I am not a mathematician. In fact I am not particularly good at maths or
calculation. But, like everyone else, I do use numbers every waking hour
of every day. Numbers invade my dreams and fantasies, my hopes and
anxieties. Perhaps because we see the world through numerical spectacles
which we never take off, even in our sleep, it is hard for us to realize
how utterly dependent we are on numbers. I am just looking at the front
page of this morning's newspaper. It is fairly typical, I suspect.
The paper costs 45p, it is published on 12 June 1998; Sport is on page
28; the Chancellor of the Exchequer will sell £12 billion of public assets
a £4 billion a year, local authorities expect to raise £2.75 billion,
public sector salaries will rise by 2.25%, Government investment has
fallen by 0.8% of GDP, net investment of infrastructure to 2002 will be
£14 billion, public sector spending will rise by 2.75%, the Government
will raise £l billion a year from selling unused assets, leader comment is
on page 19, financial analysis on page 21; Catherine Cookson dies at 91
years, 13 days before her 92nd birthday, she had 85 best-selling books at
the rate of 2 a year since 1950, they had sold 100 million copies, earning
her £14 million, she was the 17th richest woman in Britain, her husband
Tom is 87, she gave £100,000 to a charity in 1996, her neighbour Gertrude
Roberts is 78, the obituary is on page 20, and a notice of the death of
writer Hammond Innes is on page 4; Stephen Lawrence, aged 18 was murdered
in 1993, his mother is 45, 5 men summonsed to give evidence are aged 21,
22, 21, 20, and 22; the maximum charge for Cable & Wireless telephone
calls on Saturdays is fixed at 50p until the end of September, you can
call 0800 056 8182 to find out more; there's a horrifying racist murder
reported on page 3; the other sections in the paper run from Home on page
5, through pages 15, 17, 18, 20, 23, 24, and 29 to Radio schedules on page
31, and the newspaper's bar code is 9770261307354.
There are 51 separate numbers on just one page. It took me less than five
minutes to read over breakfast. I was keen to get to the sporting pages,
with World Cup results and cricket scores numbers by the bucketful. So in
the half-hour it takes me to get through the paper, I probably see, and at
least half attend to, about 300 visual numbers. Radio 4, 93.5 MHZ, was on
at the same time, and other numbers were going into my ears, occasionally
making it all the way into my consciousness (which doesn't mean that the
others were not processed at all). I had to check the numbers on my watch
- two out of the twelve - to make sure I wasn't late cooking daughter Anna
the approved number of bacon rashers - three. The two digital clocks on
the kitchen appliances advanced by 35 minutes while I was reading and
cooking. Anna's elder sister, Amy, needed £70 for a school trip. As I
walked Anna to school, we passed 73 houses, each with a number, and many
cars, also with numbers. All this before work, which of course involves
lots more numbers.
At a very, very rough guess, I would say that I process about 1,000
numbers an hour, about 16,000 numbers per waking day, nearly 6 million a
year. People whose job entails working with numbers, in supermarkets,
banks, betting shops, schools, dealing rooms, will process many more than
this.
Behind these numbers are vast systems of other numbers. The time now is
set in relation to the 1,440 minutes of a 24-hour day. Today's date is set
in relation to the number of days that have passed since 1 January, AD 1;
people's ages at death depend on this numerical system. The Chancellor's
statistics rest on other statistics of public accounts, economic growth,
and so on, which rest on the daily, weekly, and monthly transactions of an
enormous number of public bodies, private companies and individuals,
including Mrs Cookson, her publisher, its printer, and her accountant and
PR company.
Not all numbers are the same. On the front page of that newspaper there
are whole numbers and decimal fractions. There are numbers to denote how
many things there are in a collection; real visible things such as 5 men,
invisible things like 91 years, or things that are potentially visible, I
guess, like 50 pence, 100,000 pounds, or 14,000,000,000 pounds. There are
also numbers used solely for ordering things in a sequence, for example,
Cookson's 92nd year; the date 12 June falls into this category, as does
the page number 15. And there are also numbers for which both their value
and their position in a sequence are irrelevant: the telephone number and
the bar code. These are no more than numerical labels.
I confess that I didn't focus on our dependence on numbers until, as part
of my work as a neuropsychologist, I started to test people who couldn't
use them. One of the most remarkable was an Italian hotelier, who kept the
hotel's accounts until she suffered a stroke, after which she was blind
and deaf to numbers above 4. This meant she couldn't shop or make phone
calls or do innumerable other things she had previously taken for granted.
