The Name of the Number

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The Name of the Number

We’re 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.

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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 )

The name of the number

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 ISBN 978-0-86431-757-5 been designed to be accessible and rigorous, while appealing to students, educators, mathematicians and general readers.

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THE NAME OF THE NUMBER

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The name of the number Michael 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

The name of the number Michael A B Deakin

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

ACER Press

First published 2007 by ACER Press Australian Council for Educational Research Ltd 19 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 Glascodine Cover design by Mason Design Text design by Mason Design Typeset by Desktop Concepts Pty Ltd, Melbourne Printed 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

Contents List of tables Series overview About the author Preface

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Introduction

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The language families of the world

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The notion of a base

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Other aspects of number: words and symbols

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Grammar: the grammatical status of number-words

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Early history of numerical concepts

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Developed systems of number-words

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Projects 7.1 Roman numerals 7.2 Bases other than ten 7.3 Counting rhymes 7.4 The number-word game

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Resources 8.0 Introduction 8.1 The language families of the world 8.2 The notion of a base 8.3 Other aspects of number: words and symbols 8.4 Grammar: the grammatical status of number-words 8.5 Early history of numerical concepts 8.6 Developed systems of number-words

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8.7 Projects 8.7.1 Roman numerals 8.7.2 Bases other than ten 8.7.3 Counting rhymes 8.7.4 The number-word game References Index

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List of tables

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The first ten number-words in PIE

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The first ten number-words in several languages

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The first ten number-words in English and Thai

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The first ten number-words in English and ‘Mugwump’

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The first 12 numerals in Chinese notation

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The first 59 numerals in Babylonian cuneiform

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The zero and the first 29 numerals in Mayan notation

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The Kewa system of counting using parts of the body

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The first 21 number-words in Northern Fore

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Number-words once used in Motu

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2.10 The decads in French

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Measure-terms in Kwakiutl

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Parallel between pronouns and numerals

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The first thirty-five numbers in hexadecimal

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Reciprocals of the first few numbers in base ten

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Number-words in the Brythonic languages

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Two versions of the sheep-score

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The first ten numbers in six North England dialects

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The number-word game in Motu

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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 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 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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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 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.



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 number—in 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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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; Zeno’s 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. 

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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 preeminent among all of these is Linguistics. As has been written: Language fords time’s 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?

INTRODUCTION

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 today’s 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 boy’s name ‘Otto’. Rather, there is, between the first o– and the following –t­, a ‘consonant’ that, in a sense, doesn’t 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 ‘glo’al 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





THE NAME OF THE NUMBER

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 oktō, which looks very similar. The corresponding Sanskrit word was aşţa, 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.

INTRODUCTION

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 IndoEuropean 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 IndoEuropean (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 Armenian Baltic Celtic

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Iranic Italic Slavonic Tocharian

T H E L A N G UAG E FA M I L I E S O F T H E W O R L D

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 Armenian branch. 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 Baltic is 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-Slavic branch 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 (today’s Turkey) and survived as a distinct people until the 5th century CE. At the other end of the European continent, the Celtiberians inhabited today’s Spain and Portugal back in Julius Caesar’s 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





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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 IndoIranian 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 Iranic ‘clan’ descend from two older, now dead, languages: Avestan and Old Persian. This latter is the ancestor of most of today’s 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 (today’s 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 Baltic languages 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 IndoEuropean. Finally, there is the Tocharian branch, comprising two now dead languages from Mongolia. They have a theoretical importance to be discussed later.

T H E L A N G UAG E FA M I L I E S O F T H E W O R L D

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 3000–4000 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 6–8 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 Baltic and the Slavonic branches, and the other connection between the Indic and Iranian ones. Many linguists also hold that Albanian and Greek 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 śatám 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



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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 consonants Here 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: E  xcept 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 oktō. However, the Sanskrit word was aşţa, 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.)

T H E L A N G UAG E FA M I L I E S O F T H E W O R L D

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 Dickens’s 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.

11

12

THE NAME OF THE NUMBER

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 AngloSaxon 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 game’s ‘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

T H E L A N G UAG E FA M I L I E S O F T H E W O R L D

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 1–10 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.) The IE family 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 Cook’s men found that they could use a form of Pidgin

13

14

THE NAME OF THE NUMBER

Number

Lockwood

Szemerényi

Watkins

1

*oykos, *oynos

—————

*oi-no

2

*dwo(w)

*dwoi

*dwo–

3

*treyes

*treyes

*treyes

4

*k etwores

*k etwores

*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

w

w

Table 1.1.  The first ten number-words in PIE

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 Uralic known as FinnoUgric; 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 Indigenous Australian languages 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.

T H E L A N G UAG E FA M I L I E S O F T H E W O R L D

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, Illič-Svityč, 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. Illič-Svityč 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-Tibetan and 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 CavalliSforza, 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

15

16

THE NAME OF THE NUMBER

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/ö (in this context, the slanting lines indicate that the t– is regarded as more securely known than the following vowel—as we would expect from Rule 1) and the second is *to-ńće, this latter having the derivative meaning ‘second’.

5 7

2 18 93 36

3

7

4 6 8

C hapter two

The notion of a base

5

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 Peano’s axioms, which set out to formalise the fundamentals of our numerical concepts. These tell us that: 1 2 3 4

There is a number called 1. Every number x (say) has a successor xl. 1 is the successor of no number. If yl = xl, 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 17

2

1

18

THE NAME OF THE NUMBER

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,

THE NOTION OF A BASE

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

fünf

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

9

10

English

one

two

three

four

five

six

seven eight

8

nine

ten

Thai

neùng sāwng sāam

sìi

hâa

hòk

jèt

kâo

sìp

pàet

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

Numeral

1

2

3

4

5

6

7

9

10

English

one

two

three

four

five

six

seven eight

nine

ten

beb

cic

dod

efe

fuf

gyg

iji

jej

Mugwump aba

8 hah

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

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 can’t go on inventing arbitrary words and symbols forever and ever. We would soon lose track!

19

20

THE NAME OF THE NUMBER

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 n0 , n1 , n 2 , …, n r - 1 , n r , each represented by a symbol representing either zero or else some other number less than b. Then

THE NOTION OF A BASE



N = n r b r + n r - 1 b r - 1 + f + n 2 b 2 + n1 b + n0 

(2.1)

gives the representation of N in base b as the string of symbols n r n r - 1 fn 2 n1 n0 . 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

一

二

三

四

五

六

七

八

九

十

十一

十二

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 二十, 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 百 for a hundred and 千 for a thousand. Each new power of ten requires a new symbol. But now we can have 百一 for 101, and so on and 千一 for 1001, and so on. To write 4957, for example, we would put 四千 九百五十七. 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 四千五十七. 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.

21

22

THE NAME OF THE NUMBER

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:

1 000 000 = 4 # 603 + 37 # 60 2 + 46 # 60 + 40 = 4, 37, 46, 40 . (2.2)

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

THE NOTION OF A BASE

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

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

Table 2.5.  The first 59 numerals in Babylonian cuneiform

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

3601 = 1 # 60 2 + 0 # 60 + 1 = 1, 0, 1

(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.

23

24

THE NAME OF THE NUMBER

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

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 Guinea’s 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’

THE NOTION OF A BASE

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

25

26

THE NAME OF THE NUMBER

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

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

THE NOTION OF A BASE

Numeral Number-word

Meaning

1

káne

one

2

tarawe

two

3

kakágawé, tarawe'kánakíné

one-one-one, two-one

4

tarawa'tarawakíné

two-two

5

naya 'ka’amúné

hand one

6

to náentisa ká 'umaemawé

from another hand one add

7

to náentisa tara umaemawé

from another hand two add

8

to náentisa kakága umaemawé, to náentisa tara mégasimawé and others

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

9

to náentisa tarawatarawakí to náentisa (age) ká– 'mégasimawe

from another hand four, from another hand one cast off

10

naya– 'tára'múne nagisa–rísa ká tumpaemawé

hands two

11 12

nagisarísa tara tumpaemawé

from a foot two add

13

nagisarísa kakága tumpaemawé

from a foot three add

14

nagisarísa tarawa ‘tarawakí ‘tumpaemawé from a foot four add

15

nagisá ká’a’mú

add one foot

16

to nagisarísa ká ‘umaemawé

from another foot one add

17

to nagisarísa tara umaemawé

from another foot two add

18

to nagísarísa kakága úmawaemawé, to nagísarísa tara mégasimawé (etc.)

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

19

to nagísarísa tarawatarawakí 'umaemawé, from another foot four add to nagísarísa (age) ká 'mégasimawe from another foot one cast off

20

ká 'kinane and others

one person’s fingers and toes

21

ká’kina 'puma káne

one person plus one

from a foot one add

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

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

27

28

THE NAME OF THE NUMBER

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

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 triwec’h, 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.

5 7

2 18 93 36

3

7

4 6 8

C hapter T H R E E

5

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 8000–7500 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 29

2

1

30

THE NAME OF THE NUMBER

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 children’s 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 tokens—one 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 eggshaped 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 farmer’s 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

OTHER ASPECTS OF NUMBER: WORDS AND SYMBOLS

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 representation—see Chapter 7.1. It seems that they used the abacus as a calculator; Menninger’s 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

31

32

THE NAME OF THE NUMBER

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 person’s or family’s 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-Besserat’s 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.

OTHER ASPECTS OF NUMBER: WORDS AND SYMBOLS

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 三 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: 一 and 二 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 supplemented by a further means of distinguishing monosyllables that otherwise would seem the same. This is the Chinese system of tones. There are four distinct tones: the first is a high-pitched one represented by a horizontal line, or macron. Thus the symbol for 1 is pronounced yī, and the word for three, given above, is more properly transliterated as sān. The second tone is a rising one represented by an acute accent, so that the symbol for ten is pronounced shí. The third is a sort of ‘swooping’ tone that falls initially, before then rising. It is represented by a hacek accent. Thus the word for five is pronounced wuˇ. Finally, there is a fourth tone, a falling one, represented by a grave accent: the word for two is thus èr. Often these monosyllables are also individual words, but in other cases, a word will be bisyllabic, and in such cases, two characters must be

33

34

THE NAME OF THE NUMBER

used. Thus the word for ‘eleven’ is written as 十一, and is pronounced as shíyī. Chinese possesses another feature that in fact seems reminiscent of the Sumerian distinction of types of token, although it is almost certainly independent. This is the use of ‘measure-words’. These are words that must accompany the things being counted and which come between the numeral and the name of the thing. There are several of these, and here is how they work. Suppose we want to write ‘two people’. The word for ‘two’ is, as we have seen, èr. That for ‘person’ or ‘people’ is rén. However, we cannot, grammatically, say èr rén. We need to specify what sort of things people are. This is done using the appropriate measure-word, in this case xiē. So the way to say ‘two people’ is èr xiē rén. This feature is encountered in other Asian languages. In Indonesian there are about a dozen measure words, each with a specific category of objects. Thus, for example, we find orang for people, ékor for animals, tangkai for flowers. So in Indonesian, we do not say dua wanita for two women, but rather dua orang wanita, and for two cats, not dua kucing, but rather dua ékor kucing. It may be that these subtle differences between our own number usages and those of our neighbours signify some subtle difference in our apprehension of number concepts. I am not sure. Certainly, when I was teaching Applied Mathematics in Indonesia and came to the topic of Dimensional Analysis, I used the measure-word example as a means of introducing the subject and this seemed to be a useful way to broach the topic! Although the use of measure-words is seen as a special feature of many Asian languages, there are faint echoes in other languages as well. The way in which grammarians classify English words will be discussed much more fully in the next chapter. No list of English word-types includes measure-words, although English has at least one clear measure-word, ‘head’. We can say ‘500 head of cattle’ and mean (apart from a gender implication) ‘500 cows’. The word ‘head’ is used in its singular form, although there are 500 heads on the 500 cows. But here ‘head’ is really a measure-word, and should not be classified in any other way. It should not detain us that there are other uses of the word ‘head’. It can be, for example, part of a synonym for ‘each’ (‘how much are those cattle per head?’), and it may have other uses (‘a howling head of hungry wolves’), or it may simply be the name of a part of the body. English has, however, once had another measure-word, albeit now quite archaic and only ever of very limited use: strong. Again, it need not

OTHER ASPECTS OF NUMBER: WORDS AND SYMBOLS

1

2

3

Animate things

menok

maalok

yatuk

Round things

menkam

masem

yutsqsem

Long things

ments’a

mats’ak

yututs’ak

Days

op’enequls

matlp’enequls

yatqp’enequls

Table 3.1.  Measure-terms in Kwakiutl

be (or ever have been) exclusively a measure-word; it had and certainly now has other uses, as an adjective (descriptive term) in particular. But it once could have been said: ‘500 strong of bowmen were besieging the castle’ to mean simply ‘500 bowmen were besieging the castle’. When we look at other languages, we find a variety of devices that serve much the same purpose as measure-words. There are cases where the things being counted determine which number-words are employed. Menninger gives an example from a Canadian group (whom he does not identify, but I surmise them to be the Kwakiutl). They count differently according to four different categories of items. See Table 3.1. This system may be viewed as a version of the measure-word device, but one in which the measure-word merges with the preceding numeral. Something somewhat different was once a feature of the Tolai language in PNG. Here there were several different systems of counting depending on the objects being counted, but even the ‘base’ changed also. For the actual numerals, there was a base ten system incorporating base five elements, but for certain things, especially types of fruit, the counts proceeded in (most commonly) quartets, but sometimes in pairs, sixes or even dozens. Shell-money was usually counted in pairs, but sometimes in sixes. Menninger gives some examples of similar uses once found in German. Pen nibs were bought by the gross (lots of 144), buttons by the dozen, and eggs by the score. And we not so long ago did much the same with our currency and our weights and measures. The USA still holds to a pre-metric system in many of its weights and measures, as, in practice, does the UK (although theoretically committed to metrication). There are also examples of different words altogether depending on the objects being enumerated. Menninger gives an example from Fijian, in which the word bola means ‘ten boats’, while koro means ‘ten coconuts’. Somewhat similar is an anomaly connected with the Russian word for ‘forty’, sorok. This has no relation to the PIE word for ‘forty’, kwetwr¯ kont–, so that it must have some other origin. Menninger suggests one rooted in the specialised vocabulary of the fur trade.

35

36

THE NAME OF THE NUMBER

We have a few such words still surviving in English: week is a period of seven days; the older usage sennight is today gone, although it is most descriptive; however fortnight (for fourteen days) is still with us. Note, however, that the word week has a different status from the other two. The word week is quite clearly unrelated etymologically to the number-words. (It derives, according to Partridge’s Origins, from the same source as vicar and dates from the replacing of the Jewish Sabbath by Sunday as the day of Christian observance.) The words sennight and fortnight, by contrast, show clear linguistic affinity to their meanings: ‘seven nights’ and ‘fourteen nights’ respectively.

5 7

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3

7

4 6 8

C hapter F O U R

5

Grammar: the grammatical status of number-words

Words, when strung together into sentences, perform different functions from one another. It is common to classify them into various different types, and on most counts there are eight or nine of these types. The technical term for these types is ‘the parts of speech’. The key concept of the sentence is that it describes some action or happening. The word describing this focus is the verb, and it is central to the sentence. Take as an example: The quick brown fox jumps over the lazy dog.

The central action is one of ‘jumping’. The other words tell us the details of the ‘jumping incident’. The verb is ‘jumps’. Every sentence, with some minor exceptions that don’t really count, contains exactly one verb. The verb explains the way in which different ‘actors’ interact. Here the ‘actors’ are a fox and a dog. These words naming the actors are called nouns. They are those words we use to identify persons, animals, localities, times, abstract qualities, collections of things, or just things in general. So, in the example just given, ‘fox’ and ‘dog’ are nouns. There are other words that provide further information about these ‘actors’, the fox and the dog. The fox is described as ‘quick’ and ‘brown’; the dog is described as ‘lazy’. These ‘describing words’ are adjectives. This leaves three words in our example so far unclassified. Two of them are the same: ‘the’. This is a word in a class almost of its own; it is referred to as the ‘definite article’ (as opposed to ‘a’ or ‘an’, which is known as the ‘indefinite article’). The definite article is a relatively recent invention; although it is present in most IE languages, it is absent from Latin. It is also absent from many non-IE languages. It is close to being redundant in many instances, this being one of them. If it were omitted, there would be no 37

2

1

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THE NAME OF THE NUMBER

substantial change to the information the sentence provides. Headline writers usually do just that! (Indian standards of English on the use of the definite article are very different from Australian ones; while this makes our usage seem strange to them and theirs strange to us, no loss of meaning is entailed.) Current thinking sets the articles (definite and indefinite) in a class of their own. My edition of the Macquarie Dictionary takes this course, but my edition of Webster’s New World Dictionary follows an older convention and classifies the articles as belonging to a special variety of adjective. The final word we need to classify from this sentence, ‘over’, is classed as a preposition. Prepositions are words placed before nouns to indicate their relation to other nouns in the sentence. In this example, what is indicated is a spatial relationship between the fox and the dog. This leaves four further categories, so far absent from the example, to be described. First there are pronouns. These are words standing in for nouns, when there is no need to specify the noun, either because of the context or because it has already been given. I, me, you, he, she, it, we, they, them, and quite a few others are all pronouns. So suppose we had: The quick brown fox jumps over the lazy dog. I see them. We now have another sentence, with ‘see’ as the verb. The ‘actors’ are the speaker and the pair of animals. The speaker calls himself or herself ‘I’; no further identification is needed. Similarly, there is no need to name the animals all over again; it is clear who they are. ‘I’ and ‘them’ are pronouns. Now suppose a further modification: The quick brown fox jumps over the lazy dog. I see them clearly. The new word ‘clearly’ is a description of the quality of the speaker’s seeing. Just as the word ‘lazy’ provided additional information about the dog, so the word ‘clearly’ tells how well I see. The word ‘lazy’ goes with a noun; the word ‘clearly’ goes with a verb. It is called an adverb. Now adjust the example again: The quick brown fox jumps over the lazy dog, and I see them clearly. There are really two sentences here, as we have just seen, but they have been combined into a single compound one. The means of doing this is the word ‘and’, which is of a new type. It is called a conjunction.

G R A M M A R : T H E G R A M M AT I C A L S TAT U S O F N U M B E R - W O R D S

The final type is something of an outsider. It is called an interjection, and it covers words like ‘Wow!’, ‘Damn!’, ‘Awesome!’ that are not really parts of sentences at all. This completes the list: we have verbs, nouns, adjectives, articles, pronouns, prepositions, adverbs, conjunctions and interjections. The last two categories are mentioned here only for completeness; they will not be discussed further. The classification I have just been outlining was (apart from the new category of the articles) devised for Latin, and it does not fit English all that well. However, although grammarians have proposed radical modifications from time to time, this is still the one used by almost all dictionaries and so has widespread acceptance. There are a number of very common words that don’t fit comfortably into the scheme. In order to accommodate one quite large category, the class of adverbs has been expanded. Suppose we had: The quick brown fox jumps over the very lazy dog. The new word ‘very’ tells us not about the jumping, nor about the fox, and only indirectly about the dog. What it really tells us more about is the adjective ‘lazy’; by rights, it should be placed into a new category, but it isn’t. It is regarded as an adverb; just a different sort of adverb from that of our earlier example, ‘clearly’. With this understanding, many other troublesome words are all classified as adverbs, although this rather strains matters. ‘No’ and ‘not’ are described by the Macquarie Dictionary as adverbs, with examples such as: ‘He is not well’, ‘She is no better’. Even the word ‘yes’ is regarded by the Macquarie Dictionary as an adverb; I fail to follow the logic here, and perhaps it is wise that an example was not ventured. I would class this word as an interjection. Over and above these considerations there are others. One which is notorious is that by no means all words providing further information about nouns are adjectives like ‘quick’, ‘brown’ or ‘lazy’. Often it is another noun that provides the further description. So we have ‘window frames’, ‘bar stools’, ‘floor polishers’, and a host of others. We can even use an adjective to provide further description, and usually it will apply to the second noun; so a ‘broken window frame’ is a broken frame around a window, not a frame about a broken window. But there are exceptions: a ‘used car salesman’, is not necessarily himself ‘used’; it is the cars he sells that are used! It should, however, be noted that this use of nouns in places where we expect adjectives is not the same as using adjectives themselves.

