Numeral Types and Changes Worldwide 9783110811193, 9783110161137

Citation preview

Numeral Types and Changes Worldwide

I

1999

Trends in Linguistics Studies and Monographs 118

Editor

Werner Winter

Mouton de Gruyter Berlin · New York

Numeral Types and Changes Worldwide

edited by

Jadranka Gvozdanovic

W G DE

Mouton de Gruyter Berlin · New York

1999

Mouton de Gruyter (formerly Mouton, The Hague) is a Division of Walter de Gruyter & Co., Berlin.

© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress

Cataloging-in-Publication-Data

Numeral types and changes worldwide / edited by Jadranka Gvozdanovic. p. cm. - (Trends in linguistics. Studies and monographs ; 118) Includes bibliographical references and index. ISBN 3-11-016113-3 (cloth : alk. paper) 1. Grammar, Comparative and general — Numerals. 2. Historical linguistics. 3. Typology (Linguistics) I. Gvozdanovic, Jadranka. II. Series. P275.N86 1999 415-dc21 98-52394 CIP

Die Deutsche Bibliothek —

Cataloging-in-Publication-Data

Numeral types and changes worldwide / ed. by Jadranka Gvozdanovic. - Berlin ; New York : Mouton de Gruyter, 1999 (Trends in linguistics : Studies and monographs ; 118) ISBN 3-11-016113-3

© Copyright 1999 by Walter de Gruyter GmbH & Co. KG, D-10785 Berlin All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printing: Werner Hildebrand, Berlin. Binding: Lüderitz & Bauer, Berlin. Printed in Germany.

Contents

Jadranka Gvozdanovic Introduction

Numeral systems and patterns of change James R. Hurford Artificially growing a numeral system Werner Winter When numeral systems are expanded Carol F. Justus Pre-decimal structures in counting and metrology Bernard Comrie Haruai numerals and their implications for the history and typology of numeral systems Jadranka Gvozdanovic Types of numeral changes Walter Bisang Classifiers in East and Southeast Asian languages: Counting and beyond Jadranka Gvozdanovic Some remarks on numeral morphosyntax in Slavic

vi

Contents

Numeral reconstruction in several major language groups of Europe and Asia Eugenio Ramon Lujän Martinez The Indo-European system of numerals from Ί ' to '10' Dzoj (Joy) I. Edelman On the history of non-decimal systems and their elements in numerals of Aryan languages Ldszlo Honti

199

221

The numeral system of the Uralic languages

243

Paul Sidwell The Austroasiatic numerals Ί ' to '10' from a historical and typological perspective

253

Subject index

273

Language index

277

Introduction Jadranka Gvozdanovic

"How do numeral systems develop and how do they change?" Linguists interested in these questions gathered in a workshop at the 12th International Conference on Historical Linguistics held at the University of Manchester on August 18, 1995. Their joint efforts, together with additionally invited papers, have resulted in this book. The first group of papers discusses patterns involved in the emergence of numeral systems, as based on individuation, identification, and ordering, and also overt classification as an areal phenomenon. The papers are all, explicitly or implicitly, based on the assumption that understanding how numeral systems develop and change can help us gain more insight into cognitive processes and their relation to language manifestations. In a novel way, Hurford's paper reports on simulating the growth of numeral systems by a genetic algorithm. It makes explicit which types of cognitive evaluations may have underlied the development of the actual systems, including base combination possibilities and redundancy problems. It shows in the end that mutant lexical items will only persist in the pool if they happen to fit in with existing rules and lexical items. A practical example of an individual invention and growth of a numeral system by applying the existing principles in ways which are new in the details, is reported for Tibeto-Burman by Winter. Of particular interest is the contemporary invention of new numeral signs in relation to the number system they represent. The established principles of numeral growth appear to have formed the backbone of historical developments as well. Furthermore, investigation of language histories enables us to place them in their sociocultural contexts, showing how numeral systems have developed conceptually. As cultures come to cope with realities of increasing complexity, they become forced to evaluate token-bound properties and quantities in more general terms. The first stage in this process is one of establishing relations among commodity-bound tokens without their individuation as such. This is found in pre- and proto-literate Ancient Near Eastern token systems of the prehistoric 8th to 4th millennia BC, as discussed by Justus in this volume. It is also found in conservative Celtic farming communities, but with an addi-

2

Jadranka

Gvozdanovic

tional grouping into tetrads and pentads (of sheep), and digits indicating relational values within the variable groups, no values by themselves. Whereas the Ancient Near Eastern counting is based on factors as expressing relations only, the Celtic farming communities count in sets, but have identification of the digits only by their place in the series. Typological principles involved in the development of numeral systems are discussed further by Comrie and Gvozdanovic. Comrie shows how grouping of objects develops further into serialization, by which groups become bases. The notion of base thereby requires further clarification: as used in various discussions, it either refers to a building block explicitly involved in any mathematical operation, or a building block used in multiplication only. In view of language variation showing that grouping may be independent of multiplicative iteration, it seems advisable to distinguish between the two. If there is multiplication in a numeral system, it applies to bases (next to digits), thereby revealing what the bases are. If there is no multiplication, then derived grouping (of the type l (5+l)+l' for '7', as mentioned a.o. in Hurford's paper) may be taken to reveal the basic sets. In other words, we may distinguish between basic sets (such as '(5+1)' in the above example) and bases (such as '5', '10' etc., depending on the language): basic sets have a transparent inner structure, whereas bases are conceptualized as entities themselves. This is further supported by sign formation in languages, such that bases can be only iconic or symbolic signs, but no indexes, as discussed by Gvozdanovic in this volume. In the process of base establishment in a numeral system, the language sign expressing the base will undergo a shift (from an index, which is a contextually bound sign, into an icon or a symbol, which are 'free' signs) so as to comply with the requirement of serialization imposed upon a base. In the history of numeral systems, the process of individuation, identification, and classification can best be followed by the history of numeral classifiers in Southeast Asia, as analysed by Bisang in this volume. He shows i.a. that in Classical Chinese, classifiers occur first with names for single objects of particular cultural value. By a slow process they then extend their sphere of influence so as to classify the concrete semantics of an object, and only subsequently its class or category in terms of general semantic features. Classifiers thus make explicit how concrete tokens become viewed as entities with generalized properties in the process of counting. Beyond that, classifiers can display the function of classification without being primarily involved in counting, as shown by Vietnamese, where the pathways of classification follow either the principle of taxonomy (with the semantically higher noun

Introduction

3

developing into a classifier) or of meronomy (with the semantically lower noun developing into a classifier). A related phenomenon is found in highlighting on either the counted set as a whole, or the members of the set taken individually in their interrelations, as revealed by numeral syntax in Slavic (briefly discussed by Gvozdanovic). The principles established for language growth are in various ways applicable to language reconstruction. For one, serialization is established by Justus as the basic pattern governing pre-decimal structures in counting and metrology in the Ancient Near East, as mentioned above. Concerning Indo-European, Lujän Martinez traces the origin of the lower digits as deictic and shows how the system may have become decimal in the end. Late Indo-European was decimal, and this decimality was borrowed into neighboring languages, including Aryan as discussed by Edelman in this volume. She shows that a vigesimal system could develop in Aryan only if an etymologically common name was used for fingers and toes, thereby treating them alike and making them comparable to the digits of the counting system. Here again, the cognitive and linguistic systems appear to be closely intertwined. Honti shows in this volume that the reconstruction of Finno-Ugric etyma can be done properly within a decimal hypothesis. Also Austroasiatic requires a decimal hypothesis for a correct account of the preserved numeral forms, although, as shown by Sidwell's comprehensive survey in this volume, this may have been a late stage of a development typologically similar to the reconstructed Indo-European one. The contributions to this volume show that the wide range of variation in numeral systems and changes leading to them are based on a limited set of principles which - when applied to varied language structures - lead to the observed complexities.

Numeral systems and patterns of change

Artificially growing a numeral system* James R. Hurford

1. Introduction A special kind of language change, not centrally studied in historical linguistics, is language growth. Growth is change from a smaller system (or perhaps even from nothing) to a larger system. Grammatical growth can occur by addition of lexical items or, perhaps more interestingly, of rules. This paper describes attempts to simulate the growth of natural language numeral systems by computational techniques. Similar techniques could in principle be used to model the growth of other subsystems of grammar, and a long-term aim of this strand of research is to model the evolution of many of the organizational features of language. There are two salient factors in favour of using numeral systems as a test bed for this approach, and one salient factor against. In favour: 1 .Numeral systems are relatively simple, 2.Numeral systems are relatively self-contained. Against: 1.Numeral systems are in various ways atypical of other language subsystems. These properties of numeral systems are known well enough to need no further elaboration here.

2. Artificial Life ("Α-Life") research The research paradigm inspiring this work is that of Artificial Life ("A-Life"). Two very readable popular introductions to this field are Levy (1992) and

8

James R. Hurford

Lewin (1993), and there are now major collections stemming from workshops and conferences (Langton et al. 1991, 1994, Brooks — Maes 1994). Although terminologically reminiscent of the discipline of Artificial Intelligence (AI), Α-Life differs from it in several fundamental philosophical and methodological principles. Whereas AI attempts to model developed or mature intelligence, the products of millennia of evolution, Α-Life attempts to model the evolutionary processes leading to such products. AI systems are typically architecturally complex, embodying the programmer's analysis of the particular manifestation of intelligence under study (such as chess, parsing, visual processing, face recognition). Α-Life systems are often, by comparison, architecturally simple, populated in a homogeneous structure by model individuals with (initially) very simple internal structures. In an AI system, the programmer maintains control of as many events in the computational process as possible, little or nothing being left to chance; in an Α-Life system, the programmer defines boundary conditions for the evolutionary processes being modelled, and "sits back" to watch the interplay of random currents. AI typically models the behaviour of a single agent, whereas Α-Life typically models multi-agent systems, in which many individuals interact in evolving populations. Α-Life, being concerned with complex adaptive systems in general, is broader in its scope than AI, which is restricted to systems to which one can plausibly attribute intelligence. Some classic examples of (relatively easily understood) seminal, though not necessarily typical, work in the spirit of Α-Life are Conway's "Game of Life" (Garfinkel 1983; Seife 1994) and Ray's "Tierra" program (Ray 1991a and b). Both these systems illustrate how the application of very simple principles over many "generations" can give rise to entities of remarkable complexity. Many of the evolved entities, moreover, show behaviour that can be interpreted as lifelike. Conway's Game of Life is played out by shapes defined by shading squares on graph paper. These change their outline every cycle by simple rules dictating whether squares adjacent to shaded squares get shaded or not, and whether shaded squares stay shaded. Some simple configurations oscillate between two shapes; some shapes effectively glide across the graph paper by changing like a snail's foot. Still more complicated configurations spawn such "gliders", giving rise to waves of offspring shapes, and some configurations are capable of self-replication. All of this is a surprise, or was to its fascinated discoverers (see Levy 1992 and for accounts). Ray's Tierra system grows whole life-like eco-systems in the guts of a computer. Original minimally complex structures are defined in terms of con-

