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Part and whole numbers: an 'enumerative' reinterpretation of the Cambridge anthropological expedition to Torres Straits and its subjects.

The peoples of Melanesia have time and again forced anthropologists to re-evaluate what they considered innate human traits, including concepts of number, particularly the unitness of persons and things (c.f. Mauss 1954, Mimica 1988, Strathern 1988). This paper attempts a similarly reflexive analysis by looking at the meeting of the Cambridge Anthropological Expedition to Tortes Straits (CAETS) and Melanesian enumerative strategies. It utilises 1) an 'archaeological' approach to anthropology at the turn of the century in Foucault's sense (1970), and 2) contemporary and canonical ethnographies of Melanesia as compass points from which to triangulate otherwise disconnected 3) psychometric data generated by W.H.R. Rivers' experiments during the expedition. This reanalysis will thereby suggest an alternative explanation of the psychometric results by inserting a mid-level of interpretation centred on enumeration.

CAETS's group of intrepid anthropologists, linguists, and psychologists set off to map the diffusion of Melanesian cultures in 1898. They tried to implement a scientific method of abstraction to 'pure' numbers--which had been so successful in the more established physical sciences--to disentangle the diffusion of races (Haddon 1898). The psychometric tests used by Rivers, then a psychologist, were not designed to test numerical competence but to measure the basic motor skills of these 'savage' races in comparison with Europeans. However, this paper maintains that the tests were premised upon an appreciation of geometry which, though contrasted with that of the Torres Straits Islanders, claimed a privileged access to truth in a way which concealed the tests' worth as insights into both CEATS and their Torres Straits counterparts. Rivers had expected to confirm the prevailing Spencerian understanding of linear evolution, in which primitives were thought to have more advanced sensory and motor skills at the expense of advanced knowledge. Instead his data comprised a mixed bag of abilities that did not consistently fit any evolutionary model, and try as they might, the contributors could not draw convincing statistical conclusions from it (c.f. Kuklick 1991, 1996; Schaffer 1994). More recent anthropological discourse on Melanesian pattern, number and cosmology provides us with an alternative explanation for the results based on ethnographic knowledge rather than a-cultural notions of number concepts. They reveal the central place indigenous enumeration plays in calculation, and offer an implicit critique of anthropologists who either ignore or assume similarity in numerical comprehension. It is the aim of this paper to demonstrate that it is possible to glean new insights from existent data despite a different original point of focus via a subtle shift towards enumeration as a sphere of cultural thought.

The use of the term enumeration as opposed to ethnomathematics or counting will become apparent during the course of the paper, but to make its position clear, the paper must first define what this researcher takes enumeration to be. In normal discourse enumeration is used to mean 'the act of counting specified objects'. The potential utility of the definition lies specifically in the word 'act', pertaining to an ego-centric process of reaching a number which then defines the objects of interest in numerical terms. By dissolving the ego-centric aspect of enumeration, and analysing it as a culturally determined and thereby inherently social act, it is possible to come to an anthropological understanding of enumeration as the 'instantiation of a culturally circumscribed concept of quantity'. Understanding of the two groups enumerative strategies allows us to triangulate the seemingly unrelated psychometric tests used by Rivers in a way which reinvigorates them as anthropological rather than psychometric material.

CAETS AND A NINETEENTH CENTURY ANTHROPOLOGICAL ENUMERATIVE COSMOLOGY

No science can progress or be definite without measurements of one sort or another. What, then, are those made in anthropological inquiries, and for what purposes are they made? (Haddon 1898:1)

Alfred Cort Haddon, anthropological specialist in 'primitive' material culture, asked this question on the first page of his final book before setting off to Torres Straits as leader of CAETS. The query was timely; it reflected the need to put British anthropology firmly within the realm of the empirical scientific tradition and represented the culmination of a trend in the wider social sciences that dominated anthropology from the late nineteenth century through to the end of the First World War, finding its apogee in CAETS.

At the time CAETS took to the Pacific, there was a perceived need to mathematicise the understanding of man among the social sciences, and thus propagate a more scientific anthropology, avoiding 'the not unkindly hesitancy on the part of men engaged in the precise operations of mathematics, physics, chemistry, biology, to admit that the problems of anthropology are amenable to scientific treatment' (Tylor 1889:245). At the heart of the attempt to incorporate a scientific method into anthropology was measurement, which, once applied to human characteristics, meant "neutral' mathematical methods could be employed to objectively categorise first hand data on individuals and races from the comfort of British armchairs. This in turn had important effects on the choice of methods and documentation employed by the members of CAETS towards Melanesian numerical perceptions (c.f. Schaffer 1994).

In 1885, R.H. Codrington recalls the first accounts of Melanesian counting practices emanating from travellers' tales, which painted a picture of an almost entirely innumerate society. Sailors would hold up a number of fingers and ask the natives to say what word stood for that number. This often led to a complete misunderstanding, as it is common practice in Melanesia to count downwards on the fingers and toes, eliminating fingers as one goes rather than extending them. Hence not only would the travellers get the names for numbers the opposite way around, but would also mistakenly believe that savages could not count beyond four! As the travellers would hold up five fingers, the Melanesian would consider it as referring to nothing and would simply look at the hand in a bemused way (Codrington 1885:222). Reports thus circulated that these were a particularly primitive people in terms of mathematics, a factor which doubtlessly increased their reputation as some of the world's most savage people.

R.H. Codrington was the first to systematically document Melanesian counting systems in The Melanesian Languages (1885). An extremely detailed and comprehensive book, it focused on the grammatical rules, semantics and etymology of a wide array of groups. Codrington devised a system for describing counting systems by numerical size of base number. Sidney Ray, CAETS language specialist, adopts this system in Volume III of the Reports, where counting practices are divided into four categories: 1) quinary, or base number 5; 2) imperfect decimal, where base 10 is used following the use of quinary; 3) Decimal, with a straight base number 10 and; 4) Vigesimal, where quinary is used up to 20, upon which a primary base 20 takes over (Reports III [Ray] 1907:464). The closest to the Western decimal system is seen as the most advanced; those within the imperfect decimal category were treated with considerable care in order to rank them precisely within a graduated linear scale at a later date, showing how the progression to decimal occurs.

It is telling that in both books, numeracy is considered best understood as an aspect of language, and thus included in the Third (linguistic) Report. Their matter-of-fact treatment demonstrates that as a concept, numbers were themselves taken as universal. In line with a Spencerian philosophy, a complicated system for counting objects on the body was representative of a lack of skill at higher arithmetic. This was amply demonstrated by three traits which were ubiquitous in the peoples documented:

1) a variety of words for counting different objects; 2) the proliferation of bodily counting, and 3) the association of number words with other non-numerical aspects of life.