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Compare ‘He built himself a water clock’ with ‘He went to a watery grave’. Their import is quite different! So there really are a lot of problems with our traditional way to classify words, but we seem to be stuck with it. Nor have I entered into all the details of the various sub-classifications that arise, or the complications that can turn up. But we must address some further complications that arise with nouns and pronouns. Take again our example of the fox and the dog, and simplify the structure somewhat: (The quick brown) fox jumps (over) (the lazy) dog. That is: Fox jumps dog. The sentence tells us about the fox and what it did; it was the one actually doing the jumping. The dog did nothing that we are explicitly told about. It experienced the jumping. In such a case, we say that the fox is the subject of the sentence, and the dog is the object. In English, we make clear the distinction between subject and object by word order. Thus ‘Fox jumps dog’ and ‘Dog jumps fox’ refer to different events. Latin treated matters differently. Word order hardly mattered. The words for ‘fox’ and ‘dog’ modified and in the modification informed us which was the subject and which was the object. If the fox does the jumping, the sentence would read ‘Vulpes canem transilit’ or any permutation of these three words; if on the other hand, it is the dog that does the jumping, the sentence would read ‘Canis vulpem transilit’ or any permutation of these three words. In Latin, nouns were modified according to various rules telling their role in the sentence. There are usually said to be six categories into which a Latin noun could fall. This was a simplification of the earlier PIE system, which had eight. (A more accurate description of the Latin would be that there were five main categories and vestiges of two others from the earlier eight.) These categories are called cases. The subject of the sentence is placed in the nominative case; the object in the objective, or accusative, case. Other cases were modified forms that (in essence) did the task that we assign to prepositions. Mostly, they will not concern us. We may, however, remark that in English, we use the accusative case following a preposition. Latin was more complicated, and PIE more complicated still. In English, we do not distinguish the nominative and accusative nouns in this way, but we still make such a distinction with some pronouns. I see them is different from they see me, and it is not only the word order that indicates the difference; the forms taken by the pronouns also differ.

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Here I and they are nominative; me and them are accusative. This distinction is less marked than it once was. We still have the pairs: I–me, he–him, she–her, we–us, they–them and a few compounds, but thou–thee is today archaic, and who–whom is becoming an endangered species; you is the same in both cases. In Latin, an accusative was often marked by an –m ending, as with canem and vulpem in the earlier example. This is one of the features that is supposed to derive from Nostratic. We see it in English in whom, him, them and (perhaps) in me. Of the other cases in Latin, only one survives in English. This is the possessive or genitive. Again the pronouns provide the clearest example, but here there are two subtly different forms. Extend the above list now to read: I–me–my–mine, she–her–her–hers, we–us–our–ours,

he–him–his–his, it–it–its–its, they–them–their–theirs

and also: thou–thee–thy–thine and: who–whom–whose–whose. So we have, for example, ‘I own that dog’; ‘That dog belongs to me’; ‘That is my dog’; ‘That dog is mine’. And similarly for the other sets given above. In each of the above examples, the third and fourth members indicate possession. When it comes to nouns, we have a simple rule to do this job for us. Possession is usually indicated by use of an apostrophe, often followed by s. ‘The farmer’s dog’, and so on. On other occasions we use the preposition of; for example, ‘The reign of King John’. Again, we make distinctions by using word order. ‘The lazy farmer’s dog’ is different from ‘The farmer’s lazy dog’. In Latin, this difference is indicated by modifying the ends of words. Thus, these two descriptions are respectively ‘canis ignavi agricolae’ and ‘canis ignavus agricolae’, and again word order is not really important. Notice, however, in this example that the distinction between the two very different concepts relies entirely on the letters at the end of a

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single word, with –us interchanging with –i. Such small differences make for doubts at times in the interpretation of old manuscripts. It can be that the words were incorrectly copied. The case is similar to that of the ambiguities that can arise with the interpretation of Babylonian numbers. Grammarians, as well as classifying words by case, also distinguish them by gender and number. In English, gender is essentially the same as sex; we say he of a man, she of a woman and it of an inanimate object or something whose sex is irrelevant. (‘It’s a wretched nuisance, that dog!’) Much of the time, we make no distinction in the form of a noun whether it refers to a male or a female. The push for gender-neutral language has all but killed usages such as typiste and usherette, although heroine still retains some vogue. We sometimes have special uses such as he-man and she-wolf and perhaps words like lioness are still acceptable. We might for special situations use special ‘gendered’ words, as for instance when talking to a child, we say, ‘ A drake is a daddy duck’, but these are unusual cases, and we have largely lost such distinctions. Pronouns, however, retain gender differentiation. Latin was more complicated, and many of its descendants, such as French, retain in part this additional feature. So in French, number (both nombre and numéro) is masculine, although not male, while diagram (figure) is feminine, although not female. Gender need actually have nothing to do with sex, although it is now often used as a euphemism for just that. The word ‘gender’ derives from the French genre and, further back, the Latin genus, both meaning ‘type’. It is thought that very early in the development of PIE, there were two ‘genders’, (or types): animate and inanimate. Those nouns of the animate gender named things that could speak; the others those that could not. Developed PIE had three genders: masculine, feminine and neuter. German retains these, but they have largely disappeared from English; French, by contrast, has lost the neuter gender. Distinction by number is with most IE languages a division between singular and plural. The pronouns provide a clear example. He, she, it are singular; they is plural. Nouns are simpler and usually become plural with the addition of –s. (This is a widespread feature of IE languages.) In Latin, an adjective had to agree with the noun it qualified; it had to agree in case, in gender and in number. (This feature is still present in the Romance languages.) Thus ‘large number’ in French is grand numéro, while a ‘large diagram’ is grande figure; ‘large numbers’ are grands numéros, while ‘large diagrams’ are grandes figures. We have seen several examples of words or usages in English that do not really fit comfortably into the standard grammatical categories. The

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numerals constitute another such category. To capture an example of the confusion on this point, I looked up the word five in two different dictionaries. My edition of Webster’s lists it as an adjective; my Macquarie has it as a noun. Really neither description is entirely satisfactory. While children may speak, for example, of ‘one cow’ and ‘two cow’ as a way of describing respectively the first and second of two cows; we grow out of this usage. When we say ‘two cows’ we are not in any real sense describing the cows; two is thus hardly an adjective in the more usual sense. Rather, we think of two as a noun, albeit of a rather special type. In Chapter 2, we saw how the concept two could correspond to some word, such as ‘two’ (English), ‘deux’ (French), ‘zwei’ (German), ‘rua’ (Motu), and so on, or to some symbol such as ‘2’ (International), ‘II’ (Roman), 二 (Chinese). Because there is an exact one-to-one correspondence between the underlying concept and the word or symbol used to express it, there has arisen what is at least a very convenient fiction: that the number two has an independent existence, which these different modes of expression all in their different ways manage to capture. Now two does not exist in the same way as the cows exist when I say I can see two cows. We reach our concept of two by abstracting from pairs of objects. So it is more usual to say that rather than having a real existence it has an ideal existence. (Think of the word ideal as derived from the word idea rather than connoting perfection.) We do not meet a two if we go down the street, although it would be possible (if unlikely) that we might meet a cow. The philosophy according to which this ‘convenient fiction’ of regarding two as some sort of thing is usually associated with Plato. Although by no means all philosophers regard Plato’s account of these ideal entities as the best way to look at such matters, it is almost certainly true that most working mathematicians, when they think of the matter, even merely subconsciously, do regard numbers in this way. It is thus common to describe words like five as ‘platonic nouns’. Thus we see five as a (special) kind of thing, rather than a description of the set of elements forming a quintet. It is by no means only the numerals that are treated in this way by mathematicians. The very way we talk is pervaded by such views. ‘What’s a hemigroup when it’s at home? Is it the same as a semigroup, or is it some other kind of beast?’ Or why was it important to prove Fermat’s Last Theorem, when any counter-example that could possibly exist was already known to be so large that it could never possibly be found? Because, until we had such a proof, we had no assurance that out there, hiding in the

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platonic undergrowth of a world we had no hope of ever visiting, a counterexample might not have been lurking! My personal view is that the transition from number as adjective to number as platonic noun represented a real advance in mathematical sophistication. Once we have reified the numbers involved, we can say (for example) 87 – 51 = 36, and can assert this result as a theorem of widespread practical application and independent of any particular contextual reference. We could (although in practice we don’t) establish the result just quoted by reference to a simple experiment involving beads, pebbles, bottle tops, or whatever takes your fancy. Once one has adopted the platonic viewpoint, any such experiment is seen as equivalent to any other. It is immediately grasped that the objects exemplifying the numerals involved are themselves quite irrelevant. And we can confidently assert the same for any such calculation; there are cases where the numbers are so large that concrete verification has never been attempted. Yet we do not seek it, and if it were to be offered, we would view the endeavour as a waste of time! Or suppose that it were reported that a man in Yucatan had performed the 87, 51 experiment with a particular exotic type of seashell and ‘discovered’ a counter-example, finding that in this case 87 – 51 = 38, then our immediate reaction would be ‘What did he do wrong?’ But the platonic noun concept of number probably derives from earlier concrete concepts. In Chapter 2, we noticed the Kewa system of numeration by reference to parts of the body. This gives names to numbers up to and including 47, but if necessary a Kewa speaker can go further by saying, in essence, ‘right eye and one man is finished’. Even so, the concept would be, in large measure, independent of the physical act inherent in the counting. The relation to the parts of the body, the ‘finishing’ of one man, and so on are all symbolic. The 1–1 correspondence idea is even here at work, and we have, perhaps in a less developed form, the essence of the platonic noun view of number. So, although there are many philosophers who don’t like the account in terms of platonic nouns, it has proved useful and is unlikely to be abandoned. We use it in practice and it informs our number concepts at a very deep, even unconscious, level. However, when we look at the way in which number words were regarded in PIE, we discover a strange anomaly: The words for one, two, three and four were adjectives. The words for five, six, seven, and so on were nouns.

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Aspects of this dichotomy survive in some of the daughter languages of PIE. The pattern is preserved in Sanskrit, where the first four numberwords were ekab, dvi, trayah, chatvarah if the objects being counted were masculine, but eka, dve, tisra, chatasvarah if they were feminine. Similarly, in French, we must say un numéro or un nombre (‘one number’), but une figure (‘one diagram’). (As we have already seen, in French, numbers are masculine, but diagrams are feminine.) The numeral, being grammatically an adjective, had to agree with the subsequent noun. In Latin, the words for ‘one’, ‘two’ and ‘three’ retained this feature, but it was already lost in the case of ‘four’. However, we saw in Chapter 1 how Welsh, like Sanskrit, has different forms of the word for ‘four’. This is because their word for ‘four’ is an adjective and so must modify to accord with the status of whatever is being counted. English retains a faint echo of a time when this was also the case with us. In Anglo-Saxon, there were three words for ‘two’: twene (masculine), two or twe (feminine and neuter). Thus, the word ‘two’ is a feminine form; the masculine became ‘twain’. This word is now obsolete, but it just survives in the pseudonym of the American humorist Samuel Clemens, in Kipling’s line: ‘Oh, East is East, and West is West, and never the twain shall meet’ and in some compounds, such as ‘between’ and ‘twenty’. By derivation, ‘twenty’ means ‘two tens’, and the fact that the masculine form of the number 2 is employed means that ‘ten’ is a masculine noun. So, we have a peculiar anomaly: Two is a feminine adjective, but ten is a masculine noun! It is now no longer clear if the –ty that makes up the second part of the word ‘twenty’ is to be viewed as singular or plural, nominative or genitive; but, however, we decide these matters, the clear implication is that we are counting things called ‘tens’ and that there are two of them!

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The distinction in PIE between the first four numerals and the subsequent ones has left a number of footprints. In Latin, there were differences in the naming of children, in listing people’s ages, and in some calendric usages. And in Bangla (Bengali), the words for 2, 3 and 4 modify when they form compounds (analogues of English words like fortnight and tetrahedron), while in the main, the later numbers do not. But perhaps the best-preserved case is that of the ‘Slavic squish’. This takes somewhat different forms in different Slavonic languages. Here are two examples. In Czech, one says ‘Two and two are four’, but ‘Two and three is five’. (The form of the verb is dictated by the size of the total.) In Russian, they say ‘one house’, ‘two, three or four of house’ and ‘five, six, etc. of houses’. In Chapters 5 and 6, I shall argue that these relics of a differentiation between the first four numbers and the others tell us about the way in which the number-concept may have evolved. Another way words vary in the form they take is most germane to the topic of this book. This is the situation with number. We in English do not nowadays modify adjectives with numbers. Green things are simply green whether there are one or two or many of them. However, a French speaker has (one) objet vert, but (two or more) objets verts. We do differentiate nouns, as previously remarked, having one thing but two or more things. With a few exceptions, we always use a plural form if there are more than one. The verb must take the appropriate form, however, to agree with the number of the subject. So we say one thing is here, but two things are here. There are a few exceptions, but this is the general rule. The first possible exception concerns the so-called collective nouns. These are nouns that denote a set or collection of things. The words set and collection are both themselves collective nouns, and so are herd, flock, mob, and a whole host of others. There are even highly specialised collective nouns usually applying to types of animal, for example, to birds or fish; for example, a school of dolphins, a pod of whales, a pride of lions. It is even quite a game inventing new ones to add to the list. The general rule with collective nouns is that if we think of the set as a unity, then we use a singular form of the verb: A mob of cattle was heading down the Birdsville Track. But if we think of the individual components of the set as in some way independent of one another, then we use the plural: A mob of cattle were stampeding in all directions.

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There are some other situations where a singular form accompanies a number greater than one. The most common one is the use of a singular form when a unit of measurement is involved. This is even urged as correct usage by some pedagogues. ‘Three metre’ and the like. There are cases where this is clearly what everyone does. So we have a two-tonne truck, never a two tonnes truck. The same applies to other examples: ten-dollar bill, seven-year itch, threepenny opera, fourteen-carat gold. In these instances, the bill, the itch, the opera and the gold are all described by what in effect are compound adjectives; in one of the instances just given, the numeral becomes part of a compound, and something very similar happens in the others. It is sometimes said that a length of ‘three metre’ means three multiplied by metre. Well we may look at this way if we like, but it seems a little forced, and quite unnecessary. I once discussed this matter with a physicist, who maintained that ‘a force of ten newton’ was correct; he went on: ‘I might easily lapse into ten newtons, but I’d be wrong!’ Other physicists might well not take such a viewpoint! In popular speech, the plural may be employed or else the singular in such contexts; we often ignore even what might be the requirements of consistency: ‘How wide did you say that path was?’ ‘Two foot ten.’ ‘Sorry, I didn’t hear that. Could you repeat it?’ ‘Two foot, ten inches.’ In 1989, I heard a representative of Sotheby’s, the auction house, interviewed on radio, describe an item of furniture as being ‘Three foot by about two and a half feet.’ We have ‘two yards of cloth’, ‘ten kilometres of winding road’, but ‘two foot of clearance’, where in this last example, we might (just) perhaps regard foot as a measure-word. Generally, we are more likely to use a plural form with metric measures, than with the older Anglo-Saxon measures. So we have ‘Two litres of oil’, ‘A pressure of 180 kilopascals’, and even when we abbreviate we tend to stick to the plural form: ‘Two kilos of sugar’. Here we would not say kilo. But in the earlier units we often did, for the word thou, for ‘thousandth of an inch’, never inflected. However, in some metric contexts we may still use the singular form: a volume of 10 mL might be described as ‘ten mils’ or as ‘ten mil’, and other abbreviations are similarly variable: 5 km may be called either ‘five kay’ or ‘five kays’.

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Some poetic uses follow the singular form. Shakespeare had ‘Full fathom five’, and perhaps it could be argued that the line reads better this way than had he had ‘fathoms’. When the things being counted are numbers, then we tend to stick to the singular form. So we have five hundred (not hundreds), four score not (scores). Some grammar books advise combining such word-formations as five-hundred, and four-score or fourscore. But even here, we can find exceptions: ‘They won hundreds of dollars on the quiz show; five hundreds to be precise!’ There are a number of other such uses where a singular form follows an obvious plural numeral. Apparently big game hunters used to speak of a herd of elephant, or even of three tiger. This usage is noted by Partridge, who denigrates it as a ‘snob plural’; as the activity is now mercifully in decline, so are its peculiarities of grammar! And few will mourn! We also see some systematic divergences from strict logic. We tend to say ‘More than one is …’ or ‘One and one is two’, but ‘Fewer than two are …’. The ‘feel’ of an adjacent one or two takes precedence over strict logic. Perhaps also, as the sentence becomes more complex, we tend toward the singular form: ‘One and two and three and four makes ten’; here we seem to be seeing the sum as a unity and so say makes, not make. One recent change in our language is currently under way. This concerns the usage following the introductory ‘There’. I would routinely say ‘There is …’ if there was one thing to be introduced, but in the case of more than one, I would say ‘There are …’. This distinction is rapidly dying out. It is quite common to hear politicians or broadcasters saying (for example) ‘There is two problems with this piece of legislation’. It is still in my view (just) more correct to say are in this sentence, but if enough politicians and broadcasters go the other way, then our language will alter to accept this usage. Numbers do not usually take the genitive. ‘Twenty casualties, ten of men and ten of horse’ is archaic, but we can have ‘hundreds of dollars’, as noted above, and we can also have ‘three of the large and four of the small’. However, we could equally well say ‘several hundred dollars’, ‘three large and four small’. All in all, we are not consistent in our usage, but this really doesn’t affect matters adversely because, if any confusion does arise, we have means to resolve the ambiguity.

5 7

2 18 93 36

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7

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C hapter F I V E

5

Early history of numerical concepts

In the previous chapter, we saw a distinction between single and plural nouns, adjectives and forms of the verb. In point of fact, the situation is somewhat more complicated than that. PIE had in fact three number-based forms: singular, dual and plural. Greek, Sanskrit and Lithuanian followed PIE in this respect although the dual did not survive in the Latin. Thus in Classical Greek one had: ho philos the friend

tò philō both friends

hoi philoi the friends

Here the noun and its accompanying article both exhibit all three forms: singular, dual and plural. Similarly, we have for a single horse: híppos; if there two horses: híppo; and if there are more than two horses: híppoi. This is in fact a very common situation. It can be found in many of the Afro-Asiatic languages. It also is found in a number of Australian Aboriginal languages, especially those that follow the ‘one-two-many’ system of counting. In fact, there are quite a number of vestiges of the dual in English, even today. First there are specialised words such as both, either and neither that apply only to cases where there are two possibilities. Where there are more than two, we use other forms. So we can say: English and French are both Indo-European languages. But for more than two possibilities, we need to make the alteration: English, French and Hindi are all Indo-European languages. 49

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Or: Tamil and Turkish are neither of them Indo-European languages. as opposed to: Tamil, Turkish and Thai are none of them Indo-European languages. It is the same with the other (dual) but another (plural). These are distinctions that are slowly dying out. Even two hundred years ago, Coleridge wrote ‘Both man and bird and beast’, although perhaps we might call this poetic licence. But we could certainly say of a word-form that it could be either singular, dual or plural, and not offend any but the most pedantic of readers. The second vestige of the dual is found in the dual prefixes ambiand amphi-. These still hold a dual connotation in some contexts. For example, an amphibious military exercise involves the army and the navy. One also involving the air force was once referred to as triphibious, a rather barbarous, linguistically incorrect, coinage which had some vogue but seems not to have stuck. So, here again, the distinction is being eroded. Furthermore, if a statement was so imprecise as to carry three possible meanings, we might well still describe it as ambiguous. (Another, once exclusively dual word, alternative, has now lost this character. It is now OK, when once it would not have been, to be presented with three ‘alternatives’.) The third survival of a dual form is the use of the comparative and the superlative of adjectives. Thus, if there are two objects, we have the larger and the smaller of the two. If there are three objects, then one is the largest and another (note the plural form, as opposed to ‘the other’, dual) is the smallest. In the fourth place, we have many specialist terms referring to pairs. There is the word pair itself, and also yoke, brace and couple. In all these cases, the root word is independent of the word two. Compare the situation with quartet, for example, where the root is the Latin quattuor, which we have already seen to be a cousin of our word four. Fifth, we have a break between two and three in the list of ordinal numbers. So we start out with first, and second, but then we get onto third, fourth, and so on with thereafter the ordinal being obviously related to the cardinal form of the number. With the first two ordinals there is no such relation. The word one and the word first are not related, nor are the

