Volume 90 1981 > Volume 90, No. 4 > Counting and calculation on Ponam Island, by Achsah Carrier, p 465-480
COUNTING AND CALCULATION ON PONAM ISLAND 1
Although a number of people have written about counting and mathematical systems in Papua New Guinea, there is no published material on the counting systems used in Manus Province. In this paper, I describe the counting system of one Manus Province language, Ponam, and describe some of the ways Ponam speakers, the 500 people of Ponam, a tiny island off the north-central coast of Manus Island, count and calculate.
The Ponam language is spoken only on Ponam Island, but it is clearly related to other Manus languages, and the Ponam counting system and system of numeral classifiers are similar to those used elsewhere in the province. All the languages in the Admiralty Islands are Austronesian (Healey 1976), and their counting systems have much in common with those of other Austronesian systems (Frobenius 1899).
Somewhat unusually (Cheetham 1978: 22; Kitennis 1978: 32; Smith 1978: 56; Thune 1978: 71), the Ponam counting system is extensive. As Table 1 shows, there are terms for the numbers 1–9, 999, as well as a set of alternative terms for the thousands from one to nine. The numbers are structured in an imperfect decimal system, that is, a decimal system in which “some of the numbers between one and ten are constructed by multiplying or adding earlier numbers or by subtracting from a later one” (Wolfers 1972: 218). The numbers seven, eight and nine (ahatalof, ahaluof, and ahase) are composed of the words for three, two and one (talof, luof and si) and the prefix aha. That is, they seem to be constructed on the principle of ten minus three, ten minus two and ten minus one.
Although the counting system may have evolved this way, Ponam speakers now perceive it as a purely decimal system. The prefix aha has no independent meaning, and many are unaware that the terms for three, two and one repeat in the terms for seven, eight and nine. They perceive each of the terms for the numbers one to nine as distinct, not derived from any other. Other Manus languages share this feature with Ponam, and, for the most part, speakers of other Manus languages are also - 466 unaware of the repetition (see Table 2).
Although most Ponams are not aware of this redundancy in their language, they are aware of the decimal redundancy. They are conscious of the fact that the word for one repeats in the words for ten, one hundred and one thousand (si, sanguf, sangat, sapau), for example, and they take advantage of the base ten properties of their number system when speaking numbers, during counting and when calculating.
Melanesian languages frequently contain two or more distinct counting systems, sometimes used in different circumstances or for counting different objects (Wolfers 1972: 218). In Ponam, however, there is only one basic counting system, the one shown in Table I. The Ponam language does contain a system of numerical classifiers, but, as I show below, these are not an alternative counting system.
Ponam also has a system of birth order names. In addition to personal names, each child has a name indicating his position in the family. The names for first-born to eighth-born sons are: Tol, Ngih, Selef, Sepat, So'on, Kupe, Kuem and Kalai. The names for first-born to seventh-born daughters are: Aluf, Asaf, Siwa, Driniu, Salimet, No'on and Kahu. Ninth and subsequent sons and eighth and subsequent daughters are langan pe, ‘nameless’. Although these terms indicate the number of sons and daughters born to each woman, they are not numbers and are not an alternative counting system. They are used and thought of as names.
There are no ordinal numbers in Ponam. A sequence of three is described as “front, middle and behind” (maran, hakeo and ken). A sequence of more than three is described as “front, the one after that, the one after that ... and behind” (maran, kaliwin aran, kaliwin aran ... and ken). Now most people use the English terms “second”, “third”, “fourth” and so on, to express ordinal relationships in circumstances where the Ponam terms would be cumbersome.
There are two non-numerical terms of some importance for counting and calculating: patolim and homean. The term patolim means a set of 10 objects, and is one manifestation of the fact that Ponams prefer to think decimally, prefer to organise quantities in tens whenever possible. They often group objects into sets of ten for counting, calling them patolim si (one set of ten), patolim luof (two sets of ten) and so on. They only use the term to count objects which could reasonably be grouped in tens: mats, baskets, or coconuts, for example, but not days, islands or houses. However, objects reckoned as patolim need not be grouped in tens at the time of reckoning. Both a single pile of 30 coconuts and three piles of 10 coconuts may be described as niu patolim talof (three sets of 10 coconuts). The word patolim is also the word for “fingers”, and the etymological connection is clear, but Ponams think of these as - 467 homonyms rather than as one word with two applications.
