Growing Ideas of Number (The Emergence of Number)

5 75

2 2 0 0 9 9 8 8 1 1 3

Series Editor David Leigh-Lancaster

Growing Ideas of Number

Growing Ideas of Number explores the notion of how number ideas, and ideas of number, have grown from ancient to modern times throughout history. It looks at how different types of number and views of numbers (and their meaning and applications) have varied across cultures over time, and combines historical considerations with the mathematics. The book illustrates some of the real problems and subtleties of number, including counting, calculation, measuring and using machines, which ancient and modern people have grappled with—and continue to do so today. It includes a comprehensive range of illustrative examples, diagrams, tables and references for further reading, as well as suggested activities, exercises and investigations. John N Crossley moved to Australia in 1969 from All Souls College, Oxford, when he became

Professor of Pure Mathematics at Monash University. In the 1980s he drifted into Computer Science and in 1994 became Professor of Logic, in the Faculty of Information Technology. He has worked on the history of mathematics and on 13th century medieval history, as well as theoretical computer science. This is his 20th book.

1

3 67 4 6 3

8

2

1

5

THE EMERGENCE OF NUMBER

GROWING IDEAS OF NUMBER

5

THE EMERGENCE OF NUMBER

2

Jo hn N C ro ssl e y

3 67 4 6 8

DAV I D L E I G H - L A N C A S T E R ( S e r i e s E d i t o r )

Series Overview The Emergence of Number series provides a distinctive and comprehensive treatment of questions such as: What are numbers? Where do numbers come from? Why are numbers so important? How do we learn about number? The series has been designed to be accessible and rigorous, while appealing to students, educators, mathematicians and general readers. ISBN 10: 0-86431-709-3 ISBN 13: 978-0-86431-709-4

9

780864 317094

John N Crossley ideas of number cover.indd 1

3/11/06 2:21:48 PM

Growing ideas of number John N Crossley

090604•Growing Ideas of Number 31 1

3/11/06 13:07:17

The Emergence of Number Series editor: David Leigh-Lancaster 1. John N. Crossley, Growing Ideas of Number 978-0-86431-709-4 2. Michael A. B. Deakin, The Name of the Number 978-0-86431-757-5 3. Janine McIntosh, Graham Meiklejohn and David Leigh-Lancaster, Number and the Child 978-0-86431-789-6

090604•Growing Ideas of Number 32 2

3/11/06 13:07:17

Growing ideas of number John N Crossley

THE EMERGENCE OF NUMBER David Leigh-Lancaster (Series Editor)

ACER Press

090604•Growing Ideas of Number 33 3

3/11/06 13:07:17

First published 2007 by ACER Press Australian Council for Educational Research Ltd 19 Prospect Hill Road, Camberwell, Victoria 3124 Copyright © 2007 John N. Crossley and David Leigh-Lancaster All rights reserved. Except under the conditions described in the Copyright Act 1968 of Australia and subsequent amendments, no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the publishers. Edited by Marta Veroni Cover design by Mason Design Text design by Mason Design Typeset by Desktop Concepts Pty Ltd, Melbourne Printed by Shannon Books Cover photograph by John Crossley. The Geometric Staircase, St Paul’s Cathedral, London, with grateful thanks to the Dean and Chapter. National Library of Australia Cataloguing-in-Publication data: Crossley, John N. Growing ideas of number. Bibliography. Includes index. ISBN 9780864317094. ISBN 0 86431 709 3. 1. Numeration – History. I. Title. 513.5 Visit our website: www.acerpress.com.au

090604•Growing Ideas of Number 34 4

3/11/06 13:07:17

Contents List of figures Series overview About the author Preface

vii viii x xi

0

Introduction 0.1 A brief guide to the journey 0.2 Technicalities 0.3 Ariadne’s thread

1 2 3 5

1

Counting 1.1 Kinds of counting 1.2 Counting in groups 1.3 Number words 1.4 Extending counting 1.5 Sizes and bases 1.6 Continuing counting 1.7 Numerals 1.8 Counting forever

6 6 7 9 10 11 13 14 16

2

Calculating by hand 2.1 Kinds of calculating 2.2 Counting rods 2.3 The abacus 2.4 Roman numerals 2.5 Hindu-Arabic numerals and algorism

20 20 21 24 25 26

3

Measuring 3.1 From lines to numbers 3.2 Incommensurability 3.3 Comparing magnitudes 3.4 The Euclidean algorithm 3.5 The geometric line 3.6 Continued fractions

29 29 30 32 34 36 38



090604•Growing Ideas of Number 35 5

3/11/06 13:07:17

vi

CONTENTS

4

Extending numbers 4.1 Classifying numbers 4.2 Positive and negative numbers 4.3 Irrational numbers 4.4 Complex numbers

41 41 42 43 44

5

Extending the number line 5.1 The complete number line 5.2 Infinite numbers 5.3 Countable sets 5.4 Uncountable sets

48 48 50 52 54

6

Systematising 6.1 Formalisation 6.2 The Dedekind-Peano axioms 6.3 From practice to pure mathematics 6.4 Formal logic and set theory 6.5 Non-standard models

58 58 59 61 62 64

7

Calculating by machine 7.1 Designing machines 7.2 Turing machines 7.3 Universal machines 7.4 The incalculable 7.5 Feasible computation

66 66 67 69 72 74

8

Thinking 8.1 The psychology of number 8.2 The innateness of small numbers 8.3 Counting indefinitely

77 77 79 82

9

Resources 9.1 Counting 9.2 Calculating by hand 9.3 Measuring 9.4 Extending numbers 9.5 Extending the number line 9.6 Systematising 9.7 Calculating by machine 9.8 Thinking 9.9 A final note

86 87 87 87 87 88 88 88 88 89

References Index

090604•Growing Ideas of Number 36 6

90 102

3/11/06 13:07:17

List of figures

1.1 Two-counting

10

2.1 The vertical and horizontal representations of the numbers 1 to 9 by Chinese rod numerals

21

2.2 Counting board with counting rods

22

2.3 Adding the ancient Chinese way

23

2.4 The technique of multiplying, as in the Rhind Mathematical Papyrus, using powers of 2

23

2.5 Early (c. thirteenth century) and Modern Arabic numerals

26

2.6 Examples of how early Hindu-Arabic numerals could be fraudulently modified in accounts by changing 0, 4 and 7 into 8

27

3.1 Dividing a triangular prism into three equal pyramids (tetrahedra)

33

3.2 A geometric argument for the incommensurability of the side and diagonal of a square

35

3.3 Rotating the square in Figure 3.2

35

3.4 Constructing fractions

36

4.1 Bombelli’s rules for multiplication of imaginary numbers

47

5.1 Counting the positive rational numbers

52

5.2 The ‘back and forth’ argument

53

7.1 The essentials of a Turing machine

69

7.2 Numbers can be represented in the lambda calculus, but not without difficulty

70

vii

090604•Growing Ideas of Number 37 7

3/11/06 13:07:17

Series overview The Emergence of Number is a series that comprises three complementary texts: • Growing Ideas of Number • The Name of the Number • Number and the Child While each of these texts can be read in their own right according to interest, their complementary combination is intended to provide a distinctive and comprehensive treatment of questions such as: Where do numbers come from? What are numbers? Why are numbers so important? How do we learn about number? The series is designed to be accessible and rigorous while appealing to several audiences: • Teachers and students of mathematics and mathematics-related areas of study who wish to gain a richer understanding of number • Mathematics educators and education researchers • Mathematicians with a broader interest in the area of study • General readers who would like to know more about ‘number’ in terms of its cultural and historical conceptual development and related practices Growing Ideas of Number explores the notion of how number ideas and ideas of number have grown from ancient to modern times throughout history. It engages the reader in thinking about how different types of number, views of numbers, and their meaning and applications have varied across cultures over time, and combines historical considerations with the mathematics. It nicely illustrates some of the real problems and subtleties of number including counting, calculation and measuring, and using machines, which both ancient and modern peoples have grappled with— and continue to do today. The Name of the Number covers the development of number ideas in language, not only as we know and use it today, but as a record of the development of a central aspect of human evolution: how number has emerged as a central part of human heritage, and what this tells us about viii

090604•Growing Ideas of Number 38 8

3/11/06 13:07:17

SERIES OVERVIEW

ix

who we are in our own words and those of our ancestors—the story of number in language. The treatment is an anthropological and linguistic exploration that engages the imagination, combining phonetics, symbols, words and senses for and of number, counting and bases in a journey from ancient times to the present through the emergence and development of historical and contemporary languages. Number and the Child discusses how students learn about number concepts, skills and processes in the context of theories, practical experience and related research on this topic. It includes practical approaches to teaching and learning number, and the place of number in the contemporary school mathematics curriculum. It stimulates the reader to consider the role of number in the mathematics curriculum and how we frame and implement related expectations of all, or only some, students in the compulsory years of schooling. Each text in the series incorporates a comprehensive range of illustrative examples, diagrams and tables, text and web-based references for further reading, as well as suggested activities, exercises and investigations.

090604•Growing Ideas of Number 39 9

3/11/06 13:07:17

About the author John Crossley obtained his doctorate from Oxford in 1963. He taught at Oxford where he became University Lecturer in Mathematical Logic and a Fellow of All Souls College. He moved to Australia in 1969 when he became Professor of Pure Mathematics at Monash University. In the 1980s he drifted into Computer Science and in 1994 became Professor of Logic. Since 1998 he has been Research Mentor in the Faculty of Information Technology. He has done much work on the history of mathematics. As well as being a member of the Faculty of Information Technology, he is an Honorary Research Associate in both the School of Historical Studies and the School of Philosophy and Bioethics in the Faculty of Arts at Monash. He has published 19 books and more than a hundred papers.



090604•Growing Ideas of Number 310 10

3/11/06 13:07:17

Preface Writing the first paragraph, the first sentence, indeed the first word, of any book, whether one’s first or one’s twenty-first, remains important. But note how numbers have already entered. Not just the ordinal numbers ‘first’ and ‘twenty-first’, but also the singular ‘one’. And why? Because number has a simplicity and universal applicability—well, almost universal. But how simple is number? This book is concerned with ideas of number and how new and different kinds of numbers have developed in human society over the last ten thousand years or so. (There are many books on the forms and use of numbers, see Chapter 9, Resources.) In recent years I, and increasingly many others, have become aware of just how much mathematics is a human activity. Before the mid-twentieth century it was often thought of as a most perfect and immutable part of logic: a Platonic ideal. Influences from anthropology, archaeology and neuropsychology have helped to change those attitudes. Therefore any approach should include elements of those and, of course, of history and philosophy too. This book sketches the development of ideas of number from the simplest beginnings and, although many layers of meaning and different manifestations have accrued over time, we shall see how these ideas have not only been generated by humans, but some of them have become, quite literally, embedded in our brains. It is perhaps worth pointing out that the history of mathematics is usually approached in one of two quite distinct ways. In the first approach, the writer looks back on the achievements of earlier workers and expresses them in modern terms. In the second, the writer tries to get into the minds of the earlier workers and understand how they developed their ideas. Both have their advantages. The first approach is well exemplified in Stillwell (2002). I usually, but not always, incline to the second approach and hope thereby to enable my readers to look forward from where they currently find themselves. For mathematics does not stand still. Advances will continue to be made. This book is based on thirty years’ work on the intellectual history of mathematics, and of number in particular. It may, paradoxically, be considered a revision and expansion of an earlier book (Crossley 1987) in fewer words. While it considers a number of issues not discussed there, it xi

090604•Growing Ideas of Number 311 11

3/11/06 13:07:18

xii

P R E FAC E

also revisits a number of earlier ones. It is gratifying to note that some of the ideas in that book have subsequently been discovered by others, and some conjectures have been borne out. At the end of this book I have added a list of resources as well as a list of references. (Dates are given in the format day.month.year, with the month in lower case Roman numerals.) A number of these resources are located on the World Wide Web, and may therefore be ephemeral. More importantly, data on the Web is not always reliable and the reader is urged to carefully assess the quality of what is found there. The reader is also urged to follow up my own references, especially since in this short book I have paraphrased significantly. In trying to understand what really happened in the development of mathematical ideas (and in other fields too), the best advice was given to me by David Whiteside in Cambridge many years ago: ‘Go and look at the originals.’ I am grateful to many individuals and libraries over many years. Monash University Library (especially the Hargrave-Andrew Library and the Rare Books Collection), the State Library of Victoria, the Bodleian Library, Oxford, and all their staff have provided a great wealth of material and assistance. Particular thanks to Sara Miranda of the Hargrave-Andrew Library at Monash for help with the bibliography. John Stillwell, my colleague of many years and a renowned historian of mathematics (who also approaches that history in the second mode noted above), has read much of my work and has always been very helpful. He has read the draft of this book and made valuable comments, for which I am immensely grateful. Michael Deakin has also read parts of the draft and made valuable comments which have been incorporated. Very many thanks to Tim Brook who has read the penultimate draft with his usual flair and eagle eye. Gordon Smith, now sadly deceased, stimulated my initial interest in the history of mathematics when I started asking what I thought were simple questions. They were simple only in the phrasing. Ivor Grattan-Guinness has been a continuing strength. Tony Lun has been of immeasurable help in developing my knowledge and appreciation of Chinese mathematics, and my awareness of different cultural approaches. Special thanks to Don Herbison-Evans who made me think about the diagonal argument in a new way. As usual, the responsibility for any errors or omissions remains mine alone.

090604•Growing Ideas of Number 312 12

3/11/06 13:07:18

5 7

2 18 93 36

3

7

4 6 8

2

CHAPTER ZERO

5

Introduction

The numbers that circumscribe our lives seem always to be growing: our age, the cost of living (usually), our income (or so we hope), the population of the world. Numbers permeate every aspect of our lives. The distance of the sun (about 150 million kilometres or 93 million miles) has been known for centuries, and is now described as merely ‘1 astronomical unit’ so that we can readily account for much greater astronomical distances. Gigabytes were unknown in the middle of last century, now it is terabytes and petabytes that are less well known. National deficits are no longer measured in millions, but billions or trillions. Not only do numbers grow but the ideas of what a number is, and what numbers are, have also grown—and multiplied. The very word ‘number’ has many meanings: the Oxford English Dictionary (OED) lists 33 for the noun alone. Other words can sometimes be used, such as ‘digit’ or ‘numeral’. All have ambiguities. In general, but not always, this book is concerned with the abstract number rather than its representation. In this book, ‘number’ will generally refer to the ‘natural numbers’; rational, real and complex numbers will usually be referred to as ‘rationals’, and so on. Although these ideas of number may seem simple, their development has taken ten thousand years or more—with considerable acceleration in the last thousand years. It is the intention of this book to make this development accessible: to provide keys that will open the doors on the wonders of numbers and human development. For this reason there is a chapter on Resources (Chapter 9) as well as the usual set of References. The latter includes all the very varied sources I have used in the work. These two sections are intended to be the doorways to further research. Looking at a mere handful of the works listed will not only amplify what has been said in few words; it will, I hope, also raise new questions in the reader’s mind; questions that may extend ideas of number further, or nourish the growth of new ideas of number. 1

090604•Growing Ideas of Number 31 1

3/11/06 13:07:20

1



GROWING IDEAS OF NUMBER

0.1  A brief guide to the journey In developing the ideas, I start as early as I can, that is to say, with the simplest development of the idea of number. This is based on studies, principally by anthropologists and linguists, of what were once called ‘primitive societies’. As these number systems developed and spread, they became more sophisticated, culminating in our present system of numerals, normally known as the Hindu-Arabic numerals, and based on the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Chapter 1 ‘Counting’ traces this development. Chapter 2 ‘Calculating by hand’, then studies how we use these numbers, but without the aid of mechanical or electronic devices. ‘Measuring’ is the title of Chapter 3. Here we see the application of number and numbers. I am at pains to point out that the number line is not the geometric line, but there are strong connections between them, a point to which I return in Chapters 5 and 6. Not content with the whole, or natural, numbers, nor solely with rational and even some irrational numbers, people pushed the envelope of number further. Chapter 4 ‘Extending numbers’ deals with extensions to the basic number concepts, while Chapter 5 ‘Extending the number line’ considers what might be called the geometric extension of the number line. The development of the various number concepts caused some difficulties and led, slowly, to the formal development of various theories of number, which are dealt with in Chapter 6 ‘Systematising’. The impact of mechanical, and latterly electronic, assistance in calculating is examined in Chapter 7 ‘Calculating by machine’, but this chapter is not another recounting of the achievements of computer builders. It is concerned with the central idea behind computers: the notion of a universal machine that can calculate anything that can be calculated by any machine whatsoever. It also describes the sober reality that some things cannot be calculated by any machine whatsoever, and even if they can be calculated, they may take too long for us to benefit. Chapter 8 ‘Thinking’ takes us back to the smallest numbers, those no bigger than four. This leads into the worlds of psychology and neuropsychology. Recent work has shown that Jung’s ideas from the early part of last century are essentially correct and have a physical basis. Nevertheless there remains one simply stated and yet unexplained idea: the idea of ‘and so on’, which has recurred throughout the work. This takes us back to the simplest counting practices I commenced with in Chapter 1, but, I hope, with eyes more wide open.

090604•Growing Ideas of Number 32 2

3/11/06 13:07:20

INTRODUCTION



0.2  Technicalities I have tried to avoid technical details as much as possible—not entirely with success! The references provide all of those, but here are a few fundamental ones.

0.2.1  Familiar Natural numbers are taken to be 1, 2, 3 … In the context of doing pure mathematics, some people prefer to start the natural numbers with zero rather than one, but this has little or no effect on any of the arguments. It sometimes brings in a number of complications; on other occasions it has technical benefits. From a mathematical point of view there is no abstract difference when we are simply looking at counting or points on a line, because we are dealing with isomorphic (same shape, see below) sequences; 0, 1, 2 … looks like 1, 2, 3 …, when we are only concerned about an infinitely proceeding sequence. m Rational numbers are otherwise known as fractions: ! n , where m and n are natural numbers. Cardinal numbers are the counting numbers: 1, 2, 3 …; ordinal numbers are the ordering numbers: 1st, 2nd, 3rd … Logarithms and exponentiation are briefly mentioned below and used a little in Chapter 7. The basic laws for differentiating xn and a product yz occur once. Knowing these, one is aware of the difficulty of 1 trying to integrate (or antidifferentiate) x . Computer programming in both a high-level language, and in a simple language, is used briefly in Chapter 7. Euclid’s algorithm in its basic form, that is to say, for numbers, says that given positive natural numbers a and b there are natural numbers q and r, such that a = bq + r and 0 # r 1 b



(1)

There is a similar version for lengths. This was necessary for the ancient Greeks, because they distinguished numbers and lengths, see Section 3.3. Geometric series have sums given by:

n

a + ar + ar 2 + ar3 + f + ar n = / ari = a i=0



1 - rn + 1 rn + 1 - 1 =a 1-r r-1 3

a + ar + ar 2 + ar3 + f + ar n + f = / ari = i=0

090604•Growing Ideas of Number 33 3

a 1-r

(2)

(3)

3/11/06 13:07:20



GROWING IDEAS OF NUMBER

Complex numbers, and especially their multiplication, are used in Chapter 4 (see Figure 4.1, Section 4.4), but only the basic laws, such as i × i = –1, where i is the square root of –1.

0.2.2  Less familiar There are three or four important technical ideas that recur in the book and I list them below for convenience. The first of these is quite simple, yet has a bivalent role. The length of a number is most obviously taken as the number of digits it contains. This is entirely appropriate as an answer in our society, where we use Hindu-Arabic numerals. It turns out that it also works (to a reasonable degree) with words for numbers. If we write out a number in Hindu-Arabic numerals, that is using 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, then the length of the number 1 234 567 is obviously 7. This reflects the power of 10 we are dealing with: 107 = 10 000 000. So a number with 7 digits is between 1 000 000 and 9 999 999. In general, the length of the number n, written in the decimal system, is 1 + log10 n, where x, the floor of x, is the largest integer ≤ x. If we are dealing with numbers in base 2, where the decimal number 10 is represented as 1010, since 10 = 23 + 21, then the length of the number n will be 1 + log2 n. Now consider 1024 = 210. Decimal 1024 is written as 10 000 000 000 in binary. In base 10 its length is four, in binary it is eleven. However, the ratio of these two lengths is roughly log10 n : log2 n = log10 2 : 1 . 3 : 1 (but actually slightly more). So we have only a threefold reduction in length. Now a threefold reduction in the number of words we have to use to describe a particular number can be very useful. This is noted in Chapter 1. Homomorphisms are transformations that keep some of the original properties. Thus counting the number of sides of a polygon is a homomorphism from polygons to numbers. Under a homomorphism some properties may be lost. In this example, the angles are not preserved: a diamond and a square map to the same number, namely 4. Isomorphisms are transformations that preserve all the properties we are interested in. For example, rotating a geometric figure in the plane, without changing the lengths of the sides, preserves the angles and the area, as well as the number of sides and their lengths. Invariants are properties that do not change under homomorphisms (or isomorphisms). This term is used both as an adjective and as a noun. In the

090604•Growing Ideas of Number 34 4

3/11/06 13:07:20

INTRODUCTION



example of an isomorphism above, the lengths of the sides of the figures are invariant. When we take a single figure and rotate it, the area inside the figure is an invariant of the figure.

0.3  Ariadne’s thread There are many threads in this book and they run both through and across various topics. These threads appear and disappear like those in a woven tapestry; and, like the weaving of a tapestry, the book progresses in linear fashion. There are therefore many cross-references, some forward, some back. The index and the sectioning of the chapters are organised to make it easier both to find items and ideas, and to revisit them later. Inevitably, some ideas depend on others, but it will often be possible to read (or reread) the book in an order different from the usual linear flow. So it may be helpful to backtrack, or even meander at times, and not always best to follow the King’s grave advice to Alice: Begin at the beginning, and go on till you come to the end: then stop. (Carroll 1865, chapter XII.)

090604•Growing Ideas of Number 35 5

3/11/06 13:07:21

5 7

2 18 93 36

3

7

4 6 8

2

1

CHAPTER ONE

5

Counting

1.1  Kinds of counting It is a truism that counting only occurs when we have something to count. We do not, except in unusual circumstances, count one object on its own. Putting things together into a group both gives us the opportunity to count objects and also provides the necessity to count things. If we want to exchange our apples for our neighbour’s pears, counting is one simple way of making a just comparison. Weighing is another and, although that can be done without numbers, we usually use numbers there too. In the same way, we can see (and see quite literally if the numbers are small) that two given collections have the same number of objects in them. This idea was used by Bertrand Russell in his philosophy of mathematics as the basis for his abstract theory of mathematics, see Note 1 and the translation in Grattan-Guinness (1977, p. 169). It is not obvious, and Russell himself was at pains to explain, that ‘having the same number’ does not first require knowing or defining what a number is (or numbers are). To see this, consider a somewhat more complicated case: that of two collections having a different number of objects. Laying them out side by side will create a one-to-one correspondence that allows us to decide which collection is greater (or smaller). For example, when all the tickets to a football match are sold, and there are still people who want to get in to the game, we know there were more people than tickets, and we know this without knowing how many tickets (or people) there were. Long after Russell’s death it was established that this was indeed a way of counting in ancient Egypt (see Schmandt-Besserat 1992). There, tokens were correlated with objects in accounting procedures. As SchmandtBesserat notes (1992, vol. 1, p. 167): ‘… the exchange of goods per se seems to play no role in the development of reckoning technology, presumably because bartering was done face to face and, therefore, did not 6

090604•Growing Ideas of Number 36 6

3/11/06 13:07:23

COUNTING

require any bookkeeping.’ Counting using a one-toone correspondence does not even require naming the numbers, as Russell noted. It is rarely noted that in order to count we need to be able to remember. We need to remember not only what point we have reached, but inter alia that we are in the process of counting, although this awareness may be of a very limited nature. There is a degree of repetition that is important and this will be discussed at length later, especially in Chapter 6. Counting is a temporal process, and usually also a spatial one. The objects (or events) have to be ordered, or perhaps it is better to say ‘arranged’, either in the world or in our minds. Immediately there is a question of primacy: Which came first, the cardinal numbers or the ordinal numbers? When we count, as was just said, we do this in some order. This suggests that ordinals come first in counting. On the other hand, if we are comparing two collections, as Russell and the Ancient Egyptian accountants did, then we use certain cardinals. However, this latter style has no obvious extension to counting larger and larger collections. It only tells us whether two collections have the same number of members or not; it does not tell us what that number is, though it does allow us to reproduce it, as Schmandt-Besserat (1992, vol. 1, pp. 196–7) has pointed out.



N OT E 1 Russell first defined the idea of two collections having the same number. This he did by using a one-to-one correspondence. For example, if everyone is seated comfortably at the table then there is a one-to-one correspondence between seats and individuals. It is only later that Russell introduces the actual numbers 0, 1, 2 … This he does by abstracting what is common to all the collections that have a one-to-one correspondence with each other. In particular, for the number 1, he takes all those collections that have a one-to-one correspondence with a set with just one object in it, and similarly for larger numbers. This may seem circular and it takes some thought to disentangle the abstract notion from the concrete notions of 1, 2, 3 and so on. For further reading see Russell (1919).

1.2  Counting in groups If we are dealing with very small numbers, say up to about four, then we can picture objects in a neat configuration: a triangle or a square, for example. Animals can also recognise some of these numbers, and I shall return to this in Chapter 8. In order to progress beyond such small numbers we use counting: counting is a process—a process that uses number words. How did the process of counting arise? Some anthropologists have thought that counting is innate. Thus an early anthropologist such as Tylor quotes a famous deaf-and-dumb person, Massieu, from Sicard (1808), with approval as asserting: I knew the numbers before my instruction, my fingers had taught me them. I did not know the cyphers; I counted on my fingers, and when

090604•Growing Ideas of Number 37 7

3/11/06 13:07:23



GROWING IDEAS OF NUMBER

the number passed 10 I made notches on a bit of wood. (See Tylor, 1958, vol. 1, p. 221; footnote omitted.)

Cassirer (1953, vol. 1, p. 229), opined that counting came from body concepts and the process was then transferred to other objects. ‘The counting gesture does not serve as a mere accompaniment to an otherwise independent numeral, but fuses in a sense with its signification and substance.’ Seidenberg disputed this, saying: ‘The facts indicate that finger counting is learned, just as counting is … the gesture language is learned in quite the same way that the vocal language is’ (Seidenberg 1960, p. 258). More recently, Hale (1975) analysed matters differently, describing an Australian counting system (of the Walbiri or Warlpiri) as lacking numerals, yet not lacking the idea of counting. He argued that counting was universal. To support this he pointed out that the Walbiri rapidly mastered the English counting system. So we should distinguish between (i) being able to repeat a process, and thereby count higher when the words are available, and (ii) having the words available, or being able to fabricate words, to record the counting process. It should also be noted that Gordon (2004), in his consideration of the Pirahã people of the Amazon forest, has cast doubt on the innateness of a number sense for numbers above three, and on even the clarity of the number concepts ‘one, two’ and ‘many’. (See also Section 8.2.) We may conclude that, once the practice of counting has been established, it is possible to extend it. Just how far? is a question to which we shall return. So how did the process of counting arise? To answer this we need to go back to the Stone Ages, or more precisely the Palaeolithic (Old) and the Mesolithic (Middle), for that is where the oldest evidence has been found. Marshack has been investigating this phenomenon for many years. Marshack (1991) brings together much evidence from scratches on bones that numbers of events were being recorded over time. Marshack calls this ‘time-factored’ (p. 25) or ‘sequenced’. Marshack argues convincingly that the marks on the bones had been made at different times. His ultimate view is that the marks were a kind of lunar calendar. Some critics have questioned this conclusion (for example, Rosenfeld 1971) but we do not need to go so far. It is sufficient for our purposes to note that a number of events was recorded, that this took place over time, possibly an extended time of a month or more. Moreover, this process was repeated in a number of parts of the world, notably Africa and Europe. The first example Marshack used was from Ishango in the Nile valley. An example is shown in Marshack (1991, p. 23). There we see groups of small

090604•Growing Ideas of Number 38 8

3/11/06 13:07:24

COUNTING



scratches. These are not necessarily entirely uniform; for example, groups may consist of 7, 5, 10, or 8 scratches. However, the groups are clearly demarcated. This example of grouping is oft repeated, and it is clear that the groups were intended. What their exact purpose was is irrelevant to our present argument. We do have evidence of grouping, and what appears to be a record of a succession of events. We do not have to go all the way with Marshack and say that they form a lunar calendar. Marshack, of course, had only the artefacts from which to work. We cannot say whether there was an oral recitation that accompanied the marks. Therefore, we only know of the existence of a process of counting; we know nothing about any number words used in counting at that time or, indeed, if there indeed were such words.

1.3  Number words So let us turn to the question of number words. Over time, these words have developed and are used in the counting process. In the nineteenth century, there was a tremendous upsurge in interest in so-called primitive societies. Much of this was connected with the proselytising work of missionary societies such as the British Society for the Propagation of the Gospel and the Society for the Promotion of Christian Knowledge. In spreading the gospel the missionaries learnt about the languages and practices of unfamiliar societies. Indeed, this has led to major exercises in learning such languages, for example, by the Summer Institute of Linguistics, work that continues to this day. In the process of exploration, indigenous people were often quizzed about their words for numbers. In some cases, it was found that there was a very limited number of number words. Thus Gow (1968, p. 4), was able to record: It is probably familiar to most readers that many savage tribes are really unable to count, or at least have no numerals, above 2 or 3 or 4, and express all higher numbers by a word meaning ‘heap’ or ‘plenty’.

