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 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. Published data are also available for a number of Polynesian societies of the Pacific region, however the amount of data available for PNG appeared to be relatively small. Even though the population of PNG, being about four million at the 1990 National Census, is a small proportion of the world's total, the linguistic situation is nevertheless very complex with a total of about 750 languages, or a sixth of the world's total, spoken within its borders. To document the nature of the counting systems of even half the total number of these languages would make a significant addition to the data-base from which scholars have worked in the past.

I began the collection of data on PNG natural language numerals and counting systems early in 1968 and continued to collect these from both primary and secondary sources until 1988. The documentation of the accumulated data began in 1985 and a final revised version was completed at the beginning of 1991: this material comprises Appendices A, B, and C of this work. In all, data were acquired on the numerals, counting or tally systems of 532 languages, or just over 70% of the total. Generally speaking, the languages for which it was difficult to obtain data were those with 500 speakers or less; in the Madang Province alone some 93 languages fall into that category.

In contrast to the relatively homogeneous cultural and linguistic situation existing in Polynesia, the diversity of the cultures and languages existing in Melanesia has been commented on by many scholars.[13]  This is particularly true of PNG where the diversity appears to be greatest and in this sense it is untypical of its neighbours with the exception, perhaps, of Irian Jaya, the complementary western half of the island of New Guinea.  In order to see whether this diversity extended to the situation regarding counting systems and whether, therefore, the PNG data already gathered was untypical of the region as a whole, it was decided to extend the geographical scope of the original study to include the remainder of Melanesia, which is termed "Island Melanesia", and the islands of Polynesia and Micronesia as well.

This study, then, deals with the counting systems of "Papua New Guinea and Oceania".  For my purposes here I have taken "Oceania" to include the Irian Jaya Province of Indonesia (henceforth called Irian Jaya), the Solomon Islands, Vanuatu, New Caledonia, Fiji, and Rotuma. In addition I have included all of Polynesia, both Triangle and Outlier, and those islands of Micronesia on which Oceanic Austronesian languages are spoken: effectively that excludes the western part of Micronesia which contains the majority of the Marianas, where Chamorro  is spoken, the Palau (Belau) Islands, and Yap.  The islands to the west and south of Irian Jaya, which include Timor, Halmahera, and the Kai and Aru Islands, have also been excluded.  The other major area excluded is Australia and although some reference will be made to Australian languages these are, nevertheless, untypical of the rest of Oceania whose languages are generally classified as either Papuan or Austronesian (see below for a discussion of these terms).  In the region I have taken to be "Oceania", a further 450 languages or so are spoken, thus giving a total for the region under consideration, if we include PNG, of about 1200.  (See Map 1 which indicates this region).  Occasionally it has been useful to refer to "New Guinea" by which I mean the combined regions of Irian Jaya and PNG.

1.3  SOURCES OF DATA

1.3.1  PAPUA NEW GUINEA

The data for PNG are collected in Appendices A, B, and C.  The source, or sources, of a particular set of numerals is given at the bottom of the data table in which they appear.  The sources are basically of two types: primary and secondary; each of these is discussed below.

1.3.1.1  PRIMARY SOURCES

The primary sources are divided into four main groups.  The first group comprises my field notes which were taken in two different sets of circumstances.  There are those field notes taken from informants living in villages which I visited.  One-to-one interviews were held, wherever possible, with an older member of the village, usually, though not always, male.  The language used was invariably tok pisin and I have no records of ever having to resort to an interpreter in the case, say, where an elderly informant did not speak tok pisin.  Prior to 1980 notes were taken during the interview; subsequent to 1980 the interviews were recorded on a portable cassette recorder and the relevant material was later transcribed, normally using Roman orthography, from the tape within a day or two of the interview. From my first field trip in 1968 to my last in 1987, I have records of interviews with informants from 35 villages in eight provinces. The second set of circumstances in which I acquired field notes was when informants, particularly those living in relatively remote areas, visited Lae and I was able arrange an interview with them. I have records of 29 interviews which fall into this category.

The second main group of primary sources, which in fact forms the major part of my complete data-base for PNG, comprises questionnaires given to three different populations.  First, there are those completed by incoming students at the University of Technology during the period 1968 to 1983; these account for a total of 1200. Second, students at four National High Schools (Sogeri, Aiyura, Keravat and Passam) completed questionnaires in 1982, 1984, and 1986; these account for a further 1022. Third, in 1985, copies of a slightly revised questionnaire were sent to headmasters of 1700 Community (primary) Schools in all provinces of PNG.  In each case the headmaster was asked to assist in the completion of the questionnaire with an adult who spoke the local vernacular (in general, a single community school serves a single language group).  These account for an additional 302, giving altogether a total of 2524 usable, completed questionnaires.  Further details of the questionnaire approach are provided in the "Preliminary" section at the beginning of Appendix A.

The third group of primary sources comprises unpublished material gathered during the Indigenous Mathematics Project (IMP) during the period 1976 to 1979.  The IMP was established in 1976 by the PNG Department of Education under the directorship of Dr David Lancy.  The research phase of the project was primarily concerned with studying aspects of children's cognitive development and, in particular, possible cultural influences.  In association with the latter, linguistic and anthropological data were collected at each village test site and these data usually consisted of information about the vernacular classification (or taxonomic) systems used in the society together with information about its counting system.  Much of this information has subsequently been published.[14]  The unpublished material deriving from the IMP comprises two sets of questionnaires.  One questionnaire was devised by Lancy and copies of this were sent to a number of Provincial High Schools in several provinces and were completed mainly by Grade 10 students.  A second questionnaire was devised by Ms Anne Deibler, the daughter of a senior member of the Summer Institute of Linguistics (SIL), and copies of this were sent to SIL members working in the field each of whom was asked to provide details of the counting system of the language on which they were working.  Altogether a total of 238 IMP questionnaires were made available for this study.