Artificially growing a numeral system

9

figurations of addresses in core memory. Individual instantiations of such structures are "nourished" by the consumption of computing time, for which they have to compete by carrying out certain actions. If they don't get enough nourishment, they "die", that is they are removed from the arena of the computation. These "organisms" are also capable of self-replication, with the possibility of random mutations affecting the definition, in terms of core memory configurations, of the offspring. Allowed to run for many thousands of generations, Ray's Tierra systems evolve populations of diverse "species", which resemble ecological types in nature, such as parasites and predators. Every so often, there can be a wave of mass extinction, parallel to that evidenced by the real fossil record. The Tierra organisms are reminiscent of computer viruses, the most familiar form of artificial life. The interest in such work, and in the many more specialized and perhaps less "toy" projects that have succeeded them, is in general in the emergence of complexity and order, in often in particular in the modelling of the evolution of complex natural systems by basic Darwinian principles from simple beginnings. The thought inspiring the present work is that (the grammars of) languages are complex naturally occurring systems, and one wonders to what extent techniques from Α-Life research can be used to model their growth. The differences between AI and Α-Life sketched above are no doubt starkly simplified, but they give a true flavour of the methodological differences between the two approaches. One might further guess that whereas AI is more "application-driven" (good AI vision systems can be incorporated into useful robots; good parsers can be incorporated into useful machine-assisted translation programs), Α-Life is more "curiosity-driven". To an extent this is true, but it is now being discovered that some practical problems are so complex that broadly evolutionary, self-organizing, styles of system, in which programs are left to evolve their own solutions, are beginning to be viable in applied fields. One particular technique in the wider Α-Life armoury is Genetic Algorithms (GAs). See Koza (1992), Davis (1991), Goldberg (1989) and Holland (1975) for introductions to Genetic Algorithms. GAs mimic the biology of sexual reproduction. In sexual reproduction, the genes of both parents are randomly mixed; if there is much difference between the parents' genes, the genotype of the offspring will differ from that of both parents and will probably be a new genetic mixture unique in its population. The offspring will also differ from others in the population in the probability of its surviving and reproducing.

10

James R. Hurford

Evolving a working computer program by GA involves defining: • Ordered sets of basic genetic building blocks, which can be small scraps of computer code (roughly analogous, say, to amino acids); • A reproduction process, by which these elements are mixed into new sets; such assemblages of computer code could by chance form little subroutines of working programs which do something (analogous to synthesizing proteins). • Possible random mutations to the system's "genes"; • A fitness function determining the level of success achieved by any particular set of the basic building blocks at some task fixed by the programmer (analogous to an organism built of proteins surviving in a given environment). Clearly, in the expression "Genetic Algorithm", the term genetic is used metaphorically. The similarity between GAs and biological genetics is formal. GA's work because they exploit the same basic mechanisms of selection as biological evolution; the mechanisms themselves, abstracted away from the material on which they operate (say DNA), and the environments in which they are embedded (say the physical world), can be applied to try to model the evolution of any kind of complex adaptive system. It should be clear that GAs may not only model biological evolution, but many different manifestations of evolution, including, for example, the cultural evolution of social conventions. The key elements are selection, innovation by random mixing of the basic "genetic" units (whatever they may be, e.g. genes or memes), other random innovation (e.g. by biological mutation or small individual acts of cultural invention), faithful transmission of the "genetic" units between generations, and "superfecundity" (i.e. having a wide range of choices from which to select). The experiments described later in this paper apply GAs to grammars, sets of lexical entries and phrase structure rules, which progressively evolve to express a greater range of meanings (numbers in this case) with maximal coverage, minimal redundancy, and maximal economy of expression.

Artificially growing a numeral system

11

3. Why? What's the point? If a computational system can be host to the evolution of naturalistic grammars, resembling the grammars of known languages, without the rules being directly invented and simply written in by the programmer, then presumably the system must in some sense be emulating the forces which gave rise to the real grammars of languages. The cautionary phrase "in some sense" is to be emphasized. Objects built of metal may always lack some of the properties of objects built of protein, but neither should possible similarities and parallelisms be ignored. If naturalistic grammars can be evolved in a computer system, the parameters of the system in which this is done may be argued to shed light on the real-world conditions in which real grammars evolved. The grammars of individual languages are not themselves biological objects. The conditions that give rise to grammars are in part biological and in part social. Grammars are shaped both by the innate biological capacities of humans and by the pressures of the social environment. Some of the parameters of a computer system in which life-like grammars arise could be plausibly interpreted as mirroring bio-psychological capacities of individuals; and other parameters can plausibly be seen as parallel to social forces operative in the "Arena of Use" 1 . If, however, repeated attempts fail to construct an artificial system in which grammars simply evolve, then there would be mounting evidence that natural grammars do not arise merely by the interaction of selection, random innovation and faithful transmission across generations. What might an alternative source of grammars be? Presumably something like a creationist account is the alternative to evolution, some kind of deliberate, one-off, large scale invention, masterminded perhaps by some genius. Such an account would see the grammar of a language as something analogous to Esperanto or perhaps Euclid's geometry 2, a whole integrated system springing fully formed from the brow of its inventor, and preserved but scarcely improved by succeeding generations over millennia. The issue addressed could be expressed as: did humans (somewhat deliberately) make language? or did language just grow in humans (and human societies)? The more one tries to flesh out any non-evolutionary explanation of how languages got to be the way they are, the less plausible any creationist or macro-inventionist account seems. Yet on the other hand, the evolutionary style of explanation, which holds that (the grammars of) languages evolved gradually into their present shapes, itself stands sorely in need of fleshing out. It remains to be demonstrated in detail that there could be plausible sets of

12

James R. Hurford

conditions from which classical evolutionary mechanisms produce grammars of the familiar sort. Defining and adjusting the parameters of a GA in which naturalistic grammars evolve provides a strictly disciplined framework within which the detailed conditions giving rise to linguistic evolution can be explored. Putting the matter at its least pretentious, all a GA does is emulate a fitness-driven random search of some space. The search space is defined by the properties of the basic genetic units, and the more or less complex configurations into which they may be combined. The strategy guiding the search is defined by the fitness function which selects the more promising configurations for "survival" and "reproduction" with each generation. A specific example, simulating the growth of numeral systems, will be provided in the next sections, after which some due reservations about the complete appropriateness of GAs to the linguistic case will also be set out.

4. Natural numeral systems In this paper, we necessarily set rather humble goals, and concentrate on the cardinal numeral sub-grammars of languages. What, then, are the numeral systems of languages like? It suits our purpose to distinguish between two broad types, which I will call "primitive" and "developed"3. Naturally, there are some intermediate cases between primitive and developed, but the broad dichotomy is useful. I will characterize the two types of system in the next two subsections.

4.1. Primitive

systems

Not all languages have a numeral system4. Some languages have quite simple systems, capable of counting only to about '20' or even lower. In primitive systems, the words have not always fully lost their non-numerical meanings. So the word for '5' might also mean "(left) hand"; the expression for '+ 1' might also mean "and another"; the expression for '10' might also mean "man" or "whole" or "finished" or "right hand". In what follows, I will only mention the numerical meanings. In these systems, either all the numeral expressions are monomorphemic (or at least do not contain more than one morpheme with a numerical interpretation), or a relatively low number, such as '2', '3', '4', or '5', is used as a

Artificially growing a numeral system

13

basis of addition (or very much more rarely of subtraction or multiplication). Sometimes, after a base number appears in the counting sequence, it is used for all higher numbers. But this is not always so, and there can be what appears to be fairly random interspersing of morphologically complex numerals with monomorphemic numerals. Examples of the first few numerals in some such simple systems are given below (in some cases, the examples given apparently comprise the whole system). In these examples, a single Arabic digit indicates that the number in question has an arithmetically simple, often monomorphemic, numeral; and where several Arabic digits are given, this indicates an arithmetically complex numeral, typically reflecting addition. Only a single example of each type is given here, but most of the types are not uncommon; more examples of them could be given, and from more than one part of the world. These primitive types are widely documented and discussed in a number of works surveying numeral systems (such as Conant 1923, Hymes 1955, Kluge 1937-42, Lean 1985-86, Menninger 1969, Pott 1847, Salzmann 1950, Seidenberg 1960).

4.1.1. Various uses of '2' as a base Aomie5

1

2

2+1

2+2

5

5+1

Yareba6

1

2

3

2+2

5

5+1

Korafe7

1

2

2+1

4

5

5+1

Hunjara 8

1

2

2+1

2+1+1

Fuyuge ("Mafulu") 9

2+1 2+2 2+2+1 2+2+2 1 2 2+2+2+1 2+2+2+2 2+2+2+2+1

4.1.2. '3' as a base Mawae10

1

2

3

3+1

5x1

14

James R. Hurford

4.1.3. '4' as abase Tunisian Arabic egg-counting11 1 4+3 ...4x4

2 4x2 ...5x4

3 4 4+1 4+2 4x2+1 4x2+2 4x2+3 3x4

4.1.4. '3'-and '4'-based Motu12

1 7

2 2x4

3 2x4+1

2x3

Roro 13

1 2 2x3+1 2x4

3 2x4+1

2x3

Keapara, Hula dialect14

1 2x4-1

3

2x3

2 2x4

10-1

Hymes (1955:27) calls this type "pairing"; this pairing type, and several of the other types illustrated above and below, are found in Athabaskan languages, according to Hymes' survey.

4.1.5. '3'-, '4'-and '5'-based Ekoi (Cameroun) 15

1 4+3

2 4+4

3 4+5

3+3

4.1.6. '5'-and '6'-based Onjob16

1 2 3 4 5 5+1+1 5+1+2 5+1+3 5+1+4

5+1

4.1.7. '2'-, '5'-and '6'-based Miskito17

1 2 3 2+2 5+1+1 5+1+2 5+1+3 5x2

5 5+1 5x2+2 5x2+3

Artificially growing a numeral system

15

4.1.8. Quinary systems A purely '5'-based system is referred to as "quinary". Such systems are very common, being found in almost all quarters of the world, and taking the shape: 1 2 3 4 5 5+1 5+2 5+3 5+4 10.

4.2. Developed systems The most familiar type of numeral system in better known languages is decimal, and sometimes also partly vigesimal. This canonical type has the following characteristics: • Single words for Ί ' - Ί Ο ' , • Use of addition to '10' for '11-19', • Use of multiplication by '10' (or '20'), (and addition) for '20-99', • Single words for higher bases, typically '100',' 1000', and sometimes also '20'. Examples of such systems are very familiar. Further characteristics, common to both primitive and developed types of system, are: • Complete coverage to some limit - no gaps (although the sporadic use of subtraction suggests there can be temporary stages of a language in which there may be gaps); • No ambiguity or homonymy (examples of ambiguous numerals are extremely scarce, if they occur at all); • Little, if any, redundancy or synonymy (from a vast set of arithmetically possible combinations for any given number, typically there is only a single well-formed numeral used in the canonical counting sequence. The occasional exception to this generalization occurs, as in paraphrases like English one thousand one hundred versus eleven hundred)', • Recursion - expressions for higher numbers typically contain expressions for lower numbers nested within them;

16

James R. Hurford

• Packing Strategy - the recursive possibilities are severely constrained by a principle to the general effect that one builds on the highest valued expression available (See Hurford 1975, 1987, for details and discussion).

