It's difficult to classify Robert Kaplan's The Nothing That Is: A Natural History of Zero. You could describe this book simply as a rather nice history of the number zero. You could also describe it as nothing less than a history of the human race's philosophical struggle with the idea of nothingness.
It's in the first role that the book will be most interesting to mathematicians. In about 240 pages, Kaplan writes what is probably the only book-length history of a natural number ever written, and he does a good job of it. He discusses in depth much of the material on the history of the symbol for zero that is usually mentioned only briefly in most general mathematical histories, from the development of the Hindu-Arabic number system with its later addition of the symbol for zero in the ninth century A.D. all the way up to von Neumann's work building the natural numbers out of the empty set (the modern set-theoretic definition of zero). But this is only the tip of the iceberg. He begins his book with a discussion of the early Babylonian number system and--as is his style in much of the book--gives an intuitive, common-sense justification for how a positional number system must have evolved from it and how such a system entailed the need for a symbol to denote "nothing in this column"--the role we assign to zero. He gives the Indians credit for being the first to realize the algebraic properties of zero, but he credits the actual invention of the symbol for zero to the Greeks, "who discovered the crucial role that zero played in counting, when they invaded what was left of the Babylonian Empire in 331 B.C., and carried zero off with them, along with women and gold." In fact, the "0" symbol first appeared in third century B.C. Greek astronomical papyri.
Just how did these Greek astronomers arrive at the symbol we use today? Speculation abounds, and Kaplan walks his readers through many of the possibilities. The most obvious answer is that "o" (omicron) was the first letter of the Greek word for nothing (ouden). But this idea was dismissed long ago by Otto Neugebauer, who noted that the Greeks had already used omicron to denote the number 70. Kaplan notes in addition that the Greeks weren't really comfortable with using zero as a number per se: for the Greeks "0 indicates the absence of a kind of measure (degrees, or minutes, or seconds), but can't yet be taken with other numbers to form a number." The answer, Kaplan speculates, is not in the philosophical and geometrical writings of the Greek "leisure class", whose works have been handed down to us. Instead, it rests in the merchant class, with "a device that the philosophers never described, but whose descendents you see to this day in the worry-beads of the Greeks and the backgammon games of their taverna: the counting board." After all, Socrates drew his geometrical figures in the sand. Remove a counting board pebble (that's "calculus" in Latin) from the sand and you're left with a depression in the shape of "0".
Kaplan does his best work when he is relating these episodes in the story of zero as it travels through time, and he is not always on a path well-traveled by previous math histories. After a chapter on the Babylonians and two chapters on the Greeks, he spends four chapters discussing Hindu and Arabic contributions and how zero traveled from Europe to Asia and back again. He also includes a "Mayan Interlude" where he discusses the Mayan calendar and the role it played in their society. The Mayans, it turns out, took zero very seriously: For them, Zero was the leading god of the underworld, and to keep him appeased the Mayans would offer a human sacrifice, dressing their victim in the regalia of the God of Zero and ritually tearing off his lower jaw.
As far as the history it covers, this book is not a deep work aimed at scholars--or even amateurs--in the field. (The closest thing to a scholarly reference in the book is a pointer to a web site that contains extensive notes and references.) In tracing the history of zero, Kaplan has no trouble indulging in speculation. He relates that he has "tried to bridge a chasm on the slenderest threads of evidence" and he prefers to give an intuitively plausible story or an etymological argument to span the gap. In fact, the book is intended primarily for mathematical novices. The goal of the book is to relate in a breezy, conversational tone an interesting thread through the history of mathematics, while making many attempts to explain to mathematical novices something about the nature of mathematics. As the author notes in his introduction, if "you have had high-school algebra and geometry, nothing in what lies ahead should trouble you", and he means it. At one point, he poses to his readers the question of what to make of 0/0. Once you've written down a list of properties that any respectable number system must have (which he does), division by zero makes no sense (which he proves). But Kaplan has only just begun: the discussion of 0/0 leads to many pages explaining difference quotients, derivatives, and the ensuing problems of dividing by zero--all the while following a historical thread. As expected, he ends up at L'Hopital's rule, but along the way he also stops to discuss how to use calculus to find maxima and minima (which, after all, occur where the derivative is zero) and the meaning of 00.