The concept homean is both more interesting and more important than patolim. It is best translated as: the amount remaining after a number has been rounded down to the nearest relevant power of ten. Ponams prefer to think of numbers as round powers of ten, and for the most part they do not use or remember numbers which have not been rounded down unless there is some special reason to do so. Thus, for example, a man who buys something costing 56 toea would probably say that he spent 50 toea. If anyone wanted to know exactly how much the item cost, he would ask “Homean sabe?” (How many remaining?), and be told “Homean wonof.” (Six remaining.) Six is the number remaining when fifty-six has been rounded to fifty.
A number may be rounded down to the nearest ten, hundred or thousand, depending on the value the speaker wishes to emphasise and the degree of accuracy with which he is concerned. Thus, the number 1,246 might be thought of and spoken of as any of the following:
The examples of Ponam counting I have given so far all use the full formal terminology shown in Table I, but in ordinary speech Ponams tend to abbreviate numbers much as English speakers do. They abbreviate by mentioning only the number of the relevant powers of ten. Thus, the word si (one) may refer to the quantities one, ten, one hundred, or one thousand, and the word faf (four) may refer to the quantities four, forty, four hundred or four thousand. By extension si ne faf (one and four) may refer to the quantities fourteen, one hundred and forty or one thousand four hundred. A large number with a digit in each place (to use the English conceptualisation) such as 1,246, could be spoken of as si ne luof ne faf ne wonof, but this is very unlikely. Circumstances which require the use of large precise numbers usually require them to be fully stated as well.
Often Ponam speakers both round and abbreviate their numbers. Thus, 1,246 would be spoken of simply as si (one), or si ne luof (one and two). The curious can ask about the homean of an abbreviated number just as they can ask about the homean of a full one. If 1,246 is spoken of as si ne luof then its homean would be faf ne wonof (four and six) or fanguf ne wonof (forty-six).
Because Ponam speakers tend to round down their numbers and to use - 468 abbreviated forms, a great deal of contextual information is needed to follow any discussion of quantities. The statement “We caught two fish”, can only be interpreted properly by someone who knows how many fish were likely to have been caught. But Ponam is a small community in which information travels rapidly, and confusion is rare.
The counting system presented in Table I is not simply a formal system elicited by a researcher's request for numbers; it is the system in common use. Also, when Ponams count they use only these numbers. They do not use gestural counting, marking objects off on the fingers, except in a few circumstances. Young children are sometimes taught to count on their fingers, though more usually the process is illustrated by counting objects. Adults count on their fingers only when the fingers are needed as a sort of tally, as, for example, when someone is counting something which he is observing or dredging up from his memory.
Ponams do not associate numbers with parts of the body, even minimally. In most Austronesian languages the number five resembles the word for hand, and in many an explicit connection is made between the number and the body part (e.g. Thune 1978: 70). Although in Ponam the number five (limef) is almost identical to the word for hand (lime-), the connection is not perceived as significant. No one would ever translate limef as “hand”.
Most Ponams have considerable skill at arithmetic. Although in some communities large numbers are known and used by only a few (for example, Kitennis 1978: 32), this is not the case here. Almost everyone on the island can count from one to ten thousand in both Ponam and Pidgin. Very old people, particularly women over sixty, often are unsure of the Pidgin terms. Children less than fifteen often are unsure of the Ponam terms. Their parents say that this is because they spend so much time in school that they do not learn the traditional culture. But many people who have completed school and returned to the village count fluently in Ponam, and I think it quite possible that current students will learn to do so as well.