The fourth group of unpublished primary source material derives from survey word lists compiled by members of the Summer Institute of Linguistics.  The PNG Branch of the Institute was established in 1956 and is concerned primarily with Bible translation and literacy.  Its members, all of whom receive training in linguistics, have studied in considerable detail several hundred PNG languages; in addition, they have surveyed many more.  The SIL library, at Ukarumpa in the Eastern Highlands Province, holds an estimated 500 completed and unpublished "Standard Survey Word Lists" of various languages in the 19 provinces of PNG.  The Survey Word List in use after 1965 contains 170 lexical items and 20 standard sentences.  The completed Word Lists included data on the numerals 1 to 5, and 10 (and often further numeral data added by the surveying linguist).  A complete search of these materials in the period 1984 to 1986 yielded a total of 362 Word Lists which contained some numeral data on various languages spoken in each of the PNG provinces.

1.3.1.2  SECONDARY SOURCES

The published sources of data on PNG numerals, counting systems, or tally practices may be found listed in the Bibliography, Volume 17 of the PNG series, in Appendix C.  Altogether, these sources contributed an estimated 600 or so items of data (one item of data being, for example, a set of numerals or a description of a counting or tally system for a particular language).

In 1968, at the beginning of the data-gathering phase of this study, a search of the published literature revealed only a small number of articles which dealt specifically with aspects of numeration in PNG.  The linguist S.H. Ray had included a chapter on "Numeration and numerals in the Melanesian languages of British New Guinea" in Volume 3 of the Reports of the Cambridge Anthropological Expedition to Torres Straits published in 1907.  The German scholar Fr. W. Schmidt included some PNG numeral data in an article on "Numeral systems" in the 1929 edition of the Encyclopaedia Britannica.  Two SIL linguists working in the Southern Highlands Province of PNG, Karl and Joice Franklin, published an article on the "Kewa counting systems" in 1962.[15]  Between 1969 to 1972, Ted Wolfers, a journalist based in Port Moresby, wrote three articles on PNG counting systems,[16] one of these appearing in the newly published Encyclopaedia of Papua and New Guinea (1972).  Some years later in 1978, David Lancy, Director of the Indigenous Mathematics Project, referred to the "scanty available literature"[17] on PNG counting systems and this, at the time, was probably the prevailing view of the small number of scholars with an interest in the field.  It was therefore contrary to this view when an exhaustive search of the ethnographic and linguistic literatures of PNG revealed a large amount of first- and early-contact data about natural language numerals and counting systems. Although the details of the sources of these data are provided in Appendices A, B, and C, it is useful to provide a brief survey and summary here.

The earliest published examples of PNG numerals were collected by the Dutch explorers Le Maire and Schouten in 1616 during their voyage from the Netherlands to the East Indies.  Brief vocabularies, including numerals, were taken at "Moyse (i.e. Moses) Island" and "Nova Guinea"; the locations of these were subsequently established, by German scholars in the early 1900s,[18]  to be respectively Tabar Island, situated off the east coast of New Ireland, and Muliama, situated on the south-east coast of New Ireland.  More than 200 years were to elapse before further vocabularies were collected in the New Guinea region.  In 1827, the French explorer Jules Dumont d'Urville sailed through the New Guinea islands region and collected a vocabulary of the Siar  language at "Port Praslin" near the southern tip of New Ireland; this was published in 1833.[19]  In 1849, Owen Stanley in H.M.S. "Rattlesnake" sailed through the islands of the Milne Bay area and along the southern coast of Papua; several vocabularies were taken during this voyage and were published in 1852.[20]

With these few exceptions, the published sources of numeral data for the coastal PNG languages begin to appear in the 1870s after the settlement of Port Moresby and the Duke of York Islands and Rabaul region.  Much of the data collection for the northern part of the island took place during the period of the German Protectorate, i.e. from 1884/85 to 1914, after the administrative regions of Kaiser Wilhelmsland and the Bismarck Archipelago were established.  These regions encompassed what are now the provinces of East New Britain, New Ireland, Manus, Madang, Morobe, East and West Sepik; most of the first- and early-contact data on the languages of these provinces emanate from the work of German missionaries and scholars during this period, the publications appearing, by and large, in books and scholarly journals rather than the official colonial reports and gazettes.  Strictly speaking, Kaiser Wilhelmsland also incorporated what are now the provinces of the vast inland Highlands region; this, however, was not penetrated until the 1930s and remained a terra incognita  during the German colonial period.

The southern half of the island was called British New Guinea during the period from 1884 to 1906, and Papua from 1906 onwards.  Much of the first- and early-contact data on the languages of this region derive not so much from missionaries and explorers, as is the case for German New Guinea, but from the officers of the colonial administration: the published sources of data are not so much the scholarly Zeitschrift but rather the official Government reports, notably the Annual Reports on British New Guinea  (1889-1906) and the Annual Reports for Papua (from 1907).  That this comprehensive collection of data exists, mainly as vocabularies which appear in the appendices of the Reports, is due largely to the efforts of two men: Sir William MacGregor, Administrator of British New Guinea from 1888 to 1898, and Dr W. M. Strong, Chief Medical Officer for Papua, both of whom had an amateur interest in linguistics and who collected data themselves as well as organizing members of the colonial service to do so.  Between 1889 and 1917, 389 vocabularies were published in the Reports, some others, though not so many,  appearing in the period 1918 to 1940; the large majority of these contained some data on numerals.  Much of this material, accounting for some differences in orthography, shows good agreement with more recently collected data.  It is quite clear that some care was exercised in the elicitation and recording of vocabularies.  Dating from MacGregor's time, a standard vocabulary list existed (in fact what would now be called a "standard survey word list") which was compiled from a set of "questions on the customs, beliefs, and languages" originally drawn up by Sir James Frazer, author of The Golden Bough.  A standardized orthography was recommended, based, it seems likely, on that used by the Royal Geographical Society.