5. Growing numeral systems artificially We now bring the artificial techniques of Genetic Algorithms to bear on this diverse range of naturally occurring systems. The basic idea of GAs is applied here to grammars; the genetic units making up each "organism" are rules, which may be specifications of lexical items or rules of syntactic combination; each grammar is a set of such rules. The proposal is to selectively "breed" grammars of numeral systems from an initial randomly generated set of grammars. The interest is in seeing whether the range of systems described above simply emerges after systematic selection of grammars over many generations. The elementary components of the system, by which rules are defined, are: • A vocabulary of arbitrary monosyllables (e.g. ba, be, bi, ca, ce, ci, da, de, di, . . .);

• A set of semantic primitives. These are the concepts of certain small numbers— Ί ' - '10', which it is assumed are accessible to the mind independently of the existence of a counting system. It seems right to stipulate that some of these numbers are more accessible to the mind than others. In particular, probably Ί ' , '2' and '3' are ranked highest in order of accessibility, followed perhaps by '5' (a whole hand) and '10' (both hands). '6', '7', '8', and '9' are, intuitively, relatively inaccessible. Various possible weightings for the availability of the semantic primitives were experimented with. Numbers higher than '10' were judged to be inaccessible as semantic primitives, without benefit of some linguistic counting system, and were never included among the primitives. • Basic cognitive operations, such as addition, multiplication and subtraction. These can be grasped from such concrete operations as placing more objects in a pile, making groups of piles, and taking objects out of a pile. Again, these basic operations are not equally accessible to the mind. It was judged that addition is most accessible, and multiplication and subtraction much less accessible. Various quantitative interpretations of these different degrees of accessibility were experimented with.

Artificially growing a numeral system

17

• Arbitrary syntactic categories, here labelled sO, si, s2,. . . In order for any grammatical system to arise, the notion of word-class must be available. There is no presupposition of any particular natural connection between any such arbitrary word class and any particular semantic primitive or cognitive operation. These arbitrary syntactic labels are used for the construction of lexical and syntactic rules. In the simulations, the actual number of available syntactic categories was experimentally varied, ranging between '2' and '5'; clearly there have to be more than one syntactic category for the concept to be useful. An arbitrary rule in the system is generated by taking the following steps, in order: 1. Randomly select a "mother" syntactic category (from the set { sO, si, s2, . . . } ). 2. Decide, by the random toss of a computational coin (which may be biased), whether to generate a lexical item or a syntactic rule. 3. If a lexical item is to be generated, select a random semantic primitive, and a random monosyllable. Together, the "mother" category, the semantic primitive and the monosyllable constitute the randomly assembled lexical entry. For example, s i -> fa (3) would be a lexical entry for the word fa, giving its meaning as the number '3', and assigning it to the syntactic category si. 4. If a syntactic rule is to be generated, select two random syntactic categories (which can be the same), and a random cognitive operation. An example of such a randomly generated syntactic rule is: s2 —> s i s2 (addition) This rule states that a constituent of category s2 can be constructed from a pair of elements, of categories si and s2, respectively, and that the whole is to be interpreted by the arithmetical addition of the values of the parts. In this work, all syntactic rules were binary branching. No extra-linguistic significance being associated with the syntactic categories, which are merely a kind of "glue" for building syntactic rules, the choice of syntac-

18

James R. Hurford

tic categories was never weighted in any way. The choice of cognitive operation was weighted in some, but not all, simulations, reflecting the possible greater accessibility of addition than, say, subtraction. A grammar is a set of lexical items and phrase structure rules, of the sort just illustrated above. For numeral systems at least, this simple view of what constitutes a grammar is arguably adequate. A random grammar is a random-sized set of randomly generated lexical items and rules. The core routine of the simulations carried out consisted of the following steps: 1. Generate a large set of random grammars, as defined and illustrated above. 2. Select the "fittest" grammars as "parents" of the next generation; 3. "Breed" a new generation of grammars, perhaps with some "mutation"; 4. Return to step 2 and keep recycling. Implementing this system in detail calls for a number of decisions to be made. It is a virtue of the computational approach that it forces one to specify explicitly many parameters of grammars which are seldom, if ever, contemplated by descriptive linguists, although they are nonetheless real, and in no way artifacts arising simply out of the approach. In some cases, one can make a plausible intuitive guess at an appropriate value for some parameter, and hold it constant; in other cases, one can use the computational system to experimentally manipulate the values of parameters, within an intuitively plausible range. The relative success with which such experimental settings lead to naturalistic grammars can be taken as evidence pointing to corresponding values in the real-world conditions in which natural grammars evolved. In yet other cases, unfortunately, the settings of parameters are determined by computational convenience, although in no case did it seem that such settings were very counter-intuitive. The more salient parameters involved in the current simulations are mentioned below. The first parameters mentioned are linguistic, having to do with the nature of grammars; the ones mentioned later are "population-genetic", having to do with the particular mechanics of GAs.

Artificially growing a numeral system

19

SOME TWEAKABLE PARAMETERS • Size of initial grammars. An upper limit of 50 rules (lexical and syntactic totalled) was put on the initial population of random grammars. The initial grammars varied randomly in size between 1 rule and 50. • Lexicon to syntactic rule ratio. Different ratios of lexical items to syntactic rules were experimented with. In some simulations, the probabilities of random rules being lexical or syntactic were equal. (See step 2 in the rule generation procedure above.) In other simulations, the expected ratio of lexical rules to syntactic rules was as high as 20:1. • Weighting of semantic primitives. In some simulations, all the numbers from Τ to '10' were equally weighted. This implies equal probabilities for lexical items with meanings from Ί ' to '10'. A range of other weightings was tried, in which the numbers ' 1', '2', '3', '5' and ' 10' were given greater probability of having their own lexical items. • Weighting of semantic operations. Each syntactic rule generated is associated with a particular arithmetic operation, such as addition, multiplication, or subtraction. Sometimes these operations were assigned to rules with equal probability. In other cases, addition was favoured over subtraction and multiplication. • Fitness function. This is a crucial aspect of the simulation. Here, one has to try to make plausible assumptions about what factors might make one numeral grammar preferable, to a group of human users, over another. -

Greater coverage is, other things being equal, presumably desirable; a counting system that reaches to Ί 0 0 ' is more useful than one that reaches only to '20'.

-

The absence of gaps in the counting sequence is presumably also desirable, as natural numeral systems do not have gaps. But the existence of subtraction in a few natural systems shows that a fitness function that makes a single unbroken sequence a mandatory property for "survival" is too strict. Presumably in the evolution of natural systems with subtraction, the higher numeral (e.g. '10') pre-existed the lower one(s) (e.g. Ί 0 - Γ , Ί0-2').

-

Lack of redundancy is also typical of natural numeral systems, and

20

James R. Hurford

some penalty for providing several expressions for the same number should be built into the fitness function. Again, however, this cannot be an absolute prohibition, as one occasionally finds a natural numeral system with more than one way of expressing a particular number. -

The complexity of numeral expressions is relevant. Other things being equal, a system with shorter, or less grammatically complex, expressions must be preferable.

• Number of initial grammars. This is largely a matter of computation convenience. For a GA to work well, there needs to be a reasonably large number of initial grammars, providing an original "gene pool" with enough variants in it for there to be a chance of it containing most of the required types of rules and lexical items. Obviously, though, these rules are not at the beginning assembled into coherent sets (grammars) that generate naturalistic sequences of numeral expressions. In these simulations, the number of initial grammars varied from 25 to 100. • Percentage selected. The number of grammars taken each generation as parents of the next generation needs also to be set. In these simulations, the number of grammars selected as breeding stock varied between 5 and 10. • Mutation rate. This is the rate at which, during the mimicked evolutionary process, random rules were added to grammars or deleted from them. Too high a mutation rate prevents the evolutionary process from settling down to stable solutions, as the gene pool is excessively stirred. Too slow a mutation rate can lead to convergence on solutions which would not, with a higher mutation rate, be stable. Given that the term "Genetic Algorithm" is no more than a metaphor for the computational process outlined here, there is no need, of course, to worry about any biological verisimilitude in setting the mutation rate. These are not biological mutations, but just random innovations in the search space. • Evaluation methods. Given some fitness function, a method needs to be determined for applying it to grammars. Several alternatives are conceivable: -

From the grammar. In this syntax-driven approach, one evaluates a set of products of the grammar, produced either exhaustively or randomly. Having used the grammar to generate a set of expressions, one checks

Artificially growing a numeral system

21

to see how this set is valued in terms of the fitness function, i.e. what coverage it achieves, with how many gaps, with how much redundancy, and so on. -

From the meanings. In this meaning-driven approach, one takes a set of "target meanings", and tests whether the grammar generates expressions for them, and if so, with what redundancy, complexity of expression, and so forth.

6. Results A number of simulations were run, under different conditions. These are described in order of growing complexity.

6.1. Coverage

versus lack of redundancy:

A simple

lesson

In this simulation, the grammars consisted only of lexical entries. There were no phrase structure rules. This simple experiment teaches us a lesson about the fitness function which applies to all more complex simulations. Which should be paramount in the fitness function, coverage or lack of redundancy? Putting it in terms of real human numeral systems, does the optimal system (i.e. that type found most commonly in human languages) • favour coverage even at the expense of massive redundancy? • favour lack of redundancy, even at the expense of gaps in the system (lack of coverage)? • favour some kind of balance between coverage and lack of redundancy? Repeated simulations were run in which the fitness function always preferred a grammar generating expressions for the most numbers, regardless of how many such expressions it generated for each number. Only when the coverage of two grammars was equal, was relative lack of redundancy invoked to discriminate between them, in favour of the grammar generating fewer expressions. With this fitness function, there was never any convergence on a naturalistic numeral lexicon, with just one word for each of the numbers from Ί ' to Ί 0'.

22

James R. Hurford

Another set of simulations was run in which the fitness function was the reverse of that just described. In this case, lack of redundancy, rather than coverage, was paramount. The fitness function always preferred a grammar which had as few extra expressions as possible for each number, even if some other grammar actually provided expressions for more numbers. Only when two grammars had equally few expressions was the coverage of meanings invoked, so that the grammar with the greater coverage was, then, preferred. In this case also, after many runs, there was no convergence on a naturalistic numeral lexicon with just one word for each number up to '10'. On brief reflection, it becomes obvious why these simulations do not work to produce language-like results. Large grammars, containing more lexical items, are more likely to cover more meanings; just as a longer sequence of throws of a die is more likely to have at least one throw landing on each possible value than a shorter sequence of throws. If the selection process always favours coverage, large grammars will be selected which have enough lexical items to cover all the possible values. Almost inevitably, such grammars will be massively redundant. Conversely, if lack of redundancy is selected for, the preferred grammars will be those which manage to express meanings with only one expression each. Small grammars, containing fewer lexical items, are more likely to avoid generating several expressions for the same number; just as a shorter sequence of throws of a die is more likely to avoid repetitions than a longer sequence. But the price of selecting smaller grammars is that they almost inevitably lack some coverage of the set of possible meanings. Now a third set of simulations was run, in which the fitness of a grammar was calculated from a combination of coverage and lack of redundancy. Simply, and no doubt crudely, the fitness of a grammar in this case was inversely proportional to the sum of the gaps in the number sequence and the number of redundant expressions. To give an example, imagine a grammar which happened to provide words for numbers as follows: 1

2

do

mi

re

fa so

3

4

5 la

6

7 ti do

8

9

10

Artificially growing a numeral system

23

Here, there are 6 numbers unexpressed by any word (i.e. 6 gaps), and 4 superfluous words. The combined gap-redundancy score is thus 6+4=10. Grammars with lower gap-redundancy scores were favoured in the simulation. In these runs, there was always convergence to a naturalistic lexicon with just one word for each number from ' 1' to '10'. Sometimes this convergence was achieved very quickly, once in as few as 5 generations. In other runs, convergence took as long as 200 generations. The combined gap-redundancy fitness function took no account of homonymy, and occasionally a simulation converged on a numeral lexicon in which a single monosyllable happened to express more than one number. This homonymy is, as far as I know, never found in real numeral systems. Further simulations should also build homonymy-avoidance into the fitness function. As the method for doing this is straightforward, this line of investigation was not followed. (In the grammars that emerged in later simulations, the occasional instances of homonymy were tolerated; nothing rides on this, as far as I can see.) In the more complex simulations described below, the fitness function was always some combination of coverage and lack of redundancy, sometimes with other factors as well.

