CHAPTER 9
FINALE
9.1 SUMMARY AND CONCLUSIONS
Twenty-five years ago, on a hot March afternoon, I sailed across the Huon Gulf in a large outrigger canoe accompanied by a dozen or so people returning from market, some worried chickens, and a pig. I was going to spend some time in Labu Tali village and arrange to have a small outrigger made so that I could sail on weekends and visit the nearer villages on the margins of the Gulf. I had recently begun the habit of recording a few items of vocabulary: greetings, items around the village, numerals; I did this also at Labu and later filed away the notes in my beginning collection of numeral data. Some twenty years later I found myself back there for a weekend. The two old men who had been my informants all those years ago had both since died. To continue what had by now long become a habit, I recorded the numerals and other information from some younger informants who had some difficulty in counting in their own language beyond the first few numerals, eventually resorting to the use of tok pisin numerals instead. I had found this to be a common experience: within a generation it was apparent that the traditional ways of enumeration in many villages had atrophied and that this largely irreversible process was well advanced.
An important and primary aim of this study, therefore, has been the documentation of the natural language numerals and, where possible, other aspects of traditional enumeration in various societies in Papua New Guinea, using the data that I had collected since the late 1960s. The results are given in Appendices A, B, and C. Much of this material has not been previously published.[1] In addition to this, I have attempted to put together in one place the data acquired from a wide range of published sources, many not easily accessible, from the linguistic and ethnographic literatures as well as reports from the early colonial administrations. I have sought, in particular, information that was recorded at first or early contact: in some cases this derives from the early colonial period beginning in the 1870s, however, given that some parts of the highlands region of PNG were not opened up until after the Second World War, some first contact material derives from as late as the 1960s. It was apparent that, even in the early colonial period, changes to traditional enumeration were occurring as a result of the influence of the 10-cycle systems of English and tok pisin. However, such changes have been particularly marked in the past two to three decades and they have accompanied changes to the traditional politico-economic institutions of many societies as well as increased access to schooling. It would certainly be the case that some of the data recorded here would no longer be recoverable if this project were started today and that, in an important sense, this study is not reproducible.
While the data collection has been carried out over a long period of time, the writing up of the material obtained has occurred within the time-span of this study. In addition, the information from published sources on the counting systems of languages spoken outside PNG, in the remainder of Oceania, was collected and written up in the period from 1990 to 1992. The material thus amassed in the four volumes of appendices and which provides the data-base for this study represents the first comprehensive collection of natural language numerals for the region encompassing New Guinea and Oceania. Theodor Kluge's unpublished collection, which was compiled from published sources some fifty years ago, provides an earlier but less complete account. The survey of the various types of counting systems, the description of their structural features, and the delineation of their geographical distribution throughout the region, is a distillation of the material given in the data-base: this is the substance of Chapters 2 to 5 and provides the background for the analyses given in Chapters 7 and 8.
The total number of languages spoken in New Guinea and Oceania is approximately 1200 and data have been acquired for three-quarters of these. Generally speaking, the picture that we obtain of the tallying and counting situation in this region is consistent with that found in the other major regions of the world: the Americas, Africa, and Asia in particular. The means by which human societies enumerate their world is not infinitely varied. The 2-cycle systems, the (5,20) digit tally, and the 10-cycle system are found in New Guinea and Oceania as they are found elsewhere: so too are a number of variants of each type. This, however, is not to suggest that the process of enumeration in human societies is one of dull uniformity: we have surveyed evidence of the existence of unique and unusual methods of tallying and counting displays of wealth carried out in ceremonial circumstances which attest to the inventiveness of indigenous societies in elaborating an activity which might usually be regarded as mundane. Nevertheless, in the end, despite the richness apparent in the way counting and tallying is manifested, the most striking feature of this aspect of such societies is the shared similarities rather than the differences.