E A R LY H I S T O R Y O F N U M E R I C A L C O N C E P T S

words two and second. The word first comes from foremost, and second comes from the Latin secundus, meaning ‘following’. Similar points may be made in respect of primary and secondary, as opposed to tertiary, quaternary, and so on, or in respect of half as opposed to third, quarter, and so on when it comes to naming the reciprocals. Sixth, even when a word does derive from the same root as the word two, as for example twin, our language allows a wider usage than it does in for example triplet. Thus, we have twin-sets, twin-tubs, etc. In such situations we would not have the same resource were the word ‘triplet’ instead of ‘twin’: we have a three-piece suit or a 3-stage rocket, for example, not a triplet-set or a triplet-thruster! Seventh, there are two apparently anomalous number-words, eleven and twelve, occurring where we might expect oneteen and twoteen. These are derived from earlier Anglo-Saxon words anleofan and twelf. The second element of each of these words derives from the proto-Germanic *lihw-, itself a descendant of the PIE *leikw-, meaning ‘left over’. (Indeed the word left in this sense is itself derived from this same root. We see it most clearly in words like reliquary.) So what we see here is itself a sort of ‘relic’ in which we see what seem to be the first steps beyond ten. Finally, eighth, the word ‘twice’ is used, when we mean ‘two times’, but when we come to larger numbers, we say ‘three times’, even though ‘thrice’ is a technical possibility, and after that, we have to say ‘four times’, ‘five times’, and so on. No other choices are available. However this effect is not on the same level as the others. The fact that there is a word ‘thrice’ rather indicates to us that an archaic distinction between ‘twice’ and ‘n times’, where n > 2, is not involved here. It rather dates from a time when, as in Latin, words for one, two and three inflected, but later words did not. However we see, even in the English of today, a few vestiges from a much earlier time when we employed a number-system that went, in essence, one, two, many. In such a system, there are only two precise number-words, and the manner in which they and the vaguer word many find expression is essentially grammatical. Clearly there is no possibility of extending such distinctions to very large numbers, and the catch-all plural has tended to overwhelm the dual as the concept of number has become more sophisticated. But the one-two-many system is the simplest ever encountered, and it is described as employing two as the limit of precise counting. There are, or perhaps rather were, such systems. Some Australian languages, for example, exhibited this character. The next advance is what Menninger calls ‘the step to three’. That is to say that the next number, as we might expect, to find precise quantification is three. The next logical

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step is the one taken by the Kiwai and the Gumulgal of Chapter 2, in which the words for one and two are pressed into service as elements of a basetwo system in order to extend the limit of precise counting to 4. Look back to Table 2.8 and notice that the Fore system also exhibits such beginnings. The extension of precise counting to 4 allows a grammatical system in which we could find different forms of the verb (for example) depending upon whether there were one, two, three, four or many (more than four) elements in the subject. This is probably as far as it is practical to take such distinctions (and we do have the case of the Kwakiutl system of Table 3.1), but we have already noted the tendency of plural forms to swallow specialist dual (let alone therefore specialist treble or quadruple ones). Menninger remarks of the ‘step to three’ that it ‘is the decisive one … [t]wo is stripped of its unique position and is recognised as a number just like any other; the grammatical dual … disappears’. It is not entirely clear (at least not to me) that Menninger’s remarks along these lines are convincing. I do not find it obvious, for example, that the step to three suddenly opens up the way to the entire number-sequence. The existence of one-two-three-four-many systems would seem to militate against him here. What we can say, however, is that the step to three is a necessary condition for the development of the apprehension of the entire sequence. Perhaps we could go further than this (after all minimal) concession, by allowing the step to three to be the decisive psychological breakthrough paving the way to the subsequent development of the full sequence. As we pass beyond the very simplest systems, therefore, we can recognise a possible stage in which there are definite words for one, two and three; and a stage in which there are definite words for one, two, three and four. In the case of the former type, it is claimed that the Australian language Pitjantjatjara has (or once had) this character. Menninger regards the widespread occurrence of this stage of language development (the one, two, three, many system) as proven fact, although he offers no really convincing example. Perhaps his best evidence lies in a parallel he draws between the grammar of the pronouns and the first three numerals. Grammatical description

Pronoun

Numeral (PIE)

First person

I (one)

*oynos

Second person

thou

*dwo(w)

Third person

they

*treyes

Table 5.1.  Parallel between pronouns and numerals

More will be said on this in Chapter 6.

E A R LY H I S T O R Y O F N U M E R I C A L C O N C E P T S

However, once we progress to the next step and consider one-twothree-four-many systems, we are on firmer ground, certainly in respect of the influence on our own number-words. Menninger notes the major difference between the first four numerals and those that follow in many Indo-European languages and aspects of culture. These have already been noted in Chapter 4 in relation to the ‘Slavic squish’, and other examples will be given below. What these demonstrate is a system in which four has become the limit of precise counting. This may have been achieved via a system in which two was a base, as in Kiwai or Gumulgal, or possibly in some other way (but it seems most likely that the intermediate base two stage was involved, because these languages provide actual examples of just such a process). The best-preserved ‘fossil’ of such a stage in the development of the Indo-European languages is indeed the Slavic squish. Other PIE-derived languages, however, also show such a ‘break’ between the numbers 1, 2, 3, 4 on the one hand and the numbers 5, 6, … on the other. Some of these others were also remarked in Chapter 4. Perhaps the best documented of these is that involved in the naming of children in ancient Rome. A son could be called Quintus (fifth), Sextus (sixth), Septimus (seventh), and so on; however, no one was ever called Tertius (third) or Quartus (fourth); Secundus (second) was used, but only as a sort of ‘nickname’, somewhat like the American ‘Junior’, never as a personal name. So we have seen that all cultures have an independent concept for 2, and that the next steps are the progress to 3 and 4. Where 4 is the limit of precise counting, we often see 2 as being pressed into service as a ‘base’, thus allowing the new limit of precise counting to move out to 4. Can this process be repeated, and 4 become the ‘base’ with a new limit of precise counting now moved out to 8? I believe that we have some evidence that exactly this did happen with our remote ancestors. I will present the evidence in the next chapter, but it should be said here that many authors do not agree with me in postulating this step. Mine is one among many competing stories, and there is really very little hope of making any final decision between them. It should be said also that if my theory is correct, then there must have been a time of transition between a base-four system and today’s base ten system. This to my mind is not an insuperable obstacle. We have already seen an example of a system using a mix of base four and base ten. This is the case of the Motu language whose number-words were displayed in Table 2.9.

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Others have postulated that there was once a base five precursor before base ten became established. This also runs into the difficulty of a transition from base two or base four to base five. Certainly there are languages in which base two elements mingle with base five ones. Fore, introduced in Chapter 2, shows strong influences in this direction, and so, even more strongly, does another PNG language, Kuman. This proceeds exactly on the same pattern as Kiwai or Gumulgal, but then there is a change. The word for five is the same as that for one hand, with, further along, the word for ten being the same as that for two hands. It has been claimed that two languages of the Western half of New Guinea (now a part of Indonesia), Ormu and Yotafa, exhibit mixes of base four and base five elements. To my mind, the case for either has not been made, but perhaps enough has been said already to demonstrate that the situation in which two bases co-exist is not at all impossible. Quite the reverse, in fact; the situation is actually quite common. If we look back at the Babylonian and the Mayan systems described in Chapter 2, we see what may be viewed as further manifestations of the same phenomenon. Even we ourselves use mixed base in areas of very limited application. We might perhaps cite in this context the old system of weights and measures, still in use, especially in the USA. But cricket commentators speak of ‘2.3 overs bowled’, when they mean ‘2 overs of 6 balls each and 3 balls of a third such over’. The number of overs referred to is, strictly speaking, 2.5. I am told that a somewhat similar convention applies to aspects of baseball commentary. These matters will recur in the following chapter, where, however, the focus will be somewhat different.

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C hapter si x

Developed systems of number-words

5

When a culture finds some concept important, it develops a word to express it. As a general rule, the more important a concept is to that culture, the more likely it is that that concept will find such expression in a word which is short, stable and unmotivated. The meaning of the word ‘short’ is obvious. The description ‘stable’ means that the word is likely to retain its form over time and will continue to refer to the same concept. A word is said to be ‘unmotivated’ if it has no obvious antecedents; we do not immediately see what more basic ideas motivate it. The number-words are a particularly apt focus of attention in such study. Other words have rather more flexible reference, but a numeral such as eight retains its meaning completely unaltered. Compare this situation with other common words. Here is one example, a much-quoted one. Learning high-school French, we are told that the word fleuve means ‘river’, while rivière means ‘rivulet’ or ‘creek’. However, this is not quite an accurate picture of the true state of affairs. French makes a distinction that English does not. A fleuve is a watercourse emptying directly into the ocean while a rivière is one that does not; it is a tributary of either a fleuve or else of another rivière. Thus Bream Creek, near the Victorian beachside resort of Torquay, and famous as the haunt of the ‘wild white man’, the escaped convict William Buckley, is a fleuve although a fit athlete could jump over it. The mighty Murrumbidgee, by contrast, is a rivière. Another such example of a shift in the meaning of a word has a more mathematical reference. Paul Dupuy, in his life of Galois, recounts a notorious incident occurring at a banquet in the course of which that hot-headed young revolutionary proposed a ‘toast’ to the then king LouisPhilippe, doing so with an unsheathed dagger in his hand. At this point, the novelist Alexandre Dumas and several of his associates passaient par la fenêtre, a phrase that ET Bell in his Men of Mathematics rendered ‘passed 55

2

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by the window’. In fact, it means ‘departed [hastily] through the window’, in order (as Dumas himself wrote) ‘not to be compromised’! So, as words evolve, so also do their meanings. But number-words are an exception. Numbers have simple fixed values: two means two, and so on. Concentrating on number-words allows us to follow changes in words themselves independent of any changes in their referents. Applying our basic set of criteria (short, stable and unmotivated) to the number-words, we are able to contrast our own number-words with (say) those of the Northern Fore, as given in Table 2.8. In our own culture, among the names of the first ten numbers, there is only one (seven) that extends even to two syllables, we do not encounter a three-syllable word until we reach eleven, and so it goes. By contrast, Fore employs a twosyllable word for one and a three-syllable word for two. By the time we reach the numeral six, we require ten syllables to express the concept. This leads us to suppose that the Northern Fore had little use for the concept ‘six’. They could express it if they had to; they could do so precisely if required, but they never felt the need to do so in a short and efficient way. The Fore number-words are not short, whereas ours are. Similarly, we can trace the ancestry of our own number-words, as in Chapter 1, because their forms have altered very little over thousands of years. With very few exceptions (mostly referring to the concept two as detailed in the previous chapter), there is only one word for each number, and even where there is more than one possible way to speak of a number, the variation is usually slight (‘two and twenty’ versus ‘twenty-two’, etc). Yes, there are exceptions: score and dozen for example, but these have only limited usage. The case of the Russian word for forty (see Chapter 3) is noteworthy precisely because such anomalies are extremely rare. Furthermore, we can follow these single number-words through relatively minor and quite systematic alterations over thousands of years. So, hearing that the extinct IE language Tocharian had a number-word okät, we might be willing to guess that this meant ‘eight’, and we would be right! Again contrast the Fore. Already there are two forms for the number three. By the time we reach the number eight, the number of possible forms has got even larger. Nor have these forms persisted. As noted several times already of the New Guinean languages, our own forms have almost completely supplanted the traditional ones. The Fore number-words are not stable, whereas ours are. Finally, where our number-words are only subject to the most speculative etymologies (to be discussed below), all but two of the Fore words relate directly and obviously to other concepts, those detailing the process of counting on one’s fingers and toes. Some linguists see our

DEVELOPED SYSTEMS OF NUMBER-WORDS

number-words as having a similar origin, but the case is far from clear, and when it may be so, we never advert to it in everyday usage. Even where the etymology is known, as with the word eleven, we do not immediately relate that word to ‘one left (over)’. The Fore number-words are not unmotivated, whereas ours are. The deduction from this analysis is that the Fore in their traditional society found only scant use for precise enumeration, or if they did find a need for it they employed means other than linguistic ones. It should not, however, be surmised that our own language does not contain similar lacunae. English is notoriously lacking in words describing precise relations of kinship. If we have to describe someone as a ‘second cousin three times removed’ we are employing a very clumsy (and actually imprecise) descriptor, every bit as clumsy as the Fore to náentisa ká ‘umaemawé for six! Those cultures that have developed such requirements are termed cultures of advanced numeracy. Apart from our own, there have been others. We inherit a legacy that derives from the Greeks, the Arabs and the Romans, as well as (more distantly) from the Babylonians, the Egyptians and the Hebrews. Besides these, there have been advanced systems of numeration and traditional Mathematics among the Chinese, the Indians, the Japanese, the Javanese, the Koreans, the Maya, the Persians and the Tibetans. Some people might extend the list even further by including (say) the wonderful navigational feats of the Polynesians as a developed form of Applied Mathematics. Our own number-words are taking over from these others in large measure because they are more efficient. Even the Chinese system is giving place to our own, at least as far as the signs 1, 2, 3, … go. (However, they continue to use the traditional Chinese names for these symbols.) Despite the difficulty of ever establishing the origins of our numberwords, for the reasons outlined above, there have been several attempts to do so, at least in respect of some of them. Even Szemerényi, whose analysis I am following in the main, and who in some passages seems to regard the search for etymologies for the PIE numerals as a futile exercise, does in fact advance or report a few (really quite a few) tentative suggestions. Almost all such suggestions, whether by Szemerényi or by others, share some points in common. All systems of advanced numeracy now use base ten, and there is very probably a connection with counting on the fingers. This leads to the suggestion that there may at some stage have been a subsidiary base five system, but this point is not universally agreed. It is, however, agreed that there must have been a ‘base two’ system at an early stage in the development, essentially for the reasons outlined in the previous chapter.

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Menninger is perhaps the most speculative of the authors I have quoted. He draws attention to the likely connection between the personal pronouns and the first three numerals. This was summarised in Table 5.1, but now it is time to look at it in more detail. The pronoun I is spoken of as being in the first person. Its Latin equivalent is ego and this seems mighty close to the Sanskrit ekab, meaning one. A strange echo of this possible connection is a usage employed by upper-class English in which the term I is not used but is replaced by the word one. This is not the same as the impersonal pronoun. Compare ‘One (meaning I) went to Eton; other chaps went to Harrow’ with ‘One (impersonal) must always wear a tie in the club’. This is probably a relatively recent development, however. The Oxford English Dictionary (OED) fails to notice it, although they do actually provide two (misplaced) examples, dating from 1649 and 1959. The Shorter OED (SOED) does better but labels the usage as ‘colloquial or affected’; several examples are given beginning with one from Oscar Wilde. We also speak in denigration of (say) an allegedly corrupt politician: ‘He knows very well how to look after Number One’. If we consider the two alternatives for the PIE word for one: *oykos and *oynos, as given by Lockwood in Table 1.1, the words ego and ekab would seem to relate to the first form, the English word one to the second. There are other suggestive pieces of evidence as well. The Old Persian form of the word was adam, and this is the name given in the Hebrew Bible as the name of the first man. (However, there have been many other suggestions as to the etymology of the word in this context.) The close analogy continues when we consider the second person, thou. The PIE is thought to have been *tu. Compare the various forms of the PIE for two: *dwo(w), *dwoi, or *dwo– from Table 1.1, and the supposed Nostratic **to. Again, there seem to be grounds for postulating a correspondence. However, by no means all authors adopt this suggestion, and Menninger does not advance it with as much enthusiasm as one might expect. When he comes to the number three, we find several different ideas on offer. In the first place, he considers a theory that the words for one, two and three relate to the words this, the and that. To my mind, this has little to recommend it. Then there is the variant put forward in his Chapter 6, relating *treyes to they. This may be so, but there is precious little evidence for it. Indeed few will follow him here. The PIE is supposed to have been *te–, and this is a different root from *tre–. Then he considers another possibility. Reflecting that the ‘step to three’ was, in his view, the principal advance in the development of the

DEVELOPED SYSTEMS OF NUMBER-WORDS

number-concept, he considers the possibility that the word *treyes relates to the notion of trans–, meaning ‘beyond’. He makes comparison between the French très (very) and trois (three) on the one hand, and the English through and three on the other. This is an imaginative suggestion with a certain plausibility; however, it strikes me as unlikely. There seems to be very little speculation on the origin of the word w *k etwores, for four. If we accept one version of the Nostratic hypothesis, there are possible connections to Eskaleut words *qäзä (with a supposed pronunciation something like kozho, and meaning ‘palm’) and *зet'w^ (with a pronunciation something like zhetwa and meaning four). These may be related, but the idea is by no means forced upon us. Menninger does propose, as do I, that there was a time when four was the limit of precise counting, and this leads to the phenomenon of the Slavic squish and the other such differences between the numbers up to four and those beyond it. The suggestion implicit in the supposed similarity between *kwetwores and the two Eskaleut words just given is that there is a possible connection between the notions of palm and of four. Menninger, but without alluding to the Eskaleut words, makes much play with this. To my mind, he is here asking rather too much credence from his readers. Certainly, however, there is a resemblance between *зet'w^ and *kwetwores, but such resemblance may arise by quite other means than by common descent, and this is why the Nostratic hypothesis remains controversial. However, even if we accept all this, by itself it takes us little closer to a possible meaning for *kwetwores. Without adverting to the (more recently published) Eskaleut data, Menninger advances an extremely speculative hypothesis, but I doubt that anyone has ever taken it very seriously. When we come to the word for five, there is also much disagreement. There is a similarity of a sort, between the PIE *penkwe and the word finger. However, this is attested almost exclusively in the Germanic branch of the IE family tree. Indeed so much is this so that Lockwood doubts that it is even a PIE-derived word. He lists ‘finger’, ‘hand’ and ‘arm’ among a number of such words that possibly entered the Proto-Germanic from a non-IE source. There is rather more support for the idea that the word relates ultimately to the word fist. The PIE for fist is thought to have been *pno k(w)sti–, with a pronunciation rather like puhnkwsti–. Allowing for the loss of part of the final syllable, there is a considerable resemblance to the word *penkwe. Szemerényi notes this possible connection, and, although stopping short of endorsing it, he clearly has a lot of time for the suggestion involved. He takes the view that the same word *pno k(w)sti– was used to mean either fist or hand. All of this accords well with Szemerényi’s overall belief that the PIE base ten system arose from an earlier base five system.