6.2. Effects of other variables In further experiments with versions generating only simple lexicons, the situation after initial convergence on optimal fitness was studied, to see whether this fitness is maintained for long stable periods. Both mutation rate and population size were "tweaked". Clearly, too fast a mutation rate (e.g. every cycle) stirs up the pool of grammars too much, and optimal fitness is only reached sporadically, with many lapses. But also important is population size. With 20 initial grammars and 5 grammars selected each cycle, there was frequent lapsing from optimal fitness over time, with few periods of stable optimal fitness. With 100 initial grammars and 10 grammars selected each cycle, the situation improved considerably, with long stable periods of optimal fitness, but still the occasional lapse, which was almost always immediately recovered from. With this population size, a mutation rate of 1 mutation per 15 grammars bred was, impressionistically, slightly more stable than a rate of 1 per 5.

24

James R. Hurford

The lesson one quickly learns, if one did not know it before, is that there is an intractably large number of possible variables, both linguistic and non-linguistic, which affect the evolution of grammars in this framework. By no means all of the possible variations are mentioned here, for readability's sake. Anyone attempting to replicate these experiments will be (or become) aware of the vast number of theoretical possibilities. It is conceivable that the evolution of real grammars is in fact subject to a small number of real variables. The computational approach described here might in principle lead to the discovery of some very parsimonious "magical combination" of settings of a few variables, which give rise to naturalistic grammars. Alternatively, lack of progress in finding such a combination might suggest the conclusion that the evolution of real-life grammars is subject to a great many chance vagaries, both psychological and social.

6.3. "Peano-type" results In the next series of experiments, the random grammars could contain both lexical items and simple phrase structure rules, as described above. In the weightings for these runs, the random generation of a syntactic rule was made as probable as the random generation of a lexical item. Many of these simulations converged on numeral systems of a mathematically very elegant type, but a type which is quite unnatural, in the sense of being unrepresentative of natural languages. The systems arrived at were highly economical of both lexical and syntactic resources. They also achieved complete coverage of the number sequence up to any limit, sometimes with no cost in redundancy. One such extreme outcome was:

si

mi (1)

s i —> s i s i (addition)

This grammar generates a single structure for the number Ί ' , as in

Artificially growing a numeral system

25

sl

mi and a single structure for the number '2', namely sl /

\

sl

sl

mi

mi

but at '3', (structural) redundancy sets in, with two possible structures for '3', namely sl

/ sl /

sl

\

/ sl

\

sl

sl

/ \

\

sl

sl

mi

mi

mi

sl

sl

mi

mi

mi

The redundancy increases dramatically as one proceeds to higher numbers. In the simulation that gave this result, only binary branching phrase structure rules had been allowed. Without the option of unary, or non-branching, phrase structure rules (or of binary rules which can include a simple lexical item, rather than a syntactic category, on the right hand side), there exists no grammar which can avoid this burgeoning redundancy.

26

James R. Hurford

With the possibility of unary, or non-branching, phrase structure rules, a non-redundant grammar exactly parallel to Peano's axiomatic system for defining the number sequence can arise, as follows: s i —> mi (1) s2

si

s2 —» s2 s i (addition) This grammar generates a single structure for each number, as shown below for the numbers T , '2', and '3': si

s2

s2

/

/ \ mi

s2

si

s2

\ si

/\ si

mi

s2

mi

si

si

mi

mi

mi

For the short term memories of humans, such a system is highly dysfunctional, and even self-defeating. The value of a counting system is in defining compact expressions which compress information. The psychological burden of counting the instances of the monosyllable mi in an expression for a higher number in this system renders it unusable. Such systems do not occur in spoken human languages. (But note the beginnings of such a system in written Roman numerals, I, II, III; such iconic representations of numbers can also be found in the ideographic cuneiform writing of the Sumerians and in Egyptian hieroglyphs.)

Artificially

growing

a numeral system

27

Such Peano-style grammars could result from various different combinations of circumstances. Factors which favour the evolution of such unnatural Peano-type solutions include: • Fitness defined primarily as economy of storage of grammar, i.e. the fittest grammar has the fewest rules and lexical items; • Relatively favorable weighting for combinatory syntactic rules, as opposed to lexical items.

6.4. Evolution of "primitive" systems One combination of variables was found to give rise to many systems closely resembling the natural "primitive" systems described earlier. Recall these were typically found in relatively small, and relatively isolated communities. In these runs, a grammar's fitness was defined as the highest number it could "count to" with no gaps, and not too much redundancy. Here, the algorithm to determine fitness is, basically: count from Ί ' upwards, and stop at the first gap, or where there are more than 2 (an arbitrarily chosen limit on redundancy) expressions for some number. The expected ratio of syntactic rules to lexical items was set at 1:8, and the number of abstract syntactic categories permitted was 2. Starting with a population of 100 random grammars, the fittest 10 grammars were selected as parent grammars for the next generation. These parents bred (all with each other) a generation consisting of a further 100 offspring grammars. From this new generation, the fittest 10 were again selected as parents, and so on. The simulations ran for 1000 generations each, at which stage the fittest (usually the only) remaining grammar was inspected to see what numeral expressions, up to '20', it generated. Of the 164 runs, over half (86) yielded lexicon-only solutions, summarized as follows: Words for numbers

Instances of this solution

1 1 1 1 1 1

5 52 21 4 3 1

2 2 2 2 2 2

3 4 5 6 3 4 5 6 7 3 4 5 6 7 8 3 4 5 6 7 9 (gap at 8) 3 4 5 6 7 8 9 3 4 5 6 7 8 10 (gap at 9)

28

James R. Hurford

Such simple lexicons may perhaps be regarded as the basis on which somewhat more elaborate systems, with some syntactic rules, are founded. In the remaining 78 runs, all solutions in some way resembled the natural primitive systems illustrated earlier. Solutions included base-'2', base-'3', base-'4\ base-'5', base-'6', and various mixed base systems. Some examples are given below (note the occasional gaps and redundancies in these systems): BASE-'2' (The first five expressions here are the same as those of Yareba, cited above.) 1 2 3 4 5 6 6 8

= = = = = = = =

1 2 3 [22] 5 [2 [2 2]] [[2 2] 2] [[2 2] [2 2]]

MIXED '2'- '5'- and l 6'-BASE (The first four expressions here are the same (in mirror image) as those of Hunjara, cited above.) 1 = 1 2 = 2

3 4 5 6 7 8 8 8 9 9 12 13 13 14

= = = = = = = = = = = = = =

[1 [1 5 6 [5 [1 [5 [6 [1 [6 [5 [5 [6 [6

2] [12]]

2] [5 2]] [12]] 2] [6 2]] [12]] [5 2]] [6 2]] [5 2]] [6 2]]

Artificially growing a numeral system

29

MIXED '2'- '7'- and '9'-BASE (The first six expressions here are the same as those of Fuyuge ("Mafulu"), cited above.) 1 2 3 4 5 6 7 8 9 10 10 10 11 12 12 13 14 14 15 16 16 17 17 18 18 19 20 20

= = = = =

= = = = = = = = = = =

= = =

= = = = =

= = = =

1 2 [2 1] [2 2] [2 [2 1]] [2 [2 2]] 7 [7 1] 9 [2 [7 1]] [7 [2 1]] [9 1] [7 [2 2]] [2 [9 1]] [9 [2 1]] [9 [2 2]] [2 7] [7 2] [7 [7 1]] [2 [2 7]] [2 [7 2]] [7 [9 1]] [9 [7 1]] [2 9] [9 2] [9 [9 1]] [2 [2 9]] [2 [9 2]]

MIXED BASE-'3' and-'10' 1 2 3 4 5 6 7

= = = = = = =

1 2 3 [13] [2 3] [3 3] [[1 3] 3]

30

James R. Hurford 8 9 10 11 12 13 13 14 14 15 15 16 16 16 20

= = = = = = = = = = = = = = =

[[2 3] 3] [[3 3] 3] 10 [1 10] [2 10] [3 10] [10 3] [[1 3] 10] [[1 10] 3] [[2 3] 10] [[2 10] 3] [[3 3] 10] [[3 10] 3] [[10 3] 3] [10 10]

MIXED '3'- '4'- and '5'-BASE (The first nine expressions here are the same as those (or their mirror images) of Ekoi, cited above, labelled by Hymes the "pairing" type.) 1 2 3 4 5 6 7 7 8 8 8 9 9 10

= =

= —

= =

= —

= = =

= =

=

1 2 3 4 5 [3 3] [3 4] [4 3] [3 5] [4 4] [5 3] [4 5] [5 4] [5 5]

BASE-'4' (The first seven expressions here are the same as the Tunisian egg-counting system, cited above.) 1 = 1 2 = 2

Artificially growing a numeral system 3 4 5 6 7 8 9 10 11 12

= = = = = -

= =

= =

3 4 [14] [2 4] [3 4] [4 4] [[1 4] [[2 4] [[3 4] [[4 4]

31

4] 4] 4] 4]

MIXED BASE-'4' and-'8' (This solution (or its mirror image, in which, for example, 11 = [[3 4] 4]) occurred 8 times. The first seven expressions are the same as the Tunisian egg-counting system, cited above.) 1 2 3 4 5 6 7 8 9 10 11 12 16

= = = = = =

= = = —

—

= —

1 2 3 4 [4 1] [4 2] [4 3] 8 [4 [4 [4 [4 [4 [4 [4 8] [4 [4

1]] 2]] 3]] 8]]

BASE-'5' (quinary/decimal) (The first ten expressions here are the same as those of typical quinary systems, mentioned above.) 1 2 3 4 5 6 7

= = = = = = =

1 2 3 4 5 [5 1] [5 2]

32

James R. Hurford

8 9 10 11 12 13 14 15 20

= = = =

= =

= —

=

[5 3] [5 4] 10 [5 [5 1]] [5 [5 2]] [5 [5 3]] [5 [5 4]] [5 10] [5 [5 10]]

'6' 1 2 3 4 5 6 7 8 9 10 11 13 14 15 16 17

—

-

= =

= = -

—

=

= —

—

= = = =

1 2 3 4 5 6 [6 1] [6 2] [6 3] [6 4] [6 5] [6 [6 1]] [6 [6 2]] [6 [6 3]] [6 [6 4]] [6 [6 5]]

The similarities between these artificially generated systems and the "primitive" systems described earlier, are striking.