This was certainly true of many societies, but the motivation for having larger numbers was often absent. In many cases, it would have been possible to extend the system to larger numbers but there was no need. (See below for a delightful example from Tonga in Section 1.8.) The context and the pressure to count is an important factor here. Perso (2001) wrote: For some [Australian] Aboriginal children, particularly in remote communities, experiences with numbers used in an ordinal (ordering)

090604•Growing Ideas of Number 39 9

3/11/06 13:07:24

10

GROWING IDEAS OF NUMBER

Gumulgal Australia

Bakairi South America

Bushman South Africa

1

urapon

tokale

xa

2

ukasar

ahage

t’oa

3

ukasar-urapon

ahage tokale (or ahewao)

’quo

4

ukasar-ukasar

ahage ahage

t’oa-t’oa

Figure 1.1.  Two-counting

sense, a cardinal (counting) sense or a labelling sense may be rare. To facilitate the learning of these, real life experiences in and around the school community may need to be ‘created’ so that there is an immediate practical purpose in learning them.

Even the five-volume work of Dixon and Blake (1979–2000) lists few number words in the survey of Aboriginal languages of Australia. For example, for the language Guugu Yimidhirr, they list only words for ‘one’, ‘two’, ‘three’ or ‘four’ (the same word works for both), and ‘five or a few’ (Dixon and Blake, vol. 1, p. 77). Before we look at how the process of creating number words continued, let us look at the counting process a little further.

1.4  Extending counting Because counting, as a practical process, takes place over time, we have to start somewhere, and then, as we have seen, groups are formed. Obviously the smallest such groups that are distinguishable from the individual objects or events have size 2. When the end of such a group is reached, we cycle back to the beginning. Therefore many authors refer to these groups as ‘cycles’. We find these in very many languages, including English. In our process of counting, we conventionally start with one. (See Section 0.2.1.) One of the simplest examples we have, at least in terms of the ideas, is the counting system that starts 1, 2, 2 + 1, 2 + 2, though the ‘+’ may not be pronounced, and it is important to note that we use the ‘+’ for convenience only; there is no evidence of any notion of ‘addition’ in this context. Such systems can be found all over the world. As Seidenberg (1960, p. 216) has noted, the Gumulgal in Australia, the Bakairi in South America, and the Bushmen of South Africa all used such a system or something very close to it (see Figure 1.1). However, Michael Deakin (personal communication) has pointed out that Seidenberg’s extension beyond four has no support in recent work such as that of Dixon and Blake (1979–2000).

090604•Growing Ideas of Number 310 10

3/11/06 13:07:24

COUNTING

11 N OT E 2

When we start to count, we soon run out of number words. This imposes some restrictions on our arrangement, or assembly, if we are to be able to continue counting. We therefore put objects into groups. In the above examples, the group size was two. Of course there are many choices for the size of the group; indeed, there are infinitely many such choices: 2, 5, 10, 12, 20 are all popular and have been used in different contexts. Lean (1992) says about language systems in Papua New Guinea:

Glen Lean spent many years in Papua New Guinea collecting and commenting on many languages there. His work reveals two kinds of developing counting by groups. One is similar to our own base 10 counting, but based on 2s and 5s. The word for ‘5’ is lima or something phonetically close to that in many PNG languages. It means ‘hand’ as elsewhere in Asia

… the counting systems and tallies which are found among the NAN [non-Austronesian] languages of

(see the main text). The other method is based on

New Guinea, the islands lying to the east of the

identifying many more points on

mainland, and the Solomon Islands, comprise mainly

the body. The Ipili language, for

the various 2-cycle variants, the body-part tallies, the

example, uses 27 body parts: 10

(5, 20) digit tally, several 4-cycle systems, and a small

fingers plus 17 along the arms,

number of 6-cycle systems. There is also a number of

shoulders, neck and head. Other

languages which have counting systems possessing

languages use a variety of cycle

either a primary or secondary 10-cycle.

sizes, ranging from 18 to cycles as large as 37, 47 or 68 (see Lean

Conant (1931) provides many examples, including a decimal-quaternary system from Hawaii (on p. 116), though his examples are otherwise almost all based on 5, 10 and/or 20. Codrington (1885) provides a wealth of examples from Melanesia. Although, in most parts of the world, the groups are of the sizes noted above, in Papua New Guinea many other and varied sizes of groupings have been used. Many of these seem strange to someone brought up in a British education system (see Note 2). The actual choice of the size of group in all cases seems to be a product of the environment in which the counter finds him- or herself.

1992, chapter 2). Owens (2001) provides a very brief summary of Lean’s results, including a useful table of the range of cycles. Such counting, using not just the fingers, is also recorded on the Lower Tully River in Queensland, where many different points on the hand were used to record the number of days between pruns (meetings, see Roth 1908, p. 80). Interestingly, in this counting, some points were in the cycle twice, so this was not one-to-one

1.5  Sizes and bases

counting.

The size of group, or cycle, usually is used in two different ways. First, it determines the number of objects we can consider before we have, in some sense, to start again. Secondly, it is used for looking at groups of groups (or groups of groups of groups, and so on). In this case the group size becomes

090604•Growing Ideas of Number 311 11

3/11/06 13:07:24

12

GROWING IDEAS OF NUMBER

the base of the number system. Thus in our familiar base ten system, we first count up to ten, and then we count groups of objects each of size ten, giving them numbers which reflect what we have done, as noted below (Section 1.6). There are many good arguments for using base 10, including perhaps inertia and our own physical constitution: the fact that most of us have ten fingers tends to make us think that ten is a natural grouping—or five, if we consider only one hand. The Inuit in Canada used 20; Cassirer (1953, vol. 1, p. 230) notes that in Eskimo (Inuit) ‘twenty’ is expressed by ‘a man is completed’. In parts of South-East Asia the word for ‘five’ and the word for ‘hand’ are the same. In the Philippines, Malaysia and Indonesia, the word is lima. This is also essentially true of the Tongan language where the word is nima as recorded by Mariner (see Martin 1827, vol. 2, p. [369]). Mariner asserts in the Vocabulary that this ‘is derived from the hand having five fingers.’ Five is often eclipsed and ten dominates. Nevertheless, there has been some resistance to moving to ten in some contexts. Pounds, shillings and pence survived until the middle of the last century with their mixture of groupings: in twelves and then in twenties. Even in the twentyfirst century, a megabyte is not, or should not be, a million bytes but 220, but it is used as a standard group size (cf. Quinn 1992). In addition to the above reasons, there is the question of economy of utterance. When we use larger groups we can use fewer words. Thus using base 10 is more economical than using base 2 (see Section 0.2.2). The use of what is now the standard unit, ten, appears to be a product of economic and political pressure, perhaps aided by the fact that we have ten fingers. The same sort of thing seems to happen in the case of number words in a language. When one geographical region becomes subservient to another, as in the process of colonisation, the dominant force also provides the dominant language. As Sasse (1992, p. 13–14), says in the case of language death, though the present case is not quite language death, perhaps just wounding: [Studies] indicate that there is always one common element, viz. the presence of socio-economic and/or socio-psychological pressure phenomena which move the members of an economically weaker or minority speech community to give up its language.

Sometimes such pressure is resisted and the native language is used as a weapon against the invader; in other cases, there may be a greater or lesser acceptance of the new language. In particular, when a new language replaces, or even partly replaces, an indigenous language, number words

090604•Growing Ideas of Number 312 12

3/11/06 13:07:24

COUNTING

13

can be taken over into common speech. Thus Hale (1975, p. 3), reports that the Walbiri of Central Australia introduced the system wani, t.uwu, tjiriyi, puwa, payipi, tjikitji, tjipini, yayit.i, n.ayini, t.ini, l.ipini … although they had their own words for these numbers. Pronouncing these will reveal their origin. Likewise, today in the Philippines, which was taken over by the Americans in 1898, one will usually hear English words used for numbers over, say, twenty, while the local Filipino (or Tagalog) words for one to five are very commonly used in everyday conversation when the local language is being spoken. It is therefore not surprising that, with the new language, the idea of counting with one’s own size of group may be displaced by counting using the group size of the incoming language. A similar change happened last century in the way we measure (see Section 3.1).

1.6  Continuing counting Once we have our groupings, we can then regard each group as an entity and start counting these new entities. Thus we count ‘ten, twenty (= two tens), thirty (= three tens), and so on.’ This is more evident in German: zehn, zwanzig, dreissig …, where zehn (ten) has become -zig or -sig. There are similar but less obvious formations in French and Spanish. On the other hand the groups are explicit in Chinese: shi, ershi, sanshi …, meaning ‘ten, two ten(s), three ten(s) … ’ (There is no lexical distinction between singular and plural for Chinese nouns.) At this point we can see (though it is not clear that people in the past always saw) that we can repeat the process and start grouping tens together to make hundreds, and so on. Let us pause here to raise the question: Exactly what does ‘and so on’ mean? I shall return more than once to this question (in particular, in Section 1.8, and in Chapter 8). In English we have the word ‘thousand’, and then we reuse ‘ten’ in ‘ten thousand’, ‘hundred’ in ‘hundred thousand’ before the new word: ‘million’. The large groupings after that were first given names by Chuquet (see Marre 1880) shortly before 1500. He used ‘million’ for 1 000 000, ‘billion’ for 1 000 000 000 000, ‘trillion’ for 1 000 000 000 000 000 000, and so on. In recent years, the meaning of these words has changed and we now follow the American lead, referring to ‘1 000 000 000’ and ‘1 000 000 000 000’ as ‘billion’ and ‘trillion’. However, there is no clear extension of the process for generating such words, even though Johnstone (1975) does use ‘vigintillion’ for 1063 and quintoquadrogintillion for 10138, extending the use of Latin roots begun by

090604•Growing Ideas of Number 313 13

3/11/06 13:07:24

14

GROWING IDEAS OF NUMBER

Chuquet but following the American style. Thus ‘vigintillion’ from the Latin for ‘twenty’, viginta, means twenty blocks of three zeroes (000) after the initial 1000, and ‘quintoquadrogintillion’ means forty-five, literally ‘five and forty’, such blocks. One purpose of these new names is to allow us to attach number names to larger and larger groups. However, if we were to introduce a new name each time it would be very hard to communicate. At any time in history we only use a finite number of words to do our counting and so we shall eventually need to repeat at least one of them, increasingly many times. Brainerd (1968, p. 40, n. 6), says that the ‘ability [of speakers of English] to coin names for high powers of 1000 is limited by the fact that only a finite set of names in Latin are [sic] available.’ This is a step forward from the situation of the Romans. They had to say mille mille …, that is thousand thousand … (See Exercises 1 and 2.) In Chapter 2, I shall show how the use of Hindu-Arabic numerals allows us to provide number words (or rather phrases) for arbitrarily large numbers.

E x ercise 1 Prove that, if we only have words for the first ten numbers, then we need to repeat at least one of them arbitrarily often in describing larger and larger numbers.

E x ercise 2 Show that this works when ‘ten’ is replaced by any finite number.

1.7 Numerals Although we know how to create numerical descriptions of arbitrarily large groups, it is not clear how to proceed without using a system such as that of (the number) words corresponding to Hindu-Arabic numerals. Many systems in the world simply run out of representations. Long ago Archimedes (c. 287–212 BC), in the Sand Reckoner (see Archemides 1976, 8 p. 347), developed descriptions for numbers not exceeding A = ]108g10 (alternatively, 108 to the power 108). In the ancient Chinese classic Yi Jing (see, for example, Wu Jing-Nuan 1991), Leibniz (1646–1716) ‘discovered’ the binary system. However, the Chinese had only produced representations for sixty-four numbers: those from 1 to 64 (or 0 to 63, if you prefer). Sixty-four was the limit imposed by using six yarrow (milfoil) stalks, each of which determined a zero or a one according to the way up that it fell. So

090604•Growing Ideas of Number 314 14

3/11/06 13:07:24

COUNTING

15

there were 26 possibilities corresponding to numbers N OT E 3 with six binary places. Leibniz saw the potential of Conant (1931, p. 103) says of continuing the system beyond 26 = 64 and therefore Leibniz’s binary number system, providing a different way of counting. He even had a which has become such a basic medal made on which was inscribed Omnibus ex aspect of modern computing nihilo ducendis sufficit unum: ‘One suffices to produce systems: all [numbers] from nothing’ (Leibniz 1768, vol. III, ‘This curious system seems to p. 348). have been regarded with the Finger counting, as exemplified in the work of greatest affection by its inventor, the Venerable Bede (c. 673–735 AD), allowed who used every endeavour in his representations using various finger positions for power to bring it to the notice of (individual) numbers up to 1 000 000 (see Science and scholars and to urge its claims. But Society Picture Library 2004), thus demonstrating it appears to have been received another way of representing large numbers. However, with entire indifference, and to like all other such processes, it has a finite end. Even in have been regarded merely as a our own day we can see a system still developing which, mathematical curiosity.’ although not for counting in the most obvious way, is obviously a counting system. I refer to the labelling by the ISO (International Organization for Standardization) of ever larger, or ever smaller, multiples. Consider how our prefixes for thousandfold multiples have grown. A similar history can be found for how nomenclature for smaller and smaller divisions has developed. See Crossley (1987, pp. 17–20). The ancient Greeks had the word χίλιοι (khílioi) for ‘thousand’ from which we get ‘kilo’. Mega (from μέγας) is again Greek, but simply means ‘great’. This prefix was first recorded in print in 1868 by L. Clark according to the OED. Greek also gives us giga from γίγας, which has an equally simple meaning: ‘giant’. It was first recorded only in 1951 (according to the OED). Notice that there is no connection with numbers in these last two. The same applies to tera, again from Greek τέρας, but this time the original meaning is ‘monster’. As the new prefixes have come into relatively common use, a direct connection with numbers seems to have been reintroduced. Thus ‘peta’ for ‘1015 times’, which means ‘thousandfold repeated five times’, appears to come from the Greek for ‘five’: лέντε (pente). Likewise exa meaning ‘1018 times’ probably comes from the Greek εξά (hexa). Although we could continue this process using the Greek words for seven (hepta), eight (octo) and so on, we would again run into difficulty, since there is a very limited supply of ordinary Greek words for numbers, just as for Latin ones. In fact, the names in use in some quarters are ‘zetta’ and ‘yotta’, which do not come from the Greek letters ‘ζ’ (zeta) and ‘ι’ (iota, which is pronounced ‘yotta’ by Greeks), but

090604•Growing Ideas of Number 315 15

3/11/06 13:07:25

16

GROWING IDEAS OF NUMBER

from re-spellings of ‘sept-’ and ‘octo-’, the Latin-based prefixes for seven and eight. These correspond to having seven and eight blocks of zeroes, respectively, after the initial one (see Quinn 1992). In all of this invention of prefixes for higher and higher (and similarly for smaller and smaller) multiples, we are finding ourselves with exactly the same kind of problem that the Romans had. The only thing that has changed is that now the numbers are much larger. There is another way to attack the problem. Many years ago, Denvert and Oakland (1968, p. 311), asked: Is there some exotic language which provides readily voiced prefixes suggesting 15, 18 and so on with initial letters acceptable for use as abbreviated forms?

They then went on to suggest (p. 311): As an interim and somewhat retrograde step pending international agreement on such further prefixes, we can revert to the use of compound prefixes: the teraterametre (TTm: 1024 metres), for example, would be unambiguous and could cope immediately with cosmic distances.

Is this not exactly what happened with two-counting in Figure 1.1? Note, further, that we have no systematic way of advancing, all the new words have been ad hoc. We can nevertheless educe certain principles: • • • •

Terms are added as need arises. There is no obvious stopping place (until we run out of words). There are attempts at a regular continuation. There appears to be the possibility of continuing the process indefinitely.

I have already noted that peta and exa could progress to (h)epta and octo, but the idea of continuing indefinitely is, I claim, not so clear.

1.8  Counting forever How much further ahead are we than the people described by Gow (in Section 1.3)? In the process of learning to count, there comes a stage at which a child will be fascinated by the idea of larger and larger numbers. One often teases, asking, ‘What is the largest number?’ Although over a

090604•Growing Ideas of Number 316 16

3/11/06 13:07:25

COUNTING

17

period of time this may elicit answers ranging from ten to a hundred, to a thousand and then, perhaps, a million, we know that whatever number the child produces, we can always add one, and thereby get a larger number. Of course, that remark is not quite true. It is what we have always experienced, but if our last breath comes before we answer, we may fail. Likewise, there is, at any point of time, a number that is the largest named number. Philosophers of mathematics have even used such reasoning to question the use of infinity. Yesenin-Volpin (1960) was not particularly concerned with what the largest number might be; he simply took (an upper bound for) the number of heartbeats in a human lifetime. Clearly, we could count no further than that if we did not miss out any number in our counting. Such an attitude does not sit well with mathematicians. We assume that we can repeat a process that has been N OT E 4 performed before, at least in principle. Yesenin-Volpin’s approach (1960) A beautiful example comes because of the search led to the idea of ‘feasible’ by Dumont D’Urville for La Pérouse, who was lost numbers, i.e. numbers that are in while exploring the Southern Pacific in the nineteenth some sense reachable. Parikh century. Dumont D’Urville’s ship was beached on the (1971) developed an elegant island of Tonga for an extended period and his formal theory of such feasible botanist, La Billardière, quizzed the local population numbers. (These should be endeavouring to get them to count to higher and higher distinguished from the ‘feasible numbers. After exhibiting some patience, the Tongans functions’ discussed in ultimately took to inventing words. Finally, they Section 7.5.) appear to have become rude. Martin gives a secondhand account (which has been re-edited many times since), which reads in part: M. Labillardiere, however, has had the perseverance to interrogate the natives, and obtain particular names for numbers as high 1,000,000,000,000,000!! … 1,000,000,000 liaguee, which we take for liagi, and is the name of a game played with the hands, with which probably he made signs; … 10,000,000,000 tolo tafai (tole ho fáë), for which see the Vocabulary. 1,000,000,000,000 lingha (linga), see the Vocabulary: for a higher number they give him nava (the glans penis: for a still higher number, kaimaau (ky ma ow), by which they tell him to eat up the things which they have just been naming to him; but M. Labillardiere was not probably the first subject of this sort of Tongan wit, which is very common with them. In the other numbers he is tolerably correct, … (Martin 1827, vol. 2, p. [370]. The Vocabulary at the end of the book coyly uses Latin words for intimate parts of the anatomy.)

090604•Growing Ideas of Number 317 17

3/11/06 13:07:25

18

GROWING IDEAS OF NUMBER

In the twentieth century, the distinguished anthropologist, LévyBruhl wrote: It is usually admitted as a natural fact, requiring no examination, that numeration starts with the unit, and that different numbers are formed by successive additions of units to each of the preceding numbers. This is, in fact, the most simple process, and the one which imposes itself upon logical thought when it becomes conscious of its functioning. (Lévy-Bruhl 1979, English translation, p. 192.)

In the late 1970s I asked myself, ‘Where and when did the idea of an unending sequence begin?’ First I distinguished between concrete and abstract counting (see Crossley 1987, p. 14, and compare SchmandtBesserat, 1992, vol. 1, p. 196). In concrete counting one uses a physical representation, and the objects are tied to the numbers. Indeed many languages use classifiers to distinguish between the kinds of objects being counted. Here, as for Russell (see above, Note 1 in Section 1.1), number names are not essential. I then moved on to what I consider is the main problem: unending repetition. Consideration of the question led me to distinguish between the development of words for larger and larger numbers and the process of going to the next number, indeed the idea of such a process; for it is quite possible to be able to proceed in a certain way without being able to describe what is being done. An artisan or a musician, for example, may produce wonderful work without being able to describe the process. The problem has two facets: the idea of infinity; and the idea of order or arrangement of some kind, of placing one thing after another. (See Crossley 1987, p. 22 ff.) I therefore produced the terminology definite finite, bounded finite and unbounded finite; terms which all apply to collections that are not infinite. Definite finite indicates a collection of a specified finite size, for example seven or a million; bounded finite is used when the actual size of the collection may not be known but it is less than some definite number; and unbounded finite is used when there is some definite number bounding its size but we do not know what that number is. When we apply these concepts in the context of counting we have to consider repetition a specified number of times, repetition for no more than a specified number of times, and repetition for a number of times which ultimately has a bound. If, in the last case, we take Yesenin-Volpin’s bound (see above, in this section) then we can see that it is possible to distinguish between being able to go as far as we like and being able to go on forever. The latter is, as a matter of harsh reality, not practicable.

090604•Growing Ideas of Number 318 18

3/11/06 13:07:25

COUNTING

19

In order to go on forever we have to leave the world of practical matters and go into the world of the abstract. The basic ideas go back as far as the ancient Greeks, notably to the writings of Aristotle and Euclid, but their culmination is in deliberations of Dedekind and Peano at the end of the nineteenth century. I shall treat these in Section 6.2. In the next chapter, I shall look at how people of the past calculated with numbers.

090604•Growing Ideas of Number 319 19

3/11/06 13:07:25

5 7

2 18 93 36

3

7

4 6 8

2

1

CHAPTER TWO

5

Calculating by hand

2.1  Kinds of calculating Counting was clearly an essential ingredient in trade but, because objects are of different worth, calculation is also needed: if one banana is worth six apples, how much are seven bananas worth (the Rule of Three). Thus one principal stimulant to calculation was commerce. The other was the computation of the calendar. In China, this focused on the computation of the New Year, since the Chinese use a lunar calendar as opposed to the Western solar calendar. Calculations were also developed to compute other astronomical events such as eclipses. In Christian Europe, the calculation of the date of Easter was crucial. This led to the development of computus, which was the whole process of calculating the calendar (see GómezPallarès 1998). Counting on one’s fingers, or using Bede’s representation of numbers (see Section 1.7), provides a temporary record of a number. Having such a record is essential in the process of calculating (with numbers). In the simplest calculations, it is true that the mere repetition of the simple addition of numbers less than ten is sufficient. Nevertheless, societies in both East and West have always made use of databases of basic facts. Thus in teaching anyone to add, one has constant recourse to the results of adding together two numbers less than ten. Indeed this has become so familiar that I have never encountered addition tables being taught as such, although multiplication tables have been taught diligently for centuries. Nowadays electronic calculators can do much of the work. In this chapter we shall restrict ourselves to calculating by hand, in the sense of using one’s hands to move counters or other items, without other mechanical or electronic assistance. Other methods will be dealt with in Chapter 6. 20

090604•Growing Ideas of Number 320 20

3/11/06 13:07:28

C A L C U L AT I N G B Y H A N D

21

2.2 Counting rods 1 2 3 4 5 6 7 8 9 There are two very different ancient systems for calculating the results of the basic arithmetic operations: the abacus, which is recorded as long ago as 1500 BC in Egypt; and counting rods, which were in common Figure 2.1.  The vertical and horizontal representations use in China long before the beginning of of the numbers 1 to 9 by Chinese rod numerals the Christian Era. Ancient Chinese classics refer to counting rods. The earliest plausible mention is from the Warring States period (481–221 BC). The Chinese classic Dao De Jing (see, for example, Lao-tzu 1963) says: ‘Those well versed in calculation use neither counting rods nor texts’, implying that they were doing mental arithmetic. Pre-Christian era counting rods were found in August 1971 in Shaanxi province in China. Such rods were about 100 mm long and 2.5 mm thick (Shen et al. 1999, p. 12). They seem to have been used in conjunction with a counting board, a rectangular grid of squares but only very late examples of such boards have been found (see Figure 2.2). The columns of the grid correspond to the powers of 10. In 1980 Jock Hoe, an expert on ancient Chinese mathematics, showed me how to use them, on our dining table. With only the minimum amount of care, the rods can be kept to their columns without columns being drawn—at least for numbers that have no more than half a dozen or so digits, that is for numbers up to about one million. In order to distinguish the various powers of ten, the groups of rods are arranged alternately vertically and horizontally. Master Sun’s Mathematical Manual (Sunzi Suanjing, see Li and Du 1987, p. 10), of about the fifth century AD, says: Units are vertical, tens are horizontal, Hundreds stand, thousands lie down; Thus thousands and tens look the same, Ten thousands and hundreds look alike.

A number below five was represented by a group of rods; from six to nine, one rod was placed in the alternate orientation to represent five, with the remaining rods in the official direction. Thus the numbers 1 to 9 were represented as shown in Figure 2.1. Zero was not notated, but an empty space was left in the column. For example, 1024 would be written as – =  ||||. Of course there was still the possibility of confusion with

090604•Growing Ideas of Number 321 21

3/11/06 13:07:28

22

GROWING IDEAS OF NUMBER

100 024, 10 000 024, and so on, but not with 10 024, 1 000 024, and so on, because of the orientations of the rods. Unlike our present practice of working from right to left when adding, the Chinese worked from left to right (see Figure 2.3). This meant that any carried number was added after the addition of the numbers in a particular column had been performed. The same applied to the other basic operations.

Figure 2.2.  Counting board with counting rods Note that this is actually a Japanese counting board, but it is believed to be essentially identical with the Chinese boards, even including the characters for multiples of ten. This figure is based on an illustration in Smith 1923-1925, vol. 2, p. 172, which in turn comes from the Japanese work Sampõ Tengen Rokuo of 1714 by Nishiwaki Richyū. The characters across the top are not on the counting board; they simply indicate decimal places in the usual way: from left to right: thousands, hundreds, tens, units, tenths, hundredths, and so on. The characters in the right hand column indicate the roles of the numbers. For example, the result is developed in the top row, and the given number is originally placed in the second row. The remaining rows are initially for auxiliary numbers. If a root of a polynomial equation were being found, these would start out as the coefficients. All of these numbers, of course, would be expected to change in the process of calculation. The diagonal rod on the last character in the second row indicates that the whole row is a negative number: – 4351.25222. This is the convention for the diagrammatic representation. In practice the row would comprise red rods, since red ones were used for negative numbers, and black ones for positive ones (see Section 4.2).

090604•Growing Ideas of Number 322 22

3/11/06 13:07:29

C A L C U L AT I N G B Y H A N D

+

23

{ 74 85 96 + { 74 5 6 + { 1 1 85 6 + { 1 2 3 96 1 1 5 6

1 2 3 6

1 2 4 5

Figure 2.3.  Adding the ancient Chinese way

Subtraction on the counting board is clearly the reverse of addition so we shall not treat it here (see Li and Du 1987, pp. 12–13). Although we regard multiplication as being the obvious next arithmetic operation after addition, there is evidence that there were intermediate processes. The Rhind papyrus from about 1550 BC, which is now in the British Museum, presents such an intermediate process (see Robins and Shute 1987; Peet 1970, and Figure 2.4). In order to multiply, for example, 49 by 41, first write down, successively, doubles of one of the numbers; we use the 49. Then select those powers of 2 that add up to the second number, 41. Finally, add the corresponding multiply-doubled numbers. Now 41 = 32 + 8 + 1, so we add the corresponding numbers, 1568 + 392 + 49, obtaining 2009. Thus, multiplication has been reduced to doubling and adding. Later, in the work of al-Khwarizmi from about 800 AD (see Section 2.5), we find a treatment of doubling also, and although multiplication is treated immediately thereafter there is no indication of a connection as in the Rhind papyrus. In both al-Khwarizmi’s work and in ancient China,

We add the corresponding doublings for 41 = 25 + 23 + 20 = 32 + 8 + 1. Doubling table

Now add

2

0

1

49

49

21

2

98

–

2

2

4

196

–

23

8

392

392

24

16

784

–

25

32

1568

1568 2009

Figure 2.4.  The technique of multiplying, as in the Rhind Mathematical Papyrus, using powers of 2

090604•Growing Ideas of Number 323 23

3/11/06 13:07:29

24

GROWING IDEAS OF NUMBER

multiplication proceeds as today, except that it starts from the leftmost digit of the multiplier. In the process of addition using counting rods, there is no need to remember addition tables. Counting the rods themselves provides the answers for adding two numbers less than ten. Tables were used for multiplication. The Chinese version is the ‘Nine nines’ rhyme. This comprises the multiplication tables starting at ‘Nine nines are eightyone’ and descending from there.

2.3  The abacus The same techniques for the abacus applied in the West as in China. Early abaci seem to have used dust as the background. The OED says the word comes from the Greek word for ‘slab’ and the Hebrew word for ‘dust’. In the West, pebbles (apices) were used as the counters. The techniques employed were very similar (apart from the direction of the process of addition, and so on, as noted above). Chinese records suggest that the use of the abacus, which today comprises beads on wires, started in the thirteenth or fourteenth century AD and that its use was widespread in the fifteenth century. The process of moving from counting rods to the abacus was aided by the large number of mnemonic verses developed to simplify calculations. For example, in Yang Hui’s ‘converting to decimal’ rhyme in his Arithmetic methods for Daily Use (see Lam 1972) we find the way to compute sixteenths: Finding 1, omit a place 625; Finding 2, go back a place 125; Finding 3, write 1875; Finding 4, change it to 25; Finding 5 is 3125; Finding 6 liang, price is 375; Finding 7, put 4375; Finding 8, change it to 5

(.0625) (.125) (.1875) (.25) (.3125) (.375) (.4375) (.5)

In this algorithm, we see that 161 = .0625 , 162 = .125 , 163 = .1875 and so on. Note that the use of a decimal point in our description is anachronistic. The decimal point we have shown should be thought of as simply marking a specific place between columns. Thus 161 of 10 000 is 625, while 161 of 1 000 000 is 62 500. The Chinese did not use a decimal point as such, but measured divisions in units, tenths of units (fen), hundredths of units, and so on (see Note 14 in Section 5.4).

