CHAPTER 6
NUMBER AND COUNTING IN CONTEXT
6.1 INTRODUCTION
The foregoing survey of the counting system situation in New Guinea and Oceania has concentrated on delineating the various types of systems, their structural features, and their geographical distribution throughout the region in question. This focus on the nature of the systems themselves, abstracted from the cultures in which they are embedded, has been deliberate and is preliminary to addressing the chief questions with which this work is concerned, that is how and when the counting system situation, which is apparent now, came about. Any student of traditional numeral systems, however, is acutely aware that the description of their structural features does not in itself do justice to the richness of the notions of number held in a given culture: the human face of number. The highly abstract view of number, which is now held by mathematicians and logicians in the modern technological societies, is a relatively recent development, deriving from the work of Frege, Russell, Whitehead, Gödel, and others.[1] This exceedingly refined view of number is regarded by Wilder[2] as the end result of a long process of cultural evolution and indeed it is far removed from the views of number which we may infer existed in traditional societies. In order to obtain some idea of what these views are, in this chapter I shall attempt to look at various aspects of what Crump calls "the anthropology of numbers".[3]
In the sections which follow I shall address a number of major questions, which are given below, together with some subsidiary questions which are raised in each section. The main questions are as follows:
(1) Are there traditional societies in which number has a privileged position and, conversely, do societies exist in which there are alternative organizing systems and principles which relegate number to a relatively unimportant position?
(2) What is, and what is not, counted in various traditional societies and in what circumstances does counting occur? Are there particular types of traditional exchange systems or economies which emphasize counting more than others?
(3) To what extent is the phenomenon of numeral classification found in the languages of New Guinea and Oceania? Are the ways in which the traditional societies of this region categorize their universe of countable objects similar? Is there any evidence that numeral classification may have been present in Proto Oceanic?
(4) What, in the traditional societies of this region, are the terms for large numbers? Do the languages of New Guinea and Oceania differ with respect to the resources available for expressing large numbers? For the Oceanic AN languages, do we have evidence to suggest what the historical development of terms for large numbers might have been and whether such terms were likely to have been present in Proto Oceanic?
6.2 THE IMPORTANCE OF NUMBER IN TRADITIONAL SOCIETIES
In modern technological societies, number plays a central and important role and pervades virtually all aspects of everyday life: there is a pronounced emphasis on enumeration as well as the use of the arithmetical processes and their physical applications. Traditional societies, however, differ from these, and indeed from each other, in the degree to which number holds a privileged position in everyday life. Between modern and traditional societies we have the obvious distinction, respectively, of numbers being used primarily in their written forms as opposed to their spoken forms. The former has clear advantages in terms of the efficient recording of totals, the carrying out of calculations using place-value notation, and the minimization of the effect of the constraints of memory capacity. We would expect those societies which use numbers in their spoken, unwritten, forms, even if the results of enumeration are recorded by some form of tally, to differ in significant ways from those which use, primarily, the written forms. However, it is also apparent that traditional societies differ from each other in the degree to which number plays a role as an organizing construct within each society. These differences will be illustrated by a series of brief case studies drawn from the linguistic and anthropological literatures.
6.2.1 The Adzera People of the Morobe Province (PNG)
The languages of the Markham Valley in the Morobe Province (PNG), which includes that of the Adzera people, have been studied by Holzknecht.[4] The Markham Family are all AN languages descended from POC for which a 10-cycle numeral system has been reconstructed. These language groups have moved away from the POC homeland (to the east in the coastal region of the Willaumez Peninsula of the West New Britain Province) and are now located inland in areas which are predominantly NAN-speaking. Holzknecht notes that "it appears from the lack of terms for maritime objects and activities that the Markham languages moved away from the sea early in their history."[5] One important accompaniment of this shift away from a maritime culture to an inland cultivation culture with a strong NAN influence is the change in the numeral systems of the languages of the Markham Family. Holzknecht says that "under the influence of their Papuan-speaking neighbours the system became eroded to what we find today, a system with two numerals, 'one' and 'two', combined with body-tallying."[6] We thus have evidence that various factors, such as a change of environment, of economy, and of the influence of the non-maritime NAN-speaking language groups, has brought about a change in the means by which the Adzera, and other members of the Markham Family, enumerate their world.
In Smith's study of the traditional methods of enumeration in the Morobe Province, he notes that "in many Morobe societies, as in other parts of Melanesia, status was achieved by distribution of wealth. Often this distribution implied a reciprocal obligation on the part of the recipients to pay back on some future occasion as much or more than the amount received. Hence it was important to know how much was received."[7] Smith indicates that for the Adzera there were certain ceremonial occasions during which huge quantities of banana bunches were given away. The counting of the exact number of bunches, however, was less important than the way in which the bunches were displayed: "a broad ladder is sometimes built to the top of a coconut tree, and this must be completely covered with bunches of bananas, or the sponsors of the distribution will be scorned."[8] Even in the case of relatively small-scale gift-giving there is reciprocal obligation and, for example, on receipt of a joint of pork an Adzera man would measure it with a piece of bark string and this tally of its size would be kept for future reference. The main point, however, is that while the Adzera are able to count, precise enumeration is less important than satisfying the ceremonial obligation of providing an impressive display. Also, in the absence of written numerals, physical tallies such as a piece of knotted bark string are adequate substitutes for maintaining a record of gift prestation.
According to Holzknecht, Adzera trade was usually conducted between individuals in a trading relationship and such partners were either true kin or quasi-kin. However, trade transactions often transcended language boundaries: "For example, many Musom people have kin ties with Nabak (Papuan) neighbours, because trade ties were so advantageous between those two groups. In exchange for mountain food and game from the Nabak, the Musoms gave salt, grindstones for sharpening axes and bamboo for bowstrings."[9] The Adzera and their neighbours did not possess monetarized economies and goods were exchanged rather than purchased.
6.2.2 The Grand Valley Dani of Irian Jaya
The Grand Valley Dani live in the central mountain ranges of New Guinea to the west of the PNG border in Irian Jaya. They speak a NAN language which belongs to the Trans-New Guinea Phylum.[10] The Dani were studied by the American anthropologist Karl Heider in the early 1960s. Heider noted that some of the features of this horticultural society were "the complexities of a moiety-based kinship system; a political organization of confederations and alliances of great size but little power; and in the psychological realm, such traits as a five-year postpartum sexual abstinence without stress, a counting system which goes up to three, and tremendous conservatism in the face of great pressures to change their way of life."[11] Data on the Dani numeral system, other than Heider's, indicate that it is a 2-cycle variant with the basic numerals 1, 2, and 3 and 4 = 2+2, after which digit-tallying may occur.
Heider indicates that the enumeration of objects, even in ceremonial gift-exchange, is not regarded as important in Dani society: "the point is that the Dani get along very well without quantifying their environment. In other cultures, numbers are useful for tallying masses of identical things like monetary units. But the Dani do not use money. Or numbers can be useful for dealing with masses of similar things if one is willing first to consider them as identical and then tally them. If a man has a herd of pigs, it is rarely more than a dozen or two, and he can keep in mind the age, sex, and markings of his individual pigs. At a funeral, the people who were concerned could keep track of what sort of pig was brought by whom and add it to their knowledge of the current state of gifts and debts within the group. I would go to a funeral and sit counting the number of pigs, and in the end I would say 'this was a 24-pig funeral'. A Dani would know other, richer things about the funeral, but he would not be interested in such a tally."[12]
The circulation of goods and the occasions for exchange occur in three circumstances: family consumption, ceremonies (both within-society exchanges), and the between-society exchange of outright trade. With the first two, the commodities exchanged are those things which everyone possesses and thus the exchange of these is an affirmation of the social rather than the economic basis of exchange. External trade, on the other hand, is carried out to obtain imported goods which are not present in the society; these are "adze stones, exchange stones, fine furs, feathers, and wood in exchange for goods like salt, pigs, and nets which are produced in the Grand Valley." Such transactions are normally carried out by barter in the absence of traditional money.
6.2.3 The Loboda of the Milne Bay Province (PNG)
Loboda village is situated at the northern end of Normanby Island in the Milne Bay Province (PNG). The people of Loboda speak Dobu, an AN language belonging to the Papuan Tip Cluster and, even though this is a descendant of POC, it possesses a numeral system with a (5, 20) cyclic pattern rather than the POC 10-cycle pattern.[13] In the mid-1970s, the Loboda people were studied by the anthropologist Carl Thune as part of the Indigenous Mathematics Project. Thune says of the traditional counting system "the highest number I ever heard mentioned in the Loboda numerical terminology was tau nima, that is 'five twenties' or 'one hundred'. However it should be clear that it is in theory possible for the numerical system to go somewhat higher than this ... Nevertheless, today at least, use of such higher numbers is only a theoretical possibility. ... Outside of situations in which people were teaching me the traditional counting system, I never heard it used for numbers above five."[14]
Thune characterizes the Loboda as an example of a "non-numerically oriented culture" and adds that "although enumeration and counting are possible for the Loboda people, in fact they occur only rarely and are, for the most part, unnecessary."[15] In the traditional spheres of life, people might report, in discussing a feast, the number of pigs killed or the number of yams distributed; also, the number of days in a journey or the number of months to a particular festival might be stated, "nevertheless, today higher numbers are rarely if ever used, and I suspect this was always the case."[16] Thune attributes this lack of interest in the enumeration of totals to "the fact they do not think of their world in terms of units which are easily organized or represented with numbers. There are, for example, no traditional units for measuring distance or height which could be counted in the way the English (sic) units of meters or kilometers can be counted."[17] A length of shell money is described, not against some absolute standard, but by relative and uncountable measures, for example comparison against a particular person's arm. In the description of people's ages, broad time-periods are used rather than a numerical specification; thus there are terms for infant, child, adolescent boy or girl, and so on. In ceremonial gift-giving, a large amount of yams may be given to a recipient group who will, at some later date, give back an equivalent amount. "In fact", Thune points out, "no one ever counts the yams of the first gift to ensure that a numerically equal repayment can later be made. Rather, all the yams making up the gift will be heaped together in one pile and it is this collective gift ... which must later be repaid."[18] The gift which is repaid must be the same visually judged size as that given.
In any feast exchange, there are usually several different categories of gifts. Thus, in addition to yams, there may be betel nuts, tobacco, a pig, and so on. These are non-interchangeable and distinct categories which cannot be subsumed into a single total. Each category of gift must later be repaid by the same category and an equivalent amount. The amounts to be repaid are recalled by naming the individuals among whom the original gift was divided. " By thinking on the one hand", Thune adds, "of the gift as composed of a number of categories of items to be repaid in kind, and on the other as divided among a series of specific individuals, counting is rendered not only unnecessary but irrelevant."[19]
One final point made by Thune is that the nature of Dobu, the language of Loboda, to some extent precludes the extensive use of numerals as adjectives to qualify nouns. The essential feature is that many nouns are grammatically treated as mass nouns which are neither counted nor pluralized. Thune points out, for example, that the word for yam, bebai, is treated as a mass noun and is, for grammatical purposes, unable to be modified by a number or an indefinite article. Indeed, this grammatical treatment of the yam noun mirrors the way in which yams are treated in real life, that is as an indivisible mass rather than as a collection of countable, discrete yams, a fact that has mildly Whorfian overtones.