Firstly the pronounced use of different counting words with different objects seemed to display a lack of familiarity with (the concept of) abstraction. Though there were some words for general counting purposes in all the groups, each had particular words or addends for counting particularly important objects such as pigs, men, or betel nut (Reports III [Ray] 1907:474). This was evidence for a magical conception of objects rather than a rationalistic and abstractive use of numbers to refer to all objects. The natives therefore denied the primacy of number as a separable attribute of things, seen as the first great leap in the creation of mathematics, hence showing the subjects 'primitive' concept of numbers.

Secondly the fact that counting in Melanesia was primarily done on the body further reinforced their 'inability' to think about numbers abstractly. Codrington and Ray both note that though in most Papuan languages the base system for counting did contain the possibility of words denoting high numbers, people nevertheless quickly reverted to using their bodies to count with. Indeed it seemed that none of the populations was particularly bothered with high numbers, often using a blanket term for many, such as the Banks Islanders phrase tar mataqelaqela, literally 'eye-blind thousand' which describes the visual confounding of persons when confronted with many beyond count (Codrington 1885:250). When people were persuaded to count to a high number beyond the point where they applied a blanket term for many, they appeared to be working out the counting system as they went. Melanesians often reverted to what appeared babyish or dismissive tactics when dealing with difficult large numbers. Early anthropologists thus believed that the base system was an aspect of culture that created the conditions from which further advance could be fostered. The naturally most advanced was a priori the decimal system, hence the appointment of an imperfect decimal category.

Thirdly, the lack of separation of words for numbers from those used in other life practices showed their indistinction as a sphere, further denouncing their ability to abstract and look past the immediate. Ray was able to confirm Codrington's assertion that the general meaning for words that act as a completor (equivalent to a decimal place or primary base number) is 'one man' or 'one man is dead' (Reports III [Ray] 1907:466). The word dead itself also meant 'finished', so death, finishing and completing may be equated within the body of one man during the counting process. So at the same time as counting upwards in number, the person is also counting 'out' one complete person, which cohered logically with the Melanesian practice of counting their body parts "out' as they count upwards, suggesting a more subtractive rather than additional system of counting, which must have seemed overly complex.

Another aspect of Melanesian culture following this is their material culture and especially weaving and ornamentation, due to the way it was treated as possessing simple systematic technical operations analogous to--but not directly linked with--their mathematical knowledge. Melanesian 'arts' were very well documented, but their connection to mathematical expertise was overlooked because nothing which they produced appeared to demonstrate any mathematical ability in a Western abstract sense. At the time there was an assumption that all ornamentation or embellishments on functional artefacts would naturally evolve towards naturalistic representation and thus their predominantly patterned 'arts' were considered rudimentary at best (c.f. Connelly 1995; Trilling 2001).

Volume IV on the material culture of the Torres Straits Islanders includes a protracted section on textiles described by A. Hingston Quiggin as 'the most important of the native arts' (Reports 1912:63). Of these the native baskets are the most complex, and involve the most technical expertise, with 28 different terms for particular weaving techniques. These are often painted in a manner which emphasises the patterns of the weave (ibid:67). For Hingston Quiggin it was 'in this domain, more perhaps than in any other, that uncivilised fingers shew their superiority over mechanical science.' (ibid:78-79). Basket weaving was seen as a primitive mechanics, in which they were considered particularly skilled, to the detriment of a higher aesthetic and geometric ability. Nevertheless basketry was seen as essentially restrictive 'since, in woven examples, it limits all designs to straight-line effects' (ibid:82). The boldest effect, obtained by a junction of horizontal and vertical lines, was symmetrical zigzags, which was unimpressive to the Western observer despite their seemingly endless variety, due to their simple construction (ibid).

As far as technical ability was concerned, then, indigenous basketry was seen as expressing a refined understanding of an essentially primitive art form. Haddon, having documented the Islanders 'art' on a previous voyage, was careful enough to deny any parallels between modern science and this primitive art form:

In the foregoing Memoir I have repeatedly alluded to designs or patterns as "Geometrical" in style: in no case do I wish to imply that the natives have the least idea of geometry, or of any formulated knowledge of the properties of space, or of the principles of design as studied in our schools of art. Their art work is dictated by their artistic feeling: it is the product of sentiment, not of rules (Haddon 1894:270).

As this paper will argue, Haddon could not have been more right and at the same time more wrong to disconnect formal geometry with indigenous technical expertise. The team was nevertheless not in the habit of making connections between the different aspects of the Islanders' culture, as they were in the business of creating precise and generalisable categories by which cultures could be compared. Such is evident in the encyclopaedic categorisation of cultural and physical attributes in individual volumes of the Reports, and the structure of the Reports as a collective. This is not to say that Haddon had no mathematical agenda for this material, he was a great believer that 'ornamental designs could ostensibly be collected, like so many cultural fossils, and arranged in a chronological order of progression toward the presumedly universal goal of naturalism' (Connelly 1995:62).

Such was CAETS understanding of their own and Melanesian number systems; their presupposed ideas of the natural progression of mathematical knowledge resulting from their cosmological acceptance of abstract number, led them simply to document the insufficiencies of the Islanders in terms of bodily attribution, base system, and recording numbers. Through comparison this would be placed on the linear evolutionary scale. The members were part of a society that wrote numbers, and thus extracted them from their contextual designation of 'things'; they assumed that this was necessarily a higher rung on the ladder of mathematical advancement toward truth, which in turn gave them the tools from which to examine the objective world through science. This achievement was to be emulated by anthropology to increment its advance, and chart the success of other races along this scale. So the task was to see which were the more primitive counting systems by measuring the recording system which the savages used, and this would generate all the information they needed as to the stage of native mathematical proficiency. More so than any other aspect of culture, the a-cultural designation of mathematics put it on a simple scale of advance, unable to develop in any other direction except the slow discovery of the primary axioms of mathematics.

A CONTEMPORARY UNDERSTANDING OF MELANESIAN ENUMERATION

Here we shall use the term 'enumeration' to expand the scope of our investigation into an indigenous Melanesian cosmology of number. Some anthropological work has been done on culturally diverse number systems, usually labelled ethnomathematics and seen as a defence of traditional counting practices, providing a counterpoint to 'Western mathematics' and its increasing global hegemony (c.f. Ascher 1991,2002; Bishop 1990; Crump 1990). This has remained largely separate from the critical philosophers and historians of science who have produced powerful critiques of a trend in 'Western mathematics' (and associated disciplines like physics) to claim objective truth in the last two decades (c.f. Hacking 1990; Smolin 2006). 'Western mathematics' is not at issue to these thinkers, or here, rather the issue is a particular Platonic claim to universal truth and their feedback into how certain social groups claim truth over others (c.f. Hacking 1986). The very term ethnomathematics is an inherently problematic one, which a priori assumes all systems of ethnically numerical logic are to be understood as a form of mathematics, rather than on their own terms. As stated above, this paper attempts to adjoin the two potentially powerful critiques of mathematical truth claims by distancing itself from the term mathematics, in favour of the notion of enumeration, which this researcher finds a more useful way of engaging with both the foregoing exploits of early anthropology and the following account of an aspect of Melanesian thought.

