Michael Deakin the Name of the Number 2007

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Michael A B Deakin

T H E E M E R G E N C E O F N U M B E RD A V 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 )

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The Name of the NumberWere used to the idea that closely related languages have words that are similar to English. For example, the word for three in Latin, French, Italian and German is tres, trois, tre and drei. But did you know that the word for three in Sanskrit is trayah? How can words from completely different languages and cultures be so similar? Why do unrelated languages like English, Japanese and Chinese all possess a base ten counting system? Did you know that the Latin root of the word calculate means pebble?

The Name of the Number looks at the history and anthropology of the expression of numbers throughout the ages and across different cultures. It deals with the different ways that number representation has been structured, the history and prehistory of number concepts, and the evolution of numerical representation (in word and symbol). These themes are explored through the various expressions of number-concepts in different cultures in different places and times.

Michael A B Deakin has interests in the History of Mathematics, applied Mathematics (especially Biomathematics) and Mathematics Education. He taught at Monash University from 1967 until 1999, and has also taught in the USA, the UK, PNG and Indonesia. He was the editor of Function, a journal of School Mathematics (now incorporated into Parabola, for which he contributes a column on the History of Mathematics). He has authored over 100 technical papers and over 200 popular expositions. He is an honorary research fellow at Monash University.

Series OverviewThe 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.

T H E E M E R G E N C E O F N U M B E R Series Editor David Leigh-Lancaster

ISBN 978-0-86431-757-5

9 7 8 0 8 6 4 3 1 7 5 7 5

The name of the numberMichael A B Deakin

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

ACER Press

The name of the numberMichael A B Deakin

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

First published 2007by ACER PressAustralian Council for Educational Research Ltd19 Prospect Hill Road, Camberwell, Victoria 3124

Copyright 2007 Michael A. B. Deakin 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 Carolyn GlascodineCover design by Mason DesignText design by Mason DesignTypeset by Desktop Concepts Pty Ltd, MelbournePrinted by Shannon Books

Cover photograph Musee du Louvre, Pyramid by Will & Deni McIntyre/Stone Collection/ Getty Images

National Library of Australia Cataloguing-in-Publication data:

Deakin, Michael A. B. (Michael Andrew Bernard).The name of the number.

Bibliography.Includes index.ISBN 9780864317575.

1. Numeration History. I. Leigh-Lancaster, David. II. Title. (Series: Emergence of number).

513

Visit our website: www.acerpress.com.au

vContents

List of tables viiSeries overview viiiAbout the author xPreface xi

0 Introduction 1

1 The language families of the world 6

2 The notion of a base 17

3 Other aspects of number: words and symbols 29

4 Grammar: the grammatical status of number-words 37

5 Early history of numerical concepts 49

6 Developed systems of number-words 55

7 Projects 647.1 Roman numerals 647.2 Bases other than ten 687.3 Counting rhymes 747.4 The number-word game 78

8 Resources 828.0 Introduction 828.1 The language families of the world 838.2 The notion of a base 878.3 Other aspects of number: words and symbols 908.4 Grammar: the grammatical status of number-words 928.5 Early history of numerical concepts 948.6 Developed systems of number-words 96

C O N T E N T Svi

8.7 Projects 988.7.1 Roman numerals 988.7.2 Bases other than ten 988.7.3 Counting rhymes 998.7.4 The number-word game 100

References 101Index 106

vii

List of tables

1.1 The first ten number-words in PIE 14

2.1 The first ten number-words in several languages 19

2.2 The first ten number-words in English and Thai 19

2.3 The first ten number-words in English and Mugwump 19

2.4 The first 12 numerals in Chinese notation 21

2.5 The first 59 numerals in Babylonian cuneiform 23

2.6 The zero and the first 29 numerals in Mayan notation 24

2.7 The Kewa system of counting using parts of the body 26

2.8 The first 21 number-words in Northern Fore 27

2.9 Number-words once used in Motu 27

2.10 The decads in French 28

3.1 Measure-terms in Kwakiutl 35

5.1 Parallel between pronouns and numerals 52

7.1 The first thirty-five numbers in hexadecimal 71

7.2 Reciprocals of the first few numbers in base ten 73

7.3 Number-words in the Brythonic languages 74

7.4 Two versions of the sheep-score 75

7.5 The first ten numbers in six North England dialects 77

7.6 The number-word game in Motu 80

viii

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 its 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 withand 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

S E R I E S O V E R V I E W ix

who we are in our own words and those of our ancestorsthe 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.

xAbout the author

Michael Deakin has interests in the History of Mathematics, in applied Mathematics (especially Biomathematics) and Mathematics Education. He taught at Monash University from 1967 until 1999, but has also taught Mathematics in the USA, in the UK, in Papua New Guinea and in Indonesia. He was for many years the editor of Function, a journal of School Mathematics (now incorporated into a sister journal Parabola, for which he continues to contribute a column on the History of Mathematics). He has authored over 100 technical papers and over 200 popular expositions of Mathematics.

xi

Preface

This book collects material from a number of papers I have written over the past twenty-five years or so, dealing with aspects of numberin particular the influence of language on the evolution of the number concept. I am glad of the chance to revisit this material and to put it all together, and thank ACER and David Leigh-Lancaster for the opportunity to do so.

In a few instances, I have revised opinions I once held, so that there may in places be some inconsistency between what I wrote earlier and what I say here. There are only a few such cases, and in no case are they very important. For this reason, I have not drawn attention to them. However, the reader is advised that the views expressed here are to be preferred to those earlier expressions wherever any such conflict arises.

My aim is to be scholarly and authoritative. However, in an attempt also to be accessible, I have consciously avoided jargon, technical language and specialist notations. Where some technical detail seemed to me to be unavoidable, I have attempted to explain the concept in non-technical language. A book such as this necessarily depends heavily on the work of others, and all such debts are duly acknowledged in the notes supplied for each individual chapter. It is important in historical writing to say not only what we know about the past, but how we know it, and where opinions are expressed, not only to present those opinions, but to say why we hold them. This I have done throughout but, in order not to interrupt the flow of the story, these details are collated together in Chapter 8. The list at the end of the book of works cited provides a convenient summary.

In the preparation of this book, I have benefited greatly from the comments of John Crossley. There is much in common between this book and the one that he has written for publication in parallel with it. I do not draw specific attention to points of correspondence between our separate contributions, but the reader will see many such connections. The two different approaches should be seen as complementary.

It is also apposite to record that John was one of the sources of my own interest in the subject of Number and its history and in the history of Mathematics in general.

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C H A P T E R z E R O

Introduction

When we seek to find out what happened in the past, we are usually said to be studying History, and this is how most people would understand the term. History is the study of the past. However, the word history has a narrower but more precise technical meaning besides this popular usage. Professional historians confine their researches to written records or nowadays written records as supplemented by further material such as tape recordings, photographs and other products of our technological age. This restriction greatly narrows the scope of History properly so called, because written records can only go back so far in time; before then, the possibility did not exist either because writing was not yet invented, or else the relevant documents have not managed to survive.

Here are two examples. The first concerns the work of a philosopher we know as Zeno of

Elea. He lived in Greece in the 5th century BCE, and is remembered for a set of four paradoxes all concerned with the nature of space and time. These have ever since loomed large in the thoughts of mathematicians and indeed have only achieved satisfactory resolution within the last 300 years. What Zeno seems to have been concerned to show is that space can neither be discrete, nor can it be continuous, with the same going for time. Whatever view we take of space and time, we reach a contradiction; Zenos complete set of contradictions would therefore seem to support the view that space and time do not really exist, but rather are illusions.

However, we cannot be entirely clear on this because we have no record of what Zeno actually said; we only know what other (later) philosophers said that he said. (We owe the most complete account to Aristotle, who lived about a century later.) So it is open to modern scholars to dispute how accurately Zeno has been reported and what his purpose actually was. Zeno is a historical figure, but only just; we would like to have more precise evidence of his life and thought.