6.2.4 The Ekagi of the Wissel Lakes (Irian Jaya)
The counting system of the Ekagi was described previously in Chapter 4, particularly because of its possessing an unusual (10, 20, 60) cyclic pattern. Ekagi, or Kapauku, is a NAN language belonging to the Trans-New Guinea Phylum although, as discussed earlier, its numeral system shows indications of AN influence.[20] The Ekagi inhabit the Wissel Lakes region of the west-central part of the highlands of Irian Jaya. In the mid-1950s they were studied by the American anthropologist Leopold Pospisil. Ekagi culture, Pospisil indicates, is wealth oriented: "This means that the highest prestige in this society and the highest status of political and legal leadership are achieved not through heritage, bravery in warfare, or knowledge and achievements in religious ceremonialism, but through the accumulation and redistribution of capital. The major and often the only source of capital, generally in the form of shell money, is successful pig breeding."[21] The Ekagi subsist by the cultivation of sweet potatoes which are also used as food for pigs and, thus, agriculture in an indirect way "not only creates wealth but also provides a basis for acquiring political and legal powers."[22] The cowrie shell money is a scarce resource which is not produced by the people themselves but has to be acquired through trade with the lowland, coastal people.
Unlike the societies discussed in the previous case histories, the Ekagi, according to Pospisil, have a quantitative rather than a qualitative world view. "Their highly developed decimal counting system enables them to count even into thousands ... Not only is counting used in their economic transactions, but the people show a peculiar obsession for numbers and a craving for counting. They count their wives, children, days, visitors at feasts, and, of course, their shell and glass bead money."[23] Value is placed on large numbers, whether of people, pigs, or objects. Pospisil says that a "pig feast is an impressive affair. On such occasions there may be as many as 2000 visitors, and the slaughtered pigs may be counted in the hundreds. The trade turnover, in terms of shell money, may be quite considerable."[24] On another occasion, an Ekagi youth, who had counted the glass beads acquired by Pospisil, reported that "You have 6722 beads in your boxes. That means that you have spent 623 beads since Gubeeni counted your money three days ago. I would suggest that you order more beads in about thirty days so that you do not run out of funds."[25]
6.2.5 The Melpa of the Western Highlands Province (PNG)
The people speaking the Melpa dialect of the Hagen language live south-east and north of the township of Mount Hagen in the Western Highlands Province (PNG). Hagen is a NAN language belonging to the Trans-New Guinea Phylum. The Melpa people were studied by the anthropologist Andrew Strathern in the 1960s and, in particular, he has published an account of the complex ceremonial exchange system called the moka, "an institution linking groups together in alliances and ... a means whereby men try to maximise their social status.[26] The Melpa, like the Ekagi, subsist "by sweet potato horticulture, supplemented by cultivation of bananas, sugarcane, taro, yams, maize and cassava."[27] The wealth of a Melpa man is largely determined by his possession of two commodities, pigs and pearl shells, both of which are used in the moka, although other forms of wealth may also supplement these: live cassowaries, long bamboo tubes of oil for decorating the body, packs of salt, cassowary eggs, pandanus fruits, and, nowadays, live steers and cash.[28]
In the moka itself, and in the preliminaries leading up to it, the counting of both pigs and shells is important. However, during the moka, particular attention is paid to the display of wealth items, the shells, for example, being arranged in specially decorated display houses. Prior to the moka, the donor and recipient groups meet to negotiate the number of pigs and shells to be given. Wooden stakes are placed in the ground, each one representing a pig which, during the moka, will be tethered to the stake. The stakes are counted "in twos, making these into sets of eight or ten, and a supporter keeps a record on his fingers of how many sets are taken: for each set he bends one finger down."[29] The Melpa counting system is a somewhat complex one showing features which arise from treating the four fingers as a "hand", the thumb being tallied separately as 5. There is a distinct term for 8, engaka or "man", that is the eight fingers of a man. The word for 10 is "two thumbs", that is the 8 fingers augmented by 2 thumbs.[30] In the moka, the chief donor aims at giving a "grand set" of 8x8, 8x10, or 10x10 pigs. Shells are counted in a similar way in "sub-sets of two or four making up sets of eight or ten."[31] The moka, then, as a central feature in the social, economic, and political life of the Melpa, requires that the ongoing process of exchange be accounted for numerically: the number of pigs and shells involved must be counted and recorded by tally by both the donor and recipient groups for future reference. However, while the counting of gifts is important, so are other aspects of the ceremonial exchange such as the visual display of wealth, a feature shared with the Adzera in their gift prestation of bananas.
6.2.6 The Mountain Arapesh of the East Sepik Province (PNG)
The Mountain Arapesh are located in the East Sepik Province (PNG) in villages situated along the coast and in the region stretching southwards to the mountains. Mountain Arapesh is a NAN language belonging to the Torricelli Phylum and its counting systems were discussed in the previous chapter; The main system has a (2', 4, 24) cyclic pattern but there is also a special system possessing a 3-cycle.[32] The Mountain Arapesh people were studied in the 1930s by the anthropologist Reo Fortune who published a study of their language in 1942.[33] Prior to this, Fortune had also studied the Dobu-speaking people in the Milne Bay Province of which the Loboda people, discussed earlier, are a part.
Fortune notes that in using their 2-cycle variant numeral system, with its secondary 4-cycle and tertiary 24-cycle, tallying of each 24-cycle is recorded with a stick or peg before proceeding to the next. Only certain types of objects are counted using this system: "coconuts, small yams, taro, arm-rings, dogs' teeth, house poles, house posts, breadfruit, sago frond sheaths, pots plates, spears, arm-bands, bamboo lengths, sugarcane lengths, eggs, birds, lizards, grubs, fish, [etc.]." However, a 3-cycle system is used for counting "betel nuts, thatch shingles, coconut, sago, betel nut palms, big yams, packets of sago, wild game, sheets of sago bark, packages of vegetable greens, braids of tobacco, packets of tobacco [etc.]."[34] It is not uncommon, particularly in the nineteenth century literature on traditional numeral systems, to find somewhat disparaging remarks about low-cycle systems and of the ability of the people possessing such systems to count. Fortune, however, indicates that with the Mountain Arapesh "Counting is done with great facility and ease with this conventionalisation of very few special roots. To suppose that the paucity of the Papuan languages in root words for numerals makes counting difficult to the Papuan is quite incorrect. The Arapesh people count rather more quickly than the Melanesian Dobuans, who use a decimal system with many more root terms. An unambigious conventional method, with or without many root terms, is all that is necessary."[35]
The Arapesh engaged in three types of exchange. Oliver says: "first were the random and informal transactions - barter, loans, 'gifts', and so forth - whereby goods passed between kinsmen and acquaintances ... Second were the trade expeditions ... undertaken by small parties of men to exchange local goods for those not obtainable at home ... These exchanges were by barter, accompanied by haggling ...The third type of external exchange took place through formally constituted partnerships; and far from bartering, each such transfer was viewed ... as a spontaneous 'gift'."[36] None of these transactions, however, involved the use of traditional money although rates of exchange between various commodities were established.
6.2.7 The Woleai People of Micronesia
The previous case studies have derived from societies located on or near the New Guinea mainland. With the exception of the Loboda, none of the societies dealt with has a primarily maritime culture. We deal here, then, with a maritime culture par excellence, that of the people of Woleai atoll in Micronesia[37] who speak an AN language and whose way of life probably approximates some aspects of its ancestral POC culture rather more closely than those of AN groups, like the Adzera, which have largely abandoned their maritime culture and economy.
The people of Woleai, part of the Caroline Islands, were studied by the American anthropologist William Alkire in the 1960s. The Woleai, in common with other groups in Micronesia, make use of numeral classification and categorize their universe of countable objects into at least 22 classes; the defining characteristics of the classes are, for the most part, related to certain physical attributes of the objects counted or to particular ceremonial uses to which the objects are put. A delineation of various members of the classes gives some indication of the wide range of countable objects: fish of many kinds, turtles, domestic animals, people, canoes, roof thatch, chickens, eggs, stalks of bananas, garlands, shell belts, paddles, shells, leaves, stones, palm fronds, and so forth.[38] While the enumeration of these objects is a common part of everyday life, Alkire indicates that counting, and indeed counting in rather special ways, acquires a particular importance in ceremonial circumstances, as we have found with several of the highlands societies on mainland New Guinea.
Alkire writes that "On Woleai the ceremonial accumulation and redistribution of mature coconuts (cho) is part of all funeral rites." People from several islands are usually involved in the funeral rite of "exchanging" coconuts. "The exchange, however, is purely symbolic for the nuts are never moved from one island to another ... There is no need to transport the 'real' nuts from the respective islands across the lagoon since the totals are the same and one would simply end up with the same number of nuts."[39] The Woleai possess a 10-cycle counting system. In counting coconuts in normal, everyday circumstances, the nuts are grouped in tens and a "group of ten" numeral classifier is used in their enumeration. However, in the ceremonial funeral exchange, this procedure changes: "In this context a Group of Ten in reality contains only eight nuts ... The nuts are segregated into lots of eight and each of these is called a Group of Ten."[40] Alkire gives the example of 1529 Groups of Ten being counted and which, under normal circumstances, would mean that 15,290 nuts had been accumulated, but in these special cicumstances the actual total is 1529 x 8 or 12,232. This unusual treatment of the number 8 in Woleaian ceremonials is one example of the special significance attributed to that number in many Pacific cultures. Indeed, Bruce Biggs, the noted Polynesian scholar, has described the large number of instances in which the number 8, and its association with sacred matters, supernatural power, high rank, and large size or totality, is a common feature of Pacific societies.[41] Alkire also observes that "on many ceremonial occasions the number four (its multiples and divisions) is of great importance."[42]
6.2.8 Commentary
These case studies provide a representative survey of the various ways in which the traditional societies, existing in the region under consideration, place importance on number and enumeration. It is clear from the examples given that there is considerable diversity in the degree to which number holds a privileged position in these societies. In the NAN-speaking, horticultural society of the Ekagi, for example, number and counting play a prominent part indeed, whether in normal, everyday life or in ceremonial occasions. It will be recalled that their numeral system with its (10, 20, 60) cyclic pattern is probably not their original system in that it shows evidence of borrowing from an AN source and it may well be that this augmentation of their numeral lexis has been prompted by the prominent position that the Ekagi accord to number.
The other NAN-speaking societies in our sample do not show the same degree of preoccupation with number as the Ekagi do. The Melpa's complex exchange system, the moka, provides the context for the counting and display of wealth to enhance the status and prestige of highlands Big Men. Even though the Melpa numeral system is a 2-cycle variant augmented by digit tallying, this is sufficient for the task of counting the moka gifts, even in the case where the "grand sets" of 64, 80, and 100 are achieved. The Grand Valley Dani, who possess a 2-cycle variant system, do not, according to Heider, place any special emphasis on enumeration, even in ceremonial contexts. On the other hand, the Mountain Arapesh also have a 2-cycle variant numeral system which, as Fortune points out, is sufficient for counting a wide variety of objects, if necessary up to multiples of 24. Thus, while the Ekagi may well have adopted a 10-cycle system (and thereby abandoning a lower-cycle one) in order to satisfy their preoccupation with number, the possession of a 2-cycle system, or a variant, is not necessarily an indication that the societies having such systems are unconcerned with number, or that its members are unable to count with facility.
The AN-speaking societies discussed above, the Adzera, Loboda, and Woleai, each derive from a common ancestor, the society of POC-speakers. The maritime Woleaian culture probably resembles the culture of its distant ancestor more closely than those of the other two do. Number and counting play an important part in both the everyday and ceremonial aspects of Woleai life and large numbers of objects are counted with facility using their 10-cycle system. The Loboda, however, while still retaining some aspects of a maritime culture, have not retained a POC-derived 10-cycle system and now have a (5, 20) one. Even though this system would be quite sufficient for the enumeration required of a maritime economy, in fact the Loboda place little importance on counting and instead organize their transactions in largely non-numerical ways. In ceremonial circumstances, the Loboda place emphasis on the visual display of wealth; while the repayment of gifts has to satisfy quantitative requirements, quantities are judged by visual inspection rather than by counting. The tendency to regard quantities as indivisible masses is reflected in the grammar of their language in which many nouns are treated as mass nouns which cannot be modified by numerals. By contrast with the other two societies, the Adzera have long ago left their original maritime environment and have acquired an economy which is heavily influenced by their inland, NAN-speaking neighbours. The Adzera numeral system has also undergone a change; the POC-derived 10-cycle system is no longer evident and a 2-cycle system (with digit-tallying) is used. In ceremonial circumstances, the counting of gifts is less important than providing an impressive visual display of wealth.