090604•Growing Ideas of Number 324 24

3/11/06 13:07:29

C A L C U L AT I N G B Y H A N D

25

In the Middle Ages in the West, we have much N OT E 5 more information about the use of calculations for the calendar rather than for commerce. Arithmetic was Gerbert’s numerals may appear to very limited and before that time basic addition and be different from both Arabic and subtraction were supplemented by the use of ready Hindu-Arabic numerals. However, reckoner tables for multiplication and division. Beaujouan (1991) has concluded Unfortunately these tables were bedevilled both that this is because, on the abacus by errors in computation and errors introduced by counters, the orientation of the copyists (Murray 1978, p. 156). The use of the abacus numerals was irrelevant. Compare did not became popular in Europe until the tenth our 2 and 3 with the Modern century (Murray 1978, p. 155). Gerbert (who became Arabic ٢ and ٣. (Try turning the Pope Sylvester II in 999), used apices (counters) on his modern Arabic numerals through abacus, which had signs on them representing the ninety degrees.) numbers from one to nine. These signs came from the forerunners of the Hindu-Arabic numerals that we use today (see Section 2.5), but no calculations (on paper or parchment) were made with the numerals themselves. In this case, addition tables provided the answer to sums of two numbers below ten. It was not until the twelfth century that the Hindu-Arabic numerals 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 began to be used in Europe. (I have written them in this order because that is how they appeared in manuscripts. See also Note 5.) Before that time, the abacus or ready reckoner tables were used.

2.4  Roman numerals What about Roman numerals? Mediterranean peoples had various systems of numerals that did not use place notation. This applies not only to the Romans but also to the Greeks and the Jews. Such numbers were not used to perform calculations; they were used for recording numbers, including the results of calculations. We have very little information on how the calculations themselves were performed before about 1000 AD. Bede (see Science and Society Picture Library 2004) appears to have had methods of working out results on his fingers. However, as there are clear rules for how to write down a number in Roman numerals, it follows that there are, abstractly speaking, rules for calculating with Roman numerals. General algorithms for working with Roman numerals were provided last century by Detlefsen et al. (1976). Even a brief study of their paper will show how much easier it is to use an abacus.

090604•Growing Ideas of Number 325 25

3/11/06 13:07:29

26

GROWING IDEAS OF NUMBER

13th Century Modern Arabic Hindu-Arabic

٠١٢٣٤٥٦٧٨٩ 0

1

2

3

4

5

6

7

8

9

Figure 2.5.  Early (c. thirteenth century) and Modern Arabic numerals

E x ercise 3 Write down rules for adding and multiplying Roman numerals less than (a) ten and (b) one hundred. Do not forget about L (50) and expressions such as IV and XC.

Clearly, the abacus has persisted as is evident in many Chinese shops. Why then did our present system take over?

2.5  Hindu-Arabic numerals and algorism N OTE 6 There is a certain irony about the direction of Hindu-Arabic numerals. Arabic writing goes from right to left as opposed to English writing. But the numbers go left to right from largest to smallest. Therefore, when writing down a large number in an Arabic text, the writer has to predict how much space to leave in which to insert the number. Thus

١٩٢٣ is Arabic

for ‘1923’ and appears as it would be written in an Arabic text. (The forms of some numbers in Modern Arabic are somewhat different from the forms we use in the West today. See Figure 2.5.)

090604•Growing Ideas of Number 326 26

We have already mentioned the numerals we use today. These are properly known as Hindu-Arabic numerals since they originated in India (see Needham 1959, vol. 3, part 1, p. 10) and travelled west, eventually arriving in the Arab lands of the Maghreb (North Africa) before crossing the Mediterranean into Spain and Italy. This section is called ‘Hindu-Arabic numerals and algorism’. What is the difference? The HinduArabic numerals are symbols (or characters or digits). They are used in combination with place notation, which allows us to write down indefinitely large numbers. Unlike Roman numerals, Hindu-Arabic numerals change their meaning. The ‘2’ in 1234 means ‘two hundred’, while the ‘2’ in 4321 means ‘twenty’. Consequently, learning the way to write down numbers was quite arduous in the Middle Ages. The word ‘algorism’ comes from the name of alKhwarizmi who lived c. 780–850 AD in Khwarezm (now Khiva), Uzbekistan. He wrote a book on algebra, which has been preserved in Arabic (in the Bodleian

3/11/06 13:07:29

C A L C U L AT I N G B Y H A N D

27

Library in Oxford) and in English translation (Rosen 1831), and 0 4 7 one on arithmetic, which is only preserved in Latin versions (see Crossley and Henry 1990; Folkerts 1997). The beginning of his book on arithmetic takes a significant amount of space to present the place notation, and to describe how to write down the strange new symbols. The rest of the work is devoted to algorism, the art Figure 2.6.  Examples of of calculating with the new numerals, and how to manage the how early Hindu-Arabic numerals could be place notation in the calculations. fraudulently modified in The numerals can be used independently of the place accounts by changing 0, 4 notation. Indeed, they were used by Gerbert shortly before the and 7 into 8. year 1000. (They also appear in two Spanish manuscripts from 974 AD (see Cordoliani 1951) and these are their earliest occurrences in Europe.) Gerbert only used them to label the apices of the abacus. Just as the numerals can be used without algorism, so too N OT E 7 algorism can be used with any ten (mutually different) To see how hard it is to learn how signs that we might use to represent 9, 8, 7 … 2, 1, 0. to work with the new place Learning to use the new algorism and the Hindunotation, try the exercises below, Arabic numerals required a significant investment of which are due to Richard Platek in time (see Exercises 4 and 5). In addition, there was a graduate mathematics seminar at opposition to the new numerals in accounting circles Cornell University in 1972. We do (see Menninger 1969, especially pp. 424–31), because some numerals could easily be modified to look like arithmetic ‘backwards’. others by adding an extra small mark (see Figure 2.6). The consensus view is that Hindu-Arabic numerals took two centuries, that is until about 1400 AD, to become widely accepted. (See any book on medieval palaeography, for example, Bischoff 1979; Bretholz 1926; Prou 1924.)

E x ercise 4 To add 456 and 678, first write the numbers down backwards: 654 and 876. Now put one under the other in the obvious way. Perform the addition as usual, remembering that in carrying you will carry to the right instead of to the left. That was not very hard. Now try the following exercise.

E x ercise 5 To multiply 456 and 678, again write the numbers backwards as for Exercise 4. This may prove surprisingly more difficult than Exercise 4.

090604•Growing Ideas of Number 327 27

3/11/06 13:07:30

28

GROWING IDEAS OF NUMBER

N OTE 8 A similar situation obtains today. If a handheld calculator is used, then all that remains is the answer unless the calculator has a paper tape. In that case, a record of the calculations performed may be retained. This was a major problem when such calculators did not work to as great a degree of precision as they do today. Twenty years ago, performing a calculation in a different order might produce a different result because of rounding errors. A simple test is to take the square root of a number, say, twenty or more times and then square the result the same number

The advantage of algorism over the abacus was that the whole calculation could now be recorded on paper, or rather a slate or parchment. The calculation could then be checked by a third party. This was not true of an abacus calculation, where usually all that remained at the end of a calculation was the answer (see Note 8). Of course, in a complicated calculation using Hindu-Arabic numerals it may sometimes be difficult to see the exact order in which all the steps were performed. Knowing the actual algorithm that has been used is a great help in such cases. Despite the effort required, Hindu-Arabic numerals and algorism did come to dominate Europe by the fifteenth century AD. In hindsight, it is clear that they greatly facilitated calculating numbers to a high degree of precision. This was particularly important for the computation of the calendar, one of the two principal uses of calculation at that time.

of times. On an old calculator of mine from about 1980, taking the square root twenty-three times, starting from 2, yielded the answer exactly 1. See also Deakin (2004).

090604•Growing Ideas of Number 328 28

3/11/06 13:07:30

5 7

2 18 93 36

3

7

4 6 8

2

CHAPTER THREE

5

Measuring

3.1  From lines to numbers Measuring transforms lengths and other quantities into numbers. Therefore, on the one hand, it allows us to reproduce lengths and so on; on the other hand, just like counting, measuring is a homomorphism from (aspects of) the world to numbers. In this chapter, I shall consider how the idea of measuring, using numbers, developed. In particular I shall investigate certain logical discrepancies that arose and were, perhaps, more apparent in the past than they are today. Once we have counted a collection of objects and determined the number of objects in the collection, it is not too hard to reproduce that number. We can do it by writing down the numeral, or we can use Russell’s method (see Note 1, Section 1.1) and give another collection of objects of the same number: tokens for those objects. Reproducing lengths requires having some sort of standard. Parts of the human body gave rise to such standards, though there was much more variation in them than we would accept today. Such measures are the cubit and the foot. The OED gives as etymology for the cubit, Latin cubitum, the distance from the elbow to the fingertips. The foot (12 inches or 304.8 mm) was originally based on the length of a man’s foot. Clearly there was plenty of room for variation in such standards. The French are responsible for a fundamental contribution: the introduction of the standard metre (literally meaning ‘measure’) for length measurements, but that was very late (1791, just after the French Revolution) and it is still not universally adopted since the USA has not taken it up. (The only other countries that have not officially taken it up are Liberia and Myanmar, see US Metric Association 2006.) Originally a metre was intended to represent one ten-millionth of the length of a quadrant of the meridian, and defined by reference to a platinum-iridium standard [rod] kept in Paris (OED). 29

090604•Growing Ideas of Number 329 29

3/11/06 13:07:32

1

30

GROWING IDEAS OF NUMBER

Later it was defined in terms of the wavelength of a certain orange-red light. For other measures, counting was again used: an area is determined by the number of unit squares that will fit into it, a volume by the number of unit volumes that will fill it. Happily for us there is a natural notion of addition of quantities of the same kind, and this extends to multiplication by a number, which is regarded as repeated addition (but see Note 9). However (see Section 3.3), the arguments to establish the volumes of quite simple objects, such as a tetrahedron, take us into considerations of the infinite, in this case, the infinitely small. The identification of the geometric line with the N OTE 9 real numbers (or the ‘real number line’ as it is often We have other ways of adding called) is chiefly associated with Descartes (1596– lengths. Suppose we travel 400 1650, see Descartes 1979). The ancient Greeks had no kilometres due north from such identification, nor did the ancient Babylonians Adelaide, and then 300 kilometres (see Crossley 1987, p. 260), and the ancient Chinese due west (thereby arriving did not need, nor did they use, such an identification. somewhat to the south of Uluru), They simply calculated to the required degree of what is our distance from precision, naming each tenfold division of a unit as Adelaide? The answer is 500 need arose (see Section 2.3). This does not mean to say kilometres by Pythagoras’s that they were unaware of the possibility of subdividing theorem. indefinitely far. Indeed Gongsun Long and his coauthors who were Mohists (followers of Master Mo) made the following pronouncement in the Warring States period (481–221 BC) (see Li and Du 1987, p. 21): ‘A one foot [chi] long stick, though half of it is taken away each day, cannot be exhausted in ten thousand generations.’ That is to say 1 - 12 - 14 - 81 f - 21n never reaches zero, however large n may be. They were also able to calculate square and cube roots to any required degree of precision. Their methods are described in the Nine Chapters on the Arithmetical Art, which appears to have been written, originally, before 100 BC (see Li and Du 1987, p. 50; Shen et al. 1999, pp. 204 ff.).

3.2  Incommensurability The ancient Greeks rigorously distinguished between different kinds of quantities, in particular, numbers, lengths, areas and volumes. They therefore distinguished between arithmetic and geometry. There is a certain asymmetry here. While the Greeks recognised that one length can be measured by another, and therefore that a number can be associated with this process, they did not assume, nevertheless, that for each length there

090604•Growing Ideas of Number 330 30

3/11/06 13:07:32

MEASURING

31

was a corresponding number. The first place where this becomes apparent is in the relation between the side and the diagonal of a square: they are incommensurable. On this subject, two very different historical interpretations are current. The older one is that the discovery of the incommensurability was a disaster, the other that it was an opportunity. Why should incommensurability be a disaster? Aristotle (384–322 BC), who lived considerably later than Pythagoras, is our principal source of knowledge of the Pythagorean School (of about the sixth century BC). The Pythagoreans believed that numbers were the fundamental constituents of everything. But he [Plato] agreed with the Pythagoreans in saying that the One is substance and not a predicate of something else; and in saying that the Numbers are the causes of the reality of other things he agreed with them [the Pythagoreans]. (Aristotle, Metaphysics, 987b, see Aristotle 1924.)

Therefore, to find a length that could not be numbered was a disaster. Legend has it that Hippasus, who is reputed to be the first to discover irrational numbers, was thrown overboard to drown (see Crossley 1987, p. 110). However, works last century by Knorr (1975) and his follower, Fowler (1987), contend that the discovery of incommensurable lengths was a stimulus to the growth of geometry. The evidence for either case is meagre, but it certainly shows how different the approach of the ancient Greeks was from the arithmetised version of geometry that has been taught in the West, starting from Descartes. (For a clear and full discussion of this separation see GrattanGuinness 1996.) The classic example of incommensurability is given by the side and diagonal of a square. Consider the two statements: There is no fixed length (however small) that will measure both the side and the diagonal of a square (as an exact whole number of units each).

2 is irrational.

(3.1) (3.2)

Notice the difference. Statement (3.1) belongs to the language of geometry and statement (3.2) to number theory.

090604•Growing Ideas of Number 331 31

3/11/06 13:07:32

32

GROWING IDEAS OF NUMBER

3.3  Comparing magnitudes The basic problem is how to compare lengths. Since we measure one length in terms of another, we obtain a ratio, not a number. This simple-looking sentence needs some analysis. There is a hidden assumption here, which is now known as Archimedes’s axiom, though Archimedes himself calls it a ‘lemma of Eudoxus’. (Eudoxus lived c. 408 BC, Euclid c. 300 BC, and Archimedes c. 287–212 BC.) The word ‘lemma’ is usually used in modern mathematics to mean ‘auxiliary theorem’ (Hilfsatz in German). Eudoxus, Archimedes and Euclid all took this particular lemma as an axiom—something assumed. Indeed, that is what the word ‘lemma’ means in Greek: ‘something assumed or taken’. What is the axiom? In modern terms it is the following: given two (positive) magnitudes a, b then there is a number (positive integer) n such that na > b. More precisely, when a is added to itself a sufficient number of times, it eventually exceeds b. I shall discuss this axiom further in Section 6.5. I have just used the word ‘magnitude’ and this is consistent with Euclid: we compare lines with lines, numbers with numbers, solids with solids, and so on. Between two magnitudes of the same kind there is a relation which is called a ratio. The Greeks did not allow the equating of ratios between different kinds of quantity. Thus (apart from the obvious anachronism), one could not say that the ratio of one metre to one centimetre is equal to the ratio of 100 to 1. Indeed the very question of what equality of ratios means was difficult. The modern interpretation of ratio is to take the equivalence class. Thus, in the modern view, the ratio 12 : 6 is the set of all pairs 〈2a, a〉, where the as are of the same type). (Compare Russell’s approach to ‘having the same number’ in Chapter 1, see Note 1 in Section 1.1.) Next, there is the problem of comparing ratios. For whole numbers this is easy enough as each is a multiple of 1. However, when we look at incommensurable lengths, how do we compare the length of the side of a square to that of its diagonal? The solution was provided by Eudoxus and is found in Euclid (1956) Book XII. He does not say directly what it means for a ratio a : b to be equal to a ratio c : d. Instead this is defined by saying that the first is neither greater than nor less than the second. Translated m into modern terms, a : b < c : d if there is a rational number n such that a c m < n ≤ . There are two anachronisms here. The first is that we are b d a makes sense), and the looking at fractions not ratios (and assuming b second is that the ancient Greeks did not compare ratios of different kinds

090604•Growing Ideas of Number 332 32

3/11/06 13:07:32

MEASURING

33

(see above and Grattan-Guinness 1996). When we cross-multiply the above inequalities we arrive at the condition that Eudoxus and Euclid used; namely, a : b < c : d if, and only if, there are multiples m and n such that na < mb and md ≤ nc. Finally, a : b = c : d if, and only if, a : b # c : d and a : b $ c : d. These definitions allow for any two of a, b, c and d being incommensurable, and do not mix ratios of different kinds of magnitudes. Two millennia later Dedekind used essentially the same idea to define real numbers in terms of those rational numbers that are less than and those that are greater than or equal to the required real. (See Section 5.1.) There is no great difficulty in extending from the line to the plane. Any rectilinear figure can be divided up into triangles and it is easy to show, as Euclid did, that the area of a triangle is half the area of a rectangle of the same base and altitude. However, this kind of comparison becomes a serious issue in determining volumes, even one as apparently simple as that of a tetrahedron. The method of exhaustion is necessary to prove the result, and this is what Euclid uses in his Book XII (see Euclid 1956, vol. 3). Euclid’s Proposition XII.7 is disarmingly simple looking. In it he proves that a triangular prism can be divided into three pyramids (tetrahedra) of equal volume (see Figure 3.1). This immediately gives the volume as 1 3 × base area × the altitude. However, this proposition depends on Proposition XII.5, which says that ‘pyramids that are of the same height and have triangular bases have volumes that are to one another as the bases.’ (In modern language this states that the volumes are the same multiples of each other as their bases.) To prove this result Euclid uses Eudoxus’s lemma and the method of exhaustion (see Euclid 1956, vol. 3, p. 373). That is, Euclid takes smaller and smaller parts and then shows that a given volume cannot be greater than, nor less than, a given one. (Heath’s explanation, in Euclid 1956, pp. 374–8, is well worth studying.) It was not until Dehn (1900) that it was established that such infinite con­ C siderations (as in the method of exhaustion) were essential. Indeed, in 1900, Hilbert A B listed this as the third in his list of the twenty-three great unresolved problems of mathematics (see Hilbert 1901–02). Dehn D introduced a condition (involving the Dehn invariant) that determined when two solids could be cut up into rectilinear pieces and F E re-assembled to make the other. A cube and a tetrahedron have different Dehn Figure 3.1.  Dividing a triangular prism into three equal invariants, so they cannot be re-assembled pyramids (tetrahedra)

090604•Growing Ideas of Number 333 33

3/11/06 13:07:33

34

GROWING IDEAS OF NUMBER

appropriately. (It was even later, see Sydler (1965), that it was proved that Dehn’s condition was also sufficient.) Note that Dehn’s work solved the geometric problem by algebra, not by geometry. An excellent account of this work can be found in Stillwell (1998, pp. 159 ff.). Incidentally, the Chinese had the same difficulty as Euclid in proving the volume of a tetrahedron, and this was eventually done by Liu Hui in the fourth century AD (see Li and Du 1987, pp. 71 ff.; Shen, Crossley and Lun 1999, pp. 256 and 269 ff.). Liu too had to deal with similar infinite considerations.

3.4 The Euclidean algorithm Let us now return to statements (3.1) and (3.2). From the above discussion it should be clear that the Greeks would not have regarded these statements as presenting the same question. The solution to (3.1) given in Aristotle (Prior Analytics i. 23.41a, see Aristotle 1928–52) refers to ‘odd numbers being equal to even numbers’, and Aristotle has generally been regarded as presenting the standard number theoretic argument for (3.2), which we usually encounter as: p 2 p Let q be a fraction in its lowest terms such that d q n = 2. Then 2q2 = p2, and therefore 2 divides p, say p = 2r. Therefore 2q2 = 4r2

and q2 = 2r2, so 2 divides q. But then 2 divides both p and q. p Therefore q was not in its lowest terms, which contradicts our assumption.

Another argument hinges on the fact that the number of prime factors of 2q2 must be odd, while that of p2 is even. Therefore even must be equal to odd, which is impossible. Notice that this latter argument is of a rather different nature. It does not discuss the magnitude of p and q but rather the number of factors that the two numbers have. There is no extant geometric argument for showing that sentence (3.1) holds. However it can be proved more geometrico. I use the version of the Euclidean algorithm given for number in Euclid Book VII, Proposition 1, and transferred to magnitudes in Book X, proposition 3 (see Euclid 1956, vol. 2, p. 296 and vol. 3, p. 14). Euclid requires the magnitudes to be commensurable, that is to say, to be (possibly different) multiples of the same basic length. Variations of the argument can be found in Heath’s notes in Euclid (1956, vol. 3, p. 19), who quotes Chrystal (1886–89, vol. 1, p. 270) and also in Fowler (1987, p. 95).

090604•Growing Ideas of Number 334 34

3/11/06 13:07:33

MEASURING

Consider the diagram in Figure 3.2 in which the larger square has side S = s + d, where s and d are the side and diagonal, respectively, of the small square CFEG. The diagonal of the larger square is then easily seen to be d + 2s. Therefore D + S : S = ]d + 2sg + ]s + d g : d + s .

A

E G

F C

(3.3)

Writing these ratios, in an anachronistic way, as fractions, we have

B

35

D

Figure 3.2.  A geometric argument for the incommensurability of the side and diagonal of a square (from Euclid 1956, vol. 3, p. 19)

]D + Sg 2d + 2s + s s = = 2+ . S d+ s d+ s

(3.4)

s The last fraction, , is inverted from the original one so, if we continue d+ s the process, we get

s1 d+ s s = 2 + d1 + s1 ,

(3.5)

where s1 and d1 are formed by rotating the large square so that CD lies along CA and mimicking the construction of the small square to obtain the diagram in Figure 3.3. Clearly, this process never terminates A so D + S and S, and therefore D and S, are incommensurable. The process I have just described also gives continued fractions to which I turn in Section 3.6. B The process of the Euclidean E algorithm can, nevertheless, be applied even F when the two magnitudes are not com­ mensurable. Given two magnitudes a and G C b, where we assume b < a, we can subtract b a finite number of times, q0, such that

D

Figure 3.3.  Rotating the square in Figure 3.2



a = bq0 + r0 where 0 # r0 1 b

(3.6)

If r0 ≠ 0, we can reciprocally subtract r from b and obtain a new positive integer, q1, such that

090604•Growing Ideas of Number 335 35

b = r0 q1 + r2

(3.7)

3/11/06 13:07:33

36

GROWING IDEAS OF NUMBER

and successively obtain

r0 = r1 q 2 + r2 r1 = r2 q3 + r3 f rn = rn + 1 q n + 2 + rn + 2 .

(3.8) (3.9) (3.10)

The process terminates if an ri is zero. The process of reciprocal subtraction was called anthyphairesis in Ancient Greece. Fowler (1987) also uses the term for the sequence of remainders (see Section 3.6). However, von Fritz has an argument based on the pentagon, rather than the square, which uses the same unending repetition (see von Fritz 1945, p. 257; or Crossley 1987, p. 131).

3.5  The geometric line This leads us to the question of the appearance of the geometric line. It is easy to take some measure as a unit, and then to measure off all integer multiples of that unit. I shall ignore the question of where negative numbers come from. That is a difficult question to which I shall only briefly return in Section 4.2. Then it is easy to construct fractions of these multiples by drawing similar triangles. See Figure 3.4. The numbers that I have now represented on the geometric line are all rational numbers. However, we already know that there are other lengths that have not yet been represented, for example, 2 . This takes us to the question of continuity. Intuitively there is no difficulty in thinking of the geometric line as being continuous. One draws a line in a continuous fashion. Nevertheless, we have n no evidence that the actual fabric of our m world is continuous. While it is logically possible that there is no smallest length, it is equally logically possible that there is. Obviously, such a length might be very hard 1 to determine and it probably would be even harder to establish that there was no shorter m 1 1 n n length. In the case of there being a shortest length (in the world) the diagonal of a Figure 3.4.  Constructing fractions

090604•Growing Ideas of Number 336 36

3/11/06 13:07:33

MEASURING

37

particular square could be commensurable with the side of that square or, much more likely, we should not be able to find an exact (Euclidean) square complete with diagonal in the real world. The mathematical consequences are interesting. We could rework all of our mathematics in terms of discrete intervals. This would be an enormous enterprise. Moreover, it would be largely unprofitable. In the case of relativity theory, the places where answers are obtained that are significantly different from those obtained using ordinary physics are relatively esoteric. In the case of using a theory of discrete space with a tiny shortest distance, they would be even more difficult to detect. Further, given our success with analysis and calculus, by going to a discrete theory we would be giving up a vast fund of useful techniques and results. The continuous theoretical world of our ordinary mathematics is, at the very least, an excellent approximation to our putatively discrete world. Clearly, the ancient Greeks were concerned about the nature of space and matter, as the well-known paradoxes of Zeno (c. 490 BC) show (see Salmon 2001). It was not until the nineteenth century that continuity of the real line became a mathematically important question. The seventeenth century had seen the introduction of calculus by Leibniz and Newton. Leibniz, in particular, used infinitesimals in establishing some of his results. The problem lying in wait was that infinitesimals cannot exist because of Archimedes’s axiom (see Section 3.3 and Section 6.5). The rational number line has gaps. It was Dedekind who produced the solution to this problem that we use most often. I shall treat this at the end of Section 5.1.

N OT E 1 0 A curious fact about rational numbers m n : The real numbers that can be expressed as repeating infinite decimals are exactly the rational numbers. First note the use of the word ‘infinite’ here. Every rational number that we regard as being represented by a finite decimal can also be represented by the same decimal with an infinite string of zeroes at the end. Therefore it is expressible as a repeating infinite decimal. A little thought shows that 1 we only need to consider numbers of the form . Suppose k has m digits then, in the long division k process, the largest number we have to consider dividing into is 1 followed by m zeroes. This division has only a finite number of outcomes in terms of the dividend and the remainder. Therefore if the long division continues indefinitely, one of those positions must be repeated (and so must all the subsequent divisions). At that point we have determined the repeating part of the decimal.

090604•Growing Ideas of Number 337 37

3/11/06 13:07:34

38

GROWING IDEAS OF NUMBER

N OTE 10 (continued) For the other direction consider an infinite decimal that does not end in a string of repeated zeroes and view the decimal as a geometric series. Clearly it is sufficient to consider a number r between zero and one. I shall reduce considerations even further. (The dots above the digits indicate the segment that repeats.) r = .a1 a 2 fa m bo 1 b 2 fbo n =



.a1 a 2 fa m + 10- m + 1 `.bo 1 b 2 fbo nj

(3.11)

So consider just the part in brackets. .bo b fbo 1

2

n

= 10- n b1 b 2 fb n + 10- 2n b1 b 2 fb n + 1003n b1 b 2 fb n + f



= 10- n b1 b 2 fb n ]1 + 10- n + 10- 2n + 10- 3n + fg.





(3.12) (3.13)

Notice there is no decimal point in equations (3.12) and (3.13). Now sum the geometric series (see Section 0.2.1) in the brackets as 1 + 10- n + 10- 2n + 10- 3n + f =



1 , 1 - 10- n

(3.14)

which is a rational number. Therefore our original r is indeed rational. In particular, it follows that 0.9o = 1, and similarly for any other number ending in repeated 9s.

3.6  Continued fractions I conclude this chapter with a further application of the Euclidean algorithm, introduced in Section 3.4. This gives rise to continued fractions. John Wallis, in his book Opera Mathematica (1972), laid some of the basic groundwork for continued fractions. It was in this work that the term ‘continued fraction’ was first used. An excellent treatment can be found in Chrystal (1888–89, vol. 2). If we consider the Euclidean algorithm sequence

a = bq0 + r0 b = r0 q1 + r1 r0 = r1 q 2 + r2 r1 = r2 q3 + r3 f rn = rn + 1 q n + 2 + rn + 2

(3.15) (3.16) (3.17) (3.18) (3.19)

and divide through to isolate the qi we get, equivalently,

090604•Growing Ideas of Number 338 38

3/11/06 13:07:34

MEASURING



r a = q0 + 0 b b r b 1 r0 = q1 + r0

39

(3.20) (3.21)

r0 r2 r1 = q 2 + r1 r3 r1 r2 = q3 + r2 f rn rn + 2 rn + 1 = q n + 2 + rn + 1

(3.22) (3.23) (3.24)

Define the anthyphairesis of a and b as EA(a; b) = [q0, q1, q2 …]. The process involved is essentially that for finding the continuous fraction a corresponding to , though that is usually expressed as b

a = q0 + x0 b 1 x0 = q1 + x1

(3.25) (3.26)



1 x1 = q 2 + x 2

(3.27)



1 x 2 = q3 + x3 f

(3.28)

where 0 ≤ xi < 1, so each qi, for i > 0, is > 0. Therefore

a = q0 + b q

1 1

+

1



(3.29)

1 q2 + q f 3

a If this sequence of qs is finite, then is rational and the converse is b also true. (See, for example, Fowler 1987, p. 311.) Lagrange (1770) showed that the continued fraction of a ratio of the form a +  b  : c where a, b, c are non-negative integers (that is, of a quadratic surd) is eventually periodic (and so, of course, is the anthyphairesis of the corresponding ratio). For example, EA^ 2 : 1h = [2, 1o , 1, 1, 4o ] where the sequence 1, 1, 1, 4 repeats. The proof Fowler gives is by means of a complicated geometric construction. (Fowler even showed that EA^ m : n h is periodic.) On the other hand it is not very difficult to see that any repeating EA characterises a pair of the form a + b  : c where a, b, c are non-negative integers, that is to say, a quadratic surd.

090604•Growing Ideas of Number 339 39

3/11/06 13:07:34

40

GROWING IDEAS OF NUMBER

a is of the form b h! k v (I shall show how to obtain the h, k and v below). For at the m stage where the EA repeats we have First observe that if EA(a : b) is periodic, then

x = 6 po , q, f, z, xo @



(3.30)

At each stage x is of the form

linear expression1 in x ÷ linear expression2 in x

that is,

x=

so

L1]xg L 2]xg

(3.31)

xL 2]xg = L1]xg



(3.32)

which is a quadratic equation and, since all the coefficients are natural numbers its root will be of the form equivalent to q0 +



u! v w . The original fraction is then 1

q1 +

1

q2 +

1

q3 + f

1 ^u ! v h /w

This easily rationalises to the form



d! e v which, by multiplying f! g v

top and bottom lines by f " g v leaves us with an expression of the form

^h ! k v h , where h, k, m, and v are all natural numbers. This completes m

the proof. (For further details see Chrystal 1888–9, vol. 2.) Unfortunately, the basic arithmetic operations on continued fractions are very difficult to calculate if the result is to be a continued fraction. Indeed, algorithms for them were introduced only at the end of last century. Gosper (see Beeler et al. 1972) developed some algorithms, but see also Vuillemin (1990), and Liardet and Stambul (1998). Therefore, although continued fractions are useful for describing certain numbers (see, for example, Weisstein 1999), their practical use is presently limited.