T H E N A M E O F T H E N U M B E R2

The second example concerns the stories of King Arthur and his knights. These have some factual basis in the wars fought by the Romanised Celts of Britain against the Anglo-Saxon invaders of the 5th century CE. The Roman colonisers, having ruled Britain for some half a millennium, departed, and Britain was in consequence left weakened in its ability to repulse these new invaders, despite the efforts of Celtic commanders like Arthur to do so. The Arthurian legend, however, has acquired so many clearly impossible elements (swords in stones, ladies in lakes, and so on) that it cannot be regarded as a factual account in any but the most vague and general sense. Arthurian Britain lies rather beyond the fringe of History, as the historical elements of the story have become overlaid with much that is clearly myth. All the written records date from much later times.

The lack of written records as we delve further and further into the past means that other tools must be used to try to get at the truth of what happened back then. We are now in the realm of Prehistory. One such tool is Archaeology, which is concerned with unearthing (literally) a record of the ancient peoples by discovering the artefacts they left behind. (In very rare and fortunate cases, we find in this way actual documentary records and so extend the scope of History, properly so called; this has happened with the records of the mathematics performed in Babylon and in ancient Egypt.)

Rather distinct from Archaeology is Palaeontology, which, by and large, digs deeper and looks further back still. Yet other tools of Prehistory are the collection of folklore, legend and oral tradition. But perhaps pre-eminent among all of these is Linguistics. As has been written:

Language fords times swollen river,

It leads to our ancestral home,

But they will arrive there never,

Who fear the deep and threatening foam.

Much of this book will explore the consequences of this view, which will occupy later chapters. Here I will give a single example to show the general approach of the method. Because much of the material of this example is largely checkable against actual historical material, it allows independent verification of a kind not possible in the case of the later examples to be adduced.

Consider the number-word eight for the numeral 8. We pronounce this as ate, or eit. Yet it contains those apparently extraneous letters gh. Spelling reformers have long sought to remove such anomalies, which, however, persist despite their best efforts. Why, we can ask, were these extra letters ever put there in the first place?

I N T R O D U C T I O N 3

We learn from written history that this word came to Britain in the 5th century CE with the Anglo-Saxon invaders. (They won and so their records are much better preserved than those of the Celts, who lost!) The invading tribes came from what is now Germany, and the language they spoke was Germanic. If we look up the word for 8 in modern German, we find that it is acht.

Now look at the similarities between eight and acht. Both begin with a vowel. This is followed by a guttural (back of the throat) consonant, which is pronounced in the German, although it is silent in the English. Then, both words end in t. This reinforces our knowledge that the words are historically related, for their linguistic structures are also related.

We may extend this example by looking at other languages. Take Latin. This is now a dead language; no one speaks it as their mother tongue. But it has left behind an enormous literature, which is still with us, and we still have amongst us lots of people with the skill to read it. Because of this, we are able to say with complete confidence that the Latin word for 8 was octo. This also has the structure (vowel, guttural consonant, t), followed in this case by a second vowel. For the moment, forget this second vowel; it will be dealt with later.

The most direct descendent of Latin in todays world is Italian, so let us look at that. The Italian word for 8 is otto, and this we can also analyse. It is not to be pronounced as we in English would pronounce the boys name Otto. Rather, there is, between the first o and the following t, a consonant that, in a sense, doesnt exist as a true consonant, but is heard nonetheless. It appears as a sort of interruption to the natural flow of the sound. Linguists refer to it as a glottal stop, and we make it by momentarily closing the back of the throat between two sounds, in this case, the o and the t. This sound is not used in Australian English, but it is evident in other English dialects, notably Cockney. (A Cockney would call it a gloal sto .) So the upshot of all this is that the Italian preserves the form of the Latin, but has modified the second element.

There are many other Latin-derived languages, of which the principal ones are Spanish, Portuguese and French. The corresponding words in these languages are respectively ocho, oito, and huit. If we compare the Spanish with the Latin, we see that the Latin ct has slurred to a ch, but otherwise the structure remains intact. A different modification occurred with the Portuguese: the first consonant of the pair ct modified (essentially, as pronounced, to a wi), but the second remained intact. This now enables us to understand the further evolution that produced the French, which is pronounced like the English word wheat. In this case, the beginning and the end of the word have both


disappeared. These various modifications will be examined in much more detail in Chapter 1.

All this is very well attested. We know a very great deal about how Italian, Spanish, Portuguese and French evolved from their ancestral Latin. These and the other less well-known Latin-derived languages form a grouping, or clan, known collectively as the Romance languages. But German is not a Romance language, and neither is English (although it does contain many Romance elements, as a result of the invasion of Britain by the French-speaking Normans in 1066 CE). Both of these languages are classified as Germanic languages (along with the Scandinavian languages, and others).

But just as the different Romance languages are related to one another, and the different Germanic languages are themselves interrelated, so too the clan of Romance languages and the clan of Germanic languages are related to one another. And other such clans are also related to these two. We will look into this whole story in much more detail in Chapter 1, but for the moment, accept that all these clans form part of a large language family, which is named Indo-European. The three classical Indo-European languages, each of which has left behind a large body of literature, are Latin, (ancient) Greek and Sanskrit.

Because we still have this large body of literature, we know what the words for 8 were in all these languages. In Latin, as we have seen, the word was octo, in ancient Greek, it was okt, which looks very similar. The corresponding Sanskrit word was aa, which looks somewhat different. (A complication arising in both these cases is that these languages use different alphabets from our own; when different authors attempt to render words in these languages into our familiar alphabet, they may use different conventions, and so the results may look different from one another.)

The chief difference between the Latin and the Sanskrit is that a c or k sound in the former is replaced by a form of s. The various alterations that occurred as the independent daughter languages developed, all conform to relatively simple patterns, which will be described in much greater detail later. For the moment, let us merely note that that all three of these classical languages are themselves derived from an even earlier language, now lost, and never written down.

This ancestral language has been called Proto-Indo-European, PIE for short, or sometimes more simply, Indo-European, IE for short. Here I will refer to it as PIE. PIE left us no written records, so when we speak of PIE, we are well and truly in the realm of Prehistory. Nonetheless, the rules by which languages evolve are now so well known and widely accepted that we confidently assert a lot about it.

I N T R O D U C T I O N 5

In order to be up-front and to make clear the distinction between what is History and what is Prehistory, the convention has been adopted that where a word has actually been attested (in the written record), then it is simply spelt out as I have done with all the words so far listed as being in use, either now or in the past, for the number 8. However, when we go beyond this realm and enter the world of reconstructed forms, we precede the word by an asterisk. Thus the PIE word for 8 is presented as *octo(u) or as *okto(w) by different authors. The asterisk means that, instead of discovering the rules of linguistic evolution by going forward in time, as from the Latin to the later Romance languages, we now apply those rules in reverse to reach a time before the written record commences.

But we, looking at the results of such work, can appreciate the result by going forwards. Thus we can see that English and German both dropped the final vowel, and now employ a shortened form; in French, the shortening went further and both of the original vowels were dropped to be replaced by an intermediate vowel, whose insertion is also seen in the Portuguese. The guttural consonant c or k has been reduced to a fossil in English, to a glottal stop in the Italian, has slurred in the Spanish and has modified in the Portuguese and the French. We can see the family relationships involved.

Thus, linguists specialising in PIE would claim that (perhaps with a small number of possible exceptions) all languages of the entire Indo-European family have words for 8 that derive according to fixed rules from *octo(u) or *octo(w). Later chapters of this book will look in much more detail at the underlying principles involved here, and will show that what applies for one number, 8, is also true of others, and will go on into even deeper waters.

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The language families of the world

As outlined in the last chapter, a great many of our familiar languages are related to one another. English and German are both seen as members of the Germanic clan within the Indo-European family. The Romance languages are seen as belonging to another clan, and we now recognise others. The discovery of these relationships owes its origin to the work of Sir William Jones, who in the 18th century posited a connection between the three classical languages: Latin, ancient Greek and Sanskrit, and, besides these, Gothic and Old Persian. Later, in 1820, the German linguist Franz Bopp published an English translation of his major study under the title Comparative Grammar of Sanskrit, Zend (Avestan), Armenian, Greek, Latin, Lithuanian, Old Slavic, Gothic and Germanic. This work marked the transition of the theory that these languages display a family resemblance from controversial hypothesis to accepted fact.

