Abstract of http://www.uog.ac.pg/glec/thesis/ch1web/ABSTRACT.htm Abstract of Counting Systems of Papua New Guinea and Oceania by Glendon A. Lean In modern technological societies we take the existence of numbers and the act of counting for granted: they occur in most everyday activities. They are regarded as being sufficiently important to warrant their occupying a substantial part of the primary school curriculum. Most of us, however, would find it difficult to answer with any authority several basic questions about number and counting. For example, how and when did numbers arise in human cultures: are they relatively recent inventions or are they an ancient feature of language? Is counting an important part of all cultures or only of some? Do all cultures count in essentially the same ways? In English, for example, we use what is known as a base 10 counting system and this is true of other European languages. Indeed our view of counting and number tends to be very much a Eurocentric one and yet the large majority the languages spoken in the world - about 4500 - are not European in nature but are the languages of the indigenous peoples of the Pacific, Africa, and the Americas. If we take these into account we obtain a quite different picture of counting systems from that of the Eurocentric view. This study, which attempts to answer these questions, is the culmination of more than twenty years on the counting systems of the indigenous and largely unwritten languages of the Pacific region and it involved extensive fieldwork as well as the consultation of published and rare unpublished sources. The outcome was that data were obtained on the counting systems of almost 900 languages spoken in Papua New Guinea, Irian Jaya, the Solomon Islands, Vanuatu, New Caledonia, Polynesia, and Micronesia. This material appears in 23 volumes and forms the data base of the thesis. The picture which emerges is that there is a rich and diverse collection of counting systems with, for example, 250 base 2 systems, 335 base 5 systems, 203 base 10 systems not unlike the European types, as well as some rare base 4 and base 6 types. The study indicates for the first time how these systems are distributed geographically throughout the region. Material obtained from a sample of societies enabled a number of counting ethnographies to be constructed. These indicate that societies do vary in the importance that they accord to number and counting, the degree to which they use large numbers, and the types of objects counted. Thus while existence of numbers and the practice of counting appear to be universal in the Pacific region, the variation between cultures with regard to these is considerable. Of more theoretical interest, however, is that this study challenges the prevailing view of the prehistory of number. This says essentially that numbers were invented in one or more of the agricultural city states of the Middle East and thereafter diffused throughout the world. The alternative view of the prehistory of number developed here sees numerals as an integral and ancient part of the languages of early indigenous societies and that they were carried into the Pacific region with each of the major migrations over some tens of thousands of years. 1 de 1 13/12/2015 02:16 p. m. CHAPTER 1 http://www.uog.ac.pg/glec/thesis/ch1web/ch1.htm CHAPTER 1 PRELIMINARY MATTERS 1.1 � INTRODUCTION Over the past 150 years the study of systems of numeration as they occur in natural languages has had a chequered career. � During the nineteenth century, we find that the published literature of the explorers, administrators and missionaries of the European colonizing powers provides a rich source of information on the exotic languages and customs of the peoples of Africa, the Americas, the Far East, and Oceania. � Indeed this material still serves today as a major source of data on non-European counting systems. � The study of numerals and counting systems was initiated largely by scholars with an interest in linguistics and anthropology and yet it is apparent that, during the first half of the twentieth century at least, there has been a marked waning of interest on the part of both linguists and anthropologists and that the subject has been relegated to one having only marginal status. [1] �� Numerals, it would seem, are strange animals which do not possess, for the linguist, the same intrinsic interest as other aspects of linguistic analysis, or, for the ethnographer, the same intellectual challenge as unravelling a complex kinship system or social structure. � They are, perhaps, best left to the historians or philosophers of mathematics. Philosophers of mathematics tend to take the view that the study of natural language numerals has little bearing on the understanding of the nature of number, an abstract concept independent of nominal linguistic vagaries. � The standard histories of mathematics, [2] � in those sections which deal specifically with number, usually begin with the advent of written numerals, perhaps for no other reason than that historians prefer to work from written records. � Also, until