First it is important to stress the variety of Melanesian counting systems, the latest count confirms over 1,400 in Papua New Guinea alone, a country that contains approximately 700 languages (EDL 2005). Nevertheless Melanesianist anthropologists generally agree on a particular 'Melanesianness' to the cultures of this geographic area, despite much time being devoted to the boundaries of this distinction (c.f. Thomas 1989). Within the category itself, there is another commonly accepted axis of comparison between Highlands, Lowlands, and Islands (c.f. Hays et al. 1993). We must therefore be extremely careful not to subject Melanesia to sweeping categorisation which would be tantamount to a similar classification to that used by CAETS and its contemporaries in the years of diffusionism.

Within ethnomathematical documentation, as within CAETS, the major features which distinguish number systems are their vocabulary, its etymology, and the base numbers it uses, as well as some of their cultural components and effects (c.f. Ascher 1991). Here we shall consider commonalities between systems in logical terms, rather than narrowly within vocabulary differences, unless there is some uniformity to the etymological association. As well as considering the cultural elements of number systems, the following also discusses their material cultural constituents, which are particularly pertinent to societies which seem not to abstract number from its composing material elements. Thus we shall explore the 'enumeration' of Melanesian counting systems, looking at their variety, contextual instantiations and roots. This paper does not claim that all the features described below are common to all groups, rather it uses a number of ethnographic vignettes to look at some of the relatively uniform aspects of enumeration in Melanesia. In this sense the following section follows a Strathernian (1988, 1992) approach to the ethnographic material. By attempting to draw out certain common elements of Melanesian enumeration, this material is brought to bear on CAETS own interpretation of Torres Straits counting systems. Strathern figures strongly in the following, which is both a matter of analytical strategy and ethnographic necessity given her focus on the subject and the limited empirical evidence available. Combining this sometimes polarising anthropologist with a material cultural approach is by no means the only way these data might be interpreted within Melanesian ethnography, but this researcher finds them together to be the most congruent to the material.

COUNTING AND COSMOLOGY IN MELANESIA

As was described at the time, counting in Melanesia is done primarily on the body, using most often the fingers, but many groups also use toes, and/or palms, wrists, arms, facial features, nipples, and genitals (c.f. Wassman & Dasen 1994:82; EDL 2005). These follow a specific order, usually either from one side of the body to the other, or symmetrically on one side of the body then the other (c.f. Biersack 1982 for a particularly inventive version of this). This is complicated often by a primacy accorded to digits, being counted before using other parts of the body. As discussed above, Codrington and CAETS note the practice of counting downwards on the body parts, effectively eliminating parts of the body as one counts upwards until the person is 'finished' or 'dead'. To this Euro-American mind this represents a simultaneous mirroring of counting upward while counting downwards. However this is actually an unnecessary step to translate the system, making their system appear inefficient by analogy. Actually it is much easier to conceptualise the system at this point as subtractive, where a person attempts to count objects by eliminating them from the remainder, resolving the counting problem when all objects are eliminated and the actor is left with a numerical value which represents the amount taken out of the world by counting. This is expressed through the bodily counting system, where progressively parts of a person are eliminated until they are finished entirely, at which point they reach their primary base number, a person (c.f. Biersack 1982:813). Usually beginning with the hand therefore, 5 (4 if the thumb is not counted or 6 if the palm and thumb are counted) objects are literally equal to a hand taken from the body, which is usually given the same word (Codrington 1885; Reports III [Ray] 1907). Thus both person and counted object are subtracted from the rest of the world/body and given a numerical value. (1) What happens then when a person is finished, at let us say 20 or 28, and there are still objects left to count?

For the Iqwaye people studied by Mimica (1988), the counting continues after 20 on another person, to which the actor points. (2) The words for this are one person plus however much of the next person has been used. As one continues to more than two persons, the persons are represented upon the digits of the counter, such that a person might have a hand of persons and a hand of units (= (5x20) + 5 = 105). Interestingly though, when twenty persons' digits have been counted (400), this also returns to a single person being finished. Mimica notes that in the case of both 20 and 400, they are literally equal to one again, as opposed to the mathematical representations here used to illustrate the process. (The same applies for the Paiela of Enga Province studied by Biersack (1982) at 28. The Paiela see their bodies as utumane 'whole or undivided', and their bodily counting as thus progressive segmentation rather than intrinsically distinctive units (ibid:813)). The system is logically capable of continuing indefinitely, returning to a nominal one upon reaching 20". However this becomes progressively more futile for the Iqwaye for the reason stated above, that this is not a mirroring system, whereby we can switch to seeing a person as equal to one only nominally, but really 20,400 or 8000, by juxtaposing the decimal system. Instead this literally is one, a more important one surely, but one nonetheless, a problem which we shall now address from another angle, cosmogony.

The Iqwaye creator of the cosmos and the first man, known as Omalyce, started off self-contained (Mimica 1988:75), with hands and toes clasped together and sucking on his penis which was his umbilical cord. After the creative rupture, sky and earth were formed as he opened up, his eyes became sun and moon. He took some mud, shaped it into figurines, inseminated them, thus making them sons, and from these all Iqwaye people came. Therefore for them, from the 1 there came all the multitude of things in the world, and when the Iqwaye count upwards, they are actually counting downwards to the original totality of 1. Hence the many are equal to the one in the eyes of Omalyce, and if the Iqwaye were to count all the people and things of the world they would in the end equal 1 : Omalyce. (3) As this myth forms the basis of Iqwaye ontology, there is no double mindset at play here when units are accruing; there is only the returning of things to their original state of oneness via counting. (4) From here we may discard the concept of a subtractive system, as the elements being counted are not being taken out and abstracted, they are simply nominated as a part of the wholeness which is Omalyce. Roy Wagner draws a useful analogy when he points out that if one were to do a census of all the Iqwaye people, and tell them that they had so and so number of people, this would be a meaningless statistic to them, as they are only parts of what is collectively a one, not a collection of individuals (1991:167). It follows then that for these people it would be impossible to explain a counted number of objects in terms of absolute unit quantity at all, as each unit is just a fraction of the whole. (5) This explains the ability of the Islanders to explicate words for their numbers when pressed, but appearing difficult for them to CAETS. It is not difficult however, merely illogical, as one gets larger and larger, it just becomes a more significant fraction, and as one cannot say how big everything is, only that it is one, how large a thing is in numbers progressively matters less and less.