Nowadays, we list a number of branches all within the Indo-European (IE for short) family, and it is widely accepted that these relationships are all real, although some branches are more closely related than are others. There are now believed to be about a dozen such branches, with some details still not universally agreed. The Germanic clan, or branch, has already been noted; the Romance clan is a subset of a larger Italic branch of the IE family, which also includes some other now dead languages that were related to, but not derived from, Latin. Lists of the branches vary in points of detail, but the following list or some close variant of it is very widely accepted:

Albanian Germanic Iranic Armenian Greek Italic Baltic Hittite Slavonic Celtic Indic Tocharian

C H A P T E R O N E

T H E L A N G U A G E F A M I L I E S O F T H E W O R L D 7

Let us look at these in turn. The Albanian branch of Indo-European is a small one, comprising a

single member among the living languages, and the same is true of the Armenianbranch. Both branches preserve some interesting features of the original PIE, but both have also been affected by interaction with non-Indo-European languages. For our purposes here, they are less relevant than the larger branches that testify in greater detail to the structure and vocabulary of PIE.

The Balticis a larger clan and has greater significance for the study of PIE. It comprises Lithuanian and some of its relatives. Some authorities combine the Baltic and Slavonic branches into a single Balto-Slavicbranch as the two are seen as quite closely related. Lithuanian preserves many features of the original PIE, and for this reason is much studied. Had it developed a written form earlier than it did (only in the past three centuries!), it would have ranked beside the classical languages as a source of data concerning the ancestral form.

The Celtic branch is an interesting and important one also. Celtic languages were once very widely spoken throughout Europe and Western Asia. The Galatians of the New Testament were Celts (as indeed their name implies). They lived in Asia Minor (todays Turkey) and survived as a distinct people until the 5th century CE. At the other end of the European continent, the Celtiberians inhabited todays Spain and Portugal back in Julius Caesars time. The Church Father St Jerome, an authority on languages who translated the Bible into Latin, noted the resemblance of Galatian to the language of the Treveri, a group living in Switzerland and almost the last remnant of a once widespread continental Gaulish culture. The branch now survives only as a small group of minority languages, from the Western fringes of Europe, and of which the only one not today an endangered species is Welsh. The Celtic languages will be the subject of further attention in Chapter 7.3.

The Germanic clan or branch has already been noted; besides English, German and the Scandinavian languages (but excluding Finnish and Saami (Lappish), which are actually not Indo-European languages at all), it also includes Dutch and some lesser-known languages.

The Greek branch is now reduced to a single member. Modern Greek is related to several now extinct forms of ancient Greek, of which Attic is the best known, and the branch also includes some other less closely related dead languages.

Hittite is an interesting case. The Hittites were a once mighty empire centred in Anatolia (modern Turkey). Hittites are to be found throughout much of the Hebrew Bible (Christian Old Testament). Abraham was buried


in land bought from the Hittites (Genesis 25:10) and Uriah (whom King David sent to his death in order to get his hands on his wife Bathsheba, 2 Samuel 11) was a Hittite. Their language was undeciphered for many years, but once the key to its pronunciation was grasped, it became clear that it was a member of the IE family and this led to its decipherment. Other possibly related dead languages are also known and some authorities group these together with Hittite in a larger Anatolian branch.

The Indic branch is a large one. Most of the living members are direct descendants of Sanskrit. There are, however, a few somewhat more distant relatives collectively known as the Dard languages. Some authors combine the Indic clan with the next, the Iranic, into one large Indo-Iranian branch. Yet others regard the Dardic languages as constituting a separate branch or clan. Thus, on one account, there is one branch combining three clans: Iranian, Indic and Dardic. On another, there are two branches, Iranian and Indic, of which the second comprises two clans: the Sanskrit-derived languages and the others (the Dardic). The surviving members of the Iranicclan descend from two older, now dead, languages: Avestan and Old Persian. This latter is the ancestor of most of todays Iranian languages, but Pashto, a language of Afghanistan, is usually regarded as deriving from Avestan.

The Italic branch has no living members other than the Latin-derived Romance languages, although Latin had some (now dead) cousins which are included in the branch, but which have left no descendants. Besides the Romance languages already noted (French, Italian, Portuguese and Spanish), there is a considerable number of minor Romance languages and one that is not so minor: Romanian. This last arose from the actions of the Roman Emperor Constantine I, who in the 4th century CE moved his capital from Rome to Byzantium (todays Istanbul), modestly renaming it Constantinople. Romanian has absorbed many influences from the neighbouring Slavic countries. So, just as English is a Germanic language with many Romance elements, Romanian is a Romance language with many Slavonic features.

The Slavonic clan is another large one, even without the Balticlanguages with which it is sometimes combined into a single even larger branch. Most of the languages of Eastern Europe (Russian, Polish, Czech, Serbo-Croat, Bulgarian, Ukrainian, etc.) are Slavonic. The main exceptions are Romanian (already noted) and Hungarian, which is not even Indo-European.

Finally, there is the Tocharian branch, comprising two now dead languages from Mongolia. They have a theoretical importance to be discussed later.


It is now universally accepted that there was a definite language, PIE, from which all these branches and the languages that constitute them developed. By looking at the rate at which languages diverge from one another, we can attempt to estimate when this language held sway. However, there still remains considerable divergence of opinion as to the time when PIE was spoken. Most authorities opt for a time around 30004000 BCE, but other estimates have also been given.

Nor is it agreed where the heartland of PIE was located. One theory has it that a tribe called the Kurgans invaded Europe and brought their language with them from their home range north and east of the Black Sea. Another says that the PIE-speakers were predominantly farmers and occupied Anatolia (modern Turkey), and that the language spread out from there as agriculture took over from earlier hunter-gatherer societies. And there are others also.

A surprising recent suggestion is that the language was first spoken in an area that is today under the Black Sea. This theory has it that the area in question was a large basin below sea level, but separated from the Mediterranean by a sill of higher ground that acted as a dam. This was eventually breached and so the Black Sea flooded, in the process giving rise to the legends of the great flood that is recorded in both the Sumerian Epic of Gilgamesh and in Chapters 68 of the Book of Genesis.

Various attempts have been made to reconstruct a family tree showing how these different branches relate to the PIE trunk. Two points are almost universally agreed: the connection between the Balticand the Slavonicbranches, and the other connection between theIndicandIranian ones. Many linguists also hold that Albanian andGreek are more closely related to each other than they are to other members of the IE family. Beyond this, little is agreed.

A once widely held theory split the family at a point early in time into two main groupings: a Western or centum group and an Eastern or satem group. The technical words for these branches come from their words for a hundred. The PIE word is *kentom, and from this derive the Latin centum, and many relatives. (We still have per cent for example, and the word hundred itself is a more distant descendant.) In Sanskrit, the word is atm and in Avestan it is satem. These words derive from the original *kentom according to rules to be outlined below. The theory was that this change preceded the subsequent changes, which were seen as occurring within these two major groups.

This theory once held great sway, and it still commands some acceptance today. However, most current opinion is against it. The reason is that the Tocharian languages, which were the geographically most eastern


branch of the IE family, are centum languages, not satem ones. It thus seems more likely that the *kentom-satem shift was not as basic as formerly thought; rather it is a change that has occurred several times independently.

Thus many matters remain unresolved, but it is no longer the case that any reputable linguist doubts the existence of PIE and the descent of later languages from it. Linguists also agree on the overall picture of its vocabulary and grammar. In particular, the words for the numerals are (apart from minor disagreements) now considered well known.

The evolution of languages proceeds according to fixed rules that enable us to reconstruct such aspects of PIE from the subsequent daughter languages. Here I will concentrate on two of these rules, although there are also other more subtle ones.

The first rule is:

Rule 1: Vowels are much more variable than consonantsHere is an example of this rule in action. The deprecation of broad Australian accents amounts to an objection to the pronunciation of the vowels, because (apart from some very rare exceptions) the consonants are unaffected.

Or think of the old music-hall joke that has us saying The rhine in Spine stize minely in the pline. About twenty years ago, a prominent overseas politician thought to use this to poke fun at the Australian accent, but it was only a matter of time before some wag pointed out that he said The ren in Spen stezz menly in the plen. Again, it is the vowels and the vowels alone that have altered.