Following the story of zero is a clever angle to lure novices into the basic ideas of calculus, and there's plenty more of this strategy in other parts of the book: discussing the meaning of raising a number to the zero power leads to modular arithmetic, Fermat's Little Theorem, and some of the central ideas of modern cryptography; focusing on just the numbers zero and one leads to a discussion of binary numbers. The list goes on, and these lengthy mathematical asides are another great strength of the book.
But the other side to this book--the human race's philosophical struggle with the idea of nothingness--I didn't find nearly as satisfying. After he's said just about everything he can say regarding zero as a mathematical object, Kaplan moves on to what he calls the "physical and psychological embodiments" of zero. This leads to many wandering passages where the connection to mathematical zero is metaphorical at best. Consider:
... How easy still to dismiss those we reject as mere zeros, sinks of energy, black holes in which all that matters, the singular and the memory of the singular, disappear without a trace.
It is as if that hollow oval stood for anonymity, mirroring our fear of making no difference to others--to anyone--to the world: "...to pass beyond and leave no lasting trace", wrote William McFee in Casuals of the Sea (which left little enough trace itself; for me, only the colors that shimmered in the beveled glass of the school's specimen case).
How many zeros it turns out you've met and even kidded around with: those faces from old yearbooks you now can't put a name to, those names in old address-books that summon no features. They made up the crowd, into which sometimes you yourself loved to plunge as a mere spectator--until the chilling thought came of your own name as unrecognizable in tattered address-books lying in basement drawers.
To live as a zero: the superfluous man, the man without qualities, the person who, like Henry James' John Marcher, finds out only too late that the beast in his jungle was that there was no beast, no response to passion: this figure haunts our fiction and our fact, the salaryman of Japanese society, the company man, the fungible folk of our office culture, who retreat to virtual reality games at home. The worst of it is that, unlike Marcher, most never awake to the truth that they haven't lived.
What's troubling about discussing rhetorical uses of zero in a book about a mathematical object is that a mathematical novice won't know the difference between a mathematical idea and a clever analogy. Modular arithmetic, differential calculus, binary number systems, and existentialist philosophy--I know that one of these things is not like the others, but will a mathematical novice be able to tell the difference? After doing a wonderful job of entertaining and informing his readers with stories of some of the most elementary yet profound mathematical observations ever made, Kaplan is treading on shaky ground when he treats metaphysical speculation and the rhetoric of analogy like a mathematical art.
It doesn't help matters that at the same time Kaplan chooses to pile on the literary references. Kaplan sprinkles brief allusions throughout much of his writing--usually with much success. But the tempo picks up as he nears the end of the book. In one chapter less than six pages in length he throws in displayed quotes from John Bunyan, Sylvia Plath, Dostoyevsky, Ford Maddox Ford, and "a book on the latest cyberspace cryptography". Here's how he puts Sylvia Plath to use:
These circles of negative nothingness may hang for years about us--yet as Sylvia Plath wrote:
How did I know that someday--at college, in Europe, somewhere, anywhere--the bell jar, with its stifling distortions, wouldn't descend again?
Could there be any zero more negative than this?
Someone unfamiliar with Sylvia Plath will get nothing out of this. In the same chapter he also mentions Sartre (and his "neat reversal of the Thomistic formula that essence precedes existence"), John Donne (who "addressed man as that Nothing, infinitely less than a mathematical point..."), Keats ("How could your name possibly be writ in water, as Keats, dying, said of his?"), the Roman Emperor Hadrian ("Animula vagular blandula"), and Hermann Weyl (who "suggested that when the ego is extinguished, the unmarked grid of coordinate space remains"). Throw in some references to Hamlet and Theseus, a story about the Sultan Abdul Hamid the Second (a highlight actually), the motto over the entrance to the Harvard philosophy building, and you have almost the entire chapter. None of these references are really explained, and at times it felt like they were designed more to impress than to enlighten. I sometimes wondered if there should be an additional warning in the introduction stating that the literary prerequisites for this book are higher than the mathematical prerequisites.
Many readers will enjoy keeping pace with Kaplan's erudite mind, but some (including the people to whom Kaplan feels obligated to explain in detail, say, the fundamentals of differential calculus) might find all these allusions distracting, if not confusing. For most of the book the excellent mathematical story-telling makes them endurable--and even enjoyable!--but in the last fifty pages of the book--where philosophy has largely replaced mathematics as the object of study--it will take quite a bit more effort.
Andrew Leahy (firstname.lastname@example.org) is Assistant Professor of Mathematics at Knox College.