About the author
Daniel Tammet is the critically acclaimed author of the
worldwide bestselling memoir, Born on a Blue Day, and
the international bestseller Embracing the Wide Sky.
Tammet's exceptional abilities in mathematics and
linguistics are combined with a unique capacity to
communicate what it's like to be a savant. His
idiosyncratic world view gives us new perspectives on the
universal questions of what it is to be human and how we
make meaning in our lives. Tammet was born in London
in 1979, the eldest of nine children. He lives in Paris.
Thinking in Numbers
Daniel Tammet
www.hodder.co.uk
First published in Great Britain in 2012 by Hodder & Stoughton
An Hachette UK company
Copyright © Daniel Tammet 2012
The right of Daniel Tammet to be identified as the Author of the
Work has been asserted by him in accordance with the Copyright,
Designs and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any
means without the prior written permission of the publisher, nor be
otherwise circulated in any form of binding or cover other than that
in which it is published and without a similar condition being
imposed on the subsequent purchaser.
A CIP catalogue record for this title is available from the British Library
ISBN 978 1 444 73742 4
Extract from The Lottery Ticket by Anton Chekhov; Extracts from Lolita
by Vladamir Nabokov © Vladamir Nabokov, published by Orion Books is used
by permission; Extract of interview with Vladamir Nabokov was taken from the BBC
programme, Bookstand and is used with permission; Extracts by Julio Cortazar
from Hopscotch, © Julio Cortazar, published by Random House New York;
Quote from The Master’s Eye translated by Jean de la Fontaine;
Quote from Under the Glacier by Halldor Laxness, © Halldor Laxness,
published by Vintage Books, an imprint of Random House New York.
Every reasonable effort has been made to acknowledge the ownership of
the copyrighted material included in this book. Any errors that may have occurred
are inadvertent, and will be corrected in subsequent editions provided notification is sent
to the author and publisher.
Hodder & Stoughton Ltd
338 Euston Road
London NW1 3BH
www.hodder.co.uk
‘To see everything, the Master’s eye is best of all,
As for me, I would add, so is the Lover’s eye.’
Caius Julius Phaedrus
‘Like all great rationalists you believed in things that were twice as incredible as theology.’
Halldór Laxness, Under the Glacier
‘Chess is life.’
Bobby Fischer
Contents
Acknowledgements
Preface
Family Values
Eternity in an Hour
Counting to Four in Icelandic
Proverbs and Times Tables
Classroom Intuitions
Shakespeare’s Zero
Shapes of Speech
On Big Numbers
Snowman
Invisible Cities
Are We Alone?
The Calendar of Omar Khayyam
Counting by Elevens
The Admirable Number Pi
Einstein’s Equations
A Novelist’s Calculus
Book of Books
Poetry of the Primes
All Things Are Created Unequal
A Model Mother
Talking Chess
Selves and Statistics
The Cataract of Time
Higher than Heaven
The Art of Maths
Acknowledgements
I could not have written this book without the love and encouragement of my family and friends.
Special thanks to my partner, Jérôme Tabet.
To my parents, Jennifer and Kevin, my brothers Lee, Steven, Paul, and my sisters, Claire, Maria, Natasha, Anna-Marie, and Shelley.
Thanks also to Sigriður Kristinsdóttir and Hallgrimur Helgi Helgason, Laufey Bjarnadóttir and Torfi Magnússon, Valgerður Benediktsdóttir and Grímur Björnsson, for teaching me how to count like a Viking.
To my most loyal British readers Ian and Ana Williams, and Olly and Ash Jeffery (plus Mason and Crystal!).
I am grateful to my literary agent Andrew Lownie; and to Rowena Webb and Helen Coyle, my editors.
Preface
Every afternoon, seven summers ago, I sat at my kitchen table in the south of England and wrote a book. Its name was Born On A Blue Day. The keys on my computer registered hundreds of thousands of impressions. Typing out the story of my formative years, I realised how many choices make up a single life. Every sentence or paragraph confided some decision I or someone else – a parent, teacher or friend – had taken, or not taken. Naturally I was my own first reader, and it is no exaggeration to say that in writing, then reading the book, the course of my life was inexorably changed.
The year before that summer, I had travelled to the Center for Brain Studies in California. The neurologists there probed me with a battery of tests. It took me back to early days in a London hospital when, surveying my brain for seizure activity, the doctors had fixed me up to an encephalogram machine. Attached wires had streamed down and around my little head, until it resembled something hauled up out of the deep, like angler’s swag.
In America, these scientists wore tans and white smiles. They gave me sums to solve, and long sequences of numbers to learn by heart. Newer tools measured my pulse and my breathing as I thought. I submitted to all these experiments with a burning curiosity; it felt exciting to learn the secret of my childhood.