It is apparent, then, that there is no simple way of characterizing the traditional societies dealt with here with regard to the way in which they accord importance to counting and number. Also, the degree of sophistication of the numeral system of a given society is not necessarily an index to the extent to which enumeration is important or can be carried out with facility. The possession of a large primary cycle system may reflect this, however the converse is not true: the possession of a low primary cycle system does not necessarily indicate a lack of interest in number, nor in the ability to count. Finally, it is apparent that, in each of the societies discussed here, ceremonial institutions involving displays of wealth play an important social, economic, and political role. The ability to amass large quantities of wealth items accords status and prestige to clans and individuals. The judgment of quantity, however, varies in each society from the impressionistic to the precise. With the Melpa and Woleai, certain numbers such as 8 or 10 are particularly significant in ceremonial situations. However, to invest numbers in this way with sacred and power and metaphysical connotations is not necessarily a feature of all societies in New Guinea and Oceania. In this respect, the evidence does not universally support Crump's assertion that "as to the differences between traditional and modern thought, it is not so much that the former is concerned with application, and the latter with internal consistency, but rather that there is, as between traditional and modern societies, a pronounced preference for metaphysical as opposed to physical applications."[43]
6.3 WHAT IS, AND WHAT IS NOT, COUNTED IN TRADITIONAL SOCIETIES?
The primary focus of the previous section was the degree of importance with which number and counting are held in a sample of traditional societies. In this section, we focus instead on the following questions:
(1) What are the countable objects in a particular society?, and
(2) in what circumstances are they counted?
In addition, various other questions related to these will be considered, inter alia, where data are available; these are:
(3) Do societies differ in the type of economies that they have and the nature of exchange in which they engage, and do these affect the degree to which counting is carried out?
(4) Are different types of objects counted in different ways?
(5) Are there ways of recording totals?
(6) Are there certain things which are not counted?
In order to address these questions, I will continue the practice of the previous section of using brief case studies to illustrate the ways in which a sample of societies deal with enumeration. A series of NAN-speaking societies, located in both the highlands and in the coastal/islands region of New Guinea, will be considered first, followed by a number of AN-speaking societies located in both New Guinea and Oceania.
6.3.1 The Kaugel Valley People of the Western Highlands Province (PNG)
The people of the Kaugel Valley in the PNG highlands speak the Gawigl or Kakoli dialect of the NAN Hagen language.[44] Their "systems of reckoning" were the subject of an article by Bowers and Lepi published in 1975. The Kaugel people have an agriculture-based economy not unlike that of the Melpa, discussed earlier, another sub-group of Hagen-speakers. Also like the Melpa, the Kaugel people have a complex exchange system in which the counting and tallying of wealth items is of central importance. Bowers and Lepi indicate that "Kaugel kin groups ... formally distribute large numbers of goldlip pearlshells, pigs, game animals, and other valuables. There are two forms of pig distributions; in both, thousands of animals may change hands in a single day."[45] During the ceremony, pigs are tied to stakes in long rows, sometimes half a mile long: "the entire prestation is recorded by one or more core donors: important men who publicly run along the rows in a stylized manner, counting objects in the collective gift."[46] A particular set of terms is used in carrying out this formal, ceremonial count.
The terms used in the formal count are also used to count large numbers of objects in normal, everyday circumstances: the posts needed to build a house or the number of people coming to a feast. These formal counting terms are not, however, the only ones used. There is, for example, a related but somewhat different set of terms used for counting pandanus fruits. Also, Bowers and Lepi note that "different, mainly sequential, numbering systems are used in the Kaugel for ordering or reckoning certain qualities, quantities or social features."[47] There are terms for designating the birth-order sequence of siblings, particularly males. Months are, similarly, counted as a sequence of six pairs of "first born" and "last born". The number of days before or after the present can be reckoned even though the days themselves are not assigned numbers. Finally, Bowers and Lepi make the important point, valid of many of the traditional societies discussed here, that "counting does not exist in isolation. It quantifies and qualifies relations between people, objects and other entities."[48]
6.3.2 The Kewa of the Southern Highlands Province (PNG)
The Kewa people of the Southern Highlands Province (PNG) speak Kewa, a NAN language with three major dialects: East, West, and South.[49] Their language has been studied extensively by two SIL linguists, the Franklins, who, in particular, published an article on "the Kewa counting systems" in 1962.[50] Each of the Kewa-speaking dialect groups possesses a 4-cycle numeral system (an example of which was given in Chapter 5) together with a body-part tally method of the type discussed in Chapter 2; the East and West groups have tallies which utilize 47 body-parts and the South dialect group has a tally utilizing 35 body-parts. While the 4-cycle numeral systems are used in both everyday and ceremonial circumstances for counting pigs, shells, horticultural produce and other items, the body-part tallies are not used for specifying exact numbers: their main function is calendrical. The Kewa have festival and dance-cycles, the preparations for which take place over many months in a strictly ordered sequence: the passing of each month, that is full moon, is tallied on the body-parts and thus these function as ordinal systems rather than counting systems. The Franklins give examples of such festival cycles for the East and West Kewa groups which extend over periods in excess of 40 months.
6.3.3 The People of Kiwai Island, Western Province (PNG)
Kiwai Island, about 100 kilometres long, is situated in the mouth of the Fly River in the Western Province, on the south coast of PNG. The people of the Island speak a dialect of Southern Kiwai, a NAN language,[51] and possess a"pure" 2-cycle numeral system. The numerals of this system are used in the enumeration of relatively small numbers of objects. Smith, in an article on Kiwai counting and classification, says that "the most common method of keeping count of large numbers in traditional Kiwai society was the use of tallies. Tally sticks each representing an object could be kept in a bundle or tied to a string. Tallies were used to represent the number of heads captured by a man in battle, or the pigs killed in the bush, or any other significant number."[52] The Kiwai had ceremonial feasts in which prestations of food and various kinds of wealth items were made. In these, "presents were often displayed on the wooden structure called a gaera and one old informant told me how he once kept a tally of the number of bunches displayed at a feast so that his group could repay slightly more when the return feast was made."[53] In tallying the number of days before a return feast was to be held, the donor and recipient groups would part, each having a bundle with the same number of sticks. Each day, both groups would discard a stick until the bundle was exhausted and the pre-arranged day for the feast had thus been reached.
6.3.4 The People of Rossel Island, Milne Bay Province (PNG)
Rossel Island lies south-east of the New Guinea mainland in the Milne Bay Province (PNG). The people of the Island speak Yele, a NAN language belonging to the East Papuan Phylum, and possess a 10-cycle numeral system (discussed previously in Chapter 4) which shows some AN influence.[54] The Rossel Islanders were studied by the Assistant Government Anthropologist for Papua, W.E. Armstrong, in the 1920s and who, in particular, published an article on the Rossel Island economy and its unique monetary system.[55] Armstrong indicates that there exist two types of shell money: "The one, known as ndap, consists of single pieces of Spondylus shell, ground down and polished ... The other kind of money, known as nkö, consists of ten disks of shell, perforated and strung together."[56] These two types are divided into a number of "denominations": the ndap has 22 main values and the nkö has 16 values. Armstrong says that "Payments of money are, perhaps, the most important constituents of marriage rites, mortuary rites, and many other ceremonial activities. I use the term 'money' advisedly, for the objects [described] are systematically related as regards value; and any commodity or service may be more or less directly priced in terms of them."[57] There is a fixed and limited amount of both types of shell money on the island and there is a good deal of lending and borrowing, sometimes carried out by an intermediary broker. The rules for repayment of loans are complex and involve, say, a given denomination of ndap being lent and then subsequently repaid by a higher denomination according to the time period of the loan. Armstrong notes that, in keeping account of such transactions, "it became obvious that an amount of counting was done ... The frequent ceremonial of stringing nkö together, so as to make long ropes of money, is the reason for the Rossel Islanders' proficiency in counting, for counting up to the thousands would hardly be required in any other department of life ... An investigation of this latter point showed that the native had no difficulty in counting up to 10,000, or even further."[58]
6.3.5 The Ponam Islanders of the Manus Province (PNG)
Ponam Island is located immediately to the north of Manus Island in the Manus Province (PNG). The islanders speak Ponam, an AN language belonging to the Admiralties Cluster.[59] They possess a 10-cycle numeral system of the "Manus" type, discussed in Chapter 4, and also possess numeral classification in which numeral roots are suffixed by a classifier. The Ponam Islanders were studied by Achsah Carrier in the late 1970s and who, in particular, published an article on "counting and calculation on Ponam" in 1981 as a contribution to the Indigenous Mathematics Project. Carrier notes that "In Ponam ... there is only one basic counting system ... The Ponam language does contain a system of numeral classifiers, but ... these are not an alternative counting system."[60] Carrier also indicates that the Ponam have considerable skill in arithmetic: "Almost everyone on the island can count from one to ten thousand in both Ponam and Pidgin."[61] Generally speaking, counting, by adults, is not accompanied by digit-tallying: "Young children are sometimes taught to count on their fingers, though more usually the process is illustrated by counting objects."[62] In many NAN languages it is often the case that certain number words are derived from the names of body-parts. Even in the POC 10-cycle system the word for 5, *lima, is identical to the word for hand. Carrier notes, however, that "Ponams do not associate numbers with the body, even minimally ... Although in Ponam the number five (limef) is almost identical to the word for hand (lime-), the connection is not perceived as significant. No one would ever translate limef as 'hand'."[63]
Carrier reiterates the point made by Bowers and Lepi about the Kaugel, that is that counting does not exist in isolation, divorced from social relations. Carrier indicates that "all important social events on Ponam are the occasion for an exchange."[64] In the exchange of gifts, the Ponam, in common with many other traditional societies, place importance on the visual display of wealth items. Counting of individual contributions made within the donor group is carried out; the total amount of the gift given to the recipient group is, however, not usually announced and often no one knows it. Carrier says "the size of a gift reflects the donating group's sense of its relation to the focal donor and also reflects the group's individual members' sense of relation with their various kinsmen ... the numbers announced in a formal count thus expresses the quality of relationships between groups and individuals rather than the quantitative strength of particular groups or individuals."[65] While the Ponam do count a wide variety of objects, in both everyday and ceremonial circumstances, they do not, however, count people nor do they show any interest in doing so: "apparently these quantifications tell them nothing interesting about social relations."[66]
6.3.6 The Mengen People of the East New Britain Province (PNG)
The Mengen, or Maenge, people live in the Pomio region of the south coast of East New Britain. They speak an AN language, Mengen, which belongs to the North New Guinea Cluster and, in common with all the members of this Cluster, no longer have a POC-derived 10-cycle numeral system, but now possess a system with a (5, 10, 20) cyclic pattern.[67] The Mengen were studied by the French anthropologist Michel Panoff who published an article on "numeration and counting in New Britain" in 1969. Panoff indicates that "in the old days, informants insisted, all things were counted - the booms of outrigger canoes as well as the pegs inserted in the float below, the sticks used in gardening magic as well as the leaves collected for divination purposes, and of course pigs and the shell rings recognized as ceremonial money."[68] In addition to these items, Panoff mentions the counting of coconuts in groups of four, taro tubers in groups of twenty, a large catch of fish distributed in fives, tens, and twenties, and cured tobacco leaves.