The actual size of the universe is not specified outside of being one, so a person can never be sure what exact fraction of the whole a person has at any particular moment. In order to deal with this uncertain fractionality of the cosmos, numbers must return to one at a given point, though this is an essentially arbitrary fragment used to make the process easier by containing it within a one, and matters less the larger the number. Thus blanket terms for many are a more useful way of dealing with uncommonly large quantities. What becomes apparent here is that the material world, including the body, is an essential part of Melanesian enumerative strategies, all things are interconnected by their finitude, their oneness, and are thus incapable of being abstracted.

A MATERIAL COSMOLOGY OF ENUMERATION

So Melanesians do not abstract ideas from substance (c.f. Henare, Holbraad & Wastell 2007 for a discussion of this); in fact they perceive the world as constituted in temporal flows of bodily substance through things and persons. This is not a new idea. It was one of the central tenets of Marcel Mauss' The Gift, that whereas Europeans entertain a marked distinction between persons and things, Melanesians do not (1990 (1954):46). It is thus said that in the giving of a gift (the primary means of exchange) a person is literally giving a part of him/her. The implication (though not made explicitly by Mauss) was that in so far as Melanesians were concerned, people were divisible, and the idea of the person was not constituted therefore in the body as unit. The fact that one could cut oneself up and portion oneself out, as in counting, interrogated the individuality and unit-ness of persons, causing anthropologists to revise their Eurocentric assumptions of individual personhood (c.f. Gell 1999).

According to Strathern (1988), the logical principle that makes up society in Melanesia is that of relations, rather than objects/persons, which simply provide the visible signs of the relations which underlie them. In The Gender of the Gift, objects/persons are a conduit for the sum of all the relations of exchange that have made them, hence when a gift is exchanged between persons, the size of the gift is an outward expression of the inner quality of that person. Being able to instantiate their inner fibre into a large number or high quality of pigs or shells proves the size of their inner worth. Importantly though, this can only be done in comparison with the other actors in the exchange system, one's exchange partners and relatives. As such it is competitive. As the world is made up of relations, of which there are a limited (though not specified) amount, then by giving a large gift, one which an exchange partner is unable to repay, one eclipses their opponent into a temporally subservient position of owing. Hence reciprocal competitive exchanges comprise men constantly trying to indebt each other in a hierarchy of gift-giving, attempting to encompass other relations by upscaling the potential of the body to encompass increasing amounts of the relational world, literally becoming a 'big-man' over time. This principle is the same as that of counting, as one counts further parts of the world (made up of relations), one recursively, at particular moments of assessment (in exchange, reaching recursive numbers), comes to survey and individualise one's ability to contain.

[FIGURE 1 OMITTED]

Roy Wagner (1991) elaborates upon this idea by inserting the notion of fractality, a mathematical term meaning an irregular or fragmented geometrical shape that can be repeatedly subdivided into parts, each of which is a smaller instantiation of the whole. If each relation is seen as corresponding to a particular shape within the fractal, we can see that some exchange relationships encompass others. It is important here though to explicate the distinction Gell (1999) makes, which is that this is taken not in a Western mathematical sense. In the mathematical sense there is a quantifiable proportionality and similarity to the subdivided copies of the whole, however in a Melanesian world some constituting relations are regarded as of more importance than others, outside of their degree of subdivision. Furthermore, each relational part is of different weight depending on the particulars of the current exchange that one is participating in, and as such is subject to being 'eclipsed' in another situation of exchange or by new relationships, for a Melanesian fractal is a fluid fractal. Each relational person is constantly aiming to enlarge itself in comparison to the other relational persons by encompassing their relationships, and thus containing a larger portion of the fractal. However accomplished a participant is at gift-exchange, they can never elevate themselves outside of the system. One can eclipse all others at one point, but one is only the sum of those other relations, and thus unable to exist without them, and hence it is closed. Mirroring this interpretation, Wassman and Dasen (1994) note that the actual number reached at the point of completion often varies dependent upon who you speak to in any given community. Therefore the numerical extent represented by a person is not set in stone, allowing for the possibility of negotiation in numerical exchanges, where a person is as much as they can convincingly count on themselves in relation to others. Nevertheless, Wassman and Dasen's informants insist that their totals are in fact the same, being a full person.

[FIGURE 2 OMITTED]

Both Strathern (1999) and Wagner (1991) cite Mimica's (1988) ethnography of the Iqwaye counting system to illustrate their point of a closed system, or rather a recursive one. In a closed system, when one person quantifies their own position within society in the moment of exchange, they make the whole of society quantifiable. For if they create themselves as a discrete fraction of the relations within society, say IA, then there must be a remainder which is also quantified in the process = 3A. So it is only in the moment of social exchange, of determining ones position, when one can abstract and quantify persons/objects. At all other times objects/persons are inert in a sea of relations. For Melanesians the status quo of fractal similarity is not a desirable state of affairs. With reference to Baruya gender relations, Godelier demonstrates that separation from an originary androgynous state to achieve maleness is an often painful but very necessary task which must constantly be maintained (1982:7-10). Parting oneself from the totality is necessary to achieve gender, and hence prevent sickness and gain power. As we shall immanently see, the concern with part-whole relations is also present in some of the most important Melanesian materials: cloth and basket weaving. In both of these the individual weave, stitch or engraving is suggestive of the whole, containing within it the whole design, the one and the many.

THE MATERIAL CULTURE OF MELANESIAN ENUMERATION

If we can now take Melanesian enumeration as materially constituted in a radically different way to European geometry, we are no longer looking for an understanding of 'advanced' abstractive geometrical laws as CAETS were when they tackled Torres Straits counting, but an expression of multiplicity and oneness coexisting within the properties of material objects as they are in persons. As Biersack, echoing Levi-Strauss (1966), would call it, a 'science of the concrete' (1982;813). From this point we are now able to reintroduce the Melanesian material products which, though denounced by Haddon and CAETS as 'ungeometrical', provide the most complete picture of Torres Straits enumeration. The practice and products of weaving and basketry has interested both the members of CAETS and more recent theorists of Melanesia. Rather than CAETS' focus on the products, let us now examine the process of basket weaving as praxis in Melanesia.