It was during MacGregor's administration that the eminent Cambridge anthropologist  A.C. Haddon organized two expeditions to the Torres Strait region, the first in 1889 and the second, the famous "Cambridge Anthropological Expedition to Torres Straits" in 1898, in which he was accompanied by the linguist Sidney H. Ray. Ray's published works on PNG languages, and indeed many other languages of Melanesia and Polynesia, which appeared over the period 1891 to 1938, establish him as the most important linguistic scholar of the early colonial period, rivalled only by Fr. W. Schmidt.  The results of the expedition were published in seven volumes, one of which is Ray's contribution on linguistics.

Excluding the vocabularies published in the Annual Reports, something like 200 publications on linguistics and ethnographic studies of PNG appeared in the period 1875 to 1940, many of which contained data on natural language numerals and counting systems.  During the 1930s, a German scholar, Dr Theodor Kluge, assembled data, mostly from published sources, on the numerals of a large number of languages spoken throughout the world.  Several substantial typescripts were prepared during the period 1938 to 1941; two of these contain data on the numerals of languages of PNG and Melanesia generally.[21]  Due to the outbreak of the Second World War these materials were never put into final form and published; however the uncorrected typescripts survive and were consulted for the PNG part of this study.

1.3.2  OCEANIA

The sources of data for the six volumes on the Counting Systems of Oceania, which comprise Appendix D of this study, are the published literatures of linguistics and ethnography of the geographical regions of Melanesia, Polynesia, and Micronesia.  As was the case for the PNG data, there is only a small number of publications which deal specifically with numerals and counting systems.  For Irian Jaya, for example, there is a survey of counting systems by Galis,[22]  together with an article by Briley,[23]  an SIL member, which deals with several counting system types.  For the Solomon Islands there is a study by Fox[24] of "numerals and numeration" of the Arosi  language.  MacDonald[25] has an article on the "Asiatic origin of the Oceanic numerals", although this deals in fact with a small number of Vanuatan languages.  More recently, we have two further articles on numeration in the Vanuatan languages, one by Lynch[26] on Tanna  and another by Charpentier[27] on the languages of south Malekula.  For the whole of the Polynesian region we have only several studies which deal specifically with numeration.  Audran[28] published in 1930 a one-page summary of the numerals of several Polynesian languages.  Best [29] has some data on the counting system of New Zealand Maori  and Fraser [30] has some comparative data on the Polynesian numerals 1, 5, and 10.  Heider[31] has material on the numerals and counting system of Samoan, Metraux[32] has an article on Easter Island numerals, and Lemaître[33] has a survey of the numerals of mainly the languages of eastern Polynesia.

As was also the case with the collection of the PNG data from published sources, particular care was taken to try to obtain as much first- and early-contact data as possible.  Some of this material derives from vocabularies taken during the three expeditions under the leadership of Captain Cook in the years 1768 to 1780.  Many previously unpublished vocabularies obtained during these voyages were published by Lanyon-Orgill[34] in 1979.  While most of these deal with various languages of Polynesia, for example New Zealand Maori, Pa'umotu, Tahitian, Easter Island, Tongan, Marquesan, Hawai'ian, and Cook Islands Maori, there are several languages of the Melanesian region included as well which were recorded in New Caledonia and Vanuatu.  During the nineteenth century much more information about the Pacific region became available as the islands were explored and settled by the Western colonial powers.  Between  the years 1838 to 1842, the United States Exploring Expedition under the command of Captain Charles Wilkes visited many of the islands in both Polynesia and Micronesia and the ethnographic and linguistic results of the expedition, written by Horatio Hale,[35] were published in 1846.  A memoir by Turner,[36] a missionary who had spent 19 years in parts of Polynesia, was published in 1861 and this includes some comparative lexical data, including numerals, on six Polynesian languages.  The German linguist Hans von der Gabelentz[37] published his two-volume Die melanesischen Sprachen  in 1861 and 1873, a survey of a number of languages in the Solomon Islands, Vanuatu and New Caledonia.  In 1885, Codrington's The Melanesian Languages  was published and this contains grammatical and lexical data on some 35 languages in the non-New Guinea Melanesian region, in particular it has a chapter devoted to "numeration and numerals".[38]  A significant contributor to the study of the languages of the region was Codrington's protégé Sidney Ray.  Ray published an article[39] on "the languages of the New Hebrides" in 1893 and followed this with a series of articles written between 1912 and 1920 on "Polynesian linguistics"[40] and the "Polynesian languages of Melanesia".[41]   These all contain some numeral data as does his major work A Comparative Study of the Melanesian Island Languages, published in 1926.[42]

With a few exceptions, much of the modern linguistic and ethnographic data derive from sources published after the Second World War.  As these are listed in the bibliographies of each of the volumes on the Counting Systems of Oceania (Appendix D) they will not be discussed further here.  There are, however, a few major sources of data which have been indispensable for this study and which should be noted. For the example, Tryon and Hackman's Solomon Islands Languages: An Internal Classification, published in 1983, has lexical data, including numerals, on all the languages spoken in the Solomon Islands. Similarly, Tryon's New Hebrides Languages: An Internal Classification, published in 1976, has lexical data on all the languages of Vanuatu. Information on the languages of New Caledonia has been harder to access than the data on other languages of Melanesia, however a general survey by Leenhardt, published in 1946, contains lexical, phonological, and grammatical data on all the languages and, for some of these, these are the only data available.  For Polynesia, a series of anthropological studies published by the Bernice P. Bishop Museum in Hawaii, appeared during the period 1920 to 1950, several of these by the eminent Polynesian scholar Sir Peter Buck (Te Rangi Hiroa); some, though not all of these, contain useful data on numerals and counting.[43]  Finally, for Micronesia, a series of publications of the University of Hawaii Press[44] have been indispensable and have provided valuable information on Marshallese, Mokilese, Kosraean, Ponapean , and Woleaian.