Any discussion of mathematics on Ponam cannot ignore the effects of Western mathematics. Ponams began attending a Pidgin language mission school in the 1920s, and, except for a 10-year interruption during the war, the majority have attended school since that time. English language education began in Manus in the early 1950s, and all Ponams who came of school age after that date have completed six or more years. Thus, almost all Ponams are numerate, have been taught to read and write numbers and to do at least basic arithmetic, and many have gone on to secondary and tertiary education and consequently are more than basically numerate.- 469
It is obvious, however, that Ponams were skilful in the use of numbers before the introduction of schooling. Both the young people who have been to school and the old who have not are confident and accurate at mental arithmetic. This can be observed most easily when people play cards. “Lucky” and “500” are popular games requiring the use of numbers, and people of all ages play them well. “Lucky” is a men's gambling game in which each player is dealt three cards. He adds the value of the cards together, disregards the first digit in any two-digit number and takes the second or only digit as his score. Thus, the cards four, seven and ten add up to twenty-one, yielding a score of one. The highest score wins. “500” is a form of gin in which all cards have a value of five, ten or fifteen points. A player receives positive points for cards taken as tricks and negative points for cards remaining in his hand. The scores for each hand are kept cumulatively until someone reaches a score of 500.
Most lucky players seem to recognise at a glance the sum value of their cards without having to add them. “500” players add and subtract the large numbers used in scoring this game accurately and quickly. Elderly unschooled people keep score mentally as accurately as young people do with pencil and paper.
Multiplication and division were not known before the introduction of schooling. But even those who have not been to school are so familiar with the base ten properties of the counting system that they can perform calculations with multiples of ten easily. Everyone simply knows that, for example, there are nine tens in ninety, that patolim ahase equals ahasanguf.
Schooling also has led Ponams to abandon the traditional forms of measurement and mathematical record keeping. They now use pencil and paper to calculate and record the mathematics necessary for running the village council, church committee, trade stores, for making bride-price payments or for planning a child's school fees. Also, Ponams now use the Western calendar and speak of time in hours and minutes. They weigh and measure things with Western implements and speak of weights in pounds or kilograms and length in feet or metres. They still estimate lengths in spans, but when precise measurements are needed they use a tape measure.
Although these calculations are made in Western style, English numbers have not replaced Ponam ones as has happened in much of Papua New Guinea (Lancy 1978: 10). Instead, Ponam numbers are used to replace English ones in Western style calculations. Thus, people speak of feet luof (two feet) or sanguf February (February tenth). Many people, particularly the young, seem to use Ponam and Pidgin numbers - 470 interchangeably; and there is a tendency to announce quantities given in exchange in both languages, particularly if the quantities are large. This may indicate that Pidgin numbers will replace Ponam ones in the future, as many people fear. But this has not happened yet.
In addition to the basic counting system Ponam also has many numeral classifiers, terms used to refer to quantities of objects of different classes. In Table 3, I provide a list of some of the commonly known and used classifiers, rough English translations when a simple translation is possible, examples of objects in each class and a translation of the term's non-numerical root when that root is obvious.
The numeral classifiers are derived from the basic counting system, combining a numeral prefix with a class marking root. Thus, objects in the class best described in English as “bundles”, “bunches” or “packets” are numbered sabis, lobis, tulubis, fabis, terms obviously derived from the basic counting system terms si, luof, talof, faf.
Ponams do not use numeral classifier terms for all objects; the majority are enumerated by the basic counting system terms. But, if asked, Ponams often willingly generate a set of numeral classifier terms to count objects normally counted with the basic counting system, indicating both that they are conscious of the rules by which numeral classifiers are constructed and that the number of such classifiers in principle may be infinite.
Some numeral classifiers refer to a very limited class of objects. Sahou, for example, refers to cutting blades such as knives or axes. But most refer to more abstract qualities and can be applied to a great range of objects possessing those qualities. Similarly, many objects can be described by several numeral classifiers, depending on which of the object's qualities the speaker wishes to emphasise.