Panoff reports that the Mengen, unlike the Ponam, engage in digit-tallying: "Their word for 'counting' is sising ... which primarily denotes the process of tallying off a number with the help of fingers and toes ... Mental counting was practised and called lau (long, lona ...) -sisi, that is 'inside counting', but it was much less frequent than the use of the finger and toe method."[69] In carrying out enumeration "the most common operations were, of course, addition and subtraction, but multiplication was by no means exceptional." These operations were carried out mentally "at least when the result arrived at was not higher than 20 ... In contrast with these three operations, division, which is called tavoanga ('portioning'), can only be done by counting with the aid of markers."[70]
6.3.7 The Arosi of the Solomon Islands
The Arosi live on the island of San Cristobal in the Solomon Islands and speak the AN language Arosi, a member of the South-East Solomonic Group.[71] The Arosi were studied by C.E. Fox in the 1920s and their "numerals and numeration" were the subject of an article by Fox, published in 1931. The numeral system, like all those of the AN languages of the Solomons, is a 10-cycle one and there are distinct words for 100 and 1000. Fox says that, in addition to the basic numeral system, "almost every children's game has its counting song with quite different numerals, for example they may all of them be names of trees; and also many special objects are counted in a different way ... It would appear that objects were once divided into a number of categories; with some things counting only proceeded as far as ten, with some to a hundred, and in the case of coconuts to twenty million."[72] In one category we have yams, taro, bananas, stones, and mangoes, all of which are usually counted in groups of five: there are distinct words for groups of 5, 10, 25, 50, 100, 1000, 10,000, 100,000 and 1,000,000. The members of other categories which are counted somewhat differently include coconuts (counted in pairs), banana shoots, sago palm fronds for thatching, pigs and dogs, opossums, fish, eels, breadfruit, dogs' teeth, bats' teeth, porpoise teeth (the various teeth are counted in fours), and shell money. The last is strung and a length of four fathoms is regarded as a unit; there are special terms for groups of 10, 25, 50, 100 and 1000 units of shell money.
6.3.8 The People of Rennell and Bellona, Solomon Islands
Rennell and Bellona are two islands in the southern part of the Solomon Islands. The inhabitants are Polynesian and speak Rennellese, a Polynesian Outlier.[73] The American linguist Samuel Elbert has produced both a dictionary and a grammar of Rennellese, the latter having some cultural notes on counting. Elbert notes that, traditionally, much of a chief's life "consisted of fishing and raising fine gardens, and presenting the fruit of the land and the sea, carefully counted, first to the gods with impressive rituals and then to relatives and allies. A chief's prestige was gauged by the size of the offerings he was able to amass."[74] Prior to the Second World War, the Rennellese often engaged in competitive gift giving which, in one recorded instance, resulted in the distribution of 10,000 coconuts and 7,600 banana bunches. Another use of counting was in establishing the dimensions of houses, canoes, gardens, and mats. Not everything, however, was counted. Elbert says "No one knew or was at all interested in his own age ... Years and generations were not counted at all. Time was told by looking at the sky."[75]
The Rennellese, as is common with all Polynesians except those speaking Faga-Uvea, possess a 10-cycle numeral system. There are 15 categories of countable objects and the terms for 10, 100, 1000, 10,000, and 100,000 are not all identical for each category. A listing of some of the members of the categories provides an indication of the wide range of countable objects: fish, birds, crustaceans, coconuts, taro, sweet potatoes, bunches of bananas (counted in fours), trees, fathoms of shell money, canoes, and spears. A variety of measurements, each relative to a person's body, exist and are countable: the fathom, for example, is the distance between the fingertips of a person's outstretched arms. Elbert notes that the large numbers, that is those greater than a thousand, "used in food distributions have never been taken too literally but symbolize unfathomably large quantities, which are so admired in counting food."[76]
6.3.9 The Pukapuka of Central Polynesia
The final example in this series of brief counting ethnographies concerns the people of Pukapuka which is located in the central part of Triangle Polynesia, north-east of Samoa. In the 1930s, two American anthropologists, the Beagleholes, studied the culture of this representative Polynesian group and published a monograph in 1938 which includes some data on counting and measuring. They report that the Pukapuka numeral system is a 10-cycle one with distinct terms for 100, 1000, and 10,000.[77] The Beagleholes indicate that there are terms for numbers larger than these and one word, ye, that corresponds to infinity; they add, however, that such terms "indicate not so much a definite number as a definite progression of increasing greatness that is more sensed or felt than definitely apprehended."[78] As with the Rennellese, the Pukapukans subdivide their universe of countable objects into at least 14 categories. Some of the members of these categories are: people, coconut shells, fishhooks, oven stones, fishline sinkers, pandanus leaves, mats, fish, taro, crayfish, crabs and shellfish. Coconuts are counted in pairs and the terms used for this are in some cases identical to the ordinary numerals so that, for example, lua, that is 2, means 4 coconuts. A special term for 5 pairs of coconuts is used, yepulupulu, which is not identical to the numeral 5, lima. As is also common with the Rennellese, the Pukapukans have a series of countable length measurements which are made relative to a person's body and which are commonly used to measure fishlines: a fathom, for example, is ngawa and 10 fathoms is kumi.
6.3.10 Commentary
In discussing the place of counting and number among the Ikwaye people of the Eastern Highlands Province (PNG), Mimica says "It is erroneous to think that the structure of any counting system, its possibilities to generate a series of natural numbers, is somehow explainable in terms of the pragmatic exigencies of social existence. Though people have to count for practical reasons, the primordial roots of numerical expression and its meanings are not determined by that necessity."[79] It seems possible that, on the New Guinea mainland, some traditional societies have existed in relative isolation over several millennia, more or less untouched by outside influence, and whose languages have been subject only to the inevitable and gradual effects of normal linguistic change. In such societies it is likely that the basic structure of their counting systems will remain unchanged and that "numerical expression and its meanings" may well be traceable to its primordial roots. Yet it is clear that, for some of the societies discussed here, "the pragmatic exigencies of social existence" have had an effect on the structure of their counting systems and the degree to which these generate a series of natural numbers. Various AN groups, for example, no longer possess numeral systems with the same cyclic pattern as that of their POC ancestor: this is true of the Adzera, Mengen, and Loboda people. Similarly, there are NAN groups such as the Yele and the Ekagi people who have augmented their numerical lexis and have 10-cycle systems generally untypical of NAN languages. Changes to traditional economies, to the manner in which goods are exchanged, and the concomitant changes in the attitudes as to whether it is important to count marketable commodities, appear to be important factors which may influence the structure of a particular society's counting system, inducing innovations of various kinds.
The counting of important items of wealth serves a social function in providing a numerical indication of prestige, power, and obligation. Many of the traditional societies discussed here engage in within-group ceremonial exchange of valued commodities: the prestige of the donor individual or group depends on the quantity of goods amassed; similarly, the obligation of the recipient group must be determined. Societies vary, however, in the way in which quantities of wealth items are determined. The visual display of wealth is an important element in the exchange ceremonials of virtually all the traditional societies of Melanesia, Micronesia, and Polynesia. In certain of them, the Adzera, Loboda, and Kiwai for example, no particular importance is attached to the precise numerical determination of quantity which is, instead, judged by visually apprehended, impressionistic means. Other societies, however, such as those of the Kaugel, Melpa, Woleai, and Rennellese people, place particular importance on the precise enumeration of wealth items, whether these are pigs, shells, coconuts, or yams. Indeed this is usually achieved by carrying out a formal, ceremonial count of each category of wealth item. It is these totals on which the prestige of a group may depend and on which reciprocal obligation is determined. With the Ponam people, such totals reflect the important kinship interrelationships within the donor group.
Within-society exchange of commodities in ceremonial circumstances is but one example of conditions which may occasion counting in a particular society. The non- ceremonial, but still within-society, marketing of goods is another example. However, in this case, we need to distinguish those societies which rely primarily on the bartering of goods, as opposed to those which have economies that are, to some extent, monetarized. Normally, market bartering does not occasion the counting of commodities but involves instead the establishment of a one-to-one correspondence between different collections of commodities. Salisbury describes the way in which bartering occurs among the Tolai women of Rabaul: "the manner of trade among women was then [traditionally] and is now [1961] a silent one. Each woman sits demurely near her produce, which is divided into separate units - heaps, bundles, packages, or strings. When a trade is made the other party puts down a standard unit of similar value and picks up the exposed goods ... The units are related to one another numerically, most of them on a one-to-one ratio."[80] The Tolai, like many other AN-speaking groups in Melanesia, have a monetarized economy and, in marketing, goods could be purchased with shell money (tabu) as well as obtained by barter. Salisbury indicates that "traded items had standard values in terms of shell money - for example, a kure of taro equaled a string of shell money about 6 inches long."[81] The rates of exchange, however, between various commodities and shell money varied according to the supply of, and demand for, the commodities; Tolai men, though not women, engaged in trading (a nivura) in order to make a profit. In such transactions, shell money could be counted as individual shells or as multiples or subdivisions of the "standard" length, the fathom (a pokono).[82]
All of the traditional societies of New Guinea and Oceania, for which ethnographic data are available, engage in within-society marketing and the most common form of exchange involves bartering in which, on the whole, counting plays little, if any, part. There are, however, societies in which traditional varieties of money, usually shells of some kind, play an important role in market exchange as well as in ceremonial rites connected with birth, initiation, marriage, and death. In market transactions and ceremonial distribution, the counting of individual shells or lengths of strung shells, is an essential feature. Broadly speaking, those societies in which shell money constitutes a principal medium of exchange are AN-speaking groups located away from the New Guinea mainland. Those societies which rely largely on bartering in their within-society marketing and which have economies that are not monetarized to any great degree, are, by and large, the NAN-speaking groups located on the New Guinea mainland. This, then, suggests a dichotomy between the AN- and NAN-speaking groups so far as the degree to which counting occurs during non-ceremonial, market exchange.
Such a broad generalization must necessarily have important exceptions. For example, two societies discussed above, the Ekagi and the Yele, are both NAN-speaking but are such that shell money plays an important and central role in their economies. Oliver, in discussing the Ekagi (Kapauku), says "Like all Oceanians, the Kapauku engaged in several types of exchange including services for services, objects for objects, and objects for services, but the noteworthy thing about Kapauku exchange was the large degree to which it was monetarized - money having been used to purchase not only objects, but labour, use-rights in land, and so forth ... Moreover, while most of the goods purchased had fairly fixed monetary values, immediate factors of supply and demand were on occasion influential enough to encouraging haggling."[83] Similarly, the Yele economy, as described by Armstrong in the 1920s, was also monetarized to a high degree and was such that it was common for money to be lent and repaid with interest under certain fixed rules. In both of these societies, the counting of shell money is of central importance and both, as has been remarked earlier, have well-developed 10-cycle numeral systems generally untypical of NAN-speaking groups.
In traditional societies, it is not uncommon to find that there exists more than one method or set of numerals for counting various objects. The Kaugel people have a primary counting system which is used for most purposes but use somewhat different terms for counting pandanus leaves. The Kewa people have a 4-cycle numeral system which is used to count various wealth items but, in determining the number of months which must elapse before a festival occurs, they use instead a body-part tally method. The AN-speaking Arosi and Rennellese each possess a 10-cycle numeral system similar to that of their POC ancestor. However, in counting certain categories of objects the terms for large numbers, for example 100, 1000, and so on, vary according to the category being counted. The Arosi also have a set of "numerals" used in children's counting games consisting of the names of trees. Such counting-game sets are common in AN-speaking societies and have been recorded by the author for the Tolai and Duke of York (PNG) languages and by Codrington for Mota in Vanuatu.[84] It is also common among AN-speaking groups for there to be a basic numeral system for counting objects singly but that, in counting collections of objects, for example pairs of coconuts or fish strung in fours, somewhat different terms may be used.
