The intellectual activity known as "mathematics" was developed in certain areas of the world in which a tradition of writing had already been established. However, most cultures did not, and still do not have such a tradition. If we state that the mathematical skills or the systems of numeration and calculus in these cultures are rudimentary, we make a judgement from the wrong perspective. Indeed, we use the exception -- cultures with a written tradition, especially a western intellectual tradition -- as a measure for all others. If we do not base our judgement on traditional western standards, can we still look for mathematical skills in cultures without a written tradition? In my opinion we can, but we have to redefine the object of our search. What should we look for and which concepts should be considered relevant? Should we base our search on an operative comprehension of the concept of number? Or, should we consider most relevant an ability to establish one-to-one relations between objects, fingers, stones, or shells?1
A basic hypothesis of this paper is that there is internal variation in all human groups and cultures. In cultures where we cannot observe "mathematics" as we traditionally define it, we should look for basic capacities on which a mathematical intellectual activity can be built. However, as some degree of variation is always present, we should look for both the presence of an operative comprehension of numerical concepts as well as a use of number which does not imply such a comprehension.
It seems to me that the postulate of variability is necessary. A uniformitarian perspective of so-called "primitive" societies, so often implicit in the works of anthropologists and linguists, prevents us from understanding many aspects of those societies and encourages us to view them as static entities. In reality, we find some degree of individual variability in skills and intellectual achievements in even the smallest human groups. We find a dynamism that is related to the existence of variability. While this perspective might be obvious for many readers, it is not very wide-spread. For example, one encounters statements such as "the people X do or say such and such" even in the most recent anthropological literature. Nevertheless, in anthropological tradition the assumption of variation in intellectual skills is implicit. Some persons are more reflexive, more knowledgeable, or more curious than others. These are the informants each anthropologist in the field would like to find as a reliable associate in his or her research. To attribute the same capacities of reflection to all the members of a community is an unfortunate oversimplification. A uniformitarian and static perception of "primitive" cultures is an inadequate one. It is my claim that the most significant perspective in approaching the problem of mathematical skills in cultures without a tradition of literacy is one based on synchronic variability and dynamism.
In most cultures without a written tradition a capacity for abstract reflection is achieved by only a few individuals and cannot be accumulated and passed on from one generation to the other. The reflection of a single Thales starts and dies in his or her lifetime, without a chance of spreading or of being fixed for future reflection. Each Thales begins again on the base of his or her own culture.
Recently the Piagetian anthropologist C.R. Hallpike (1979, p.62) wrote:
"it seems insufficient to argue that wherever there is a demand for a cognitive skill to become general in a society, that demand will be satisfied. [...] Merely because a society manages to get by with its existing repertoire of collective cognitive skills it does not demonstrate that these are wholly adequate to its needs, or that cognitive demand will be met by cognitive supply.
The fact that a society does not display a particular cognitive skill in a high proportion of its population is therefore likely to be the result of two factors - that it is not developed in many of them as a result of general environmental factors, and that it is not relevant in that society, though as we have just noted, 'relevance' is not at all easy to define."
Although I agree with the general view expressed in these statements, I would argue that a 'demand' for cognitive skills can become widespread in a society and some members of it can satisfy that demand in some way even if their intellectual achievement does not become general and is not preserved for other generations. Without a written tradition, innovative knowledge and reflection can hardly become a seed for future elaboration.
Yet, spoken language can be a very indirect 'register' by which some concepts find their way into a cultural tradition. This is a slow process of lexical change and innovation. However, with a perspective based on variation and dynamism we cannot claim that language is a mirror of the shared and unshared cognitive representations of its speakers. The use that many western anthropologists and linguists have made, and still make, of linguistic data is a consequence of a uniformitarian and static view of 'primitive' cultures. Many of them have considered language as a primary indicator able to reveal the degree to which the speakers could or could not deal with mathematical concepts. An idea still alive today is that limited skills or lack of interest in quantification and numbers are represented by a very limited numerical terminology.
On the other hand, a somewhat different and very engaging position can be found in the research of Tylor and Levy-Bruhl, concerning numeration in 'primitive' societies. Tylor (1903 [1871], vol.I, pp.242, 246), in his chapter on "the art of counting", noted that:
"Among the lowest living men, the savages of the South American forests and the deserts of Australia, 5 is actually found to be a number which the languages of some tribes do not know by a special word."
But, the same author went on to say:
"men counted upon their fingers before they found words for the numbers they thus expressed [...] in this department of culture, Word-language not only followed Gesture-language, but actually grew out of it."
Levy-Bruhl (1926 [1912], p.205), some forty years later, also stated that the existence of counting skills could be possible without number names in a language and claimed that:
"It is a mistake to picture the human mind making numbers for itself in order to count, for on the contrary men first of all counted, with much effect and toil, before they conceived the numbers as such."