I have suggested that the 2-cycle and the 10-cycle counting systems, together with the (5,20) digit tally, share a primary status. In addition it seems likely that the complex body-part tally methods which appear to be unique to Australia and New Guinea may also be assigned primary status: there is no reason to assume that these are the more complex ancestral forms of the digit tally. It is the interaction of these primary means of enumeration which produce the secondary, or hybrid, types. Thus the (5,10) and (5,10,20) systems are secondary derivatives resulting from the interaction of the 10-cycle numeral system with the (5,20) digit tally. The (2,5) or (2,5,20) systems, together with their variants, denoted 2' and 2" in Chapter 2, are secondary derivatives of the 2-cycle numeral system and the (5,20) digit tally. The users of the body-part tallies usually have, in addition, either a 2-cycle variant numeral system or, as we find in the Southern Highlands Province (PNG), 4-cycle systems. Occasionally we find examples of hybrids of the 2-cycle systems and the body-part tallies which are such that the numerals 1, 2, 2+1, and 2 + 2 displace the first four names of the tally points. With the exception of this last case, the various systems that I have suggested belong to the hybrid class are found in many indigenous societies in other parts of the world. The most economical interpretation for their existence is that the same mechanism applies for the generation of such systems as applies in New Guinea and Oceania, that is that the (5,10), (2,5), and other systems found, say, in the Americas and Africa, are also hybrid systems rather than primary systems in their own right.
In the works of some nineteenth century scholars writing on the counting systems of indigenous societies, it is not uncommon to find disparaging or dismissive remarks which compare such systems unfavourably with the highly abstract number systems used in modern technological societies. This is particularly noticeable when reference is made to systems with a relatively small primary cycle. Conant, for example, says that "at first thought it seems quite inconceivable that any human being should be destitute of the power of counting beyond 2. But such is the case; and in a few instances languages have been found to be absolutely destitute of pure numeral words".[2] Similar statements can be found in modern text on the history of number: Struik's Concise history of mathematics, reissued in 1987, for example, speaks of number systems which essentially comprise the words "one, two, many". The evidence gathered for this study does not provide any basis of support for such statements. While it is certainly the case that the expression of grammatical number in the personal pronouns of many NAN and AN languages involves the use of singular, dual, and plural forms, that is a "1, 2, many" construction, I have been unable to find a single instance of a language which has only the numerals 1 and 2 and which terminates precise enumeration at 2. As noted in Chapter 2, pure 2-cycle systems are found in New Guinea although they comprise only about ten percent of the 2-cycle variants. Pure 2-cycle systems, however, do not terminate at 2: higher numerals are formed as compounds of the basic numerals. It is also the case that 2-cycle systems are often associated with tallies: either the body-part tally or the digit tally. The possession of a relatively small primary cycle system does not imply that the users are unable to count to any extent or that they find counting difficult. As Fortune has noted with respect to the Mountain Arapesh people: "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 and better than the Melanesian Dobuans, who use a decimal system with many more root terms".[3]
A further criticism of certain indigenous counting systems asserts that they are tied to the concrete or qualitative aspects of the objects counted and do not possess the quality of abstraction characteristic of true number systems. This type of criticism has its origins in two different phenomena found in certain languages. The first is numeral classification, discussed in Chapter 6, in which a set of numeral roots is affixed by a classifier signifying the class to which the objects being enumerated belong. It is, however, normally the case that languages which employ such classifier constructions also posssess a set of numerals which are used in the serial counting of any class of objects. The second phenomenon which some writers suggest indicates a lack of an abstract number system is the use of different names for multiples or standard collections of objects. Thus while one term may be used to denote 100 coconuts, a different term may be used to denote 100 dogs' teeth. The assignment of names to specific quantities of objects is most commonly found in the AN languages with well-developed counting systems having terms for large numbers. Barnes makes the point that "in branding the use of special objects or words for mensuration as primitive, writers such as Cassirer and Levy-Bruhl neglected to consider the respective requirements of oral versus written arithmetic. In the absence of writing, the employment of heaps or pairs of objects, as well as mnemonic aids such as standard multiples, serves purposes similar to the marking of figures with pencil or stylus".[4]
The concept of number and its practical applications play such an important and integral part of modern technological societies that it is often used as a means for judging the degree to which a given society, particularly an indigenous society, has attained intellectual sophistication. In Chapter 6 I have argued that indigenous societies differ in the degree to which number is accorded a privileged position. Thus, while a particular society may possess the linguistic means for enumeration it does not follow that the members of that society value, or constantly engage in, the process of precise quantification. There are many societies in which individuals or groups engage in the accumulation of wealth items which are, in special ceremonial circumstances, displayed and distributed. While it is the case that certain wealth items which are accorded significant value, such as pigs, are counted precisely, other items such as collections of bananas, yams, taro, and the like, tend to be judged impressionistically. Ceremonial prestations involve displays of wealth which accord status and prestige to the donors. The recipient group must retain, in memory or tally, an account of goods distributed so that, at some subsequent time, a reciprocal prestation is made to the original donor group. Such accounting, however, varies from one society to another: some place importance on precise quantification of the goods involved while others do not. The brief counting ethnographies given in Chapter 6 provide examples of societies which differ according to the importance that they accord number and counting, ranging from the Ekagi, with a virtual obsession for counting, to the Loboda who have, according to Thune, an essentially non-numerically oriented culture.