Then there was an intelligent young man with a degree and professional
qualifications. He was even good at statistics, provided he could use a
computer, yet he was unable to do even the simplest numerical things in a
normal way. Arithmetic was a disaster area for him, but it went further
than that. Most of us can tell how many things there are at a glance,
without counting, as long there aren't more than about five. This young
man had to count when there were just two things! This wasn't a failure of
education - it was something of a very different order.
In the 1980s reports began to appear in the learned journals of
experiments showing that newborn babies, who certainly hadn't learned to
count, were able to do what this young man could not: recognize the number
of things they saw at a glance. Naturally, I tried a version of one of
these experiments on our first daughter, when she was just four weeks old.
She was sat inside a large cardboard box in which industrial quantities of
nappies were delivered, to watch one, two, three, or four light green
rectangles appear and disappear on a dark green computer screen. Her
response to what she saw was measured by how often she sucked on a rubber
teat attached to a pressure transducer connected to the computer. The more
she sucked, the more interested she was. We were getting very promising
results until the subject decided that she wasn't going to suck the teat
any more. She has remained averse to doing things she can't see the point
of ever since.
It crossed my mind at the time, though without really registering, that
if you are born with the ability to recognize the number of things you
see, then maybe you could also be born with a handicap that prevented this
ability from developing normally. I remember thinking that perhaps there
is a numerical equivalent of colour blindness. Ten years later, I began to
wonder whether this was what was wrong with people like the young man who
needed to count two items. From this it was a relatively short conceptual
step to wondering whether the human genome normally contained instructions
for building circuits in the brain that are specialized for numbers. But
were these circuits built for just this purpose, or had they evolved for
some other purpose and been coopted by the need to cope with numbers?
Colour vision is universal. Everyone, save those with particular
identifiable genetic abnormalities, sees the world in colour. But does
everyone see, or think about, the world in terms of numbers, save those
with some genetic abnormality? If thinking in numbers is something that
has to be taught, then there should be people who had not been taught and
couldn't do it.
We, in our advanced technological, trading society, need to be able to
use numbers, so numeracy has emerged as a key ingredient of our
educational system. But what about "stone-age" societies with little
technology and little trade? Do they use numbers? Do they count? Is number
ability really universal?
Finding out is by no means as easy at might at first seem. Let me give an
example. One way to tell whether a culture uses numbers is to see whether
they have symbolic representations for numbers, either written or spoken.
English has special words for numbers, and a syntax that allows us to name
numbers as large as we like, but most indigenous Australian languages have
words only for "one", "two", and "many". Users of these languages -
particularly the Aborigines of the Central Desert, who are traditional
hunter-gatherers - have little to trade. Their technology, although
exquisitely adapted to their way of life, is limited to a few implements
such as boomerangs, and bark shields and containers. If anyone is likely
not to use numbers, not to think about the world in terms of numbers, it
is them. The problem is how to tell. Nowadays, all Aborigines have
encountered Western money culture, and the English language with its
number words and written numerals. So the question turns into a historical
one: before contact, did they use numbers? If they didn't, this would be
one strike against the universality of a specialized number ability.
An obvious objection to this idea is that, even within a society, some
people are really good with numbers while others approach numbers with
fear and loathing. Surely, if we are all born with much the same brain
circuitry for numbers then our abilities should be much the same, just as
almost all of us are born with nearly identical abilities to see colour,
or to use language (also thought to depend on special genes, as yet
unidentified). But perhaps that's like requiring everyone to show the same
sense of colour in the colour coordination of their clothes, or the way
they decorate and furnish their homes, or maintaining that we should all
be equally good at putting words together to create narratives or poetry.
It may turn out that there are basic capabilities that are indeed innate
and universal, and that the differences in the level of adult performance
will depend on experience and education.
Thinking along these lines, I began to wonder what these basic
capabilities could be. The things that babies can do without instruction
seemed to me a good place to start. How do babies think of the world? Do
they see it in terms of numbers of things, just as they see it in terms of
colour? Another line of attack was to see which numerical ideas seemed
natural and easy to grasp. For example, I noticed in my own children that
what I was taught to call "proper fractions" (1/2, 1/4, 7/8) were easy,
while "improper fractions" (3/2, 5/4, 8/7) were hard. Most people find
probabilities obscure. Can calculus be made easy? Are the ideas that seem
natural and easy those we are born with, or are they just learned earlier,
or taught better?
