that it would at least take an exceedingly long time before the exact same date would roll around. The millennium held a bit of that old Mayan terror for us; it was easy to think that the coming of 1/1/00 meant the end of time. But, of course, we're not rusty old mainframe machines, and we knew that the date would really be 1/1/2000. Thanks to zero, our position-based numbering system never runs out of fresh dates.

Rudy Rucker is an author and computer-science professor in San Jose, Calif. His most recent book is "Saucer Wisdom," a novel of millennial speculation.