1.4  THE LINGUISTIC SITUATION

Generally speaking, I have taken the linguistic situation of all regions under consideration to be that given in the Language Atlas of the Pacific Area, Part 1: New Guinea Area, Oceania, Australia, edited by Wurm and Hattori[45] and published in 1981. The languages of New Guinea and Oceania are classified into two distinct groups: (a) the Papuan languages, sometimes referred to as non-Austronesian (NAN), and (b) the Austronesian (AN) languages. Each of these groups is considered below.

1.4.1  THE PAPUAN LANGUAGES

Wurm notes that "the Papuan languages are generally regarded as radically different from the Austronesian languages and for that matter from all other known language groups in the world.  There may only be a remote possibility of some link existing between the languages spoken by the aboriginal population of the Andaman Islands, to the south of Burma, and some Papuan language groups in the extreme west of the New Guinea area".[46]  Wurm, however, also indicates that several groups of NAN languages show strong influence by AN languages although it seems likely that such influence was from Oceanic, or Pre-Oceanic, AN, rather than a more archaic form.[47]  Wurm also notes that, in general, the NAN languages are not thought to be related to the Australian languages.

The majority of the NAN languages are located on the New Guinea mainland. Some are also found to the west, in north Halmahera and east Timor; both of these lie outside the region under consideration here.  They also extend to the east of the mainland and are found in Rossel Island (located in the Milne Bay Province), in the New Britain area, New Ireland, and Bougainville Island. Seven NAN languages are scattered throughout the largely AN-speaking Solomon Islands, although that is the limit of their eastward extension.  We do not find NAN languages in the remainder of Melanesia, although Wurm indicates that some of the eastern Vanuatan languages may display some Papuan influence.[48]

Of the two types of languages spoken in the New Guinea area, by far the majority are NAN.  According to Wurm and Hattori there are 722 NAN languages spoken in the region under consideration: 543 in Papua New Guinea, 172 in Irian Jaya,[49]  and 7 in the Solomon Islands.[50]   Since the 1950s a great deal of work has been carried out in investigating and classifying the NAN languages which, at one time, were thought to be largely unrelated.[51]  While it is unnecessary to provide the fine details of the classification here (an individual language's classification is given in each data table in Appendices A, B, C, and D), it is nevertheless useful to indicate the major classificatory phyla for the NAN languages together with the number of languages assigned to each group (arranged in descending order of magnitude).[52]  Those are given in the first column.  The numbers in the second column indicate the number of languages in that group for which data have been acquired for this study.

Table 1

The Number of Languages in Each NAN Phylum for Which Data Have Been Acquired

NAN  Language Phyla	Languages in Phylum	Data Acquired
1.  Trans-New Guinea Phylum	>500	298
2.  Sepik-Ramu Phylum	98	44
3.  Torricelli Phylum	48	26
4.  East Papuan Phylum[53]	23	23
5.  West Papuan Phylum[54]	13	10
6.  Sko Phylum-level Stock	8	6
7.  Arai Phylum-level Family	6	6
8.  Geelvink Bay Phylum	6	3
9. Kwomtari Phylum-level Stock	5	4
10. East Vogelkop Phylum-level Stock	3	2
11. Amto-Musian Phylum-level Family	2	2
12.  Isolates	8	6

Thus out of a total of about 720 NAN languages, data have been acquired for 430 or about 60%.  (The distribution of the NAN languages and their phyla are shown in Map 2).

1.4.2  THE AUSTRONESIAN LANGUAGES

While the large majority of the NAN languages are spoken on the New Guinea mainland and especially in the interior highland regions, the AN languages are dispersed over an enormous geographical area stretching from Madagascar in the west to Easter Island in the east.  They are found in Formosa, the Philippines, southern Vietnam, Indonesia and Malaysia. In the region under consideration here, the AN languages are spoken mainly in the coastal areas of the New Guinea mainland, throughout almost all of Island Melanesia as well as the atolls and islands of Micronesia and Polynesia.[55]  All the AN languages are regarded as descendants of Proto Austronesian (PAN), thought to be spoken in Formosa at about 7000 B.P.[56]  At some time after this date the Austronesians dispersed, some eventually migrating into the Borneo and western Indonesian area. A migration into the New Guinea region, between 5000 and 4000 B.P.,[57] brought the ancestors of all Austronesians of the Pacific area and who established, somewhere in the northeast of Papua New Guinea (possibly the Willaumez Peninsula of West New Britain), the homeland of Proto Oceanic.[58]

In the region under consideration there are two major distinct groups of AN languages:  the Oceanic and non-Oceanic.  Ross notes that "there is today widespread acceptance among Austronesianists of the 'Oceanic hypothesis', according to which all the Austronesian languages of Oceania east of a line drawn from north to south through the western Pacific are descended from a single proto language, today named 'Proto Oceanic' (POC).  Its descendants are known simply as the 'Oceanic languages'.  This line divides Chamorro (Marianas Islands) and Belau (formerly Palau) from the rest of Micronesia and bisects the north coast of the island of New Guinea at 138^˚ longitude".[59]  This essentially means that virtually all of the AN languages dealt with here are Oceanic, with some exceptions in the western part of Irian Jaya.  (The distribution of the AN languages throughout the New Guinea mainland is shown in Map 3).