So now we have a handle on the vowel changes noted in the Introduction in the case of the words for 8. Vowels change easily and also, as a corollary, may also disappear easily.

The second rule tells us how consonants change.

Rule 2: Except in rare and special circumstances, consonants ride forward in the mouth as time goes by

To see this rule in action, consider again the case of the number 8, discussed in the Introduction. We saw that the ancestral PIE word was *octo(u) or perhaps *octo(w). This form is preserved quite well in both the Latin octo and the Greek okt. However, the Sanskrit word was aa, and here we see that the original c or k sound has been replaced by a form of s. Here the original consonant is a guttural consonant, made in the back of the throat, while the later Sanskrit word uses the sibilant s, made in the front of the mouth by the tongue and the teeth. (Exactly this same change gave rise to the Sanskrit and Avestan words for a hundred.)


We see the same direction of sound-shift in the change from the Latin to its daughter languages. The Italian has only modified slightly, but the Spanish replaces the guttural c by a ch, a frontal consonant. In the Portuguese and the French, the process has gone even further and the consonant is w, made with the lips. We can also look at the Romanian. Here the word for 8 is opt, and we now recognise that the original guttural consonant c has mutated into a labial (made with the lips) p in accordance, again, with Rule 2.

Now consider a more difficult case, the number 4. In order to expedite matters, I will start the discussion of this number-word with the PIE and travel forward in time. The PIE word for 4 is given as *kwetwores or as *kwetwor by different authors. The initial consonant is believed to have been intermediate between k and kw, hence the somewhat unusual form given.

If we now look at the three classical languages, we find the Latin quattuor, which is quite close to the PIE, with essentially only a vowel change in accordance with Rule 1. However, in ancient Greek, we have tettares, and we see an application of Rule 2, as the initial kw has moved forward to become the dental consonant t, made between the tongue and the teeth. (The second consonant, tw, has morphed into a t; we see this same shift in our pronunciation of the word two.) The Sanskrit was chatvaras or chatasvarah, and the initial consonant has also slid forward in this case. We will see later why there are different forms.

The other consonantal change involved in the Sanskrit is the replacement of a w by a v before the final syllable. These two consonants are in fact very closely related. If you are familiar with Charles Dickenss novel The Pickwick Papers, you may recall a scene in which one character, a Sam Weller, instructs a judge on how to spell his surname: Put it down a we, my Lord, put it down a we. Elsewhere, he advises: Be wery careful o vidders all your life. Another example is more recent. During the beatification ceremony for Mother Mary McKillop, Pope John-Paul II, reading the order of service in a language not his own, was faced with the words in accordance with church law; this he initially pronounced (before correcting himself) in accordance with church love.

If we now come further forward in time, we reach the French quatre, the Spanish cuatro, the Italian quattro and the Portuguese quarto, all of which are very clearly derived from the Latin. Rather more divergent is the Romanian patru, and here the second vowel has altered, which should come as no surprise because of Rule 1 (this also happens in the French), but the initial consonant qu has mutated into a p, in accordance with Rule 2. Here the consonant has moved even further forward than in the Greek; as we saw before, p is made with the lips.


The same shift has taken place in the Celtic languages. Here we have in Welsh, pedwar or pedair. (Again, we will see later why there are two forms.) Breton has peuar or peder, Cornish has pajer, and in each of these cases, we can follow the lines of descent from the PIE.

In the Germanic languages, we see a related shift in the initial consonant. Old German has fidwor. The initial consonant here has become an f (and the later t has become the voiced sound d. The Anglo-Saxon word was feower, and this shows clear affinity to the Old German. The d sound has disappeared, being replaced by a modification to the preceding vowel. (This is the same process that gives us hard, as opposed to had; Australians do not pronounce the r as a consonant, although most Americans do.)

But now feower is clearly related to the English word four. We compress the two syllables into one, although there are American dialects that do not. However, we now have in front of us a full enough story to validate the descent of our word four from the PIE *kwetwores. Had this relationship been proposed without the intermediates, we might well have been inclined to dismiss it out of hand, but now we have the fuller picture and so are better placed.

Just to finish off one annoying detail, however, let us return to the full statement of Rule 2, which allowed exceptions in rare and special cases. One such occurs with the number five. The PIE is universally agreed to be *penkwe, and so this must be ancestral to the Latin quinque. The initial p has been replaced by a qu, which is a trend in exactly the wrong direction. When such exceptions occur, they demand explanation. They have to be explained away.

The explanation given for this particular anomaly is that in counting: , quattuor, penque, , the ancestral Romans adjusted the initial p sound to accord with the qu sounds around it. The effect is a quite common one. We often hear even educated people speak about honing in on when they mean homing in on. What they say is in strict language a complete nonsense, but the adjacent n sounds exert a profound influence on the m sound and work to alter it. Similarly, we frequently hear sporting commentators speak about a games stastistics, or describe the person who records them as our stastistician. This is the same effect.

While we are talking about the words for 5, look at the derivatives of that Latin word quinque. The French is cinq, the Spanish and the Portuguese both have cinco, the Italian is cinque and the Romanian is cinci. Very clearly all derive from the Latin. In pronunciation, the initial


consonant has altered in all cases in accordance with Rule 2. An initial guttural qu has become in the French and the Portuguese the equivalent of our s; in the Spanish our th; in the Italian and Romanian our ch. All these shifts are in accordance with Rule 2. All five of these languages preserve the n in their written forms, and in Spanish, Portuguese, Italian and Romanian, it is still pronounced. However, in the French, strictly speaking, it is not. What has happened is that the i sound, which in normal circumstances would approximate our ee, has been modified into a nasal vowel approximated, but only approximated, by our ang. Nasal vowels do not exist as such in English, although they actually do occur in some dialects, notably many American ones; speakers of these are often said to talk through their noses.

As mentioned above, there are other cases of consonants morphing into vowel modifiers: the English r and the suggestion on d in the genesis of Anglo-Saxon. Of course, once a consonant has been lost in this way, then the affected vowel is free to mutate at the much faster vowel rate.

It could be that something of this process underlies the *kentom-satem change noted above between the PIE and the Avestan. Everything else is just as we would expect. So the suggestion that the change was not a one-off but occurred several times independently is not really very far-fetched.

Table 1.1 shows the PIE words for the numbers 110 according to three different authorities. It will be seen that there is excellent broad agreement, but that some details differ. Yet other authorities produce further slight variants, but these matters are very much secondary to the great thrust of the overall agreement.

Now to move on. The IE family of languages (those derived from PIE) is by no means the only one; there are others. (It has already been noted that Hungarian and Finnish are not Indo-European.) TheIEfamily is the largest such language family and it is also the most studied and best understood. Counts of the different language families vary between the different linguists who study such problems. There is a widespread (but far from universal) agreement that there are about 20 of them, and even among those who accept this count there is much difference in detail.

After IE, the most widespread group is the Austronesian. This is a family extending in a semicircular arc from Madagascar, through Indonesia and Malaysia and across the Pacific all the way to Easter Island. Its major clan is Malayo-Polynesian, and because much of the Polynesian dispersal is relatively recent, the various Polynesian languages are still closely related. (Captain Cooks men found that they could use a form of Pidgin


Maori to make themselves understood all over the Pacific.) There are quite a lot of agreed bases for the genealogy of this family, although it is not known in the same detail as the IE, in large part because of a lack of early written forms.

The largest language family, if we use a headcount of native speakers as the basis, is the Sino-Tibetan. This is well described by its title, and it covers all the major languages of what is now China.

Hungarian and Finnish both belong to another group known as Uralic. They form part of a clan, a branch of Uralicknown as Finno-Ugric; Estonian and Saami (Lappish) are also members of this clan. As well as the Finno-Ugric clan, there is another, the Samoyedic.

Other European languages that lie outside the Indo-European family are Turkish, which is part of the Altaic family, Maltese, which is classified as Afro-Asiatic, various languages of Georgia in the Transcaucasus, which are given the label Kartvellian or Caucasian, and finally Basque and perhaps a couple of others, which have no known relatives and are thus described as orphan languages.