While, in each of the societies discussed above, counting of horticultural produce, wealth items, and so on, occurs to a greater or lesser degree, it is also the case that societies vary according to what is not counted. The Rennellese, for example, do not enumerate time periods. The Ponam, on the other hand, do not count people: whether this is the result of a tabu against counting people is uncertain. The tabu against counting various objects has been noted for traditional societies in other parts of the world. Frazer, for example, gives many instances from African societies in which there are tabus against counting men or animals: to count either of these would bring misfortune to them.[85] Seidenberg also lists various societies in Africa, North America, and Europe in which counting prohibitions of certain kinds exist. Indeed, Seidenberg suggests that non-verbal digit-tallying may have originated as a means of circumventing tabus on the verbal counting of people or objects.[86] The evidence for the existence of counting tabus in the societies of New Guinea and Oceania is far from conclusive. It is certain, however, that there are many societies, both AN and NAN, in which there is no prohibition on counting people; whether or not this is generally the case is unknown and further data are required.
In various types of ceremonial exchange it is necessary that both donor and recipient groups are able to recall, at some future time when the exchange is to be reciprocated, the quantities of commodities involved in the original prestation. How, in the absence of written records, is the memory of such transactions kept alive? While we have recorded two basic ways of tallying using either the fingers and toes or other body-parts, such tallies are the means of enumerating quantities or of determining the number of days or months elapsed from some initial starting point. It would seem that some other forms of recording tallies which are to act as a means of recording totals are clearly necessary and indeed there is some evidence for the existence of these in several societies. Wolfers, for example, says that "methods of tallying and recording quantities include cutting notches in a stick, piling sticks or stones together, or tying knots in lengths of twine. Each notch, stick, stone or knot then represents a certain number of units of whatever is being counted ... The Parevavo, inland from the Gulf of Papua, tie a knot in a length of twine for each man killed in battle and unravel the knots as each man is avenged. Some Chimbu groups reputedly kept scoreboards in red pigment on the walls of rockshelters, one mark for each important man killed in battle."[87] Gwilliam reports an unusual method of recording the number of voyages made by a Motuan man in the hiri trading system which involved the tattooing of his wife: "the tattoos covered the whole of her body including her eyelids."[88] It needs to be observed, however, that the keeping of such physical tallies was not necessary for the subsequent recall of totals. Strathern tells us, with regard to the Melpa, that "As a record of the shell-moka gift sets which they have amassed and given away men wear tallies made out of slats of bamboo or cane. Each slat represents a set of eight or ten shells given. The tallies are a sign or demonstration to the public in general of the extent to which their wearers have made moka. They are not an aide-memoire." The moka exchange transactions are remembered not by these physical tallies but by regular recall and discussion: "Such gifts are the subject of frequent conversations at intervals of time in between bouts of public activity, so the memories are kept alive."[89]
6.4 NUMERAL CLASSIFICATION
The linguistic phenomenon of numeral classification is found in many languages on all the major continents of the world. In this section, I will discuss some of the salient features of numeral classification and the ways in which countable objects are categorized. The distribution of this phenomenon in various parts of the world is summarized prior to an investigation as to its occurrence in New Guinea and Oceania, first among the NAN languages, and second, among the AN languages of Melanesia, Micronesia and Polynesia.
6.4.1 What is numeral classification?
How quantity is expressed varies in important ways in different languages, in particular in the way nominal phrases are constructed. In languages like English, the expression of quantity usually has the construction "numeral + noun" as in "three men" or "five houses". There are other languages, however, in which such quantifying expressions contain not only a numeral or quantifier and a noun but also an obligatory "classifier" which indicates the specific class or category to which the noun belongs. In such languages, the universe of countable nouns is categorized into a number of classes, each noun being assigned to a class according to some criterion, although some nouns may be members of more than one class. In quantifying a given noun both the numeral or quantifier and the classifier, which indicates the class to which the noun belongs, must be stated. Thus suppose in a particular language the noun "man" belongs to the class of "animate beings" then an expression quantifying the noun "man" might take the form "quantifier (Q)+classifier (C)+noun (N)", for example "three + [classifier for animate beings] + man", the English gloss of which is, of course, "three men". The word order in this construction, i.e QCN, which occurs in various Amerindian languages, is not the only possibility: Japanese, for example, uses the order NQC. Allan, in his survey of more than fifty classifier languages, indicates that these two constructions, together with an additional two, CQN and NCQ, appear to be the only permissible ones and that the noun never interrupts the nexus of quantifier and classifier so that, for example, the orders CNQ and QNC do not occur.[90] Not all the permissible constructions have the same frequency of occurrence, indeed for all the classifier languages of east and south-east Asia for which I have been able to acquire data, the classifier invariably follows the quantifier, a construction which we shall find exists in certain of the Oceanic AN classifier languages as well.
After identifying the classifier languages of New Guinea and Oceania, I will consider the following questions:
(1) To what extent is numeral classification used in the act of counting the members of a particular class?
(2) What is the nature of the "quantifiers" which occur in quantifying expressions and, in particular, are these the same as the counting numerals?
(3) Is numeral classification employed in all quantifying expressions or only some, for example for a limited range of numerals or quantifiers?
(4) For various languages, what is the word order used in quantifying expressions? Is there a typical word order, say, in the AN classifier languages?
(5) Does the use of numeral classification imply that there is a different "counting system" for each class of countable objects?
(6) Do different classifier languages categorize their universe of countable objects in similar ways?
6.4.2 The geographical distribution of classifier languages
There is a widespread distribution of languages which employ numeral classification throughout the major continents of the world. Allan, for example, reports their existence in Africa and cites the languages Bantu, Swahili, Loka, Luyana, Luganda, Fula, Tiv, and a further ten or so located in different parts of the continent.[91] Various authors have reported on the classifier languages of east and south-east Asia; some examples are Thai, Burmese, Vietnamese, Chinese, Japanese, and the languages of the Mon-Khmer, Nicobarese, and Aslian sub-families.[92] Also, various non-Oceanic AN languages such as Malay, Iban, and those of Indonesia display numeral classification.[93] Among the classifier languages of North America, Allan cites as examples Nootka, Ojibway, and Navajo, among others. In Central and South America there are a number of examples, notably the languages belonging to the Mayan Family such as Tzeltal, Chontal, and Yucatec.[94] Allan also indicates the existence of classifier languages in northern Australia.[95]
6.4.3 Women, fire, and dangerous things
Lakoff, in his study on systems of human categorization and what they reveal about human cognition generally, draws attention to the importance of investigating the categories which exist in classifier languages, and in particular the similarities which exist in the way widely separated and diverse language groups categorize their world.[96] Allan concludes his survey of classifier languages by remarking on the similarities which do appear to exist between the classificatory systems of quite disparate cultural groups. The number of classes employed differs considerably between languages. For example, the Australian Aboriginal language Dyirbal, studied by Dixon and cited in Lakoff, has four: (1) human males and animals; (2) human females, water, fire, fighting; (3) non-flesh food; and (4) everything else not in the other classes. On the other hand, Keller's study of Chontal lists 78 classes and Berlin indicates that Tzeltal may have several hundred.[97] Despite this wide variation regarding the number of classes employed, Allan's analysis of some 50 languages yields that seven major categories of classifiers can be identified, most languages having some or all of these. He describes the categories as being primarily based on: (1) material, (2) shape, (3) consistency, (4) size, (5) location, (6) arrangement, and (7) quanta (including groups, collections, measurements).[98] While it is possible to categorize a large number of classifiers into a relatively small set of semantic domains, it is not always possible to determine precisely what the criterion is for class membership on the basis of semantics alone. For example, in the Dyirbal case, the members of the class of women, fire, and dangerous things hardly appear to have a common semantic basis and indeed Lakoff indicates that it is necessary to understand Dyirbal mythology in order to see the connections between these apparently disparate things and why they are members of a common class.[99]
6.4.4 The occurrence of numeral classification among the NAN languages
There is evidence for the existence of a number of NAN classifier languages. At least four languages belonging to the East Papuan Phylum and which are located in central and southern Bougainville (PNG) possess numeral classification; these are Nasioi, Nagovisi, Siwai, and Buin.[100] Foley says of these that "the most extensive system of nominal classification in Papuan languages is found in Nasioi and perhaps other Papuan languages of southern Bougainville. The system of Nasioi ... parallels in certain respects the numeral classifier systems of south-east Asia ... The classifiers number over 100, and are very specific semantically."[101] In Nasioi the classifiers are suffixed to numeral roots (the quantifiers): 1 is na-, 2 is ke-, 3 is bee-, and 4 is kare-; 5 is invariant in any counting sequence and is unsuffixed: panoko. Hurd's analysis of the Nasioi semantic categories together with the number of classifiers in each category are: (1) social groupings of people (22); (2) body-parts (14); (3) animals and food (8); (4) trees, wood, leaves, feathers (13); (5) ropes and vines (3); (6) bamboo (4); (7) bananas (7); (8) taro (3); (9) coconuts (4); (10) fruit (2); (11) houses, furniture, building materials (14); (12) containers (7); (13) clothes, bags, nets (3); (14) money (2); (15) implements, weapons (7); (16) physical or geographical features (16); (17) locations, areas, paths (8); (18) temporal periods (7); (19) fractions, parts, sections, groups (16).[102]
Other languages which exhibit classification are located mainly in the East and West Sepik Provinces, several belonging to the Torricelli Phylum (Mountain Arapesh, Southern Arapesh, Monumbo, Olo ), the Upper Sepik Stock (Abau ) and the Lower Sepik Family (Yimas). Foley says of these that "The noun classification systems of the Torricelli and Lower Sepik languages ... are among the most complex in the world and represent an extreme development in New Guinea."[103]
6.4.5 Numeral classification among the AN languages of PNG and Oceania
Among the AN languages of PNG, numeral classification occurs most noticeably in the languages of the Admiralties Cluster and the (Peripheral) Papuan Tip Cluster, located respectively in the Manus and Milne Bay Provinces. With regard to the Admiralties Cluster it seems likely that all 24 languages belonging to this group and for which data were acquired exhibit numeral classification.[104] The number of classes which are employed varies between languages: Ponam, for example, has at least 28 while the Gele' (or Kele) dialect of Ere-Lele-Gele'-Kuruti has at least 43.[105] Two features which appear to be common among all the languages of the Admiralties Cluster for which data exist are that, first, the word order in all classifier constructions is "quantifier + classifier" (that is QC), where the quantifier is a numeral root, and, second, that such quantifying expressions are used to state quantities but are not used in serial counting. In determining a quantity by serial counting, a basic set of unsuffixed numerals is used; once the number of objects is known, this may be stated using a classifier construction. The languages differ in regard to the number of quantifiers that must take the obligatory classifier: in Gele', for example, classifiers are suffixed to the numeral roots for 1 to 9 while, in Ponam, classifiers are suffixed only to the numeral roots for 1 to 4, after which the counting, unsuffixed, numerals are used.[106]
The classifier languages of the Peripheral Papuan Tip are Kilivila, Muyuw, Budibud, Sud-Est, and Nimowa, all of which are spoken on islands eastwards from the south-east tip of the mainland.[107] The languages vary according to the number of classes employed: Kilivila has at least 42 while Sud-Est has about 22. The word order used in quantifying expressions is different from that found in the Admiralties Cluster and takes the form CQ, i.e the classifier is prefixed rather than suffixed to the numeral root. However, Kilivila, like the Admiralties languages, has a set of numerals used for serial counting; these do not have a classifier prefix and are used for the counting of yams. Kilivila, as well as the other languages of this group, have a "general" classifier which is frequently used instead of specific classifiers and numeral roots prefixed by this are usually the means by which serial counting of any class of objects (excluding yams) is carried out.