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I, however, have advanced another suggestion, which I will detail later. I don’t claim that my idea is any better than the ones just described. It is simply another possibility with some points to recommend it. It is, of course, highly plausible that the word for five should be related to that of hand. We use base ten almost certainly because we have ten fingers, and when we find independent words for five, they most commonly relate to the word for hand. One example will suffice, but it is an important one. Almost the only case of a number-word having clear ‘motivation’ (that is, clear etymology) in a culture of advanced numeracy is the word lima found in almost all Austronesian languages. It has exactly this form in Indonesian, and it is the clear basis of the word ima in Motu (Table 2.9). Maori has rima, and Tongan has nima, which are both clearly closely related. Many other Austronesian languages can be found to use the same root-word, whose original meaning was ‘hand’. When we come to the word six, we find Szemerényi again tending to favour an explanation that involves a base five precursor to the base ten one. He quotes with some approval a suggestion that the PIE *(H)weks, meaning six derives from a word ‘to grow’, a suggestion based on a Greek word aexo with this meaning. Thus, he has that ‘“6” would be “the increase” after the first “hand”’. There would seem to be very few suggestions put forward as to possible derivations for *septm, meaning seven. When it comes to *octō(u), the PIE word for eight, there are few developed suggestions either. The word has the form of a grammatical dual. Thus in Lithuanian, we have the form judu, meaning ‘you two (women)’, we see the same structure. So *octō(u) should mean ‘two ocs’. This is where most speculation stops. Szemerényi says: ‘It is often asserted that*oktō is the dual of ‘4’ but, however much one would like this to be the case, the attempts to reconcile this wish with the actual form remain unconvincing.’ In fact he goes on: ‘The remaining numerals still defy all rational analysis.’ This despite his discussions of some possibilities for them! He does go on to look at non-IE languages where the word for 8 is a dual form of the word for 4. He finds this to be the case for the Finno-Ugric language Ostyak. If we adjoin the Nostratic hypothesis to this conclusion, then we have stronger grounds for our conclusion in respect to the PIE word for 8. I will give my own further thoughts on this matter later on. When we come to *newn, meaning nine, there is considerable agreement on one basic point. The word *newm, to give Lockwood’s version from Table 1.1, bears a close resemblance to the Latin novum, meaning ‘new’. Indeed Watkins lists the PIE word *newn as meaning both ‘nine’ and ‘new’. Menninger notes the similarity and makes quite a lot of

DEVELOPED SYSTEMS OF NUMBER-WORDS

it, and even Szemerényi accords it some weight: ‘If we accept the old suggestion that, in the primitive IE system of counting, “9” represented the “new” unit after counting had stopped at “8”, then the original can hardly have been anything else but *newom, which later became *newm.’ As Szemerényi regards *newn (rather than *newm) as the ancestral form, and as he also tends to regard the derivation of the base-ten system as derived from an earlier base-five one, he is reporting a point against himself here. Finally, we come to the word for ten. This is given as either *dekm or *dekmt, this last form being the one favoured by Szemerényi. If we look up the etymology for the word hand, we discover that it comes from the Germanic, but has been speculatively derived from a possible PIE *kent. This, following the discussion on the m-n question just seen in relation to the number 9, looks very like the second element of Szemerényi’s form *dekmt. The initial de– can with reasonable plausibility be related to the Nostratic **to, meaning two. Thus, there is a possible original of ‘two hands’. I have proposed a theory that incorporates some of these elements, but combines them in a way that seems to me to be more satisfying, and more coherent than the various other tentative suggestions so far advanced. All the same, I offer it very tentatively as it is extremely speculative. However, I regard the earliest version to appear to have been a onetwo-many system, as attested by the survival of the various vestiges of the dual along with the singular and plural nouns of nouns and other parts of speech. In time, this extended to a one-two-three-four-many system, with two, formerly the limit of precise counting, now being pressed into service as a base. The new limit of precise counting then became four. Both the Kiwai and the Gumulgal systems show linguistic evidence of just such a change, and we see in the ‘Slavic squish’ and other such ‘linguistic fossils’ evidence that there was such a period in the development of PIE when four was the limit of precise counting. Just as the first extension of the number-system (from two to four) requires the ‘step to three’, so will this next development require a ‘step to five’. This is the new feature of my proposed theory. I offer an alternative suggestion as to the origin of the PIE *penkwe. As well as the words *oykos, *oynos or their variants noted in Table 1.1, there is another PIE word for ‘one’. That word is *sem(s). We still see its descendants today: words like simple (which quite literally means ‘one-fold’), same, single and singular. In Greek, *sem(s) turns up as *hen-s, deriving from an earlier *hem-s. This in its turn has given us words like hendecagon and hendiadys. This s–h shift is a violation of Rule 2 from Chapter 1, but only in an apparent sense. If a consonant pushes so far forward in the

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mouth that it falls out the front, it begins a new such journey, going to the very back once more! But now suppose that the sound travels forward, but not as drastically as this, and stops at the lips. This produces *pem–. The –kwe that makes up the remainder of the word is very reminiscent of the Latin –que. This means ‘and’; it is adjoined to a word to indicate that this word is to be added to what went before. The Roman legions fought on behalf of Senatus populusque Romanum—meaning ‘the senate and people of Rome’. Putting this construction on the –kwe produces *pemkwe, with the meaning ‘and one’. This is almost the given PIE form. The only difference is that and m turns into an n and this is a natural shift for a nasal consonant before a guttural. Think of the way children frequently corrupt the word pumpkin into punkin! Less flippantly, recall that *sem– gives rise to single and singular via exactly this shift in the consonant. Thus, I suggest that the PIE word for ‘five’ reflects exactly the ‘step to five’ when the limit of precise counting passed beyond four. It means ‘and one (more)’. Next consider the word for ‘eight’ *octo–. Here also I have a suggestion to make. As noted above, it is a dual form, and means ‘two ocs’, whatever ocs might be. Now the PIE word *ok– has the meaning ‘eye’. We see it preserved in words like oculist and ogle. Thus, we may consider for *okto–, or *octo–, the meaning ‘two eyes’, or perhaps ‘seeing double’. We saw earlier that the limit of precise counting moved out from two to four. Imagine this process being repeated with the limit now moving out to eight. This can account for the forms for the PIE words for both five and eight. It also accounts (as indicated above) for the word for nine as the ‘new number’, a possibility that has attracted quite a lot of support. Thus I envisage an earliest stage in which the limit of precise counting was two, with this followed by a later stage in which the limit moved out to four, possibly making use of two as a ‘base’, as in Kiwai or Gumulgal. This stage was followed by a third stage in which the limit moved out to eight, with four playing a role as a ‘base’, as evidenced in the PIE words for five and eight. After that, the limit must have moved out yet again, as evidenced in the PIE word for nine. This process, however, does not in any way account for the rise of our base ten system. That, almost surely¸ derives from our ten fingers. Indeed the word digit, used for both the ten basic number-symbols and for the fingers of the hand, is strong evidence of precisely this. (Our word digit and the PIE *dekm are both supposed to derive from a Nostratic root **tik.) The base ten system must have had a separate origin, possibly

DEVELOPED SYSTEMS OF NUMBER-WORDS

involving five as an intermediate base. But just as the Motu numerals in Table 2.9 follow a base ten pattern, but with well-preserved elements of a base four one, so I suggest that there may have been such an intermediate stage in the development of PIE before the base ten system became firmly established. I do stress that this suggestion is highly speculative, but so are all other suggestions. I advance it not as being ‘more right’ than others, but rather in a spirit of demonstrating a plausible alternative.

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Projects

I conclude by discussing in some detail four topics that could be used for further investigation or as the basis of student projects.

7.1  Roman numerals There are actually a number of different versions of the Roman numerals, but they differ in only minor details, at least until quite large numbers are encountered. What I describe here is a version of the most widespread one. Every Roman numeral is a string of letters written one after another and read from left to right. I will refer to the individual letters in the string as the elements. There are five basic elements: • • • • •

I, meaning 1 X, meaning 10 C, meaning 100 M, meaning 1,000 W, meaning 10,000.

In this last case, I have inserted a modern convention in place of the ones the Romans actually used. The inclusion of this fifth symbol takes us beyond the simpler account presented above in Chapter 2. It will be obvious to the reader that the system is thus a base ten one. In its overall character it is not unlike the Chinese one described in Chapter 2. However, there is a complication. Besides these basic elements, there are four auxiliary elements: • • • •

V, meaning 5 L, meaning 50 D, meaning 500 K, meaning 5000. 64

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Here also, the last entry is a modern simplified convention, and again it was not listed in Chapter 2. It is the existence of these auxiliary elements that has led to the suggestion that the Roman numerals provide evidence of an earlier base five system. This is consistent with Szemerényi’s linguistic suggestions. If, as I suggest, there was also a base four one, it has not survived in the notation of Roman numerals. Strings are constructed from these elements in accordance with certain rules. These rules constitute a ‘grammar’ for the strings. They determine which strings are meaningful, and what those meanings are in such cases. If a string is so constructed that the rules are obeyed, then that string has a meaning, and we can determine what that meaning is. For any string or element a, say, let n(a) be the number represented by a. Here are the rules.

Rule 1: If a, b are elements such that n(a) < n(b), then a may precede b in a string (i.e. be written to its left) only if a immediately precedes b. Thus, one may write IV or IX, but not IXV, because the I here does not immediately precede the V. Nor can one write XCM, because the X does not immediately precede the M. Similarly, XXL is outlawed because the first X does not immediately precede the L. Rule 2: If a is an auxiliary element, and b is any element, then a may precede b only if n(a) > n(b). This rule ensures that each auxiliary element occurs at most once in any string, and, if any are present, they must appear in the order W, D, L, V. Thus LX is permissible, but VX and WK are not. Rule 3: If a, b are elements and n(a) > n(b), then the combination aba is not permitted. Thus CXC is permissible, but XLX is not. Rule 4: If a, b are elements, then a may not precede b if n(b) > 10. n(a) This rule allows XC, XL, XX, XV and XI, for example but excludes XK, XW, XM or XD. IC, which is allowed in some variants, is here excluded. Because of Rule 2, Rule 4 imposes further restriction only in the case where a is a basic element.

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Rule 5: If a is an element, the combination aaaa is disallowed. Here too, there is only further restriction when a is a basic element. This rule outlaws strings like XXXX and CCCCC. The combination IIII, allowed in some variations, is here outlawed. Rules 1 and 2 are the most important. It is they that ensure that a string has a meaning. The remaining ones ensure that there is a unique representation for any integer. Strings forbidden under Rules 3, 4 or 5 are not meaningless, but are ‘ungrammatical’; rather like ‘I go’d Saturday football’. We know what it means, but it is incorrectly expressed. With the rules laid down, we now need a Canon of Interpretation. This enables us to translate Roman numerals into our familiar ones and vice versa. This proceeds as follows. An element a may be used in one or other of two ways. To decide between these, look at the element immediately following a, b let us say. Then we have: If n(b) > n(a), then a is used in the subtractive mode. Otherwise, it is used in the additive mode. The final element in a string is always used in the additive mode. Rule 1 tells us that if an element is used in the subtractive mode, then its successor must be used in the additive mode. Rule 2 tells us that only basic elements may be used in the subtractive mode; auxiliary elements may not. It is now possible to say unambiguously of every element in any string whether it is used in the additive or in the subtractive mode. Notice that we have only used Rules 1 and 2 here. This is the reason for saying that these two are more important than the others. The Canon of Interpretation may now be stated: Add the numerical equivalents of all the elements used in the additive mode, and from this sum subtract the sum of all the elements used in the subtractive mode.

Here is an example. Consider the string MDCCCXLIX. Here the first X and the I are used in the subtractive mode; all the others are used in the additive mode. So now we can carry out the translation: n]MDCCCXLIXg = 6n]Mg + n]Dg + 3n]Cg + n]Lg + n]Xg@ - 6n]Xg + n] I g@ = 51000 + 500 + 300 + 50 + 10? - 510 + 1? = 1860 - 11 = 1849. (7.1) It is equally simple to go in the opposite direction. A single example will suffice to illustrate the method. We seek the string a such that n(a) = 24 197. Proceed as follows:

PROJECTS

20 000 is 2 # 10 000 4 000 is 5 000 - 1 000 100 is just 100 90 is 100 - 10 7 is 5 + 2

or KK or MW (M in subtractive mode) or C (7.2) or XC (X in subtractive mode) or VII

Now simply adjoin the letters from the right-hand column, to find KKMWCXCVII. This is the required string. (Notice that here we have made fundamental use of the fact that Roman numerals constitute a base ten system!) The upshot of all this may be summarised by a theorem: Every sufficiently small number corresponds uniquely to a permissible string and vice versa.

The question is often raised as to whether the Roman numerals can be used in practical computation. The answer is that they can; there are several published algorithms for the basic arithmetic operations. Whether the Romans themselves used them is uncertain. However, it is known that they made use of the abacus. Two examples have survived. One is in a museum in Rome, the other in Paris. Both are small hand-held devices about the same size as a pocket calculator of today. Because the physical evidence for these devices exists, whereas no evidence at all has come down to us of written computation (which we do have in the case of the ancient Greeks), it seems likely that, not unlike students today, the ancient Romans were hooked on their calculators! Among the projects that could be built around this material, consider the following problems. 1 What is the largest number that can be expressed by a permissible string? That is, how small is ‘sufficiently small’ in the theorem stated above? 2 Write computer programs to decide if strings are permissible or not, and to interconvert between Roman and familiar numerals. 3 What is the longest permissible string? What fraction of strings of this length or shorter are permissible? 4 Devise algorithms for addition, subtraction, multiplication and division in Roman numerals. Write computer programs to implement these.

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THE NAME OF THE NUMBER

7.2  Bases other than ten As noted in Chapter 2, there is no particular mathematical reason for choosing ten as the base of our system of numeration. Any integer greater than 1 can serve perfectly well. We can interconvert between bases quite easily by means of simple algorithms. I will use the notation nb to represent the number n expressed in base b. So the number n will be represented by a string of digits each drawn from the set 0, 1, 2, …, b – 1. Take as an example the case of base eight, which is known as octal, and which has applications in computing. Consider the (base ten) number 12 345 678, and seek to convert it to octal. Proceed as follows, dividing successively by 8 and recording the remainder at each step. Here is the result.



8 12 345 678 8 01 543 209 8 00 192 901 8 00 024 113 8 00 003 014 8 00 000 376 8 00 000 047 8 00 000 005 8 00 000 000

:6 :1 :5 : 0 :6 :0 :7 :5

(7.3)

To recover the octal representation of the number, read the final column from the bottom up: 57 060 516. So we are saying that

12 345 678ten = 57 060 5168

(7.4)

or

1 # 107 + 2 # 106 + 3 # 105 + 4 # 10 4 + 5 # 103 + 6 # 10 2 + 7 # 10 + 8 = 5 # 87 + 7 # 86 + 0 # 85 + 6 # 8 4 + 0 # 8 4 + 0 # 83 + 5 # 8 2 + 1 # 8 + 6.



(7.5)

This last line shows how to proceed in the opposite direction. Here is the calculation in full.

PROJECTS

5 # 87 = 10 485 760 7 # 86 =

1 835 008

6 #8 =

24 576

5 #8 = 1 #8 =

320 8

4 2

6



=

(7.6)

6 12 345 678 

This also constitutes a check on the accuracy of the original calculation. There are active, from time to time, various duodecimal (or dozenal) societies, who urge the reform of the number-system, by replacing base ten with base twelve. This is most unlikely to happen (and it is far from clear that base twelve is really any better). I discuss such questions a little later on. Here simply note that the use of base twelve requires the use of two new number-symbols (digits). There are a number of possibilities here. Most usual are A, for ten although X is also used, and B, for eleven (although H is also used); in the duodecimal system, of course, 10 stands for our twelve, and 11 for our thirteen. Another possibility with a base larger than ten is the hexadecimal one, using base sixteen. This has some application to computing, as the powers of two find ready expression in the binary (base two) arithmetic basic to computer logic. Perhaps somewhat surprisingly, however, it had an advocate before this ever became an issue. In 1936, Joseph Bowden, a mathematician at Adelphi College in New York State advocated it in a book (now a very rare book) called Special Topics in Theoretical Arithmetic. I have not managed to view a copy of this, and so I here rely on a brief account by the Scientific American columnist Martin Gardner. Bowden retained the names of the numerals up to ‘twelve’, either not knowing or else ignoring the fact that the words ‘eleven’ and ‘twelve’ do make reference to the base ten (see Chapter 5). However, he altered the names for our thirteen, fourteen and fifteen to thrun, fron and feen respectively. Our sixteen, he called wunty. From this last word, I deduce that Bowden not only set out to reform arithmetic, but spelling also. So he probably counted: wun, too, three, fore, five, six, seven, eit, nine, ten, eleven, twelve, thrun, fron, feen, wunty. His ‘hexadecads’ then would have gone: wunty, tooty, threety, etc. up to feenty. When it came to digital representation, he used the numerals 2, 3, 4, 5 reversed to represent the numbers twelve, thrun, fron, feen respectively. This device is not available in the case of ten and it is not advisable in the

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THE NAME OF THE NUMBER

case of eleven. I once suggested using two symbols from the Cyrillic (Russian) alphabet: Ю, for ten and И for eleven. However, here I will use the letters A, B, C, D, E and F for the extra digits. (Some calculators use essentially this convention, but because their digital displays cannot distinguish between B and 8 or between D and 0, these symbols are replaced there by b and d respectively.) Thus we may write the first few numerals in hexadecimal as in Table 7.1. It is not difficult to convert between decimal and hexadecimal. The methods are exactly the same as those demonstrated above for octal. We may also teach ourselves to count in base sixteen. Indeed my two daughters learned to do exactly this when they were still in primary school. They used to occupy themselves in the course of longish car journeys in this activity, using Bowden’s presumed names, and very soon knew that after ‘ninety-nine’, you do not say ‘a hundred’, but rather ‘ninety-ten, ninety-eleven, ninety-twelve, ninety-thrun, ninety-fron, ninety-feen, tenty, tenty-wun,’ and so on. There was even the secret thrill of knowing that it was really all right to say things like ‘fivety-eleven’ or ‘tenty-twelve’, without providing evidence of innumeracy, but rather of its opposite. When we come to examine the relative merits of the various bases, there are a number of matters to consider. First, there is the length of the string of digits that represents a numeral. The rule is this: if the base is b, then: • numbers N in the range 0 < N < b have strings of length 1, • numbers N in the range b ≤ N < b2 have strings of length 2, and so on. The general rule is that the length of the string is 7logb N A + 1, where

5 x? is the ‘floor function’ of x, the greatest integer less than or equal to x.

For purposes of simple illustration, it is enough to approximate this by logb N . So if we go to a base other than ten, we alter the length of the 1 string by a factor of about . In base two, for example, the length of log10 b a string is about 3.3 times the length in base ten; in base sixteen, it is about 0.83 the length we know. Small bases lead to impractically long strings. The next consideration is the size of the multiplication table. Recollect that a # 0 = 0 , a # 1 = a , which are trivial, and a # b = b # a , to ]b - 1g]b - 2g see that the size of the multiplication table to be memorised is 2 Our table thus contains 36 number facts to be learned. Binary has none,

PROJECTS

Standard English name

Our usual symbol

Base sixteen symbol

Bowden’s presumed name

one

1

1

wun

two

2

2

too

three

3

3

three

four

4

4

fore

five

5

5

five

six

6

6

six

seven

7

7

seven

eight

8

8

eit

nine

9

9

nine

ten

10

A

ten

eleven

11

B

eleven

twelve

12

C

twelve

thirteen

13

D

thrun

fourteen

14

E

fron

fifteen

15

F

feen

sixteen

16

10

wunty

seventeen

17

11

wunty-wun

eighteen

18

12

wunty-too

nineteen

19

13

wunty-three

twenty

20

14

wunty-fore

twenty-one

21

15

wunty-five

twenty-two

22

16

wunty-six

twenty-three

23

17

wunty-seven

twenty-four

24

18

wunty-eit

twenty-five

25

19

wunty-nine

twenty-six

26

1A

wunty-ten

twenty-seven

27

1B

wunty-eleven

twenty-eight

28

1C

wunty-twelve

twenty-nine

29

1D

wunty-thrun

thirty

30

1E

wunty-fron

thirty-one

31

1F

wunty-feen

thirty-two

32

20

tooty

thirty-three

33

21

tooty-wun

thirty-four

34

22

tooty-too

thirty-five

35

23

tooty-three

Table 7.1.  The first thirty-five numbers in hexadecimal

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THE NAME OF THE NUMBER

ternary (base three) has exactly one ]2 # 2 = 11g, but base twelve has 55 and base sixteen has 105. Given the difficulty primary students have with our own base ten ‘times table’, these are probably impractically large. Another argument that is easy to appreciate has to do with packaging. If the base is rich in divisors, then there are various options for compact packaging. This argument is used by proponents of duodecimal systems: a dozen items can be packaged as one layer of 3 # 4 , or as two layers of 2 # 3, for example. Our only option is 2 # 5 , which seems unhandily long for some purposes. Even bases allow ready determination of evenness and oddness. We merely inspect the final digit. Compare base three for which 2201 (our 73) and 2210 (our 75) are both odd, but this is not immediately apparent. (We can see immediately that the second of these numbers is divisible by 3, but this is a less important property than its oddness.) In base ten, we can use this same device to check divisibility by 5; in base six or base twelve, it may be used to test for divisibility by 2 and 3. In base ten, we thus have simple tests for divisibility by 2 and 5, and there is a different rule to test for divisibility by 3. We simply add the digits in the decimal representation, and test this sum. This rule works because 10 / 1 (mod 3). For 11, the test for divisibility is a little more complicated: alternate the signs of the digits in the representation, and add the results. Here is an example: consider 204 765. Form the number - 2 + 0 - 4 + 7 - 6 + 5 = 0, which is divisible by 11, so 204 765 is also divisible by 11 as we may easily check with a calculator. More generally, consider a number representation in base b.