In the Indian subcontinent there is a group of related languages which are not Indo-European. (Tamil is the largest.) Such languages are referred to as Dravidian. The IndigenousAustralianlanguages constitute a further family, and there are also families in America, Sub-Saharan Africa and elsewhere. Some believe that Japanese and Korean are orphan languages (they are not Sino-Tibetan); others think they are related to one another, but perhaps only to one another; yet others regard them as Altaic. This list is not exhaustive, nor is it agreed in all its detail.

Number Lockwood Szemernyi Watkins

1 *oykos, *oynos *oi-no

2 *dwo(w) *dwoi *dwo

3 *treyes *treyes *treyes

4 *kwetwores *kwetwores *kwetwor

5 *penkwe *penkwe *penkwe

6 *seks* (H)weks*s *s(w)eks, *seks

7 *septm *septm *septm

8 *okto(w) *octo(u) *octo(u)

9 *newm *newn *newn

10 *dekm *dekmt *dekm

Table 1.1. The first ten number-words in PIE


However, just as the various clans or branches combine to form the families, so too do some linguists think that the various families are themselves related in super-families. This is a much more controversial subject and by no means all linguists accept it. Even among those who do, there is much disagreement over detail.

The most widely accepted super-family goes by the name Nostratic. This super-family comprises IE, Uralic, Altaic, Kartvellian and Dravidian. When first proposed, it was supposed to include also the Afro-Asiatic, but recently there has been a rethink on this point. The initial proposal came from two researchers working independently of each other in the then USSR. One, Illi-Svity, was a specialist in IE, Altaic and Kartvellian. The other, Dolgopolsky, was an Indo-Europeanist with a large knowledge of Afro-Asiatic. After a period of working independently, they joined forces.

For many years this work was little known outside the Soviet Union. Illi-Svity died prematurely, killed in a road accident, and Dolgopolsky emigrated to Israel. Another prominent Nostraticist, Shevoroshkin, left for the USA. It was only after these latter two developments, that Nostratic theory became known in the West. There have since been attempts to merge other language groups into the Nostratic picture. One of these is Eskimo-Aleut (or Eskaleut) from Alaska, northern Canada and Greenland, and there are others sometimes added to the list.

Another attempt to define a super-family merges somewhat different groups and produces a Eurasiatic super-family. Yet other researchers claim to find relationships between Sino-Tibetanand some smaller families.

All this work is very speculative, but has received some independent verification from an unexpected source. The idea that languages and their relationships mirrored genetic relationships was long regarded as discredited. However, it has now reappeared with the work of Cavalli-Sforza, as an outgrowth of the Human Genome Project. His results seem to demonstrate correlations between the proposed linguistic super-families and genetic affinities (mostly as revealed by frequencies of various blood groups). This work also, however, remains controversial, not least because, to some, any such research seems to carry racist overtones.

Of all the proposed super-families, Nostratic has the most widespread support, and there are certainly many linguists, who, while not accepting the theory in all its detail, would be sympathetic to the idea of a remote relation between IE and Uralic.

When we go to the first level of language families and reconstruct (for example) a PIE root, we signal the enterprise by an asterisk. The


convention has arisen of signifying a supposed Nostratic root with a double asterisk. One example will suffice, but it is relevant here. The Nostratic word for 2 is supposed to be **to. This corresponds to the PIE *duo and to reconstructed forms in proto-Altaic and proto-Uralic. The first of these is */t/ (in this context, the slanting lines indicate that the t is regarded as more securely known than the following vowelas we would expect from Rule 1) and the second is *to-e, this latter having the derivative meaning second.

17

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C H A P T E R T W O

The notion of a base

It is important to distinguish two different aspects of the concept of number. Here and in all that follows, I will say number to mean natural number. In the first place, there is an unsophisticated view that must have informed the earliest attempts at counting; but as well as this there is a mathematically informed point of view that has become second nature to all of us who have grown up with the system of natural numbers from the beginnings of our education. In fact, so ingrained in our thinking is this more developed idea that we find it somewhat difficult to imagine ourselves back to a time before it came to be developed.

It is therefore best here if we begin with the more sophisticated notion and later to go back and recover the earlier more amorphous concepts.

Our notions of number very early in our childhood find expression in the activity of counting. Young children in our culture, and in many others also, learn to count in their first few years of life. The insights so formed find expression in the first four of Peanos axioms, which set out to formalise the fundamentals of our numerical concepts.

These tell us that:

1 There is a number called 1.2 Every number x (say) has a successor xl.3 1 is the successor of no number.4 If y x=l l, then y x= .

There are other formulations, but all are equivalent to this one.Axiom 1 gets us started. Some accounts begin with the number 0,

which leads to rather more elegant mathematics, but at the expense of taking us further from the underlying motivation in the counting process. So here I start with the number 1. Once we are started, we can continue, and it is important to realise that we can continue indefinitely. If we speak of x as being the predecessor of the number xl, then Axiom 3 tells us that


the number 1 has no predecessor. Axiom 4 tells us that it is impossible for two different numbers to have the same predecessor.

A full account of the Peano axioms includes a fifth one which is perhaps the most important of all. It is the Principle of Mathematical Induction, which enables us to define operations such as addition and multiplication and to prove their properties. Here we need not venture into this territory, except to note that by its use, it is easy to show that the number 1 is unique in having no predecessor. There is no other such initial number.

What we have here is a sequence, which begins with an initial member, 1, and then progresses through the numbers 1l, 1ll, 1lll, , and this is the only such sequence we can have.

So, there is exactly one system of numbers, and each of its embodiments is really just a re-expression of the others. So, for example, in English we say:

one two three four five six seven eight nine ten

A Spaniard, however, would have:

uno dos tres cuatro cinco seis siete ocho nueve diez

The actual words are different, but there is an exact pairwise correspondence between them. That is, it is possible to draw up a table of precise correspondences so that each English word pairs with just one Spanish word and vice versa.

This same point may be made in respect of every language that has developed a precise means of representing the sequence of counting numbers. The same point may also be made of the symbols for the numbers: 1, 2, 3, 4, 5 and so on.

So we may set up a table in which the same numerical concept is represented in many different ways. See Table 2.1.

Of course, given the material of the previous chapters, we see linguistic connections between the various words used to represent the numbers, but the point being made here is a different one. We can take a quite unrelated language, Thai say, and find an exact correspondence as before. The correspondence is entirely independent of the linguistic relations. See Table 2.2.

Even though there is no linguistic connection (that we know of) between the different words for any given number, the correspondence between that word and that number is precise. As if this were not enough,

T H E N O T I O N O F A B A S E 19

we can make up words to the same effect. There is nothing to stop us proposing a language called, say, Mugwump, and having the first ten numerals as shown in Table 2.3.

As long as the translation is clear, a Mugwump speaker can be understood to be referring to exactly the same number as we are.

But this last example makes it abundantly clear that the actual word, or for that matter the symbol also, is completely arbitrary. There is no need for the word one to correspond to the symbol 1. It is an agreed convention that these two do indeed go together when we speak and write English. Likewise, it was an agreed convention that the word unus and the symbol I went together for a Latin speaker. And so on.

But alert readers will note that in the tables above is contained a feature that I have not so far mentioned. The symbol for ten is 10, which is a compound of two other symbols: 1 and 0. This is a concession to the limited power of human memory. We simply cant go on inventing arbitrary words and symbols forever and ever. We would soon lose track!

Numeral English Spanish French German Latin Roman numeral

1 one uno un ein unus I

2 two dos deux zwei duo II

3 three tres trois drei tres III

4 four cuatro quatre vier quattuor IV

5 five cinco cinq fnf quinque V

6 six seis six sechs sex VI

7 seven siete sept sieben septem VII

8 eight ocho huit acht octo VIII

9 nine nueve neuf neun nouem IX

10 ten diez dix zehn decem X

Table 2.1. The first ten number-words in several languages

Numeral 1 2 3 4 5 6 7 8 9 10

English one two three four five six seven eight nine ten

Thai neng swng sam si ha hk jt pet ko sp

Table 2.2. The first ten number-words in English and Thai

Numeral 1 2 3 4 5 6 7 8 9 10

English one two three four five six seven eight nine ten

Mugwump aba beb cic dod efe fuf gyg hah iji jej

Table 2.3. The first ten number-words in English and Mugwump


It follows that there must be a finite (in fact relatively small) number of basic words or symbols, and that there must be some way of generating others beyond the basic list. A simple such example is the system we use to establish our number representation by means of positional notation. We employ exactly ten basic symbols: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. All the members of the infinite sequence of numbers can be represented in terms of strings of these numbers. Because there are ten basic symbols, we say that our system of number representation employs base ten.