Ceremonial exchange within a given society is but one example of circumstances which may evoke the counting of goods. Another example is the trading of commodities. Here again, however, societies vary according to the degree to which marketing may involve counting. The bartering of goods is based on establishing equivalence between different types of commodities and and requires a sophisticated sense of quantity and relative value: in practice, however, it does not usually evoke the counting of the individual objects involved. By contrast, those societies which have a monetarized economy in which traded items have a standard value in terms of the traditional currency, usually shells of some kind, do count in marketing and in ceremonial circumstances which involve the distribution of money: either individual shells or standard lengths of strung shell money may be counted. Such societies are most commonly found among the speakers of AN languages although there are a few instances of NAN-speaking groups having monetarized economies as well. Generally speaking, it is in these societies that we also find terms for large numbers of the order of a thousand or more.
The similarities in the various types of counting systems that are found in widely dispersed languages located in the world's major continents gave rise, in the late nineteenth and early twentieth centuries, to speculation on why such similarities should exist. The independent inventionist view was that people everywhere invented their numeral systems: their similarities were due to common responses to the human condition. By contrast, there was the view that there was a time when language groups ancestral to those existing today did not possess the means of enumerating their world. Counting was invented under certain special circumstances in one of the ancient centres of civilization from which it was diffused all over the world. This occurred not once but several times: after one primary system of counting was invented and diffused, further primary systems were developed and diffused as well. In some cases those systems which were diffused earlier were unaffected by subsequent diffusions and remained intact. In other cases the earlier systems were displaced by subsequent ones or were modified in a way which produced secondary or hybrid systems. This interpretation of how counting systems came to exist in human societies and which might be termed the "strong" diffusion hypothesis was first enunciated in detail by Seidenberg in 1960. Seidenberg's account of the prehistory of number has not been seriously challenged and still remains the prevailing view which finds its expression in recently published texts.
There are several components to the Seidenbergian hypothesis. First, there is his argument that counting had its origin in special ritual circumstances: this is elaborated in an article published in 1962. This aspect of his work has not been addressed here as I am not persuaded that the data collected for this study can be used to support or deny speculation on the origin of counting. Second, there is that aspect of Seidenberg's work which deals with the type and nature of the counting systems and tallies which were invented and diffused, the essence of which is set out in his "genealogy" as given in Figure 1 in Chapter 7. Seidenberg's view is that the pure 2-cycle system was invented first. Subsequently the neo-2 or neo-2-10 system, of which there are two types, was invented; the diffusion of this system resulted either in the displacement of the original pure 2-cycle system or in its being left intact.