090604•Growing Ideas of Number 340 40

3/11/06 13:07:35

5 7

2 18 93 36

3

7

4 6 8

2

CHAPTER FOUR

5

Extending numbers

4.1  Classifying numbers There are two natural approaches when building a system: bottom-up and top-down. This is also the case for our number systems. In the West there has always been a great deal of effort invested in trying to classify numbers. I have already discussed some of the impact of the discovery of incommensurable numbers. In the period from the beginning of the Christian era until the middle of the sixteenth century, systems were developed to classify numbers further. In the work of Nicomachus (fl. AD 85, see D’Ooge 1972, pp. 241 ff.) we find a great deal of attention being paid to square numbers, rectangular numbers, triangular numbers, pentagonal numbers and so on. These are the so-called ‘figurate numbers’ (see Definition 1 below). This practice continued up to the sixteenth century. Thus we find Maurolico (1575) producing a huge table of his DEFINITION 1 classification at the beginning of his first volume. From a twenty-first century point of view it seems inevitable A figurate number is a number that such a classification project would be unending. It that can be represented by a was useful in trying to work out an abstract theory of regular geometrical arrangement numbers, but seems to have been of little practical use. of points, equally spaced. When It is interesting to note that the Chinese had none of the arrangement can be made into these classifications. The differences between East and a regular polygon, the number is West can be epitomised, not too unfairly, by saying that called a polygonal number. Thus the West was concerned about the theory and the East we have triangular, square, about the practice of number. pentagonal, and hexagonal As well as figurate numbers, perfect and numbers. amicable numbers were considered from an early date, Rectangular numbers may have indeed from the time of Pythagoras (late sixth century sides of different lengths, and, in BC). A perfect number is one that is the sum of its particular, prime numbers form a proper divisors (including 1). Two numbers a and b sub-class of rectangular numbers. 41

090604•Growing Ideas of Number 341 41

3/11/06 13:07:37

1

42

GROWING IDEAS OF NUMBER

are amicable numbers if each is equal to the sum of the factors of the other. The sum excludes the number itself. The numbers 220 and 284 form the first amicable pair. The first perfect number is 6, as is noted by the first century AD Pythagorean follower, Nicomachus (see D’Ooge 1972, p. 209), since 6 = 1 + 2 + 3. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next two are 496 and 8128. All of these were known (in the West) before the Christian era. The search for larger perfect numbers generated a great deal of mathematical activity. The first major result was: Theorem 1 k is an even perfect number if, and only if, it has the form 2n – 1(2n – 1), and 2n – 1 is prime. N OTE 11 Marin Mersenne (1588–1648) stated, incorrectly as it turned out, in the preface to his Cogitata Physica-Mathematica (1644) that the numbers 2n(2n – 1) were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and composite for all other positive integers n < 257. The correct list is 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127.

The proof of the ‘if’ direction follows easily by considering the geometric series (see Section 0.2.1) 2n - 1  = 2n – 1. For the 1 + 2 + 22 + 23 + … + 2n – 1 =  2 -1 reverse direction see, for example, Caldwell (1994– 2006). The problem of classifying numbers remains unsolved, or to be more precise, the classifications become more and more esoteric. Matters will probably continue thus, since the theory of numbers has its own fascination. However there were strong motivations to develop numbers from the bottom up and this introduced new kinds of numbers.

4.2  Positive and negative numbers The ancient Chinese added negative numbers to the positive ones long ago. In chapter 8 of the Nine Chapters on the Mathematical Art (see Shen et al. 1999, p. 404; Li and Du 1987, p. 50) which dates from about 100 BC, we find black counting rods being used for negative numbers (fu: negative, in debt) and red rods for positive numbers (zheng: positive, affirmative). So this was exactly the opposite convention to our using red for deficits. (Although zero had not yet been introduced, that did not cause a problem. See Section 2.2 and below.) The numerals were used as would be expected. In the West, negative numbers came into acceptance only very slowly. According to Montucla (1798–1802), the eighteenth-century historian of mathematics, Cardano (1501–1576) was the first to distinguish positives and negatives. (See Cardano 1968.) This was in the context of solving quadratic equations. Very slightly later, Viète (1540–1603) was still refusing

090604•Growing Ideas of Number 342 42

3/11/06 13:07:37

EXTENDING NUMBERS

43

to give the negative roots of such equations (see Crossley 1987, p. 99; Viète 1970, p. 86). However Bombelli (1526–1572) gave clear rules for the manipulation of positive and negative numbers, both for addition and multiplication. He went even further, being the first to use complex numbers accurately (see below Section 4.4). Although the Chinese initially simply left an empty space on their counting boards, a symbol for zero was introduced from India in about 500 AD (see, for example, Needham 1959, volume 3, Part 1, p. 10). It was then a simple circle. As the Hindu-Arabic numerals developed in the Arab world it became a dot, and a small circle was used for 5, as it still is today (see Figure 2.5). Zero, as a symbol, entered the West through the HinduArabic numerals (see Section 2.5) and its uptake was as slow as that of the other Hindu-Arabic numerals. The concept of zero is a fascinating one and too complicated to go into here. There have been two recent books on the subject: Seife (2000) and Kaplan (2000). Both are of a relatively popular nature, and Kaplan’s is the more focused. Hindu-Arabic, as opposed to Roman, numerals were used more by the abacist schools: the schools where people learnt practical arithmetic, as opposed to the theoretical approaches in the universities from the twelfth to the end of the sixteenth centuries. However, the two approaches slowly came together, particularly in France, and L’Arithmétique (1549) by Peletier (1517–1582) presents the integration of a learned and a ‘vulgar’ tradition (see Cifoletti 1992, p. 73). Subsequent major developments in mathematics focused on Italy. The most important of these, in my opinion, was the introduction of complex numbers. As noted above, negative (and other kinds of) numbers had an unresolved status in the sixteenth century. Negative numbers were not in common use and could usually be avoided, as we do in ledgers, by putting them on the other side of the ledger or, in the mathematical case, on the other side of the equation (where they become positive numbers). In the Algebra of al-Khwarizmi (c. 780–c. 850) (see Rosen 1931) this is called ‘restitution’ (almucabala): by adding the same (positive) quantity to each side of an equation, the deficiency on one side is removed and, of course, we have an additional amount on the other side. In modern notation, if we restore b to the equation a – b = c we obtain a – b + b = c + b, that is a = c + b.

4.3  Irrational numbers Irrational numbers were also a cause for suspicion. These had been a thorn in the side of the Pythagoreans when they were first discovered as noted

090604•Growing Ideas of Number 343 43

3/11/06 13:07:37

44

GROWING IDEAS OF NUMBER

above (in Section 3.2), for they meant that it was impossible to measure the side of a square as a rational fraction of the diagonal of that square. By the end of the sixteenth century such styles of number were beginning to be accepted as mathematically respectable entities. Stevin (1548–1620), writing in 1585 (after the death of Bombelli, whom I treat below), has a section (Struik, 1958, vol. IIB, p. 532): That there are no absurd, irrational, irregular, inexplicable or surd numbers. It is a very vulgar thing among authors of arithmetics to treat numbers such as 8 and the like, which they call absurd, irrational, irregular, inexplicable, surds, etc. What we deny to any future number: By what reason will the adversary disprove it?

Now the method of solving quadratic equations was by applying an algorithm, or recipe. The algorithm in the case of quadratic equations is easily applied, but requires mathematical understanding to see why it works. The algorithm, as such, was known (at least for certain kinds of these equations) in the Mesopotamian era about 1500 BC, as is evidenced by cuneiform texts (see Neugebauer 1973; Neugebauer and Sachs 1945). It N OTE 12 is important to remember that the problems were, of ‘Algorism’ comprises the course, not expressed in terms of equations at that time. algorithms for working with HinduThe idea of an equation is a sixteenth century invention Arabic numerals, as noted above; (see Cifoletti 1992) but it is easy to translate the ancient ‘algorithms’ are general procedures formulations into equations. for producing results. The word is The necessary steps for solving a quadratic most commonly used today in the equation with real roots can be performed on an context of computer programs. unsophisticated calculator today, and requires only the Such programs encode the presence of a square root facility. Now some quadratic algorithms; the algorithms equations, for example x2 + 1 = x, do not have solutions themselves are abstract procedures. in the ordinary, so-called, ‘real’ numbers; they require The word ‘algorithm’ developed the use of imaginary or complex numbers. Surprisingly through association with the Greek it was not in the context of solving quadratic equations, word arithmos (see the OED entry but of solving cubic equations that imaginary numbers for ‘algorithm’). first came into evidence.

4.4  Complex numbers Scipio dal Ferro (1465–1526) is generally accepted to be the first person to give a general method for solving cubic equations. (There is a question as to whether Scipio dal Ferro could solve all kinds of cubic equations, but

090604•Growing Ideas of Number 344 44

3/11/06 13:07:37

EXTENDING NUMBERS

45

this is not relevant to the present discussion.) His dispute with Cardano (1501–1576) about priority is well-known, see for example, the classic text of Paul Rose (1975) or Crossley (1987). However, Cardano claimed that he had produced a mathematical proof (of the solution) whereas dal Ferro had only produced the solution. It seems worthwhile to distinguish between the technique for solving a cubic equation, which is in fact an algorithm, and the mathematical theory behind the solution. In the cubic case, the equation is first transformed into a special form and then the problem of solving it becomes one of solving a different, this time quadratic, equation. After that, cube roots have to be extracted. What happens here is that a different kind of number is inevitably involved. These new numbers are what we now call complex numbers. The reason for this is quite complicated technically and I shall not give the details here. They may be found in, for example, Birkhoff and MacLane (1965). In the same way that a quadratic equation has at most two roots, we expect a cubic equation to have at most three. For example, consider x3 – 6x2 + 11x = 6. A little work will show that this is equivalent to (x – 1)(x – 2)(x – 3) = 0, which clearly has the solutions x = 1, 2, 3. Ironically, in the case where there are three real roots, the algorithm of dal Ferro inevitably involves taking the square roots of negative quantities (see Birkhoff and MacLane 1965). Since all numbers, positive and negative, always have positive squares, the idea of taking the square root of a negative number appeared impossible: Cardano uses the word subtilitas (subtlety) in dealing with such numbers. His calculations led him to 5 + 25 - 40 and 5 - 25 - 40 which he has to multiply together. He says: Dismissing mental tortures, and multiplying 5 + - 15

5 - - 15 we obtain 25 - ]- 15g. Therefore the product is 40.

by

But he concludes: ‘… and thus far does arithmetical subtlety go, of which this, the extreme, is, as I have said, so subtle that it is useless.’ (See Cardano 1968, p. 220, and Crossley 1987, p. 86.) Bombelli (1526–1572) was the first to give a consistent exposition of how to manipulate complex numbers. However it should be noted that he did not provide a theory and such a theory had to wait until much later (as I note below). The publication of Bombelli’s work was long delayed and it did not appear until 1572, although the work had been achieved in the mid-1550s. So its publication was long after the work of dal Ferro and Cardano which took place in the 1540s. Bombelli was an engineer, at one time engaged in trying to drain the Val di Chiana marshes, which he did

090604•Growing Ideas of Number 345 45

3/11/06 13:07:38

46

GROWING IDEAS OF NUMBER

successfully. Subsequently he was also engaged to drain the Pontine marshes at Rome but this was unsuccessful and the task was not completed until last century. Bombelli’s L’Algebra (1966) draws heavily on the newly discovered Vatican manuscript of Diophantos for its problems, but the technical treatment is Bombelli’s alone. He systematically treats ordinary numbers, educing the rules for the manipulation of what we now write as + and − signs, though he used ‘p.’ for ‘plus’ (più) and ‘m.’ for ‘minus (meno)’. He also used ‘L’ and its reverse as brackets:  . For the square ) he used an R with a line through the tail as we do for a root sign ( prescription: . It is therefore very easy to read his notation and transcribe it in a thoroughly modern way. In treating cubic equations he describes the complex numbers that arise in the computation of solutions as follows: ‘I have found another kind of tied cube root very different from the others …’ (Bombelli 1966, p. 133.) ‘Tied cube root’ refers to taking the cube root of a square root which is already a complex number. The details are not important here but may be found in Bombelli’s work and in précis in Crossley (1987, chapter 4); the point is that he has new expressions to deal with. With these he is then able to present formal rules for the manipulation of quantities such as - 1 or - 15 . Now there are two square roots of any number, even a negative one; thus, the two square roots of –3 are + - 3 and - - 3 , which Bombelli notates as ‘p. di m. 3’ and ‘m. di m. 3’ (più di meno 3 and meno di meno 3). He then went on to show, by examples, that these manipulations produce numbers that are solutions of the given cubic equation. After giving examples of such solutions he adds (Bombelli 1966, p. 225): And although to many this will appear an extravagant thing, because even I held this opinion some time ago, since it appeared to me to be more sophistic than true, nevertheless I searched hard and found the demonstration, which will be noted below. So then even this can be shown by geometry, which indeed works for these operations without any difficulty, and on many occasions one can find the nature of the unknown as a number. But let the reader apply all his strength of mind, for even he will find himself deceived [otherwise].

There is some question about how justifiable Bombelli’s ‘shown by geometry’ (in the original Italian: mostra in linea) is from a geometric point of view, though algebraically there is no problem at all, given his rules (as listed in Figure 4.1). It is important to note that Bombelli had no clear idea of what complex numbers might be. He simply followed the rules—and

090604•Growing Ideas of Number 346 46

3/11/06 13:07:38

EXTENDING NUMBERS

47

Bombelli’s rules for addition and subtraction are simple and obvious: like goes with like. His rules for multiplication are as follows: Plus of minus times plus of minus makes minus Plus of minus times minus of minus makes plus Minus of minus times plus of minus makes plus Minus of minus times minus of minus makes minus

6+ - . + - =-@ 6+ - . - - =+@ 6- - . + - =+@

6- - . - - =-@

Figure 4.1.  Bombelli’s rules for multiplication of imaginary numbers

they worked. There is no suggestion here of a metaphor (as Lakoff and Núñez 2000 imply); that would only come later when the complex plane was introduced. Indeed the full formal treatment of complex numbers was recognised considerably later, in 1806, where they were given a geometric representation in the work of Argand (1768–1822). In fact this had been accomplished some years earlier by Wessel (1745–1818) (see Argand 1971; Wessel 1799). Nevertheless Bombelli’s treatment of complex numbers, and in particular of imaginary numbers, took mathematics out of the dark swamp in which Cardano had struggled into a clear arena. Bombelli made imaginary numbers concrete. In doing so he had used a purely formal approach, manipulating, according to what have now become familiar rules, symbols and combinations of symbols which, although we now regard them as designating a special kind of number and including signed numbers, were mystifying at that time. The approach was very much that of the practical engineer; the benefits for mathematics were enormous. Thus the blind, but observant, pursuit of the standard rules for operating with plus and minus led Bombelli to a formal procedure that solved all cubic equations. Finally I briefly mention quaternions. These were introduced by Hamilton in the nineteenth century. They may be regarded as an extension of the complex numbers and it is true that the complex numbers form a subset of the quaternions. Complex numbers are usually written (by mathematicians) in the form a + ib, where a and b are real numbers and i2 = –1. (Engineers use a + jb.) Quaternions are the fourfold version, as the name implies. Thus the standard representation is a + ib + jc + kd. Here i2 = j2 = k2 = –1. Quaternions obey all the basic laws of complex numbers with one important exception: multiplication is not commutative, for we have ji = –ij and similarly for the other pairs j, k and k, i (see Joly 1969).

090604•Growing Ideas of Number 347 47

3/11/06 13:07:38

5 7

2 18 93 36

3

7

4 6 8

2

1

CHAPTER FIVE

5

Extending the number line

5.1  The complete number line One of the most spectacular developments in the West, although Euclid had tangled with it (see Crossley 1987, chapter 6), was Descartes’s correlation of numbers with geometric objects, which eventually led to the development of the calculus. I could have used the word ‘identification’ instead of ‘correlation’ but I prefer not to, because a geometric line or other object is not a set of numbers nor any sort of construct from numbers. Nevertheless, in order to work out what happens in geometry it is often much easier to deal with Cartesian geometry rather than pure geometry, that is to say, with numbers rather than points, lines and so on. Descartes’s view provides a metaphor. (In this context see Lakoff and Núñez 2000 for an extended discussion of how mathematics depends on metaphor.) The number line and the geometric line are not the same object, but our view of the number line allows us to bring geometric lines into a context where we feel comfortable, and therefore find ourselves able to reason about such lines. The number line, beloved of many, is a metaphor, not a reality. But it works wonderfully well. Using the metaphor, Descartes succeeded in changing paradigms. He used a structure we know, that of the real numbers, to unlock the unknown, or at least largely unknown, world of parabolas and other conic sections and to go far beyond them. In Chapter 3, I showed how to construct a point on the (geometric) line for every rational number. However, there are also numbers, such as 2 , which are not rational. So the rational number line has gaps. Cantor (1845–1918), in a paper in 1872, considered sequences of rational numbers a1, a 2, a3 f

(5.1)

48

090604•Growing Ideas of Number 348 48

3/11/06 13:07:40

EXTENDING THE NUMBER LINE

49

such that the difference am+n – an becomes arbitrarily small as n increases, whatever m may be. Then he introduced the notion of limit (see Cantor 1962, p. 93) saying that sequence (5.1) has a limit. The limit is thought of as a (possibly new) number, b, such that b – an becomes arbitrarily small as n increases. He then has to define what it means to say that two limits (from two sequences) are equal to, or greater than, or less than each other. The limits just defined are of level 1. Next he goes on to consider sequences of numbers of level 1. These give numbers of level 2, and so on. He continues to repeat the process, obtaining, successively, numbers of levels 2, 3 … Later it was discovered that the process does not need to be repeated: the numbers of level 1 already give all the real numbers. Notice that Cantor is developing points on the geometric line as a result of considering numbers. What is the distance of the point constructed? He says (see Cantor 1962, pp. 96–7), ‘… the distance of the point to be determined from point o is equal to b, where b is the numerical quantity corresponding to sequence [(5.1)]’, but then goes on to add: In order to complete the connexion presented in this [section] of the domains of the quantities define … with the geometry of the straight line one must simply add an axiom that simply says that inversely every numerical quantity also has a determined point on the straight line … I call this proposition an axiom because by its nature it cannot be universally proved.

Cantor was clearly recognising the difference between the number line and the geometric line. Stolz (1885, p. 80) expresses a similar view. The solution we use today is due to Dedekind. A sense of how recent this is was given to me by the late Professor E. Pitman of Tasmania who recounted how, when he was a student at the University of Melbourne in about 1910, the new theories of Dedekind were being taught there as the latest thing. Although Dedekind’s paper ‘Continuity and irrational numbers’ (see Dedekind 1963) did not appear in print until 1872, he says in the preface that he originally had the ideas in 1858, that is, before Cantor’s work. His approach was to consider not sequences but sets of rational numbers. He started from the fact that every rational number, r, divides the number line into two sets: the set of numbers less than r and the set of numbers greater than r. (There is a slight difficulty if the number is rational so we actually take, say, ‘greater than or equal to’ instead of ‘strictly greater than’.)

090604•Growing Ideas of Number 349 49

3/11/06 13:07:41

50

GROWING IDEAS OF NUMBER

Dedekind then takes this as a definition of a real number, saying that if we have so divided the straight line ‘then there exists one and only one point which produces this division’ (Dedekind 1963, p. 11), that is, produces a section (cutting) of the rational numbers. It is then necessary to prove that if one takes sections of the real numbers the same thing happens. However, this theorem establishes the supremely important result that the (number) line is complete: there are no more gaps. At this point we have a shift from the bottom-up approach, which was mentioned at the beginning of Chapter 4, to the top-down approach. The real numbers are a system with a closure condition: they are the smallest set of numbers including the rational numbers and closed under taking Dedekind sections. Subsequent to Dedekind’s work mathematicians have generally regarded the real numbers as constituting the real line, even in the geometric sense of ‘line’. Hilbert (1971) in giving axioms for the foundations of geometry explicitly remarks that the Archimedean axiom is needed (see Section 3.3). We can go further, the Archimedean axiom is equivalent to the statement that every Dedekind section defines a real number. In numerical terms the Archimedean axiom means that, given any two positive (real) numbers a and b, there is a positive integer n such that na > b. This axiom rules out the possibility of infinitesimals, since it implies a a 1 there is an n such that  >  n and that for any positive fraction b b infinitesimals are ‘numbers’ that are less than every positive fraction of the 1 form n yet greater than zero (see Section 3.3). The number line, as defined by Dedekind, is complete in the sense that any Dedekind section of it defines a number which is already there. This is equivalent to the mathematicians’ sense of ‘complete’: any strictly ascending sequence with a least upper bound has a limit. Further, this number line is unique in the sense that any completion of the rational line is isomorphic to it. Notice that what has now been established is that it is the Archi­ medean number line (that is, the line of numbers satisfying the Archimedean axiom) that has been shown to be complete. In Section 6.5, I shall show how this number line can be further extended if we do not adopt the Archimedean axiom.

5.2  Infinite numbers The next really spectacular advance in the development of larger number systems was in the serious consideration of infinite numbers. Long ago in the seventh century, Isidore of Seville (d. 636, see Isidore of Seville 1911)

090604•Growing Ideas of Number 350 50

3/11/06 13:07:41

EXTENDING THE NUMBER LINE

51

had written an encyclopaedia. The title of chapter IX of Book III may be translated as: ‘That numbers are infinite.’ (The original Latin is Quod infiniti numeri sunt, which is rather difficult to translate accurately.) In the chapter he concludes that there are infinitely many numbers (meaning natural numbers), but each individual one of them is finite. Immediately a problem arose: there seem to be twice as many numbers as there are even numbers when we look at the first few natural numbers, yet we can number them off in a one-to-one correspondence: 1, 2, 3, 4 f 2, 4, 6, 8 f



N OT E 1 3

This kind of equivalence was noted long ago by Galileo, in his Discorsi e Dimostrazioni Matema­tiche Intorno a Due Nuove Scienze (1966) (which I have not seen), in which he showed there is the same number of square numbers as there are numbers. The problem was eventually resolved by Cantor in the nineteenth century, and led to the development of set theory. Again it is surprising that the motivation for his work did not come from looking at infinite collections as such but from the study of Fourier series (see Note 13). Let us take a simpler example. Consider a sequence of nested intervals on the line that get smaller

and smaller as n increases, say the intervals :- n , + n D for n = 1, 2 … When we take the intersection of all these intervals we get the single point 0. However, if we remain a little further out, so to speak, and take the 1

1

intervals :- 12 - n1 , + 1 + n1 D, and then take the 2 1 1 intersection, we get the whole interval :- 2 , + 2 D. Call

Regular waveforms can, with some reservations, be represented by sums of, usually infinite, series of terms of the form ancos(nq) + bnsin(nq). Naturally such series repeat when the angle q increases by 2p radians. For continuous waveforms the reservations arise because there are certain places where there are discontinuities in the Fourier series. Removing these sequentially provided a sequence of processes that could be counted by 1, 2, 3 … but after all these finite removals there were still points to remove. So the list could be continued as in the sequence in (5.2) below. This was Cantor’s inspiration.

this stage 1. We can now repeat such a process inside the resulting interval using :- 1 - n1 , + 1 + n1 D and we get :- 1 , + 1 D. Call this stage 2. How many 4

4

4

4

intervals did we consider? If we start counting them, we run through all

the finite numbers 1, 2, 3 … in the first stage, so how do we count the steps in stage 2? Notice that here we are really thinking of ordinal numbers, the numbers we use for counting things in order. Cantor introduced the notation ω for the first infinite ordinal number. He counted

090604•Growing Ideas of Number 351 51

1, 2, 3, f w, w + 1, w + 2, f

(5.2)

3/11/06 13:07:42

52

GROWING IDEAS OF NUMBER

(a) 1 2 3 4 h

1 2 3 4 f 1 2 3 4 1 1 1 1 f 1 2 3 4 2 2 2 2 f 1 2 3 4 3 3 3 3 f 1 2 3 4 4 4 4 4 f f

(b) 1 2 3 4 h

1 1 2 4 7

2 3 4 f 3 6 10 f 5 9 ff 8 f f f

which give the list in the second row below:

1

2

3

4

5

6

7

8

9

10

…

1 1

1 2

2 1

1 3

2 2

3 1

1 4

2 3

3 2

4 1

…

Figure 5.1.  Counting the positive rational numbers (a) the array of rational numbers (including many repetitions) and (b) the numbers assigned to the corresponding positions in array (a)

And this process could be carried much further to w # w or ww , or even to ww where the ωs continue infinitely. This ordinal is now known as ε0 and even that is not the end. There is no end: given any infinite ordinal number α we can very simply construct a larger one as α + 1. Moreover, given any strictly increasing infinite list of ordinals α1, α2, α3 …, then there is an ordinal that is larger than all ordinals in the list. Such a number can be chosen uniquely if we take the smallest such ordinal. (In fact we can prove there is a smallest such.) This is usually written as limn αn and if the αn are strictly increasing it is called a limit number. For example, w = lim n n and ww = lim n wn , where w1 = w and wn + 1 = w_wni . So now we know how to construct ever larger infinite ordinals. h

w

h

w

5.3  Countable sets Sets that one can (in principle) count are called countable (and are called countably infinite if they are countable but not finite). Obviously the natural numbers are countable, but so too are all the integers: slot each negative integer into the list next to the corresponding positive integer. The rational numbers are also countable as is clear from the two arrays in Figure 5.1 and the previous sentence. Array (b) shows how to number off the rationals in array (a). Of course we shall have counted several rationals more than once. That does not matter: if a set is countable, so is any subset.

E x ercise 6 The set of rational numbers, which is sometimes called Q, is a linearly ordered set: that is to say, given two distinct rationals x and y, we have x < y or y < x (and not both). (We also

090604•Growing Ideas of Number 352 52

3/11/06 13:07:42

EXTENDING THE NUMBER LINE

53

have x < y and y < x implies x < z.) Q is unbounded or unbordered, that is, there is no least and no greatest rational. Between any two distinct rationals x and y, there is a third, different y from both, for example x + , so Q is dense. Finally, as was just seen Q is countable. This 2 makes Q unique in the sense that any other countable, dense, linearly ordered set looks the

same as Q; technically speaking, is order-isomorphic to Q (see Section 0.2.2). To see both of these properties consider the lines in Figure 5.2. Count the two sets out (since they are countable) as a1, a2, a3 … and b1, b2, b3 … Suppose that a1 has been mapped to b1. We now go back and forth between the sets, putting the next element in the ‘right’ place. Suppose we have mapped a1 … an – 1 across and b1 … bn – 1 as shown. (Note that a1 probably does not map to b1, and so on.) We have only mapped a finite number of as so far, therefore its closest neighbours (or it may just have one) are a certain a and an a‘. Find a b from the other set, that has not necessarily been dealt with yet, and which sits in the corresponding relation to the bs that have indeed been dealt with. Since the sets are dense, there is such an in-between b. Map it to that. Now take the next b, this is where the ‘back and forth’ comes in, and find an a from the other set. Show that, eventually, everything (on both lines) gets mapped across.

E x ercise 7 Using just the ‘forth’ part of the argument it is just as easy to prove that Q is also universal in that it contains a copy of any countable linearly ordered set. (Note that here I have dropped the word ‘dense’.)

Now I just said we can count any particular subset of a countable set, but how many subsets are there? First notice that this is a different sort of ‘how many’? I am now talking about cardinal numbers. What is the difference between cardinals and ordinals? a38 a3 a1 a17 a2 The cardinal number of a set can be defined as the least ordinal that will count it. The finite cardinal numbers are 0, 1, 2 … Since 4 5 1 2 3 no finite number can count an infinite set, the first infinite cardinal number is ω. This is often written as ℵ0 and read ‘aleph zero’. b3 b23 b1 b2 b15 (Note that ordinals such as ω + ω still have cardinal number ℵ0, since if we interlace the Figure 5.2.  The ‘back and forth’ argument two ω sequences we can get a single ω sequence. For example, we can rearrange 1, a17, a38 are points in the top ordering that match the points b2, b3 in the bottom ordering, that is, they are chosen to match 3, 5, … 2, 4, 6 … as 1, 2, 3, … The cardinals the prescribed ordering. succeeding ℵ0 are ℵ1, ℵ2, … ℵω, … The Note that ‘preserving the order’ means that none of the first of these, ℵ1, is larger than all countable lines with arrows must cross each other.

090604•Growing Ideas of Number 353 53

3/11/06 13:07:43

54

GROWING IDEAS OF NUMBER

ordinals: if we could count a set of size ℵ1 with a countable ordinal β, say, then we could rearrange it so that we could count it with ω, that is to say, its cardinal number would be ω = ℵ0. Therefore ℵ1 is the first uncountable cardinal number. (‘Uncountable’ simply means ‘not countable’.) There are also limit cardinals, cardinal numbers of the form ℵα where α is a limit ordinal. Thus ℵω is the first limit cardinal. Ordinal numbers, and cardinal numbers, can be added and multiplied, though the results do not always come out exactly as might be expected. For example, in the case of ordinal numbers we have the result that n + ω = ω holds for any finite n, but ω = ω + n is false for all finite n > 0. In this case, the addition of ordinal numbers imitates laying out objects representing n and w in a line and counting the objects. Lay out first n and then ω, or vice versa and note the difference. Exponentiation can also be defined for cardinal and ordinal numbers. When the n comes first, the whole sequence still has ordinal number ω. For cardinal numbers, all of the operations give the usual results for finite numbers. To find out more of the fascinating basic arithmetic of cardinal and ordinal numbers see Sierpiński (1965).

