The Nothing That Is: A Natural History of Zero

Natural History, Dec, 1999 by Keith Devlin

The Nothing That Is: A Natural History of Zero, by Robert Kaplan. Oxford University Press; $22; 225 pp.

As cognitive psychologist Karen Wynn demonstrated some years ago, all humans are born with an innate ability to count and to add and subtract small numbers. To go beyond that, we use numerical language--symbols--a major advance that was introduced at least 5,000 years ago. By manipulating symbols, we can handle numbers of any magnitude with total accuracy.

A few special and very powerful numbers have been identified. The best known and most frequently encountered of these mathematical "constants" is pi [Pi], the ratio of the circumference of a circle to its diameter, approximately equal to 3.14159. Two other crucial constants are e (the base of natural logarithms, approximately 2.71828) and i (the square root of-1). And deep in the heart of elementary arithmetic is a special number that. leaves unchanged whatever it is added to or subtracted from: zero.

A quarter century ago, Petr Beckmann wrote a book called A History of [Pi]. In 1994 Eli Maor followed up with e: The Story of a Number. Last year came the publication of An Imaginary Tale: The Story of the Square Root of Minus One, by Paul J. Nahin. Now, not one but two books turn the spotlight on zero: Robert Kaplan's The Nothing That Is: A Natural History of Zero (published in October) and Charles Seife's Zero: The Biography of a Number (coming in February).

Both books start out in much the same way, by tracing the long and tortuous (but fascinating) path that led to the recognition, now taken for granted, that zero is a legitimate number, meaning a character that designates an arithmetical value. The historical, intellectual, and cultural strands of the story are woven into the development of the very concept of numbers--and, in particular, the decimal system that predominates today.

The idea of representing all numbers by using just the ten symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 was developed in India, picked up by the Arabs during their sixth-century conquests, and subsequently introduced to the West. The Hindu-Arabic number system (generally known as Arabic numerals) quite simply changed the world forever. Without an efficient system for representing and computing numbers, Galileo would have been unable to begin the mathematical study of nature that we now call science.

In Hindu-Arabic notation, the position occupied by a digit in a number expression is critical. This is where zero comes in. The Indians introduced the symbol 0 to act as a placeholder when there is no entry in a particular position. In the number 304, for example, the 3 denotes three hundreds, the 0 denotes no tens, and the 4 denotes four units. Because of position, "nothing" suddenly becomes informationally important.

But is this symbol 0 purely a position marker, or does it denote a number, as do all the other digits, 1 through 9? The ancient Greeks had a positional number system using a marker to denote an empty place, but the Indians were the first to regard the zero as a genuine number. Accepting zero as a number required mathematicians to change their conception of a number.

Does zero merit, or can it sustain, a whole book---or two books coming out at the same time? I don't think so. Both authors have to work hard to make the story of zero fill their allotted pages. Since they write well, the padding is agreeable enough (if slow at times). But why do we suddenly get two books about, well ... nothing?

I see three factors here. The first is that, since Dava Sobel's runaway success with Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time, followed by David Ewing Duncan's Calendar: Humanity's Epic Struggle to Determine a True and Accurate Year, publishers have sniffed a new genre: popularizations of the history of science. The two new books about zero have their sights clearly set on the same target.

In addition, the imminent Y2K problem provides us with an unavoidable lesson on the importance of zero in modern life. Because society as a whole has failed to truly understand zero, we shall celebrate the start of the new millennium exactly one year before it actually arrives!

Finally, if books on pi, e, and i have illuminated these mathematical constants, why not do one on 0 as well? I'll tell you why not. Whereas pi, e, and i provide their authors with plenty to write about, the same cannot be said of 0. Once Kaplan and Seife have traced the historical developments leading to the Hindu-Arabic system, both authors turn to developments in mathematics and science that arose from attempts to avoid the thorny problem of division by zero. Kaplan uses zero as an excuse to discuss calculus, and Seife heads off into physics. Because mathematicians had wrapped up calculus by the end of the nineteenth century, whereas physicists have yet to nail down the ramifications of division by zero, Seife's book provides the more compelling read. The trouble is, in both calculus and modern physics, zero is no more than a bit player. As a result, I finished both books feeling that they make much ado about nothing.