N = n r b r + n r - 1 b r - 1 + f + n 2 b 2 + n1 b + n0 , where 0 # ni 1 b  (7.7)

Define:

A]N g = n r + n r - 1 + f + n 2 + n1 + n0,

S]N g = ]- 1gr n r - ]- 1gr - 1 n r - 1 + f + n 2 - n1 + n0



(7.8)

It is now straightforward to see that N / A]N g (mod b - 1), and almost straightforward to see that N / S]N g (mod b + 1). Thus the rule given above works because 3 is a divisor of 9, and the rule for eleven works because eleven is one greater than ten, the base. ‘Good’ bases therefore are such that b, b ! 1 are all rich in small prime divisors. In our base ten, we have good rules for 2, 3, 5 and 11, but lack a simple rule for divisibility by 7; however because 7 # 11 # 13 = 1 0001, the rule with S(N) works in base a thousand. This may readily be adapted to

PROJECTS

1 = 0.5 2

1 = 0.333f 3

1 = 0.25 4

1 5 = 0.2

1 = 0.166f 6

1 7 = 0.142857142f

1 = 0.125 8

1 = 0.111f 9

1 = 0.1 10

1 = 0.0909f 11

1 = 0.0833f 12

1 = 0.0769223076f 13

Table 7.2.  Reciprocals of the first few numbers in base ten

base ten by grouping the digits in trios. So, suppose that we wanted to know whether the number 1 031 426 859 314 is divisible by 7. Perform the calculation 1 - 031 + 426 - 859 + 314 =- 149 / 5 mod 7 so that 5 is the remainder if 1 031 426 859 314 is divided by 7. This same procedure also works for 13, and tells us that if 1 031 426 859 314 is divided by thirteen, there is a remainder of 7. A (mathematically related) question concerns the representation of the reciprocals. In base ten, see Table 7.2. 1 Here we have the length of the repetend (repeating pattern) in is N 1 necessarily a divisor of N - 1. Thus the expansion of 7 is as bad as possible, but we strike lucky in the case of 13. I am far from convinced that the duodecimal system is any better than our own in regard to these various requirements. Base sixteen has a very good set of reciprocals, and this was one of the arguments that Bowden advanced in favour of it. However, a very good case may be made out to the effect that, when all these matters are weighed up, base ten is as good as we are likely to find. These questions I leave to the reader to explore further. Among the possible activities are the following. 1 Find the divisibility rules applying in other bases. Compare the simplicity of these in the different cases. 2 Find the reciprocals of the smaller numbers in the various bases. Compare the simplicity of these. 3 Find out about ‘Fermat’s Little Theorem’ and use it to explore the relationship between the divisibility rules and the expressions for the reciprocals. 4 Investigate the base-three system using 0, 1 and 1 (meaning –1) as the digits. 5 Consider the representation of numbers in terms of negative bases.

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THE NAME OF THE NUMBER

7.3  Counting rhymes There are some interesting relics of the Celtic numerals still preserved (at least until very recent times) in parts of the British Isles and elsewhere. Very few of the Celtic languages are still with us. Welsh is going strong, and its closest relative, Breton still survives, but is seen as endangered. These two are the only survivors of the Brythonic branch of Celtic, although a third, Cornish, finally died out only in relatively recent times. The situation is similar, but rather worse, in the case of the other ‘clan’ of the Celtic branch, the Goidelic. Two Goidelic languages are clinging precariously to life; they are Scots Gaelic and Irish Gaelic (Erse). A third, Manx, died out but not so very long ago. Manx and Cornish, however, both cling to a sort of after-life; they are kept half alive by the efforts of societies devoted to their preservation. But no-one speaks either as their native tongue. Here I will look at some of the features of the Brythonic branch and its number-words as shown in Table 7.3. Number

Welsh

Breton

Cornish

1

un

un

idn

2

dau/dwy

dou/diu

deu

3

tri/tair

try/teir

try/teir

4

pedwar/pedair

peuar/peder

pajer

5

pump

pemp

pemp

6

chwech

huech

wheh

7

saith

seiz

seyth

8

wyth

eiz

eyth

9

nau

nau

now

10

deg

dec

deg

11

un ar deg

unnec

idnak

12

deuddeg

douzec

dawdhak

13

tri/tair ar deg

trizec

tôrdhak

14

pedwar/pedair ar deg

peuarzec

peswôrdhak

15

pymtheg

pempzec

pempthak

16

un ar pymtheg

huezec

whedhak

17

dau/dwi ar pymtheg

seizdec

seydhak

18

deunau

eizdec, triwec’h

eydhak

19

pedwar/pedair ar pymtheg

nauntec

nownjak

20

ugain

uguent

igans

Table 7.3.  Number-words in the Brythonic languages

PROJECTS

Number

West Cumbrian

Isle of Man

1

yan

yan

2

tyan

tan

3

tethera

tethera

4

methera

pethera

5

pimp

pimp

6

sethera

sethera

7

lethera

lethera

8

hovera

hovera

9

dovera

covera

10

dick

dik

11

yan-a-dick

yan-a-dik

12

tyan-a-dick

tan-a-dik

13

tethera-dick

tethera-dik

14

methera-dick

pethera-dik

15

bumfit

bumpit

16

yan-a-bumfit

yan-a-bumpit

17

tyan-a-bumfit

tan-a-bumpit

18

tithera-bumfit

tethera-bumpit

19

methera-bumfit

pethera-bumpit

20

giggot

figgit (sic, ?jiggit)

Table 7.4.  Two versions of the sheep-score

However, these number-words also appear in other guises. The so-called sheep-score was used by shepherds in counting sheep and by farmers in trading them. The numbers were marked off on a tally-stick and when twenty was reached, a score (notch) was cut in the stick. (Hence the use of the word ‘score’ to mean ‘twenty’.) Two versions are shown in Table 7.4. Many dialects from the North of England and into Scotland display (or until recently displayed) versions of the sheep-score. Like these ones, they show greater affinity with the Welsh than with the Cornish or the Breton. However, they live on in children’s counting rhymes. In a classic account of these, Iona and Peter Opie list six versions of the sheep-score as shown in Table 7.5. These authors name the underlying pattern the AngloCymric Score. (Cymru is the Welsh name for Wales.) They see these as forming the basis of many such counting rhymes. My grandmother, during her childhood in the rural Tasmania of the 1880s, used such a rhyme that began: indy tindy alligo Mary. Indy and tindy are very plausibly related to yan and tyan, and could Mary really be methera? The Yarmouth dialect count from Table 7.5 starts out very like

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THE NAME OF THE NUMBER

the now taboo eeny meeny miney mo. Of the counting rhymes quoted by the Opies, the closest to the Anglo-Cymric Score goes: Ya ta tethera pethera pip Slata lata covera dovera dick. Another goes: Inty tinty tethera methery Bank for over dover dick. This is close to the PIE in many ways. The first five entries relate very plausibly, with ‘bank’ taking the place of *penkwe. Then the correspondence breaks down; there are no real equivalents of 6 or 7, but after that the pattern resumes with garbled versions of 8 and 9 and finally a clear PIE derivative in dick for ten. Looking at other counting systems, we see the breakdown occurring with other (psychologically) less important numbers. An example is afforded by yet another such rhyme that crossed the Atlantic and ended up in America: Een teen tether fether fip Sather lather gather dather dix. It is clear that the counting rhyme further garbled the already garbled words of the sheep score, but the origins are clear! In fact, we can see faint echoes of such counting systems in other places too. One further garbled version of the Westmoreland hevera devera dick turns up as Hickory dickory dock! In Australia, there have been even fainter echoes of these. There is, for instance, the one I learned from my grandmother and quoted above. Perhaps the best example, although the origins are rather obscured in it, is: Enden deena tucka lucka teena Sucka lucka ticky tacky enden boom. This has, anyway, ten elements, if we count ‘enden boom’ as a compound element, just as the numeral representation 10 is a compound element. ‘Enden’ and ‘deena’ may be plausibly related to Celtic words for ‘one’ and ‘two’. The others, however, would seem to be interpolated ‘nonsense words’.

PROJECTS

Dialect

High Furness

West Riding Yorks.

Yarmouth

Northumberland

Westmoreland

North Riding Yorks.

1

aina

eina

ina

eën

yan

yan

2

peina

peina

mina

tean

tyan

tean

3

para

paira

tethera

tether

tethera

tithera

4

peddera

puttera

methera

mether

methera

mithera

5

pimp

pith

pin

pimp

pimp

mimph

6

ithy

ith

sithera

citer

sethera

hitter

7

mithy

awith

lithera

liter

lethera

litter

8

owera

air-a

cothra

ōva

hevera

over

9

lowera

dickala

hothra

dōva

devera

dover

10

dig

dick

dic

dic

dick

dick

Table 7.5.  The first ten numbers in six North England dialects Note: The Northumberland is the West Cumbrian under another name.

When children use these rhymes in their playgrounds to decide who will be ‘it’ in a game of chasies or the like, they are echoing the use of counting for such purposes. This is not immediately obvious, because some children, in their use of the rhymes, point to the members of the circle on each stress, others on each syllable, others on each word and yet others in some combination of these patterns. If we reinstate the actuality of a count into such a process of selection, we encounter the mathematical algorithm describing the ‘Josephus problem’. Given a group of n persons arranged in a circle under the rule that every mth person will be removed going around the circle until only one remains, find the position L(n, m) in which you should stand in order to be the last survivor.

The problem takes its name from Flavius Josephus, a Jewish historian of the 1st century. According to legend, he and 40 of his fellow soldiers were trapped in a cave, surrounded by besieging Romans. They chose suicide rather than capture and decided that they will form a circle and start killing themselves using a step of three. As Josephus did not want to die, he was able to find the safe place, and stayed alive, later joining the Romans who captured him. This is the case n = 41, m = 3. In this case L^n, mh = L^41, 3h = 31. More typically in a playground situation, m > n, but the same ideas apply. Take a typical example. Suppose seven children form the circle and that

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THE NAME OF THE NUMBER

the count chooses every tenth child. In the first round, the children are numbered in the order to which the counter points goes 1, 2, 3, 4, 5, 6, 7, in turn and then continues to 1, 2, 3; as this is the tenth number called, child number 3 is eliminated. The count then begins again with child number 4, going 4, 5, 6, 7, 1, 2, 4, 5, 6, 7. As this is now the tenth number in the sequence, child number 7 is eliminated. The process continues until only two children are left. These are numbers 1 and 5, with the count beginning on child number 5. This results in the elimination of child number 1, and so child number 5 is selected: L^7, 10h = 5 . There is no known closed form for L^n, mh in the general case, although there are a number of efficient algorithms available and one explicit formula can be given: it is known that L^n, 2h = 1 + 2n - 21 +5lnn? where 5 x? is the ‘floor function’ introduced in the previous section. In the actual playground situation, the rules may vary somewhat. One child performs the count, and there are a number of different conventions relating to the way in which this child is counted. Some counting out rhymes introduce an element of choice. A colour, for example, may be chosen by a member of a circle with the count now proceeding (for example) ‘B L U E spells blue and you are out of this game’. Of course, the fact that many such rhymes find their origins in counting rituals does not restrict us from applying the same analysis of the Josephus problem even where there is no such linguistic connection. Think of: ‘Mickey Mouse bought a house. What colour was it? [Blue!] B L U E spells blue and you are out of this game.’ Whether we point by words, stresses, syllables or whatever, we might as well be counting. For example, with words, Mickey = 1, Mouse = 2, bought = 3, etc. Several possible projects suggest themselves, mostly of an interactive class nature. They involve the collection and analysis of actual rhymes and practices. It would also be possible to collect and compare different cultural traditions. Students could be encouraged to see if they can apply the Josephus algorithm to actual playground practice.

7.4  The number-word game Start by writing out any number in (English) words. For example, choose 1 234 567 891, which is ‘one thousand two hundred and thirty-four million, five hundred and sixty-seven thousand, eight hundred and ninety-one’.

PROJECTS

Now count the letters in this number. The count will differ depending on whether we count the comma, the hyphens and the spaces, but this will turn out not to matter. If we include everything, we find 117 characters. Now write out the number 117 in words: ‘one hundred and seventeen’. Repeat the process, this time to find 25 characters (including spaces). Write ‘twenty-five’ and repeat the count (including the hyphen). There are now eleven characters, all of them letters. The word ‘eleven’ contains six letters, the word ‘six’ contains three letters, the word ‘three’ contains five letters, and the word ‘five’ contains four letters. From now on, we continue to obtain the number 4. Moreover had we used different counting conventions in respect to spaces and hyphens, it would have made no difference. Had we written ‘a hundred’ in place of ‘one hundred’, again it would have made no difference. Although the route by which we arrived at the number 4 would have changed, the end result would have been unaffected. If we start with the same initial number, but count only letters, we reach 97. ‘Ninety-seven’ contains 11 actual letters. From here, we are back on track to come to rest at 4. No matter what number we start with, and whichever counting convention(s) we care to adopt, the end result is always 4. Indeed we could do the same starting with the decimal expansion of the number. In the example just given, there were ten digits and three commas, so we call this 13. ‘Thirteen’ contains 8 letters; ‘eight’ has 5 letters and so once again we end up with 4. However, if we consistently used decimal digits rather than letters of the alphabet, the result would be different. 13 has 2 digits; 2 has 1, and thereafter we reach 1, and this applies wherever we begin, and also if we include commas, spaces or both, in the count or not. This last example leads us to consider the same game played with languages other than English. Take French for example. Let F]ng be the number of characters in the French name for the number n. Consult Table 2.1. It is relatively simple to check that if n 2 10 , then F]ng 1 n . Thus the application of the operation F reduces any number n until it lies in the range 0 1 n 1 10 . Indeed from the table we see that our bound of 10 can be improved. We have F]ng 1 n for all n 2 5 . Now look at the numbers up to 5. We have:

F]1g = 2 , F]2g = 4 , F]3g = 5 , F]4g = 6 , F]5g = 4 

(7.9)

and we note that F(6) = 3. Thus, once we get into the range 0 < n < 5, we find ourselves sent into the cycle 4 " 6 " 3 " 5 , and this pattern persists

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n

M1(n)

M2(n)

1

2

2

2

3

3

3

3

3

4

4

4

5

3

3

6

8

8

7

4

4

8

9

9

9

11

12

10

6

6

11

8

9

12

9

10

13

9

10

14

10

11

15

9

10

Table 7.6.  The number-word game in Motu

thereafter. Whereas English possesses a single stable equilibrium, French has a single stable 4-cycle. Now look at a more complicated case: that of Motu (Table 2.9). Because of the hyphens in some of the numbers, there are two possible counts. Denote the number of characters in the Motu word for n by M1 ]ng if we ignore the hyphens, and M 2 ]ng if we include them. Then we may set up a table like Table 7.6. We see that if n 2 9 , then both M1 ]ng and M 2 ]ng are less than n. Thus we need only examine what happens for n 1 10 . We see that both operations M1 and M 2 have stable equilibria at n = 3 and n = 4 . On top of this the operation M1 has a stable 3-cycle 8 " 9 " 11, whereas the operation M 2 has, in place of this, a stable 5-cycle 6 " 8 " 9 " 12 " 10 . These operations provide simple examples of deep and powerful results in an area of Mathematics known as Dynamical Systems Theory. Take an operation L acting on (in this instance) the set of positive integers. Suppose that for the operation L there exists an N such that, for all n 2 N , L]ng 1 n . Thus for all such n, application of operation L results in a lesser number, until a number is reached which is less than N. Now consider the set S of all numbers less than or equal to N. It is possible that for some n ! S , L]ng 2 N , but, if this occurs, then L]L]ngg = L2 ]ng 1 L]ng, and so the argument may be repeated. The action of L eventually produces a number in the set S.

PROJECTS

Let M be the maximum of all L(n), where n ! S . Now let T be the set of all numbers n such that 1 # n # M . For all such n, we have not only n ! T , but also L]ng ! T as well. Thus the repeated operation of L will eventually send any n into the set T, and once a number is inside T, it will be constrained to stay there when operated on by L, no matter how many times L is applied. In the examples so far considered, for L = F, we had N = 5, M = 6. For L = M1 : N = 10 , M = 11. For L = M 2 : N = 10 , M = 12 . For L = E (for English), it is easy to deduce that N = 4 , M = 5 . But now, whatever member n of the set T we choose, repeated application of L will produce a member of the set T. If we then have L]ng = n , then n is a stable equilibrium; if not, because the set T has only finitely many members, there must be a number k such that Lk ]ng = n . If we choose the minimal such k, then n is a member of a stable k-cycle. The very simplest case is given by Chinese. See the list of Chinese numerals given in Chapter 2. Now the game requires some modification in the case of Chinese, because letters as such do not exist. However, if we consider characters in place of letters, then we have a very simple situation: N = 2 , M = 1, and so there is a unique stable equilibrium at n = 1. This is exactly as we found using decimal digits in place of English letters. If, however, in the Chinese we consider, not characters, but strokes, things are somewhat more complicated, and I leave the exploration of this case to the reader. The proof that the result of repeated application of L must result in stable cycles is similar to another result. When we examined in Chapter 1 7.2 the length of the repetends in the decimal expansion of reciprocals , N we saw that the length of this repetend was bounded by N. After at most N - 1 digits, the string must repeat; indeed the length of the repetend must be a divisor of N - 1. The reader is invited to explore this result in the light of the analysis just given here. It is necessary for the success of the number-word game depends upon the language chosen being a language of reasonably advanced numeracy. If we tried it with a version of (say) Kiwai that went beyond the limit of counting at four, we would find that K]ng, let us call it, was a monotonic increasing function of n and so the basis of our proof of the existence of stable cycles would not exist. It is probable that something similar would happen in Fore if we tried to extend this counting system beyond the first few numbers. Readers may care to examine the way the number-word game goes in languages other than those discussed in the text.

81

5 7

2 18 93 36

3

7

4 6 8

2

1

C hapter eight

5

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8.0  Introduction A most useful book for many purposes relevant to this entire work is Karl Menninger’s Number Words and Number Symbols (1969). Another is GG Joseph’s The Crest of the Peacock: Non-European Roots of Mathematics (1990). In the material presented here, I make frequent reference to Menninger, because his focus is closer to that of my discussion. However, Joseph provides a useful supplement. Readers are urged to seek out his somewhat different emphases for themselves. A very easily accessible website for topics on the history of Mathematics is the MacTutor website maintained by St Andrew’s University in Scotland. Easiest is to go to: http://www-groups.dcs.st-and.ac.uk/~history/ and then follow the prompts. Here, unless there is a need to be more explicit, I simply refer to this as the MacTutor website. I do likewise in the cases of other generally useful sites such as the Mathworld site: http://mathworld.wolfram.com and the Math Forum site: http://mathforum.com At all of these you will find quite full accounts of Zeno of Elea and his paradoxes.

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The electronic (CD) version of the Encyclopædia Britannica gives 10 relevant entries on Arthurian Britain. In the hard copy, look up ‘Arthurian Legend’, which will point you towards further articles. Much of the linguistic material presented here is discussed in greater detail in Chapter 8.1, and fuller references will be given there. The verse quoted is based on one written in Nostratic (also to be more fully described in Chapter 8.1) by the linguist Illič-Svityč. A literal translation reads: Language is a ford through the river of time It leads us to the dwelling of ancestors. But he doesn’t arrive there Who fears deep water.

The version quoted in the text is my own. At the expense of some imprecision of meaning, it preserves the meter and the rhyme scheme of the original. More background will be given in Chapter 8.1. Other more detailed references will also be given in Chapter 8.1, which will go into these matters in much greater detail. It should be said that the word ‘clan’ is not a standard usage. Rather, it is a coinage of my own and is always signalled as such by enclosing it in quotation marks. More standard would be, for example, ‘the Celtic branch of the Indo-European family’. My word ‘clan’ is adopted for brevity and ease of reference. In Chapter 1, I use the word ‘branch’ in the standard sense, but refer to as ‘clans’ those groups of languages that do not make up an entire branch. Thus the Romance ‘clan’ is usually seen as part of a larger Italic branch of the Indo-European family of languages.

8.1  The language families of the world The best single reference on languages and their classification is the Encyclopædia Britannica article ‘Languages of the world’. There are several condensed versions of this on the Internet, but the original (printed or CD) version is much fuller and is not available in this form. Many dictionaries also include useful summaries of this material. The first three editions of the Macquarie Dictionary had a page devoted to this subject, but the later fourth does not. Some editions of Webster’s do carry accounts, others not. A particularly useful dictionary is Eric Partridge’s Origins (1983), dealing with individual English words and their etymology.

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For lists of number-words in various languages, Menninger’s book is a convenient place to look, and of course specialist dictionaries relating to specific languages can always be consulted. It should however be noted that Menninger did not accept the doubts on satem theory. It should also be said here that my Rules 1 and 2 greatly simplify (indeed perhaps oversimplify) a complex subject. Indeed, the very notion that such rules exist is not now universally agreed. The position that there are such rules is known as the neogrammarian hypothesis. What I judge to be a modern consensus on this question is aptly put by PL Trask (1996, p. 70). sound change is normally regular and the cases that are not are puzzles calling for an explanation. This policy has proved to be of great benefit in historical linguistics, and it will provide a firm foundation … in spite of the fact that it is not strictly true … [but although it is not always true] it is none the less an excellent working hypothesis.