My autobiography opens with their diagnosis. My difference finally had a name. Until then it had gone by a whole gamut of inventive aliases: painfully shy, hyper sensitive, cack-handed (in my father’s characteristically colourful words). According to the scientists, I had high-functioning autistic savant syndrome: the connections in my brain, since birth, had formed unusual circuits. Back home in England I began to write, with their encouragement, producing pages that in the end found favour with a London editor.
To this day, readers both of the first book and of my second, Embracing the Wide Sky, continue to send me their messages. They wonder how it must be to perceive words and numbers in different colours, shapes and textures. They try to picture solving a sum in their mind using these multi-dimensional coloured shapes. They seek the same beauty and emotion that I find in both a poem and a prime number. What can I tell them?
Imagine.
Close your eyes and imagine a space without limits, or the infinitesimal events that can stir up a country’s revolution. Imagine how the perfect game of chess might start and end: a win for white, or black or a draw? Imagine numbers so vast that they exceed every atom in the universe, counting with eleven or twelve fingers instead of ten, reading a single book in an infinite number of ways.
Such imagination belongs to everyone. It even possesses its own science: mathematics. Ricardo Nemirovsky and Francesca Ferrara, who specialise in the study of mathematical cognition, write that, ‘Like literary fiction, mathematical imagination entertains pure possibilities.’
This is the distillation of what I take to be interesting and important about the way in which mathematics informs our imaginative life. Often we are barely aware of it, but the play between numerical concepts saturates the way we experience the world.
This new book, a collection of twenty-five essays on the ‘maths of life’, entertains pure possibilities. According to the definition offered by Nemirovsky and Ferrara, ‘pure’ here means something immune to prior experience or expectation. The fact that we have never read an endless book, or counted to infinity (and beyond!) or made contact with an extraterrestrial civilisation (all subjects of essays in the book) should not prevent us from wondering: what if?
Inevitably, my choice of subjects has been wholly personal and therefore eclectic. There are some autobiographical elements but the emphasis throughout is outward looking. Several of the pieces are biographical, prompted by imagining a young Shakespeare’s first arithmetic lessons in the zero – a new idea in sixteenth-century schools – or the calendar created for a Sultan by the poet and mathematician Omar Khayyam. Others take the reader around the globe and back in time, with essays inspired by the snows of Quebec, sheep counting in Iceland and the debates of ancient Greece that facilitated the development of the Western mathematical imagination.
Literature adds a further dimension to the exploration of those pure possibilities. As Nemirovsky and Ferrara suggest, there are numerous similarities in the patterns of thinking and creating shared by writers and mathematicians (two vocations often considered incomparable). In ‘The Poetry of the Primes’, for example, I explore the way in which certain poems and number theory coincide. At the risk of disappointing fans of ‘mathematically-constructed’ novels, I admit this book has been written without once mentioning the name ‘Perec’.
The following pages attest to the changes in my perspective over the seven years since that summer in southern England. Travels through many countries in pursuit of my books as they go from language to language, accumulating accents, have contributed much to my understanding. Exploring the many links between mathematics and fiction has been another spur. Today, I live in the heart of Paris. I write full-time. Every day I sit at a table and ask myself: what if?
Daniel Tammet
Paris
March 2012
Family Values
In a smallish London suburb where nothing much ever happened, my family gradually became the talk of the town. Throughout my teens, wherever I went, I would always hear the same question, ‘How many brothers and sisters do you have?’
The answer, I understood, was already common knowledge. It had passed into the town’s body of folklore, exchanged between the residents like a good yarn.
Ever patient, I would dutifully reply, ‘Five sisters, and three brothers.’
These few words never failed to elicit a visible reaction from the listener: brows would furrow, eyes would roll, lips would smile. ‘Nine children!’ they would exclaim, as if they had never imagined that families could come in such sizes.
It was much the same story in school. ‘J’ai une grande famille,’ was among the first phrases I learned to say in Monsieur Oiseau’s class. From my fellow students, many of whom were single sons or daughters, the sight of us together attracted comments that ranged all the way from faint disdain to outright awe. Our peculiar fame became such that for a time it outdid every other in the town: the one-handed grocer, the enormously obese Indian girl, a neighbour’s singing dog, all found themselves temporarily displaced in the local gossip. Effaced as individuals, my brothers, sisters and I existed only in number. The quality of our quantity became something we could not escape, it preceded us everywhere: even in French, whose adjectives almost always follow the noun (but not when it comes to une grande famille).
With so many siblings to keep an eye on, it is perhaps little wonder that I developed a knack for numbers. From my family I learned that numbers belong to life. The majority of my maths came not from books but from regular observations and interactions day to day. Numerical patterns, I realised, were the matter of our world. To give an example, we nine children embodied the decimal system of numbers: zero (whenever we were all absent from a place) through to nine. Our behaviour even bore some resemblance to the arithmetical: over angry words, we sometimes divided; shifting alliances between my brothers and sisters combined and recombined them into new equations.