Outside of PNG, numeral classification is exhibited most noticeably in the languages of Micronesia, although Marshallese and, to some extent Kosraean, appear to have lost it.[108] The way in which the classificatory system systems operate in the Micronesian (MC) languages more closely resembles that of the Admiralties languages than that of the Papuan Tip group. While the number of classes employed varies between the MC languages, from 62 in Trukese to only several in Kosraean, we find that all quantifying nominal expressions have the the word order QCN, that is the numeral roots are suffixed by classifiers (and are followed by the noun). Each of the MC classifier languages also has an unsuffixed set of numerals which are used for serial counting; once the number of objects in a set has been determined, this may be stated using a classifier expression.
While these three widely separated groups of AN classifier languages vary considerably in the number of classes into which their universe of countable nouns is subdivided and while some of these classes are clearly culture-specific, this does not necessarily mean that the way in which things are categorized differs fundamentally between languages or groups of languages. The categories which occur in all three groups of the classifier languages are listed here: (1) animate entities (Kilivila distinguishes two animate categories: males versus females and other animals; (2) various body-parts; (3) blades and cutting instruments; (4) long wooden objects, trees, canoes; (5) flat, two-dimensional objects; (6) round, three-dimensional objects; (7) ropes, strings, belts; (8) containers and hollow objects; (9) pools, streams, water courses; (10) paths, tracks; (11) collections of food, bundles; (12) fractions, parts, pieces; (13) measurements, particularly of length; and (14) countable bases, that is tens, hundreds, thousands, etc.[109] It is interesting to observe that all of these categories are mentioned by Allan in his summary of the categories which occur in classifier languages in other parts of the world and of which he says "the recurrence of similar noun classes in many widely dispersed languages from separate families, spoken by disparate cultural groups, demonstrates the essential similarity of man's response to his environment."[110]
The fact that three separate groups of Oceanic AN languages exhibit numeral classification and that their systems have a number of salient features in common, raises the question as to whether they have independently developed their systems as quite separate innovations which have occurred subsequent to the breakup of their common POC ancestor and that the common features which are apparent are indeed due, as Allan suggests, to "the essential similarity of man's response to his environment". Alternatively, it might be suggested that the similarities between these groups of daughter languages of POC are due to numeral classification being originally present in POC and that, while the majority of the descendants of POC have largely lost this feature, these three groups have retained it. The question as to why they should be conservative in this respect may perhaps be partially explained by the fact that they are all island communities which have undergone a degree of separation and isolation from outside influence. If numeral classification was an original feature of POC it would seem reasonable to expect to find vestiges of it remaining in some daughter languages other than those of the three groups in which it is fully displayed. And indeed there is evidence of such vestigial remains among a number of the Oceanic AN languages. Several of the Polynesian languages including Nuguria, Takuu, Nukuoro, Kapingamarangi, Pukapuka, and Rennellese show evidence of possessing "group of ten" classifiers which are prefixed to numerals or numeral roots and which are used to construct the decades from 20 onwards.[111] Similarly a number of the AN languages of the North Solomons Province (PNG), for example Solos, also appear to exhibit numeral classification to a limited extent and are such that the classifier is prefixed to the quantifier.[112] I have also suggested earlier in Chapter 4 that the phenomenon of base-suppletion, which occurs in some AN 10-cycle numeral systems, appears to be due to the use of "group of ten" classifiers which are affixed to numerals. This is apparent in a number of Polynesian languages and in several languages of the Papuan Tip Cluster which have "Motu" type numeral systems.[113] Indeed, according to Lawes, Motu itself has a number of classifiers, including one each for males and females and another for long wooden objects such as spears; these are prefixed to numerals.[114] The validity of the conjecture that numeral classification existed in POC and the question as to whether such classifiers are reconstructable, can only be settled by the methods of comparative linguistics and is beyond the scope of this work.
6.4.6 Summary
Numeral classification, a sub-category of the more general phenomenon of noun classification, occurs among several groups of NAN and AN languages in New Guinea and Oceania. Among the NAN languages it is found most noticeably among the members of the East Papuan and Torricelli Phyla, the ancestral languages of which were probably established in New Guinea prior to the migration of the speakers of the Trans-New Guinea Phylum. This tends to suggest that noun classification found in the NAN languages is likely to be an archaic feature rather than an innovation, although the degree to which it has been elaborated may be innovative. Similarly, I have suggested that numeral classification may have been present in POC and this is, thus, a general archaic feature rather than an innovation which has occurred among a few groups of languages. While the majority of the Oceanic AN languages have subsequently lost numeral classification as a linguistic feature, a few groups of largely island communities have retained it either fully or in part.
Returning to the questions raised in 6.4.1 above, it is apparent that at least among the AN classifier languages, numeral classifier expressions are not normally used in the counting of relatively small numbers of objects and for this purpose a set of serial counting numerals is used. These numerals are not affixed with a classifier. However, classifier constructions are employed when a nominal phrase is used to state the number of objects of a particular class once these have been counted. In counting a larger number of objects, in fact numbers greater than or equal to 10, it is necessary to use classifier constructions in that decades, hundreds, and usually thousands, are expressed by means of classifiers. In the MC and Admiralties languages, for example, 40 is expressed as "numeral root for 4 + decades classifier" while 500 is expressed as "numeral root for 5 + hundreds classifier". Also, in both of these language groups, the serial counting numerals are similar but not identical to the quantifiers which appear in classifier expressions; the quantifiers are a set of numeral roots which remain largely invariant from one class to another. Thus, in Ponam, the first four serial counting numerals are si, luof, talof, and faf, while the corresponding numeral roots, to which a classifier is suffixed, are sa-, lo-, tulu-, and fa-. In fact these four numeral roots are the only ones which are suffixed. In Gele', however, classifiers are suffixed to each of the numeral roots for 1 to 9. It is not the case, then, that the AN classifier languages can be thought of as having a large number of different counting systems, one for each noun class; for most purposes, class-invariant counting numerals are used. Thus, there are no grounds for the sort of criticism, apparent in the nineteenth century literature, which suggests that those traditional societies which have numeral classification have been unable to develop an abstract concept of number, freed from the context in which counting takes place.[115]
From examples given previously it is clear that, at least among the AN classifier languages, that there is a degree of similarity between them in the nature of the noun classes that they possess, and indeed we might expect this if their common POC ancestor also exhibited numeral classification. However, it is also apparent that differences exist between and within the various AN groups, particularly with regard to the numbers of classes each language possesses: within the Admiralties Cluster these vary between 20 to 40-odd while in the MC languages they vary between 3 or 4 to 68. There is, in addition, a basic difference between groups in the word order used in quantifying expressions, the order QC being universal in the Admiralties and MC languages while the order CQ appears to be common among the classifier languages of the Papuan Tip and Polynesian groups. Of the two exhibited word orders, it seems reasonable to suggest that one was present in POC and that the other is an innovation. Ross has indicated that his view is that it is among the Admiralties and MC languages that the innovation has occurred: "the possibly innovative feature is not the use of classifiers, which are reconstructible in POC, but the sequence of numeral + classifier, rather than the reverse".[116] While this may be the case, it is relevant to note that, among the AN languages of the Indonesian region and south-east Asia, the most common form which occurs in classifier constructions is QC which tends to suggest that this may be the more archaic form.
6.5 LARGE NUMBERS IN THE NAN AND AN LANGUAGES
We have seen, in the earlier sections of this chapter, that various societies within PNG and Oceania differ in significant ways according to the importance that they attribute to the enumeration of objects as well as to the types of objects counted and the circumstances in which counting is carried out. These societies also differ according to the extent to which they count: all have the resources enabling them to count at least to 10 and many have the capability of counting in an efficient way to higher orders of magnitude. In this section we consider:
(1) the evidence for the existence of terms for large numbers in the counting systems of the NAN and the Oceanic AN languages;
(2) whether these languages differ in significant ways in the resources available for the expression of large numbers;
(3) whether there is any indication of the origin of terms for large numbers, and in the case of the AN languages, which terms were likely to have been present in Proto Oceanic; and, finally,
(4) whether such terms are true numerals or whether they are largely descriptive or impressionistic in nature.
6.5.1 Large numbers in other indigenous cultures
In the nineteenth century literature on natural language numeral systems, it is not uncommon to find reference to societies which are such that their entire means of enumeration is encompassed by the terms "one, two, many". The numerical horizon of people possessing such "systemless" numeration, it was inferred, was very limited indeed: the scholars of the time, under the sway of social Darwinism, had little compunction in assigning such people to the very bottom rung of the hypothetical ladder of cultural evolution. It is worth noting here that with the data collected for this study, in particular the data relating to 2-cycle systems as summarized in Chapter 2, there is no evidence of the existence of systems which terminate at 2: even in the case of languages possessing "pure" 2-cycle systems, there exists the syntactic resources for the expression of numbers larger than 2. Setting aside, however, the question of the validity of claims that "one, two, many" numeration systems exists and whether such claims were an artifact of the way in which data were elicited, there still persists today, in popular texts on the history of number, the view that many indigenous peoples possessed, prior to the influence of European numeration, only the most rudimentary linguistic means of enumerating their world. There were, nevertheless, nineteenth century scholars such as Tylor and Conant[117] who were aware that there were significant exceptions to this view and that there existed examples of indigenous cultures in Africa, the Americas, and the Pacific, which possessed terms for large numbers ranging from tens of thousands to millions. Conant, for example, notes that "the Tonga Islanders have numerals up to 100,000, and the Tembus, the Fingoes, the Pondos, and a dozen other South African tribes go as high as 1,000,000."[118]
Among the Amerindians of North America, Schoolcraft (quoted by Closs[119]) indicates that "the Dakota, Cherokee, Objibway, Winnebogo, Wyandot, and Micmac could all count into the millions, the Choctaw and Apache to the hundred thousands and many tribes to 1000 or more". Similarly, in Central and South America, "the Aztec, Inca and Maya all counted into the millions". Evidence therefore exists which indicates that various indigenous cultures do have the linguistic resources for the expression of large numbers and we shall consider now whether this is also the case for the cultures of New Guinea and Oceania. In order to do this we shall consider first the evidence for the NAN language groups and, second, the evidence for various sub-groupings of the Oceanic AN languages.
6.5.2 Expressing large numbers in the NAN languages
Relatively simple monomorphemic terms for the expression of numbers greater than or equal to 1000 are rarely found in the NAN language of the New Guinea region. We have seen earlier, in Chapter 5, that the Yele of Rossel Island in the Milne Bay Province (PNG) have distinct terms for the numbers 1000, 2000, 3000 and 4000, and that the thousands from 5000 to 8000 are formed from these.[120] Yele is a member of the East Papuan Phylum and it is largely among other members of this phylum that we find terms for 1000 or more. In southern Bougainville, for example, there appears to be a unique local development among the East Papuan languages Nasioi, Buin and Siwai in which the term for domestic fowl is used to represent the number 1000. This is also true for several neighbouring AN languages such as Banoni, Uruava and Torau.[121] Immediately to the south of Bougainville, in the most northern of the Solomon Islands, the AN language Mono-Alu also represents the number 1000 by the term for domestic fowl while, in the AN language Central-East Choiseul, the same term represents 10,000 rather than 1000.[122] The actual term used is kokolei, (kokoree, kokorako, and its various reflexes) and is AN in origin; thus, for the East Papuan Phylum languages mentioned above, the term for 1000 is an AN loanword. In Nasioi, the reduplicated form kokokokorei is used to represent 1,000,000.