Rules 1 and 2 themselves also are themselves in need of some qualification. Rule 1 is my distillation of the gist of many discussions of Historical Linguistics. However, it is often implicit in such discussions, rather than being stated explicitly. Rule 2 is even more controversial. It essentially summarises a table in Trask’s book (see his p. 234), which gives most of the sound-shifts here involved with the number-words. However, the full story is considerably more complicated. A more complete version is given on page 154 of WP Lehmann (1973). One clear exception to Rule 2 as I have stated it is the progression of the so-called bilabials. Bilabials are made by the two lips coming together. The consonants m, p and b are bilabials. In the case of the last two of these, the consonants are made when the lips separate. A class of bilabials that we no longer have in English is the class of bilabial fricatives. If we put our lips together and blow through them, so that the breath causes them to part, we produce a sound akin to the consonant p, but different from it. It is represented by the spelling –ph–. This survives in words like ‘pharmacy’, but we now pronounce this word as if it were farmacy. In the case of the word ‘fantasy’, the current spelling has replaced the older ‘phantasy’. This has been a relatively slow replacement, as far as spelling is concerned. The change from ‘phantasy’ to ‘fantasy’ was completed in my lifetime, but the related words ‘phantom’ and ‘phantasmagoria’ retain the initial ph–. ‘Gulph’ was long ago replaced by ‘gulf’, but ‘graph’ and ‘morph’ are still with us, as are

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many initial instances of ph–; ‘pharmacy’ has already been mentioned, but there are many others: ‘phyto–’, ‘phenol’, ‘philatelist’, and scores more. Lehmann expresses this change as a progression from a bilabial to a labio-dental. The role played by the upper lip in the original ph– is taken over by the upper teeth, when we make the sound f–. The f– sound is made slightly further back in the mouth than is the ph–, and so we have an exception to my Rule 2. Something similar has happened with the word two, whose second letter is no longer pronounced. The change from a w– sound to a v–, noted previously, constitutes another example. For fuller accounts, consult books such as Trask’s or Lehmann’s or any of the many other good accounts available. For Indo-European, there are several good sources. Lockwood’s two works, Indo-European Philology (1969) and A Panorama of IndoEuropean Languages (1972), are accessible and authoritative. Rather more specialised is C Watkins’s American Heritage Dictionary of Indo-European Roots (1985), and considerably more specialised is Otto Szemerényi’s Studies in the Indo-European Systems of Numerals (1960). Also more specialised, but in a different direction, is Lehmann’s Theoretical Bases of Indo-European Linguistics (1993). (The first three of these authors are the source of my Table 1.1.) Watkins, incidentally, lists another branch of the IE family: Phrygian. The Phrygian languages are all extinct, but are thought to have been possibly IE. Most authors taking this route include the Phrygian ‘clan’ along with Hittite among the Anatolian languages. Lockwood lists the various branches of IE somewhat differently. In particular, he divides the Italic family into a number of separate branches, reserving the term Italic for the Romance ‘clan’, and dividing the other ‘clans’ among two families, all of whose members are now dead. He also combines Phrygian and Armenian in a Thraco-Phrygian branch and uses the term Illyrian to describe Albanian and its (now dead) relatives. In Table 1.1, I have somewhat simplified some of the spellings. I also do this at other times. Several articles in Scientific American cover material either alluded to here or else related to it. See P Thieme, ‘The Indo-European language’ (October 1958), Colin Renfrew, ‘The origins of Indo-European languages’ (October 1989), TV Gamelkridze and VV Ivanov, ‘The early history of Indo-European languages’ (March 1990), Peter Bellwood, ‘The Austronesian dispersal and the origin of languages’ (July 1991), LL CavalliSforza, ‘Genes, people and language’ (November 1991), Joseph Greenberg and Merritt Ruhlen, ‘Linguistic origins of native Americans’ (November 1992) and Colin Renfrew’s ‘World linguistic diversity’ (August 1994).

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Renfrew’s first article gives two different possibilities for the origins of the PIE language (the two described here), but omits the ‘bottom of the Black Sea’ theory. For a full account of that, see Noah’s Flood, by William Ryan and Walter Pitman (1998). Renfrew espouses a version of Nostratic Theory in which IE, Uralic and Altaic form one branch of Nostratic, Dravidian another and Afro-Asiatic a third. (He does not consider Kartvellian.) More recently, doubt has been cast on the inclusion of Afro-Asiatic in the Nostratic fold. Accounts of Nostratic theory in the published literature can be found in the article ‘Nostratic’ by Kaiser and Shevoroshkin in Annual Review of Anthropology 17 (1988), pp. 309–329, and the book Reconstructing Languages and Cultures (ed. V Shevoroshkin, 1989), the proceedings of an international conference on language and prehistory. (There have been several further such conferences, the proceedings of most having been edited by Shevoroshkin.) The first of these references is my source for the verse quoted in the Introduction. A more recent discussion is Nostratic: Sifting the Evidence (1998), the proceedings of a conference giving arguments on both sides of the debate. Other recent discussions on Nostratic and related matters can be found on the World Wide Web, but see the comments below. The doubts concerning Nostratic concentrate on the possibility that similarities between words may arise accidentally. A spectacular example of such coincidence concerns the Latin deus and the Greek theós, both of which mean ‘god’. However they are independent (see Partridge’s Origins), although deus is related to the Greek Zeus! There are a number of general books with much useful and interesting material. Among them are JR Hurford’s The Linguistic Theory of Numerals (1975) and Language and Number (1987). Another is Thomas Crump’s The Anthropology of Numbers (1990); particularly relevant to this work is his Chapter 3. In this context also note Chapter 1 of JN Crossley’s The Emergence of Number (1987) and Growing Ideas of Number in this series. I have myself written a number of studies, published in the journal literature. The two major ones are ‘Number-words and their significance’, in The Mathematical Scientist 15 (1990), 1–6 (a condensed version of a preprint entitled ‘What number-words tell us about number concepts’ released as Monash University History of Mathematics Pamphlet no. 45, 1989) and ‘The origins of our number-words’ in the Australian Mathematical Society Gazette 23 (1996), 50–66 (an edited version of an invited address to the 1995 annual meeting of the Society). The full text of the address constitutes Monash University History of Mathematics Pamphlet 62, 1996. An earlier study, also published in the Australian Mathematical Society Gazette, is ‘Five and four’, 16 (1989), 125–129.

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There are also more popular articles in the school mathematics journal Function for February 1984, February 1991 and October 1991. There is much about this material on the Internet. The Internet provides a useful way to keep up to date. It is widely accessible and it does not suffer the same publication delays as the material appearing in books or journals. On the other hand, there is not the same sifting for quality, and so a certain vigilance is needed. Furthermore, the material available changes very frequently. This means that it is not especially efficient to cite individual websites. A better strategy is to institute a Google search for a key word or phrase. Look for ‘proto-indo-european’, ‘nostratic’, ‘language families’, ‘satem theory’, and so on to discover some very good sites. In particular, the Wikipedia sites are generally of good quality. They share a common feature of many websites in that they are not vetted for quality, so that the general caution just given applies here also. However, there have been a number of studies that claim to show that their overall quality compares quite well with that achieved by more standard encyclopedias such as the Encyclopædia Britannica. The Wikipedia site may be accessed directly via: http://en.wikipedia.org/wiki/Main_Page This site will simply be referred to as ‘Wikipedia’ in what follows, unless there is a need to be more explicit. A search through the different sites will reveal, in a quite dramatic way, where differences of opinion lie. For example, while most authorities list about 20 different language families, counts of up to 200 have been given. However, one needs to exercise a certain caution in the interpretation of these apparently different numbers. Often they simply reflect different methodologies and definitions, so that the apparently vastly divergent counts are not really in fundamental disagreement. Some authors make the definition of ‘family’ a narrower concept, while others ‘lump together’ the different categories rather more freely. When a topic is controversial, as with the work of Cavalli-Sforza, then there is much more scatter in the quality of the sites. Search in Google for Cavalli-Sforza, but exercise careful judgement on what you find there!

8.2  The notion of a base The Peano axioms are discussed in many books and on many websites. Perhaps the best treatment is that in HA Thurston’s The Number System (1956), but there are many other good accounts.

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The aspect of ‘base’ as understood by mathematicians is discussed in many books and websites also. A Google search under ‘number base’ will reveal a lot. Special cases will be revealed by looking for specific bases with (for example) ‘binary numbers’; base three is called ternary, base eight is referred to as octal and base sixteen as hexadecimal. A very good site to visit for much material of a mathematical nature is the Mathworld one. A search there for ‘number base’ will show the way to many relevant pages. Both the MacTutor website and the Wikipedia give good accounts of the Chinese symbols for numbers. Symbols like those used in Chapter 2 can be viewed at: http://en.wikipedia.org/wiki/Chinese_numerals Table 2.5 has been adapted from: http://en.wikipedia.org/wiki/Babylonian_numerals Table 2.6 has been adapted from: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Mayan_ mathematics.html My source for the Thai numerals is the Lonely Planet’s Thailand Phrasebook by Joe Cummings (1984), but other Thai dictionaries would serve equally well. The general point can be made that specialist dictionaries are a good source of information on specific points and individual languages. When it comes to the less precise notion of base, it is possible to distinguish a number of different underlying patterns. Specialists in this area of Anthropology or Linguistics sometimes avoid the word ‘base’ in favour of other supposedly more precise terms. However, I will not follow this route here. The comments on indigenous PNG counting systems are mostly drawn from the article ‘Counting and numbers’ in Encyclopædia of Papua and New Guinea (1972). This is the source for Table 2.7. The author of the relevant article (Edward Wolfers) neglects to say whether the count begins on the right-hand side or the left-hand side of the body. However, Menninger (his page 35), in his discussion of a somewhat similar system, is explicit in starting the count on the right and ending it on the left. On the

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other hand, several websites (to be found by means of the Google search detailed below) claim the reverse convention. A much more detailed account of the number-systems of PNG, but harder to come by, is contained in a PhD thesis by the late GA Lean (PNG University of Technology, 1992). Lean, before his death, collected a vast number of number-words from all over Papua New Guinea and beyond. The theoretical aspects of his thesis are best ignored, but the collection of primary material is impressive in the extreme. The bulk of this work is now available on the web at: http://www.uog.ac.pg/glec/thesis/thesis.htm At the time of writing (July 2006), the website omitted the appendices. There are four of these, of which the first three provide the data on the PNG counting systems. These, to my view, constitute the strength of the work; much of the theoretical discussion is badly flawed. However, the extremely detailed list of references is available at this site. A project named in Lean’s honour is also on the World Wide Web, and there are plans to expand what is now there. Go to: http://www.uog.ac.pg/glec/index.htm Follow the prompts to ‘Counting Systems’. Rather more can be found by instituting a Google search for ‘Glendon Lean’. The Motu words are taken from Percy Chatterton’s A Primer of Police Motu (n.d.). The table giving the number-words in Northern Fore derives from Scott’s Fore Dictionary (1980; Pacific Linguistics, Series C, no. 62). The careful reader will notice a number of apparent inconsistencies in the use of the diacritical marks; these inconsistencies are features of the original. Most written Fore in fact omits the diacritical marks. Australian languages are discussed by (inter alia) RNW Dixon in The Languages of Australia (1980) and Blake in Australian Aboriginal Languages (1981). Implicit in these sources is the correct account of Gumulgal; many sources commit the error of continuing the count beyond 4. Both Dixon and Blake subscribe to the view that all Australian languages are either of the ‘one, two, many’ type, or else of the ‘one, two, three, four, many’ type. This belief has since been challenged by John Harris. See ‘Australian Aboriginal and Islander mathematics’ in Australian Aboriginal Studies (1987, no. 2, pp. 29–37). Sadly, this is another area of

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research where unfounded accusations of racism are all too easily and frequently made. Blake (op. cit., p. 33) has things almost right when he says: The Aborigines did not find the need to count. They had numbers for ‘one’ and ‘two’; some had a word for ‘three’, while others used a compound of the words for ‘one’ and ‘two’. … Where there was a word for ‘four’ it was always literally ‘two-two’. They always had a word to mean ‘a number of’ or ‘many’, usually translated into Aboriginal English as ‘mob’.

This, however, is one of the passages that Harris finds objectionable, and indeed it is not entirely correct. Harris’s study goes on to list a number of counter-examples, mostly from groups who did find a need to count, especially in the maintenance of trade tallies. More balanced, but no less sympathetic to the aspirations and dignity of Aboriginal peoples, is a short popular reader intended for primary school students, J Rudder’s An Introduction to Aboriginal Mathematics (1999). Rudder’s view may be summarised as siding with those who do not find precise quantitative notions in most Aboriginal traditional languages. This whole matter was considered in Chapter 5. There is more about Australian languages available on the World Wide Web. A good place to start is: http://www.dnathan.com/VL/eMUlg_A.htm#73 This source lists number-words up to 5 in the Anjumarla language. More generally, the reader is once again recommended to consult Menninger’s classic, and again there is complementary material in Joseph’s work.

8.3  O ther aspects of number: words and symbols The work of Denise Schmandt-Besserat is usefully summarised in a Scientific American article she wrote many years ago (see ‘The earliest precursor of writing’, June 1978, pp. 38–47). A Google search under her name will yield a lot more references, some printed, others on the World Wide Web. She has continued to write extensively on the subject, in both the technical and the popular literature. Her major work is Before Writing (1992), a two-volume account of her research.

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For Petersen’s account of Schmandt-Besserat’s work, see: http://www.sciencenews.org/articles/200603121/mathtrek.asp The contest between an abacus and an accounting machine was witnessed by my father, then a lecturer at RMIT. For completeness, it should be stated that the officially appointed judges declared the result to be a tie. However, the student organisers of the event awarded the win to the abacus, because it was worked by an amateur, whereas the accounting machine was run by a professional comptometer on loan from a major retail store. The students claimed that a true professional would have manipulated the abacus much more dexterously. For an overview of Chinese, a good start is the book About Chinese (2nd edn) by Richard Newnham and Tan Ling-tung (1987), or, among websites, the Wikipedia site referred to in Chapter 8.1. There is also interesting material in Joseph’s The Crest of the Peacock (1990), as detailed in Chapter 8.0. For Indonesian, see (among many possibilities) JB Kwee’s Teach Yourself Indonesian (1965), Chapter 22. For Tolai, see Mosel’s Tolai Syntax and its Historical Development (1984). Once again, I have drawn heavily on Menninger and refer again to the relevant websites referenced previously. Menninger’s discussion of the origin of the Russian sorok is on his page 185. There is an excellent introduction to the underlying principles of Chinese notation in ‘Zhu Shijie and his Jade Mirror of the Four Unknowns’ by Jock Hoe and published in History of Mathematics: Proceedings of the First Australian Conference (ed. JN Crossley, 1981). As Hoe remarks on the point made in the text concerning the identity of the symbol and the ideogram: ‘it is a debatable point whether the table [being discussed at this point] is written in words or in figures’. The principal point of his interesting discussion lies beyond the representation of numbers themselves, and proceeds to the solution of systems of simultaneous nonlinear algebraic equations. This the text manages by means of an ingenious algorithmic approach, whose details lie outside the scope of this present book. However, it is very much to the point to note that Hoe gives a ‘feel’ for the impact of the text on a native Chinese speaker by rewording an English translation of sample passages in terms of the way in which a computer programmer might express matters. A somewhat similar point applies to the way in which the Greek mathematician Diophantus, worded his Arithmetic (actually a pioneering

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work in Number Theory). Here we can see the first steps taken towards our modern approach of representing unknowns or parameters by means of letters, although Diophantus uses notations that seem very clumsy by today’s standards. For an account of these and related matters, see TL Heath’s Diophantus of Alexandria (1964). Especially relevant is the discussion in his Chapter III.

8.4  G rammar: the grammatical status of number-words The best Internet references on the material of Chapter 4 and the classification of words are to be found by using a Google search for ‘parts of speech’. This will lead to several good sites. This works better than searching under ‘English grammar’, for example. This is because much of what’s on offer at these other websites is either instructional material on how to write in a formally correct way or else is some highly technical account of other means of classifying English words. When it comes to describing English usage, there are many good books. Two that come readily to mind are ICB Dear’s compilation Oxford English: A Guide to the Language (1991) and Eric Partridge’s Usage and Abusage (1987). This latter is the source of my information on the peculiar speech-habits of big game hunters. When it comes to Latin, I follow the older (and to my mind still standard) usage employing the letter v. There is today a revolt among scholars against this, and it seems to me that in strict logic this revolt is correct. The best of modern Latin Dictionaries do not use the letter v, but rather u. Thus, my example involving the fox ‘vulpes’ should strictly speaking be ‘uulpes’. It is now firmly established that the ancient pronunciation of the letter we represent as v is that of our letter w. (Compare in this connection, the discussion on v and w in Chapter 1.) However, for ease of reading I have stayed with the older convention. For Latin grammar, there are many accounts. A Google search under ‘latin grammar’ will direct the reader to many good sites. Most university libraries hold quite a few texts on the subject. For PIE, we are not quite so fortunate. However, one very good source is WP Lehmann’s Proto-Indo-European Syntax (1974). It should however be noticed that Lehmann, although giving the details of the eight ‘cases’ in PIE proceeds to the discussion of gender. He reports it as the general opinion that the origin of gender was a distinction between animate and inanimate objects or creatures (as detailed in my account above). However, Lehmann himself does not accept this account and argues against it. (The details

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are given on pages 190–200, with this particular discussion to be found on page 199). There is an interesting sidelight on the loss of the neuter gender in French. Because of this loss, there is no French word corresponding to our word ‘it’. In French a thing must be either a ‘he’ or a ‘she’. The French mathematician Laplace was so struck by the power of the Newtonian revolution in science that he wrote (Théorie analytique des probabilités, 1896, pp. vi–vii): An intelligence which, at any given instant, knew all the forces by which the natural world is moved and the position of each of its component parts, if as well it had the capacity to submit all these data to Mathematical analysis, would encompass in the same formula the movements of the largest bodies in the universe and those of the lightest atom; nothing would be uncertain for it, and the future, as also the past, would be present to its eyes.

This passage is a famous one, which has had a considerable impact on philosophical thought. However, the words here rendered as ‘it’ and ‘its’ actually refer to the subject ‘intelligence’. This word is feminine in French, so a literal translation would replace these words by ‘she’ and ‘her’. However, one translator, to whom I will here extend mercy and refrain from naming, found this such an affront to masculine pride that he used instead ‘he’, ‘him’ and ‘his’! On the question of platonic nouns, see Davis’s article ‘Fidelity in mathematical discourse: Is one and one really two?’ American Mathematical Monthly 79 (1972, pp. 252–263). Davis opposes the platonic account of mathematics, but he nonetheless succeeds in documenting extremely well the general acceptance of this view among the mathematical community. A more recent discussion, also opposing the view, is contained in G Lakoff and RE Núñez’s Where Mathematics Comes From (2000). On the other hand, the utility of the platonic approach is brilliantly exemplified by a passage in JL Borges and A Bioy-Cesares’ Chronicles of Bustos Domecq (trans. NT di Giovanni, 1976): the exact sciences [are not] based on an accumulation of statistics. In order to teach the young that three plus four makes seven, you do not add four cakes plus three cakes nor four bishops plus three bishops nor four cooperatives plus three cooperatives nor four patent leather buttons with three wool socks. Once the principle has been intuited, the youthful mathematician grasps that three plus four

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invariably make seven and he does not have to prove it over and over again with chocolates, man-eating tigers, oysters or telescopes.

We may even extend this example to apply to hundreds for example and so deduce that three hundred(s) plus four hundred(s) makes seven hundred(s), and so, by generalising the example, arrive at the distributive law. The different words for 2 and their gender relevance are attested both by the Oxford English Dictionary and Partridge’s Origins (detailed in Chapter 8.1). On the ‘Slavic squish’, see Hurford’s Language and Number. Menninger gives a lot of examples, but without using the word ‘squish’. A Google search sends the inquirer to a summary of one of my own papers, which appeared in the Australian Mathematical Society Gazette and is also listed more fully in Chapter 8.1. For the similar effect in Bangla, see, for example, NB Halhed’s A Grammar of the Bengal Language. AG MacLeod’s Colloquial Bengal Grammar (published in 1967 without naming the publisher) lists another feature of Bangla: ‘There are distinct words for the ordinal numbers (first, second, third, etc.) but in colloquial speech these are not used with the exception of the first three … For the rest, the Cardinal numbers are used in the Possessive [genitive] case.’ This is somewhat reminiscent of the case of Latin, for which the first three numbers inflect, but quattuor does not. I am indebted to Professor DK Sinha, formerly of the University of Calcutta, for bringing this ‘Bangla squish’ to my attention.