The Oceanic AN languages are themselves classified further. Recent work by Ross[60] on the AN languages of "Western Melanesia", and by Lynch and Tryon[61] on eastern Melanesia and the Pacific region, provide the following picture (the implications for migrations are discussed subsequently).  The Oceanic languages spoken in Papua New Guinea and the northwest Solomon Islands[62] may be classified into just two first order sub-groups:  (1) the Admiralties "Cluster" (located in the Manus Province of PNG),  and (2) "Western Oceanic".  The latter comprises three clusters: (1) the North New Guinea Cluster, which includes the languages of the Huon Gulf-Markham Valley area, parts of New Britain and the north to northeast coast of the New Guinea mainland;  (2) the Papuan Tip Cluster, consisting of the languages of the Milne Bay and Central Provinces; and (3) the Meso-Melanesian Cluster, comprising the languages of the New Ireland Province (with the exception of the St Matthias group), several New Britain languages, those of the North Solomons Province of PNG, and the northwest Solomon Islands group.[63]  A third sub-group of Oceanic languages has been delineated by Lynch and Tryon. This is the Central/Eastern Oceanic sub-group which comprises, according to Ross "all the languages of north and central Vanuatu, Fiji, Polynesia and Micronesia ... together with the southeast Solomons, Utupua and Vanikoro, the south Vanuatu, and possibly the Loyalties and New Caledonia groups".[64]

To summarise the situation regarding the AN languages spoken in the region encompassing New Guinea and Oceania, the information  given below shows the total number of languages spoken in each country (shown in the first column), together with the number of languages for which data were acquired for this study (shown in the second column).[65]

Table 2

The Number of AN Languages in Each Country Dealt With in This Study for Which Numeral Data Were Acquired

Country	No. of AN Languages in Country	No. of AN Languages For Which Data Were Acquired
1.  Papua New Guinea	206	188
2.  Irian Jaya	39	38
3.  Solomon Islands	56	56
4.  Vanuatu	105	105
5.  New Caledonia	28	28
6.  Fiji	2	2
7.  Rotuma	1	1
8.  Polynesia	24	23
9.  Micronesia	18	12

Of the 38 languages shown in column 2 for Irian Jaya, 33 are non-Oceanic and the remaining 5 are Oceanic.  With this exception, all other languages are Oceanic.  The total number of languages regarded as Polynesian is actually 36. The 24 languages shown in column 1 comprise 22 languages spoken in "Triangle Polynesia" and 2 others which are situated in Micronesia.  The remaining 12 languages are dotted throughout Island Melanesia and are included in the figures for Papua New Guinea (3), Solomon Islands (5), Vanuatu (3), and New Caledonia (1).  These Polynesian languages found in Melanesia, plus the two in Micronesia, are known as the "Polynesian Outliers".  Of a total possible 479 AN languages, data have been acquired for 453, or about 95%.  Finally, if we include the NAN languages as well, we have a total of 1201 languages spoken altogether in the region under consideration and data have been acquired for 883, or about 74%.

1.5  THEORIES OF MIGRATION AND PREHISTORY

The complex linguistic situation existing in New Guinea and the remainder of Island Melanesia is paralleled by the obviously apparent cultural and genetic heterogeneity of the region.  While it might  seem to be an impossible task to unravel the prehistoric conditions which have led to the contemporary diversity, it is precisely in this region of the world where a marriage between archaeology and the reconstructive methods of historical linguistics is producing fruitful results.  The reconstruction of the prehistory of the Oceanic languages and their speakers is rather more advanced, and probably more accurate, than that of the NAN languages.  Nevertheless I will summarize here the current theories regarding the prehistoric migrations of the groups ancestral to today's speakers of the NAN and Oceanic AN languages insofar as they are relevant to my subsequent arguments.

1.5.1  PAPUAN LANGUAGE MIGRATIONS

The picture of the putative migrations of NAN speakers into the New Guinea area that is given here is that first put forward by Wurm, Laycock, Voorhoeve, and Dutton,[66] reiterated by Wurm[67] and by Tryon,[68] and investigated further by Voorhoeve.[69]

1.5.1.1  About 60,000 years ago the first immigrants arrived into the New Guinea region. These were probably Australoids who came from the west and entered the northern part of what was then the single New Guinea-Australian continent.  They spread into New Guinea and south into Australia.  Approximately 10,000 years ago, after the last ice age, the increased sea level brought what is now the Torres Strait into existence and New Guinea and Australia were separated.  The theory put forward by Wurm is concerned with the establishment of the NAN languages in New Guinea and does not deal with the origin of the Australian languages and whether in fact there were several Australoid migrations.  Whatever remnants of an Australoid population there were in New Guinea have either disappeared or have been largely engulfed by subsequent Papuan migrations.  Wurm notes, however, that it may be "possible to suggest that Sepik-Ramu Phylum speakers may constitute some remnants of the earlier Australoid population in the New Guinea area who have been strongly affected by thousands of years of contact with Papuans".[70]

1.5.1.2  Between about 10,000 to 15,000 years ago, i.e. shortly after, or not long before, the separation of New Guinea and Australia, the first NAN migration came from the west of New Guinea and spread in an easterly direction right across the mainland overlaying the extant Australoid population.  Wurm says "it seems possible that languages descended from ancestral language forms spoken by these first Papuan immigrants are still surviving today in the form of the isolates and the members of at least some of the small phylic groups ... there is a possibility that members of the East Papuan Phylum, or at least an element in them, may also be derived from the languages of the first Papuan immigrants".[71]  Other archaic language groups which may be a survival from this migration are the West Papuan Phylum and the Torricelli Phylum.[72]

