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## Overview

“A lively new book . . . [that] readers will enjoy sprinting through.”—Jordan Ellenberg, New York Times Book Review

What Lynn Truss did for grammar in Eats, Shoots & Leaves, Andrew Hodges has done for mathematics. In One to Nine, Hodges, one of Britain’s leading biographers and mathematical writers, brings numbers to three-dimensional life in this delightful and illuminating volume, filled with illustrations, which makes even the most challenging math problems accessible to the layman. Starting with the puzzle of defining unity, and ending with the recurring nines of infinite decimals, Hodges tells a story that takes in quantum physics, cosmology, climate change, and the origin of the computer. Hodges has written a classic work, at once playful but also satisfyingly instructional, which will be ideal for the math aficionado and the Sudoku addict, as well as the life of the party.

## Product Details

ISBN-13: | 9780385665766 |
---|---|

Publisher: | Doubleday Canada Limited |

Publication date: | 06/03/2008 |

Pages: | 336 |

Product dimensions: | 5.57(w) x 8.17(h) x 1.06(d) |

## About the Author

Andrew Hodges is the author of Alan Turing: The Enigma, described by The New Yorker as “one of the finest scientific biographies ever written.” He is a lecturer at Wadham College, Oxford University.

## Read an Excerpt

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 One-ness 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 x 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 x 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 x 3 Sudoku puzzle. The grid must be completed so that every row, column and subsquare contains just the numbers One to Nine.

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 o*rdinal* 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 Thermodynamics (the conservation of energy) had been established, the Zeroth Law (temperature exists) was recognised as more primitive. Oxford University has a Noughth Week before the first week of term, but no Noughth Class degrees better than a First. There are other grey areas where language calls for starting a list with zero: the Dewey decimal classification, the Noughties for this first decade. However, having to explain the ordinal ‘noughth’ probably causes more problems than it solves.

## First Chapter

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 One-ness 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 x 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 x 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 x 3 Sudoku puzzle. The grid must be completed so that every row, column and subsquare contains just the numbers One to Nine.

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 o*rdinal* 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 Thermodynamics (the conservation of energy) had been established, the Zeroth Law (temperature exists) was recognised as more primitive. Oxford University has a Noughth Week before the first week of term, but no Noughth Class degrees better than a First. There are other grey areas where language calls for starting a list with zero: the Dewey decimal classification, the Noughties for this first decade. However, having to explain the ordinal ‘noughth’ probably causes more problems than it solves.

## Table of Contents

The Unloved One 1

To Be or Not to Be 37

Trinidad 71

It's a Square World 101

Fifth Freedom 147

The Joy of Six 175

The Wonder of it All 215

Sound Bytes 239

End Times Table 283

Hands-on Experience 321

Index of Main Subjects 327

## Reading Group Guide

The Unloved One 1

To Be or Not to Be 37

Trinidad 71

It's a Square World 101

Fifth Freedom 147

The Joy of Six 175

The Wonder of it All 215

Sound Bytes 239

End Times Table 283

Hands-on Experience 321

Index of Main Subjects 327

## Interviews

The Unloved One 1

To Be or Not to Be 37

Trinidad 71

It's a Square World 101

Fifth Freedom 147

The Joy of Six 175

The Wonder of it All 215

Sound Bytes 239

End Times Table 283

Hands-on Experience 321

Index of Main Subjects 327

## Recipe

The Unloved One 1

To Be or Not to Be 37

Trinidad 71

It's a Square World 101

Fifth Freedom 147

The Joy of Six 175

The Wonder of it All 215

Sound Bytes 239

End Times Table 283

Hands-on Experience 321

Index of Main Subjects 327