As Ingold (2000) argued, counter to the standard view of making artefacts in which form emanates from a pre-existent idea in the mind of its creator, weaving involves an autopoiesis of material and idea. The growing basket forms the template for its own construction as each band is attached to the preceding one. Furthermore, the materials involved in weaving have tensile properties which resist and mould the finished artefact and eventually lend it its strength. In an important way every stitch in a weave, in as much as it is a symbiosis of idea and material, contains the possibility of the next, and so the finished product. As we shall see, not only does this have important implications for the artificial way that 'We' think of artefact producing, but is a recognised and integrated part of Melanesian cosmology in which making, patterning and enumerating are not individual processes, but part of an overall differentiating process which contains within it the possibility of both continuity and differentiation.

Basketry and weaving in Melanesia are not only extremely prevalent, but are often seen as an important aspect of groups' and individuals' identities, each employing various techniques which, when combined in particular ways, identify not only the skill of the maker, but also their clan and/or tribe (Kuchler & Were 2005). These objects provide a readily identifiable indicator of status, carrying a wealth of information on its visible outside, and at the same time having the power to keep its contents and construction technique secret. Drawing in part on Gell's (1993) analysis of the power of tattoos to trap the mind in Polynesia, Kuchler and Were argue for a wider recognition of the agency of pattern to constitute and transmit ideas: 'patterning, far from being a neutral or unremarkable artistic activity, has effects much more enduring than the often ephemeral media that serve as its vehicle, because it casts ideas about who we are into highly communicable forms' (2005:172). Such woven containers literally wrap things, persons and ideas in images which provide a whole made of individual lines bound together to give relational strength. More than this though, each weave and knot contains within it the possibility of all others. The choice of which motif to use in a band limits what can then follow, both physically through the materials and in culturally circumscribed ways. Each weave in a band must be representative of its fellows, and follow a regular, uniform system in order to generate the desired patterned effect and give maximum strength. Nevertheless bands can vary in their form and express the individual skills of their maker, as shown in Figs.5 and 6 below. This not only resonates with but also forms a possible material instantiation or a 'map' of Melanesian enumeration in that the regulated mass is at once interconnected and delineable/personable as a result of individual action. As a person weaves a basket, their knowledge of basket-making and tactile sense of the material allows them to identify how it will take shape, when it will cease being only a number of knots and become both many knots and one basket. This too appears fractional, as one can use only the length and strength of the starting material to make the basket, each band is a negation of the totality of the material in all its qualitative form.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

In Melanesian society, where pattern is preponderant in the material culture of weaving, the forms the weavings generally take are as containers of other objects. It is a common trait for groups to consider any object contained within string bags or other containers to be untouchable, and this is used to great effect by ambitious persons who intend to keep, or profess to have, powerful secret objects. Thus the many-ness of the knots, culminating in the oneness of the basket or bag has its own agency to contain and encompass other objects contained within. The temporal act of weaving and basket making contains in each movement the completed object, its creator, and an emergent potentiality for further encompassment. This aspect is again shared with Melanesian enumeration, both achieving a certain separation (and hence creation) of the self from the mass of the vast oneness by the act of creating a personal (in this case patterned) objectification of myriad relations, and gaining the ability to extend this further by showing one's skill at encompassing and containing a portion of that totality by its achievement.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Basket weaving also demonstrates that while a person may express individual skill in patterning, new patterns must always be built on existing styles. The very idea of patterning is inherently conservative due to the limiting properties of its constituent component shapes and materials, and thus it provides a stable medium from which innovation can be incorporated. The innovative act of creating new patterns by elaborating on existing schemes is a building block for putting new ideas into practice. Thus, as is noted by CAETS member Hingston-Quiggin: 'the distinctive character of each basket is most intimately bound up with its weaving pattern, which is, as it were, the expression of its maker's individuality, and the device of [his/]her peculiar skill' (Reports IV [Hingston-Quiggin] 1912:78). There is thus an individuality resulting from the exercise of skill in the enacting and personalising of patterns which are centrally conservative in their mechanical properties, and yet may result in vastly diverse manifestations. This is also the nexus of Melanesian enumeration, that there is a mass of undifferentiated possibilities from which it is possible for persons to individuate themselves, making it an end product, an achievement. Both Melanesian pattern and counting contain the same sense of a myriad undifferentiated whole, from which one must create difference, detachment, and thus unitness, by encompassing parts of the myriad and thereby making them discrete. Unitness is considered in quite an opposite way to our own system which takes it as a starting point from which relations must be achieved and maintained- here it is the relations themselves that pre-exist unitness, which instead must be achieved and maintained. From the example of basket weaving we come to understand the possible centrality of pattern in Melanesian enumeration as a material and temporal expression, and begin to see how it might be used as a vehicle for understanding numerico-geometric operations.

Returning to exchange and this time focusing on its materiality, as Strathern (1992) observed, and the above evidence suggests, unitness can be regarded as the finishing outcome of a transaction rather than the basis from which it proceeds in PNG. The act of exchanging qualitatively different items is done by comparison in order to achieve a correspondence of 1 to 1, or a paired totality of 1. For instance if a man had been saving pigs and cowrie shells to attain a bride, he would then enter into negotiation with the clan of the potential spouse. This wife-giver would look at the overall quality, including the amount, of the items on offer, and they would argue over their equivalence. Again this has a mirror in counting practices, this time among the Paiela. Biersack's Paiela informants pair sticks with the 28 pigs given during bridewealth payments, and say they are unable to count objects unless they are brought into a paired alignment. In an experiment derived by Biersack involving randomly distributed potatoes, the difference in their countability is whether the 'counter is able to perceive relationship' between them (1982:824). As the objects themselves come from a person or clan who are subject to the same qualitative assessment, there is no prior numerosity to the objects (including the wife) themselves outside of the quality of both items and exchange partners (Strathern 1992:172). The numerosity of the group of objects is achieved only at the point where the two parties decide that their relational qualities and the material qualities of the items offered, taken together, are of equal value. Among the Paiela, this is signified by the gestures of the pig counters, who show their approval of the animals with open or folded arms, eye contact and audible/inaudible speech. According to Strathern, the two parties then achieve a numerical status that can be equated 1 to 1. From this we see that tying qualities to each other through negotiation is a communicative way to achieve abstract unitness. Thus just as a woven item, made up of relational parts, only becomes 'one' upon completion, so a person should stabilise the materials and relations he/she brings to the exchange before it can be considered as 'one'. If this analysis is correct, and these persons cannot be separated from the objects with which they quantify, then it follows that materials contain both number and vital substance within them, which are inseparable until their unitness is achieved.

It can be seen therefore that patterns and the temporal-material actions associated with them may provide a model for achieving unitness in spheres such as exchange which appear to Western eyes as inherently calculative and consistent with a priori unitness. We shall now search for analogy with this evaluation in the data brought back by CAETS psychometric tests.