For example, one of the most frequently used classifiers is sabeh, best translated as “one half”. One part of any object which has been broken in two is sabeh, two parts are lofeh, and three parts of two objects broken this way are tulufeh. But these classifiers are not always used to refer to parts of objects which have been broken in two. They are only used when the speaker wishes to emphasise the object's “half-ness”. Thus, one section of a stick that has been broken in two could be described as paran (stick) sabeh, but if that same stick were to be used in a context which makes irrelevant the fact that it was once part of a longer stick it would be referred to simply as paran sai (one stick). Most numeral classifiers apply to classes of concrete objects, sticks, bundles and so on, but many can be applied to more abstract concepts as well. Thus, a section of a speech, an item on an agenda and an interval of work are all sabeh.- 471
The numeral classifiers quantify objects, but for several reasons they are not alternative counting systems. First and most simply, they are used only up to the number four, beyond that the basic counting system numbers are used instead. Thus, one bundle is sabis but five bundles are limef. People recognise the logic of a term like limebis, but they never use it. Second, for the most part people use these terms to state quantities, but not to count. For example, they are most likely to count bundles using the basic counting system terms si, luof, talof, faf, but then state that the number of bundles is fabis. Finally, Ponams themselves think of these terms primarily as classifiers rather than as numbers. Thus, they translate sabis into English as “a bundle”, and sabeh as “a half”. They translate classifiers as numbers only when there is no simple English word for the Ponam class. Thus, sehek, which refers to flat objects or sections of objects like coins or orange segments, is translated reluctantly as “one”.
So far I have described the structural properties of the Ponam counting system, but this allows only a partial understanding of the use of numbers. As Bowers and Lepi (1975: 309) have argued, the real nature of counting systems is obscured when one simply lists the numbers according to English glosses. Their paper illustrates two important reasons for this. First, people may vary their counting system in interesting and important ways when they actually use it. I have shown some of the ways that Ponams manipulate and abbreviate their system and some of the mathematical concepts they use, concepts which would not be revealed by a simple elicitation of counting terms. The second and more complex reason is this: “counting does not exist in isolation. It quantifies and qualifies relations between people, objects and other entities” (1975: 322). I want now to look at one of the most important ways that Ponams use quantities and numbers to represent and reveal relations between people.
One of the most striking things about Ponams is that they do not count people. Despite obvious skill with numbers, no one has any idea how many people live on the island, how many households there are or how many children are attending the primary school. Even more surprising, many parents of large families do not know how many children they have without stopping to think about it. And almost no one knows that there are 14 clans on the island, although everyone knows their names and can calculate the number in a few moments. Ponams simply are not interested in counting people; apparently these quantifications tell them nothing interesting about social relations. But other sorts of quantifications do, most importantly those used in exchange.
All important social events on Ponam are the occasion for an - 472 exchange, almost always between affines. A man makes a gift to his wife's or his child's spouse's nearest relative and later receives a return gift from him. These men, whom I call the focal donor and focal recipient, do not make their gifts by themselves, but with the help of their cognates. The focal donor receives help from his cognates in making the initial gift. The focal recipient distributes this to his cognates who will later help him amass the return gift which the focal donor will redistribute to those who helped him in the first place.
The focal donor's cognates do not give to him as individuals, but rather as members of cognatic stocks descended from a sibling of one of the focal donor's lineal ancestors. Figure 1 shows the way the gifts are collected.
1 and 2 on the diagram join together to give a gift to 5, the focal donor, in the name of A, the relative who connects them most nearly with one of 5's lineal ancestors. Following the same principle, 3 and 4 unite in E's name, 6 and 7 in F's name and 8 and 9 in D's name. Also following the same principle, when 1 or 2 is the focal donor in an exchange then 3, 4 and 5 will unite to present a gift in B's name. In all cases the focal donor's true siblings will give to him in their own names.
On the day appointed for the presentations the gifts amassed by each cognatic stock are brought to the focal donor's house and laid out on the ground in a formal display. The pattern of the display is intentional and its principles are invariate, for the display is a kinship diagram, an illustration of the relations between the focal donor and the contributing cognatic stocks. The goods given by each stock are piled separately. Piles given by paternal relatives are arranged in a line slightly to the right side of the front door of the house and those given by maternal relatives are placed in a line to the left of the door. The piles given by the most closely related cognatic stocks are placed immediately in front of the door, and - 473 those given by more distant ones are placed further away. A possible display based on the very simplified kindred in Figure 1 is this:
A real display in which the focal donor received gifts from 14 different cognatic stocks looked like this (letters indicating the kinship relations between each person in whose name a gift was given and the focal donor are placed in the diagram where the gift was placed in the display):
Arranging the display to everyone's satisfaction is difficult and often time consuming, and in fact it is not often that everyone is satisfied. Because the arrangement of gifts represents relations between groups, and because relations between groups are the subject of much dissension, the display is the subject of dissension as well.