relatively recently, there has been a marked Eurocentric bias in the Western accounts of the development of mathematical ideas [3] and even the contributions of India, China and Islam have been given scant attention. � The occurrence of mathematical ideas as they existed in "primitive" tribes, is dealt with in a speculative or dismissive paragraph or two. There are several points which should be made concerning this lack of interest in the numerals and counting systems which exist in the societies outside the usual ambit of historians. � First, those societies which do not possess a tradition of historical documentation are those in which the majority of the world's languages are spoken; their exclusion from the historical record means that the numerically greater part of the counting systems used by humankind is ignored. � Second, those societies which are the focus of historical scholarship possess a relatively small range of counting system types. � Generally speaking, their counting systems possess a base of 10, although several systems have features or irregularities which suggest vestiges of base 12 or base 20 systems: English and German being examples of the former, and French and Dutch being examples of the latter. � The inference which may be drawn from this highly selective view is that humankind's response to the enumeration of its world is largely consistent across both time and cultures. � Third, it may be useful to consider whether the absence of historical documentation for these societies means that we are unable to construct a prehistory � of number. � Given the information � that has accumulated in the disciplines of anthropology and archaeology, an extensive data-base of natural language numerals and counting systems, together with the methods developed by comparative linguists, it may be theoretically possible to construct a tentative outline of the prehistoric development of the concept of number in human societies in the millennia preceding the advent of written numerals. Fortunately, since the mid-1960s, there has been a renaissance of interest in documenting the mathematical ideas existing in non-Western societies. � This has been partly due to the work of cross-cultural psychologists, for example Cole, Gay, Glick, and Sharp [4] and Gay and Cole, [5] who have been interested in situated cognition and the cultural context of learning and thinking, especially as it relates to the development of mathematical concepts. � In particular, a literature of "ethnomathematics" has been established which has dealt, for example, with the mathematical ideas existing in African societies [6] and in North and Central American Indian societies. [7] � The ethnomathematical approach has even been extended to include the study of identifiable cultural (and sub-cultural) groups within Western societies as well as non-Western societies, and is concerned, for example, with the mathematics used by gamblers or supermarket shoppers. [8] The renewed interest in counting systems of non-Western societies is making its mark in the more recent histories of number and mathematics as may be found in works by 1 de 20 13/12/2015 09:40 a. m. CHAPTER 1 http://www.uog.ac.pg/glec/thesis/ch1web/ch1.htm Flegg, [9] Ifrah, [10] and van der Waerden and Flegg. [11] � Much of the material, however, on which these studies are based derives from the nineteenth century. � There has not been an increase in the data-base commensurate with the increase of interest in the systems themselves. � This is particularly true of the large geographical region which encompasses Melanesia, Polynesia and Micronesia and which has received little attention in the ethnomathematical literature and virtually none in the historical literature, even though this region accounts for over a quarter of the world's languages. [12] � This study, in part, attempts to redress the situation by, first, providing documentation of the natural language numeral systems used in the cultures of the New Guinea area and the Pacific, and, second, by investigating the implications of these for the history, or prehistory, of number. 1.2 � SCOPE AND PURPOSE OF THE STUDY The original aims of this study were, firstly, to elucidate the number and nature of the counting systems used in the traditional societies of Papua New Guinea (PNG) and, secondly, to investigate whether the accumulated data had any implications for several theoretical questions which arise in the literature of the history and development of the concept of number in human societies. � For example, are the structural/cyclic features of natural language numeral systems invented independently in various societies in different geographical locations, or are they diffused from a single source? � Is there any evidence for a chronological developmental sequence of counting system types; that is to say, can we construct something approaching a prehistory of number? Much of the existing data on numerals and counting systems derives, as noted above, from information collected during the nineteenth century in the traditional societies of Africa and the Americas.