This position appears to represent a historical perspective but was not extended to a reflection of synchronic data. It represented, at least for Tylor (1903 [1871], vol.I, p.251), an important step toward the historical understanding of the origin of numerical terminology. Tyler wrote:
"The theory that man's primitive mode of counting was palpable reckoning on his hands and the proof that many numerals in present use are actually derived from such a state of things, is a great step towards discovering the origin of numerals in general."
From a synchronic perspective a limited set of numerals in a language (and in a culture) does not imply that all the members of that group can do only what the lexicon permits. At least some members of the group can count and calculate beyond the 'limits' of the language. On the other hand, I agree with Hallpike (1979, p.245) when he writes:
"even where a culture possesses an extensive verbal system of numbers, we are not entitled, from the fact alone, to deduce that the members of that culture have an operational grasp of number. Such a conclusion must rest on empirical evidence of the way in which their numbers are used."
The problem of expressing numbers and quantities is a challenging one for a conservative view which considers language as merely a verbal system of communication, because the use of parts of the body, mainly the hands, is very clearly associated with verbal expression. Both Tylor (1871) and Levy-Bruhl (1912) pointed this out when they referred to the widespread use of signals which in a systematic way integrate the verbal numeration system in many languages and cultures. In many cases body movement and hand movement play a role in communication which is far more relevant than is generally recognized. Gestuality is present not only in 'contextual' or 'deictic' communication but also in discussion of highly abstract topics.2 In many cultures the use of fingers, toes or other parts of the body is an essential component of numeration. In some groups of Melanesia we find numeration systems which involve the use of the whole upper part of the body to count up to 47 (Hallpike 1979, pp.240-241). This systematic use of the body in numeration seems to me worthy of research and reflection. Because the use of hands or body signals is relevant in numeration, many linguists and anthropologists had to include the description of non-verbal systems which, together with words or utterances, were used to express numerals.
With the foregoing ideas in mind, I will discuss the case of a culture and a language which has been considered from a traditional perspective without any real appreciation of the mathematical skills manifested by its members. I will analyze some aspects of the language and culture of a native society of the Upper Amazon, the Shuar of Ecuador.3 By observing the available data we gain insight into the slow growth of a set of numerals and, consequently, into the dynamic process of the language and of the culture.
A good account of the way in which the Shuar used to count can be found in the first ethnographic monograph on their culture (Karsten 1935, pp.548-549) written on the base of field data collected between 1916 and 1917, and between 1928 and 1929. I will quote the somewhat lengthy description because it illustrates in a very clear way the complementarity of verbal and non-verbal communication.
[Quotation begins]
The majority of the Jibaros are able to count to 'ten', but only for the five first numerals have they proper names. They always count with the fingers, beginning with those of the left hand, and then also with the toes.
Among the Upano tribes the numerals are as follows:
chikíchi, one
hīmera, two
menéindu or kámbatama, three
eínduk-eínduk, four
wéhe amúkei (= "I have finished the hand"), five.
If the Jibaro Indian is obliged to count more than five, he seizes the fingers of his right hand with those of the left; then, beginning with the thumb, he goes on counting, using the following expressions:
huíni wéhe, six, (here I have a [finger from the other] hand)
himera wéhe, seven (two fingers [from the other hand])
menéindu wéhe, eight (three fingers)
eínduk-eínduk wéhe, nine (four fingers)
mai wéhe amúkahei, ten ("I have finished both hands").
If it is necessary to continue counting the Jibaro seizes the toes of one foot, one by one, and counts chikichi, hīmera, menéindu (one, two, three), etc. When he arrives at the fifth toe he says: huini náwi amúkahei, "here I have finished one foot" (Huíni = here, náwi = foot), it being understood that he has begun with the hands. The said phrase, therefore, is equal to fifteen.
Thereupon he may continue counting with the toes of the other foot, and when he arrives at the last toe he says: mai náwi amúkahei, "I have finished both feet"; this means 'twenty'. Twenty is the absolute limit for the Jibaro counting, as far as it is expressed in words.
'Ten' may also be expressed by joining both hands closed, without using any particular expression, and if the Jibaro wants to indicate 'twenty', 'thirty', 'forty', etc., he does it by joining his closed hands two, three, four, etc. times. It is characteristic of the Jibaro Indian that he cannot indicate a number in the abstract, but always does it by signs, even when he possesses a particular word for it, and nearly always he begins from 'one', counting on his fingers until he arrives at the number he wants to indicate.
[Quotation ends]
In one early grammar of the Shuar language (Ghinassi 1938) written by a missionary who spent many years among the Shuar, we find a description which is very similar to Karsten's.