1.5.1.3  Wurm says "a few millennia later, a second NAN migration appears to have entered the New Guinea area and spread through much of it with the languages carried by this migration overlaying the language picture brought in by the first migration".[73]

1.5.1.4  The main NAN migration is thought to have taken place about 5000 B.P. when a group immediately to the west of the New Guinea mainland moved into the Vogelkop-Bomberai Peninsula and moved eastward right through the island.  The languages introduced by this migration are those that eventually formed the current Trans-New Guinea Phylum.  Because the Trans-New Guinea Phylum languages appear to have adopted some archaic AN loanwords it is assumed that they were in contact with AN groups prior to the migration across New Guinea.  The Trans-New Guinea Phylum migration took perhaps about 1000 years to spread right across the mainland and eventually to the southeastern Papuan tip where East Papuan Phylum languages may have been displaced eastwards to Rossel Island, New Britain, New Ireland and to Bougainville and the Solomon Islands, reaching as far east as Santa Cruz.  Tryon notes that "the existence of Papuan languages southeast of Santa Cruz has not been demonstrated, although there is some speculation that they may have once extended as far as southern Vanuatu and New Caledonia".[74]  (See Map 4 for some of the main migrations, according to Wurm, occurring in the New Guinea mainland).

1.5.2  OCEANIC AUSTRONESIAN LANGUAGE MIGRATIONS

The linguistic reconstruction of the prehistory of all the Oceanic AN languages was outlined above (1.4.2).  The migrations within Papua New Guinea will be discussed first, followed by the putative situation for the remainder of Oceania.

1.5.2.1  The view given here derives essentially from Ross[75] and may be summarized as follows.  Speakers of "Pre-Oceanic" moved from the eastern Indonesia area and settled in what is now New Britain, probably in or near the Willaumez Peninsula.  This migration may have occurred about 5000 B.P.,[76]  or by 4500 B.P. at the latest.[77]  After a period of stability in which the innovations characteristic of Proto Oceanic  developed, two early migrations of Proto Oceanic speakers out of the homeland occurred:   1) a movement eastward and probably southwards into the Solomon Islands.  This resulted in the establishment of the homeland of Proto Central-Eastern Oceanic.[78]         2) a movement via New Ireland and the St. Matthias group northwest to the Admiralty Islands, the homeland of Proto Admiralty.  Kennedy[79] notes that the archaeological evidence implies "settlement in the Admiralties in the period 4500-5000 B.P." and that the immigrants came "from the New Britain-New Ireland area".

After these two early migrations, there was a further period of stability in the homeland region where the innovations characteristic of "Western Oceanic" occurred.  After a period when Western Oceanic developed east and west "linkages", or perhaps dialect chains, the west linkage gradually dispersed giving rise to the North New Guinea Cluster of languages extending from the Huon Gulf region north and northwest to the Madang and Sepik coasts.  A migration southwards by a Western Oceanic dialect established the homeland of Proto Papuan Tip somewhere in the D'Entrecasteaux Islands or the nearby mainland.  The eastern linkage of Western Oceanic was ancestral to the Meso-Melanesian Cluster and one or perhaps several migrations to southern New Ireland gave rise to that region as a centre of dispersal, some groups spreading north and east to occupy the rest of New Ireland and its offshore islands, and other groups moving south to establish the Northwest Solomonic group of languages.  (See Map 5 for the main Oceanic AN migrations in the New Guinea region.)

As indicated earlier, the AN languages of Irian Jaya comprise 30-odd non-Oceanic and 5 Oceanic languages. It is possible that the Oceanic languages represent the furthest western migration of the North New Guinea Cluster.  It is unclear how long the non-Oceanic languages have been established in western Irian Jaya.

1.5.2.2.  It appears to be the case that the peopling of the remainder of Oceania began with an early migration out of the Proto Oceanic homeland and into Bougainville and the Solomon Islands.  Further migrations from this region filtered down through Island Melanesia, that is Vanuatu and New Caledonia and eventually Fiji, by about 1500 B.C.  "At about this time", says Tryon,[80] "a set of migrations apparently began in the northern/central Vanuatu region, one moving north, spreading the Austronesian languages throughout Micronesia (for which there is evidence of an east to west spread), another moving southeast to the Fiji group.  From there, after a period of consolidation, the Polynesian languages evolved, moving out from the Tonga-Niue area sometime around 1000 B.C."  The great Polynesian migrations spread throughout Triangle Polynesia as well as back into Island Melanesia to form the Outliers.  There was an eastward flow from Samoa to the Marquesas, the Society Islands, Mangareva and Easter Island; there was also a north-eastward movement to Hawaii.  From the Tahiti area there were migrations to the Cook Islands and eventually to New Zealand which was settled by 900 A.D.[81]

1.5.3  PREHISTORIC TRADE AND TECHNOLOGY

Allen puts the view that "before white contact Papua New Guinea communities were small and very localized.  With few exceptions they could provide for themselves all their subsistence requirements from within their own territory ... Nevertheless, the country was criss-crossed with well established trade routes."[82]  Hughes,[83] in his New Guinea stone age trade, indicates the existence of an extensive trade system which involved the exchange of various goods such as salts, pigments, mineral oils, pottery, stone tools, and sea shells.  In addition to these interior trade routes there were also the well-known trading cycles of the coastal regions such as the Vitiaz Strait cycle,[84] the Kula Ring[85] in the Trobriand Islands-Milne Bay area, and the Hiri exchange system between the Port Moresby area and the Gulf of Papua.[86]  That there is considerable time depth to the existence of trade routes is supported by archaeological evidence.  Bulmer[87]  indicates that the introduction of new technology, in the form of a particular type of stone axe/adze, occurred in the highlands of Papua New Guinea between 11,000 and 6000 years ago.  Evidence for widespread trade occurring during the period 6000 to 3000 years ago is clear.