E x ercise 8 Show by mathematical induction (see (6.1) in Section 6.1) that the number of subsets of a set with n elements is 2n, and that 2n > n for all (finite) n.

5.4  Uncountable sets Cantor showed that there are always more subsets of a set than there are elements of the set. This applies to infinite sets as much as to finite ones. So if we take a countable set such as the rationals, and then take all its subsets we shall get more than a countable set. Cantor’s proof was by the so-called ‘diagonal argument’. One special case will be treated here. As said above in Section 5.1, every Dedekind section determines a real number. But every Dedekind section is defined by a pair of sets of rational numbers. Therefore, to every real number there corresponds, in a unique way, a set of rational numbers: the set of rationals less than it. I shall now show that the set of real numbers is uncountable, from which it follows that the set of all subsets of the rational numbers is also uncountable. This has the consequence that the number of Dedekind sections is also uncountable, but I shall show this directly. First, if I can show that any set of real numbers is uncountable,

090604•Growing Ideas of Number 354 54

3/11/06 13:07:43

EXTENDING THE NUMBER LINE

then the whole set of reals is also uncountable, but it is at least as big. I shall produce a contradiction by supposing that the set of real numbers between 0 and 1 is countable. For convenience we take these real numbers as represented by infinite decimals. We can avoid duplicates. We do not allow decimals to end in repeated 9s; we delete these since they are equal to numbers ending in repeated 0s, as follows from the last line of Note 10, Section 3.5. Since we are assuming the reals between 0 and 1 are countable, we can enumerate them as number1, number2, and so on. So we have a list:

…

1 2 3 4

what has become our decimal notation in his book De Thiende of 1586 (Struik 1958, pp. 386 ff). Minutes, seconds, and even thirds, fourths and fifths had been used for fractions of angles, for example, 30°15’25’’49’’’. Stevin imitated this and designated the decimal places by writing the number of the decimal place in a

wrote as

0 1 2 3 3 1 4 2

where the ai, bj, ck and so on are digits between 0 and 9. Now take the digits in the places corresponding to the diagonal places in the following array and form a real number between 0 and 1: 3

Stevin (1548–1620) introduced

we would write as 3.142, Stevin

.b1 b 2 b3 f .c1 c 2 c3 f f

2

N OT E 1 4

circle above the digit. Thus what

.a1 a 2 a3 f

1 1

55

4

…

or simply 3142 3 indicating that there are three decimal places. Nowhere,

however,

in

De

Thiende is there any suggestion of infinite decimals.

2 3 4 …

so that we get .a1b2c3 … Now change each of these digits according to the following rule: If the digit is between 0 and 7 inclusive, write down the next greater digit. If the digit is an 8, write down 7. If the digit is a 0, write down 5.

090604•Growing Ideas of Number 355 55

3/11/06 13:07:43

56

GROWING IDEAS OF NUMBER

We then get a new number .x1x2x3 …, which differs from the first real number .a1a2a3 … in the first decimal place, from the second real number .b1b2b3 … in the second decimal place, and so on. This implies that it differs from the nth real number in the nth decimal place. But the number is supposed to be in the list since, by its construction, it cannot end in a repeated 9, and the list is supposed to contain all real numbers between 0 and 1. (The purpose of the second clause of the rule was to avoid repeated 9s.) This is a contradiction to our assumption that all the real numbers between 0 and 1 were in the list. So the real numbers cannot be counted: they are uncountable.

E x ercise 9 Recall that the number of subsets of a set of n elements is 2n (see Exercise 8). Now extend this to any set and show that the cardinality of the set of all subsets of any set is strictly larger than the cardinality of the original set. This can be expressed another way as saying: there is no one-to-one map of a set onto the set of all its subsets. Hint: Use Cantor’s diagonal argument, but this time do not worry about a list. Consider an assumed one-to-one map from elements to subsets: Ax is the subset corresponding to the element x. So Exercise 9 becomes Exercise 10.

E x ercise 1 0 Show that {x : x z A x} is not equal to any of the A x . Note: The same argument was used by Russell to create his paradoxical set of all sets that are not members of themselves. (See, for example, Crossley 1973; Anellis 1984.)

Immediately the question arises: What is the cardinal number of the set of real numbers? One unilluminating answer is that it is 2 "0 , where here we have exponentiation of cardinal numbers as mentioned above. The number 2 "0 is the number of subsets of a countably infinite set (cf. Exercise 9). But where does 2 "0 fit in among the sequence of alephs: ℵ0, ℵ1, ℵ2 …? Cantor thought that 2 "0 = "1 and this conjecture is known as ‘Cantor’s Continuum Hypothesis’. The most famous logician, Kurt Gödel, opined that 2 "0 = " 2 but was unable to prove it. Currently opinion is divided between Cantor’s view and Gödel’s, but no-one has a proof of either. In the 1960s, Paul Cohen showed that the Continuum Hypothesis was independent of the usual axioms of set theory. (See Cohen 1963, 1964, 1966. For more on the axioms of set theory see Section 6.4.) Solovay (1964) proved that 2 "0 could be anything it ought to be. The one clear result is that 2 "0 cannot be

090604•Growing Ideas of Number 356 56

3/11/06 13:07:43

EXTENDING THE NUMBER LINE

57

"w , nor any other limit cardinal (see Section 5.3). One recent approach to this problem (Woodin 2000) leads to extremely complicated mathematics, and even to new ideas of what proving such a result might mean. There still remains a gulf between the abstract treatment of sets and what we intuitively think of as a set. (See also Note 16, Section 6.4.) To put it another way, the question ‘What is a set?’ has still not been adequately resolved.

090604•Growing Ideas of Number 357 57

3/11/06 13:07:44

5 7

2 18 93 36

3

7

4 6 8

2

1

CHAPTER SIx

5

Systematising 6.1  Formalisation

The approaches to systematising numbers in the East and the West have been very different. In China the Nine Chapters on the Mathematical Art (Shen et al. 1999; Li and Du 1987) was a book that was used for civil service examinations and became rote learning. Nevertheless it epitomised major techniques of calculation and survived relatively unchanged for much more than a millennium. It was definitely a book aimed at producing results. By contrast the approach of the ancient Greeks was to produce a theoretical system. Euclid (c. 300 BC) developed his axiomatic approach and Aristotle (384–322 BC) developed both a logic, that of syllogisms, and started classifications of the natural and scientific world. The early books of Euclid were known in the medieval world, but most of the works of Aristotle did not become available until about 1200 AD. The work of Nicomachus (fl. 85 AD) was transmitted through Boethius (480–524/5 AD). Boethius himself wrote a very abstract treatment of number incorporating much of Nicomachus’s work (see D’Ooge 1972), and including a treatment of figurate numbers (see Note 1, Section 4.1). Maurolico (1575, see Section 4.1) continued this, but was also the first to consider anything like what we might consider as mathematical induction in a general setting, although Fibonacci (Leonardo of Pisa) had considered particular cases (see Crossley 1987, pp. 37–8). The idea of mathematical induction is central to all later formulations of, and formalisations of, number systems. Mathematical induction is often formulated as follows: If 1 has the property P and whenever a number n has the property P, then n + 1 has the property P, then every (natural) number has the property P.

(6.1)

(I started with 1 but I could equally well start with 0.) 58

090604•Growing Ideas of Number 358 58

3/11/06 13:07:46

S Y S T E M AT I S I N G

59

Maurolico considers the sequence of squares: he shows that, for the first few numbers, the nth square plus the nth odd number is the next square number. In modern notation we would write: n 2 + ]2n + 1g = ]n + 1g2



(6.2)

Maurolico does not have a general variable and merely gives particular cases. But the argument is persuasive and he then adds ‘and so on successively to infinity.’ In the West it was Pascal (1623–62) who discovered the triangle 1 1 1 1 1

1 2 3 4

1 3 6

1 4

1 N OT E 1 5 The fact that the Peano axioms are not called by Dedekind’s name is

which gives the coefficients in the binomial expansion of (x + y)n if one reads diagonally up from left to right (Pascal 1963). In China this was discovered much earlier by Jia Xian (eleventh century; see Li and Du 1987, p. 126, and the figure on p. 127, or Shen et al. 1999, p. 226). However Jia did not characterise induction as Pascal did. Pascal establishes the triangle by considering what he calls the first ‘base’ and then proceeds from one base to the next, just as we do today. Bernoulli (1686; see Crossley 1987, pp. 45–6) clearly enunciates the principle of induction. He establishes a result from one number to the next. So by the end of the seventeenth century the idea of mathematical induction was firmly established.

interesting. (A similar naming has certainly happened to a large number of mathematicians’ discoveries.) There is a question as to the extent to which Peano’s work was independent. Peano does explicitly note in Peano (1957–59), p. 93, for example, that the axioms are due to Dedekind, but elsewhere he merely says there are similarities or that there is a ‘substantial coincidence with the definition of Dedekind’. See Grattan-Guinness (2000) for further discussion of this point.

6.2  The Dedekind-Peano axioms The fundamental significance of mathematical induction was brought out by Dedekind (1963, pp. 29ff.) in what are now know as the ‘Peano axioms’. Dedekind endeavoured to capture this notion of indefinite repetition or ‘and so on’. He characterised the natural numbers in terms of what he called ‘systems’. A ‘system’ can be defined by starting from an initial object, in our case, 1, and then having a procedure for going to the next, distinct, object.

090604•Growing Ideas of Number 359 59

3/11/06 13:07:47

60

GROWING IDEAS OF NUMBER

For numbers, the next number after x may be denoted by Next(x). We say Next(x) is the ‘successor’ of x and x is the ‘predecessor’ of Next(x). In order to avoid circularity, which would give us only a finite number of objects, Dedekind imposes the restriction that 1 is not the successor of any object, and that each object has a unique successor and a unique predecessor (except for 1). The natural numbers clearly form such a system. However, there are other systems that do not look like the natural numbers and yet form a system. For example, consider the natural numbers arranged in the order 1, 3, 5 …, 2, 4, 6 …, where all the even numbers come after all the odd numbers. Here the successor is formed by adding 2. We do not have circularity, but we have an extraneous tail of numbers (starting at 2). To avoid this Dedekind added what has become known as the ‘axiom of induction’. This has the form essentially given in (6.1) above: For every property P, if 1 has the property P and whenever a number n has the property P, then n + 1 has the property P, then every (natural) number has the property P. This raises the question: What does ‘property’ mean? The axioms can be stated as follows: 1 2 3 4 5

1 is a number. If n is a number, then so is Next(n). For all numbers x, we have 1 ≠ Next(x). Next(x) = Next(y) implies that x = y. (Next is one-to-one-one.) If 1 has property P and whenever n has property P, then n + 1 has property P, then every number has property P (induction axiom).

E x ercise 1 1 What is wrong with the following argument? 1 is small. If the number n is small, then n + 1 is small. Therefore, by induction, every number n is small. See also Sudbury and Wright (1977).

090604•Growing Ideas of Number 360 60

3/11/06 13:07:47

S Y S T E M AT I S I N G

61

6.3  From practice to pure mathematics The first problem that arises is that the axioms are intended to ensure that every number is connected with the starting number, 1, in a finite number of steps. Expressed in this way a circularity becomes apparent. We are trying to characterise (finite) numbers, but we are using the phrase ‘finite number of steps’. The second problem is that, although Dedekind has provided a theory, there is no guarantee that it is consistent. One way to ensure that it is consistent is to provide an actual model. Then, since the model actually exists, it is evident that the theory cannot be inconsistent. So how does one provide a model? First let us note that we cannot take the natural numbers as we know them in the world, for we cannot guarantee that we can keep on adding 1 forever (in the world, see Note 4 in Section 1.8). The standard strategy in mathematics is therefore to provide a mathematical proof. Dedekind follows this path and produces the most remarkable ‘proof’ I have ever seen. The theorem he ‘proves’ begins as follows (see Dedekind 1963, p. 64): There exist infinite systems. Proof. My own realm of thoughts, i.e. the totality S of all things, which can be objects of my thought, is infinite.

And for the Next(x) function he uses ‘the thought of x’. Is this a mathe­ma­ tical proof? If so, it is very strange indeed. One expects mathematical proofs to refer to abstract objects in a mathematical world, not the psychological aspects of the mathematician! The question immediately arises as to whether it is possible to provide a truly mathematical proof. Dedekind’s ‘proof’ was harshly criticised by Brouwer in his thesis in 1907 (see Brouwer 1975, p. 73) saying: … his proof, which introduces ‘meine Gedankenwelt’ [world of thoughts] is false, for ‘meine Gedankenwelt’ cannot be viewed mathematically, so it is not certain that with respect to such a thing the ordinary axioms of whole and part will remain consistent. Consequently DEDEKIND’s system has no mathematical significance; in order to give it logical significance, an independent consistency proof would have been needed, but DEDEKIND does not give such a proof.

090604•Growing Ideas of Number 361 61

3/11/06 13:07:47

62

GROWING IDEAS OF NUMBER

I agree, provided one understands ‘Dedekind’s system’ as simply the one of his world of thoughts. The reason for this view is bound up with the earlier point: the circularity. In order to present an infinite system one has to have, already existing (at least in the mathematical world), a process that can be repeated indefinitely. But this is exactly what we are trying to establish, namely, that it is possible to repeat some operation indefinitely. An actual consistency proof had to wait until the 1940s when Gentzen provided one. However this had its own difficulties, see below, Section 6.5. The reaction of mathematicians to this work of Dedekind was somewhat akin to the reaction of biologists to their first acquaintance with kangaroos and other Australian mammals. When a specimen of a platypus (Ornithorhynchus anatinus) was produced in London it was held to be a fake and ignored (see, for example, Agnew 2005), subsequently new classifications were made and then the platypus fitted in with the (new) conventional nomenclature. In the case of mathematics, the Peano axioms, as formulated by Dedekind, changed status from being characteristics of the natural numbers that we use in our daily life, to being the means of definition of the natural numbers. This, as I have previously argued in Crossley (1987, p. 50), transformed the study of the natural numbers from a study of the world to an abstract study in the realm of pure mathematics. The axioms given by Dedekind capture the natural numbers precisely. By this I mean that if one takes any two realisations (models) of the axioms they are isomorphic. We say that the axioms are categorical. The proof of this uses mathematical induction both inside and outside the two models. One proves that the obvious map, which takes the starting point of one model to the starting point of the other, and proceeds to map the ‘Next’ element by looking at where the predecessor has been mapped, is one-toone and onto. A full discussion involves second-order logic and I shall avoid this here. When attention is restricted to first order logic, there are unintended, so-called ‘non-standard’ models of the system, which I treat in Section 6.5. Further evidence for the point of view, which argues that Dedekind’s axioms have actually become a definition of an abstract system, comes from a consideration of the axiomatisation of set theory to which I shall now briefly turn.

6.4  Formal logic and set theory Set theory developed out of the work of Cantor (see Section 5.2) and was ultimately formalised about the beginning of the twentieth century. The idea of formalising parts of mathematics goes back to Leibniz (and the

090604•Growing Ideas of Number 362 62

3/11/06 13:07:47

S Y S T E M AT I S I N G

63

beginnings of formalising logic to Aristotle in his study of syllogism). Leibniz began a formal treatment of set theory (see Couturat 1961, pp. 264–9) in the seventeenth century, and also of real numbers, in a manuscript in the Niedersächsische Landesbibliothek, now the Gottfried Wilhelm Leibniz Bibliothek, which I have seen. His treatment of real numbers was complicated and became, as far as I can ascertain, essentially lost until the twentieth century. In the nineteenth century, Boole (1958) developed his laws of thought. There is no explicit reference by Boole to Leibniz’s work, though Dipert (2006) in the Encyclopaedia Britannica article on the History of Logic suggests that he may have known of it. It was natural that Boole should refer to ‘thought’ since, at that time, logic was in the domain of psychology. Then, in 1879, Frege published his Begriffsschrift (Frege 1964), whence came Whitehead and Russell’s Principia Mathematica (1908, second edition 1925–27). In set theory, in order to settle the question of whether there is indeed an infinite set (such as that of the natural numbers), a new axiom, the axiom of infinity, was N OT E 1 6 introduced (see Note 16). This was modelled on Dedekind’s notion of what I have called the Next Russell tried to deduce function. (All of this is unaffected by the debate mathematics from (pure) logic. between potential and actual infinities which arises in In this endeavour he was largely Aristotle. I am only concerned with being able to successful but there were two fatal generate arbitrarily large, but finite, numbers.) stumbling blocks. One was the At about the same time as axioms for set theory axiom of reducibility which is a were being developed, Dedekind’s axioms for technical matter and not relevant arithmetic (Dedekind 1963) were formalised by Peano here. The other was the axiom of (see, for example, Peano 1957–59, and Section 6.2). infinity. This was of an entirely Unfortunately, the formalised versions of the different character. The axiom of Peano axioms are not categorical. Thoralf Skolem infinity in effect stated that there (1967) showed that there were (countable) models of exists an infinite number of the formalised Peano axioms that are not isomorphic objects. This axiom therefore to the natural numbers. The reason this could happen appears to require empirical is as follows. We saw earlier that the number of subsets knowledge, though there are of the rational numbers, a countably infinite set, is insurmountable difficulties in uncountable (see Section 5.4). Likewise the set of all verifying the existence of infinitely subsets of the natural numbers is uncountable. Now many objects in our world. Thus, each property, P, of natural numbers corresponds from the point of view of the exactly to the set of natural numbers having the property working mathematician this axiom P. Therefore there are uncountably many properties to is more akin to an unverifiable consider for the induction axiom. However, when we empirical statement, than a purely work in a formalised language (with a finite number logical one.

090604•Growing Ideas of Number 363 63

3/11/06 13:07:47

64

GROWING IDEAS OF NUMBER

of symbols) we can only write down a countable number of formulae (see Exercise 12). Therefore there are many properties without names in our formalised language.

E x ercise 1 2 Show that if we have a language with a finite number of symbols, then we can only construct a countable number of (finite) words. Hint: Write down all the words with one letter, then all those with two, and so on.

E x ercise 1 3 Show that the set of all (infinite) subsets of the natural numbers is uncountable. Hint: Use the style of the diagonal argument in Section 5.4.

6.5  Non-standard models Skolem exploited this result and produced models of the Peano axioms that have ‘infinite’ numbers. Many years later Skolem’s idea was developed in what, at first sight, looks to be a completely different area. The natural numbers had been formalised, so it was natural to think of formalising the real numbers too. When that is done, we can apply Skolem’s idea and construct models of the theory of the real numbers that are not isomorphic to the real numbers: they are non-standard models of the reals. This was done by Abraham Robinson (1966). He discovered a whole class of nonstandard models. (An excellent and brief treatment of non-standard models for the layperson can be found, perhaps somewhat surprisingly, in Dehaene 1997, p. 238.) Note that there is not just one non-standard model: there are very many that are not isomorphic to each other. All of these models have all the real numbers that we have on the Archimedean number line (see Section 5.1), but they also contain infinitesimals. Thus the line in such a model does not satisfy the Archimedean 1 axiom. Moreover, since every non-zero element, x, has an inverse, x , the models contain infinite elements. Note that these are neither ordinal nor cardinal numbers but extended reals, sometimes called ‘hyperreals’. Robinson’s technique can then be used to do calculus and analysis in a new way. The one problem is the technicality involved in setting up the non-standard models, which requires an excursion into mathematical logic

090604•Growing Ideas of Number 364 64

3/11/06 13:07:47

S Y S T E M AT I S I N G

65

(see Robinson 1966; Keisler 1976, 1986). After that, the subject is no more difficult than it is in conventional treatments. I cannot leave this area without a few words on Gödel’s incomplete­ ness theorems. None of the above follows from Gödel’s famous theorems. Gödel conducted an analysis of Russell’s formalisation of Peano arithmetic and proved two major results. The first was that there are statements in formalised arithmetic that are true, but cannot be proved in the formal system. This is his first incompleteness theorem. The argument hinges on the diagonal style of argument (see Section 5.4), but also exploits the syntax of the language in a way that was strangely new and still causes difficulty in understanding today. The second theorem showed that it is not possible to prove that the Peano axioms are consistent, if one is restricted to using only the means available within the formal system. Later Gentzen in the 1940s (see Gentzen 1969) proved this by using a stronger system including an extension of induction to transfinite induction up to the ordinal ε0. (For this ordinal see Section 5.2.) That is to say, the means for establishing consistency had to include a method significantly more powerful than the induction axiom of arithmetic. Finally, moving away from formal logic, the real number line was further extended by Conway (1976). Recall that a field is an algebraic system (like the rationals or the real numbers). Specifically, it is a set closed under +, × , – and ÷ except by zero, satisfying certain standard axioms. The ordered field of the real numbers is (up to isomorphism) the unique homogeneous universal Archimedean ordered field (compare the way in which the rational numbers are universal, as mentioned in Exercise 7, Section 5.3). On the other hand, Conway’s ‘surreal’ numbers form (up to isomorphism) the unique homogeneous universal ordered field. Inevitably this system, which Conway denotes by No, includes infinitesimals. Notice that since the complex numbers cannot be ordered as a field, it follows that Conway’s numbers do not extend the complex numbers. Extensions of Conway’s system are still being investigated.

090604•Growing Ideas of Number 365 65

3/11/06 13:07:47

5 7

2 18 93 36

3

7

4 6 8

2

1

CHAPTER SEVEN

5

Calculating by machine

7.1  Designing machines Leibniz (1646–1716) is generally regarded as the leader in developing calculating machines in the early 1670s, but in fact Pascal had developed a machine for doing addition and subtraction a little earlier; Leibniz then designed a machine that would also perform multiplication and division. Leibniz’s machine was, however, never built. Although these machines were only for doing calculations with the natural numbers, Leibniz had a wider view of calculation. He wished to develop a formal system for doing logic so that the work of deciding logical arguments would be reduced to a routine that could be mechanised. This was the basis of his universal characteristic (see Leibniz 1999), which aimed to reduce argumentation to calculating with symbols. Witness his famous remark: … when controversies arise, there will be no more dispute between two philosophers, than between two [human] computers. For it will suffice to take their pens in their hands and sit down at abaci, and say to each other (if it pleases his summoned friend): ‘Let us calculate.’ (Gerhardt (ed.) 1978, vol. 7, p. 200; Couturat 1961.)

Leibniz’s dream had to wait a long time for even partial fulfilment. The shift, or perhaps I should say ‘expansion’, from arithmetic to logic was a major factor. It was Babbage in the nineteenth century who designed machines that went far beyond simple calculators. However, his engineering demands, in particular the precision required for the cutting of the gear wheels, were not met by the standards of the time, and the actual building of his Analytical Engine was never completed. It is only in recent years that a working version of even his Difference Engine has been built. It resides in the Science Museum, South Kensington, London. Babbage, and later 66

090604•Growing Ideas of Number 366 66

3/11/06 13:07:50

C A L C U L AT I N G B Y M A C H I N E

67

Turing in the 1930s, made analyses of the basic principles of calculation, and then built them into their machines.

7.2  Turing machines The following analysis describes the situation of Turing’s machines. 1 2 3 4

Calculation is algorithmic. Any result of steps in a procedure is a change of number or no change. The semantics of the operations of +, × and so on are clear. Cantor’s diagonal argument changes the last of these: it allows other interpretations.

The first point was indubitably true of all calculations actually made. The second clarifies what we mean by ‘calculation’. The third went without question as numerical expressions were thought to have no ambiguity. Cantor’s diagonal argument changed that situation dramatically in that it allowed numbers to take two different roles simultaneously. In the case of sets (see Exercise 10, Section 5.4), for Cantor’s diagonal argument one considers the expression ‘n z A n ’, so that element number n is both a possible set element and simultaneously an index of a set. Likewise, in the case of the real numbers (see Section 5.4), numbers can be used in dramatically different roles. Apart from the actual decimal numbers in the list we have the following two roles: (i) the number of the decimal place, i.e. the k where a whole number multiple of 10–k is represented, and (ii) the ordinal number of a specific (decimal) number in the putative list of all decimal numbers (between 0 and 1). There is a sharp conceptual difference between the two roles: the number (call this Number1) of a place in a decimal number (d1d2 …, say) on the one hand, and the ordinal number N OT E 1 7 (call this Number2) of .d1d2 … in the list of all decimal numbers (between 0 and 1) on the other. Thus any For an amusing differentiation (natural) number N can be interpreted in both senses: between what something is, and Number1 and Number2. But it is the same number N: what it denotes, see the White the number has two different denotations (Bedeutungen, Knight’s Song, in chapter 8 of to use Frege’s term). Carroll (1927). Now the same style of argument is used in demonstrating each of the following: 1 that the real numbers are uncountable (see Section 5.4), 2 that the set of all subsets of a set is bigger than the original set, and 3 Gödel’s incompleteness theorem (see Section 6.5).

090604•Growing Ideas of Number 367 67

3/11/06 13:07:50

68

GROWING IDEAS OF NUMBER

Each demonstration uses the diagonal argument and all exploit the syntax of the language being used. Consider telling someone who knows very little D EFINITION 2 mathematics (but can add, subtract and multiply) how to extract a square root of a certain number, which is Syntax concerns the arrangement known to them but not to you. All that is needed of the letters (and numbers) in a is moving one or two places to the right or left, copying language; certain numbers from one place to another, and Semantics is concerned with knowing how to double numbers. We can give simple the meaning. syntactic instructions. When we look at a sentence Alan Turing was not only a theoretician, I such as understand he actually constructed mechanical ‘The quick brown fox jumps machines to solve differential equations but cannot over the lazy dog.’ give a reference for this. However he certainly was we do not wonder about such familiar with differential analysers, which were used a strange fox (and dog), we think for solving differential equations and which he of the syntax as using all the letters mentions in Turing (1950). On the theoretical side of the English alphabet. (A he made a deep analysis of this and other kinds of common mistake is to use computation. About the same time, Emil Post in the ‘jumped’, which defeats the United States, independently made a similar kind of purpose.) The fact that the analysis (see Davis 1965, pp. 289–91). Post’s ultimate sentence makes sense, even if it is model had a clerk writing characters on pieces of odd, is a bonus and indeed an paper. I should also mention Zuse, in Germany, who aide-mémoire. worked in the opposite direction to Turing, that is going from the practical to the theoretical. He was N OTE 18 working at about the same time, namely 1936. He constructed an automatic calculating machine, and it Actual computers are based on was only later that the theoretical aspects of it were theoretical ideas due to Turing, but considered. Indeed Zuse does not seem to have learnt much has been said about von about Turing’s work until a number of years later (see Neumann’s role. It seems clear that Rojas 1994, 1998). von Neumann used Turing’s ideas Here is a version of Turing’s model (see Figure of a universal machine, and of 7.1). A paper tape is divided into squares, on each of treating a program as data (see which one character, at most, can be written. The tape below Section 7.4), but von can be extended in either direction as far as necessary, Neumann never said this explicitly but at any stage there is only a finite part of it on which in print. For a thorough discussion characters are written. This is the tape of the Turing of this issue see Davis (2001, machine. In addition there is a head that does the writing chapter 8, pp. 177 ff.). and can also read what is written on the (unique) square under scrutiny. The tape can move, one square at a time only, to left and right. The machine has only a finite number of states, which will

090604•Growing Ideas of Number 368 68

3/11/06 13:07:51

C A L C U L AT I N G B Y M A C H I N E

Si

1

Si

• 2

•

•

Si

• k

•

Si

69

m

The arrow indicates the position of the reading and writing head

Figure 7.1.  The essentials of a Turing machine

affect how it behaves according to a certain program. The program is a set of instructions of the form: If the machine is in state q and the scrutinised square has a character S on it, replace S by S´ and move one square to the left [or right].

7.3  Universal machines

N OT E 1 9

Any particular calculation we can write down can be We only need to consider natural simulated by such a Turing machine. I say ‘simulated’ numbers when dealing with Turing because there is a question of how we represent items machines because, just as in a on the tape. Such representations sometimes present modern computer, any syntactic technical challenges but no more than faced by any expression can be coded as a writer of mathematics. Even to represent π or e (the (binary) number. (Think of the base of the natural logarithms) is possible, in the sense ASCII code for any item on the that we can write a Turing machine program that will keyboard, or a bitmap for a pixel give us as many decimal places of numbers such as π on the screen.) Moreover, when we or e as we choose. want to deal with infinite decimals Because Turing had analysed the execution of we either have a name for the calculations so carefully and thoroughly, and had number, such as π or e, or else, at arrived at such amazingly simple building blocks, he any one time, we only need and his successors were able to show that all other consider a finite approximation to approaches that could describe and perform all that infinite decimal. (See also mechanical calculations, had exactly the same power. Section 7.4.) The first system that Turing showed equivalent was the lambda calculus of Church (1941), which had been developed in the 1930s. (More recently the lambda calculus has formed the basis for the programming language LISP; see, for example, Allen 1978.) Church’s system took the idea of abstracting a function from an expression as its basic notion. Consider the following:

090604•Growing Ideas of Number 369 69

3/11/06 13:07:51

70

GROWING IDEAS OF NUMBER

N OTE 20 Despite Conant’s prediction (see Note 3 in Section 1.7), the binary system plays an essential role in

What is the function denoted by x + y? We have several choices: as a function of two variables, as a function of x only with y held constant, and as a function of y only with x held constant. These are usually denoted by

computers. Clearly we can write any algorithm to work with the binary system or the decimal

lxly.x + y (often written as lxy.x + y ), lx.x + y and ly.x + y

system. However the arithmetic of the binary system is simpler and, in the case of building physical machines, it is much more easily implemented than the decimal system. Thus Zuse (see Section 7.2) used relays which are either on or off; modern computers use transistors, which again can easily be made to move between just two states but are difficult or impossible to make move into ten distinguishably different states. So the physical system directly mirrors the binary system. Curiously Leibniz did not use this fact in designing his machine, though he knew about the binary system (see Section 1.7).