The number ten holds a privileged place in the way we choose to write our numbers. There is no particular mathematical reason to choose ten for this role. Any number could be used. (This is not to say, however, that there are not extra-mathematical considerations involved also, for example, practicality. See Chapter 7.2.)

Let us take some time off to look at the different possibilities. Simplest of all is base one. Here we could write the first few numerals as 1, 11, 111, 1111, 11111, and so on. This is quite a logical system; indeed we have a name for it. It is referred to as a tally. It is exactly the same as the representation mentioned earlier:

1, 1l, 1ll, 1lll,

It works fine for small numbers, but it soon becomes unwieldy as the numbers increase.

In practice, when we tally, we tend to group the tally-marks in some systematic manner. One very popular method is the gatepost tally, in which the numbers are grouped as follows: |, ||, |||, ||||, ||||, followed by |||| |, |||| ||, and so on. This makes the tally more convenient and easier to read, but introduces, in effect, a further symbol, ||||, having the meaning five. We have here the beginnings of a base five system.

The simplest practical base is base two, which uses just two symbols: 0 and 1. This is the well-known binary system, and it commands great theoretical importance, not least for its connection with computer logic. For our purposes here, note that the new symbol, 0, has a different status from the 1 of the tally system. Now that two different symbols are in use it can make a difference if the order of their appearance is changed. For example, 110 represents the number we know as six, whereas 101 corresponds to our five.

The mathematical notion of base is now firmly linked to the positional notation. Take b to be the base. To represent a number N in base b means that we find a number r and a finite sequence of r 1+ numbers n

0, n

1, n

2, , n

r 1-, n

r, each represented by a symbol representing

either zero or else some other number less than b. Then


N n b n b n b n b nr

rr

r1

12

21 0

f= + + + + +-

- (2.1)

gives the representation of N in base b as the string of symbols n n n n nr r 1 2 1 0

f-

. The representation is unique for each N.Any number larger than 1 may be used as such a base (the trivial case

b 1= has already been dealt with). Base three has some nice properties. Base eight and base sixteen have connections with computer logic and there are from time to time calls to reform our number-system by employing base twelve. Some of this material is further explored in Chapter 7.2. In a different category are suggestions that other societies have used bases other than ten. The claim that the ancient Babylonians used base sixty will be examined a little later, as will the claim that the Mayans used a base twenty system.

The suggestion that other bases have been used by different societies, however, depends on a subtly different notion of the meaning of base. A good place to start is with the Chinese numerals. There are in fact several different versions of these, and it is also true that all of them are rapidly being supplanted by our own familiar symbols. However, the most widely used traditional system is shown in Table 2.4.

1 2 3 4 5 6 7 8 9 10 11 12

Table 2.4. The first 12 numerals in Chinese notation

The sequence continues in the obvious way until we reach 20. This can be written as , or also in other ways, but this is the one I will use here. The same principle invoked here can now be pressed into service to give expressions for all the numbers up to 99. Then a new symbol is needed. We have for a hundred and for a thousand. Each new power of ten requires a new symbol. But now we can have for 101, and so on and for 1001, and so on. To write 4957, for example, we would put . There is no need for a special symbol 0; where we would use one, the relevant symbol is simply omitted. So to write 4057 we would just put .

This is clearly a base ten system, although, strictly speaking, it does not employ positional notation. Its drawback is that each new power of ten needs a new symbol. This same weakness also besets our system of spoken number-words, as opposed to the strings of digits: for example, 103

is a thousand, 106 is a million, 109 is a billion, 1012 is a trillion, and this sequence continues by pressing into service the Latin names for the natural numbers. However, by the time we reach 1036, an undecillion, things are getting rather difficult. We finally run out shortly after 103003, which would be a millillion.


Almost all cultures of advanced numeracy use base ten (presumably because we have ten fingers). Even the Roman system of counting is essentially a base ten system, although it is less tractable than the Chinese, with which it has, however, some affinities. For more on this, see Chapter 7.1.

There are two partial exceptions to this rule. It is said, and said often, that the ancient Babylonians used a base sixty system, and the Mayans used a base twenty system. Neither statement is precisely accurate, although there is, nonetheless, some truth in both of them.

First consider the Babylonian system, which was written in a script called cuneiform. In order to have a base sixty system in the positional notation that we understand, there would need to be sixty different arbitrary symbols (digits), and this is far too many for comfort. Rather, there was a further distinction made so that the first fifty-nine symbols are, in fact, produced by the adjoining of a mere two symbols.

The first of these is and has the meaning one; the second is , meaning ten. A full list of the symbols is given in Table 2.5. It is apparent that each of the other symbols is made up of geometric arrangements of these two. The system may thus also be seen as a mixed base ten and base six system. The base six enters because the system of representation involves the use of sixty, the first number without a distinct symbol.

There was no symbol among the ancient Babylonians for zero. This has caused some confusion in the reading of the clay tablets on which they inscribed their mathematics. Some scribes used a clear space for this purpose; others were less careful. Thus, when it comes to writing the number sixty, one should write , with a clear space following the symbol for a unit. When we represent this system today in our more familiar numbers, we resort to an artifice, and use our own system with the numbers separated by commas. For ease of reading and to ensure precision of representation, we use a zero when we write Babylonian numerals, although this is not true to the way the ancient scribes would have proceeded.

So we write sixty as 1, 0. This same convention applies for larger numbers. So, for example, if we write a million in these terms, we have:

, , ,1000 000 4 60 37 60 46 60 40 4 37 46 403 2# # #= + + + = . (2.2)

In order to work out how a Babylonian would have written it, use Table 2.5.


As another example, consider the Babylonian representation of our number 3601, we see that:

, ,3601 1 60 0 60 1 1 0 12# #= + + = (2.3)

A careful scribe would now write , with a clear space between the two s, but there is always the possibility that the space will get lost, so that the number would read as 61, by mistake. (The problem is compounded if two consecutive spaces are needed!)

The Mayan system shares some of these general characteristics. Refer to Table 2.6. It has a zero: the rather elaborate symbol in the top left-hand corner of the table. It does not, however, have nineteen arbitrary symbols for the next nineteen digits. Rather these are made up from two other symbols, a dot for one and a horizontal line for five. Because there is a zero we have a clear positional system, although the positions are separated vertically rather than horizontally as is the case with standard decimal notation. This feature is most clearly exhibited in the symbol for twenty in the table.

91 2 3 4 5 6 7 8 10

11 12 13 14 15 16 17 18 19

21 22 23 24 25 26 27 28 29

31 32 33 34 35 36 37 38 39

41 42 43 44 45 46 47 48 49

20

30

40

50

51 52 53 54 55 56 57 58 59

Table 2.5. The first 59 numerals in Babylonian cuneiform


Although both the Babylonian and the Mayan traditions in mathematics reached high achievements, their systems of numerals have not lasted. Base ten now reigns supreme. However, there have been claims that various other bases have been used and in some places are still being used. In order to appreciate the strength of such claims, we need to move away from the usual understanding of the word base, via the positional notation. We already saw something of this with the Chinese and the Babylonian systems, but now we proceed even further down such paths.

The term base is used by anthropologists in a rather different sense from the way in which mathematicians use it. When so used, it refers to a privileged number, which is the basis of some recurring pattern. Indeed this usage is typically less precise than that applying to positional notation. Gatepost tally could be called a base five system in this more general but less precise sense.