"1) After twenty the Jíbaro Indian does not have any other number but he can count as much as he wants repeating with the closed hands the value for 'ten' and adding with the fingers the unities he needs. Between one value for ten and the other he says the adverb - atáksha (and again) - or júsha - (and this).
2) Ordinarily the Jíbaro Indian does not count more than fifteen or twenty values for 'ten' because it would be difficult to remember them; so that for a bigger number he uses the word untsúri (very much) - píshi (plenty) or a comparison - uéka núke (as ants)." (Ghinassi 1938, p.85).
These two descriptions4 show how non-verbal communication plays a central role in numeration and quantification and that quantities up to an undetermined limit are easily recognized. It is clear that the main problem for both authors is a traditional western point of view which places emphasis on what people can say. Both authors have to admit that the Shuar, when they have to, are able to count beyond the limits that would be represented through verbal expression. This hypothesis was implicit in the above citations: "If the Jibaro Indian is obliged to count more than five...", "If it is necessary to continue counting...", "Thereupon he may continue counting..." (Karsten), and "After twenty [...] the Jibaro Indian does not have any other number, but he can count as much as he wants..." (Ghinassi). With this perspective it is possible to edify an image of the "primitive". In relation to numerical terminology, this perspective was explicitly stated by Tylor (1903 [1871], vol.1, pp.243-244):
"It is not to be supposed, because a savage tribe has no current words for numbers above 3 or 5 or so, that therefore they cannot count beyond this. It appears that they can and do count considerably farther, but it is by falling back on a lower and ruder method of expression than speech - the gesture-language."
This perspective is the same one that leads some linguists to state that in a language "there is not" a comparative construction, but that if the speakers really have to verbalize a comparison they can use "some other" syntactic construction of the language. It is quite obvious that, in a case like this, a specific syntactic model -- the comparative construction present in most of the Indo-European languages -- is taken as an arbitrary measure for the other languages. We will go back to this specific example when we mention some recent developments in the Shuar mathematical terminology. The main point here is the arbitrariness of similar judgements. In the anthropological literature it is easy to find such arbitrary interpretations which do not contribute to the understanding of cultures. To mention a case in point, I quote a passage from the most recent anthropological monograph on the Shuar (Harner 1972, p.30):
"In actual practice trading partners do not keep a strict accounting of transactions. Since a variety of valuables is exchanged by two 'friends' at one time and since the Jivaro do not value numeration, the exchange is often slightly uneven."
First of all, it is only if we base our judgement on explicit linguistic data (the number names in the language) that we can say that the Shuar "do not value numeration". The fact that the exchange is often "slightly uneven" is only distantly related to mathematical skills proper. Such a statement, moreover, is an example of a judgement made with a western conception of trade and exchange in mind. It is likely that in most trading systems, as among the Shuar and other Upper Amazon peoples, the participants know that one of them receives some advantage. This is a central characteristic of the trade system which establishes and preserves social relations. If this is the case, a conclusion that the Shuar "do not value numeration" is based on a concept of trading which is deeply divergent from the traditional Shuar view of it.
Another aspect of the statement we are analyzing here is that of the relevance of a concept of "numeration" in trading patterns. The problem here concerns the relevance of quantifying skills and establishing relations between quantities as cues of operational capacities in general. The relevance of these concepts is not at all clear. Hallpike (1979, pp.98-99) writes:
"All estimates of size, or length, or height, or quantity, or duration thus immediately conjure up sensory images or associations of familiar activities and forms of behaviour, of procedures and customary modes of coordination, and do not stimulate or require quantitative analysis or dimensional abstraction. The sensations of size, duration, weight, and heat, among many others, are thus necessarily subjectified in primitive experience. Without units of measurement and quantification it is very difficult to separate out particular dimensions and to compare objects in terms of them alone, or to become aware of relations of compensation between different dimensions.
[...]
In a world of gourds, pots, bamboo tubes, baskets, hollowed-out tree-trunks, string bags, and sewn-up animal skins, which are used for transporting and storing things and not for measuring them, it is extraordinarily difficult accurately to perceive displacements and conservations of quantity or area."
While this may be "extraordinarily difficult" it often happens, and when western observers find such skills in a "primitive" people they are struck by such abilities. One recent example can be found in the account of the success of the Kpelle people of Liberia in evaluating the number of cups of uncooked rice which could be contained in a bowl (Gay and Cole 1967, p.8). In that case the skills of the Kpelle people became even more evident when compared to the failure of some American Peace Corp Volunteers in the same test. From this example we could conclude that the operative use of the concept of number (which we should attribute to the American Peace Corps members) can be irrelevant to the skills of practical evaluation of volumes and quantities. On the other hand, when we find some good practical skills they should not be used as evidence of mathematical abilities.