In Island Melanesia, there is evidence of continued contact between fairly widespread Oceanic groups.  For example, Kennedy indicates "that more than haphazard and accidental contact between the Admiralties and other parts of Melanesia continued for most of the prehistoric period, marked by pottery styles widespread in Melanesia, first Lapita then incised-impressed relief, and by distributions of Lou Island obsidian."[88]  Obsidian from the Talasea region of West New Britain has been found in various sites in Melanesia, the earliest recorded example, found in another part of West New Britain, is dated 11,400 B.P., and a further sample, from New Ireland, some 600 km away, is dated 6800 B.P.[89]

Various prehistorians[90] have associated Proto Oceanic and its early daughter languages with the Lapita cultural complex.  There is evidence of a period of widespread ceramic production throughout Oceania which has some shared features and which suggests the spread of an identifiable cultural complex from, possibly, the Proto Oceanic homeland right through to Polynesia.[91]  Davidson notes, for example, that "the archaeological evidence suggests that most or all of the Western Polynesian islands as well as Fiji were settled during the second millennium B.C. by related people and thereafter were in sufficiently regular contact to share in a similar sequence of ceramic change".[92]

1.5.4  SUMMARY

The overall picture we obtain from linguistic reconstructions and from the archaeological record is a complex one.  The main points that have a bearing on the subsequent argument are summarized here.

1.5.4.1  There was at least one or perhaps several migrations by Australoids into the New Guinea-Australia continent beginning about 60,000 years ago.  Remnants of these Australoid groups may still have descendants in New Guinea and these may be language groups belonging to the Sepik-Ramu Phylum.

1.5.4.2  There were at least two, perhaps more, migrations of NAN language groups into the New Guinea mainland, the first between 10,000 to 15,000 years ago.  The main migration was that bringing speakers of the language(s) of the Trans-New Guinea Phylum and this is thought to have occurred about 5000 years ago.  East Papuan Phylum speakers were, at some time, displaced from the mainland and moved into the islands to the east.

1.5.4.3  Perhaps about 4000 to 5000 years ago, AN speakers settled in the New Britain region which became the homeland of Proto Oceanic.  Various migrations originating from here brought AN speakers to the various coastal areas of Papua New Guinea, the remainder of Island Melanesia, Micronesia, and Triangle Polynesia.

1.5.4.4  A significant amount of contact occurred between groups in Island Melanesia.  Trade routes were established throughout various coastal regions and also between the coast and the interior highlands, and within the highlands region itself.  Thus, in the prehistoric period, within the New Guinea mainland as well as in the islands of Oceania, groups were not entirely isolated from one another and there existed varying degrees of contact and flow of trade goods.

1.6  THE DESCRIPTION AND CLASSIFICATION OF DATA

In order to present a coherent picture of the very large amount of number data given in the Appendices, it has been useful to adopt a number of descriptive terms which may be used to classify the various systems into a relatively small number of types.  In some of the older linguistic literature concerned with the description of natural language numeral systems, it was common to use the descriptive term "base" when discussing the cyclic nature of the system.  Thus we find counting systems variously termed "binary" (base 2), "ternary" (base 3), "quinary" (base 5), "decimal" (base 10), and "vigesimal" (base 20).  Using a single number to characterise a counting system is reasonably adequate when we are dealing with, say, the English counting system which, with some irregularities, is essentially a base 10 one.  The cyclic structure of many of the counting systems found in Melanesia is often more complex than the English system, in that a single system may have elements of base 2, base 5, and base 20; others have a structure in which we can discern elements of base 5, base 10, and base 20.  This was recognized in the older literature in which we find reference to "mixed base" systems and such terms as "incomplete decimal" systems (that is, one which had elements of both base 5 and base 10). 

The descriptive terms which were adopted for this study, which have been used throughout the PNG and the Oceania volumes, and which were found to be generally adequate, are those coined by Salzmann,[93] namely frame pattern, cyclic pattern, and operative pattern.  These terms are defined in the "Preliminary" section of Appendix A. They will be briefly summarized here.  Suppose we have a sequence of natural language numerals which we can associate, in a one-to-one correspondence, with the symbolic numerals of mathematical notation.  This sequence will normally contain a unique set of number morphs together with a complementary set of complex number words which are analyzable into component number morphs. For each complex number word which can be broken down into its components, it is usually possible to infer a number sentence involving operations on two or more numbers.  To take an example, let us suppose that we have analyzed a sequence of numerals to have the form:

1, 2, 3, 4, 4+1, 4+2, 4+3, 2x4, (2x4)+1, (2x4)+2, (2x4)+3, 3x4, ..., (4x4)+3, 20, 20+1, 20+2, 20+3, 20+4, (20+4)+1, (20+4)+2, ..., 2x20, (2x20)+1, (2x20)+2, ...

i.e there are distinct number morphs for 1 to 4, and 20, and all other members of the sequence are composed of these. The numeral "5", for example, may be analyzed as having the component number morphs "4" and "1".  We infer that "5" is thus a complex number word which is represented by the number sentence "5 = 4+1".  Similarly "8" is analyzable as having the component number morphs "2" and "4" and we infer that "8" is a complex number word representable by the number sentence "8 = 2x4".  Using Salzmann's terminology, the set  (1, 2, 3, 4, 20), i.e. the number morphs from which all other numerals in the sequence are generated, is called the frame pattern  of the sequence.  Once the sequence has been analyzed into its symbolic form, its cyclic structure becomes apparent.  For the example above, the sequence has a cycle of 4 and a superordinate cycle of 20.  Salzmann calls this the cyclic pattern  of the numeral sequence and denotes it by the set (4, 20).  It is often the case that there is not always sufficient data available to determine the complete cyclic pattern of a numeral sequence.  Finally, the operative pattern  of a numeral sequence is essentially a summary of the various number sentences which indicate how the complex number words in the sequence are composed.