Seidenberg's explanation of how body parts came to be used for the act of enumeration is somewhat complex. He suggests that it has its origins in the practice of "parceling out various parts of the body to various gods" and was not initially associated with counting at all. Counting was essentially a linguistic act rather than a physical one. However, in certain societies there was, or there developed, a tabu on the verbal counting of particular things: it is often the case that the counting of people, for example, is proscribed. It was in circumstances such as these that verbal counting could be circumvented by the use of non-verbal, gestural tallying: the practice of parceling out the body to various gods and the act of counting came together in order to be able to enumerate objects under the verbal counting tabu. Two main types of tallying resulted from this: the complex body-part method and the simpler digit tally, the former being historically prior to the latter. After the diffusion of the (5,20) digit tally, Seidenberg suggests that when the users of the tally came in contact with people possessing 10-cycle systems there developed such hybrids as the (5,10) system.
The third component of Seidenberg's work is that which places the invention of the primary counting systems in an historical context. His view of the invention and diffusion of counting and tallying has its origins in the work of Lord Raglan. Raglan places the locus of the invention of various "civilized" practices in the ancient city-states of Sumeria which developed more than 5000 years ago. Seidenberg suggests, following Raglan, that it it is this time and place that we find the beginnings of counting and that in the succeeding millennia the various counting systems and tallies were invented and diffused throughout the world although the details of the diffusion process are not elaborated. In Chapters 7 and 8 this study has addressed certain issues relating to the diffusionist view of counting, Seidenberg's in particular, and by an analysis of the current counting system situaition in New Guinea and Oceania attempts to outline a prehistory of counting for this region. The conclusions reached provide some measure of support for Seidenberg's views in certain respects; however, in other respects there are some significant departures from his views.
The reconstruction of the prehistory of counting in New Guinea and Oceania has been based on results deriving from work in the disciplines of archaeology and historical linguistics. Broadly speaking, the peopling of the region resulted from a series of migrations beginning with the Australoids about 50-60,000 years or more ago. These moved into the New Guinea region and southwards into Australia at a time prior to their geographical separation after the last ice age. At a much later date, perhaps 10-15,000 years ago, the first Papuan (NAN) language groups moved into New Guinea. I have suggested that both the 2-cycle system and the (5,20) digit tally were present in the early history of the NAN languages and indeed were likely to have been introduced as part of the cultural baggage of these early immigrants. It may also be the case that the 2-cycle system entered Australia with the early Australoid migrations. The implication that the 2-cycle system has historical priority over other systems is in agreement with Seidenberg's view; however I have suggested that its origins lie much further back in time than than posited by Seidenberg. The possession of a basic numeral stock and the means by which the basic numerals may be compounded to form larger numerals is seen in this study as an archaic feature of human language and not as a relatively recent invention. The presence of the 2-cycle system in South America, Africa, and parts of Asia may be interpreted as being the result of its being historically ancient and of its being carried into the major continents with the original migrations rather than being introduced at a much later time as a result of diffusion.
I have suggested that the (5,20) digit tally was present at an early time in the NAN languages of New Guinea and indeed may have entered the region with the original NAN immigrants. The more complex body-part tally, however, may have been introduced to New Guinea by the migration of the speakers of languages ancestral to those which now constitute the Trans-New Guinea Phylum. It is possible that this tally was also diffused southwards into the Australian continent. Seidenberg has surmised that of the two types of tally, the complex body-part method is the older. Whether this is the case is probably impossible to settle with certainty. What is suggested here, however, is that the digit tally was introduced into New Guinea and Australia prior to the introduction of the body-part tally.
Subsequent to the introduction of these primary tallies and the 2-cycle system, several developments occurred in the New Guinea region. Many people possessing a 2-cycle system also made use of one of the two types of tally. Over time this resulted in the formation of hybrids so that, for example, the (2,5) or (2,5,20) systems developed as well as a number of variants. The 4-cycle system developed from the digit tally due to the practice of certain groups which regarded the four fingers, but not the thumb, as constituting "one hand". It is possible that the rare 6-cycle system also developed from one of the tallies, perhaps by the practice of augmenting the five fingers with an additional tally point such as the thumb joint. These developments which occurred in the numeration of the NAN languages prior to the arrival of the AN immigrants is given in the genealogy in Figure 2 in Chapter 8.