We are, my brothers, sisters and I, in the language of mathematicians, a ‘set’ consisting of nine members. A mathematician would write:
S = {Daniel, Lee, Claire, Steven, Paul, Maria, Natasha, Anna, Shelley}
Put another way, we belong to the category of things that people refer to when they use the number nine. Other sets of this kind include the planets in our solar system (at least, until Pluto’s recent demotion to the status of a non-planet), the squares in a game of noughts and crosses, the players in a baseball team, the muses of Greek mythology and the Justices of the US Supreme Court. With a little thought, it is possible to come up with others, including:
{February, March, April, May, August, September, October, November, December} where S = the months of the year not beginning with the letter J.
{5, 6, 7, 8, 9, 10, Jack, Queen, King} where S = in poker, the possible high cards in a straight flush.
{1, 4, 9, 16, 25, 36, 49, 64, 81} where S = the square numbers between 1 and 99.
{3, 5, 7, 11, 13, 17, 19, 23, 29} where S = the odd primes below 30.
There are nine of these examples of sets containing nine members, so taken together they provide us with a further instance of just such a set.
Like colours, the commonest numbers give character, form and dimension to our world. Of the most frequent – zero and one – we might say that they are like black and white, with the other primary colours – red, blue and green – akin to two, three and four. Nine, then, might be a sort of cobalt or indigo: in a painting it would contribute shading, rather than shape. We expect to come across samples of nine as we might samples of a colour like indigo – only occasionally, and in small and subtle ways. Thus a family of nine children surprises as much as a man or woman with cobalt-coloured hair.
I would like to suggest another reason for the surprise of my town’s residents. I have alluded to the various and alternating combinations and recombinations between my siblings. In how many ways can any set of nine members divide and combine? Put another way, how large is the set of all subsets?
{Daniel} . . . {Daniel, Lee} . . . {Lee, Claire, Steven} . . . {Paul} . . . {Lee, Steven, Maria, Shelley} . . . {Claire, Natasha} . . . {Anna} . . .
Fortunately, this type of calculation is very familiar to mathematicians. As it turns out, we need only to multiply the number two by itself, as many times as there are members in the set. So, for a set consisting of nine members the answer to our question amounts to: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512.
This means that there existed in my hometown, at any given place and time, 512 different ways to spot us in one or another combination. 512! It becomes clearer why we attracted so much attention. To the other residents, it really must have seemed that we were legion.
Here is another way to think about the calculation that I set out above. Take any site in the town at random, say a classroom or the municipal swimming pool. The first ‘2’ in the calculation indicates the odds of my being present there at a particular moment (one in two – I am either there, or I am not). The same goes for each of my siblings, which is why two is multiplied by itself a total of nine times.
In precisely one of the possible combinations, every sibling is absent (just as in one of the combinations we are all present). Mathematicians call such collections without members an ‘empty set’. Strange as it may sound, we can even define those sets containing no objects. Where sets of nine members embody everything we can think of, touch or point to when we use the number nine, empty sets are all those that are represented by the value zero. So while a Christmas reunion in my hometown can bring tog
ether as many of us as there are Justices on the US Supreme Court, a trip to the moon will unite only as many of us as there are pink elephants, four-sided circles or people who have swum the breadth of the Atlantic Ocean.
When we think and when we perceive, just as much as when we count, our mind uses sets. Our possible thoughts and perceptions about these sets can range almost without limit. Fascinated by the different cultural subdivisions and categories of an infinitely complex world, the Argentinian writer Jorge Luis Borges offers a mischievously tongue-in-cheek illustration in his fictional Chinese encyclopaedia entitled The Celestial Emporium of Benevolent Knowledge.
Animals are classified as follows: (a) those that belong to the Emperor; (b) embalmed ones; (c) those that are trained; (d) suckling pigs; (e) mermaids; (f) fabulous ones; (g) stray dogs; (h) those that are included in this classification; (i) those that tremble as if they were mad; (j) innumerable ones; (k) those drawn with a very fine camel’s-hair brush; (l) et cetera; (m) those that have just broken the flower vase; (n) those that at a distance resemble flies.
Never one to forego humour in his texts, Borges here also makes several thought-provoking points. First, though a set as familiar to our understanding as that of ‘animals’ implies containment and comprehension, the sheer number of its possible subsets actually swells towards infinity. With their handful of generic labels (‘mammal’, ‘reptile’, ‘amphibious’, etc.) standard taxonomies conceal this fact. To say, for example, that a flea is tiny, parasitic and a champion jumper, is only to begin to scratch the surface of all its various aspects.