6.5. Systems get stuck in local

optima

It is noteworthy that, of all the artificial grammars generated in the runs described above, not a single pure decimal system emerged. We will explore why this is so. Recall that what is simulated here is a quasi-social process whereby grammars are formed from a pool of variant lexical items and syntactic rules which

Artificially growing a numeral system

33

are somehow ambient in the community. Over time, the pool of available lexical items and rules becomes restricted so that only one grammar may be constructed from them, and this grammar is the "fittest" that the community has happened upon. After that point of convergence, only random "mutations" to the pool of lexical items and rules may disturb the situation. And such mutant lexical items or rules will only persist in the pool if they happen to fit in with existing rules and lexical items to form a grammar which is fitter than the one previously converged upon. As readers familiar with any complex adaptive system will know, there can exist many "local optima", that is solutions to a problem (such as finding an effective counting system) which are fitter than their close neighbours in the possibility space, but not the fittest solutions overall, in any global sense. It is clear that the artificial systems illustrated above are local (near-)optima, in this sense. In other words, any small mutation to one of the grammars arrived at tended very strongly to produce a grammar that was less fit. If a particular grammar converged upon by the algorithm is at a local optimum, then any random mutation applied to it, in the form of a randomly added or deleted syntactic rule or lexical item, will result in a grammar less fit (by whatever definition of fitness was used to arrive at the tested grammar) than the tested grammar. If such random mutations do not always, but nevertheless tend strongly to, result in less fit grammars, then the tested grammar is near a local optimum. The 13 first grammars converged upon by the algorithm decribed above were tested for local optimality in this way. To each of these grammars, 100 random mutations were applied (not serially, but always starting with the tested grammar). Thus for each tested grammar, 100 one-step mutants were produced. The results for the 13 tested grammars are aggregated below. 1 mutation produced a less fit grammar 1 mutation produced an equally fit grammar 1 mutation produced a fitter grammar

in 444 cases. in 776 cases. in 80 cases.

Thus the probability of a random mutation producing an improvement to one of the converged upon grammars is low, 0.062 (80/1300). These grammars are at or near local optima. To test whether the algorithm would recognize a globally optimal solution if it saw one, another series of runs was carried out in which the initial population of 100 random grammars was "seeded" with a small number (10) of copies of a grammar which (I thought) was globally optimal. There were no nasty surprises here, and the algorithm always quickly converged, after just

34

James R. Hurford

one or two cycles, on the globally optimal solution. The algorithm did however administer one sobering little lesson, in that it discovered a better grammar than the one with which I had seeded it. My "optimal" grammar was not quite optimal, as it allowed both 1-deleted and non-1-deleted variants o f ' 10' (i.e. 10 = 10 and 10 = 1x10), which occurred in all the -teen expressions, giving redundancy.

6.6. Evolution of "developed" systems The algorithm explored here never succeeded, in a limited number of trials, in converging on a complete and pure decimal numeral system such as is the basis of the counting systems of most of the world's major languages. On several occasions it got close. One of its better efforts was as follows: LEXICON: Value

Syntactic Category

Word

1 2 3 4 5 6 7 8 9 10

sO sO sO sO sO sO sO sO sO s2

re

SYNTACTIC RULES sO -> s2 sO (Addition) si —> s2 s2 (Addition)

ga ko se β ge ci ti le di

Artificially growing a numeral system

35

COMPLEX EXPRESSIONS GENERATED 11 = 10+1 12 = 10+2

13 = 10+3 14 = 10+4 15 = 10+5 16 = 10+6

17 = 10+7 18 = 10+8

19 = 10+9 20 = 10+10

As can be seen, this system is like a developed decimal system up to '20', but goes no further. Another near miss (though at first blush it may not look like it) is as follows: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

(GAP) = 2 = 3 = 4 = 5 = 6 (GAP) = 8 = 9 = 10 (GAP) = 10+2 = 10+3 (GAP) (GAP) = 10+6 (GAP) = 10+8 = 10+9 = 2x10 (GAP) = 2x10-

James R. Hurford

23 = 2x10+3 24 (GAP) 25 (GAP) 26 = 2x10+6 27 (GAP) 28 = 2x10+8 29 = 2x10+9 31 (GAP) 32 = 3x10+2 33 = 3x10+3 34 (GAP) 35 (GAP) 36 = 3x10+6 37 (GAP) 38 = 3x10+8 39 = 3x10+9 40 = 2x2x10 41-59 (GAPS) 60 = 2x3x10 3x2x10 6x10 (REDUNDANCY) 61 (GAP) 62 = 6x10+2 63 = 6x10+3 64 (GAP) 65 (GAP) 66 = 6x10+6 67 (GAP) 68 = 6x10+8 69 = 6x10+9 70-79 (GAPS) 80 = 8x10 81 (GAP) 82 = 8x10+2 83 = 8x10+3 84 (GAP) 85 (GAP) 86 = 8x10+6 87 (GAP) 88 = 8x10+8 89 = 8x10+9 90 = 9x10 3x3x10 (REDUNDANCY) 91 (GAP)

Artificially growing a numeral system 92 93 94 95 96 97 98 99

37

= 9x10+2 = 9x10+3 (GAP) (GAP) = 9x10+6 (GAP) = 9x10+8 = 9x10 + 9

The main problem with this system is the lexical gaps for values Ί ' and ' 7 \ and the gaps caused by the fact that the words for '4' and '5' are not of the appropriate syntactic category to fit into the higher-valued syntactic constructions. If those gaps were filled with appropriately categorized lexical items, the system would be close to a modern decimal system. But note the hint of a vigesimal system in the various ways of expressing '60'.

7. Conclusion In the artificial approach described here, a numeral system resembling the dominant type found in the world's languages can emerge only very rarely, though it is not actually impossible. On the other hand, "suboptimal systems", resembling the systems found in a number of isolated language communities throughout the world, emerge frequently. If a developed system is artificially imposed on a community with such a suboptimal system, the developed system is quickly adopted. If the approach outlined here has any verisimilitude, we may conclude that the natural primitive systems have an internal stability but are highly vulnerable to invasion (through language contact) by the developed decimal system which prevails throughout much of the world. This fits very well with the facts of language contact; in fact, "exotic" numeral systems as have been shown above are typically abandoned in favour of a "modern" decimal system. The simulations here show that this replacement may be due to some kind of real linguistic superiority (in coverage, lack of redundancy, and suitability to specifically human memory constraints) of the decimal system, and not just a consequence of the superior economic or military power of the invading culture.

38

James R. Hurford

Notes * Thanks to Simon Kirby and Matthew Aylett for helpful and stimulating discussion of the topic. None of it is their fault. 1. See Hurford (1990, 1994) and Kirby (1994, 1996). 2. Pretending, for the sake of the analogy, that Ludwig Zamenhof did not model Esperanto closely on existing natural languages, and that Euclid did not stand at the end of a long tradition, itself the production of evolution. 3. No insult is implied to the speakers of languages with primitive numeral systems. It will be apparent that the descriptive terms adopted are not apt. 4. For example, many Australian languages have no numeral system. See Dixon (1980: 107-8). 5. Austing — Upia (1975: 523-24), quoted in Lean, vol. 5: 40. 6. Weimer — Weimer (1974), quoted in Lean, vol. 5: 52. 7. Lean, vol. 5: 38. 8. Lean, vol. 5: 28. 9. Williamson (1912: 228-229), quoted in Lean (vol. 7: 42). Some expressions in this counting system have alternatives. 10. Smith (1984: 253), quoted in Lean (vol. 5: 16). 11. Informant, Mukhtaar ben Fraj: These forms were used by the informant's grandmother, and are (or were) used in markets. Friederici (1913, quoted in Lean, vol. 4: 92) mentions a similar '4'-based counting system for yams, taro and coconuts in Bariai (Austronesian). 12. Lean (vol. 7: 67). 13. Lean (vol. 7: 20-22). 14. Lean (vol. 7: 74-76). 15. Cauty (1986: 138). 16. Macdonnell (1917: 171), quoted in Lean (vol. 5: 70). 17. Conzemius (1929: 81-82); these data were verified in fieldwork in 1971 by Ruth Fowlks (personal communication).

References Austing, J. — R. Upia 1975 Highlights of Ömie morphology. In: Τ. E. Dutton (ed.) 1975, 513-598. Brooks, Rodney A. — Pattie Maes (eds.) 1994 Artificial Life IV (Proceedings of the Fourth International Workshop on the Synthesis and Simulation of Living Systems). Cambridge, Mass.: MIT Press.

Artificially growing a numeral system

39

Cauty, Andre 1986 Taxonomie, syntaxe et economie des numerations parlees. Amerindia 11: 87-141. Conant, Levi Leonard 1923 The Number Concept: its Origin and Development. New York: MacMillan and Co. Conzemius, Eduard 1929 Notes on the Miskito and Sumu languages of Eastern Nicaragua and Honduras. International Journal of American Linguistics 5: 57-116. Davis, Lawrence (ed.) 1991 Handbook of Genetic Algorithms. New York: Van Nostrand Reinhold. Dixon, Robert M.W. 1980 The Languages of Australia. Cambridge: Cambridge University Press. Dutton, Thomas E. 1975 Studies in Languages of Central and South-East Papua, Pacific Linguistics, C-29. Garfinkel, Simson L. 1983 The Game of Life on the IBM PC, DR Dobbs Journal of Software Tools 8, 6: 42. Goldberg, David E. 1989 Genetic Algorithms in Search Optimization and Machine Learning. Reading, Massachusetts: Addison-Wesley Publishing Inc. Holland, John H. 1975 Adaptation in Natural and Artificial Systems. Ann Arbor, Michigan: University of Michigan Press. Hurford, James R. 1990 Nativist and functional explanations in language acquisition. In: I.Roca (ed.), Logical Issues in Language Acquisition, 85-136. Dordrecht: Foris Publications. 1994 The study of language systems. Journal of Quantitative Linguistics 1, 1:43-55. Hymes, Virginia Dosch 1955 Athapaskan numeral systems. International Journal of American Linguistics 21, 1: 26-45. Kirby, Simon 1994 Adaptive explanations for language universals: A model of Hawkins' performance theory. Sprachtypologie und Universalienforschung 47: 186-210. 1997 Competing motivations and emergence: explaining implicational hierarchies. Language Typology 1,1: 5-32. Kluge, Theodor 1937-42 I. Die Zahlenbegriffe der Sudansprachen; II. Die Zahlenbegriffe der