The next major development in the New Guinea region occurred about 5500 years ago with the arrival of the AN language groups in the western part of Irian Jaya. A later migration by AN-speaking people to the east established the homeland of Proto Oceanic (POC) in, probably, the Willaumez Peninsula area of New Britain by about 5000 B.P. The reconstruction of POC by historical linguists indicates that the speakers of POC, in common with their ancestors, the speakers of Proto Austronesian (PAN), had a 10-cycle numeral system. It seems likely therefore that such a system was in use in the south-east Asian region, the homeland of PAN, by at least 7000 B.P. when PAN was thought to have existed. Although it is not clear how the 10-cycle system originated, the fact that the PAN and POC words for both "five" and "hand" are identical suggests that it may have developed originally from the digit tally. In any case the arrival of the Austronesians introduced the 10-cycle numeral system to New Guinea; subsequent migrations from the POC homeland eventually carried the system into the previously unoccupied areas of Island Melanesia, Polynesia,, and Micronesia. The view adopted here is that the 10-cycle system accompanied the AN settlers as they peopled the Pacific: it was not introduced subsequently by diffusion from some external source.
Ross's reconstruction of the linguistic history of the AN languages in Papua New Guinea has various groups moving out of the POC homeland and on to the coastal and island regions of the PNG mainland. The situation arose in which the established NAN language groups, with their 2-cycle variant numeral systems and tallies, came into contact with the immigrant AN groups with their 10-cycle system. In this situation, diffusionists such as Crump take the view that the more efficient 10-cycle system would displace systems with a smaller primary cycle according to a sort of "survival of the fittest" principle. The evidence, however, indicates otherwise: on the mainland, at least, the 10-cycle system did not arrive and overwhelm the existing NAN systems. On the contrary, most AN groups accommodated in some way with the dominant NAN cultures, resulting in modifications to their original numeral system: loss of part of the numeral lexis and a reduction in the magnitude of the primary cycle. As a result of this we find AN groups possessing 2-cycle variant systems, (5,10) and (5,10,20) systems, and other variants in which the second pentad numerals are affected. This includes the type of system which Seidenberg has termed the neo-2-10 type which I therefore do not view as an elaboration of the 2-cycle system. The summary picture of AN numeration is given in Figure 3 in Chapter 8. Given that the contact between AN and NAN groups has occurred over a period of 5000 years or so, what is surprising is how little the 10-cycle system appears to have affected the counting systems of the NAN cultures of the New Guinea region.
With regard to enumeration, then, the nature of the diffusion process is rather more complex than that which the diffusionists seem to assume. The diffusion model implicit in Seidenberg's work seems to be a "relay baton" type with a counting system being passed from a donor group to a recipient group, the latter either not possessing a system or possessing one with a smaller primary cycle than that of the donor group. The outcomes when two such groups meet are either that the recipient group will adopt the new system or, if it already possesses one, its original system will be displaced or modified in some way. The view adopted in this study, which sees the primary systems and tallies as being carried with migrating language groups, regards a given group's counting system as an integral and relatively stable part of its culture. I have suggested that in the circumstance of linguistic speciation occurring over time, the cyclic structure of the counting system will normally remain invariant despite divergence in the numeral lexis between languages. This is one aspect of stability.
A further aspect of stability relates to the circumstances under which a counting system may be diffused from one group to another. In the event of two groups, which possess different types of counting system, interacting in some way, it does not inevitably follow that one group will adopt the counting system of the other, relinquishing, say, a lower primary cycle system for one with a larger primary cycle. Instead, I have discussed evidence which indicates that when an immigrant group settles in a region already occupied by an established cultural group, the former will often accommodate its own politico-economic institutions to those of the latter. For the immigrant group this may include the adaptation of its counting system to conform with that of the dominant group and, contrary to the expectation of the diffusionists, this may result in the loss of part of its numeral lexis and the reduction in the magnitude of the system's primary cycle as is the case, mentioned above, with certain AN groups. The conditions under which this may occur are not frequently met: two disparate cultural groups living in neighbouring regions may well retain their cultural identities even when engaging in a degree of trade. In these circumstances both groups are likely to retain intact their original counting systems. The main implication to be drawn from the various ways that different cultural groups interact is that diffusion is not an automatic process.