Certainly there is much anxiety about mathematics education. Children are
distressed when they fail, as are their parents. Governments worry that
their working population is not properly equipped to compete in a highly
technological, and therefore highly numerical, world. Could we improve how
well we understand mathematical ideas if the education system were to base
its teaching more firmly on the mathematical toolkit we are born with?
These are some of the questions that led to this book - They have led me
into all kinds of fascinating subjects completely new to me:
thermoluminescent dating of rock faces and the intricacies of Venetian
house numbering, Aboriginal sign language and New Guinea bodycounting,
Ethiopian farming practices and ancient Indus Valley poetry, the origins
of number words and the Venerable Bede's system of fingercounting. I have
also had to rethink much of what I thought I knew about numbers and about
the brain.
On the way I have had help from many people. I have been struck by how
willing busy experts have been to answer naive and often stupid questions
from a complete stranger. Among them are Jean Clottes, Gordon Conway,
Josephine Flood, Les Hiatt, Rhys Jones, Deborah Howard, Alexander
Marshack, Karen McComb, Bert Roberts, Robert Sharer, Stephen Shennan, and
David Wilkins.
Over the years I have had the enormous benefit of working closely with
brilliant and knowledgeable scientists on how the brain deals with
numbers: Bob Audley, Lisa Cipolotti, Margarete Delazer, Franco Denes,
Marcus Giaquinto, Luisa Girelli, Jonckeere, Carlo Semenza, Elizabeth
Warrington, and Marco Zorzi. This work was generously supported by the
Commission of the European Union and by the Wellcome Trust.
I have also benefited enormously from discussions with Mark Ashcraft,
Peter Bryant, Jamie Campbell, Marinella Cappelletti, Alfonso Caramazza,
Laurent Cohen, Richard Cowan, Stanislas Dehaene, Ann Dowker, Karen Fuson,
Randy Gallistel, Rochel Gelman, Alessia Granà, Patrick Haggard, Thom Heyd,
Jo-Anne LeFevre, Giuseppe Longo, Daniela Lucangeli, George Mandler,
Ference Marton, Mike McCloskey, Marie-Pascale Noël, Terezinha Nunes, Mauro
Pesenti, Manuela Piazza, Lauren Resnick, Sonia Sciama, Xavier Seron, Tim
Shallice, David Skuse, Faraneh Varga-Khadem, John Whalen, Karen Wynn, and
the late Neil O'Connor. Some of these discussions were held at a workshop
on "The Concept of Number and Simple Arithmetic", sponsored by the Scuola
Internazionale Superiore di Studi Avvanzati in Trieste, organized by Tim
Shallice, the Director of Cognitive Science. Sean Hawkins and Martin Hill
provided invaluable library research. Being Editor of the academic journal
"Mathematical Cognition", published by Psychology Press, has helped me
keep in touch with the latest developments.
The philosopher Marcus Giaquinto inspired my approach, and he kindly read
all the chapters and offered penetrating but helpful comments. Designer
and film-maker Storm Thorgerson worked with me on adapting some of the
ideas in the book for other media, and tried to ensure that the written
material was as clear and gripping to the reader as it was to me. My
daughters Amy and Anna were major inspirations, not just because they have
been handy, and often unwitting, sources of data on how children construct
numerical ideas, but also because their witting insights into their own
mental processes have been invaluable. My partner, Diana Laurillard,
contributed so much and in so many ways that it is now hard to identify
her contributions, except negatively: the least sensible ideas are not
hers.
Lisa Cipolotti, Margarete Delazer, and Norah Frederickson expertly
scrutinized some of the chapters. My original editor at Macmillan, Clare
Alexander, commissioned the book and provided perceptive comments on the
first two chapters. Georgina Morley at Macmillan and Dr Michael Rodgers
offered detailed advice on many aspects of the text. Stephen Morrow at my
US publisher, The Free Press, offered strategic suggestions on the
organization of the whole book, as well as detailed comments. John
Woodruff scrupulously edited the whole text.
Probably none of this would have happened without Peter Robinson, my
agent, whose faith that people would want to read a book with the word
"mathematical" in the title reassured me, and persuaded my publishers,
that it was a viable project.
For me, this has been a wonderful adventure into the history,
anthropology, psychology, and the neuroscience of the ideas that shaped
how we all think about the world. I hope you find it so too.
(October, 1998)