0 1 2 3

= = = = f n = f

lf.lx.x lf.lx.fx lf.lx.f ^ fxh lf.lx.f _ f ^ fxhi lf.lx.f (f (fff (x)…)) n fs

Figure 7.2.  Numbers can be represented in the lambda calculus, but not without difficulty

090604•Growing Ideas of Number 370 70

This is abstraction. The other side of the coin is application. This is written in a familiar way: thus T1T2 denotes the application of the lambda term T1 to the lambda term T2. In particular fa is the application of the lambda term f to the lambda term a. These notations have the obvious interpre­tations. (Try them on x + y for specific values of x and y.) In ordinary mathematics if we apply the function λx.f to a then we get f 5 a/x?, which is read ‘f with a for x’. In the lambda calculus, however, this is not the same as the (application) term (λx.f)a, i.e. λx.f applied to a. That is to say they are syntactically different. We therefore have to introduce the notion of β-reduction. (There is also simpler operation called αreduction, which refers to the simple renaming of one variable by another without clashes.) b-reduction: ^lx.f ha 2 f 5 a/x?

(7.1)

(Here 2 is read ‘reduces to’.) Thus ^lx.x + yh 3 b-reduces to 3 + y , while ^ly.x + yh 3 reduces to x + 3. So Church’s lambda calculus is a purely syntactic calculus. Using the basic operations of the lambda calculus it is then possible to simulate Turing machines and it is also possible to simulate calculations in the lambda calculus on Turing machines. The proofs are long (see, for example, Hindley et al. 1972) but, once one kind of simulation is understood, it is usually not difficult to cope with a new kind of system. All of this led to what is now known as the Church-Turing thesis

3/11/06 13:07:51

C A L C U L AT I N G B Y M A C H I N E

71

or sometimes just Church’s thesis. This simply states that any function that can be calculated mechanically can be calculated by a Turing machine. (Turing (1950, p. 439), points out that the question of whether it is an electrical or a mechanical device is a red herring.) The only person of note who has attempted to challenge the ChurchTuring thesis is Kálmar (1959). Note that it is not at all clear what a mathematical proof of Church’s thesis might look like, since ‘calculate mechanically’ has not been defined precisely, and to take this as meaning ‘calculate by a Turing machine’ reduces matters to a tautology. Nevertheless, to date, all systems for calculating mechanically can be simulated by Turing machines. It follows at once that the functions Babbage’s Analytical Engine could (in theory) calculate can actually be calculated by a Turing machine. The converse is also true because of the simplicity of the basic operations. Perhaps it is because that is so obvious that Turing (1950, p. 450), simply asserted without further comment that: ‘the Analytical Engine was a universal digital computer’. Note that this is a theoretical statement, the fact that Babbage’s Analytical Engine was not actually completed is irrelevant to its theoretical capabilities. N OT E 2 1 The ordinary differential calculus is (largely) another syntactic system, as Leibniz pointed out in one of his manuscripts. It has meaning for us (in terms of the slope of graphs, and so on), but can also be taken as being simply a set of formal rules. (The same is not true of the integral calculus because of problems in finding antiderivatives or integrals.) In the differential calculus we have a number of purely syntactic rules, such as

dy d n d dz x = nx n - 1, +z  ^yzh = y dx dx dx dx

(7.2)

which can be applied without understanding. Indeed, I have noticed students who have persisted with using these syntactic rules, successfully enough, but without understanding, for a considerable period of time.

N OT E 2 2 Shepherdson-Sturgis machines. Turing machines are not easy to work with, although they provide the most widely used model. Perhaps the most elegant approach to characterising computable functions is that using Shepherdson-Sturgis machines or unlimited register machines (see Shepherdson and Sturgis 1963). These are much easier to work with than any of the others I have mentioned. An excellent description of these machines can be found in Davis et al. (1994).

090604•Growing Ideas of Number 371 71

3/11/06 13:07:51

72

GROWING IDEAS OF NUMBER

N OTE 22 (continued) Consider an abstract machine, very much like a modern computer. It has a finite number of registers in which natural numbers are stored. The registers are identified by variables X1, X2 … , Y, Z1, Z2 … The Xi are the input registers. There is also an output register, Y, where the answer will be found. The Zj are auxiliary registers. Initially each register has zero in it. The whole programming language is very simple and is shown below. All computable functions can be computed using programs in this language! Here is the programming language for Shepherdson-Sturgis machines. Z"0 Z " Z+1

If Z ! 0 go to L where Z stands for an arbitrary register name. Instructions are labelled L1, L2, L3 … The first instruction puts 0 in register Z; the second increments the register Z by 1; and the third is a conditional jump instruction: if register Z does not have 0 in it, go to the instruction labelled L.

7.4  The incalculable Can Turing machines calculate everything? The answer is clearly ‘no’— they cannot calculate all the decimal places in every real number. This is because the real numbers are uncountable (see Section 5.4) and, since a Turing machine has only a finite number of characters and a finite program, we can actually list all possible Turing machines, and therefore there is only a countable set of them. (Compare Figure 5.1, Section 5.1 and the discussion there.) However there are more interesting, and more surprising, items that Turing machines cannot compute. The demonstration of these depends on the notion of a universal Turing machine. Hitherto I have only considered individual Turing machines, each of which has its own program. We can always simulate a mechanical calculation by a Turing machine, so it follows that we can simulate a Turing machine by another Turing machine. In fact we can find one Turing machine which can simulate all Turing machines. The construction exploits the use of syntax (see Definition 2 in Section 7.2). Each Turing machine T has a tape with only a finite number of characters written on it. It also has a finite program. (Note that the only internal states a Turing machine uses are those already mentioned in the program and which therefore form a finite set.) It follows that, for any Turing machine, T, we have two syntactic expressions: PT′, the program, and DT ; the data on the tape. On the tape of the universal machine write (a representation of) PT followed by a marker, say *, and then (a representation of) the data DT . The tape of the universal

090604•Growing Ideas of Number 372 72

3/11/06 13:07:52

C A L C U L AT I N G B Y M A C H I N E

73

Turing machine is now set up. Next we write a program so that the Turing machine reads along PT on the tape until it finds the instruction we want to simulate. Then the universal machine changes DT to simulate what P would have done to the tape in the original machine T. (If and) when this program stops, having been started on the tape with PT * DT on it, we shall have (a representation of) the answer that was on the original machine. The word ‘halt’ is usually used instead of ‘stop’ following Turing. The writing of the program for a universal Turing machine looks very complicated, but it involves more bookkeeping problems than anything else. The realisation that a program for one machine can be treated as (part of the) data for another machine is the crucial conceptual breakthrough needed. We are reminded again of the Cantor-Gödel conceptual shift discussed at the beginning of this chapter (Section 7.2). I shall now use that conceptual shift once more to show that there are non-computable functions, that is functions that cannot be computed on any Turing machine. First notice that many Turing machines produce rubbish or do not stop (just as actual computers sometimes do, unless one pulls the plug)! I shall show that we cannot compute whether a certain Turing machine actually stops (halts) on a given input. We proceed as for showing that the real numbers are uncountable (see Section 5.4). Since the Turing machines can all be listed (ordered according to their finite programs), we list them as T1, T2, T3 … We call Tm ‘Turing machine number m’. Note also that all finite tape expressions can be numbered off. (One can think of the binary representation on the tape.) We assume that there is a Turing machine, call it H, which computes whether Tm stops (or halts) when it is started on the data n. We represent this by a function h(m, n): (7.3) h(m, n) = 1 if Tm halts on input n = 0 otherwise, i.e. if Tm keeps on running forever on input n Consider the function h(n, n). If h(m, n) is computable, h(n, n) must also be computable. (We are taking the ‘diagonal’.) So now consider a Turing machine, D say, which keeps on running if h(n, n) = 1 and stops with answer 1 if h(n, n) = 0. This Turing machine is in our list, so D is Td for some number d. What happens if we give Td input data (whose representation is) d? If D halts on input d, then h(d, d) = 1 by (7.3), but by definition D keeps on running in this case. Likewise, if h(d, d) = 0, then D stops with answer 1, by our definition of the machine D. So again we have a

090604•Growing Ideas of Number 373 73

3/11/06 13:07:52

74

GROWING IDEAS OF NUMBER

contradiction. Our assumption that we could compute h(m, n) is therefore false. To put it another way: the Halting Problem is unsolvable; it is not solvable by any machine. When we want to do arithmetic (or more complicated calculations) with real numbers, there is a problem about how we represent the real numbers. As noted above, we can only work with a finite representation of, or an approximation to, a real number. This has certainly caused problems, especially for banks. If the bank has an interest rate of something as simple as 5%, and the interest is compound, then we rapidly get down to very tiny fractions of a cent. How is this to be dealt with? (There is also the question of fraud where people have siphoned off these fractions for their own nefarious purposes.) This kind of problem is an approximation problem. It had been known for many centuries that some techniques for approximating calculations could lead to error. Perhaps the most notorious of these is Newton’s method for finding a zero of a (real-valued) function, even one as simple as a polynomial. Such problems led to the development, in the twentieth century, of the subject of numerical analysis. Because of the way in which errors can occur when calculating with finite approximations of infinite decimals, there can be a dramatic shift in attention required when one wishes to obtain accurate numerical results on a machine, as opposed to proving a theorem. When it was first developed, numerical analysis was regarded by many as being an inferior kind of mathematics, but it is vital for the proper operation not only of banks but also of all kinds of machines, including aircraft. There is also the question of how we represent a real number. Many different representations are in use. Finite approximations have already been mentioned. Another method is to take sequences of intervals [an, bn] that gradually reduce in length, tending to zero, but which always contain the required number. Different methods give different results! Moreover, it is not always easy to convert from one representation to another. Finally, even representing natural numbers can be a problem, as the recent so-called ‘millennium bug’ demonstrated. Here the problem was that memory was at a premium in early commercial computers, and the amount of storage required for the many dates that had to be stored could be halved if a date such as 1957 was simply stored as 57.

7.5  Feasible computation The dramatic growth in the power of computers, in terms not only of their memory but also their speed, has tempted people to make more accurate calculations. For example, in designing a bridge an engineer may divide the

090604•Growing Ideas of Number 374 74

3/11/06 13:07:52

C A L C U L AT I N G B Y M A C H I N E

75

length of the bridge into a thousand parts and do calculations for each. When the time required for the calculations depends linearly on the number of such divisions, doubling the number of divisions doubles the time, and multiplying by 100 multiplies the time by 100. However, if it varies as the fourth power, multiplying by 100 makes the time go up by a factor of 100 000 000, so that one second blows out to more than three years. Unfortunately, many mathematical calculations are of this nature. This is why engineers tend to use methods that are as close to linear as possible. The finite element method (see Davies 1980), for example, exploits this idea. In order to speed up calculations even further, parallel machines have been used. These do not change the class of computable functions because a parallel machine can always be simulated by a sequential machine. However this does have an impact on what is feasible. As noted above, the time for a calculation can become excessively large. Attention has therefore begun to focus on what can be achieved in ‘reasonable’ time. One way to attack this has been to look at the idea of being able to compute a result in ‘polynomial time’. P is usually used to designate the class of such functions. To be computable in polynomial time means that the time for the computation is of the order of ln where l is the length of the input data (see Section 0.2.2). (The computing models used for this have been Turing machines, but it turns out that changing to a different kind of computer, even a practical one, does not change the class of feasible computations.) The machines I have considered have been deterministic in that, at any time, there is only one instruction that has to be followed next. A non-deterministic machine is one where there is a DEFINITION 3 finite set of possible instructions available, any one of which may be followed. (And then, at the next stage, P problems are those where we there may again be a choice.) Trying to understand actually find the solution in such machines is not easy. However, it turns out that polynomial time. there is another way of characterising the functions NP problems can be they calculate (see Definition 3). These functions form characterised as those where we a class called NP. can guess a solution and then If it could be established that these two classes verify that it really is a solution in were the same (that is that P = NP) then we would be polynomial time. able to calculate results it was thought not feasible to For more detail, see Davis et al. calculate in feasible time. Proving P = NP would (1994). change our ideas of computation dramatically. It might also reveal new and different ways of carrying out computations. If, however, it turns out (as most of us expect) that P ≠ NP, then this will show that there are many common problems that are intrinsically very hard. Such problems include scheduling, and optimisation problems. In

090604•Growing Ideas of Number 375 75

3/11/06 13:07:52

76

GROWING IDEAS OF NUMBER

the meantime much effort is devoted to finding more efficient algorithms for specific tasks. The question of using the binary The big (apparent) difference between P and NP versus the decimal system for depends heavily on the idea of exponentiation (see numbers does not affect the Exercise 14). For example, the problem of whether a P = NP question. As noted in Boolean expression can get the value 1, seems to Section 0.2.2, the length of a require an exponential search: if there are n variables, number in binary is a constant each of which can take 2 values, then there are 2n multiple of its length in decimal, possible choices to sift through. We therefore expect and vice versa. that this problem cannot be solved in polynomial time. (It is definitely in NP.) If, however, it turned out that P = NP, then it would indeed be solvable in polynomial time, and so would myriads of other problems that had previously been thought to require exponential time. This would definitely change our ideas of how to compute.

N OTE 23

E x ercise 1 4 How long does it take to compute 2x? If l is the number of characters in the string representing the number x, then the time taken to write 2x is approximately proportional to 2l. Remember that the number x lies between two adjacent powers of 2, say 2m and 2m+1, so m = log2 x – 1. The powers of 2 are represented in binary notation by the numbers 100 f4 003 and the same with one more zero. 14 2 m zeroes

Therefore log2 x itself is |x| – 1 and x itself is bigger than 2|x| – 1 and therefore close to 2 , that is to say, to 2 to the power |x|. |x|

Multiplying by 2 means putting a zero on the end, so multiplying by 2x means putting x zeros on the end. So, the time required to compute 2x is approximately proportional to x itself and not to the length of x. But we are measuring time in terms of the length of x, and therefore the time required to put x zeros on the end is roughly proportional to x = 2log2 x 1 = ]2log2 x # 2g 2 1 = ]21 + log2 xg 2 1 x . 2 2



that is to say, it is proportional to 2 to the power l, where l = the length of x. The time required is exponential in the length of x. Here is another case where exponential computations occur: the number of subsets of a (finite) set with n elements is 2n. Thus, the time required to trawl through all the subsets of a set is exponential in the number of elements in the set.

090604•Growing Ideas of Number 376 76

3/11/06 13:07:53

5 7

2 18 93 36

3

7

4 6 8

2

CHAPTER EIGHT

5

Thinking

8.1  The psychology of number The natural numbers are familiar yet retain mystical or metaphysical overtones. In this chapter I shall attempt to clarify a few questions about the very smallest natural numbers. (In this chapter I specifically assume, for convenience, that zero is excluded from the ‘natural numbers’.) The first few numbers, 1, 2, 3 and possibly 4, are well established, but the innocent-looking phrase ‘and so on’ lurks like a snake on the path. When so confronted, if we do not freeze in terror, as mathematics causes many people to do, then we carefully go around the snake, keeping a respectful distance. In the nineteenth century Richard Dedekind (1963) asked the question: Was sind und was sollen die Zahlen? which may be translated as ‘What are the numbers and what do they mean?’ The German verb sollen is notoriously difficult to translate, but there is an implied problem about intention here. I try to answer his question and, in the case of 1, 2, 3 and possibly 4, the answers are available. Dedekind’s attempt to deal with ‘and so on’ is less than satisfactory and has led mathematicians to treat it as the snake is treated (see Section 6.3). Here I shall take the approach of the student of natural history. I shall observe, and give the best description that I can at the present time. Instead of looking at philosophies of mathematics, I shall consider how people actually use numbers. Of course I have been influenced by what I have read, and the ideas of Jung and other psychological and neuropsychological evidence, together with anthropological and linguistic evidence will be considered. The philosophy of mathematics, in particular Bertrand Russell’s analysis of number, will also turn out to be relevant. So this will be, in part, a Jungian chapter and, like all such endeavours, will leave questions still to be answered. However, I shall try to make these questions clear and to avoid a Jungian trap. This trap is evident in the last chapter of Jung et al. (1964), which was actually 77

090604•Growing Ideas of Number 377 77

3/11/06 13:07:55

1

78

GROWING IDEAS OF NUMBER

written by von Franz, where the number archetypes for 1 to 4 briefly appear, and in Penrose (1990), where the reader is led to believe that the answer is in the next chapter, only to find that he or she is already reading the last chapter. The trap is to think that (to change the metaphor) after peeling off all these layers of an onion we get to the essential core. We never do. ‘Can you count?’ one asks a young child. The answer varies with the child’s age. Early there is pride in answering, ‘Yes,’ and then beginning to count, ‘One, two, three.’ The supplementary question, ‘How far can you count?’ will usually elicit a specific answer such as ‘Up to ten.’ Later a game may be entered into where the child can go on counting beyond a number he or she is given. This game will eventually peter out because of lack of time or interest. However the adult tends to assume that it is possible to go on counting beyond any given number. I shall consider this later, but let us now turn to the first few numbers. It has been argued that counting, at least up to a small number, is innate and indeed that animals and birds can count up to, say, three. The evidence for this has sometimes been found to be wanting. Such was the case with the famous horse, Clever Hans (see, for example, Katz 1953, pp. 13–16). However, it has been established that animals and very young babies can distinguish between groups of two and groups of three (see Dehaene 1997; Butterworth 1999). This applies not only to groups of objects, but also to sound patterns. When the number of elements is larger than three, the discernment is less reliable. Therefore it is clear that this use of the word ‘counting’ is different from that of counting indefinitely far, which was a major concern in our Chapter 1 (in particular, see Section 1.8). When most workers in psychology and neuropsychology refer to counting, or even ‘counting by adding one’ they refer only to numbers below a relatively small number. In the works I discuss this number is around six. So this should be clearly distinguished from the idea of ‘unending repetition’ (see Section 1.8 and Section 6.2). It does appear possible simply to recognise groups of certain fixed numerical sizes, even as far as six, in the case of a raven and a grey parrot. (See Thorpe 1963, p. 390, and more recently the work of Pepperberg and Jarvis, as discussed in Scholtyssek 2006.) As noted in Note 1, Section 1.1, in his analysis of number, Russell (1937, para 109, pp. 113–14) was able to give definitions of numbers without counting. After this, Russell went on to define individual numbers such as one, two, and so on. In volume two of Principia Mathematica (Whitehead and Russell 1908), *101 is entitled ‘On 0 and 1 and 2’, and

090604•Growing Ideas of Number 378 78

3/11/06 13:07:55

THINKING

79

there he proves that these three numbers are indeed cardinal numbers according to his definition. Again as noted, in Section 1.3, in some societies it was found that small numbers were the only ones for which there were names. Nevertheless, the numbers one, two, three, and possibly four, seem to have their own existence. For Jung (see the last chapter of Jung et al. 1964) These existed as psychological archetypes. What is an archetype? ‘The archetype is a tendency to form ... representations of a motif—representations that can vary a great deal in detail without losing their basic pattern’ (Jung et al. 1964, p. 58). Jeromson (1999) points out that the voice of number as archetype is heard, principally, in Jung (1952) and (1958). These archetypes manifest themselves as psychological instincts. There is a serious question as to where these archetypes reside. Jung believed that they were in the ‘collective unconscious’, but that not only seems to posit a very metaphysical entity, it also immediately raises the question of where the collective unconscious might be. Recent work in neuropsychology has produced a simpler answer: it is in our brains. (See Section 8.2.)

8.2  The innateness of small numbers Because the simplest numbers are so omnipresent in our daily lives and we use them so often, it is natural for us to bring them into use in further situations. The very generality of applicability of the numbers one, two, three and possibly four, gives rise to further uses, in the same way that the ancient myths still strike a chord with us because, in each generation, they are reinterpreted and resuscitated. With numbers, the presence of the natural world and the use of language is essential. In this context, the recent work of Gordon (2004) raises the question of how innate precise ideas of these first numbers are. However, given the nebulousness of Jung’s archetypes, it is still possible to accommodate Gordon’s concerns with the Jungian number archetypes. Consider first the number one. In Aristotle’s time, one was not regarded as a number, but rather as the source of numbers (Metaphysics, 1087b33–1088a15, see Aristotle 1924). The one signifies a measure, evidently. And in each case there is some different underlying subject, such as in the musical scale a quartertone; in magnitude a finger or a foot or some other such thing; and in rhythm a beat or a syllable … And this is also according to reason [logos]; for the one signifies a measure of some plurality and the

090604•Growing Ideas of Number 379 79

3/11/06 13:07:55

80

GROWING IDEAS OF NUMBER

number signifies a measured plurality and a plurality of measures. Therefore it is also with good reason that the one is not a number; for neither is a measure measures, but a measure is a principle [arche], and so is the one.

Likewise in the ninth century, al-Khwarizmi, who gives his name to algorism, the science and art of working with Hindu-Arabic numerals, as noted in Section 2.5 says: ‘... every number is composite and that every number is put together from one. Therefore one is found in every number ... ’ (Crossley and Henry 1990; Folkerts 1997). Nevertheless, counting did begin with one. In our world the differentiation of objects is all pervasive, but consider a world in which everything was made of drops. Then, when two drops met, they might form into a single drop. In this way our idea that one plus one equals two would no longer be valid. In the new world one plus one would be one. In such a world there might be no need for counting at all. Salmon (2001) has a similar example: since alcohol dissolves in water, when the two are mixed the final volume is less than the sum of the two volumes. It follows that ordinary addition does not apply to such volumes. These, and other examples, cast doubt on the Pythagorean and Platonic ideas that numbers are already existing in the world. This is not to say that we, as humans, do not have both physiological properties and psychological instincts that lead us to numbers. As Huber-Dyson (2005) put it: ‘I believe that we are basically wired to perceive and handle the world according to the blueprint of mathematics.’ Maturana in Maturana and Varela (1980, p. 73) says: A universe comes into being when a space is severed into two. A unity is defined. The description, invention and manipulation of unities is at the base of all scientific enquiry.

However, the idea of the visible world being differentiated is physiologically based. Marr (1982) shows how the presence of, say, a white speck on a dark ground, triggers an electrical impulse. Indeed, the boundary between any two objects (unless of very close colours and textures) triggers such an impulse. Therefore we have a physiological tendency to distinguish objects. It is not a large jump from this to distinguishing small numbers of distinct objects. (Perhaps it is better to say ‘small groups’.) It requires some kind of memory to compare configurations. As Seidenberg (1962–66, p. 8) pointed out:

090604•Growing Ideas of Number 380 80

3/11/06 13:07:55

THINKING

81

The basic things needed for counting are a definite sequence of words and a familiar activity in which they are employed. The creation ritual offers us precisely such sequence and activity. Processions of couples in ritual are well known. ‘Male and female he created them’ [Genesis 1, v.27]. ‘There went in two and two unto Noah into the ark, the male and female, as God had commanded Noah’ [Genesis 7, v.9]. Presumably a couple is announced (in rituals the Word always accompanies the Deed) and then they make their appearance on the ritual scene. The sequence of words so used might have come to be used as the initial number-words.

Similar ritual descriptions may be found in Vaillant (1950). The continuing emphasis on rituals and the importance of small collections, such as the family, only serves to reinforce the idea of number. In the West, as in all languages that I know of, the numbers one and two are clearly present. The emphasis on the number three seems less clear. In the Middle Ages the order in the world was thought of as a reflection of the heavenly order. In the Christian view the notion of the Trinity was not evident or clear, although the debate in the Council of Nicaea and the concern over the views of Arius had taken place long before. The problem here is not just a theological one, the identification of the three persons of the Trinity as constituting one God required a conceptual view of numbers that was at odds with the normal use of the numbers one and three in everyday life. Four presents even more problems. In the East, especially in forms of Mahayana Buddhism, the mandala with its fourfold symmetry is significant and omnipresent. Although Jung argues that, in his own experience, and in that of some patients, the mandala is also relevant, I find it difficult to believe that the four number archetype is as evident as those for one, two and three. Further, since in many languages, at least before their subversion by missionaries and colonisers, there was no word for ‘four’, it seems hard to argue that there is an innate instinct of four. Thus in Pitjantjatjara, another Australian Aboriginal language: The Pitjantjatjara count only to two, thus kutju ‘one’, kutjara ‘two’ (1+1), mankurpa ‘three’—but also ‘few’. It is not strictly (1+1+1). Nowadays in school ‘four’ is made from kutjara kutjara and ‘five’ from kutjara mankurpa but that is as far as they go. Nonetheless, mankurpa is currently employed as ‘few’. (Personal communication from Miss M. Bain, 23.i.1977. See also Section 1.3.)

090604•Growing Ideas of Number 381 81

3/11/06 13:07:55

82

GROWING IDEAS OF NUMBER

This also suggests that the four archetype, and therefore possibly also the three archetype, are constructs in the psychological culture of a society. I think it would be very hard to justify an argument for a five archetype, because of the large number of languages that have no word for five. Thus the Jungian number archetypes seem to run out around the number four. This is consistent with the experiments described by Dehaene (1997) and Spelke (see, for example, Feigenson et al. 2004) on babies and other subjects. Indeed, it is consistent with a view that some animals have an innate sense of small numbers, but not exceeding about six. Spelke and Dehaene (1999) and Butterworth (1999) have shown that human beings naturally possess two distinctive qualities. The first is the ability to recognise small numbers, usually up to three or four. This correlates with Jung’s observations (see Section 8.1). This ability has been located in the left lower parietal lobe of the brain (Spelke and Dehaene 1999). The second is the ability to compare groups of discrete objects, provided they are not too large. The limit is about sixteen. This clearly ties in with Russell’s notion of one-to-one correspondence (see Note 1 in Section 1.1). I note, however, that this ability is limited to relatively small size groups. Beyond three or some small number there is the potential for further development of the appreciation of number. When a society with few number words is confronted by a society with many such words, such as that represented by missionaries, then the range of number words rapidly increases by adoption or adaptation of the incoming vocabulary. This is true of the Walbiri people referred to above (Section 1.5) where I noted how the English words had rapidly been taken up. I observed the same phenomenon in the Philippines (Section 1.5), where the Americans took over at the end of the nineteenth century.

8.3  Counting indefinitely Let us now turn to the question of counting on indefinitely. I raised this issue in the first edition of Crossley (1987) pp. 54–5, and the whole concept seems to have been largely neglected. However, Rafael Núñez (1997) wrote, in correspondence about Dehaene (1997): … when we refer to the ‘natural numbers’ or to ‘1, 2, 3, 4, etc.’ we are doing something that cognitively is much more complex than what is shown in Dehaene’s book. … the ‘etc.’ hides a very complex cognitive universe that is missing in Dehaene’s account [Dehaene (1997)]. In fact, with the ‘etc.’ we implicitly invoke a whole universe

090604•Growing Ideas of Number 382 82

3/11/06 13:07:55

THINKING

83

of entities that have never been explicitely [sic] thought or written by anybody in the whole history of humankind (e.g., really huge integers)! Still, some cognitive mechanisms allow us to know that these entities preserve certain properties, such as order or induction, that they share some kind of inferential structure such that we put them in the same cognitive category, and so on. All that is hidden when we say ‘integers’ or we write ‘...’ or ‘etc.’ after 1, 2, 3. This fact is an extraordinary cognitive accomplishment that is not in Dehaene’s account of the ‘number’ sense. That cognitive activity requires other fundamental mechanisms that are not intrinsically related to quantity and that can’t be overlooked.

None of the neuropsychologists mentioned—Spelke, Dehaene and Butterworth—has anything to say on this area. Indeed this topic seems to be conspicuously absent from the literature on the psychology of mathematics. On the other hand, in the latter stages of his life Jung moved from studying the sequence of number archetypes for one, two, three, four, to considering an archetype which was the archetype of all the natural numbers. This is discussed in Jung (1952, paras 870 and 871). To a mathematician the English phrase ‘order archetype’ seems misleading as a translation of the German Archetypus der Ordnung, which might perhaps be better rendered as ‘ordering archetype’. This would certainly be in accord with the approach, if not the technical details, of Dedekind’s work: Was sind und was sollen die Zahlen? (1963). Consider the small child who, on being tossed high into the air, requests: ‘Do it again!’ a request that is repeated until adult or child gets tired—usually, of course, the adult. In counting, the child progresses from being able to count up to, say, ten, to being able to count to a hundred. After that the debate seems to change to a game of trying to name a largest number. When the child appreciates that beyond any number there is a larger one, the debate changes again. Eventually the practice involved in, if not the idea of, generating ever larger numbers is established. This process can take place in two ways that seem to be very different from each other. First, if one uses number words, then, in any language there is, necessarily, a largest number for which there is a single word (see Section 1.8). Therefore, for very large numbers, this largest number word has to be repeated the requisite number of times. Ironically, the simplest system of numeration, using tallies, whereby we write, for example, ||||||| for seven, has exactly the same need of repetition! Second, if one uses a place notation, as in the case in the HinduArabic system commonly in use (and also in Chinese), then it seems, at

090604•Growing Ideas of Number 383 83

3/11/06 13:07:55

84

GROWING IDEAS OF NUMBER

first, as if one is avoiding the kind of repetition just noted. While it is true that with each extra symbol (Arabic numeral) one can count many more numbers, for example three figures allow us to write down 900 numbers from 100 to 999, and six allow us to go from 100 000 to 999 999 (900 000 numbers), nevertheless we have two tasks to perform in going to the next number. First we have to know how to increase a number other than 9. This is simple, since we just write down the next number in the list 1, 2, 3, 4, 5, 6, 7, 8, 9. In the case of 9, we write down 0 and ‘carry 1’. Then we repeat the exercise with the next number to the left. Finally if we find the last digit to the left is a 9, then we write down 10. Note that the repetition in which we engage is of the same nature as we had when using tallies: in that case we simply go to the last position (with tallies this could be either the leftmost or rightmost position) and write down another tally ‘|’. In either approach we are presented with the idea of indefinite repetition, which is evident in the phrase ‘and so on’. Now the archetype of ordering is not evident in the languages I referred to in Chapter 1. Beyond certain small numbers there is a vagueness which in English we refer to as ‘many’. It should also be borne in mind that Dedekind and Peano’s work (see Section 6.2) took place only relatively recently, that is to say, just before and just after the start of the twentieth century. If therefore, as I believe, archetypes evolve from our physical and psychic nature, then it should be unsurprising that the archetype corresponding to the construction of the unending sequence of natural numbers should have appeared only relatively recently. In the mathematical world the precursor of the axiom of induction only dates from the work of Maurolico (1575) (see Section 6.1). Further developments of the idea took place only slowly, culminating in the idea of mathematical induction which was made explicit in the work of Pascal and Bernoulli in the seventeenth century (as noted in Section 6.1). If, as many of us now believe, ‘Our sense of and use of number may rely on the basic properties of brains and even of neurons, but they develop through the inextricable interplay of the biological and the social’ (Rose 1998), then Jung’s archetype of ordering does seem to come to the rescue here. Since archetypes, by their very nature, are not knowable, but only manifest themselves through the human psyche, the fact that we cannot (without circularity) characterise the notion of the next larger number or, indeed, the idea of unending repetition, does not impede us from appealing to the archetype as the underlying source of our thoughts and actions. It does not solve all our questions, but it does give us a model that we can usefully apply. The archetype gives us an instinct for the unending repetition

090604•Growing Ideas of Number 384 84

3/11/06 13:07:56

THINKING

85

of an act. Discovering the neuropsychological correlate of such an archetype remains to be achieved. The mathematical solution to the problem of characterising what such a repetition might be in a mathematical context, even one as apparently simple as that of the natural numbers, remains to be found. So in counting we have the inbuilt ability to recognise small numbers up to 3 or 4, and also to compare not-too-large groups. How we extend this to continue the number sequence as ‘1, 2, 3 and so on’, remains unexplained.