Seven NAN languages are spoken in the Solomon Islands and each of these is a member of the East Papuan Phylum. All languages possess a single term for 100 and in several cases it is clear that the term used is an AN loanword: Lavukaleve, for example, has tangalu and Nanngu has te/ta lau. There are, however, exceptions to this: Mbilua has paizana and Aiwo and Santa Cruz both have tevesiki (tövisiki), neither of which appears to be AN in origin. All of the languages appear to have a single term for 1000 and in several cases these are AN loanwords: Mbaniata, for example, has tina which is commonly used in the languages of the North-West Solomonic group. Savosavo has toga which is found in ten AN Solomons languages while Mbilua has vuro which appears to have been borrowed from the neighbouring AN Ghanongga and which, in Kia, signifies the number 10,000. However not all terms appear to be borrowed: Santa Cruz has siu (jiu) and Lavukaleve has lamukas, for example.
While single terms for numbers of 1000 and larger are very rare among the NAN languages and indeed are largely confined to some members of the East Papuan Phylum (occasionally as AN loanwords), there is nevertheless a number of NAN languages which express numbers of the order of 1000 by means of complex expressions. For example, Sulka, another member of the East Papuan Phylum, has complex expressions for the numbers 400, 800, and 1600 which may be analyzed as having the constructions respectively of 20 x 20, 20 x 40, and 20 x 80.[123] The East Papuan Phylum language Pele-Ata has a distinct term for 100 and a complex expression for 1000 which may be analyzed as 100 x 10.[124] Examples such as these indicate that the existence of single terms for large numbers is not a prerequisite for the expression of large numbers: this may be achieved by the use of complex expressions which may be analyzed as having multiplicative constructions.
6.5.3 Expressing large numbers in the AN languages of Papua New Guinea
The existence of single terms which express large numbers varies considerably among the AN languages of PNG and between their various Clusters as identified by Ross. The North New Guinea Cluster languages, none of which now exhibits the "pure" 10-cycle numeral system of their POC ancestor, have a paucity of simple terms for numbers of the order of 1000 or even of 100. In the West Sepik Province, two neighbouring languages, Ali and Tumleo, have a term for 100, that is raput.[125] Kairiru, spoken in the East Sepik Province, has the term wurol for 100.[126] The languages of this cluster which possess a single term for 100 are, however, largely located in the West New Britain Province, and one, Gitua, is in the Morobe Province. Each of the former, namely Bariai, Kilenge-Maleu, Kombe-Kove-Kaliai, Lamogai-Rauto-Ivanga, Arove, and Asengseng, has a term vuno (and its cognates) for 100. Gitua has ai.[127] The remaining 68 languages of the North New Guinea Cluster show no evidence of possessing single terms for numbers above 20 although a proportion do possess complex expressions for 100 which have the form 5 x 20 or 20 x 5.
In the Papuan Tip Cluster, the 25 languages classified as "Nuclear" and for which we have data show no evidence of possessing single terms for the numbers 100 and above, although 100 is in some cases expressible as a complex multiplicative construction of the form 5 x 20. This is in contrast to most of the languages of the "Peripheral" Papuan Tip for which there is evidence of the existence of terms for 100, and in some cases, much larger numbers as well. Each of the languages Kilivila, Muyuw, Nimoa and Sud-Est possesses a single term for 100.[128] In the Central Province, this is also true of the languages Nara, Roro, Mekeo, Gabadi, Doura, Motu, Sinagoro and Keapara. Motu has single terms for each of 1000, 10,000 and 100,000, while Sinagoro and Keapara both have terms for 1000.[129] For each of the eight languages which possess a term for 100, the term used is sinahu (and its cognates: sinavu, sinau, tinavuna). This appears to be a local development without cognates in other Papuan Tip languages.
The 64 languages of the Meso-Melanesian Cluster, of which 23 belong to the North-West Solomonic group, are such that the majority possess single terms for 100. Some also possess single terms for 1000: this is true for almost all of the North-West Solomonic group. Over the whole cluster, there is little uniformity regarding the terms used for 100. Certain terms are, however, uniformly found among regional groupings of languages. In at least nine of the New Ireland languages, the term for 100 is (a )mar: this is identical to the Tolai word for 100 and I have previously suggested that it is likely that this term was borrowed by a number of New Ireland languages under the influence of Tolai missionaries working in New Ireland in the nineteenth century.[130] In the North-West Solomonic group some 11 languages have the term gogoto for 100: this appears to be a local development and the term does not have reflexes in other members of the Meso-Melanesian Cluster.[131] A further seven languages of the North-West Solomonic group have the term gobi for 100: this, too, appears to be a local development. One term, however, does appear to have an incidence which is not merely localized and is found in various forms in Nakanai in West New Britain (salatu), in seven languages of Bougainville (latus, natus), in Mono-Alu of the North-West Solomonic group (latu), and in several members of the South-East Solomonic group (for example Nggela has hangalatu). Significantly, this is also cognate with the Indonesian/Malaysian term for 100, ratus, and is thought to have been present in Proto Austronesian (PAN) as *Ratus and in POC as *Ratu. [132]
Apart from the three clusters of Ross’s Western Oceanic we need also to consider the languages of the Admiralties Cluster, most of which as we have noted earlier, possess numeral classification. Indeed the terms for which the hundreds exist in these languages are classifier constructions so that , in Papitalai for example, 100 is se-ngat, 200 is ru-ngat, and 300 is tulu-ngat where se-, ru-, tulu- are numeral roots or quantifiers, and -ngat is the "hundreds" classifier.[133] Similarly, there are in a number of the Admiralties languages terms for the thousands which also employ classifier constructions: Andra-Hus, for example, has sa-po for 1000, lu-po for 2000, and tulu-po for 3000. The term for 10,000, however, is pue-sih which does not employ a classifier construction. Similarly, in Titan, while classifier constructions are used for the hundreds, they are not employed for the thousands so that 1000 is pue-si, similar to the Andra-Hus term for 10,000. This discontinuity in the means of signalling quantification may be an example of what Hurford terms a "growth mark" of a numeral system, that is a point which at one time would have been a limit of counting but which now represents a transition point between one means of enumeration and another which extends the previous limit.[134]
The discussion of terms for large numbers in the AN languages given above has concentrated mainly on those for the hundreds: terms for 1000 and more are much less common. Indeed they appear to be virtually absent from the languages of the North New Guinea Cluster and largely absent from the Papuan Tip languages with the exception of Motu, Sinagoro and Keapara in the Central Province, for which the terms for 1000 are, respectively, daha, dagatana and ragana.[135]
Terms for 1000 are found more commonly in the languages of the Meso-Melanesian Cluster and in particular among those of the North-West Solomonic group. In the latter, nine languages have tina for 1000 while seven have toga. For the AN languages of Bougainville, reference has already been made to the use, in at least three cases, of the word for domestic fowl in representing 1000; in four other languages the terms piku, tapan, and tuku are employed. In the East New Britain Province, both Tolai and Duke of York have arip. Each of the terms used in the various clusters appear to be localized occurrences, without cognates in languages outside a relatively small region.
6.5.4 Expressing large numbers in Island Melanesia, Polynesia and Micronesia
In the Solomon Islands, the 56 AN languages spoken comprise 23 belonging to the North-West Solomonic group, 22 belonging to the South-East Solomonic group, 6 belonging to the Eastern Outer Islands group, and 5 Polynesian Outliers.[136] The North-West Solomonic group belongs to Ross’s Meso-Melanesian Cluster and was discussed above. With respect to the members of the South-East Solomonic group, all have single terms for the numbers 100 and 1000; several have terms for larger numbers as well. The terms for 100 may be divided into three groups, each with its own variants: (a) sangatu (thengetu, hathangatu), (b) talange (talanae), and (c) tangarau (tangalau). Of the last, Codrington notes that "the most common word in use in Melanesia, as in Polynesia, is rau a branch or leaf". He indicates that the explanation for the use of this term arises from the practice of using a frond from the cycas tree to tally days: "beginning on one side of it a leaflet was counted each day, one being pinched down as a tally for every tenth. The frond when treated in this way on both sides furnished tallies for a hundred ... The same practice is found in the Solomon Islands, where ... not the simple rau but tangarau is the word in use." [137] In POC, the reconstructed form is *dau which has the meanings "hundred, unit of hundreds" as well as "leaf".[138]
Codrington’s assertion that this term for 100 is the one most commonly used in Melanesia is not in fact correct: it is found only in eight languages of the South-East Solomonic group, in Fijian (ndrau) and Rotuman (ta rau). It is, however, found extensively in the Polynesian languages. A different term for 100, but one which still has the meaning of palm frond, is found in the languages of the northern islands of Vanuatu. This is meldol (melnol, medol) and was also recorded by Codrington.[139] There is no one term in the South-East Solomonic group which is predominantly used to signify 1000: toga (toha) is used in seven languages, toni (to’ani) is used in six, and several other terms (sinora, meru) are used in one or two languages. Finally, a term for 10,000, mola, is found in several languages while in Arosi the term used is husia.
In the languages of Vanuatu, the data available for terms for large numbers are less complete than those available for the Solomon Islands. It is clear, however, that in the northern islands at least most languages have terms for both 100 and 1000: these are, respectively, meldol, (melnol, medol), mentioned above, and tar (ter). In the Sakao language the latter term is used for 100 rather than 1000. In North and South Efate, the term manu is used for 1000 and this is commonly found in the Polynesian languages; indeed, the presence of this term in the Efate languages may be due to the influence of the neighbouring Fila-Mele and Emai Polynesian Outliers.
The situation in New Caledonia appears to be not unlike that occurring in the North New Guinea Cluster of Western Oceanic, that is distinct terms for 100 and 1000 are rare although more complex expressions for 100 exist which tend to have the form 5 x 20.
It is among the Polynesian and Micronesian languages that we most commonly find single terms for very large numbers. All Polynesian languages for which data exist possess terms for 100: in at least 24 languages the term is rau (and such variants as lau, selau, 'au, etc.). Similarly, all languages appear to have terms for 1000, the most common being mano. At least six languages have, instead, afe, and for several of these the term mano is used not for 1000 but for 10,000. Terms for numbers of the order of 100,000 and more are not uncommon. The American anthropologists P. and E. Beaglehole reported of the Polynesian Pukapuka that "it was a favourite jest among informants that the Pukapuka could count to a higher power than we could; proof of this they argued was not only the presence of words indicating progressions to infinity, but also the ability of the culture hero Maui to find Pukapuka words which enabled him completely to enumerate the stars in the sky, the fish in the sea, the sands on the beach, and so forth."[140] Among the Mangareva, another Polynesian group, the ethnologist Sir Peter Buck (Te Rangi Hiroa) indicates that the terms used for large numbers when counting men, houses, boats, stars, and so on, are:[141]
rau 100
ten rau mano 1000
ten mano makiu 10,000
ten makiu makiukiu 100,000
ten makiukiu makorekore 1,000,000
ten makorekore maeae 10,000,000
Terms for such very large numbers as those given in the latter part of the table are not uncommon among some of the PN languages. Indeed, in Nukuoro, one of the PN Outliers, there are terms for increasing orders of magnitude up to 1010. These are: [142]
106 : seloo
107 : sengara
108 : semuna
109 : sebugi
1010: sebaga
It should be noted, however, that terms for very large numbers are not restricted to the PN languages only but have also been reported for several languages of the South-East Solomonic group, for example Arosi and Mbughotu. Fox reports that, for the Arosi, the term for 106 is raurauni ha’aro and that for 107 is e ahusia. [143]
Among the Micronesian (MC) languages, most languages for which we have data possess terms for hundreds and thousands. The MC languages possess numeral classification and have both "hundreds" and "thousands" classifiers. In Puluwat, for example, the former is - pwukuw and the latter is -ngeray: each of these may prefixed by a numeral root from 1 to 9. Several languages have terms expressing higher orders of magnitude than these. Kiribatese, Ponapean and Woleaian, for example, have the following:[144]
Table 59
Showing Terms for Increasing Orders of Magnitude in
Three MC Languages.