I have indicated above that Salzmann's terms were found to be generally adequate.  No single typological system can do justice to the diversity and richness of the ways number words are combined to form other number words.  Salzmann introduced his terms as a reaction to the oversimplistic labelling of numeral systems as found in the work of nineteenth and early twentieth century writers.  Hymes[94] has criticized Salzmann's terminology as being inadequate for the description of the numeral systems of certain Amerindian languages, however the criticisms do not appear to apply to the situation regarding, in particular, the systems found in PNG.  As far as the typology of numerals found in the New Guinea languages is concerned, there have been several different variants applied.  Ray,[95]  in his discussion of "numeration  and numerals in the Melanesian languages of British New Guinea" used the terms, as given above, for describing the "base" of the numeral system, that is "incomplete decimal", "decimal", "vigesimal", and so on.  Galis,[96] in his study of the counting systems of Irian Jaya, arrived at a sixfold classification: (1) body-part tally systems, (2) base 2 (or "binary") systems, (3) base 6 systems, (4) base 4 systems, (5) the "digit-tally" system (with a (5, 20) cyclic pattern), (6) the "Austronesian" type, i.e a base 10 system with 10 discrete numerals.  Lancy,[97] introduced a four-type classification ranging from body-part tally systems (Type I); systems which have 3 to 4 discrete number words and a base of 2, 3, 4, or 5, and where objects may be used in carrying out tallies (Type II); the "quinary- vigesimal" system, that is a counting system having a (5, 20) cyclic pattern and which usually employs fingers and toes as an aid to tallying (Type III); and the "decimal" system, i.e a 10-cycle system which normally has no reference to body-parts and which has 6 to 10 discrete numerals.  Smith,[98]  in his study of the counting systems of the Morobe Province of PNG, distinguishes 8 varieties of "counting procedure": (1) body-part tally systems, (2) systems with only two numerals, (3) systems which employ two numerals plus finger-and-toe "digit tally", (4) systems with three numerals only, (5) systems which employ three numerals plus "digit tally", (6) systems which employ four numerals plus "digit tally", (7) systems which have a numeral for 10, and (8) systems which have numerals for 10 and 20. 

1.7  ORGANIZATION OF THE STUDY

In the chapters which follow, I will first describe the counting system situation of New Guinea and Oceania. I have grouped various types together according to their cyclic pattern rather than use any of the alternative typological systems outlined above.  Initially, I will distinguish systems by the primary cycle of their cyclic pattern.  Having used an admittedly blunt way of differentiating the systems, I will then discuss in some detail the variants which occur in a particular type. 

In Chapter 2, I provide a brief summary of the existence and nature of 2-cycle systems as they occur in other parts of the world.  I then delineate several variants of numeral systems which have a primary 2-cycle and their geographical distribution in the New Guinea area is discussed.  In addition, I will describe several varieties of body-part tally methods (as opposed to systems which use only the fingers and toes for tallying) which occur uniquely in Australia and New Guinea. 

In Chapter 3, the 5-cycle system and its variants are dealt with; this includes the common "digit tally" system with a (5, 20) cyclic pattern.  The occurrence of the various 5-cycle variants in both the NAN and AN languages is discussed and their geographical distribution in New Guinea and Island Melanesia is given. 

In Chapter 4, the 10-cycle system and its variants are discussed after providing a brief summary of their existence in other parts of the world.  The distribution of these in New Guinea, Island Melanesia, Polynesia, and Micronesia is also discussed.

In Chapter 5, the relatively unusual 4-cycle and 6-cycle systems are dealt with, and the chapter concludes with an overview of the various types of numeral system which occur in the region under consideration, together with a brief delineation of types which are not found.  While Chapters 2 to 5 are essentially descriptive they also prepare the ground for the remaining chapters. 

Chapter 6 deals with counting and number in context and the range of circumstances in which number is used or is important in the cultures under examination; this is illustrated by a series of brief case studies of several traditional societies.  In addition, I will deal with several related matters such as numeral classification, and large numbers and the limits of counting. 

In Chapter 7 I examine aspects of the diffusionist view of the existence of various counting system types and their distribution throughout New Guinea and Oceania.  I shall refer especially to a major work by Seidenberg who is a leading proponent of the diffusionist view and I shall provide a critique of his theory in the light of the data collected for this study. 

Finally, in Chapter 8, an attempt is made to construct an outline of the prehistory of number for this region.  Beginning with the NAN languages, the contemporary situation regarding the distribution of counting systems and tally methods in each of the major and minor phyla is surveyed.  From this survey I attempt to infer the nature of the proto systems which may have been present in the early history of these phyla.  Secondly, the fate of 10-cycle system of the AN immigrants is traced as they spread out from the Proto Oceanic  homeland into the coastal and island regions of PNG and into the remainder of Island Melanesia and the Pacific.

Map 1: Melanesia, Micronesia, and Polynesia

Map 2: Distribution of NAN languages in New Guinea

Map 3: Distribution of AN languages in New Guinea

Map 4: Early language migrations in New Guinea

 

Map 5: Oceanic AN migrations in Papua New Guinea