One to Nine: The Inner Life of Numbers

233 Pages·2007·1.88 MB·English

by  Andrew Hodges

Contents
1 The Unloved One
2 To Be or Not to Be
3 Trinidad
4 It’s a Square World
5 Fifth Freedom
6 The Joy of Six
7 The Wonder of it All
8 Sound Bytes
9 End Times Table
Hands-on Experience
1
The Unloved One
It is a truth universally acknowledged that a single man in possession of a
good fortune must be in want of a wife. So runs a famous first sentence, full
of statements about One. It claims a universal truth, but there is an
ironic touch in the grandeur of Jane Austen’s opening, bound as it so
obviously is by human pride and prejudice, and specific to the English
bourgeoisie of the early nineteenth century. Even so, a modern reader
finds it a good story-starter. It is well wired into human brains, tastefully
pressing the buttons labelled ‘sex’ and ‘money’ in unison.
A chapter about the number One itself cannot count on any such
welcome mat. Numbers do not make light conversation, and are
gatecrashers at the great party of the human arts. They seem devoid of
that sympathetic wit and irony which smooths the passage of letters and
life. Indeed, the numbers can hardly join in the human party at all, being
abstractions unable to shake hands or flirt. Where did Page One go to?—
this is already Page Two, and Page One exists only by implication,
possessing a Oneness only in one’s mind’s eye.
Even more annoying to the party-goers is that the numbers adopt
unappetising manners whenever they get the upper hand, dictating
bossy, tedious and incomprehensible regulations. This is a book about
the inner life of these unwelcome guests. It is, more or less, the tale as
told by the mathematicians who get to know them. I hesitate to
generalise because mathematicians are an un-unionised bunch, with the
solidarity of a plate of spaghetti. But the assumption implicit in
mathematics is one of universal truths deeper than any Jane Austen
would have countenanced as an opening gambit in polite conversation.
Numbers, to mathematicians, are the same for Cleopatra or
Moctezuma, for the Neanderthals and the dinosaurs, true in any galaxy,
past or future. They tell stories which could be shared with
extraterrestrial intelligences. When Dante wanted something for the
happy inhabitants of the Paradiso to do with eternal life, he took from
Platonic tradition the contemplation of mathematical truth. Not
everyone would welcome this connection: the British mathematician G.
H. Hardy, who as an atheist wouldn’t be seen dead in the One to Nine of
Dante’s heaven, made a famous declaration of mathematical idealism
which had nothing to do with religion.
Prime time
Hardy expressed this other-worldliness by using the concept of a prime
number. Any number n can be represented as a multiplication of 1 × n.
Put another way, 1 and n are always divisors of n. The prime numbers
are those numbers which have only those two divisors, and so enjoy a
special relationship with One. The numbers 2, 3, 5, 7 and 11, for
instance, are prime. The number 6 is not. The distinction can be seen as
a picture. Six objects can be arranged in a rectangle, but seven must lie
on a line.
Even the United States is powerless to change the régime of the
primes, and as the stars have swelled from 13 to 50, ingenious means
have been necessary to deal with this fundamental problem. In 1940, as
those then neatly 6 × 8 states were about to face their transformation
into a superpower, Hardy published a short book, A Mathematician’s
Apology, with a statement of superhuman Number: ‘317 is a prime, not
because we think so, or because our minds are shaped in one way rather
than another, but because it is so …’
What view was Hardy arguing against? His scorn was reserved for the
profilic British polymath biologist Lancelot Hogben, author in 1936 of a
major popular book, Mathematics for the Million. Holding his nose, Hardy
quoted Hogben’s characterisation of mathematics as ‘the grammar of size
and order’, needed for the planning of ‘the rational society’. Hogben was
expressing the Marxism of the 1930s, which saw all cultural
manifestations as the outcomes of relations of production, and thus
mathematics as based in human practice. Hardy was no conservative
élitist, and in fact had rather radical views. He recognised that
mathematics might be useful, but insisted that its utility did not explain
why mathematicians did it.
Mathematicians have largely maintained a discreet radio silence, and
Hardy was unusual for striking a dissonant note in public. Someone who
knew him, novelist C. P. Snow, wrote that Hardy ‘didn’t give a damn’.
Snow himself became famous in the 1960s for trying to combine Jane
Austen’s legacy with insight into twentieth-century science. It is doubtful
whether Snow had much success with his unifying vision, and his books
are neglected now. But his expression ‘Two Cultures’ has stuck as an
awkward reminder of the fact that those who run the human world and
dominate its discourse are generally unaware of its base in the physical
world, and of the mathematics that is the only language for expressing
that world. Snow used the Second Law of Thermodynamics as his
example of what ought to be well known, but he might as well have
used the First Law, or indeed the Zeroth Law, as his examples: it is as
true now as it was in the 1960s that none of these can be referred to in
polite society.
There is now an extra reason for presenting an unwelcome message
about the inescapable properties of numbers, summarised in Al Gore’s
expression of ‘inconvenient truth’. Although the question of recent,
anthropogenic climate change involves many physical sciences—
including all those laws of thermodynamics—the power of mathematics
to predict is at the heart of the developing theory. The calculation of
climate change certainly falls into Hogben’s realm of the relations of
human production, but the phrase ‘inconvenient truth’ could have come
from the inconvenient Hardy himself. The Earth’s atmosphere lacks the
simplicity of 317: there are even more factors involved in climate
science than in cosmology, where notable corrections have been made,
and there are bound to be revisions to current models. And yet those
models are essentially mathematical. Environmental campaigners who