This same general sense of the word base is used to describe the base two system of the Kiwai of Papua New Guineas Western Province. Their traditional system had two number words: nau, for one, and netewa for two. The system then continued netewa nau for three, and netewa netewa for four. In theory, it could have been extended a little beyond this to netewa netewa nau for five, netewa netewa netewa for six, and so on. In practice, the system rapidly becomes unworkable even for relatively small numbers, and some authorities regard such systems as ending with their word for four, which is called the limit of counting, but would more accurately be called the limit of precise counting. That is to say, the use of terms like netewa netewa nau and netewa netewa netewa is hypothetical only.

A similar system is to be found in the Australian language Gumulgal, where the words for one and two are respectively urapon and ukasar. Thus Kiwai and Gumulgal are described as of a one, two, three, four, many

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

Table 2.6. The zero and the first 29 numerals in Mayan notation


type. In other words, the numbers larger than four are simply described by a catch-all term meaning many. Other languages are similarly described as being of a one, two, many type.

Most Australian Indigenous languages are either of the one, two, many type, which will be further discussed later in Chapter 5, or else of the one, two, three, four, many type also discussed in more detail in Chapter 5. It was once held that all Australian languages fell into one or another of these categories or else into a third closely related category. There has, however, been a more recent re-evaluation of this belief. The entire question will be revisited in Chapter 5. These two counting systems are both of considerable theoretical importance.

Papua New Guinea (PNG) developed a variety of different numeration systems in its pre-colonial era. Some languages used base five in much the same way as the Kiwai used base two, and sometimes there is a mix of base two and base five elements. The use of base five is related to the five fingers on the human hand, and usually the word for five is the same as the word for hand. Yet others use a base twenty system, and the word for twenty is then the same as for man. An example is given a little later.

And there are other systems using parts of the body to produce bases that may strike us as quite strange. The Kewa people, for example, are said to have used a base forty-seven, by running through the various parts of the body in a symmetric pattern, up one side of the body and down the other. See Table 2.7.

When we look at bases other than ten, we usually encounter either five or twenty or some combination of these. It is clear that the use of these derives from the fact that we have five fingers on each hand and five toes on each foot. Table 2.8 illustrates the traditional system of numerals in the PNG language Northern Fore. Compared with the system just illustrated, it is more systematic and it shies away from the arbitrariness inherent in the Kewa system. (There are other systems like the Kewa, but using different body parts, and thus resulting in different bases.) The Fore system will be further discussed in Chapters 5 and 6.

The use of mixed base systems was also found. For example, the Motu language from the area around Port Moresby had the system shown in Table 2.9.

Here, clearly, we have a mix of base four features with base ten characteristics.

Two points need to be made here. The first is that the various systems described here are mainly things of the past. They have been supplanted almost entirely by our own system of numerals or else by something very close to it. This is almost certainly because we have a more efficient method


for dealing with more complex numerical ideas. This matter will reappear in Chapter 5.

The second point is that the existence of mixed base systems should not surprise us so very much. Already, we have seen such features in both the Babylonian and the Mayan systems. In Chapter 6, we will detail vestiges of a base two system in English, and, in Chapters 4 and 6, remnants of a more developed version in other Indo-European languages. Even more exotic bases surface briefly in a few other Indo-European languages. The best-known case is that of French with the decads (multiples of ten) shown in Table 2.10.

The word for eighty, quatre-vingts, means literally four twenties. We see here a relic of an old base twenty system. (And the word for seventy might be viewed as a relic of a base sixty system!)

Other Indo-European languages exhibit even stranger such fossils. In Welsh and in some other Celtic languages, fifteen has a somewhat

1 little finger 47

2 ring finger 46

3 middle finger 45

4 index finger 44

5 thumb 43

6 heel of thumb 42

7 palm 41

8 wrist 40

9 forearm 39

10 large arm bone 38

11 small arm bone 37

12 above elbow 36

13 lower upper arm 35

14 upper upper arm 34

15 shoulder 33

16 shoulder bone 32

17 neck muscle 31

18 neck 30

19 jaw 29

20 ear 28

21 cheek 27

22 eye 26

23 inside corner of eye 25

24 between the eyes 24

Table 2.7. The Kewa system of counting using parts of the body


privileged place. The Welsh for fifteen is pymtheg, which translates quite literally as five-ten (as does our own word fifteen). But next come:

16 = un ar pymtheg 17 = dau ar pymtheg or dwy a pymtheg

which mean, respectively, one and fifteen and (one or another form of) two and fifteen. But then comes a bigger surprise. The word for eighteen

Numeral Number-word Meaning

1 kne one

2 tarawe two

3 kakgaw,tarawe'knakn

one-one-one,two-one

4 tarawa'tarawakn two-two

5 naya 'kaamn hand one

6 to nentisa k 'umaemaw from another hand one add

7 to nentisa tara umaemaw from another hand two add

8 to nentisa kakga umaemaw, to nentisa tara mgasimaw and others

from another hand three add,from another hand two cast off

9 to nentisa tarawatarawakto nentisa (age) k 'mgasimawe

from another hand four,from another hand one cast off

10 naya 'tra'mne hands two

11 nagisarsa k tumpaemaw from a foot one add

12 nagisarsa tara tumpaemaw from a foot two add

13 nagisarsa kakga tumpaemaw

from a foot three add

14 nagisarsa tarawa tarawak tumpaemaw from a foot four add

15 nagis kam add one foot

16 to nagisarsa k umaemaw from another foot one add

17 to nagisarsa tara umaemaw from another foot two add

18 to nagsarsa kakga mawaemaw,to nagsarsa tara mgasimaw (etc.)

from another foot three add,from another foot two cast off

19 to nagsarsa tarawatarawak 'umaemaw,to nagsarsa (age) k 'mgasimawe

from another foot four add from another foot one cast off

20 k 'kinane and others

one persons fingers and toes

21 kkina 'puma kne one person plus one

Table 2.8. The first 21 number-words in Northern Fore

1 ta 2 rua 3 toi 4 hani

5 ima 6 tauratoi 7 hitu 8 taurahani

9 taurahani-ta 10 gwauta 11 gwauta-ta 12 gwauta-rua

13 gwauta-toi 20 ruahui 21 ruahui-ta, 30 toi-ahui

Table 2.9. Number-words once used in Motu


is deunau, which means two nines, so we have a fleeting glimpse of another exotic base: nine. Welsh has a close relative in Breton, and here one of its two words for eighteen is triwech, which means three sixes, so we have another brief appearance, this time of six as a base.

The Roman numerals display a remnant of an old base five system. The basic system is a base ten one, with the symbols I for one, X for ten, C for hundred and M for thousand. However, as well as these main symbols, there are three auxiliary symbols: V for five, L for fifty and D for five hundred. These auxiliary symbols, however, are treated differently from the main ones. See Chapter 7.1 for more detail.

20 = vingt 60 = soixante

30 = trente 70 = soixante-dix

40 = quarante 80 = quatre-vingts

50 = cinquante 90 = quatre-vingt-dix

Table 2.10. The decads in French

29

536 129421678 3 8573

Other aspects of number: words and symbols

Counting may be performed either orally or in writing. If we attend to the oral form, then Linguistics can take us so far, but we soon lose the thread of the argument, as the conclusions we draw become less and less secure. We know a lot about PIE because several of its daughter languages were written down, and have left large bodies of literature as the raw material of study. This means that the early history of number-words can be known much more securely.

But as well as operating with number-words, we also have symbols, which we can manipulate algorithmically, that is to say, in a routine and mechanical way that, if performed correctly, guarantees precise results. The relation between the words and the symbols was broached in the previous chapter.

Although there is no particular reason for us to represent the concept of one by the word one or the numeral 1, we choose to do so, and we know from an early age that all these things go together. Nonetheless, we use some representations for some purposes, others for other purposes.

Nowadays we are familiar with the idea of a one-to-one correspondence. When we count, we set up such a correspondence between the objects being counted and the standard numerals: one, two, three, and so on. Taken for granted also is the implied one-to-one correspondence between both these sets and the set of standard number symbols: 1, 2, 3, and so on.

