Growing Ideas of Number (The Emergence of Number)

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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.

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THE EMERGENCE OF NUMBER

GROWING IDEAS OF NUMBER

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THE EMERGENCE OF NUMBER

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Jo hn N C ro ssl e y

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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

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Growing ideas of number John N Crossley

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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


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Growing ideas of number John N Crossley

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

ACER Press


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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


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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

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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

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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

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CONTENTS

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

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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

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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

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

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

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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

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References Index


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List of figures

1.1 Two-counting

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

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2.2 Counting board with counting rods

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2.3 Adding the ancient Chinese way

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

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

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

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

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

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3.3 Rotating the square in Figure 3.2

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3.4 Constructing fractions

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4.1 Bombelli’s rules for multiplication of imaginary numbers

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5.1 Counting the positive rational numbers

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5.2 The ‘back and forth’ argument

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7.1 The essentials of a Turing machine

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

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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


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SERIES OVERVIEW

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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.


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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.


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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


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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.


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CHAPTER ZERO

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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


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1


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.


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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


a 1-r

(2)

(3)

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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


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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.)


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CHAPTER ONE

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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


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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


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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


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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)


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10


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).


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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


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12


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


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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


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14


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


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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


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16


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


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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.)


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18


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.


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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.


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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


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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


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22


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).


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+

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


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24


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).


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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.


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26


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.)


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

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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.


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28


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).


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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


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1

30


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


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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.


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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


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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)


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34


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).


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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


b = r0 q1 + r2

(3.7)

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36


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


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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.


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38


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,


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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.


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40


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.


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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


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1

42


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


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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


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44


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


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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


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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


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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).


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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


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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’.)


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50


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)


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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


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

(5.2)

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52


(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

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