40

James R. Hurford

Australier, Papua und Bantuneger; III. Die Zahlenbegriffe der Voelker Amerikas, Nordeurasiens, der Munda und der Palaioafrikaner; IV. Die Zahlenbegriffe der Dravida, der Hamiten, der Semiten und der Kaukasier; V. Die Zahlenbegriffe der Sprachen Central- und Suedostasiens, Indonesiens, Micronesiens, Melanesiens und Polynesiens, published by the author, Berlin. Koza, John R. 1992 Genetic Programming: On the Programming of Computers by Means of Natural Selection. Cambridge, Mass.: MIT Press. Langton, Christopher G. — C. Taylor — J. D. Farmer — S. Rasmussen, (eds) 1991 Artificial Life II (proceedings of the workshop on artificial life held February, 1990 in Santa Fe, New Mexico), Santa Fe Institute studies in the sciences of complexity v. 10. Redwood City, CA: Addison-Wesley Publishing Inc. Langton, Christopher G. et al. (eds) 1994 Artificial Life III (proceedings of the workshop on artificial life held June, 1992 in Santa Fe, New Mexico), Santa Fe Institute studies in the sciences of complexity v. 17. Redwood City, CA: Addison-Wesley Publishing Inc. Lean, Glen A. 1985-86 Counting Systems of Papua New Guinea (volumes 1-12 and research bibliography). Papua New Guinea University of Technology: Department of Mathematics. Levy, Steven 1992 Artificial Life, Pantheon Books. Lewin, Roger 1993 Complexity: Life at the Edge of Chaos. London: J. M. Dent. Macdonnell, F. 1917 Vocabularies - Cape Nelson Station, North-Eastern Division. Papua, Annual Report for 1914-15, 171-174, App.4. Menninger, Karl 1969 Number Words and Number Symbols. Cambridge, Mass.: M.I.T. Press. Pott, August Friedrich 1847 Die Quinare und Vigesimale Zählmethode bei Völkern aller Welttheile. Wiesbaden: Dr. Martin Sandig oHG. Ray, Thomas. S. 1991a An approach to the synthesis of life. In: Langton, C. — C. Taylor — J. D. Farmer — S. Rasmussen (eds), Artificial Life II, Santa Fe Institute Studies in the Sciences of Complexity, vol. XI, 371-408. Redwood City, CA: Addison-Wesley Publishing Inc. 1991b Evolution and optimization of digital organisms. In: Billingsley, K. R. — E. Derohanes — H. Brown (eds), Scientific Excellence in Supercom-

Artificially growing a numeral system

41

puting: The IBM 1990 Contest Prize Papers, The University of Georgia, 489-531. Athens, GA, 30602: The Baldwin Press. Salzmann, Zdenök 1950 A method for analyzing numerical systems. Word 6, 1: 78-83. Seidenberg, A. 1960 The Diffusion of Counting Practices, University of California Publications in Mathematics 3, 4: 215-300. Berkeley: University of California Press. Seife, C. 1994 Mathemagician - presenting the legendary Conway, John Horton - trickster, group theorist, inventor of the Game of Life. Sciences 34, 3: 12-15. Weimer, Harry — Natalie Weimer 1974 Yareba language, Dictionaries of New Guinea, Vol. 2. Ukarumpa: Summer Institute of Linguistics. Williamson, R.W. 1912 The Mafulu, Mountain People of British New Guinea. London: MacMillan.

When numeral systems are expanded Werner Winter

1. Introductory remarks In their use of numerals and numeral systems, people seem to respond to the conditions and needs of their respective societies (cf. Winter 1992) - a group of hunters and gatherers can do without much precise counting; the raising of livestock, let alone trading activities, make a mastery of a range of numerals essential, and a modern society with a fully developed technology is in great need of high (and low) numbers and the terms for them. Not all conditioning will be economy-triggered: the highest numerals that have come down to us in the Tocharian languages do not occur in financial records, but in religious texts where - in the context of Indian tradition - numbers such as 108,000 or 2,304,000 (years) or, in a slightly damaged passage, 16,777,2(1)6 (miles) (= 216; Sieg 1952: 29 is in error) make their appearance. Systems of counting may be reduced or expanded; the latter is the more common phenomenon, though large-scale reduction seems to have occurred in an area which will concern us later in the present discussion, viz., the Himalayan foothills in eastern Nepal (cf. Gvozdanovic 1985). In view of the fact that the reduction here concerns only the sets of inherited forms, of which all but the lowest ones were very commonly replaced by Nepali numerals, one cannot say that counting as an activity has been affected, but only its expression in native terms. What is found here is thus comparable to what I incurred when doing fieldwork with Yuman languages of southern California in 1956: one otherwise quite competent informant used only Spanish numerals in counting, and the only - limited - access I could gain to native terms was by asking for 'single' and 'twin'. In the present paper I will be concerned not with attrition, but with expansion of an existing system - surprisingly enough in a member of the Kiranti group of languages which normally tends to show reduction rather than growth. Before turning to the Tibeto-Burman data, it may, however, be useful to discuss in some detail a system which has properties not too different from what is found in Kulung.

44

Werner Winter

2. Α Western case Disregarding deviating patterns that survive from the past, such as English four-score-and-ten, our own system in English or German is essentially decadic (except for the lowest numerals) - that is, we have, apart from terms for multiples of '10', those naming powers of '10' (and based on them, for multiples of powers of '10'). This system includes a term for '1,000', to which multiplication is applied to form basic higher numerals. Separate terms are introduced, beginning with '1,000,000', for powers of '1,000' so that for higher numbers the decadic system is superseded. Above '1,000,000' (English one million, German eine Million), the two languages differ: the third power of '1,000' is named billion in American English, but Milliarde in German. German Billion corresponds to American English trillion. The differences between American English and French, on the one hand, and British English and German on the other, frequently cause misunderstandings, for instance, when the national debt of some country with a highly inflated currency is reported - or when the (American) Random House dictionary (1987) translates German Billion by English billion. Partly perhaps to help avoid such possible misinterpretations and partly because technical language requires a maximum of conciseness and precision, a standardized system of prefixes has been developed for units of weights and measures, which in its nucleus makes use of Latin and Greek numerals. Initially there was a neat symmetry in the system - slightly deformed Greek prefixes were used for multiples of the basic unit, Latin ones for fractions thereof: deca- < Greek deka 'ten' hecto- < Greek hekaton 'one hundred' < Greek khilioi 'one thousand' kilodeci-

< Latin decern 'ten'

centi-

< Latin centum 'one hundred'

milli-

< Latin mille 'one thousand'

Just as in actual counting as practiced in modern Western languages, items such as ΊΟ 4 ', ΊΟ 5 ', Ί Ο 4 ', Ί 0 5 ' were not represented by simple terms, but as multiples of '10 3 'or fractions of ' 10 3 ' - ten kilograms, ten kilometers, one tenth of a milligram, etc., although Greek would have provided an underived term for '10,000', viz., mürioi. ΊΟ 6 ' in its turn, was expressed by a prefix of

When numeral systems are expanded

45

its own, thus providing for a parallel to English million, etc. At this point, three things happened: one, the basic pattern of differentiating multiples and fractions was preserved; two, the prefixes were no longer numeral-based; three, the contrast between Greek for multiples and Latin for fractions was eliminated. For Ί 0 6 ' , mega- < Greek megas 'great' was chosen; its Latin equivalent, magnus, apparently seemed ill-suited semantically as a means to express a fraction Ί 0 6 ' . So micro- < Greek mikros 'small' was introduced; one wonders why Latin parvus 'small' or one of its near-synonyms was not made use of. Parallel to the pattern provided by American English million: billion (or British English million: milliard), new primitive prefixes were added to the system, based again on Greek nonnumerals: giga- < Greek gigas 'giant' and nano- < Greek nänos 'dwarf' for Ί 0 9 ' and Ί0" 9 ', respectively. Another Greek common noun yielded Ί0 1 2 ', viz., tera- < Greek teras 'monster'. At this point, however, the symmetry of the system was abandoned: pico12 Ί 0 ' does not have a Greek source form. Its etymology is not immediately clear. A proposal in the Random House dictionary (1987) makes reference to Spanish pico 'peak, beak, bit' - an unlikely comparison if one considers what appears to have led to a deliberate rejection of Latin magnus as a counterpart of Greek megas. A solution for the problem seems possible once the terms for '10 15' and '10 18' are taken into consideration: '10 15' is expressed by femtoand Ί 0 1 8 ' by atto-. The source of these prefixes is rather obvious: the Random House dictionary references to Danish femten '15' and atten '18' are immediately persuasive - the terms chosen indicate, to be sure in truncated form, the number of digits to the right of the period in 0.000,000,000,000,001 and 0.000,000,000,000,000,001, respectively. This observation provides, so it seems, a clue for the proper identification of the source of pico- ΊΟ"12': if pico could be shown to refer in some, possibly oblique, way to 'twelve', then there would be a parallelism with femto- and atto-. pico- is, of course, not of Danish origin; neither are Greek and Latin possible donor languages. What shows up in pico- < 'twelve' apparently is pica, the designation of a type size which corresponds to twelve points on the typesetter's Didot scale for letters: there is a clear reference to 'twelve' here, and pico- < pica neatly fits into the disyllabic pattern found in all other prefixes in the system. What we have here then is an artificially created paradigm of numeral prefixes with a nucleus extending from kilo- to milli- with its contrast of forms based on Greek and Latin numerals; a second layer involving Greek adjectives, mega- and micro-·, a third layer with prefixes based on Greek common

46

Werner Winter

nouns (giga- and nano-), to which a further such term, tera-, was added, which was not given a counterpart in the domain of fractions. Rather, a new principle was introduced here with a slightly veiled reference to the twelve digits of 0.000,000,000,001, and the new principle was reapplied when truncated forms of Danish femten '15' and atten '18' were added to the system. Clearly, the reintroduction of a reference, however oblique, to numerals provides a means of extending the system to ΊΟ 2 1 ', Ί0" 24 ', etc., without the necessity to come up with another principle again, should there be a need to make reference to quantities or distances smaller than what is indicated by atto-. The asymmetry between the multiples and fractions (the chain of multiples is not extended beyond tera- Ί0 1 2 ') may reflect the fact that in an area where the greatest distances are referred to, viz., in astronomy, the basic unit is not the meter, but the light-year, and that for heavy goods, it is not the gram that provides the reference point, but the metric ton - that is, not multiples of Ί 0 0 ' are named, but those of Ί 0 6 ' . 1 The system discussed is, as far as the forms are concerned, essentially a motivated one. In some domains, the motivation even was increased beyond what has been described so far. The German terms Billion, Trillion, Quadrillion neatly reflect the components of a new subsystem for very high numerals: Million is taken as a new base, and Billion designates the second power of this new base, Trillion the third power, Quadrillion, the fourth power, etc. The multipliers are expressed by onsets reflecting Latin numeral prefixes; an extension of the set to include quinqui-, sexi-, septi-, etc., seems perfectly possible. In the German system, Billiarde, Trilliarde, etc., are clearly derived forms. The American English system is much less transparent than the German one; it cannot express very high numbers without recourse to a longer list of Latin-derived prefixes - ΊΟ 33 ' in German would not require a prefix higher than 'fivefold', while American English would have reached 'tenfold' at this point. All in all, the motivations incurred are not internally consistent, but diversified. This diversity reflects layers of historical development (cf. Encyclopaedia Britannica 23: 373), albeit with relatively little time-depth involved in the extranuclear components. The overall structure not surprisingly mirrors that of natural systems of - in particular, high - numerals in Western languages; this fact serves to enhance the acceptability of the pattern in the realm of technology and modern science.2 However, conformity with the Western way of counting is not a conditio sine qua non for the development of an elaborate system for counting large quantities.

