# Much ado over nothing

DOES a new millenium have something to do with the attention that zero gets? Suddenly, we have two good books on zero. First, there is Charles Seife's "Zero, the Biography of a Dangerous Idea", brought out by Viking. Second, we have Robert Kaplan's "The Nothing that is, a Natural History of Zero", with illustrations by Robert Kaplan's wife, Ellen Kaplan. This is not a comparative review, it is a review of Kaplan's book.

"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 the 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 and all of their parts swing on the smallest of pivots, zero."

Robert Kaplan's book is like a work of historical detection, tracing the origins of zero. It is not only about zero, it is also about notions of "nothing". A note to the reader states, "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." This is not quite true. Even if the essence can be grasped, an understanding of everything in the book requires some familiarity with number theory. The note to the reader also says, "You will find the bibliography and notes to the text on the web" and the web-address is given. I am not sure that this is a good idea. In addition, the notes in the web-site are not particularly illuminating.

But let's stick to the book. What is zero? First, it is a number that signifies nothing. We begin with the Sumerians, 5000 years ago. The Sumerians didn't have a word for nothing and neither did the Greeks. Nor does zero figure in Alexandria, in Sicily or in the work of Archimedes. Several Indians believe that the notion of zero originated in India. This is what the author has to say on this claim. "It does strike me, however, that burdening actual Indian achievements with others' goods ends up diminishing them, and that it is a loss to replace a story rich in the accidents and ambiguities of time with an uplifting tale." That is, the symbol for zero is squarely ascribed to Greek origins.

However, inventing a symbol for zero (or nothing) doesn't grant zero the status of a number. For that to happen, operations of addition, subtraction, multiplication and division between zero and other numbers have to be defined. Robert Kaplan ascribes this credit to Indian mathematicians Bhaskara, Brahmagupta and Mahavira. Collectively, they established rules for adding, subtracting and multiplying zero and thus, elevated a mere symbol into a number like any other. What about division by zero? The Indian mathematicians got 0/0 and a/0 wrong, where "a" is a positive or negative number, other than zero. When did mathematicians get these right?

Robert Kaplan's book is anecdotal and full of digressions. Although the book's language is easy enough and it is difficult to see how a mathematician could have done a better job, these sudden jumps and shifts sometimes make it difficult to retain the link. Without answering the question about 0/0 and a/0, we therefore jump to the Mayas and their calendar system. From there, we jump to various names for zero. Fibonacci and the invention of double entry book-keeping in Italy are thrown in. All very interesting and the link is clear to the author. "This opened the door to granting equal status to the signs for quantities and for the operations on them, all subject now to yet more abstract operations in turn, and those to others, endlessly, each bearing the peculiar, defining mark of this language: that no matter where in its hierarchy an operation or relation stood, it was expressed by a sign of like standing with the rest, in the matrix of their common grammar." Clear?

We still have to dispose of 0/0 and a/0 and it gets worse. There is now a digression on exponents and Fermat's little theorem. This is not Fermat's last theorem, which has recently been proved. Fermat's little theorem states the following. If p is a prime and "a" is a number less than p, ap-1 1 is exactly divisible by 'p'. Bung in a digression on solutions of quadratic equations and end up with the axioms of number (field) theory. This establishes notions of an additive inverse and a multiplicative inverse and the point is simple. An additive inverse need not be the same as a multiplicative inverse. While 0 is an additive inverse, the multiplicative inverse is 1. Every number except the additive identity (0) has a multiplicative inverse. That disposes of the 0/0 or a/0 problem.

Perhaps because there isn't enough material to fill a book, Robert Kaplan now moves from the nothing to the almost nothing. This is the province of differential calculus and Liebniz and Newton. 0/0 returns, because in the context of slopes, l'Hopital's rule established that 0/0 has some meaning. There is an anecdote on this. "The Marquis was immortalized by his discovery of this general principle, which now goes under the name of l'Hopital's Rule. The only problem with our story which I told you was one of disguises is that Monsieur le Marquis had neither this insight nor its proof. Both were the work of his teacher, Johann Bernoulli, who was apparently willing to take the Marquis's cash and let the credit go."

After some philosophical digressions, we return to a slightly different problem. Under some operations, the numbers 0 and 1 can yield every positive rational number. But are both these numbers needed, or can one do with only one of them? This is a bad way of stating the problem. What the author is really after is set theory, the notion of the null set and the axioms of set theory. "This trick was pulled off in 1923 by John von Neumann, and it is one we have since gotten used to, although you may feel that for its simplicity it remains slippery. Identify zero, von Neumann says, with the empty set: it doesn't even have zero in it, it is 0. And now (nothing up our sleeves) consider the set that contains the empty set. Since this set has one element in it namely, the empty set we can identify it with the number one. There! We've gotten all the natural numbers from the empty set, and from these naturals we'll make the negatives, the fractions, the reals and the imaginaries in the well-known ways." Russell's paradox is missing. But with Wittgenstein's "Tractaus Logico-Philosophicus" thrown in, it is not surprising that these parts of the book are even more difficult to grasp than the others.

Here is the last paragraph from the book. "I write this in the midst of things, in the middle of time. The world extends away on every side, taking its coordinates from a quiet center fitfully seized as self, which, like Wallace Steven's snowman, listens and beholds Nothing that is not there and the nothing that is."

That is precisely the problem with this book. Who is this book for, what is the target audience ? The non-mathematical reader, who has been exposed only to high-school algebra and geometry? The titles states, "The Nothing that Is", whereas the sub-title proclaims, "A Natural History of Zero". In trying to do justice to both the title and the sub-title, the book falls between two stools. Of the sixteen chapters, six (by my reckoning) have to do with the sub-title, the natural history of zero. The remaining 10 are about the title, the nothing that is, and the non-mathematical reader with no more than high-school algebra and geometry, is completely lost. However, this is not meant to be entirely negative. Everyone will find a lot of interest in the first six chapters. And the mathematical reader, with more than a passing interest in algebra and number theory, from the remaining 10 as well.