W.H.R. RIVERS' PSYCHOMETRIC (MIS-)UNDERSTANDING OF TORRES STRAITS ISLANDERS

In this section the pattern-centric enumerative cosmology of Melanesians is seen to provide a grammar for interpreting the geometric concepts articulated in W.H.R. Rivers' psychometric tests during CAETS. The aim of the following section will be to demonstrate a connection between the counting systems as documented by Codrington and the patterned basketwork of the Islanders described by Haddon and Hingston-Quiggin through the lens of Rivers' psychometric tests. This, as we shall see, is an area we might now find most conducive to Melanesian enumerative expression. The British anthropological scene at the time of CAETS was looking for differences between races at what they assumed to be a very deep level. The tests Rivers was in charge of conducting included visual and audio acuity, considered outside the realm of culture by most at the time. The findings however were inconclusive and ambiguous. Absent from the members was a full awareness of their method's cultural construction and its implications when met by an entirely different cultural construction of enumeration. This distorted the results completely, and led to the unreliable conclusions which eventually played a major part in the breakdown of a unified social scientific project.

Visual Spatial Perception

Rivers noted that this area of experimentation was one of the most popular of all his tests, as well as the most difficult to interpret. In one experiment four different groups representing adults and children in Melanesia and England were given a horizontal line of 100mm and asked to draw a vertical line from different points on the horizontal of an equivalent length to said horizontal. The illusion is supposed to make the subject draw a shorter vertical line than the horizontal model due to a visual trick of depth perception. As fig.7 below shows, while the Melanesian children displayed very similar results to the English children, who were only narrowly more accurate in most cases, the adults differed markedly from each other. This demonstrated a certain universality in human perception which only served to make the results of the adults more confusing. The Melanesian men were consistently much worse than even children from their own group while English adults are the most accurate in all cases. No explanation is given by Rivers except that in the case of the first test, where the vertical is to be drawn from the centre of the horizontal, that proficient English children stated that they imagined the two halves of the line and then doubled it. To Rivers this 'observation illustrates very well one cause of difference between the results of the savage and the cultured measurements, for one may feel fairly confident that such an artificial method was not employed by the Murray Islander' (Reports II [Rivers] 1901:114). He does not speculate whether this was the case with the Murray Island children, though their results were so close. This paper agrees with the conclusion reached here in a sense, as the most obvious fact coming from this test is that the two adult groups have learnt to understand what Rivers saw as 'geometric lines' differently in terms of their defining properties. Here is statistical evidence for the learnt nature of visual perception of 'measurement' in the two groups. Whilst children are likely to take a number of approaches to a problem, like dividing lines in half, adults will become used to a particular method of perceiving such problems. In the case of the English adults, especially psychology students, theirs is to take perspective as unreliable and attempt to approach the problem in terms of objective measurement.

The question of what the Islanders were doing here remains. In this case the comparable lines represent an opportunity to draw the line which feels aesthetically similar in length. For a Melanesian it would be inappropriate to draw a line which was of exactly the same length if it would not look like it was 'equal'. The fact that perspective makes a person think a line on the vertical is longer than a horizontal is simply not the issue for the Islanders. For Melanesians proportionality in terms of overall length in measurement is only one factor in the overall quality of the pattern in a way analogous to Strathern's (1992) analysis of Melanesian deriving of equivalence. As discussed above, unitness can be regarded as the finishing outcome of a transaction rather than the basis from which it proceeds, and the act of exchanging qualitatively different items is done by comparison in order to achieve a correspondence of 1 to 1, or a paired totality of 1. Therefore, in our current example, it is much more important that the length fit as equal in the overall aesthetic than it is that they be of an exactly proportional length. In fact to them this is proportionality, as is suggested by the stubborn assertion of those tested, despite being corrected, that they were in fact right the first time, and their producing similar results when tested more than once. So it is the case that measured length is a component in a material system of quality in which enumeration is but one aspect of overall quality. This may at the moment seem speculative, yet comparison with the following two examples will make this assertion more solidly grounded.

Rivers (Reports: 106-108) found more differences testing visual spatial perception by the bisection of lines. When asked to divide a line into an equal amount of parts, for example four or eight, not one Melanesian divided the line into half and then subdivided the sections into halves again, instead they always began from one side and proceeded along the line successively portioning it out. The results tended to be extremely inaccurate. Rivers also noted that this tended to be further accentuated with age, children being much more capable at the exercise than adults, which he thought might be the result of a mission education among the children. One thing did stand out though; it was standard for Melanesians to divide from right to left, rather than left to right, as English children had almost certainly learnt from reading. (6) Rivers repeated the experiment with British subjects upon his return, and found that the results were much the same between the children in England and on the Islands, which surprised Rivers (ibid: 107), though the results closely mirror those for the last experiment. The ability to divide and subdivide 'correctly' was learnt in Britain within the school system, and hence older children and adults were able to do so and achieved more accurate results. It seemed that the two groups were distinctly different, and Rivers concludes that:
 Taking into account both the number of trials necessary before the
 lines could be subdivided successfully and the degree of accuracy
 as compared with the [British] Girton children, we may conclude
 that the Torres Straits natives were distinctly deficient in this
 operation. When, however, one remembers the difficulty in language,
 and in understanding what was to be done, and secondly their
 deficiencies in numeration, the results were surprisingly good. It
 has already been mentioned that in their own language these people
 only have definite words for one and two, and are now accustomed to
 use English numerals, and their powers of counting are still very
 defective. (ibid: 108)


When the Islanders learnt exactly what they had to do, their results were not statistically different from British subjects. So what can be gleaned from these two experiments apart from a Melanesian mental deficiency at counting and consequently measurement that can be easily rectified by a little teaching? Firstly the fact that both the sets of children had similar results in the second experiment suggests that people have learnt to subdivide in a particular manner as they age. Secondly that in both experiments it was shown (although not acknowledged by Rivers) that methods of tackling enumerative problems are learnt, placing these experiments in the realm of the cultural rather than the innate. This is a simple relativist claim, but on its own a rather bland one. It is insufficient for drawing further conclusions as it tells us nothing about the logic behind why Melanesians were predisposed to dividing differently. In answer to this more pertinent question we must take a closer look at the adults who took to the experiment with glee, though initially incorrectly.

For fully socialised Melanesians it might be simply counterintuitive to divide into equal portions by halving and then halving again. As Mimica (1988) points out, and downward bodily counting suggests, when one divides a totality which is a one, it is normal to count it in a temporal act of discarding, or rate of progressive diminishment of the available length of material in weaving. As this is not a unit but a length which starts at one point and ends at another, a person must travel along it temporally also, cutting proportions off it as one goes. Thus the subdivision into units can only be done as one progresses, rather than by cutting up first in the centre, then cutting the new unit lengths again into halves. This is precisely the point of a fractal cosmology culminating in a oneness that is everything; that one cannot know how large a thing is until one actually processually engages with it, in normal circumstances on the body or in the practice of weaving, but in this case along a line. However when they were told how it was to be done, they were just as able to cut it up as Europeans into precise amounts as they are able to use particular lengths of material in basket-making. There was thus nothing wrong with their ability to subdivide logically, simply that the logic was of a different order.

