Embedding Zero in Exposition

September 17, 2000

Book Review
Philip J. Davis

The Nothing That Is: A Natural History of Zero. By Robert Kaplan, Oxford University Press, Oxford, 2000, xii + 225 pages, $22.00.

The basic paradox of nothing---whether nothing is something---derives from meanings at multiple levels. Now "nothing" is not the same as zero, and zero is not the same as "not," nor is it the same as an approach to zero in the limit. When these several distinct notions are conflated, as easily occurs, the result can be a rich variety of ideas, constructions, opinions, ranging over the centuries from Abraham Robinson's nonstandard analysis back to the doctrine of Nargajuna, the master of Mahayana Buddhism, that the world is both empty and full (page 218). When this material is assembled under one cover, what we have is an anthology of tangentialisms, unified in that it constitutes a romanticization of zero.

"Nothing will come of nothing," said King Lear, as quoted by Kaplan. This aphorism (which predates Shakespeare by many centuries) has been cited with both approval and disapproval by metaphysicians, theologians, philosophers, mystics, poets, artists, and with approval by practically everybody in the world in moments of frustration.

How does the mathematical zero get into theology? Well, for example, the identity 0/0 = any number at all---school children can check this by multiplying up---has been claimed to bolster the case for creatio ex nihilo. Disregarding theology, out of nothing have come at least the two books on nothing reviewed in this issue of SIAM News, and I know of a third: algebraist-turned-semiotician Brian Rotman's Signifying Nothing: the Semiotics of Zero (1987).

The Nothing That Is, written by Robert Kaplan, who most recently has been teaching mathematics at Harvard and running their Math Circle, presents the history of the numerals and of their representations. Through the contemplation of 1/0, it fast forwards to Archimedes's "Sand Reckoner" and to the huge numbers of Ramsey theory, together with Knuth's notation for super-exponentials. Reference also is made to the curses some men must bear, which preacher and poet John Donne vivified by pulling in huge numbers.

Kaplan's book later becomes a mini-text on algebra and on calculus. (I have often told classes in elementary calculus that the job of the differential calculus is to give meaning to 0/0, that of the integral calculus to give meaning to 0*(1/0).) Cryptography, quantum theory, Gödel's Incompleteness Theorem, zero-knowledge proofs, the Apocalypse (page 104), snatches of general world history, are a few of the many topics that make cameo appearances---to use the terminology of movieland.

Personally, I would have tucked in my favorite among the recent (mid-1950s) zero developments: generalized matrix inverses. The Moore-Penrose inverse, now residing in all scientific computer packages worthy of the name, yields 0/0 = 0; 0*(1/0) = 0. The former identity, by the way, was anticipated by Brahmagupta (c. 630) and has been booed by later commentators, Kaplan among them.

In this recent context, division by zero is not taboo, nor does it produce zero's "twin": infinity. And these identities and their matrix cousins have been found to be convenient and useful. Apparently, how we operate with zero mathematically depends, in part, on what we find convenient. The zero of analysis is not the same as the zero of projective geometry, or of logic or of matrix theory. And how one embeds zero in the larger culture, to instruct, to entertain, and to inspire a general audience whose mathematical experience is limited to high school algebra and geometry, depends entirely on an author's imagination.

Standard English-language histories of mathematics, such as those written by mathematicians Morris Klein, Carl Boyer/Uta Merzbach, and Ivor Grattan-Guinness, devote little space to zero. They note its presence or absence in ancient times; they present a few notations, a few bits of zero algebra, and little else. These authors make no mention of zero as "the additive identity of the integers." There is no tooting of the horn for zero as one of the great inventions of all time.

To parlay into a whole book what appears slight in professional summaries, requires, as described, the infusion of a fair amount of tangential material. This will appeal to a vast readership, those who prefer sweet wine to dry. It will appeal also to cross-disciplinarians and diversificationists (to use the latest university buzzword) who keep banging on specialists to loosen up and present a larger picture.

I have to confess that reading through Kaplan's The Nothing That Is, I went through some of the same initial and subsequent feelings that Steven Krantz reports in regard to Charles Seife's book: from initial annoyance and a high-horse stance to a subsequent relenting. And I could have repeated, word for word, some of what Krantz wrote about Seife. But another side of me says "loosen up," discard the exclusivity of the definition-theorem-proof mode that has held mathematical exposition in an iron grip for more than a century.

Some of the ancillary matter in the two books under review surely resides in the minds of some mathematicians of the past---in Napier, who did Revelation and logarithms, in Newton, who did the Book of Daniel and gravity, in Gödel, of whom biographer Hoa Wang implied that it would be very difficult to separate Gödel's scientific impetus and accomplishments from his religious concerns. Who is to say, at the end of the day, that all this material is irrelevant? Who is to say that the tangentialisms collected here were of no influence in the historic development of zero in its multiple personas?

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island.


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