There is now a plausible theory as to how all this began. As the societies of the Middle East developed from hunter-gatherer bands into settled agricultural communities, the need arose for some sort of bureaucracy in order to keep track of grain and other commodities stored in shared facilities. This development took place about 80007500 BCE in the fertile valleys of the Tigris and Euphrates rivers among the Sumerian people then living there. (This is now an area in southern Iraq.) The rise of agriculture led to a great increase in the

C H A P T E R T H R E E


population of the area and also to the need to store grain, for example, for use in the periods between harvests.

When Denise Schmandt-Besserat began her archaeological work on this period and in this region, she began by collecting clay and pottery artefacts from the area. Her original thought was to discover the earliest human uses of clay, and she was surprised to find that the oldest such objects were small tokens, rather than obviously utilitarian artefacts, but rather more like what we might expect for childrens toys. These were often in the form of simple geometric shapes: cones, spheres, disks, cylinders and tetrahedra. A few others looked like stylised renderings of animal shapes, but these were very much in the minority.

As the journalist Ivars Petersen put it in Science News Online: It seems they [kept track of personal property in communal silos and other storage facilities] by maintaining stocks of baked-clay tokensone token for each item, different shapes for different types of items. A marble-sized clay sphere stood for a bushel of grain, a cylinder for an animal, an egg-shaped token for a jar of oil. There were as many tokens, or counters, of a certain shape as there were of that item in the farmers store.

Thus, at this early stage, for each type of commodity, a tally was used. The one-to-one correspondence connected the set of animals (say) with (in this case) the set of cylinders.

It is suggested that, in this earliest form of the accounting system, each token of its kind represented one item. If additional items appeared in the real world, then the corresponding number of tokens would be added to the pile the accountant had in his possession. If a real-world item was used, lost or destroyed, then the corresponding token would be removed. And finally, if an item of the real-world commodity was transferred from one owner to another, then the corresponding token would be moved from one location to another.

As Schmandt-Besserat continued her research work, she amassed a vast quantity of data on these tokens, their dates, their locations, their types and shapes. All this material was then subjected to extensive statistical analysis. For example, at one very early site, Jarmo in Iraq, the first occupation was about 6500 BCE, and the entire collection amounted to 1153 small spheres, 206 disks and 106 cones. Each token was about one centimetre in diameter. As the remains of houses were unearthed in Jarmo, the excavators found that the tokens were distributed over the floors in clusters that suggested that they had once been collected together in baskets or pouches that had long since disintegrated. The fact that these tokens were separated from the other household objects, however, suggested that

O T H E R A S P E C T S O F N U M B E R : W O R D S A N D S Y M B O L S 31

they had a separate function and an especial value. It seemed that they had been housed in dedicated storage areas in each of the houses.

It also became clear to her that the system was very widely used. Tokens of the same types appeared in archaeological sites from Sudan to Turkey to Pakistan. The whole of Western Asia and parts of North Africa all seemed to use essentially the same system of keeping accounts. Schmandt-Besserat, writing in 1978, claimed that shepherds in Iraq still used pebbles to represent animals in their flocks.

Similar simple systems of accounting have been used in much more recent times, even up to today. Schmandt-Besserat notes that counters have been used in the calculation of even quite complicated calculations. The Romans were, as their engineering feats still testify, adept in practical mathematics despite their relatively cumbersome system of number representationsee Chapter 7.1. It seems that they used the abacus as a calculator; Menningers book shows some examples. I can personally testify to the use of the abacus, for when I first visited China in 1988, the abacus was still in routine use, even in the big city hotels. A competition between an abacus and an electric calculator of the day was held in about 1960 at RMIT, and resulted in a victory for the abacus, which proved to be faster.

The abacus is essentially a set of counters or tokens nowadays held together by a frame of some sort, and this use of counters or tokens was once very widespread. The Latin word for a pebble is calculus and it is this that has given rise to our words calculate and indeed calculus. It was only in about 1800 that British tax accountants ceased to use counters in their calculations. The east Asian uses of the abacus have been supplanted by electronic devices only within my lifetime.

These recent applications of tokens are seen as a survival of the earlier system. The earlier one was more complex in that it employed a variety of different tokens, each with a different meaning from the others. Schmandt-Besserat categorised the tokens into some 15 major classes, further divided into some 200 subclasses on the basis of size, marking or fractional variation. By fractional variation, she meant the use of half or quarter tokens.

This simple system eventually evolved into more elaborate means of keeping accounts. For example, at another Iraqi site, Uruk, it is possible to follow this evolution over time as the site was occupied for many centuries. The use of actual tokens was gradually supplanted by two-dimensional representations of these same tokens, together with arbitrary signs for numerals, such as a small cone representation for the number one, a circular impression for the number 10 and a larger cone-shaped impression


for the number 60. These signs were pressed into clay tablets, which were later baked to ensure durability.

Here then is the origin of the cuneiform representation described in Chapter 2, and there referred to as Babylonian after its most famous practitioners.

This, according to the theory that Schmandt-Besserat has developed, is one origin of writing. Many of the symbols can now be read in that we know what they represented. For example, a sheep was represented by a circle enclosing a cross and a garment was represented by a circle enclosing four parallel lines.

As time went by, in particular as the Neolithic era gave way to the Bronze Age, the system became more sophisticated. The tokens themselves were impressed with markings; the number of markings increased dramatically, as did the number of shapes taken by the tokens. The tokens came to be perforated in such a way that sets of tokens could be strung together as a record of some transaction or other, or as a sort of bank account showing each persons or familys holdings.

This theory was quite novel when it was first advanced more than 30 years ago. Prior to that the view had been that writing evolved from pictures into ideographs. The writing that would emerge from such a development would be somewhat like the Chinese, with symbols, with each symbol standing for an individual word. It is now believed that the Middle Eastern system of writing as discovered by Schmandt-Besserat, although it came later than the Chinese, was a completely independent invention, and there is now widespread agreement that Schmandt-Besserats theories are correct.

Schmandt-Besserat holds that, far from developing from pictures, the symbols for words derived directly from abstract ideas. They were arbitrary in the same way that our words and number-symbols are arbitrary. For example, the use of a circle enclosing a cross to represent a sheep is quite different from drawing a sheep, even in a stylised form. Almost certainly, of course, the origin of this particular symbol as a sign for a sheep has some explanation or other, but we are no longer able to fathom it.

The relationship of this symbol to a sheep could have been as simple as a piece of happenstance, based on the whim of some single individual who lived all those years ago, and whose name and other details we will almost certainly never know. We shall see later that this is also true of our own number-words and number-symbols. This failure to recognise the original provenance of a word or a symbol in no way prevents our making a mental association between the object or numeral being represented and the sign we use to represent it. The connection is however less immediate than perhaps it once was.

O T H E R A S P E C T S O F N U M B E R : W O R D S A N D S Y M B O L S 33

Thus, in the Middle East, the thrust of development was from accounting to numbers, with writing being a later spin-off from this. In both instances, the development of the symbol was via a sign rather than through a pictorial representation, with just a few rare exceptions.

We can contrast this with the Chinese system, which many authorities (although not all) believe did develop from an initial pictorial form. If we now look at that Chinese system, however, we see some features that are not present in our own. The most obvious one is that the symbols (characters) for the numbers are not different from the written form of the numerals. In English, we can write 3 to represent the numeral three or else we can spell it out as three. We know that the two representations correspond, and in fact we probably regard the two ways of expressing the same number as completely equivalent. This, however, does not disguise the fact that the two are actually subtly different.

Compare this with the case in Chinese. The written form represents their word san for our three, but also is the same as the representation we would translate as 3. That is, word and symbol are one and the same.

This example also shows the clear derivation of this particular symbol as a straightforward representation of a tally. The same can be said for the symbols for their numbers 1 and 2: and respectively. (It is quite likely that our own numerals 1, 2, 3 developed from the same starting point: one (vertical) line for 1, two (connected horizontal) lines for 2, and three (connected horizontal) lines for 3.) However, once we enter the realm of numbers greater than 3, this simple system breaks down, as all such systems must for the reasons advanced in Chapter 2.

The underlying principle in the Chinese system of writing is the understanding that each individual character represents a monosyllable. Because the available number of individual monosyllables is rather small, this system has needed to be supplem

Michael Deakin the Name of the Number 2007

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