Is zero a number? Today, many mathematicians and math teachers will scoff at this question. Of course zero is a number—they might argue—first, zero behaves like any other number; second, it follows the rules of arithmetic; and, third, it is the origin of coordinate axes and number lines. But, zero doesn’t act like any other number: it is utterly unique in that x + 0 = x and x · 0 = 0, for all x. Further, the rules of arithmetic include special rules for zero (namely, those mentioned in the previous sentence); that is, we’ve created the rules of arithmetic so that zero follows them. Equally as circular, “zero is on the number line so it is a number” follows only from the fact that we call it a “number line” and we happen to put zero on it.
More than this: how can we have a symbol—“0”—which represents nothing? Okay, this isn’t so bad; after all, we have lots of terms, like “love,” “duty,” “hate,” “energy,” that represent nothing on which we can put our fingers (or even our microscopes), but that operate and are very useful in the world. And perhaps most importantly, you might say, why worry about all this? It’s more than a little like contemplating one’s own navel or navels, as the case may be.
If your sensibilities resonate with this last statement, then you probably don’t want to bother with Robert Kaplan’s The Nothing That Is: A Natural History of Zero. Take two parts math, two parts history, four parts philosophy, and one part literature and focus them through what Kaplan calls the “lens” of zero and you might end up with something like this book. It wouldn’t hurt to also be an erudite Harvard prof steeped in arcana and unafraid to make references that his readers don’t understand—no, unconcerned is probably more accurate. Or maybe he sees his book as motivating the reader to further study, something like Joyce’s vision of Finnegan’s Wake.
Kaplan claims, in “a note to the reader” before the book formally begins, that “if you have had high-school algebra and geometry nothing in what lies ahead should trouble you, even if it looks a bit unfamiliar at first,” but I challenge any high school senior to read and understand every word, every reference. I will admit to following only about 75% of the references, on a good day. Even so, Kaplan weaves an engaging web of ideas and is nothing if not inspiring:
If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things. From counting to calculating, from estimating odds to knowing exactly when the tides in our affairs will crest, the shining tools of mathematics let us follow the tacking course everything takes through everything else—all of their parts swing on the smallest of pivots, zero.
With these mental devices we make visible the hidden laws controlling the objects around us in their cycles and swerves. Even the mind itself is mirrored in mathematics, its endless reflections now confusing, now clarifying insight.
Zero’s path through time and thought has been as full of intrigue, disguise and mistaken identity as were the careers of the travelers who first brought it to the West.
And so he launches on his own journey.
For me there are several striking moments along the way. Kaplan’s tour of the many symbols that have been used for zero takes us through cultures in places like Babylonia, China, Egypt, Greece, India, and the Mayan empire. The way in which these symbols may or may not have turned into our contemporary is fascinating, if at times a little longer than I wished it were. The history of the way humans thought about zero and its religious implications (from the Hindu “nothingness” to the Christian “devil”) is more important to me (and to Kaplan) and leads to one of the main points he is trying to make:
While having a symbol for zero matters, having the notion matters more . . . . [W]hat is hanging in the balance . . . is the character this notion will take: will it be the idea of the absence of any number—or the idea of a number for such absence? Is it to be the mark of the empty, or the empty mark? The first keeps it estranged from numbers, merely part of the landscape through which they move; the second puts it on par with them.
Kaplan goes on to argue convincingly that zero, at least in Western culture, moves more and more toward being a number. In what he calls the “Great Paradigm,” numbers evolve beyond metonyms into objects worth studying in themselves:
If you say there are seven apples in a bowl, exactly what does that ‘seven’ belong to? Not to any one of the apples taken singly (not even the last you counted, since you could have arranged them differently), nor to the bowl that contains them, but—to there being just seven of them. Many a fine head has broken on this problem. Some have ended up saying that seven is the set of all those sets that contain seven objects. And if you eat one of the apples, where has the seven gone? Fled, presumably, to those sets that still or newly have seven members.
In such contemplation and elsewhere, Kaplan more or less follows the logical development of the foundations of mathematics as it is generally practiced today. However, when he goes on to argue that we move “from coordinating the meaning of facts to subordinating facts to their significance,” I think—well, maybe at Harvard and similar locales, but I’m not so sure elsewhere.
(It is worth noting that Kaplan’s flair for metaphor is a little more extended than is generally practiced in mathematical circles today. Zero is “estranged” from other numbers, “merely part of the landscape through which they move”; seven has “fled” to join its like; mathematics is an “organic sprawl”; the mind is “mirrored” by mathematics, creating “endless reflections.” The book is ripe with such as these, which will make it sweet for some, saccharin for others, and bewildering to more than a few.)
Oh, and there is some math: some algebra and some calculus; the Cartesian plane and binary arithmetic. In these moments, Kaplan is on firm ground, although I found myself wondering how well I would have understood it had I not already known the math. Nevertheless, these sections are interesting and certainly part of the picture he wants to paint. But, for Kaplan, the book’s 219 pages are not enough. He has so much more to say that he gives web address for the 78-page notes and bibliography. And even that is not enough; those pages conclude:
There is much more to say about nothing, both in the outer language of experience and the inner language of mathematics; and numberless pages to write about zero, from sets of measure zero, zero-sum games and reconstructing a function from its zeroes, to zero-dimensional objects and nilpotent elements—but I must take to heart Michael Stifel’s advice, clausis oculis abeam, and leave with eyes closed.
Throughout the book, Kaplan manages to both say what he believes and at the same time problematize those beliefs and leave the reader to decide. He strolls among philosophy, mysticism, mathematics, and history, with liberal sprinklings of literary reference (e.g., the book’s title is from a Wallace Stevens poem). In the end, it is a profoundly personal book, a book in which I feel he wanted to express some of his love for a subject on which he has spent many years thinking and to give a sense of some of the connections between ideas he has noticed during that time. In other words, it is a book I wish I could write, but never will for want of both ability and will.