The Muller-Lyer Illusion

This experiment too was both well-liked by the Islanders, and perplexing for Rivers to conclude upon given the test results already discussed. The Muller-Lyer test involved a slide box with a continuous line upon it and arrows on either end of the line, which is extended by pulling out the slide. An arrow divides the line into two portions, one of which is fixed and the other is extendable (Fig.8 below, shown with an image of the illusion when completed successfully). The aim is to make both sides of the central arrow of equal size, which can be done either by starting from the closed position and opening the slide box, thereby lengthening the line until it is of equal length, or alternatively reducing the length by closing the sliding mechanism from an initially fully open starting point.

The results were particularly perplexing due to the apparent similitude it has with the line bisection experiment, both being measured by accurate assessment of the relative length of the line's subdivision. While the Islanders appeared to be very poor at the line bisection experiment, they were particularly good at this exercise, better in fact than their European counterparts. Rivers attempts to downplay the difference by pointing out the (marginally) smaller variation in results obtained from Europeans, but does not deny the discrepancy between abilities its scientific validity. Instead he concludes that the difference is the result of Papuans concentrating more fully on the task at hand of making the lines of equal length, rather than being distracted by the arrows. This is seen as a result of the Papuans simplicity of thought, whereas Europeans were more likely to see the figure as a whole, and hence be confounded by the illusion (Reports II [Rivers] 1901:126). The explanation is distinctly Spencerian, as the primitive is assumed to be better at more basic mental processes. Therefore in order for a 'savage' to be better at an activity, it must be either based on simple motor skills or a result of the primitive making a complex task simple as a result of their simplistic outlook. Richards (1998:138-139) in his analysis of the 'racist' underpinnings of the psychological aspect of CAETS, comments on the double-cross of this illusion; the more accurate and clever the natives ability to see through the illusion, the less intelligent the participant actually was at conceptualising advanced spatial perception, as they could not see the figure as a whole. Though the point is certainly valid, our concern here is the actual process by which these conclusions were reached; though this conclusion may have been asserted by Rivers after the experiment, it was not his original intention to have to resort to such intellectual flippantry.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Rivers expected the Papuans who undertook the experiment to be worse at it than Europeans, and was confounded that the opposite was the case, hence post-experiment he concluded that in fact being taken-in by the illusion was more advanced than seeing through it. This blatantly derisory and counterintuitive conclusion serves to show that Rivers was deeply puzzled by the mixed results he was getting. It is not entertained that the conception of visual illusion and the complexity and ready analogy of Papuan patterned material culture which is central to other volumes, are in any way linked.

How might a patterned enumeration strategy affect a conception of this illusion? Along the horizontal, the judging of equidistance despite conflicting diagonal lines bisecting it, was easy for Melanesians so used to doing exactly that in the weaving process. In fact, moving along the horizontal line to judge the spacing of the next emanating line correctly is exactly how a weave is constructed. We must simply imagine the horizontal band the same as in a basket and the arrows as connecting pieces of material. One judges the construction of a pattern in a weave by obtaining correct distances from the previous stitch, doing so correctly is the prerequisite for an even finish and a mark of skill.

How does this explain the discrepancy between this test and the previous one then? The Muller-Lyer test has a precedent section from which to work, as such the Melanesian can conceptualise it as a stitch which needs a corresponding one- it must be equidistant in order to portray the effect required. When this principle is translated to the aforementioned line bisection test, it would be normal to start at one end and base the distance on the previous bisection rather than to halve and halve again. In short, the first test is the measured cutting up aesthetically for the participants, a more abstract form of division which was unlike their normal practice, while the Muller-Lyer is the chance to continue a pattern from a precedent and create an illusion correctly.

Pseudoptics

Rivers brought a selection of pseudoptics, (7) made by the Bradley Martin Company in Britain to the field, and found them the most fascinating of all the optical illusions for the Islanders (Figs.10, 12, 14). People would often turn up just to see them, having heard about them from test subjects, many of whom would ask to be tested again. They were used to confirm the long held assumption that for primitives 'the manipulation of pattern and material could neither break through the surface into an illusionistic space nor organize the shapes into a meaningful narrative' (Connelly 1995:56). This was to prove the 'ungeometric' assertion put forward by Haddon and others.

As was expected, the Islanders delighted in the tricks, but after having expressed answers which suggested that they had been fooled by the illusion, would more often than not gesture with glee that in fact they had seen the illusion and enjoyed its trickery. Thus they demonstrated that they were very aware of the illusory power of the image, and were not taken in by it. An anology can be drawn here to the magic eye posters of today that are enjoyed for their ability to fool, and attract prolonged interest as one attempts to 'see what it is'. Rivers weakly explains the natives very apparent competence in this regard with reference to just one of the unsuccessful visual illusions. This one (not illustrated in the Reports) consisted of two equal sized squares, one white on a black background, and the other black on a white background, which were supposed to trick the observer into seeing a difference in size in favour of the white square. The failure of natives to fall for the illusion was put down to irradiation (8) being less pronounced in primitive eyes, thus giving them increased visual acuity. It becomes quite obvious though from the figures below that some of the illusions are not based on the principle of irradiation at all. Included side by side with these illusions are Indigenous artefacts that have analogous features to them. (9) This similarity alone is scant evidence for a penchant for a pattern-centric understanding of what we would normally consider optical illusions. In conjunction with the next illusion discussed however, the significance of this similarity will become apparent.