8.5  Early history of numerical concepts The dual, as an intermediate between the singular and the plural, is discussed in many accounts of Indo-European philology, but the account in Menninger is quite enough for present purposes. See, however, some of the accounts contained in the specialist references listed for Chapter 1. Note, in the example here given of the dual form in Greek, a possible influence of the Nostratic **to. Several accounts of number-words and concepts draw attention to the remnants still preserving aspects of the dual in English. The list given here slightly extends that given in my 1996 article in the Australian Mathematical Society Gazette. That was already a longer list than any other I have seen. In the discussion I have given of number concepts, I have adopted Menninger’s view that all languages possess the concepts and words for

RESOURCES

‘one’ and ‘two’. Recently, this view has been challenged. An Amazonian group, the Pirahã, have recently adduced as a counter-example. The case is disputed. Earlier work did list Pirahã words for ‘one’, ‘two’ and ‘many’. More recently, it has been claimed that these words do not convey the same precision as our words for these same concepts. The basis for this contention is the use of these same words to use (approximately) ‘small’, ‘medium’ and ‘large’. Perhaps this provides a glimpse into an earlier concept of number than the others described in the text. For more details, see: ‘Don’t count on it’ by Annette Lessermann and Daniel Everett (2006) and the further references provided there. The dual, the ‘step to three’ and the adjectival character of the first four number-words occupy pages 9–32 of Menninger’s book. This also contains a detailed discussion of the Slavic squish and related matters. The word ‘triphibious’ seems to have passed from use. I learned of it some fifty years ago from an article by the grammarian John Medley. At that time, Medley wrote a regular column in The Age, and it was in this context that, much to my surprise, he gave the word his blessing. The account of Pitjantjatjara can be found on the web at: http://www.bri.net.au/spokenword.html For accounts of the New Guinea languages, I am again indebted to the researches of the late GA Lean. It was Lean who characterised Ormu and Yotafa as I have described. I myself, however, found his evidence unconvincing on these points. His account of Kuman, by contrast, carries conviction. The suggestion of an archaic base-four system has some further (rather weak) support from the Japanese. According to Menninger, the first few numbers in Old Japanese went:

1 hi 5 i

2 fu 6 mu

3 mi 7 na

4 yo 8 ya

It may just be that the similarities between the first four of these words and the next four provide evidence of such a stage. It is thought by some that Japanese is a Nostratic-derived language, and this may be thought to be evidence of a base-four stage before PIE was developed. However, there are a lot of ‘ifs and buts’ in this line of reasoning! (Menninger, it should be noted, makes no reference to Nostratic theory, which postdated his book.)

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8.6  Developed systems of number-words An excellent reference for some of this material is Thomas Crump’s The Anthropology of Numbers (1990). My opening statements essentially summarise his remarks along similar lines in his Chapter 3. The requirement of stability is central to the Nostratic hypothesis. The best single reference is the paper by Dolgopolsky in the collection Typology, Relationship and Time (ed. and trans. VV Shevoroshkin and TL Markey (1986). The editors, in their foreword to this collection, also discuss the matter on pages xvii–xviii. See also the paper by Kaiser and Shevoroshkin (1988). These authors also make use of the requirement that important words be short, although their discussion on this point is not as explicit as is their discussion of stability. Another source for this aspect of theory is the paper by Alexis Manaster-Ramer in his contribution to Reconstructing Languages and Cultures (1989, see pp. 104–108, especially p. 107). However, I take the description ‘unmotivated’ from Szemerényi. The description of much-used words in these terms relates to a linguistic theory known as the SapirWhorf hypothesis, which is the subject of several good discussions on the Internet. The shortening of number-words as they become more central to language is most strikingly illustrated by the descendants of the Sanskrit numerals. The first five were in Sanskrit: 1: ekab, eka   2: dvi, dve   3: trayah, tisrah 4: chatvarah, chatasvarah   5: pancha (derived from Menninger, but with slightly simplified spelling). Compare the modern forms: 1: ek   2: do   3: tin   4: cha   5: panch applying in both Hindi and Urdu. These come from AH Harley’s Colloquial Hindustani (1944), and again I have slightly simplified his spelling. The fleuve-rivière example is taken from Jonathan Culler’s Saussure (1976, pp. 23–24). This book is an introduction to the thinking of one of the greatest of modern linguists; it is also pertinent to many other points of the discussion. The account of the incident with Galois, and of ET Bell’s unfortunate mistranslation is taken from Tony Rothman’s ‘Genius and biographers: the fictionalization of Évariste Galois’, American Mathematical Monthly 89 (1982, pp. 84–106).

RESOURCES

It should be remarked that the coincidence between adam meaning one in Old Persian and Adam as the name of the first man is very likely just that: coincidence. The meaning for Adam is the subject of various theories, based on Hebrew (which is not IE). A full account of these is available in the article ‘Adam and Eve’ in the Wikipedia. (But this article carries a caution that its matter is regarded as controversial.) The words *qäзä and *зet'w^ are given on pages 121 and 124 (respectively) of the collection Reconstructing Languages and Cultures, detailed in Chapter 8.1. The connection between the Austronesian lima and the meaning ‘hand’ is ‘well known’ but surprisingly difficult to track down! However, EM Kempler Cohen’s ‘Fundamentals of Austronesian Roots and Etymology’ (Pacific Linguistics, 1999) lists it on page 101. It is described as evidenced by both Formosan and non-Formosan attestations. Bellwood’s Scientific American article (detailed in Chapter 8.1) has Formosan splitting off early from the main group Austronesian, which became the Malayo-Polynesian branches of the family. (However, not all linguists accept the classification of Formosan as Austronesian.) Lockwood, although he does not discuss the matter, would presumably take issue with Szemerényi’s derivation of the PIE word for ten, because of his doubts on the Indo-European provenance of the word ‘hand’. The way in which a consonant may ‘begin again’ at the back of the mouth, after ‘falling out of the front’ was illustrated to me in a somewhat surprising way. In 1984, I attended a UNESCO conference on Mathematics, Language and Mathematical Education held in Calcutta. In one short presentation by a local high-school teacher, the speaker waxed eloquent on the difficulty his students experienced with ‘herbal problems’. It was only toward the end of his address that I realised that he meant ‘verbal problems’! The various suggestions discussed by Szemerényi in relation to the derivation of the PIE numerals are scattered throughout his text. There is a sort of summary on page 141, and it is here that he describes the PIE words as ‘unmotivated’. He makes (partial) exceptions in the cases of 5, 6, 9, 10 and possibly 8. The discussions of these number-words are to be found on pages 113, 79, 173, 69 respectively for the first four of these and pages 103 and 141 for the case of 8. His discussion of dual forms (such as *octo(u) may be) and the material from Ostyak is to be found on page 145. My hypothesis on the basis for the numbers 5, 8 and 9 was put forward in my article in the Australian Mathematical Society Gazette

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1996, but part of the underlying idea was advanced even earlier in my 1989 article.

8.7  Projects 8.7.1 Roman numerals There are many good accounts of Roman numerals. Menninger gives a lot more detail than is provided here, and considers a number of variants. The analysis presented in this book is my own, and was first published in Function (vol. 19, Part 2, April 1995). That paper drew attention to two accounts of how arithmetic may be performed using Roman numerals. In 1975, a group of four American philosophers (Detlefsen et al.) published methods for performing addition and multiplication. Their paper appeared in the journal Archive for History of Exact Sciences (vol. 15). (It was these authors who introduced the symbols W and K.) In 1981, JG Kennedy, an engineer with interests in computational mathematics for use in the aerospace industry, came up with very elegant methods for multiplication and division, and published these in the American Mathematical Monthly (vol. 88). There have undoubtedly been others, but these are the two for which I have full details. There has been a suggestion that the Roman numerals owe their form to the use of the abacus in computation. This suggestion is discussed in Menninger’s book on pages 297–298. A Google search reveals many websites. Most of these are concerned only with the interconversion between Roman and standard numerals. However, the Mathworld and the Wikipedia accounts go into much greater detail.

8.7.2  Bases other than ten There is a large literature on the effect of using bases other than ten. Google searches for ‘number base’, ‘octal’ and ‘hexadecimal’ will direct the inquirer to many useful websites. Many will automatically convert between the various possible bases. As usual, the Mathforum, Mathworld and Wikipedia sites can all be recommended. Martin Gardner’s Scientific American column for May 1964 was principally concerned with base three, and this discusses the matter raised in the fourth suggested activity: the variant involving the digits 0, 1 and 1 (that is –1) instead of the more usual 0, 1, 2. This same article is also the source of my information on Bowden’s version of hexadecimal. I myself wrote an account in Function, (vol. 19, Part 3, June 1995).

RESOURCES

I discussed the relative merits of different bases in another Function article (vol. 9, Part 1, February 1985). This article gave somewhat more detail than I include here. It should also be mentioned that there are other methods of representing numbers over and above the use of a fixed base. One that has generated a lot of interest is the so-called Zeckendorf Arithmetic, which makes use of the Fibonacci numbers. A Google search will reveal a lot of good sites describing this system. It has a long and reputable history and has been independently rediscovered many times. A site that can be highly recommended is: http://eom.springer.de/Z/z120020.htm Some of the history is summarised in an article by Garry J Tee in the Australian Mathematical Society Gazette (vol. 30, Part 5, November 2003). This contains the information that the system is named after Edouard Zeckendorf (1901–1983), an amateur Belgian mathematician. One interesting sidelight on this history is the later reinvention of the same system by the then 12-year-old George Bergman. It was published in the journal Mathematics Magazine (vol. 31, 1957) along with an explanatory note from his mother. Bergman went on to become a professor of Mathematics at the University of California at Berkeley. Yet another approach is the use of non-integral bases. A brief account 1+ 5 f= of the use of the golden ratio f for this purpose is included in an 2 article by Christopher Stuart in Function, (vol. 1, Part 1, February 1977).

8.7.3  Counting rhymes The sheep-score is discussed in a number of places. One account of its actual use is to be found in Amy Stewart Fraser’s Roses in December, a volume of Edwardian recollections. The West Cumbrian version appears in Melvyn Bragg’s novel A Place in England (1970). The account of the Isle of Man version is to be found at: http://www.isle-of-man.com/manxnotebook/fulltext/sheep.htm The connection between the sheep-score and children’s counting rhymes is made by Iona and Peter Opie on pages 28–61 of The Oxford Dictionary of Nursery Rhymes (1951). The Opies make explicitly the connection between the sheep-score and the Celtic languages. They returned to the subject in Chapter 7 of their later work Children’s Games

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in Street and Playground (1984). Both these works and the website listed above make some comparison with the numerals as given in the surviving Brythonic languages, but fuller accounts are available in specialist dictionaries. In the second of the books just mentioned, the Opies use the term ‘dips’ to refer to counting rhymes. The term is derived from the children’s own usage. Many examples are given including a version of the one I learned from my grandmother; they source it to Victoria in the 1890s. They note that, in some instances, the relation to counting is recognised, but attributed to other cultures, for example the Chinese. For Welsh there is Y Geiriaur Mawr: The Complete Welsh-English English-Welsh Dictionary, by HM Evans and WO Thomas (1985). For Breton see Geriadur Brezhineg-Saozneg: gant Skouerioù (Breton-English Dictionary) by Remon Ar Porzh (Raymond Delaporte) (1993), or (perhaps more accessibly) Standard Breton by Ian Press (2004). For Cornish there are Cornish for Beginners, by PAS Pool (1970) and An English-Cornish Dictionary, by FWP Jago (1984). I wrote an account of this material in Function, (vol. 8, Part 1, February 1984). It also drew on Cinderella Dressed in Yella by Ian Turner (1969) and a number of other sources there given in more detail than I provide here. Discussions of the Josephus problem can be found on a number of websites. Again I recommend the Mathforum, Mathworld and Wikipedia sites. The following website calculates L^n, mh automatically if the values of n and m are entered: http://www.ship.edu/~deens/mathdl/Joseph.html Several of the websites dedicated to the Josephus problem point to the connection with children’s counting rhymes.

8.7.4  The number-word game The number-word game is one of those pieces of folklore that must be discussed somewhere in the literature, but which can be notoriously hard to pin down. I first learned of it (as applied to English) from a short communication to Function (vol. 1, Part 3, June 1977). The author was Cynthia Kelly, then a third-year Science student at Monash University. In fact, she pointed out that the starting point need not be a number-word, but could be any word. A later issue of Function (vol. 1, Part 5) considered the situation in languages other than English.

References

General works of reference Encyclopædia Britannica, various editions Macquarie Dictionary, various editions Oxford English Dictionary, 2nd edition Shorter Oxford English Dictionary, revised edition Webster’s Dictionary, various editions

Websites General sites MacTutor: http://www-groups.dcs.st-and.ac.uk/~history/ Mathforum: http://mathforum.com Mathworld: http://mathworld.wolfram.com Wikipedia: http://en.wikipedia.org/wiki/Main_Page

Specific sites For Australian Aboriginal data: http://www.dnathan.com/VL/eMUlg_A.htm#73 For Babylonian numerals: http://en.wikipedia.org/wiki/Babylonian_numerals For Chinese numerals: http://en.wikipedia.org/wiki/Chinese_numerals For Josephus problem: http://www.ship.edu/~deens/mathdl/Joseph.html For Mayan numerals: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/ Mayan_mathematics.html For Pitjantjatjara: http://www.bri.net.au/spokenword.html For PNG numerals: http://www.uog.ac.pg/glec/thesis/thesis.htm For PNG numerals: http://www.uog.ac.pg/glec/index.htm For Schmandt-Besserat: http://www.sciencenews.org/articles/200603121/mathtrek. asp For the sheep-score: http://www.isle-of-man.com/manxnotebook/fulltext/sheep. htm For Zeckendorf Arithmetic: http://eom.springer.de/Z/z120020.htm

Books and articles Anon 1977, ‘A fact about the English language’, Function 1, 14. Ar Porzh, R 1993, Geriadur Brezhineg–Saozneg: gant Skouerioù, Breton–English Dictionary, Lesneven, Mouladurioù. 101

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Bell, ET 1937, Men of mathematics, Simon & Schuster, New York. Bellwood, P 1991, ‘The Austronesian dispersal and the origin of languages’, Scientific American (July), 70–75. Bergman, G 1957, ‘A number system with an irrational base’, Mathematics Magazine 31, 98–110. Blake, BJ 1981, Australian Aboriginal languages, Angus & Robertson, London. Borges, JL & Bioy-Cesares, A 1976, Chronicles of Bustos Domecq, NT di Giovanni (trans.), Dutton, New York. Bragg, M 1970, A place in England, Secker & Warburg, St Albans. Cavalli-Sforza, LL 1991, ‘Genes, people and language’, Scientific American (November), 72–78. Chatterton, P n.d., A primer of Police Motu, Pacific Publications, Sydney. Crossley, JN 1987, The emergence of number (2nd edn), World Scientific, Singapore. Crossley, JN 2007, Growing ideas of number, ACER Press, Camberwell. Crump, T 1990, The anthropology of numbers, Cambridge University Press, Cambridge. Culler, J 1976, Saussure, Fontana Modern Masters Series, Glasgow. Cummings, J 1984, Thailand phrasebook, Lonely Planet Publications, South Yarra and Berkeley, CA. Davis, PJ 1972, ‘Fidelity in mathematical discourse: Is one and one really two?’, American Mathematical Monthly 79, 252–263. Deakin, MAB 1984, ‘Yan-a-bumfit and all that’, Function 8, 18–25. Deakin, MAB 1985, ‘What is the best base?’, Function 9, 8–12. Deakin, MAB 1989, ‘Five and four’, Australian Mathematical Society Gazette 16, 125–129. Deakin, MAB 1990, ‘Number-words and their significance’, The Mathematical Scientist 15, 1–6, a condensed version of a preprint entitled ‘What numberwords tell us about number concepts’ released as Monash University history of mathematics pamphlet no. 45, 1989. Deakin, MAB 1991, ‘The Pre-pre-history of mathematics’, Function 15, 19–21. Deakin, MAB 1991, ‘From two to three to four and so on’, Function 15, 150–153. Deakin, MAB 1995, ‘Roman numerals’, Function 19, 45–50. Deakin, MAB 1995, ‘Thrun, fron, feen, wunty’, Function 19, 76–81. Deakin, MAB 1996, ‘The origins of our number-words’, Australian Mathematical Society Gazette 23, 50–66, an edited version of an invited address to the 1995 annual meeting of the Society (the full text of the address constitutes Monash University History of Mathematics Pamphlet 62). Dear, ICB (ed.) 1991, Oxford English: A guide to the language, BCA, London. Detlefsen, M, Erlandson, DK, Heston, JC & Young, CM 1975, ‘Computing in Roman numerals’, Archive for History of Exact Sciences 15, 141–148. Dixon, RMW 1980, The languages of Australia, Cambridge University Press, Cambridge.

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Dolgopolsky, AB 1986, ‘A probabilistic hypothesis concerning the oldest relationships among the language families of northern Eurasia’ in Shevoroshkin, VV & Markey, TL, Typology, relationship and time, Karoma, Ann Arbor, 27– 50. Evans, HM & Thomas, WO 1985, Y Geiriaur Mawr: The complete Welsh-English English-Welsh dictionary, Christopher Davies, Llandul, Gwesg Gomer; Llandybie, Wales. Fraser, AS 1981, Roses in December, Routledge, London. Gamelkridze, TV & Ivanov VV 1990, ‘The early history of Indo-European languages’, Scientific American (March), 82–89. Gardner, M 1964, ‘The “tyranny of 10” overthrown with the ternary number system’, Scientific American (May), 118–124. Greenberg, J & Ruhlen, M 1992, ‘Linguistic origins of native Americans’, Scientific American (November), 62–65. Halhed, NB 1778, A grammar of the Bengal language, unnamed publisher, Hoogly. Harley, AH 1944, Colloquial Hindustani, Routledge, London. Harris, J 1987, ‘Australian Aboriginal and Islander mathematics’, Australian Aboriginal Studies 2, 29–37. Heath, TL 1964, Diophantus of Alexandria, Dover Reprint, New York. Hoe, J 1981, ‘Zhu Shijie and his jade mirror of the four unknowns’ in History of Mathematics: Proceedings of the First Australian Conference, J Crossley (ed.), Monash University, Clayton, 103–134. Hurford, JR 1975, The linguistic theory of numerals, Cambridge University Press, Cambridge. Hurford, JR 1987, Language and number, Blackwell, Oxford. Jago, FWP 1984, An English-Cornish dictionary, AMS Press, New York. Joseph, GG 1990, The crest of the peacock: Non-European roots of mathematics, Penguin, Harmondsworth. Kaiser, M & Shevoroshkin, V 1988, ‘Nostratic’, Annual Review of Anthropology 17, 309–329. Kelly, C 1977, ‘A fact’, Function 1, 30. Kempler Cohen, EM 1999, ‘Fundamentals of Austronesian roots and etymology’, Pacific Linguistics (series D, vol. 94), ANU, Canberra. Kennedy, JG 1981, ‘Arithmetic with Roman numerals’, American Mathematical Monthly 88, 29–32. Kwee, JB 1965, Teach yourself Indonesian, Hodder & Stoughton, Sevenoaks. Lakoff, G & Núñez, RE 2000, Where mathematics comes from, Basic Books, New York. Laplace, PS 1896, Théorie analytique des probabilités: œuvres complètes, vol. 7, Gauthiers-Villars, pp. vi–vii. Lean, GA 1992, The counting system of Papua New Guinea, unpublished PhD thesis, PNG University of Technology, Lae.