When numeral systems are expanded

47

3. A case from the East Among the Kiranti languages studied by Jadranka Gvozdanovic (on the basis of data from her own fieldwork and of materials of the Linguistic Survey of Nepal), Kulung shows one of the least reduced systems. (That a deeper study of an individual language may yield information far beyond that obtained by the Survey, is obvious from a comparison of Dumi data in Gvozdanovic 1985 and in van Driem 1993; on the other hand, the fairly extensive work that has been done on Bantawa indicates that the reduction of the native numeral system to Ί ' through '3' or Ί ' through '4' is the norm in this rather widely used language.) There is a fairly close agreement among all four Kulung speakers recorded for the lists given in Gvozdanovic (1985) concerning the numerals ' 1' through '4'; the omission of ngaci '5' and the concomitant demotion of tupci '6' to '5' seems to be a mistake on the part of the speaker from Pawoi. A mistake is also the use of retci for '7' rather than '8' in the Bung list; the list from Hollu yields a form bhoci for '7'. noci/nuci is provided for '9' in the lists from Hollu and Gudel. '10' appears as a complex form 'one times X' in the Hollu data, while Gudel has an unrelated simplex bOci. The Hollu list is the only one for Kulung (and, for that matter, for any of the Rai languages covered by the Survey) that provides forms for the higher decads (ngipong ' 2 0 \ s u k p o n g '30', likpong '40', ngakpong '50', tukpong '60', bhocipong '70', retpong '80', nocipong '90'); one suspects that some of these are nonceforms (why should '70' and '90' contain the nonsingular suffix -cil) so that the conclusion of Gvozdanovic (1985: 146) that the system as given here reflects old Rai numerals lost elsewhere can hardly be accepted. Such doubts receive further support from another Kulung list which has become available only recently. In 1992, Krishna Bahadur Gankhu Rai, a retired Gurkha captain, provided me with a very extensive English-Kulung vocabulary (a parallel Nepali-Kulung dictionary by the same author has since been published, cf. Rai *1993) which contains, as addenda, two extensive lists of numerals.3 All numbers from Ί ' through Ί 0 0 ' are given in the first one; it is immediately obvious that, e.g., '51', '52', '53' are expressed by '50' + Ί ' , '50' + '2', '50' + '3', respectively, with the digits appearing in their short forms. There is a remarkable disagreement between the two long lists in the case of '7' and '9': For Gyenwar Singh Rai from Hollu, bhoci is '7' and noci, '9' (the latter assignment seemingly supported by nuci '9' from Gudel), for Krishna Bahadur Rai, '7' is nu or nukci and '9', vau or vauci. The latter variant is supported by the Limbu parallel in van Driem (1987: 32), nusi

48

Werner Winter

'7' andphangsi '9'; nuusi '7' occurs also in Ingwaba's Limbu data from Phidim, while for '9' a composite form iboong is given in Gvozdanovic (1985: 162) boong denotes 'decad' in the numbers Ί 0 ' - ' 3 0 ' with van Driem and in ' 1 0 ' '50' in Ingwaba's set. The terms for 'decad' differ radically in the two long Kulung lists. In the Hollu data, -pong in ikpong '10', ngipong '20', sukpong '30', etc., is closely similar to Limbu -boong (with voicing of the intervocalic labial stop); -ka in Krishna Bahadur Rai's items is an isolated form. In both Kulung lists, there is a widespread use of an internal -k- which may have been generalized from some forms in the sets whose identity cannot be safely established. Among the multiples of '10', the most remarkable item in Krishna Bahadur Rai's data is '90', vavau, (i.e., /bhabhau/), a seemingly reduplicated form based on vau (/bhau/) '9'; as it stands apart from the system, it is not likely to have been generated as part of a pattern characterized by generalization. Ί 0 0 ' again shows no agreement between the two lists: ikbu may originally mean 'one heap'; for chhum, no analysis appears to be immediately possible. Up to this point, there is nothing truly unexpected about Krishna Bahadur Rai's material; the first list shows the application of processes of addition and multiplication so characteristic of numeral systems all over the world. There are agreements and disagreements between the two long Kulung lists, but this is not really surprising considering the degree of uncertainty about numerals and their precise values that is found throughout the Kiranti area. What does, however, come as a major surprise is the second appendix provided by Krishna Bahadur Rai in his English-Kulung vocabulary (it has also been included in the published Nepali-Kulung dictionary, Rai *1993 : [585]). The author gives an enumeration of terms for numbers with one, two, three, and more digits, iri' 1' consists o f ' 1' plus ordinal suffix; 'decad', of '2' plus suffix; for the three-digit unit, chhum Ί 0 0 ' plus -ri is found (the question arises whether chhum is related to the numeral '3' in its older form ending in -m, but no answer is possible at this point.' 1,000' is a form without -ri (so are the items following immediately in the list); no analysis of habau can be given beyond pointing to the fact that ha- matches the onset of Nepali hajär' 1,000'.' 10,000' is 'ten habau'.' 100,000' is not treated as a multiple of '1,000', but as a separate primitive counting unit; lankau thus functionwise resembles läkh in Nepali and other modern Indo- Aryan languages. '1,000,000' is 'ten lankau'. The alternation between simple unit and derived unit based on a multiplication by ten continues for karbau '10,000,000', achheng ' 1,000,000,000', shipa ' 100,000,000,000', and gol' 10,000,000,000,000'. For

When numeral systems are expanded

49

'100 goV, bungri may be explainable as consisting of the term for 'flower' plus the suffix -ri (the presumed reasons for the selection of this term will be discussed presently); pau bungri follows the pattern of the lesser high numerals; dheru bungri Ί 0 0 bung' deviates, but it is analyzable as 'large bung' - its very deviation from the pattern (starting with 1,000) A, 10 χ A; B, 10 χ B; C, 10 χ C, etc., may indicate that Ί0 1 8 ' is treated as some kind of terminal number. The system found in Krishna Bahadur Rai's presentation of high numerals in Kulung differs from that found in the international terminology used for weights and measures in that the latter might be called a decimal-millesimal system, while the Kulung pattern represents a decimal-centesimal system. The American-English paradigm contains basic numerals for ΊΟ', '100', 1,000', Ί,ΟΟΟ,ΟΟΟ', '1,000,000,000', etc., the Kulung one, for ΊΟ', Ί 0 0 ' , Ί,ΟΟΟ', '100,000', '10,000,000', etc.; in both of them, '1,000' is the cut-off point for the decimal system (with an additional cut-off point found at '1,000,000' in the German system). The Kulung pattern agrees with the Indie one where ΊΟ', Ί 0 0 ' , Ί,ΟΟΟ', '100,000', '10,000,000'are primitive numerals. The similarity of the onsets of Kulung habau and Nepali hajär recurs in Kulung lankau: Nepali lakh '100,000' and Kulung karbau: Κ. B. Rai (*1993: [585]) does not use Nepali koti- for '10,000,000', but a variant form karod there is thus a complete parallelism between the Nepali onsets ha-: lä(N)- : kar- and the Kulung ones. As concerns the pivotal numerals above '10,000,000', a suggestion can only be made concerning Ί0 1 5 ' (and, by implication, its derivatives): In Kulung, the normal meaning of bung is 'flower'; in the Indian tradition, (Sanskrit) padma- 'lotus flower' is used as a term for '1,000,000,000,000,000' - one thus can hardly escape the conclusion that the Kulung numeral is a caique based on the numerical use of the Sanskritic common noun. The derived term Kulung dheru bungri 'big bung' can also be taken to be a direct loan translation of (Sanskrit-based) mahä-padma- 'big padma-' used to express the notion of Ί0 1 7 '.

Concluding remarks The Western system discussed in the second part of the present paper was designed to meet the needs of a civilization with high demands on the part of technology and science. The compiler of the Kulung word list is a man very much concerned with the preservation of his language and its adaptation to the requirements of the rapidly changing modern society in Nepal. This seems to have led to the elaboration of the Kulung system in the domain of one- and

50

Werner Winter

two-digit numbers - an elaboration deviating in some details from that undertaken by the speaker from Hollu - and the creation of high numerals which might be useful in preparing the speakers of the language to cope more adequately with the tasks of today's life. The system developed is to some extent a replica of a pattern incurred in the Indian tradition; the details, however, seem to be almost entirely new. In a sense, Rai's system is more innovative than the Western nomenclature in the area of weights and measures, although it reflects to some extent the vocabulary (and even more so the structure of the field of numerals) of the culture at large. The likelihood that the extended Kulung system has a chance of actually being put to use seems rather remote; as the product of a fairly daring intellectual exercise, it is, nevertheless, of genuine interest.

Appendix Lists of higher numerals according to Krishna Bahadur Rai; the orthography is his.

A. Kulung numerals 7' - 700' 1 /; ibim 2 m"; nicci 3 su\ sukci 4 li\ lici 5 ngd 6 tu\ tukci 7 nu\ nukci 8 re\ rekci 9 vau; vauci 10 paw, pauci 11 paui 12 pau ni 13 pau su 14 pau li 15 pau ngd 16 pau tu

31 suk i

61 tuk i

32 suk ni 33 suk su 34 suk li 35 suk ngd 36 suk tu 37 suk nu 38 suk re 39 suk vau 40 likka 41 lik i 42 lik ni 43 l[ijk su 44 lik li 45 lik ngd 46 lik tu

62 tuk ni 63 tuk su 64 tuk li 65 tuk ngd 66 tuk tu 67 tuk nu 68 tuk re 69 tuk vau 70 nukkd 71 nuk i 72 nuk ni 73 nuksu 74 nuk li 75 nuk ngd 76 nuk tu

91 92 93 94 95 96 97 98 99 100

vavau i vavau ni vavau su vavau li vavau ngd vavau tu vavau nu vavau vau vavau vau chhum

When numeral systems are expanded

17 pau nu 18 pau re 19 pau vau

47 lik nu 48 lik re 49 lik vau

77 nuk nu 78 nuk re 79 nuk vau

20 nukkd 21 nuk i 22 nuk ni 23 nuk su 24 nuk li 25 nuk ngd

50 51 52 53 54 55 56 57 58 59 60

8 0 rekkd 81 rek i 82 rek ni 83 rek su 84 rek li 85 rek ngd 86 rek tu

26 nuk 27 nuk 28 nuk 29 nuk

tu nu re vau 3 0 sukkd

ngakkd ngak i ngak ni ngak su ngak li ngak ng[d] ngaktu ngak nu ngak re ngak vau tukkd

51

87 rek nu 88 rek re 89 rek vau 9 0 vavau

Note: In the main text of the English-Kulung word list as well as in the NepaliKulung dictionary, the form given for '20' is Kulung nissd.

B. Higher Kulung numerals

('10' and its

powers)

One digit: iry'l' two digits: niry '10'

three digits: chhumry Ί00' four digits: habau '1,000' five digits: pau habau '10,000' six digits: lankau

'100,000'

seven digits: pau lankau '1,000,000' eight digits: karbau '10,000,000' nine digits: pau karbau '100,000,000' ten digits: achheng '1,000,000,000' eleven digits: pau achheng '10,000,000,000' twelve digits: shipd '100,000,000,000' thirteen digits: pau shipd '1,000,000,000,000' fourteen digits: gol '10,000,000,000,000' fifteen digits: pau gol '100,000,000,000,000' sixteen digits: hungry '1,000,000,000,000,000' seventeen digits: pau bu[n]gry '10,000,000,000,000,000'

eighteen digits: dheru hungry

'100,000,000,000,000,000'

52

Werner Winter

Notes 1 Long after the text of the present article had been sent to the volume editor, additional information became available to me in a quite unexpected place: My pocket calendar for 1997 (Rido Technik, Bielefeld) contains under the heading "Vorsätze. Dezimale Vielfache und Teile" the following new prefixes: 1015 Peta10 lsExa1021 Zetta1024 Yotta-

10 21 zepto10"24 yocto-

The forms chosen reflect the inclusion in the respective subsystem by the use of -a- in positive powers (as in Mega-, Giga-, etc.) and of -o- in the negative powers (as in micro-, nano-, etc.). Lower case is now used for abbreviated reference to negative, upper case, to positive values, zepto- and yocto- remind one of Latin septem '7' and octo '8', respectively (or, for that matter, of Greek heptd and okto:) - '21' being, after all '3 χ 7', and '24', '3 χ 8'. A Greek basis for these forms is made a certainty by Peta-: Greek pente '5' and Exa-: Greek hex '6', respectively referring to ' 5 x 3 ' and '6 χ 3'. A greater consistency in the pattern developed would have been achieved if Zepta-* and Yocta-* had been introduced for Ί 0 2 " and Ί 0 2 4 ' , and deformed zetto-* andyotto-* for ΊΟ" 2 " and ΊΟ"24' instead. The present solution creates another set of partially motivated forms within a pattern with a rather high degree of internal inconsistency. 2 A concurrent use of Latin, French, and Greek prefixes is found in the designations of fractions of a quaver, an eighth note: a sixteenth is called a semiquaver (attested since 1576 according to the Oxford English dictionary), a thirty-second, a demisemiquaver (since 1706), and a sixty-fourth, a hemidemisemiquaver. 3 The Kulung orthography in the present paper follows closely that used by Κ. B. Rai (except that in the text, final -y is replaced by -i). His ch renders Id, his chh, /c h /; his a denotes both schwa and an unstressed /a/.