One of the illusions (Fig. 14 below) that Rivers had high hopes for, being a very popular optical trick in Britain, depicted two figures of the same size but at a distance from each other along a path, such as to create the illusion of difference in size. Rivers was bemused by the lack of interest or confusion brought by the image, and put it down to the European garb of the people depicted being unlike the dress of the natives. The Islanders simply stated that they were the same size and moved on. Looking at the image within the context of the other images, it is the only one that is pictorial, and three dimensional by virtue of the subjectivity invested in the persons in the illusion, rather than the others which appear both two dimensional and pattern-like. The fact that they showed no interest in this illusion confirms the argument above that the other illusions were taken within the context of their own aesthetic of pattern, whereas this one could not be so taken. The centrality of pattern within what was called spatial perception is a central concern in Melanesia, and in the context of indigenous patterning styles (see above), these images represented novel yet comparable forms (excluding the final example), and excited great interest. Hence the previous tests can be examined in terms of their relevance to a pre-existing Melanesian cosmology which was brought to bear on all these experiments, while the final illusion cannot.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

We have so far seen within this early ethnography support for the later understanding of pattern as an integral part of Melanesian enumerative cosmology, let us now look at the overall picture more closely. Where Rivers saw a deficiency in numerical skill, we can now see the possibility of an aesthetic of fractality which informs both spatial perception and numerical consciousness, imagined through the lens of pattern. We must however not fall into the trap of stating that pattern is a result of a Melanesian cosmology, rather to return to Ingold's (2000) weaving argument, patterning and cosmology are in an autopoietic relationship with each other, which informed the perception of these imposed illusions, and granted them the same aesthetic properties. This is why no interest was given to this final illusion; it did not contain within it the possibility of autopoietic reproduction. In other words, while the other illusions were patterns, this was simply an image.

It is now possible to see within the data an aesthetic interrelating material and size which does not rely on precise measurement but the possibility of continuing an aesthetic ideal. The crucial distinction is between a group who are intent on seeing the 'reality' behind the illusion, and a group for which the illusion itself is the thing to savour. Creative individualism within a closed yet manipulable universe of the sort discussed above allows these people to both perceive an illusion and to express themselves within the illusion. A Melanesian aesthetic is one of creating and manipulating regularity (including numerosity) so that it produces illusions by which one can gain power by trickery. As the world is made of relations, some of which are secret and illusory like the contents of baskets, then a person is working within this totalisation, but is conscious of its aesthetic qualities and can seek to manipulate this to his/her advantage. Hence the possibilities of numbers are both bounded and boundless, and we can see how this was brought out by Rivers' tests. What emerges from this section is the sense that enumeration, as a categorical imperative or as an aspect of quality expressed in pattern, played an important role in what both CAETS and its subjects actually saw in these tests. Contrasting what we can see as a unit-centred and a pattern-centred aesthetic of number using an event where they met has been useful for triangulating a stronger sense of what otherwise would be mere postulation. By taking a new approach then, such otherwise ethnocentrically projective experiments can yield valuable data about both sets of participants, a theme we will take up presently.

CONCLUSIONS

[O]bservations which have been held to show extraordinary sense acuity have been made in surroundings with which the savage is extremely familiar... [and therefore] may depend merely on a correct inference founded on specialist knowledge.

(Reports II [Rivers] 1901:12)

This quote of Rivers is particularly pertinent, as his scientific methodology had hoped to counter cultural influence, yet specialist knowledge (or 'culture') was actually causing mixed results in the majority of his tests. Though acknowledging as much, Rivers neglects a mid-level analysis of the interpretation of data by his informants. The tests rest on an understanding of the traits under investigation as universally existent in discrete forms, and then attempt to control for cultural factors influencing these forms. Having identified reliable methods for testing these in a European setting, he assumes they are not influenced by European culture and will work just as well in another cultural context. There is thus no space for local interpretation of the tasks, instead the informants either can do them well or they cannot, in direct comparison to European results. In hindsight Rivers becomes aware that the results are highly inconclusive, and that specialist knowledge played a large part in this. However he persists with his original framework of discrete categorisation, thereby trapping the Torres Straits Islanders in a double hermeneutic tantamount to racial stereotyping, and thus gives little insight into Melanesian cultures.

This reanalysis has attempted to suggest an alternative explanation of the psychometric results by inserting a mid-level of interpretation of the data using contemporary Melanesian ethnography and anthropological assessments of the two positions as regards enumeration. As such this has been an attempt to triangulate CAETS enumerative system, the results of the psychometric tests, and Melanesian ethnography. It may be noted that the triangulation need not be focussed in any particular direction, as each reveals more about itself through the lens of the other two. In this sense we have tried to work with this complex web of contingent understanding rather than factoring it out, using enumeration as a cosmological point of focus. In doing so the paper brings to the fore a particular approach, that of enumeration and its usage as heuristic which allows both systems to be teased out of Rivers' tests. What is thereby offered is another possible interpretation, but one which takes subject and anthropologist interaction seriously, by triangulating the two with ethnographic material, through the lens of a shared material context.

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NOTES

(1.) At this point we might be tempted to say that there is in fact an abstract quality to this counting method, taking person and thing out of the world would appear to abstract them, however, upon closer inspection of the cosmological properties of Melanesian societies, we shall see that this bears only a passing resemblance to abstraction which is much more the result of the narrative form of explanation than it is to do with how 'they' might think.

(2.) Though this is the only complete ethnography devoted to a Papua New Guinean counting system, this feature, and many others used hereafter, have been confirmed to be similar if not the same (Butterworth 1999).

(3.) This type of cosmogonic creator, which begins as all things self contained, both male and female, split into many via a rupturous event is a common one (c.f. Moutu forthcoming).

(4.) Here we may note that the absence of a concept of zero no longer matters, as what 'we' might consider an original nothingness, for Melanesians is simply the original state, which is only nominated as one, or rather completeness/oneness.

(5.) Though doubtless this has changed since exposure to certain Western imports such as money, gambling and wage work.

(6.) The reason why it was normal for Melanesians to work fight to left is not clear.

(7.) These are visual images designed to trick the eye of the observer into thinking the image is moving, lines are bent when they are straight, shapes/objects depicted are of the same size, and were very popular around British dining tables at the time (Kuklick 1996, p139).

(8.) Before being adopted to refer to the act of exposure to radiation, the term irradiation was defined as the apparent extension of edges of illuminated object seen against dark ground' (Fowler: The Concise Oxford Dictionary 1954, p631).

(9.) Many others might have been used in their stead, and the two selected are typical examples rather than selected for exemplary closeness.

Anthony J. Pickles

University of St. Andrews
Fig 7. Above depicts the visual spatial ability of Melanesian
and English participants as tested by W.H.R. Rivers.
Taken from Reports II (1901:113)

 No. 1
 No Average Max/Min M.V. Average

Murray Island men 20 65-7 88/49 10-13 77-0
Murray Island boys 12 79-5 97/66 7-67 84-3
English students ... 15 89-0 101/73 5-73 92-5
Girton children.. 12 78-2 92/62 8-53 88-7

 No. 2 No 3
 Max/Min M.V. Average Max/Min M.V.

Murray Island men 89/55 7-85 90-1 111/75 9-75
Murray Island boys 109/63 7-93 99-4 129/85 12-84
English students ... 112/81 4-37 94-5 106/81 3-43
Girton children.. 100/82 4-8 90-7 115/70 9-28
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Author:Pickles, Anthony J.
Publication:Oceania
Article Type:Report
Date:Nov 1, 2009
Words:12138
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