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Lehmann, WP 1973, Historical linguistics: An introduction (2nd edn), Holt, Rinehart & Winston, New York. Lehmann, WP 1974, Proto-Indo-European syntax, University of Texas, Austin. Lehmann, WP 1993, Theoretical bases of Indo-European linguistics, Routledge, London. Lessermann, A & Everett, D 2006, ‘Don’t count on it’, Scientific American Mind, (November), 74–77. Lockwood, WB 1969, Indo-European philology, Hutchinson, London. Lockwood, WB 1972, A panorama of Indo-European languages, Hutchinson, London. Macleod, AG 1967, Colloquial Bengal grammar, undocumented publisher. Manaster-Ramer, A 1989, ‘Language, history and computation’ in Reconstructing languages and cultures, Proceedings of the 1st International Interdisciplinary Symposium on Language & Prehistory, VV Shevoroshkin (ed.), 104–109. Menninger, K 1969, Number words and number symbols: A cultural history of numbers, P Boneer (trans.), MIT Press, Cambridge, MA. Mosel, U 1984, Tolai syntax and its historical development, ANU, Department of Linguistics, Research School of Pacific Studies, Canberra. Newnham, R & Tan Ling-tung 1987, About Chinese (2nd edn), Penguin, Harmondsworth. Opie, I & Opie, P 1951, The Oxford dictionary of nursery rhymes, Clarendon, Oxford. Opie, I & Opie, P 1984, Children’s games in street and playground, Oxford University Press, Oxford. Partridge, E 1983, Origins, Greenwich House, New York. Partridge, E 1987, Usage and abusage (revised edn), Guild, London. Pool, PAS 1970, Cornish for beginners, Cornish Language Board, Penzance. Press, I 2004, Standard Breton, LINCOM, Munich. Renfrew, C 1989, ‘The origins of Indo-European languages’, Scientific American (October), 82–90. Renfrew, C 1994, ‘World linguistic diversity’, Scientific American (January), 104–110. Rothman, T 1982, ‘Genius and biographers: the fictionalization of Évariste Galois’, American Mathematical Monthly 89, 84–106. Rudder, J 1999, An introduction to Aboriginal mathematics, Restoration House, Canberra. Ryan, W & Pitman, W 1998, Noah’s flood, Simon & Schuster, New York. Salmons, JC & Joseph, BJ 1988, Nostratic: Sifting the evidence, Benjamins, Amsterdam. Schmandt-Besserat, D 1978, ‘The earliest precursor of writing’, Scientific American (June), 38–47. Schmandt-Besserat, D 1992, Before writing, University of Texas, Austin. Scott, G 1980, ‘Fore dictionary’, Pacific Linguistics (series C, vol. 62), ANU, Canberra.

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Shevoroshkin, VV (ed.) 1989, Reconstructing languages and cultures, proceedings of the 1st International Interdisciplinary Symposium on Language & Prehistory, Brockmeyer, Bochum. Shevoroshkin, VV & Markey, TL 1986, Typology, relationship and time, Karoma, Ann Arbor. Stuart, C 1977, ‘Fibonacci sequences’, Function 1, 24–29. Szemerényi, O 1960, Studies in the Indo-European systems of numerals, Carl Winter Universitätsverlag, Heidelberg. Tee, GJ 2003, ‘Russian peasant multiplication’, Australian Mathematical Society Gazette 30, 267–276. Thieme, P 1958, ‘The Indo-European language’, Scientific American (October), 63–78. Thurston, HA 1956, The number system, Blackie, London. Trask, PL 1996, Historical linguistics, London, Arnold. Turner, I 1969, Cinderella dressed in yella, Heinemann, Melbourne. Watkins, C 1985, American heritage dictionary of Indo-European roots, Houghton Mifflin, Boston. Wolfers, EP 1972, ‘Counting and numbers’, Encyclopaedia of Papua and New Guinea (vol. 1), P Ryan (ed.), Melbourne University Press, 216–220.

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Index auxiliary element  64–6 Avestan  6, 8, 9, 10, 13

abacus  31, 91, 98 Aboriginal  49, 89–90, 101–4 Abraham  7 Adam  58, 97 additive mode  61, 68 adjective  35, 37–9, 42–7, 49 advanced numeracy  22, 57, 60, 81 adverb  38, 39 Afghanistan  8 Afro-Asiatic  14, 15, 49, 86 Albanian  6, 7, 9, 85 Altaic  14–16, 86 American  12–14, 53, 76 Anatolia  7, 9 Anatolian  8, 85 Anglo-Cymric Score  75, 76 Anglo-Saxon  2, 3, 12, 13, 45, 47, 51 Anjumarla  90 anthropologist  24 Anthropology  86, 88, 96, 102, 103 Ar Porzh, R  100, 101 Archaeology  2 Aristotle  1 Armenian  6, 7, 85 Arthur, King  2, 83 article, definite or indefinite  37–9, 49 Asia Minor  7 Attic  7 Australian  3, 10, 12, 14, 24, 25, 38, 49, 51, 52, 89, 90, 101–3 Austronesian  13, 60, 85, 97, 102, 103

Babylon  2 Babylonian  21–4, 26, 32, 42, 54, 57, 88, 101 Baltic  6–9 Balto-Slavic  7 Bangla  46, 94 base  17–28, 35, 52–4, 57, 59–61, 63–5, 67, 68–73, 87–90, 95, 98–9, 102 basic element  64–6 Basque  14 Bell, ET  55, 96, 102 Bellwood, P  85, 97, 102 Bengali  46 Bergman, G  99, 102 bilabial  84, 85 bilabial fricatives  84 binary  20, 69, 70, 88 Bioy-Cesares, A  93, 102 Black Sea  9, 102 Blake, BJ  89, 90, 102 body-counting  25, 26, 34, 44 Bopp, F  6 Borges, J  93, 102 Bowden, J  69–71, 73, 98 Bragg, M  99, 102 Breton  12, 28, 74, 75, 100, 101, 104 Bronze Age  22 Brythonic  74, 100 Bulgarian  8 106

INDEX

canon of interpretation  66 cardinal  50, 94 case  40–2, 94 Caucasian  14 Cavalli-Sforza, LL  15, 85, 87, 102 Celtiberians  7 Celtic  2, 6, 7, 12, 26, 74, 76, 83, 99 Celts  2, 3, 7 centum  9, 10 Chatterton, P  89, 102 China  14, 31 Chinese  21, 22, 24, 32–4, 43, 57, 64, 81, 88, 91, 100, 101, 104 clan  4, 6–8, 13–15, 74, 83, 85 Clemens, S  45 collective nouns  46 comma  22, 79 computer logic  20, 21, 69 computer program  67, 91 conjunction  38, 39 consonant  3, 5, 10–13, 61, 62, 84, 97 Constantine I  8 Cook, Captain  14 Cornish  12, 74, 75, 100, 103, 104 counting  12, 17, 18, 22–6, 29, 35, 44, 45, 49, 51–3, 56, 57, 59, 61, 62, 74–9, 81, 88, 89, 100, 103, 105 counting rhymes  74–8, 99–100 Crossley, JN  86, 91, 102, 103 Crump, T  86, 96, 102 Culler, J  96, 102 Cummings, J  88, 102 cuneiform  22, 23, 32 cycle  79–81 Cyrillic  70 Czech  8, 46

Dard  8 Dardic  8 Davis, PJ  93, 102 Deakin, MAB  102 Dear, ICB  92, 102 decad  26, 28 dental  11 Detlefsen, M  98, 102 dialects  3, 12, 13, 75–7 Dickens, Charles  11 digit  21–3, 62, 68–70, 72, 73, 79, 81, 98 Diophanus  91, 92 Dixon, RMW  89, 102 Dolgopolsky, AB  15, 96, 103 dozen  35, 56, 72 dozenal  69 Dravidian  14, 15, 86 dual  49, 50, 52, 60–2, 94, 95, 97 duodecimal  69, 72, 73 Dutch  7 Dynamical Systems Theory  80 Egypt  8 eight  2, 3, 18, 19, 21, 37, 40, 55, 56, 60, 62, 68, 71, 79, 88, 92 eleven  3, 4, 51, 56, 57, 69–72, 79 Encyclopaedia Britannica  83, 87, 101 English  3–8, 13, 18, 19, 26, 33, 34, 36, 38–43, 45, 46, 49, 51, 55, 57–8, 71, 78–81, 83, 84, 90–2, 94, 100–3 equilibrium  80, 81 Erlandson, DK  102 Eskaleut  15, 59 Estonian  14 etymology  57, 58, 60, 61, 83, 97, 103 Euphrates  29 Eurasiatic  15

107

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INDEX

Evans, HM  100, 103 evenness  72 Everett, D  95, 104 family  4, 5, 6–16, 59, 83, 85, 87, 97 feminine gender  42, 45, 93 Fermat’s Last Theorem  43 Fermat’s Little Theorem  73 Fibonacci Numbers  99, 105 fifteen  26, 27, 69, 71 Fijian  35 Finnish  7, 13, 14 Finno-Ugric  14, 60 five  12, 18–20, 23–5, 27, 28, 33, 35, 43, 44, 46–8, 51, 54, 57, 59–63, 65, 69, 71, 79, 86, 96, 102 Fore, Northern  25, 27, 52, 54, 56, 57, 81, 89, 104 Formosan  97 four  12, 18, 19, 24–7, 32, 33, 44–6, 50, 52–4, 59, 61–3, 65, 71, 79, 81, 86, 89, 90, 95, 102 Fraser, AS  99, 103 French  3–5, 8, 11–13, 19, 26, 28, 42, 43–6, 49, 55, 59, 79, 80, 93 Function  87, 98, 99–103, 105 Gaelic  74 Galatians  7 Gamelkridze, TV  85, 103 Gardner, M  69, 98, 103 gatepost tally  20, 24 Gaulish  7 gender  34, 42, 92–4 Genesis, Book of  8, 9 Georgia  14 German  3–7, 12, 19, 35, 42, 43 Germanic  3–8, 12, 51, 59, 61

Gilgamesh, Epic of  9 glottal stop  3, 5 Goidelic  74 Google  87–90, 92, 94, 98, 99 Gothic  6 grammar  6, 10, 34, 37–48, 52, 65, 92–4, 101, 104 grammarian  34, 39, 42, 95 Greek  4, 6, 7, 9–11, 49, 57, 60, 61, 67, 86, 91, 94 Greenberg, J  85, 103 Gumulgal  24, 52–4, 61, 62, 89 guttural  3, 5, 10, 11, 13, 62 Halhed, NB  94, 103 hand  25, 27, 53, 54, 59–62, 97 Harley, AH  96, 103 Harris, J  89, 90, 103 Heath, TL  92, 103 Hebrew  7, 57, 58, 97 Heston, JC  102 hexadecimal  69–71, 88, 98 Hindi  49, 96 historical linguistics  84, 104, 105 history  1, 2, 3, 5, 29, 49–54, 82, 85, 86, 88, 91, 94–5, 98, 99, 101–4 Hittite  6, 7, 8, 85 Hoe, J  91, 103 Human Genome Project  15 hundred  9, 10, 21, 28, 48, 50, 70, 79, 94 Hungarian  8, 13, 14 Hurford, JR  86, 96, 103 hyphen  79, 80 ideographs  32 Illič-Svityč  15, 83 Illyrian  85 Indian  14, 38, 57

INDEX

Indic  6, 8, 9 Indo-European  4–8, 13–15, 26, 49, 50, 53, 83, 85, 87, 92, 94, 97, 103–5 Indo-Iranian  8 Indonesian  34, 60, 91, 103 interjection  39 Internet  83, 87, 92, 96 Iranian  8, 9 Italian  3–5, 8, 11–13 Italic  6, 8, 83, 85 Ivanov, VV  85, 103 Jago, FWP  100, 103 Japanese  14, 57, 95 Jerome, St  7 John-Paul II, Pope  11 Jones, Sir William  6 Joseph, BD  104 Joseph, G  82, 90, 91, 103 Josephus problem  77, 78, 100, 101 Julius Caesar  7 Kaiser, M  86, 96, 103 Kartvellian  14, 15, 86 Kelly, C  100, 103 Kempler Cohen, EM  97, 103 Kennedy, JG  98, 103 Kewa  25, 26, 44 Kipling, R  45 Kiwai  24, 25, 52–4, 61, 62, 81 Korean  14 Kuman  54, 95 Kurgans  9 Kwakiutl  35, 52 Kwee, JB  91, 103 labial  11 labio-dental  85 Lakoff, G  93, 103

language  1–15, 18, 19, 24–6, 29, 34, 35, 37, 42, 45, 46, 48–54, 56, 57, 60, 74, 79, 81, 83–90, 92, 94–7, 99–105 Laplace, P-S  93, 103 Lappish  7, 14 Latin  3–12, 19, 21, 31, 37, 39– 42, 45, 46, 49–51, 53, 58, 60, 62, 78, 84, 86, 92, 94 Lean, GA  89, 95, 103 Lehmann, WP  84, 85, 92, 104 Lessermann, A  95, 104 limit of precise counting  24, 51–3, 59, 61, 62 Linguistics  2, 29, 84, 85, 88, 89, 97, 103–5 linguists  3, 5, 9, 10, 15, 56, 97 Lithuanian  6, 7, 49, 60 Lockwood, WB  14, 58–60, 85, 97, 104 MacLeod, AG  94, 104 Macquarie Dictionary  38, 39, 43, 83, 101 MacTutor  82, 88, 101 Malayo-Polynesian  13, 97 Maltese  14 man  25, 42, 44, 58, 97 Manx  74, 101 Maori  14, 60 Markey, TL  96, 103, 105 masculine gender  42, 45 mathematical induction  18 mathematics  4, 17, 22, 24, 31, 34, 57, 80, 82, 88–90, 93, 97–9 Mathforum  82, 98, 100, 101 Mathworld  82, 88, 98, 100, 101 Mayan  21–4, 26, 54, 88, 101 measure-word  34, 35, 47 Mediterranean  9 Medley, J  95

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Menninger, K  31, 35, 51–3, 58–60, 82, 84, 88, 90, 91, 94–6, 98, 104 Middle East  29, 32, 33 million  21, 22 mixed base  22, 25, 26, 54 Mongolia  8 Motu  25, 27, 43, 53, 60, 63, 80, 89, 102 ‘Mugwump’  19 multiplication  18, 67, 70, 98, 105 naming of children  46, 53 nasal  13, 62 negative bases  73 neogrammarian hypothesis  84 Neolithic  32 neuter gender  42, 45, 93 nine  18, 19, 22, 23, 28, 60, 62, 71 Noah  86, 104 Normans  4 Nostratic  15, 16, 41, 58–62, 83, 86, 87, 94–96, 103, 104 noun  37–46, 49, 61, 93 number  2, 5, 10–14, 17–25, 27, 29–37, 39, 42–62, 64, 65, 67–82, 84, 86–93, 94–100, 102–5 number-word  2, 11, 14, 19, 21, 27, 29, 32, 35, 36, 37–48, 51, 53, 55–63, 74, 75, 84, 86, 89, 90, 92–4, 96–8, 102 number-word game  78–81, 100 numeral  2, 10, 19–25, 27–9, 31–5, 43–8, 52, 53, 55–8, 60, 63–7, 69–88, 96–98, 100–3, 105 numeration  25, 44, 57, 68 Núñez, RE  93, 103

object (in grammar)  40 octal  68, 70, 88, 98 oddness  72 Opie, I & P  75, 76, 99, 100, 104 ordinal  50, 94 Ormu  54, 95 orphan languages  14 Ostyak  60, 97 Oxford English Dictionary  58, 94, 100 Pacific  13, 14, 89, 97, 102–4 Palaeontology  2 Papua New Guinea  24, 25, 88, 89, 103, 105 Partridge, E  36, 48, 83, 86, 92, 94, 104 parts of speech  37, 61, 92 Pashto  8 Peano’s axioms  17, 18, 87 Persian  6, 8, 57, 58, 97 Petersen, I  30, 91 Philology  85, 94, 104 Phrygian  85 PIE  4, 5, 7, 9–14, 16, 29, 35, 40, 42, 44–6, 49, 51–3, 57–63, 76, 86, 92, 95, 97 Pirahã  95 Pitjantjatjara  52, 95, 101 Pitman, W  86, 104 Plato, platonic  43, 44, 93 plural  42, 45–52, 61, 94 Polish  8, 39 Polynesian  13, 57, 97 Pool, PAS  100, 104 Port Moresby  35 Portugal  7 Portuguese  3–5, 8, 11–13 positional notation  20–2, 24 Prehistory  2, 4, 5, 86, 104, 105 preposition  38–41

INDEX

Press, I  100, 104 projects  64–81, 98–100 pronoun  38–42, 52, 58 Proto-Altaic  16 Proto-Germanic  51, 59 Proto-Indo-European  4, 87, 92, 104 Proto-Uralic  16 racism  16, 90 Renfrew, C  85, 96, 104 Roman numeral  19, 28, 64–7, 98, 102, 103 Romance Languages  4, 5, 6, 8, 42, 83, 85 Romanian  8, 11–13 Rome  8, 53, 62, 67 roots  85, 97, 103, 105 Rothman, T  96, 104 Rudder, J  90, 104 Ruhlen, M  85, 103 rule  4, 5, 9–13, 16, 22, 40, 41, 46, 55, 61, 65, 66, 70, 72, 73, 77, 78, 84, 85 Russian  8, 35, 46, 56, 70, 91, 105 Ryan, W  86, 104 Saami  7, 14 Salmon, JC  104 Samoyedic  14 Sanskrit  4, 6, 8–11, 45, 49, 58, 96 Sapir-Whorf hypothesis  96 satem  9, 10, 13, 84, 87 Scandinavian  4, 7 Schmandt-Besserat, D  30–2, 90, 91, 101, 104 score  35, 48, 56, 75, 76, 99, 101 Scott, G  89, 104 sequence  18, 20, 21, 52, 78, 105

Serbo-Croat  8 seven  18, 19, 25, 36, 44, 56, 60, 69, 71 sheep-score  75, 99, 101 Shevoroshkin, V  15, 86, 96, 103–5 Shorter OED  58, 101 sibilant  10 singular  34, 42, 45–50, 61, 62, 94 Sinha, DK  94 Sino-Tibetan  14, 15 sixteen  21, 69–73, 88 sixty  21, 22, 26 Slavic  6, 7, 8, 46, 53, 59, 61, 94, 95 Slavonic  6, 7, 8, 9 sound change  84 space  22, 23, 79 Spain  7 Spanish  3, 4, 5, 8, 11–13, 18, 19 squish, Slavic  46, 53, 59, 61, 94, 95 step to three  51, 52, 58, 61, 95 string  20, 21, 64–8, 70, 81 Stuart, C  99, 105 subject (of sentence)  40, 46, 52, 93 subtractive mode  66, 67 Sumerian  9, 29, 34 super-families  15 symbol  18–23, 28, 29–36, 43, 57, 62, 64, 69–71, 82, 88, 90–2, 98, 104 Szemerényi, O  14, 57, 59–61, 65, 85, 96, 97, 105 tally  20, 24, 30, 33 tally-stick  75 Tamil  14, 50 Tee, GJ  99, 105

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ten  14, 18–22, 24–8, 33, 35, 45, 51, 53, 54, 56–8, 60–4, 67–73, 76, 77, 79, 97, 98 tests for divisibility  72 Thai  18, 19, 50, 88, 102 Thieme, P  85, 105 thirteen  69, 71, 73 Thomas, WO  100, 103 thousand  21, 28, 47, 56, 72 Thraco-Phrygian  85 three  18, 19, 21, 24, 25, 27–9, 33, 44–6, 48–52, 54, 56, 58, 59, 61, 69, 71–3, 77, 88–90, 93–5, 98, 102 Thurston, HA  87, 105 Tibetan  14, 15, 57 Tigris  29 Tocharian  6, 8, 9, 56 tokens  30–2 Tolai  35, 91, 104 tones, Chinese system of  33 Tongan  60 Transcaucasus  14 Trask, PL  84, 85, 105 Treveri  7 Turkey  7, 9, 31 Turkish  14, 50 Turner, I  100, 105 twain  45 twelve  21, 51, 69–72

twenty  10, 21–3, 25, 26, 45, 48, 56, 71, 75 Ukrainian  8 Uralic  14–16, 86 Urdu  96 verb  37, 38, 46, 49, 52 vowel  3, 5, 10–13, 16 Watkins, C  14, 60, 85, 105 website  82, 87–9, 91, 92, 98, 100, 101 Webster’s Dictionary  34, 43, 83, 101 Welsh  7, 12, 26–8, 45, 74–8, 100, 103 Wikipedia  87, 88, 91, 97, 98, 100, 101 Wilde, O  58 Wolfers, E  88, 105 Yotafa  52, 95 Young, CM  102 Zeckendorf Arithmetic  99, 101 Zeckendorf, E  99 Zend  6 Zeno  1, 82 zero  20, 22–4

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2 2 0 0 9 9 8 8 1 1 Series Editor David Leigh-Lancaster

The Name of the Number

We’re 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.

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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 )

The name of the number

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 ISBN 978-0-86431-757-5 been designed to be accessible and rigorous, while appealing to students, educators, mathematicians and general readers.

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THE EMERGENCE OF NUMBER

THE NAME OF THE NUMBER

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THE EMERGENCE OF NUMBER

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Michael A B Deakin

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