090604•Growing Ideas of Number 385 85

3/11/06 13:07:56

5 7

2 18 93 36

3

7

4 6 8

2

1

CHAPTER NINE

5

Resources

The References provide a starting point. If you really want to know what happened follow the trails to their sources. The World Wide Web is a useful resource provided it is used with care. The quality of information there varies from impeccable to completely fanciful. The Stanford Encyclopaedia of Philosophy is excellent. On the other hand some entries in Wikipedia can be at the far end of the spread, and therefore every entry consulted should also be checked against other sources. Of course written sources, too, are not always particularly reliable. The reader should look at other sources if he or she finds any statements in the present book doubtful. The same applies to the references in this book. As a general resource book Grattan-Guinness (1994) is compendious and well informed. All of the above are secondary sources. In the References there are a number of primary sources. Some of the older works, for example Maurolico (1575), are also on the World Wide Web. Access to them is sometimes restricted, but public and university libraries may ease that access. The older works are sometimes presented as digital images, and in such cases it is usually not possible to search them for specific terms. In looking at any material, therefore, remember the advice of David Whiteside, which I included in the preface: ‘Go and look at the originals.’ National and State libraries are usually excellent. In Australia, the State libraries have excellent, and sometimes surprisingly rich, collections of early books. University libraries are also good, and Monash University has published a detailed catalogue of its books relevant to the history of mathematics including a number of ancient original works (Smith et al. 1992). Here are some specific starting points.

86

090604•Growing Ideas of Number 386 86

3/11/06 13:07:58

RESOURCES

87

9.1  Counting For further extensive general detail see Deakin (2007) in this ACER Press series. Also see chapter one of Crossley (1987; 1993). Flegg (1989) has excellent contributors and contains a wealth of information. Menninger (1969) is principally occupied with the representation of numbers but is an encyclopaedic resource. Schmandt-Besserat (1992) volume 1 describes the use of one-to-one correspondence, based on archaeological evidence from the Middle East. The Goroka Research Centre continues the work of Glen Lean and his thesis is available at , but the detailed reports of his investigations are harder to find, for they are only available in rare hard copy.

9.2  Calculating by hand For accounts of Chinese calculating see Li and Du (1987) and Shen et al. (1999). For ancient Egyptian mathematics see Gillings (1972) and for the Rhind mathematical papyrus see Robins and Shute (1987). For the history of the abacus in Europe start with Murray (1978) and the works cited there; also Pullan (1968). Murray’s splendid book (1978) devotes very many pages to mathematics in Medieval Europe, much of which is about numbers and their use. Menso Folkerts’s works (2003) are less accessible but of excellent quality.

9.3  Measuring Salmon (2001) is the definitive work on Zeno and his paradoxes. Stillwell’s work, in particular Stillwell (2002), presents a complementary view to the approach in this book, as was noted in the preface. He is very much more aware of geometry than most contemporary writers on mathematics, and his presentation is impeccable. Heath’s edition of Euclid (1956) includes copious and very illuminating notes. Chrystal (1886–89) is very readable and contains lots of illustrative examples. It is particularly good on continued fractions.

9.4  Extending numbers For early classifications of numbers, see Crossley (1987), chapter 2. I have found very little on the introduction of negative numbers in my researches.

090604•Growing Ideas of Number 387 87

3/11/06 13:07:59

88

GROWING IDEAS OF NUMBER

The best information is on their use in Chinese mathematics, see Li and Du (1987) and Shen et al. (1999).

9.5  Extending the number line Dedekind (1963) is excellent and so is Sierpin´ski (1956). There are many excellent books on set theory. Devlin (1993) is entertaining and informative.

9.6  Systematising The philosophy of mathematics and formal logic are huge subjects. The older (early twentieth century) philosophical views—Platonism, Formalism and Intuitionism—are well discussed by Black (1933) and Korner (1986). Two extremely readable books are Wilder (1968) and Waismann (1951). Views moved to a more human or anthropological approach in the second half of the century. Kitcher (1983) was one of the earlier people to have such views. More recently this area has become popular, but the scholarship is not always as good. Davis and Hersh (1981) is also useful. For non-standard analysis Keisler (1986) is on the World Wide Web.

9.7  Calculating by machine The best source for a general history of the ideas is Davis (2001). The Science Museum, South Kensington, website has excellent coverage of Babbage at . On Gödel’s theorem: see chapter 5 of Crossley et al. (1972). Hofstadter uses analogy excessively and does not back up his arguments. See the trenchant and entertaining review by Huber-Dyson (1981).

9.8  Thinking For the neuropsychology of counting and number, Dobbs (2005) gives a popular introduction. Dehaene (1997) and Butterworth (1999) deal thoroughly with the neuropsychological questions, even though they are concerned primarily only with small numbers and arithmetic concepts applied to these.

090604•Growing Ideas of Number 388 88

3/11/06 13:07:59

RESOURCES

89

Lakoff and Núñez (2000) puts a human face on the discovery, or invention, of mathematics. It is a book full of significant insights and superficial discussion. It also has significant and important omissions. For example, Cardano who is the best known of the early Italians working on cubic equations does not rate a mention. See the review in Mathematical Reviews by Michael Deakin. Davis and Hersh (1981) is useful for a new look at mathematics.

9.9  A final note In this book I have not tried to present any grand theories, I have simply tried to show how the world is, and how the very basic ideas of mathematics and number relate to it. This book is not an encyclopaedia with the final answers. Rather it is a work in progress. There has been very significant progress over the period of ten thousand years or more covered by this book. I expect this progress to continue. After all, mathematics is a human activity. The reader is therefore encouraged to follow the trails he or she may find indicated in this book or elsewhere, and to use the keys provided here to open the doors that the articles and books in the References represent. There is much more to discover, or invent.

090604•Growing Ideas of Number 389 89

3/11/06 13:07:59

References This list contains all the works referred to in the main text. It is intended as a starting point for those who wish to pursue ideas outlined in this book. I have seen all of these except the works by Galileo and Wessel. Some of the items listed below are now available on the web, but it is not always clear how long they will stay there. Dates of viewing these items are given in the format noted in the Preface. The dates against the authors’ names may sometimes appear odd. This is because the date quoted against an author’s name is the date of the edition to which I am referring. This should explain apparent anachronisms. If the edition listed in the references is in a foreign language, then the translation in the text is by me. Agnew, G 2005, Platypus, viewed 30.v.2006, . Allen, JR 1978, Anatomy of LISP, McGraw-Hill, New York. Anellis, IH 1984, ‘Russell’s earliest reactions to Cantorian set theory, 1896–1900’, in JE Baumgartner, DA Martin, & S Shelah (eds), Axiomatic set theory: Proceedings of Symposia in Pure Mathematics XIII (pp. 1–11), American Mathematical Society, Providence, RI. Archimedes 1976, The works of Archimedes, edited in modern notation, with introductory chapters by TL Heath, with The Method of Archimedes. Dover Publications, New York. Argand, J-R 1971, Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques (Essay on a way of representing imaginary quantities by geometric constructions), reprint of the 2nd edn, Albert Blanchard, Paris. Aristotle 1924, Aristotle’s metaphysics: a revised text with introduction and commentary by WD Ross, Clarendon Press, Oxford. Aristotle 1928–1952, The works of Aristotle translated into English, Clarendon Press, Oxford. Beaujouan, G 1991, ‘Étude paléographique sur la rotation des chiffres et l’emploi des apices du Xe au XIIe siècle’ (‘Palaeographic study on the rotation of digits and the use of apices from the tenth to the twelfth century’), in Par raison de nombres: l’art du calcul et les savoirs scientifiques médiévaux (Because of numbers: the art of calculation and medieval scientific knowledge), Paper IX, Aldershot, Hampshire, Great Britain: Variorum. (The pages retain the numbering from the original journals and are not otherwise numbered.) Originally published in Revue d’histoire des Sciences, vol. 1 (1948), pp. 301–12. 90

090604•Growing Ideas of Number 390 90

3/11/06 13:07:59

REFERENCES

91

Beeler, M, Gosper, RW, & Schroeppel, R 1972, HAKMEM [electronic version], MIT AI Memo 239, . Bernoulli, J 1686, ‘Excerpta ex iisdem litteris’, Acta Eruditorum, 5, 360–1. Birkhoff, G, & MacLane, S 1965, A survey of modern algebra, 3rd edn, Macmillan, New York. Bischoff, B 1979, Paläographie des römischen Altertums und des abendländischen Mittelalters (Palaeography of Roman Antiquity and the Middle-Eastern Middle Ages), E Schmidt, Berlin. Black, M 1933, The nature of mathematics: a critical survey, Routledge & K Paul, London. Bombelli, R 1966, L’Algebra. Prima edizione integrale. Introduzione di U Forti – prefazione di E Bortolotti (Algebra. First complete edition. Introduction by U Forti – preface by E Borlotti), Feltrinelli editore, Milan. Boole, G 1958, An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities, reprint of 1854 edn, Dover Publications, New York. Brainerd, B 1968, ‘On the syntax of certain classes of numerical expressions’, in HB Corstius (ed.), Grammars for number names, pp. 9–40, Reidel, Dordrecht. Bretholz, B 1926, Lateinische paläographie (Latin palaeography), BG Teubner, Leipzig. Brouwer, LEJ 1975–76, Collected works, 1. Philosophy and foundations of mathematics, A Heyting (ed.) North-Holland, Amsterdam. Butterworth, B 1999, What counts: how every brain is hardwired for math, Free Press, New York. Caldwell, CK 1994–2006, A proof that all even perfect numbers are a power of two times a Mersenne prime, viewed 5.vii.2006, . Cantor, G 1962, ‘Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen’ (‘On the extension of a theorem in the theory of trigonometric series’), Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Mit erläuternden Anmerkungen sowie CantorDedekind, in E Zermelo (ed.), pp. 92–102, Georg Olms Verlag, Hildesheim. Cardano, G 1968, The great art; or, The rules of algebra: Artis magnae, sive de regulis algebraicis, TR Witmer (trans.), MIT Press, Cambridge, MA. Carroll, L 1865, Alice’s adventures in wonderland. With forty-two illustrations by John Tenniel, Macmillan, London. Carroll, L 1927, Through the looking glass: and what Alice found there (children’s edn), Macmillan, London. Cassirer, E 1953, The philosophy of symbolic forms, Yale University Press, New Haven, CT. Chrystal, G 1886–89, Algebra: an elementary text-book for the higher classes of secondary schools and for colleges, 2 vols, A and C Black, Edinburgh.

090604•Growing Ideas of Number 391 91

3/11/06 13:07:59

92

REFERENCES

Church, A 1941, The calculi of lambda-conversion, Princeton University Press, Princeton, NJ. Cifoletti, GC 1992, Mathematics and rhetoric: Peletier and Gosselin and the making of the French algebraic tradition, PhD thesis, Princeton University. Codrington, RH 1885, The Melanesian languages, Clarendon Press, Oxford. Cohen, PJ 1963, ‘The independence of the continuum hypothesis’, Proceedings of the National Academy of Sciences of the United States of America, 50, 1143–8. Cohen, PJ 1964, ‘The independence of the continuum hypothesis II’, Proceedings of the National Academy of Sciences of the United States of America, 51, 105–110. Cohen, PJ 1966, Set theory and the continuum hypothesis, WA Benjamin, New York. Conant, LL 1931, The number concept: its origin and development, viewed 28.vi.2006, . Conway, JH 1976, On numbers and games, Academic Press, London. Cordoliani, A 1951, ‘Los textos y figuras de cómputo de los códices Aemilianensis y Vigilianus y el tratado de cómputo de Rodríguez de Campomanes’ (‘The texts and figures on computus in the Aemelianensis and Vigilanus codices and the tract on computus by Rodríguez de Campomanes’), Revista Bibliográfica y Documental, 51, 117–52. Couturat, L (ed.) 1961, Opuscules et fragments inédits de Leibniz: extraits des manuscrits de la Bibliothèque royale de Hanovre (Lesser works and unedited fragments of Leibniz: extracts from the manuscripts from the Royal Library of Hanover), reprint of 1903 edn, Georg Olms Verlag, Hildesheim. Crossley, JN 1973, ‘A note on Cantor’s theorem and Russell’s paradox’, Australasian Journal of Philosophy, 51, 70–71. Crossley, JN 1987, The emergence of number, 2nd edn, World Scientific, Singapore. Crossley, JN 1993, ‘The beginnings of counting and number’, in I Grattan-Guinness (ed.), Encyclopaedia of the history and philosophy of the mathematical sciences, vol. 1, pp. 150–59, Routledge, New York. Crossley, JN, Ash, CJ, Brickhill, C, Stillwell, JC, & Williams, NH 1972, What is mathematical logic? Oxford University Press, Oxford. Crossley, JN, & Henry, AS 1990, ‘Thus spake al-Khwarizmi: A translation of the text of Cambridge University Library Ms. Ii.vi.5’, Historia Mathematica, 17(2), 103–31. D’Ooge, ML (ed.) 1972, Introduction to arithmetic by Nicomachus of Gerasa; translated into English by Martin Luther D’Ooge, with studies in Greek arithmetic by Frank Egleston Robbins and Louis Charles Karpinski, reprint of 1926 edn, Johnson Reprint Corp., New York. Davies, AJ 1980, The finite element method, Oxford University Press, Oxford. Davis, M 1965, The undecidable; basic papers on undecidable propositions, unsolvable problems and computable functions, Raven Press, Hewlett, NY.

090604•Growing Ideas of Number 392 92

3/11/06 13:07:59

REFERENCES

93

Davis, M 2001, Engines of logic: mathematicians and the origin of the computer, Norton, New York. Davis, MD, Sigal, R, & Weyuker, EJ 1994, Computability, complexity, and languages: fundamentals of theoretical computer science, 2nd edn, Academic Press, Boston. Davis, PJ, & Hersh, R 1981, The mathematical experience, Birkhäuser, Boston. Deakin, MAB 2001, Review of Lakoff, G, & Núñez, RE 2000, Mathematical Reviews, MR1794854 (2001i:00013), viewed 8.ix.2006, . Deakin, MAB 2004, ‘The Kahane [sic] function’, Function, 28(2), 55–6. Deakin, MAB 2007, The name of the number, ACER Press, Melbourne. Dedekind, R 1963, Essays on the theory of numbers, WW Beaman (trans.), Dover Publications, New York. Dehaene, S 1997, The number sense: how the mind creates mathematics, Oxford University Press, New York. Dehn, M 1900, ‘Über raumgleiche Polyeder’ (‘On equal volume polyhedra’), Nachrichten von der Königleichen Gesellschaft der Wissenschaften zu Göttingen, 345–54. Denvert, FW, & Oakland, WH 1968, Letter, Nature, 220, 311. Descartes, R 1979, La géométrie (Geometry), M Mahoney (trans.), Dover Publications, New York. Detlefsen, M, Erlandson, DK, Heston, JC, & Young, CM 1976, ‘Computation with Roman numerals’, Archive for History of Exact Sciences, 15(2), 141–8. Devlin, KJ 1993, The joy of sets: fundamentals of contemporary set theory, 2nd edn, Springer, New York. Dipert, RR 2006, ‘Logic, history of [electronic version], Encyclopædia Britannica online, viewed 1.vii.2006, . Dixon, RMW, & Blake, BJ (eds) 1979–2000, Handbook of Australian languages Canberra, Australian National University Press. Dobbs, D 2005, ‘Big answers from little people’ [electronic version], Scientific American Mind, 16, 38–44, viewed 30.v.2006, . Euclid 1956, The thirteen books of Euclid’s elements, 2nd rev. edn with additions, translated from the text of Heiberg, with introduction and commentary by Sir Thomas L Heath, 3 vols, Dover Publications, New York. Feigenson, L, Dehaene, S, & Spelke, ES 2004, ‘Core systems of number’, Trends in Cognitive Sciences, 8, 307–14. Flegg, G (ed.) 1989, Numbers through the ages, Macmillan in association with The Open University, Basingstoke, Hampshire. Folkerts, M 1997, Die älteste lateinische Schrift über das indische Rechnen nach al-H wārizmī/Edition, Übersetzung und Kommentar von Menso Folkerts; unter ˇ Mitarbeit von Paul Kunitzsch (The oldest Latin text on Indian reckoning

090604•Growing Ideas of Number 393 93

3/11/06 13:07:59

94

REFERENCES

according to al-Khwarizmi/ edition, translation and commentary by M Folkerts; in co-operation with P Kunitzsch), Neue Folge vol. 113, Munich, Bayerischen Akademie der Wissenschaften. Folkerts, M 2003, Essays on early medieval mathematics: the Latin tradition, VT Ashgate, Burlington. Fowler, DH 1987, The mathematics of Plato’s Academy: a new reconstruction, Clarendon Press, Oxford. Frege, G 1964, Begriffsschrift und andere Aufsätze (Concept writing and other essays), with remarks by E Husserl and H Scholz, I Angelelli (ed.), 2nd edn, Georg Olms Verlag, Hildesheim. von Fritz, K 1945, ‘The discovery of incommensurability by Hippasus of Metapontum’, Annals of Mathematics, 46, 242–64. Galilei, G 1966, Discorsi e dimostrazioni matematiche: intorno a due nuove scienze, attenenti alla mecanica e i movimenti locali; con una Appendice del centro di gravità d’alcuni solidi (Mathematical discourses and demonstrations: relating to two new sciences, with respect to mechanical and local movements; with an appendix on the centre of gravity of some solids), (Facsimile edn), Culture et Civilisation, Brussels. Gentzen, G 1969, ‘Consistency of arithmetic’, in, The collected papers of Gerhard Gentzen, ME Szabo (ed.), North-Holland, Amsterdam. Gerhardt, CI (ed.) 1978, Die philosophische Schriften von Gottfried Wilhelm Leibniz (The philosophical writings of Gottfried Wilhelm Leibniz), Georg Olms Verlag, Hildesheim. Gillings, RJ 1972, Mathematics in the time of the pharaohs, MIT Press, Cambridge, MA. Gómez-Pallarès, J 1998, Estudios sobre manuscritos latinos de cómputo, Madrid (Essays on Latin manuscripts of computus in Madrid), viewed 28.vii.2006, . Gordon, P 2004, ‘Numerical cognition without words: evidence from Amazonia’ [electronic version], Science, 306, 496–9, viewed 30.v.2006, . Gow, J 1968, A short history of Greek mathematics, revised edn, Chelsea Pub. Co, New York. Grattan-Guinness, I 1977, Dear Russell, Dear Jourdain: A commentary on Russell’s logic, based on his correspondence with Philip Jourdain, Columbia University Press, New York. Grattan-Guinness, I (ed.) 1994, Companion encyclopedia of the history and philosophy of the mathematical sciences, 2 vols, Routledge, London. Grattan-Guinness, I 1996, ‘Numbers, magnitudes, ratios, and proportions in Euclid’s Elements: How did he handle them?’ Historia Mathematica, 23, 355– 75. Grattan-Guinness, I 2000, The search for mathematical roots, 1870–1940: logics, set theories, and the foundations of mathematics from Cantor through Russell to Gödel, Princeton University Press, Princeton, NJ.

090604•Growing Ideas of Number 394 94

3/11/06 13:08:00

REFERENCES

95

Hale, KL 1975, Gaps in grammar and culture, P DeRidder Press, Lisse. Hilbert, D 1901–02, ‘Mathematical problems’, Bulletin of the American Mathematical Society, 8, 437–79. Hilbert, D 1971, Foundations of geometry, 2nd edn; taken from the 10th German edn, Open Court, La Salle, IL. Hindley, JR, Lercher, B, & Seldin, JP 1972, Introduction to combinatory logic, Cambridge University Press, Cambridge. Huber-Dyson, V 1981, Critical notice of DR Hofstadter, Gödel, Escher, Bach: an eternal golden braid, book review, Canadian Journal of Philosophy, 11(4), 775. Huber-Dyson, V 2005, Gödel and the nature of mathematical truth II [electronic version], Edge, viewed 29.vi.06 . Isidore of Seville 1911, Isidori Hispalensis Episcopi Etymologiarvm sive originvm libri XX (Isidore, Bishop of Seville, Twenty books on etymologies or origins), WM Lindsay (trans.), Oxford University Press, Oxford. Jeromson, B 1999, Jung and mathematics in dialogue: a critical study, PhD thesis, University of South Australia. Johnstone, WD 1975, For good measure, Rinehart and Winston, New York, Holt. Joly, CJ (ed.) 1969, Sir William Rowan Hamilton’s elements of quaternions, 3rd edn, Chelsea, New York. Jung, CG 1952, ‘Synchronicity: an acausal connecting principle’, RFC Hull (trans.), in The structure and dynamics of the psyche, vol. 8, CG Jung: collected works, H Read, M Fordham & G Adler (eds), Routledge and K Paul, London. Jung, CG 1952–91, The collected works of CG Jung, Pantheon Books, New York. Jung, CG 1958, ‘Flying saucers: a modern myth of things seen in the skies’, RFC Hull (trans.), in Civilization in transition, vol. 10, CG Jung: collected works, H Read, M Fordham & G Adler (eds), Routledge and K Paul, London. Jung, CG, Henderson, JL, von Franz, ML, Jaffé, A, & Jacobi, J 1964, Man and his symbols, Doubleday, Garden City, NY. Kálmar, L 1959, ‘An argument against the plausibility of Church’s thesis’, in Constructivity in mathematics, A Heyting (ed.), North-Holland, Amsterdam. Kaplan, R 2000, The nothing that is: a natural history of zero, Oxford University Press, New York. Katz, D 1953, Animals and men: studies in comparative psychology, H Steinberg & A Summerfield, (trans.), Penguin, London. Keisler, HJ 1976, Elementary calculus, Prindle, Weber & Schmidt, Boston. Keisler, HJ 1986, Elementary calculus: an approach using infinitesimals, electronic edition; based on the 1986 print edition, viewed 5.vii.2006, . Kitcher, P 1983, The nature of mathematical knowledge, Oxford University Press, New York.

090604•Growing Ideas of Number 395 95

3/11/06 13:08:00

96

REFERENCES

Knorr, WR 1975, The evolution of the Euclidean elements: a study of the theory of incommensurable magnitudes and its significance for early Greek geometry, D Reidel, Dordrecht, Holland. Korner, S 1986, The philosophy of mathematics, an introductory essay, Dover Publications, New York. Lagrange, JL 1770, ‘Additions au mémoire sur la résolution algébrique des équations numériques’ (‘Additions to the memoir on the algebraic solution of numerical equations’), Mémoires de l’Académie royale des Sciences et Belleslettres de Berlin, 24. Lakoff, G, & Núñez, RE 2000, Where mathematics comes from: how the embodied mind brings mathematics into being, Basic Books, New York. Lam L-Y 1972, ‘The Jih yung suan fa (Arithmetical methods for daily use): An elementary arithmetic textbook of the thirteenth century’, Isis, 63, 370–83. Lao-tzu 1963, Tao te ching [Dao de jing], translated with an introduction by DC Lau, Penguin, Harmondsworth, Middlesex. Lean, GA 1992, Counting systems of Papua New Guinea and Oceania, PhD thesis, Papua New Guinea University of Technology. Leibniz, GW, Freiherr von 1768, GW Leibnitii Opera Omnia (GW Leibniz’s collected works), vol. 3, Apud Fratres de Tournes, Geneva. Leibniz, GW, Freiherr von 1999, Samtliche Schriften und Briefe; Herausgegeben von der Berlin-Brandenburgischen Akademie der Wissenschaften und der Akademie der Wissenschaften in Göttingen (Collected writings and letters; edited by The Berlin-Brandenburg Academy of Sciences and the Academy of Sciences Göttingen), Series 6, Philosophische Schriften (Philosophical writings), vol. 4, 1677–Juni 1690, Akademie-Verlag Berlin. Leibniz, GW, Freiherr von 2000, ‘Fundamenta calculi ratiocinatoris’ (‘Fundamentals of the calculus of reasoning’), in Die Grundlagen des logischen Kalküls. Lateinisch-Deutsch (Foundations of the logical calculus. Latin-German), F Schupp (ed.), Felix Meiner, Hamburg. Lévy-Bruhl, L 1979, How natives think (in French), authorized translation, LA Clare (trans.), Arno Press, New York. Li Y, & Du S 1987, Chinese mathematics: a concise history, JN Crossley & AW-C Lun (trans), Oxford University Press, Oxford. Liardet, P, & Stambul, P 1998, ‘Algebraic computation with continued fractions’, Journal of Number Theory, 73, 92–121. Marr, D 1982, Vision: a computational investigation into the human representation and processing of visual information, WH Freeman, San Francisco, CA. Marre, A 1880, ‘Notice sur Nicolas Chuquet et son Triparty en la science des nombres’ (‘Notice about Nicolas Chuquet and his Trilogy in the science of numbers’), Bulletino di Bibliografia e di Storia delle Scienze matematiche e Fisiche, 13, 555–659, 693–814. Marshack, A 1991, The roots of civilization: the cognitive beginnings of man’s first art, symbol and notation, revised and expanded from 1972 edn, Moyer Bell, Mount Kisco, NY.

090604•Growing Ideas of Number 396 96

3/11/06 13:08:00

REFERENCES

97

Martin, J 1827, An account of the natives of the Tonga Islands, in the South Pacific Ocean: with an original grammar and vocabulary of their language, compiled and arranged from the extensive communications of Mr. William Mariner, several years resident in those islands (pages in the appendices are unnumbered), 3rd edn, Constable, Edinburgh. Maturana, HR, & Varela, FJ 1980, Autopoiesis and cognition: the realization of the living, D Reidel, Dordrecht. Maurolico, F 1575, ‘Arithmeticorum libri duo’ (‘Two books of arithmetics’), in Opuscula mathematica (Mathematical opuscules), Francisci, Venice. Menninger, K 1969, Number words and number symbols, a cultural history of numbers, MIT Press, Cambridge, MA. Mersenne, M 1644, Cogitata physico-mathematica (Physico-mathematical thoughts), Sumptibus Antonii Bertier, Paris. Montucla, JE 1799–1802, Histoire des mathématiques: dans laquelle on rend compte de leurs progrés depuis leur origine jusqu’à nos jours; où l’on expose le tableau et le développement des principales découvertes dans toutes les parties des mathématiques, les contestations qui se sont élevées entre les mathématiciens, et les principaux traits de la vie des plus célébres (History of mathematics: containing an account of their progress since the beginning up to our day; presenting the picture and the development of the principal discoveries in all parts of mathematics, the controversies that arose between mathematicians, and the principal characteristics of the lives of the more celebrated), nouvelle édition, considérablement augmentée, et prolongée jusque vers l’époque actuelle ed., Henri Agasse, Paris. Murray, A 1978, Reason and society in the Middle Ages, Oxford University Press, Oxford. Needham, J 1959, Science and civilisation in China, vol. 3. Mathematics and the sciences of the heavens and the earth. With the research assistance of Wang Ling, Cambridge University Press, Cambridge. Neugebauer, O 1973, Mathematische Keilschrift-Texte. Mathematical cuneiform texts, with translation and commentary in German, Springer-Verlag, Berlin. Neugebauer, O, & Sachs, A (eds) 1945, Mathematical cuneiform texts, American Oriental Society and the American Schools of Oriental Research, New Haven, CT. Núñez, RE 1997, Message: Commentary on S. Dehaene’s La bosse des maths (The number sense: how mathematical knowledge is embedded in our brains). What are numbers really? A cerebral basis for number sense, [electronic version], Edge: The Reality Club, viewed 19.vi.06, . Owens, K 2001, ‘Indigenous mathematics: A rich diversity’, in Mathematics: shaping Australia, Proceedings of Australian Association of Mathematics Teachers, Adelaide, pp. 157–67, viewed 28.vii.2006, .