References Duden 1989

Duden. Deutsches Universalwörterbuch. 2. Auflage. Mannheim / Wien / Zürich: Dudenverlag. Encyclopaedia Britannica 1972 Chicago, etc.: Encyclopaedia Britannica.

When numeral systems are expanded

53

Gvozdanovic, Jadranka 1985 Language system and its change. On theory and testability. Berlin / New York / Amsterdam : Mouton de Gruyter. Oxford English dictionary 1992 Second edition, reprinted with corrections. Oxford : Clarendon Press Rai, Krsnä Bahadur Gähkhu * 1993 Nepali kulun räi Sabdakosa (Nepali - Kulung dictionary). Kathmandu. Sieg, Emil 1952 Übersetzungen aus dem Tocharischen 2. (Ed. by Werner Thomas). Berlin: Akademie-Verlag, van Driem, George 1987 A Limbu grammar. Berlin / New York / Amsterdam : Mouton de Gruyter. 1993 A Dumi grammar. Berlin / New York : Mouton de Gruyter. Winter, Werner 1992 Some thoughts about Indo-European numerals, in: Jadranka Gvozdanovic (ed.), Indo-European numerals. Berlin / New York : Mouton de Gruyter, 11-28.

Pre-decimal structures in counting and metrology1 Carol F. Justus

1. Introduction For the better part of this century, faith in Neogrammarian sound correspondence has ignored numeral data from pre-technological societies or cognitive studies of early childhood development which suggest alternative hypotheses. The fact that regular sound correspondences could be found to plot the forms for decimal numeral forms back into a prehistoric Indo-European protolanguage reinforced the assumption that the Indo-Europeans used an abstract numeral (decimal) system to count their sheep and cattle. Tokens and clay tablets from the Ancient Near East now reveal that prenumerical systems quantified cattle and food stuffs as late as the fourth millennium BC. The token system, attested as early as the eighth millennium, used one-to-one correspondence between symbols for smaller and larger units like a sheep or a herd of sheep, a small jug or a large jug. A token for a herd of cattle or a jug of oil thus represented a unit of goods. Smaller and larger units came to be rank ordered in a series based on size. Such serialized units are the ancestors of bases in abstract numeral systems (Justus ms.). Where then did the digit-like entities (such as 1 to 9) that subdivide bases come from? Celtic sheep-counting systems generally dismissed by scholars in fact preserve at least two pre-numerical digit-like forms that subdivided pre-decimal bases: lethera 'the half; half again as much/many; second unit' and methera 'the more; next to last unit'. The first part of this paper differentiates numerical powers of our decimal system from the pre-numerical base-like units of the traditional British pound system. The second recalls late fourth and early third millennium BC systems of numeration that demonstrate the non-numerical value of early serialized base-like units. Part three then analyzes digits lethera and methera of North English and Welsh sheep counting as similarly relational in value, not numerical. While serialization of different-sized measure-like/base-like units foreshadowed the concept of numerical base, digits result from reckoning the space between such units, first as 'the half' and 'the more (of a base unit)', then as 'second member' and 'next to last member'.

56

Carol F. Justus

2. Base units, factors, and abstract numerical values The mathematical 'factor' that converts traditional measure units such as the inch and foot conceptually precedes abstract numerals. In length measure, the factor 12 converts inches to a foot, and the factor 3 converts feet to a yard: 12 inches thus make a foot, 3 feet a yard. Measure units inch, foot, and yard differ from the powers of numeral systems in that a measure unit is a baselike entity without numerical value. By contrast, powers 10, 100, 1000 of the base 10, the results of raising the base 10 to successively higher powers, have numerical values. Between non-numerical measure units and decimal powers lie base-like units that are neither powers nor bases, nor even quite the same as measure units. While 10 is the base of the decimal system and 100 one of the powers of base 10, units of the Ancient Near Eastern protoliterate systems of numeration, often using arbitrary units that are not confined to one system of measurement (Justus ms.), lack abstract numerical value. As such they are not bases, but base-like. 'Base unit' here refers to such units that, as the descendents of tokens, are not properly measure units nor yet bases or powers of a numeral system.

2.1. Numerical decimal factors Words 'ten, hundred, thousand, million', if they did not refer to powers of 10, might (or might have) constitute(d) a set of arbitrary base units, with or without abstract numerical values. For the sake of argument, let us view words 'ten, hundred, thousand, million', not as exponential powers of 10, but as arbitrary, serialized units related by repeated factors, first of 10, then 1000: Figure 1. Modern exponential factors of 10

10

1

10

10

1000

>TEN

»HUNDRED

»THOUSAND

101

102

1 03

»MILLION

1 06

Pre-decimal

structures in counting and metrology

57

Here 10 is both the first base unit to result from conversion by a factor and the first factor. Subsequent factors of 10 then relate the unit 'ten' to the unit 'hundred' and so on. As word forms, 'ten, hundred, thousand, million' are arbitrary and unrelated, although mathematically numerals, 10, 100, 1000, and 1,000,000, are exponential powers of 10 or 1000. Our knowledge of the mathematics underlying such word forms makes it difficult to view 'ten, hundred, thousand, million' as arbitrary, serialized units. Units of the traditional British pound system by contrast are clearly serialized base-like units without numerical values.

2.2. Pre-decimal factors Unlike units of the decimal system, related by repeated factors 10 or 1000 , units in the traditional British pound are related by single factors 12 and 20: Figure 2. The traditional English pound 12

20

Pence

Shilling

Pound

= "1"

= 1 2 pence

= 20 shillings

If 'ten' is the first base unit and 10 a first factor, shilling here is the first baseunit and 12 the first factor. 2 A different factor, 20, then relates the shilling to the pound. Unlike ten, hundred, thousand, units shilling and pound have no independent numerical value, although values of the units in both sets result from rank-ordered positions in the series. While 12 pence make a shilling and 20 shillings a pound, the shilling did not refer to an abstract numerical '12', nor pound to '20' or '240', even though a pound might be replaced by 20 shillings or 240 pence. The value of the shilling is both 12 pence and one twentieth of a pound. Similarly, 'foot' and 'yard' or 'mile' have no independent numerical meaning, nor originally any standard size other than the king's foot perhaps. Values for non-numerical base units result from relational order (and arbitrary factor conversions), not from any independent numerical value. Serialized base units,

58

Carol F. Justus

moreover, allow calculation of large quantities with a few digit-like repetitions. The 36 inches in a yard, for example, results from, at most, 12 digit-like repetitions of the inch and the even fewer 3 repetitions of the foot. But different commodities use different systems of factors. While factors 12 and 3 quantify commodities to which length is appropriate, shilling and pound quantify monetary units. Such pre-numerical counting of money, moreover, exemplifies systems of numeration in which the distinction between counting and measuring was once not nearly so discrete. Coming full circle now the metric system, replacing traditional systems of measure, unifies the conceptual bases of quantification systems for both counting and metrology.

3. Preliterate and protoliterate numeration The earliest evidence for human (ac)counting comes from the Ancient Near Eastern token systems of the prehistoric eighth to fourth millennia BC (ca. 8000-3500 BC, 3500-3100 BC). Small clay tokens with different forms for different quantity-commodity units were used in a process of one-to-one correspondence that fused the concept of quantity with the concept of the commodity quantified. Impressions of those token shapes onto clay envelopes containing the tokens, then onto clay tablets independent of the tokens, led to developments in counting that culminated in the Sumerian sexagesimal numerical system of ca. 2500 BC that attests number words with abstract numerical values, values independent of the commodity counted.

3.1. Quantity-commodity units: tokens and concrete counting In the protoliterate period (ca. 3500-3100 BC), the period just preceding the onset of literacy, both clay tokens and their impressions on clay tablets accounted for goods using concrete counting techniques. The impressed signs are the stylized successors of one-to-one impressions of token shapes. Impressions were often augmented by incised pictographs that captured more of the detail of the token's three dimensional character. The impressed and pictographic renderings of a token shape thus began to separate the idea of quantity from the kind of commodity being accounted for (Schmandt-Besserat 1992: 162-165; 189-192). 3 Out of the token impressions and incised pictographs on clay tablets arose protoliterate systems of numeration characterized by sets of base-like units.

Pre-decimal

structures in counting and metrology

59

Studies determining the factor relations for the units of each system (Damerow — Englund 1987; 1989 passim) show that the impressed shapes, although they were predictably rank ordered, did not yet have numerical value.4 Like the inch and foot or shilling and pound, values of sets of impressed shapes were determined by factor sets that varied depending on the commodity that they quantified. For example, the characteristic system for quantifying units of grain used as many as six different impressed shapes, rank ordered in relation to each other by factors 10, 3, 10,6, and 5 where the left-most shape was the largest, the right-most the smallest:

Figure 3. A protoliterate grain system with factors 10-3-10-6-5 10

3

ß>

D

N-48 < "9000"

N-34 "900"

10

· N-45< "300"

6

· N-14 < "30"

5

d N-l< "5"

^ N-39 "1"

A pictographic sign for grain accompanying an impressed sign made it clear which system and consequent rank-ordered value the impressed shape referred to, as the values of impressed shapes depended for interpretation on a position in a rank order in a commodity-linked system of numeration somewhat as foot has the value 12 only in relation to the inches of length measurement. We don't know what names their users gave to the impressed signs of the protoliterate tablets, only their role in the system. Sumerologists' sign lists use number codes (Green — Nissen 1987: 335-345; Nissen et al. 1990: 62 passim) to refer to them while archaeologists give names to the shape types (Schmandt-Besserat 1992: 134-142) that reflect the technique of impression into the clay. The small circular impression listed in the sign list as N-14 ( · ) has a value in the grain system that results from its rank order between the small short horizontal wedge (N-l D) and the large circular (N-45 · ) . That the numerical value "30" given in Figure (3) is not the meaning of the small circular (N-14 · ) is immediately clear by comparison of its value in the system of field measure with number-signs rank ordered on the basis of factors 6, 10, 3, 6, and 10:

60

Carol F. Justus

Figure 4. Protoliterate field measure with factors 6-10-3-6-10

6 •

10 ®

3 ·

6 B>

10? D

Ό

N-45