____________________________________________________________________
KIR PNP WOL
__________________________________________________
104 terebu nen sen
105 tekuri lopw selob
106 teea rar sepiy
107 tetano dep sengit
108 tetoki sapw sangerai
_____________________________________________________________________
6.5.5 The origin and development of terms for large numbers
The data presented in the previous sections suggest that the degree to which the languages of New Guinea and Oceania possess the resources for the expression of large numbers varies both between and within the NAN and AN language groups. Generally speaking, the NAN languages rarely possess single terms for the expression of numbers of the order of 100 and there is no evidence to suggest that any NAN language possesses terms for the expression of numbers of high order of magnitude of the sort found in the PN and the MC languages. Those NAN languages which do have single terms for the order numbers of 100 and 1000 belong, generally, to the East Papuan Phylum. Furthermore, not all East Papuan Phylum languages exhibit this property: only certain languages possessing a 10-cycle numeral system have the resources for the efficient expression of large numbers. I have suggested earlier, in Chapter 4, that those NAN languages which do possess 10-cycle numeral systems have developed these as a result of the influence of neighbouring AN speakers. Indeed, in some cases, though not all, the terms for large numbers are AN loan words. The borrowing of terms is only one manifestation of influence, however. A more fundamental aspect of influence is the adoption of the principle of denoting numbers, which might otherwise be expressed in complex multiplicative constructions, by a simple expression or a single word. Thus we have, for example, the replacement of a complex expression of the form 10 x 10 by a single term denoting 100 and, by extension, the replacement of expressions of the form 10 x 10 x 10 or 10 x 100 by a single term for 1000. It is this principle and not merely the possession of a 10-cycle system, which enables numeral systems to be extended indefinitely in an efficient way. It is important to point out, however, the replacement principle is most commonly a feature of 10-cycle systems and while, in theory, it could be applied in other systems, for example those with a (5, 20) cyclic pattern, this does not generally appear to be the case.
It is among the Oceanic AN languages that we most commonly find single terms for large numbers. Yet even among the AN languages there is a considerable variability: some groups such as certain Solomon Islands languages together with the PN and MC languages have resources for the expression of numbers of high orders of magnitude; other groups, such as the North New Guinea Cluster, do not have these resources and many indeed lack terms for numbers of the order of 100. For those AN languages which do have terms for 100 and 1000, there is no obvious uniformity in the terms used. This raises the question whether terms for these numbers were present in POC. As indicated above, two different words for 100 have been reconstructed for POC: one is *Ratu, present in PAN as *Ratus, and the other is *dau. The fact that two words are reconstructible, rather than one unique word, reflects the lack of uniformity among the daughter languages of POC in the terms used to denote 100. There is, among the PN languages, a greater degree of uniformity in this respect: lau, and its cognates, are most commonly found. But even among certain of the PN languages the word lau denotes not 100 discrete objects but refer, when counting collections, to 100 pairs or 100 quartets of objects, that is to 200 or 400 items. The Polynesian Outlier Rennellese has the term gau for 100 which is used when counting various classes of objects; it is not used universally, however: mano is used when counting piles of bananas, huata is used in counting panels of thatch, and kauhusi is used in counting pairs of yams or breadfruit.[145] This sort of variability in the terms used for numbers when enumerating various collections of objects is not uncommon among the Oceanic AN languages. In Arosi, in the Solomon Islands, for example, Fox indicates that tangarau (a reflex of POC *dau) is seldom used except for counting men and coconuts. In counting houses, quartets of bats’ teeth and pairs of yams the term for 100 items is ‘arangi. In counting banana shoots the term used is umuumu; for pigs and dogs the term is nahomera.[146]
A similar situation applies when we consider the range of terms used to express 1000 in a given AN language. In Rennellese, Elbert indicates that generally noa is used in the enumeration of various objects or collections of objects. However, in counting pairs of yams and breadfruit, ahe is used.[147] In Arosi, Fox reports that meru is normally used to denote 1000 but that the term for 1000 yams, taro, bananas, stones or mangos is wawaibe’o (the same term is used to denote 1000 units of shell money where one unit comprises four fathoms.) Coconuts are counted in pairs and 1000 pairs is bwera. The term for 1000 pigs or dogs is hagahaga.[148] Among the PN languages, we have indicated earlier that the most common term used for 1000 is mano. The same term may be used to denote 1000 pairs or quartets: in Hawai’i, for example, mano is used for denoting 4 x 1000 objects while, in Tongareva, mano denotes 2 x 1000 objects.[149] In at least four PN languages, mano is used to denote not 1000 but rather 10,000: this is the case in both Tongan and Tokelau, for example, in which the term afe is used to denote 1000.
The foregoing discussion of the Oceanic AN terms used for 100 and 1000 highlights the difficulties in attempting to reconstruct the historical development of the AN numeral systems and, in particular, the nature of the POC numeral system and the extent to which terms for large numbers were present in POC. We have, for example, the possibility that the practice of counting various categories of objects existed in which category-specific terms were used for enumeration rather than one unique set of numerals. This practice is widespread among the AN-speaking societies of Melanesia, Polynesia and Micronesia and it is not unreasonable to assume that it existed in the POC community. If this were the case then it is likely that there existed more than one term for the expression of each of the large numbers of the order of 100, 1000 and more. Thus, for example, the term for 100 used in enumerating yams or taro may have been different from those used in enumerating units of shell money or pairs of breadfruit or quantities of eggs. The fact that at least two terms for 100 are reconstructible in POC, as indicated earlier, may well derive from this. Similarly, it is possible that more than one term for 1000 existed in POC. The reconstruction of such terms, however, is complicated by at least two factors. The first is that it is difficult to nominate even a single candidate from the reflexes occurring in the POC daughter languages from which a term for 1000 might be reconstructed. For example, while the term mano is common among the PN languages it is not commonly found among the other Oceanic AN languages. The second factor arises from the uncertainty as to whether terms for numbers of the order of 1000 and more actually denoted precise values or whether they were used to describe indefinitely large quantities. Elbert, for example, in discussing the Rennellese, says that "the large numbers ... used in food distributions have never been taken too literally but symbolize unfathomably large quantities, which are so admired in counting food."[150] Two types of evidence suggest this possibility. First, in certain languages a given term is used to denote 1000, while in other languages the same, or a similar, term is used to denote, say, 10,000. Thus in the PN languages, mano, is commonly used to denote 1000 but in several cases it denotes rather 10,000. The second type of evidence derives from the semantics of certain terms used for large numbers for which the general meaning is "countless" or "indefinitely large". Harrison and Jackson, for example, provide instances from the MC languages in which the nominal interpretation of terms for large numbers includes such meanings as "end", "limit", "sand", "soil", and so on.[151]
In summary, then, the languages of New Guinea and Oceania do not present a uniform picture with regard to the presence of simple terms for large numbers of the order of 100, 1000, and so on. Among the NAN languages such terms are not generally found although there are a few exceptions: among members of the East Papuan Phylum, for example. These languages also possess 10-cycle numeral systems which thus have (10, 100, 1000) cyclic patterns. Several of these languages have clearly borrowed their terms for large numbers from AN sources, however this is not true in all cases. Other NAN languages clearly have the resources for expressing numbers of the order of 100 but this is often done by the use of complex expressions rather than of single terms. The conclusion which might be drawn is that, generally speaking, the languages ancestral to those belonging to the various NAN phyla now found in the New Guinea region did not possess single terms for large numbers and that those languages which now possess such terms have acquired these by either direct or indirect AN influence.
It seems likely that POC possessed single terms for large numbers at least of the order of 100 and 1000. Also, as is common among many of its daughter languages, POC possessed a variety of terms for counting collections of objects. The outcome of this is that there may exist not just one unique term denoting 100, say, but rather several terms, depending on the objects being counted. This possibility needs to be taken into account while attempting to reconstruct the POC terms for 100 and 1000. Among the daughter languages of POC the evidence tends to suggest that some terms for large numbers are found only in certain regional groups: the inference which may be drawn is that such terms have been invented and have gained local currency rather than their having been derived from POC. It is also apparent that there have been at least two other post-POC developments with respect to the expression of large numbers. One of these is that the languages of the North New Guinea and "Nuclear" Papuan Tip clusters have lost their terms for numbers of the order of 100 and 1000 as indeed they have lost their POC-derived 10-cycle numeral systems. The second development appears to have occurred among certain Oceanic AN languages located largely outside the New Guinea region, most noticeably the PN and MC languages. This is the extension of their 10-cycle numeral systems to include single terms for numbers of the order of 104 and more. Harrison and Jackson review data which "suggest that the higher ten-power bases have to some degree a history distinct from that of the lower. In our view, they may have developed as numbers at a more recent historical period." They also suggest that "the systems found are the result of a number of independent innovations."[152] If this view is correct, then it seems possible that POC did not possess terms for numbers of this magnitude. The other, less parsimonious, possibility is that POC did possess such terms but that these were largely lost and that the PN and MC languages have subsequently invented new terms.
It seems likely that certain terms now used for large numbers in the Oceanic AN languages did not originally denote numbers but rather had meanings which referred to imprecise amounts, collections of objects, or limitless quantities. Thus in the MC languages we have the Kiribatese words for 107 and 108 have the meanings, respectively, "sand or soil" and "end".[153] Elbert, in discussing the Rennellese word nimo, indicates that this "theoretically means a million, but it is sometimes used for impossibly high numbers, such as the national debt. It ordinarily means 'to forget' or 'to disappear'."[154] A number of languages possess a term for 100 (lau, meldol) which originally referred to a palm frond and had the meaning leaf: the implied quantity being the leaflets of the palm frond. Within a particular group of languages, a certain word which originally had the meaning of some large but unspecified quantity was eventually use to denote a specific number. The number denoted, however, was not necessarily the same across all languages in the group. Thus, in the PN languages, mano is not commonly used to denote 1000 but in several languages it denotes 10,000 instead. Similarly in several languages of the North Solomons Province (PNG), kokolei (and cognate terms) is used to denote 1000 but in Central-East Choiseul (Solomon Islands) a similar term denotes 10,000.
It may be that such terms are used to denote numbers which represent the approximate practical limits of counting in these cultures: in some cultures this may be of the order of 1000 while in others the limit may be larger. While it is certainly the case that some Oceanic AN societies have terms for much larger numbers than these, it seems unlikely that, even for ceremonial occasions in which large quantities of food or shell-money might be involved, counting up to the order of 105 or more would take place. As Beaglehole and Beaglehole observe about the presence of terms for large numbers among the Pukapuka: "it is a little hard to see the function of high numerical concepts in an atoll culture ... It is likely, however, that such words ... indicate not so much a definite number as a progression of increasing greatness that is more sensed or felt than definitely apprehended."[155]
Map 16. Location of societies cited in text (PNG)
Map 17. Location of societies cited in text (Irian Jaya)
Map 18. Location of societies cited in text (Island Melanesia and Polynesia)
Map 19. Location of societies cited in text (Micronesia)