»  The New Criterion

October 1999

  The Conquering Zero

The Nothing That Is
        by Robert Kaplan


Suppose I ask you to step into the next room, count the people in there, and report the answer back to me. What is the smallest number you can report? Obviously the answer is zero, corresponding to the case when there are no people at all in the next room. Thus it is plain that zero is the smallest of the counting numbers. It is also the smallest of the unsigned measuring numbers: the least possible distance between two points is zero, when the points coincide.

These simple facts, so easily stated, did not become generally understood until modern times — well into the 16th century in Europe and even later elsewhere. There is nothing like the history of mathematics to leave you thinking that people in the past were less intelligent than ourselves. That is, of course, an illusion. The correct conclusion to be drawn from Robert Kaplan's book, which is accurately subtitled A Natural History of Zero, is that until words and symbols have evolved to denote a thing, that thing cannot be discussed or used; and words and symbols for abstract entities evolve terribly, terribly slowly.

Mathematical entities — numbers, lines, functions — are of course the ultimate abstractions. The millennia-long struggle to get to grips with zero testifies to the deeply unnatural nature of mathematical thinking. Every mathematical truth is like that: extruded after unimaginable intellectual effort and lifetimes of frustration, against all the grain of ordinary human thought and language processes. In the preface to Principia Mathematica Bertrand Russell noted that an everyday object like a whale is immensely more complex, on any scale of complexity that makes sense, than a number like "five"; yet the whale is much easier for the mind to name, encompass and manipulate than the number. Any group of humans that was in regular contact with whales would certainly have a name for them; yet there are plausibly said to be primitive peoples that have no word for any number larger than three, though five-ness is right there, literally, at their fingertips.

The Nothing That Is ably documents humanity's long and weary groping toward the domestication of zero, with many curious and illuminating asides drawn from the author's wide field of interest. He knows Sanskrit, for example, and relates with gusto the astonishing array of names in that language for very large numbers — tallakchana, should you ever encounter it, means a hundred thousand trillion trillion trillion trillion. Yet the Indians of the early first millennium who deployed these terms still had no proper word for zero, nor any real concept of it except as a "place-holder," a sort of punctuation mark. So we stumbled along right through the Middle Ages, the inability to comprehend zero's true status as a number contributing to countless errors and confusions — some of which, like the tedious quibbling about the true starting date of the new millennium, are still with us.

It was the Arabs, picking up where India left off, who developed a regular symbol for zero. They passed it, with the other nine digits, to Europe, where these Arabic numerals were regarded with deep suspicion until the late Middle Ages. For one thing, they lent themselves to forgery: zero especially was all too easily transformed into 6 or 9. Arabic numerals were also slower and more cumbersome to compute with than the abacus, a skilled user of which can still beat anyone calculating with paper and pencil. Bookkeepers did their accounts on the abacus and published them as Roman numerals — which are useless for arithmetic — or with the figures written out in words. (One finds oneself wondering, not for the first time, how on earth medieval society managed to stay afloat.) Not until the invention of double-entry bookkeeping in the early 14th century did the use of Arabic numerals for computation begin to gain favor over the counting-board, supplying necessary power for the lift-off of science, capitalism and modern statecraft.

In the latter part of his book Mr Kaplan gives over much space to pondering the thing that zero denotes, which is of course nothing. Here his ruminations must inevitably face comparison with one of the great classic essays in popular mathematics, Martin Gardner's "The Significance of 'Nothing'," which can be found in his book The Night Is Large. Mr Kaplan does not emerge well from the comparison. He has nowhere near Gardner's literary sensibility or fluency of style. The reader who wants to know something about nothing would be much more profitably engaged with Gardner. (Synchronicity alert: Amongst other topics, Gardner mentions the all-black canvases of 1960s minimalist painter Ad Reinhardt and names various critics who praised them, including one Hilton Kramer.)

Oxford University Press are launching The Nothing That Is with an initial print run of 40,000 — this at a time when the average literary novel is lucky to get 5,000. It is a grim paradox that publishers should be turning out popular books on mathematics with such enthusiasm just as the actual profession of mathematics is heading for a major crisis. Simon Singh's Fermat's Enigma was the darling of reading circles last year and the passing of Paul Erdős occasioned two biographies and a TV program. Yet an article titled "The Sky Is Falling" in the February 1998 Notices of the American Mathematical Society reports that enrollment in advanced math courses at four-year colleges and universities is dropping like a pebble (Latin: calculus ) down a well, implying major job losses in college math teaching. The reason for this decline is not hard to fathom. Get an average degree in pure math and your career opportunities are pretty much restricted to commencing study for the Society of Actuaries very stiff exams, after passing which you will start work at around $35,000. Get an equivalent degree in computer science and you can walk right into a job paying $42,000 even, apparently, in Kansas City. Since college math departments and computer science departments draw from essentially the same pool of applicants, the result (as mathematicians say) follows.