In conclusion, the picture that has been reconstructed here of the counting and tally situation in New Guinea and Oceania supports the view that the prehistory of number covers a period of some tens of thousands of years, considerably longer than the 5-6000 years suggested by Seidenberg. There was a long period of time when the dominant system in use was the 2-cycle one: the historical span of its use, far exceeding that of the 10-cycle system, attests to its success as a means of enumeration. While there were periods of relative stasis in the counting system situation, there were also periods of flux and change when new systems were introduced or invented. In the era prior to the arrival of the European colonists there is evidence to suggest that various changes occurred to the counting system situation and that a degree of internal diffusion took place within the region. It is, however, in the colonial and post-colonial period that major and rapid changes have been induced in the traditional means of enumeration in many societies as a result of changes to their political and economic institutions and of the establishment of the pre-eminence of the introduced decimal system of enumeration of the colonial powers. It is in these circumstances that we see the operation of diffusion most decisively and its effect has been to set in train the largely irreversible process of the gradual decline and extinction of many of the region's counting systems which, until recently, have survived as a link to humankind's earliest intellectual history.
9.2 AFTERWORDS
One of the difficulties of a study such as this is in trying to locate it as belonging to a particular intellectual discipline. While I perceive it as being essentially a contribution to the History of Number or, more generally, the History of Ideas, it might also be conceived as being situated at the intersection of such other disciplines as Anthropology, Linguistics, and the Philosophy of Number. Indeed I would anticipate that the data on which it is based is likely to be of interest to certain workers in each of these fields. Nevertheless I do not claim that the study is , for example, essentially an anthropological one in that I have not used the theoretical constructs or approaches to theoretical analysis which an anthropologist might use. Similarly, a linguist might justifiably claim that in the collection of data I have not, for example, attended to a rigourous methodology in dealing with the recording and transcription of phonological material; nor have I generally obtained instances of how numerals function in sentences.
While these could be regarded as legitimate concerns, my focus here, however, has been to investigate the distribution of certain types of counting system, as defined by their cyclic structures, among a broad range of indigenous societies and to see what implications this may have on how this distribution came about. The data available are sufficient for this purpose. There is nevertheless scope for detailed studies in which the focus is on an individual society or language in order to investigate the role of number and counting in that society, or on how numerals function in the language. The number of studies in both Anthropology and Linguistics in which this is a central focus is quite small: number and counting are more often than not accorded a peripheral status in these disciplines. I would argue, as indeed Hurford has, that such alleged marginality is a strength rather than a weakness and that subjects that cross disciplines are often of considerable importance: "They can provide windows through which the central doctrines of any one domain may be viewed from the perspective of another."[5]
There are several limitations to this study which are self-imposed. The temptation to be encyclopaedic, to be definitive, is all too strong. There are pleasures I have had to forgo: there are, for example, certain issues associated with the Philosophy of Number which are worth pursuing. There are other issues of interest to those of us working in the field of Mathematics Education of the sort first raised in Gay and Cole's A new mathematics and an old culture. It is certainly the case that the data available for this study could be exploited for purposes other than those on which I have chosen to focus.
Finally, there might well be scholars, particularly anthropologists, who may be bemused by my choosing to deal with the issue of the diffusion of counting practices. These days, the diffusion of cultural institutions and technologies is an area of study which is largely neglected by anthropologists although it did enjoy a period of popularity in the 1930s. However the issue of diffusion still holds some credence among historians of number largely due to the fact that Seidenberg's diffusionist theory has not been seriously challenged. That it has not been challenged is possibly due to the lack of any additional data to that on which Seidenberg originally based his views. The data which is provided here considerably enlarges the overall data base on numeral systems and it seemed timely to carry out a revaluation of the prevailing view of the prehistory of number which, in the last analysis, is essentially a Eurocentric one which denies the intellectual contribution of indigenous societies to our common history.