Once the display has been completed, the focal donor, or his appointed spokesman, will “count” it (rienni). Starting with the gift given by the focal donor's closest paternal relatives, he announces what is in each pile and in whose name it was given. Thus, he might say, “Selef [the father's brother], two bags of rice; Aluf [the father's sister], one bag of rice”, and so on, moving from the most closely related stocks to the most distant. The gifts in Figure III are numbered in the order in which they were announced.
Once the gifts have been “counted” they are carried to the house of the focal recipient and placed in a single pile before his front door. The focal donor makes a presentation speech and the focal recipient thanks him, but usually they do not announce the total amount of the gift, and often no one knows it. When the speeches are complete the focal recipient rearranges the goods into a new formal display following the same principles as the previous one, but indicating the cognatic stocks to whom he will distribute his gift. The focal recipient now “counts” this - 474 gift, his kinsmen pick up their shares and depart. The work is finished. Later similar, though smaller, displays will be made. The members of each cognatic stock divide their gift among themselves.
This method of displaying and counting formal gifts is interestingly different from that used in the pig kills described by Bowers and Lepi. There the pigs and other gifts are arranged in long rows and “in addition to reckoning ... individual transactions, the entire presentation is recorded by one or more core donors: important men who publicly run along the rows in a stylised manner, counting the objects in the collective gift” (1975: 312). This gift is presented by one group (“clans, segments of clans and occasionally combinations of clans” (1975: 312)) to another, and the formal count “not only validates the name and prestige of the core donors, but the viability of the kin group itself” (1975: 312). The more pigs and valuables a group can amass, the greater its strength, and the formal count provides an index of that strength.
On Ponam, on the other hand, the contributions which make up the focal donor's gift are counted in public before an audience of the contributors when they bring their gifts to his house, but a total count is not made then or at the time of the final presentation. In order to understand why the contributory gifts rather than the main gifts are stressed in this way we need to understand some of the principles that determine the size of gifts and thus the way that Ponams interpret the meaning of quantities as they are announced.
Ponam exchange is not a straightforward competition in which larger gifts are better gifts. Instead, individuals who contribute to the focal donor's gift strive to give gifts of the appropriate size. The gifts presented to the focal donor by closely related cognatic stocks should be larger than those presented by more distant ones, so that the size of gifts as well as their position in the display indicates the degree of relationship. Furthermore, all of the individuals who contribute to any given cognatic stock also contribute to other stocks through other kinship links. These individuals must judge the size of the gifts they give to each one appropriately. They should give larger gifts to stocks to which they are closely related than they give to more distant ones. Thus, the size of a gift reflects the donating group's sense of its relation to the focal donor and also reflects the group's individual members’ sense of their relations with their various kinsmen.
One can see then that the formal display of gifts in exchange is a kind of graph in which the arrangement of gifts and their size indicate the current relationships between kinsmen. Individuals and groups take advantage of this to make statements about the way they perceive these relationships by giving larger or smaller gifts and by placing them in - 475 different positions in the display. Here the relative size of gifts is more important than their absolute size. The fact that a particular group gives three bags of rice to the focal donor is meaningless unless one also knows how much all the other groups give and how they are related. The numbers announced in a formal count thus express the quality of relationships between groups and individuals rather than the quantitative strength of particular groups or individuals. For Ponams, then, counting absolute numbers of people or quantities of goods does not reflect on social relations and is uninteresting, but counting the relative quantities of goods which people give to one another does reflect social relations in a very precise way and is of the greatest possible interest.
These terms were collected by Mrs Agnes Paliau
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1 Based on 15 months’ field work between November 1978 and January 1981. I want to thank the University of London Central Research Fund and the Papua New Guinea Department of Education for their financial support and the governments of Papua New Guinea and Manus Province for permission to do the field research and Dr David Lancy, previously Principal Research Officer of the Department of Education and Director of the Indigenous Mathematics Project.