## The Conquering Zero

The Nothing That Is

by Robert Kaplan

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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.