want to disrupt airports say that ‘the science tells us’ in terms that they
feel put their case above the law, words that recall Hardy’s ‘it is so’.
Whether one takes Hogben’s viewpoint or Hardy’s, the question of the
potential of mathematics is newly vital.
One and I
In the 1990s, I wrote a weekly column on mathematical topics for a
British newspaper, the Observer. The editor, Rebecca Nicolson, contacted
me again in 2005 and suggested a short book with the title One to Ten. I
saw a point of departure. For this had been done before, in a classic
book by Constance Reid, From Zero to Infinity (1956), still in print 50
years later. Her ten chapters were headed by the numbers from Zero to
Nine, each used to spark off an elegant exposition of some substantial
feature of the theory of numbers. Anything I wrote was bound to
resemble her plan.
I remembered her book well because I learnt so much from it, perhaps
most of all because through Constance Reid I had absorbed Hardy’s view
of other-worldly mathematics, communicated with all his charm but
without his awkward-squad agenda. Her succinct and lucid pages now
convey a 1950s America, fortress of world culture, but not much given
to airing controversies. It is compelling in its commitment to the
fascination of numbers for their own sake. Questions of motivation or
usefulness hardly arise in her account. These, however, poured from
another book of my childhood: this was Man Must Measure (1955) by
none other than Lancelot Hogben. In the 1950s he blazed a new trail,
using extensive graphics and photographs to pour out an encyclopedic
knowledge. In his enthusiastic account, mathematical advance was as
anthropogenic as carbon dioxide, the fruit of human industry,
interwoven with every kind of human skill and argument. So between
them, these books exposed me to two different points of view, echoing
the 1930s political conflict. Fifty years later, I find myself having to re-
echo them.
Given the publisher’s suggestion of One to Ten, and the classic example
of Reid’s sequence from zero to nine, my decision was for One to Nine.
The clinching factor was, of course, the worldwide popularity of the
Sudoku square. In case you’ve been away from the solar system for a
while, here is an example of the basic 3 × 3 Sudoku puzzle. The grid
must be completed so that every row, column and subsquare contains
just the numbers One to Nine.
A TOUGH problem, with no obvious first step. Hint: there is only one
place where the 3 can go in the central square.
The numbers entered this book saying that they made no light
conversation. But if numbers did make light conversation, Sudoku shows
how they would talk. What’s more, it shows they can tell a joke. It is
this: although The Times of London and other newspapers insist that the
puzzle ‘needs no maths’, the process of solving it is an elegant miniature
version of the experience of real, adult, mathematics. What The Times
means is that it needs no school maths, reflecting the legacy of fear and
anxiety generated by schools, which leaves most of their victims with a
lifetime of mumbling apologetically about ‘my worst subject’.
Sudoku problems are Hardy-ish in having no use whatsoever, apart
from helping to keep Alzheimer’s at bay. They do nothing for the once-
planned society of the future, nor for the market economy which has
replaced it. In fact they must have subtracted millions of person-hours
from the duties of profit-making. They have none of the cosy linguistic
clubbiness of cryptic crosswords. The problems need up to an hour of
concentrated thought. Yet the demand for them appears insatiable.
In contrast, school mathematics teaching seems to be in a particular
state of crisis. The Guardian, the leading British newspaper for the
education business, describes it as ‘needing a makeover, to make it sexy
again’. Mathematicians, its writer explains, ‘are bald, overweight with
beards and glasses and eternally single, leading little or no social life’.
Jane Austen’s efforts would clearly have been doomed had she opened
with such a character, and curiously, the writer for this progressive
newspaper—long a leading voice for feminism—tacitly and
unrealistically assumes mathematicians to be male. Another Guardian
Education feature helpfully describes all science as ‘boring, prohibitively
hard, too abstract and too male, in a spoddy, won’t-get-a-girlfriend kind
of way’. These artless articles reveal an unlovely truth: that mass,
compulsory, school maths has enjoyed about the same success as the
War on Drugs.
In stark contrast to these effusions, Sudoku gains cool customers
without any need for sexing up, and the Times champion is, as it
happens, a young woman. The reason, perhaps, is that Sudoku
encapsulates some of the most fascinating elements of adult
mathematics: elegant geometry and pure logic. You can try to solve a
puzzle by guessing—but pencilling in trial solutions and rubbing them
out again is likely to be a losing battle. Faith, hope, fantasy and bluster
are prominent in the planning of world domination, but are powerless to
solve the Sudoku square—or mathematical problems.
Just say Nought
Although we are following Sudoku with One to Nine, we cannot ignore
the Zero with which Constance Reid chose to begin. There is a difficulty
about her choice. She says that Zero, ‘first of the numbers, was the last
to be discovered.’ This makes One her second number, and Two her
third. This seems very odd, but it makes sense if you distinguish cardinal
and ordinal numbers. Word roots show how human culture has thought
of ‘one’ in more than one way. The root one, un, uno, ein … answers the
question of ‘how many’. It is the cardinal number which counts. But the
root that shows in prime, prince, Fürst, first, evoking the social
relationships of primitive primates, with their prima donnas and prime
ministers, is the ordinal number. It is the difference between ‘one page’
and ‘Page One’. So the apparent contradiction can be explained;
Constance Reid took the cardinal Zero to correspond to the ordinal ‘first’.
She could have avoided this by describing Zero as the noughth number.
The words ‘noughth’ or ‘zeroth’ do have a shadowy existence in
language, used for something that has to be placed before something
that has already been called ‘first’. After the First Law of