Zero: The Biography of a Dangerous Idea by Charles Seife, Viking,
$24.95, ISBN 067088457X
The Nothing That Is: A Natural History of Zero by Robert Kaplan, Oxford
University Press, $22, ISBN 0195128427
WHEN zero was introduced to Britain, it was the subject of a postgraduate
course at the University of Oxford. Now, the notion of a notation for nothing is
so entrenched in the culture that everyone鈥攅xcept for a few tiresome
pedants鈥攁ssumes that a new century must start with an 鈥�00鈥�.
Advertisement
But, curiously, when you begin to examine how nothing went from specialist
topic to commonplace knowledge in such a short time鈥攐nly 200
years鈥攜ou may be surprised to find that the logical development of the
idea of zero does not match its history. How did this come about?
A good starting point would be the two books featuring zero as hero: Charles
Seife鈥檚 Zero and Robert Kaplan鈥檚 The Nothing That Is. Both
insist that we travel right back in time to find the roots of nothing.
In the ancient world, we can discern two main classes of numbering. The
Romans, for example, used a different symbol for each order of magnitude. We are
familiar with it, if only from obfuscated movie dates: I for a unit, X for a
group of ten, C for a group of a hundred, M for a thousand. (A subtle ploy to
conceal the age of a film from viewers by showing the date in Roman numerals is
likely to fail with MM as this year鈥檚 date.)
The ancient Egyptian system was more versatile, with symbols going up to one
million, while the Greek and Hebrew systems were based on the alphabet. Today,
we might write: A = 1, B = 2 up to J = 10; then K = 20, L = 30 up to S = 100;
then T = 200 up to Z = 800.
In all systems of this class, the drawback is clear: sooner or later you run
out of symbols. Those who needed more extended the system in various ways. One
variant of the Roman system used ((I)) for ten thousand, (((I))) for a hundred
thousand and so on. A variant of the Greek used dashes to indicate values
multiplied by a thousand.
The second class of systems used a small fixed set of symbols in ordered
strings, in which the magnitude denoted by a single symbol is determined by its
position in the string. Our decimal system is, of course, one of these, but it
was not the first: a system based on sixty rather than ten was in use in Sumeria
around 3000 BC.
Such systems characteristically need a symbol to denote an empty space, to
fix positions and distinguish 11 from 101 or 1001. Our system emerged in India
and arrived in Europe via the Arab world. Doing calculations with this
new-fangled system came to be called 鈥渃iphering鈥�, after its most notorious
symbol鈥� cipher, zifr or zero.
Once zero is accepted as a symbol, we have to face the question of whether it
counts as a number. The statement that there are two cats in my study sounds
rather similar to the claim that there are no dogs there. But is it quite the
same to say that the number of dogs is zero? What about the number of
unicorns?
Accepting zero as a counting number was not much easier than accepting
negative numbers. These became familiar to the adventurous pioneers of banking
and venture capitalism. Practical and logical necessity combined, and a number
system evolved capable of answering questions in forms such as these: 鈥淲hat do
you get if you subtract five from five?鈥� Zero. 鈥淔rom three?鈥� Overdrawn by two.
No logical problems here, although possibly some embarrassment in practice.
The next great leap involving zero came in the development of the calculus.
How, for example, do we define the speed of a moving object at a particular
instant? In terms of the infinitesimal distance travelled in an infinitesimal
period of time, of course.
But before the concept of the limit emerged in full rigour, mathematicians
were subject to the withering scorn of philosophers. Bishop Berkeley derided the
鈥済hosts of departed quantities鈥�. He argued that to claim that working with
infinitesimals gave the right answer could always be countered with the
accusation that calculus could be trusted only when its answers were verifiable
by other means.
Not quite zero, but not quite anything bigger, infinitesimals disappeared,
apparently for good, in the 19th century. It came as a surprise when, in 1961,
Abraham Robinson proposed an alternative set of axioms on which to base
calculus. These brought back Newton鈥檚 infinitesimal as a rigorously logical
entity, rather than as a mere numerical placeholder.
So what does zero count? What does it measure? Is the answer 鈥渘o thing鈥� or
鈥淣othing鈥�? If zero can be a number, surely Nothing must be a thing.
The ancient maxim that nature abhors a vacuum is an attempt to deduce the way
things must be from the way they are. We now know that vacuums do exist, but the
ancients were not entirely wrong. Paul Dirac suggested that the vacuum is full
of spontaneously appearing particle-antiparticle pairs resonating in
confined empty spaces, and in 1948 Hendrik Casimir was able to measure the force
exerted by Nothing. So even Nothing counts for something.
We are, it appears, still far from understanding Nothing. But as Seife and
Kaplan show, we are now confused at a higher level. Valuable lessons also emerge
from looking at how our predecessors approached challenging problems. First, we
might take heed how easily the way we do things today might have been different:
the role that chance plays in the development of a field. Secondly, even when we
do know more about zero than our predecessors, it does not make us smarter.
So how do these two books differ? What are their unique selling points for a
reader in search of Nothing? Seife鈥檚 title gives a flavour of his approach. He
takes us on a breathless tour of the 鈥渄angerous idea鈥� of zero. He associates it
with the even more perilous notion of infinity, although the association seems
rather forced and the exposition becomes quite hectic as he approaches the 21st
century via transfinite set theory, relativity and quantum physics. His analysis
is marred by a failure to engage with the reasoning of the historical people he
is describing. To him, they are merely 鈥減rimitive鈥�, his favourite term of
abuse.
Kaplan takes a more leisurely pace; he strolls through mathematical history
with a more philosophical point of view. Although he has talked to
professionals, he wisely avoids technical details and takes pains to sympathise
with the ancients.
So is Nothing all just sound and fury? There is a lesson to learn, and
re-learn in every generation. That we can safely ignore what went before is
truly the 鈥渄angerous idea鈥�.