090604•Growing Ideas of Number 397 97

3/11/06 13:08:00

98

REFERENCES

Oxford English dictionary 2006, viewed 30.v.2006, . Parikh, RJ 1971, ‘Existence and feasibility in arithmetic’, Journal of Symbolic Logic, 36, 494–508. Pascal, B 1665, Traité du triangle arithmétique (Treatise on the arithmetic triangle), Guillaume Desprez, Paris. Pascal, B 1963, Oeuvres complètes (Complete works), Éditions du Seuil, Paris. Peano, G 1957–59, ‘Sul concetto di numero’ (‘On the concept of number’), in Opere Scelte: a cura dell’Unione matematica Italiana e col contributo del Consiglio nazionale delle ricerche (Selected works: edited by the Italian Mathematical Union and with contributions from the National Council for Research), U Cassina (ed.), pp. 80–109, Edizioni Cremonese, Rome. Peet, TE 1970, The Rhind mathematical papyrus, British Museum 10057 and 10058, Facsim. reprint of edn pub. Liverpool, 1923, Kraus Reprint, Nendeln. Peletier, J 1549, L’algébre de Iaques Peletier du Mans, départie an quatre liures (The algebra of Jaques Peletier of Mans, presented in four books), Théodore Debesze à l’Enseigne du Pelican, Poitiers. Penrose, R 1990, The emperor’s new mind: concerning computers, minds, and the laws of physics, Vintage, London. Perso, T 2001, ‘Aboriginal numeracy’, in Mathematics: shaping Australia, Proceedings of Australian Association of Mathematics Teachers, Adelaide, pp. 357–62, Adelaide: AAMT. Prou, M 1924, Manuel de palëographie latine et française du VIe au XVIIe siècle (Manual of Latin and French palaeography from the sixth to the seventeenth century), 4th edn, Auguste Picard, Paris. Pullan, JM 1968, The history of the abacus, Books That Matter, London. Quinn, TJ 1992, ‘News from the BIPM’, Metrologia, 29(1), 1–7. Robins, G, & Shute, C 1987, The Rhind mathematical papyrus: an ancient Egyptian text, British Museum Publications, London. Robinson, A 1966, Non-standard analysis, North-Holland, Amsterdam. Rojas, R 1994, On basic concepts of early computers in relation to contemporary computer architectures, IFIP Transactions A (Computer Science and Technology), vol. A-52, pp. 324–31. Rojas, R 1998, ‘How to make Zuse’s Z3 a universal computer’, IEEE Annals of the History of Computing, 20(3), 51–4. Rose, PL 1975, The Italian Renaissance of mathematics: studies on humanists and mathematicians from Petrarch to Galileo, Droz, Geneva. Rose, S 1998, ‘Downloaded at birth’, review of The number sense: how the mind creates mathematics’ by S Dehaene, 8 February, New York Times. Rosen, F 1831, Muhammad ibn Musa al-Khwarizmi: Algebra, Oriental Translation Fund, London. Rosenfeld, A 1971, A review of A Marshack’s Notation dans les gravures du paléolithique supérieur, (Notation in scratchings in the Neopaleolithic), Antiquity, 45, 317–19.

090604•Growing Ideas of Number 398 98

3/11/06 13:08:00

REFERENCES

99

Roth, WE 1908, ‘Counting and enumeration’, Records of the Australian Museum, Sydney, 7, 79–82. Russell, B 1919, Introduction to mathematical philosophy, George Allen and Unwin, London. Russell, B 1937, Principles of mathematics, 2nd edn, George Allen and Unwin, London. Salmon, WC (ed.) 2001, Zeno’s paradoxes, Hackett, Indianapolis, IN. Sasse, HJ 1992, ‘Theory of language death’, in Language death: factual and theoretical explorations with special reference to East Africa, M Brenzinger (ed.), pp. 7–30, Mouton de Gruyter, Berlin. Schmandt-Besserat, D 1992, Before writing, volume I: From counting to cuneiform, University of Texas Press, Austin TX. Scholtyssek, C 2006, ‘Bird brains? Hardly’, Scientific American Mind, April/May, pp. 51–5. Science & Society Picture Library 2004, Bede’s system of finger counting, viewed 30.v.2006, . Seidenberg, A 1960, ‘The diffusion of counting practices’, University of California Publications in Mathematics, 3, 215–300. Seidenberg, A 1962, ‘The ritual origin of counting’, Archive for History of Exact Sciences, 2, 1–40. Seife, C 2000, Zero: the biography of a dangerous number, Viking, New York. Shen K, Crossley, JN, & Lun, AW-C 1999, The nine chapters on the mathematical art: companion and commentary, Oxford University Press, New York. Shepherdson, JC, & Sturgis, HE 1963, ‘Computability of recursive functions’, Journal of the Association for Computing Machinery, 10, 217–55. Sicard, RAC 1808, Théorie des signes pour l’instruction des sourds-muets (Theory of signs for the instruction of deaf-mutes), l’Imprimerie de l’institution des sourds-muets, Paris. Sierpin´ski, W 1965, Cardinal and ordinal numbers, J Smółska (trans.), 2nd, rev. edn, Polish Scientific Publishers, Warsaw. Skolem 1967, ‘Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by L. Löwenheim and generalizations of the theorem’, in From Frege to Gödel; a source book in mathematical logic, 1879–1931, pp. 252–263, J van Heijenoort (ed.), Harvard University Press, Cambridge, MA. Smith, DE 1923–25, History of mathematics, 2 vols, Ginn & Co., Boston. Smith, GC, Radvansky, S, & Chiba, M 1992, History of mathematics and related sciences: an annotated bibliography of sources held by Monash University Library; JN Crossley (ed.), Monash University Department of Mathematics and Monash University Library, Clayton, Victoria. Solovay, RM 1964, ‘ 2 "0 can be anything it ought to be’, in The theory of models, JW Addison, L Henkin, and A Tarski (eds), North-Holland, Amsterdam, p. 435.

090604•Growing Ideas of Number 399 99

3/11/06 13:08:01

100

REFERENCES

Spelke, ES, & Dehaene, S 1999, ‘Biological foundations of numerical thinking’, Trends in Cognitive Sciences, 3, 365–6. Stillwell, JC 1998, Numbers and geometry, Springer, New York. Stillwell, JC 2002, Mathematics and its history, 2nd edn, Springer, New York. Stolz, O 1885, Vorlesungen über Allgemeine Arithmetik. Erster Teil (Readings on general arithmetic. Part one), Teubner, Lepzig. Struik, DJ (ed.) 1958, The principal works of Simon Stevin, vol II A: Mathematics, Swets & Zeitlinger, Amsterdam. Sudbury, A, & Wright, C 1977, ‘The paradox of the unexpected examination’, Australasian Journal of Philosophy, 55, 41–58. Sydler, J-P 1965, ‘Conditions nécessaires et suffisantes pour l’équivalence des polyèdres de l’espace euclidien à trois dimensions’ (‘Necessary and sufficient conditions for the equivalence of polyhedra in Euclidean space of three dimensions’), Commentarii Mathematici Helvetici, 40, 43–80. Thorpe, WH 1963, Learning and instinct in animals, Methuen, London. Turing, AM 1950, ‘Computing machinery and intelligence’, Mind, 59(236), 433– 60. Tylor, EB, Sir 1958, Primitive culture, Harper, New York. US Metric Association 2006, FAQ: Frequently asked questions about the metric system, viewed 27.vii.2006, . Vaillant, GC 1950, The Aztecs of Mexico: origin, rise and fall of the Aztec nation, Penguin, Harmondsworth. Viète, F 1970, Opera mathematica. Recognita Francisci à Schooten. Vorwort und Register von Joseph E. Hofmann (Mathematical works. Ed. by F à Schooten. Foreword and index by JE Hofmann), Georg Olms Verlag, Hildesheim. Vuillemin, J 1990, ‘Exact real computer arithmetic with continued fractions’, IEEE Transactions on Computers, 39(8), 1087–1105. Waismann, F 1951, Introduction to mathematical thinking: the formation of concepts in modern mathematics, Ungar, New York. Wallis, J 1972, Opera mathematica (Mathematical works), Georg Olms Verlag, Hildesheim. Weisstein, EW 1999, e. Wolfram MathWorld, viewed 5.vii.2006, . Wessel, C 1799, ‘Om Directionens analytiske Betegning’ (‘On the analytic representation of direction’), Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter, Köbenhavn, 469–518. Whitehead, AN, & Russell, B 1908, second edition1925–27, Principia mathematica, 3 vols, Cambridge University Press, Cambridge. Wilder, RL 1968, Evolution of mathematical concepts: an elementary study, John Wiley & Sons, New York. Woodin, WH 2000, The Continuum Hypothesis, Lectures at the Paris Logic Colloquium, viewed 28.vii.2006, , , .

090604•Growing Ideas of Number 3100 100

3/11/06 13:08:01

REFERENCES

101

Wu, J-N 1991, Yi Jing, J-N Wu (trans.), The Taoist Centre, Washington, DC. Yesenin-Volpin, AS [Ésénin-Volpine], 1960, ‘Le programme ultra-intuitionniste des fondements des mathématiques’ (‘The ultra-intuitionistic programme for the foundations of mathematics’), Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics 2–9 September 1959, pp. 201–23, Pastwowe Wydawnictwo Naukowe, Warsaw, and Pergamon Press, Oxford.

090604•Growing Ideas of Number 3101 101

3/11/06 13:08:01

Index

ℵ0  53 2 "0   56 α-reduction  70 β-reduction  70 e  69 ε0  52, 65 π  69 ω  51 2   36 2-counting  10, 16 2-cycles  11

Arithmetic Methods for Daily Use  24 L’Arithmétique  43 Arius (256–336 AD)  81 ASCII  69 Australia  8–10, 13, 81 axiom of Archimedes  32, 37, 50, 64 of Cantor  49 of Eudoxus  32, 33 of induction  60–3, 84 of infinity  63 of reducibility  63 axioms, Dedekind-Peano  59, 60, 63–5

abacist schools  43 abacus  21, 24–8, 66, 87 Aboriginal  9, 10, 81 abstraction  70 Agnew, G  62 alcohol  80 aleph  53, 56 aleph zero (ℵ0)  53 algebra  26, 43 L’Algebra  46 algorism  26–8, 44 algorithm  44 Allen, JR  69 almucabala  43 Amazon  8 and so on  13, 77, 83, 85 animals  78, 82 anthyphairesis  36, 39 apices  24, 25, 27 application  70 archetype  78, 79, 81–3 archetype, order(ing)  83, 84 Archimedes (c.~287–212 BC)  14, 32, 37 Argand, Jean Robert (1768–1822)  47 Aristotle (384–322 BC)  19, 31, 34, 58, 63, 79 arithmetic backwards  27

Babbage, Charles (1791–1871)  66, 71, 88 babies  78, 82 Babylonians  30 back and forth argument  53 Baikiri  10 Bain, Margaret  81 base  4, 11, 12, 33, 59, 69 Beaujouan, Guy (1925–)  25 Bede, Venerable (c.~673–735)  15, 20, 25 Beeler, M  40 Begriffsschrift  63 Bernoulli, Jacob (1654–1705)  59, 84 billion  13 binary system  4, 14, 69, 70, 73, 76 Birkhoff, Garrett (1911–96)  45 Bischoff, Bernhard  27 Black, Max (1909–98)  88 Blake, Barry J  10 body parts  11 Boethius, Anicius Manlius Severinus (480–524/5 AD)  58 Bombelli, Rafael (1526–72)  43, 45–7 bones  8 Boole, George (1815–64)  63 Brainerd, Barron  14 brains  79, 84 102

090604•Growing Ideas of Number 3102 102

3/11/06 13:08:01

INDEX

Bretholz, Berthold  27 Brouwer, Luitzen Egbertus Jan (1881– 1966)  61 Bushman  10 Butterworth, Brian  78, 82, 88 calculator, handheld  28, 44 calculus  48, 64, 70 (see also lambda calculus) Caldwell, Chris K  42 calendar  8, 9, 20, 25, 28 Canada  12 Cantor, Georg (1845–1918)  48, 49, 51, 54, 56, 62, 67, 73 Cardano, Girolamo (1501–76)  42, 45, 89 cardinality  56 Carroll, Lewis [Charles Lutwidge Dodgson], (1832–98)  5, 67 Cassirer, Ernst (1874–1945)  8, 12 categorical  62, 63 China  20–4, 58, 59 Chinese  13, 20ff., 30, 34, 41ff. Chinese mathematics  88 Chrystal, George (1851–1911)  34, 38, 40, 87 Chuquet, Nicolas (1445–88)  13, 14 Church, Alonzo (1903–95)  69ff. Church’s thesis  70ff. Church-Turing thesis  70ff. Cifoletti, Giovanna Cleonice  43, 44 Clever Hans  78 closure  50 Codrington, Robert Henry (1830– 1922)  11 Cogitata Physica-Mathematica  42 collective unconscious  79 complete  50 computus  20 Conant, Levi Leonard (1857– 1916)  11, 15, 70 continued fractions  38ff., 87 continuity  36ff. Continuity and irrational numbers  49 Continuum Hypothesis  56 Conway, John  65 Cordoliani, Alfred  27 Council of Nicaea  81 countable  52ff., 63ff. countably infinite  52

090604•Growing Ideas of Number 3103 103

103

counting idea of  13ff. in groups  7 indefinitely  78, 82 kinds of  6 process of  18 counting board  21 counting rods  21ff., 42 Couturat, Louis (1868–1914)  63, 66 cube root, tied  46 cubit  29 cycles  10ff. dal Ferro, Scipio (1465–1526)  44ff. Dao De Jing [Tao te ching]  21 data  72, 73 Davis, Martin  68, 71, 75, 88 Davis, Philip J  88, 89 De Thiende  55 Deakin, Michael AB  10, 28, 87, 89 decimal system  76 Dedekind, Richard (1831–1916)  33, 37, 49ff., 59–63, 77, 84, 88 Dedekind section  50, 54 Dehaene, Stanislas  64, 78, 82, 88 Dehn invariant  34 Dehn, Max Wilhelm (1878–1952)  33 denotations  67 dense  53 Denvert, FW  16 Descartes, René (1596–1650)  30, 48 Detlefsen, Mic  25 Devlin, Keith  88 diagonal argument  54, 56, 64, 67 differential analyser  68 differential calculus  71 differential equations  68 Diophantos (3rd century AD)  46 Dipert, Randall R  63 Discorsi e dimostrazioni matematiche intorno a due nuove scienze  51 Dixon, Robert Malcolm Ward  10 Dobbs, David  88 drop world  80 Dumont D’Urville, Rear Admiral Jules Sébastien César  17 Egypt, Ancient  6, 21 Egyptian mathematics, ancient  87 electrical impulse  80 Encyclopaedia Britannica  63

3/11/06 13:08:01

104

INDEX

Engine Analytical  66, 71 Difference  66 equality of ratios  32 equation, cubic, quadratic  44ff. equivalence class  32 Eskimo  12 Euclid (c. ~300 BC)  32–4, 58, 87 Euclidean algorithm  3, 34, 38 Eudoxus (c.~408 BC)  32, 33 exotic language  16 exponential  76 exponentiation  3, 54, 56, 75 feasible computation  74, 75 Feigenson, Lisa  82 Fibonacci (Leonardo of Pisa, 1170– 1250)  58 field  65 Filipino  13 finite bounded  18 definite  18 unbounded  18 Flegg, Graham  87 Folkerts, Menso  27, 80, 87 formalisation  58 formalised language  63ff. Fourier series  51 Fowler, David Herbert (1937– 2004)  31, 36, 39 von Franz, Marie-Louise (1915– 98)  78 Frege, Friedrich Ludwig Gottlob (1848–1925)  63, 67 French  13 French Revolution  29 von Fritz, Kurt  36 fu, negative  42 Galileo, Galilei (1564–1642)  51 gap  37, 48 Gentzen, Gerhard (1909–45)  62, 65 geometry  30, 31, 34, 46, 48–50 geometry, Cartesian  48 Gerbert of Aurillac (c. 946–1003)  25, 27 Gerhardt, Carl Immanuel (1816–99)  66 German  13, 77 giga  15

090604•Growing Ideas of Number 3104 104

Gillings, Richard J  87 Gödel, Kurt (1906–78)  56, 65, 73 Gómez Pallarès, Joan  20 Gongsun Long (3rd century BC)  30 Gordon, Peter (1951–)  8, 79 Goroka Research Centre  87 Gosper, RW  40 Gow, James (1854–1923)  9 Grattan-Guinness, Ivor  6, 31, 33, 59, 86 Greece  36 Greek  15 Greeks  15, 30–2, 58 group  6ff., 21, 78, 82 Gumulgal  10 Guugu Yimidhirr  10 Hale, Kenneth L (1934–2001)  8, 13 halt  73 Halting Problem  74 Hamilton, Sir William Rowan (1805– 65)  47 Hawaii  11 heartbeats  17 Heath, Sir Thomas Little (1861– 1940)  33, 34, 87 Henry, Alan Sorley  27, 80 Hersh, Reuben  88, 89 Hilbert, David (1862–1943)  33, 50 Hindley, Roger  70 Hippasus (c. 500 BC)  31 Hoe, Jock  21 Hofstadter, Douglas  88 homomorphism  4, 29 Huber-Dyson, Verena  80, 88 hyperreals  64 incommensurable  30ff., 41 Indonesia  12 infinite  30, 33, 37, 50ff., 61ff. infinite decimals  37, 38, 69 infinitesimals  50, 64, 65 International Organization for Standardization  15 Inuit  12 invariant  4, 33 Ipili  11 irrational  2, 31, 43ff., 49 Isidore of Seville (c. 560–636)  50 ISO see International Organization for Standardization

3/11/06 13:08:01

INDEX

isomorphism  4, 50, 62ff. Italy  26, 43 Jarvis, Erich  78 Jeromson, Barry  79 Jia Xian (11th century)  59 Johnstone, William D  13 Joly, Charles Jasper (1864–1906)  47 Jung, Carl Gustav (1875–1961)  77, 79, 81ff. Jungian number archetypes  79, 82 Jungian trap  77 Kálmar, Laszlo (1905–1976)  71 Kaplan, Robert  43 Katz, David (1884–1953)  78 Keisler, H Jerome  65, 88 khílioi (χίλιοι)  15 Khiva  26 Khwarezm  26 al-Khwarizmi, Abu Ja’far Muhammad ibn Musa (c. 780–c. 850)  23, 26, 43, 80 Kitcher, Peter  88 Knorr, Wilbur (1945–97)  31 Korner, Stefan  88 La Billardière, Jacques-Julien Houtou de (1755–1834)  17 La Pérouse, Jean François Galaup, comte de (1741–88)  17 Lagrange, Joseph Louis (1736– 1813)  39 Lakoff, George  47, 48, 89 Lam Lay Yong  24 lambda calculus  69, 70 language death  12 Lao-tzu [Laozi] (4th century BC)  21 Latin  13ff., 27, 29, 51 Lean, Glendon Tolele Angove (1943– 95)  11, 87 Leibniz, Gottfried Wilhelm Freiherr von (1646–1716)  14, 37, 62, 66, 70 lemma  32, 33 lengths  3, 29ff. lengths, comparison of  32 Lévy-Bruhl, Lucien (1857–1939)  18 Li Yan (1892–1963)  21, 30, 34, 42, 58, 59, 87 Liardet, Pierre  40

090604•Growing Ideas of Number 3105 105

105

Liberia  29 lima  11, 12 limit  49, 52, 54, 57, 82 line, geometric  30, 36, 48ff., 64 linearly ordered  52, 53 LISP  69 Liu Hui (220–280)  34 logarithms  3 Lower Tully River  11 machine, non-deterministic  75 MacLane, Saunders  45 Maghreb  26 Mahayana Buddhism  81 Malaysia  12 mandala  81 Mariner, William  12 Marr, David  80 Marre, Aristide  13 Marshack, Alexander (1918–2004)  8, 9 Martin, John (fl. 1817)  12, 17 Massieu, Jean (1722–1846)  7 Master Mo (c. 470 –c. 390 BC)  30 Master Sun’s Mathematical Manual  21 mathematical induction  58, 60, 62 Maturana, Humberto (1928–)  80 Maurolico, Fransisco (1494– 1575)  41, 58, 59, 84, 86 Mega  15 Megabyte  12 Melanesia  11 memory  80 memory, computer  74 Menninger, Karl (1898–1963)  27, 87 meno di meno  46 Mersenne, Marin (1588–1648)  42 Mesolithic  8 Metaphysics  31, 79 method of exhaustion  33 metre  29 Middle Ages  25, 81 millennium bug  74 million  13 model  61ff., 71, 75, 84 modern Arabic numerals  25 Mohists  30 Montucla, Jean Étienne (1725–99)  42 Murray, Alexander  25, 87 Myanmar  29

3/11/06 13:08:02

106

INDEX

nested intervals  51 Neugebauer, Otto (1899–1990)  44 neuro-psychology  77ff. Newton, Sir Isaac (1642–1727)  37 Newton’s method  74 Nicomachus of Gerasa (fl. 85 AD)  41, 58 nima  12 Nine Chapters on the Arithmetical Art  30, 42, 58 Nine nines rhyme  24 non-standard models  64 NP  75 number Dedekind’s definition of real  50 feasible  17 figurate  41 innateness of  7, 79, 82 largest  16 length of  4, 76 limit cardinal  54, 56 limit ordinal  52 perfect  41 small  60, 82 triangular, square, pentagonal, hexagonal  41 number line  48 complete  48ff. rational  37 number names  14ff. number one  79, 80 numbers absurd, irrational, irregular, inexplicable, surd  44 amicable  41, 42 cardinal  3, 7, 53ff., 79 complex  4, 44 imaginary  44 infinite  50ff., 64 irrational  31, 43 natural  3, 51, 60ff., 74, 77 negative  42 ordinal  3, 7, 9, 51ff., 64ff. positive and negative  42ff. prime  41 rational  32, 37, 48ff. real  30, 33, 49ff., 63ff., 72ff. representation of real  74 surreal  65 numerals  2, 14 Hindu-Arabic  4, 14, 25ff., 43, 80

090604•Growing Ideas of Number 3106 106

Roman  25, 43 numerical analysis  74 Núñez, Rafael E  47, 48, 82, 89 Oakland, WH  16 one  79ff. one-to-one correspondence  6, 7, 11, 51, 56, 60, 62, 82, 87 Opera Mathematica (Wallis)  38 Owens, Kay  11 P = NP  75ff. Palaeolithic  8 Papua New Guinea  11 Parikh, Rohit  17 parrot  78 Pascal, Blaise (1623–62)  59, 66, 84 Peano, Giuseppe (1858–1932)  59, 62ff., 84 Peet, Thomas Eric (1882–1934)  23 Peletier, Jacques (1517–82)  43 Penrose, Roger  78 Pepperberg, Irene  78 Perso, Thelma  9 Philippines  12, 13, 82 physiology  80 Pirahã people  8 Pitjantjatjara  81 Pitman, Edwin James George (1897– 1993)  49 più di meno  46 Platek, Richard  27 Plato (c. 427– c. 347 BC)  31, 80 Platonism  88 platypus  62 polygon  4, 41 polynomial  22, 74 polynomial time  75ff. Post, Emil (1897–1954)  68 Principia Mathematica (Whitehead and Russell)  63, 78 Prior Analytics  34 program  68, 69, 72, 73 programming, computer  3 programming language  69, 72 proof  39ff., 45, 54, 61ff., 70, 72 Prou, Maurice (1861–1930)  27 psychology  63, 77, 83 Pullan, JM  87 pure mathematics  61, 62 pyramid see tetrahedron

3/11/06 13:08:02

INDEX

Pythagoras (c. 569–c. 475 BC)  30, 31, 41ff., 80 quaternions  47 Queensland  11 Quinn, Terry  12, 16 quintoquadrogintillion  13 ratio  4, 32, 39 reciprocally subtract  35 registers  72 repeating infinite decimals  37 representation  14, 20ff., 47, 69, 72, 73 binary  76 Rhind Mathematical Papyrus  23, 87 Robins, Gay  23, 87 Robinson, Abraham (1918–74)  64 Rojas, Raúl  68 Romans  14, 25 Rose, Paul  45 Rose, Steven  84 Rosen, Frederic  27, 43 Rosenfeld, Andrée  8 Roth, Walter Edmund, (1861– 1933)  11 Rule of Three  20 Russell, Lord Bertrand Arthur William (1872–1970)  6, 7, 18, 32, 56, 63, 65, 77, 78, 82 Russell’s paradox  56 Sachs, Abraham J (1915–83)  44 Salmon, Wesley C  37, 80, 87 Sand Reckoner  14 Sasse, Hans-Jürgen  12 Schmandt-Besserat, Denise  6, 7, 18, 87 Scholtyssek, Christine  78 Science and Society Picture Library  15, 25 Science Museum, South Kensington, London  66, 88 scratches  8 Seidenberg, Abraham (1916–88)  8, 10, 80 Seife, Charles  43 semantics  67ff. sequences  48, 53, 74 series, geometric  3, 38, 42 series, Fourier  51

090604•Growing Ideas of Number 3107 107

107

set theory  52ff., 62ff. sets  49 sets, countable and uncountable  52ff. Shaanxi  21 Shen Kangshen  21, 30, 34, 42, 58, 59, 87 Shepherdson-Sturgis machines  71, 72 shortest length  36ff. Shute, Charles  23, 87 Sicard, Abbé Roch-Ambroise Cucurron (1742–1822)  7 Sierpin´ski, Waclaw (1882–1969)  54, 88 Skolem, Thoralf Albert (1887– 1963)  63ff. Smith, David Eugene (1860–1944)  22 Smith, Gordon Charles  86 Society for the Promotion of Christian Knowledge  9 Society for the Propagation of the Gospel  9 Solomon Islands  11 Solovay, Robert M  56 South Africa  10 South America  10 Spain  26 Spanish  13 Spelke, Elizabeth  82 Stambul, Pierre  40 Stanford Encyclopaedia of Philosophy  86 Stevin, Simon (1548–1620)  44, 55 Stillwell, John Colin  34, 87 Stone Ages  8 Struik, Dirk Jan (1894–2000)  44, 55 subset  47, 52ff., 56, 63ff., 67, 76 subtraction  23, 36, 47, 66 successor  60 Summer Institute of Linguistics  9 surd  44 surd, quadratic  39 Sydler, Jean-Pierre  34 symmetry  81 syntactically different  70 syntax  65, 68, 70, 72 Tagalog  13 tallies  11, 83ff. tetrahedron  30, 33ff.

3/11/06 13:08:02

108

INDEX

theorem  61 of Gödel  65, 88 of Pythagoras  30 Thorpe, William Homan (1902– 86)  78 time-factored  8 Tonga  12, 17 Trinity  81 Turing, Alan Mathison (1912–54)  67ff. Tylor, Sir Edward Burnett (1832– 1917)  7, 8 unbordered  53 unbounded  53 uncountable  54ff., 63ff., 67 unending repetition  18, 36, 78 universal characteristic  66 universal linearly ordered set  53 universal machine  68, 72 universal Turing machine  72 unlimited register machines  71 unsolvable problem  74 US Metric Association  29 Uzbekistan  26 Vaillant, George Clapp (1901–45)  81 Val di Chiana  45 Viète, François (1540–1603)  42 vigintillion  13

090604•Growing Ideas of Number 3108 108

Vuillemin, Jean  40 Waismann, Friedrich (1896–1959)  88 Walbiri  8, 13, 82 Wallis, John (1616–1703)  38 Warlpiri  see Walbiri water  80 waveforms  51 wavelength  30 Weisstein, Eric W  40 Wessel, Caspar (1745–1818)  47 White Knight’s Song  67 Whitehead, Alfred North (1861– 1947)  63, 78 Whiteside, David  86 Wikipedia  86 Wilder, Raymond Louis (1896– 1982)  88 Woodin, Hugh  57 World Wide Web  86 Wu Jing-Nuan  14 Yang Hui (1238–1298)  24 Yesenin-Volpin, AS  17, 18 Yi Jing  14 yotta  15 Zeno (c. 490–c. 430 BC)  37, 87 zetta  15 zheng, positive  42 Zuse Konrad (1910–95)  68, 70

3/11/06 13:08:02

5 75

2 2 0 0 9 9 8 8 1 1 3

Series Editor David Leigh-Lancaster

Growing Ideas of Number

Growing Ideas of Number explores the notion of how number ideas, and ideas of number, have grown from ancient to modern times throughout history. It looks at how different types of number and views of numbers (and their meaning and applications) have varied across cultures over time, and combines historical considerations with the mathematics. The book illustrates some of the real problems and subtleties of number, including counting, calculation, measuring and using machines, which ancient and modern people have grappled with—and continue to do so today. It includes a comprehensive range of illustrative examples, diagrams, tables and references for further reading, as well as suggested activities, exercises and investigations. John N Crossley moved to Australia in 1969 from All Souls College, Oxford, when he became

Professor of Pure Mathematics at Monash University. In the 1980s he drifted into Computer Science and in 1994 became Professor of Logic, in the Faculty of Information Technology. He has worked on the history of mathematics and on 13th century medieval history, as well as theoretical computer science. This is his 20th book.

1

3 67 4 6 3

8

2

1

5

THE EMERGENCE OF NUMBER

GROWING IDEAS OF NUMBER

5

THE EMERGENCE OF NUMBER

2

Jo hn N C ro ssl e y

3 67 4 6 8

DAV I D L E I G H - L A N C A S T E R ( S e r i e s E d i t o r )

Series Overview The Emergence of Number series provides a distinctive and comprehensive treatment of questions such as: What are numbers? Where do numbers come from? Why are numbers so important? How do we learn about number? The series has been designed to be accessible and rigorous, while appealing to students, educators, mathematicians and general readers. ISBN 10: 0-86431-709-3 ISBN 13: 978-0-86431-709-4

9

780864 317094

John N Crossley ideas of number cover.indd 1

3/11/06 2:21:48 PM