The interaction between the activities of measuring, quantifying and finding mnemonic devices for quantities can be relevant to the growth of numerical terminology. We could assume, as a working hypothesis, that words in general, and numerals and quantitative expressions in particular, are "indices to pre-linguistic cognitive schemata, according to which we organize and remember our experience" (Kay 1979, p.1).
In the traditional culture of the Shuar and of other Jivaros we find some activities in which good skills of exact execution and measure are requested. On the other hand, it is hard to imagine any activity in which the capacity of counting and calculating could be particularly helpful. We mention three activities: the construction of a blow-gun, the digging out of a canoe, and the building of a house.
Only some of the Achuar men5 are able to produce long blow-guns. They are constructed from two pieces of hard wood which have to fit together perfectly. The internal part of the gun cannot be even minutely uneven.
Only a few of the Achuar and Shuar men are able to manage the work of digging out a big log of selected wood to produce a canoe. This work requires great skill in evaluating the volume of wood needed to yield a dugout canoe that will be perfectly balanced once it will be put in the water.
Almost all the Shuar and Achuar men are able to build their house, with the help of other men. The Achuar build a new house every six to eight years. The old house is left when the new one is ready. In general many men participate in this activity. Consequently, this is a relatively common activity in the life of each Achuar man, who builds his own house several times and who has to help other men many times in his life. In this frequent and collective activity a standard measure is used. The name of the measuring unit is nekapek or nekapmatai and in general it is represented by a stick. The length of the stick is based on the human body. It is the segment from the girdle to the ground. A relevant question is why such a standard measure is in use in one of the most frequent and socializing activities in the Achuar culture.
The root of the name of the measure is nekap-. This root, from our point of view, can be glossed with a set of meanings. It is possible, however, that for an Achuar or a Shuar this root identifies a single conceptual area which includes meanings such as 'to show', 'to indicate' (as in hintya nekap-,'to indicate the path'), 'to demonstrate', 'to measure' and 'to quantify'. In the recent process of implementation of a Shuar mathematical terminology6 this root has been used to express the concept of 'number' and 'to count'. The same root is related to the root neka-, 'to know, to be aware'. From this root are derived forms such as nekas, 'right, true', or followed by the negative morpheme (-ca), nekasmianča, 'valueless, fake'. We could say that meanings such as 'to measure' and 'to know' are related through two varieties of a basic root. I am not claiming that these meanings are 'the same' for a Shuar or an Achuar, but I would like to claim that some common component exists in such meanings as 'to know, to show, to measure, to quantify'.
When we look at the few numerals of the Jivaroan languages, from a comparative and etymological point of view, we can reach some insight into the historical growth of the set. Although my main concern here is with the Shuar language, I will also use data from two other strictly related Jivaroan languages, Achuar and Aguaruna. The first four numerals in the three languages are as follows.
|
Shuar |
Achuar |
Aguaruna |
| 1 |
čikíčik |
kíčik |
bakíčik |
| 2 |
hímyar |
hímyar |
hímaη |
| 3 |
manáintyu |
kampátam |
kampátum |
| 4 |
áintyuk áintyuk |
učínyuk učínyuk |
ipáksumat |
The first fact we can point out is that while the forms for 'one' and 'two' are basically the same in the three languages, for 'three' we find two distinct forms and for 'four' three distinct forms, one for each language. This fact can be interpreted as revealing that the first two numerals -- common to the three languages -- are older than the third, and that the third numeral is older than the fourth.
The etymological analysis leads us to the same conclusion because the first two numerals are much more opaque in etymological terms than the third and the third more than the fourth.
In the forms for 'one' we find that kíčik of the Achuar represents the common segment of the morpheme. I am unable, however, to establish whether the forms with initial či- and ba- of Shuar and Aguaruna are expansions of the kíčik form. In the Shuar language we find other similar morphemes such as išíčik, 'a few, a little', číkič or tíkič, 'other', and čikyá-s-, 'to separate, to stay alone'. I am unable, however, to relate any of these morphemes to the form for 'one', in a more significant way.
The forms for 'two' are exactly the same in Shuar and Achuar. The difference in Aguaruna is due to systematic phonological correspondences between this language and the other two. I am unable to propose any etymology for these forms. I can only remember the existence in Shuar of the expression himyámpramu, 'twin', which seems derived from the morpheme for 'two'.
Things become a little more transparent with the Shuar forms for 'three' and 'four'. The Shuar form for 'three' can be related to a whole set of morphemes such as ména, 'left, left side, left hand', menánt-, 'to stop, to stay at a side', men-ká-ka-, 'to miss, to loose', men-á-k-, 'to miss the path, to be unable to find the path', main-ηka-, 'spoiled' (used for manioc beer or for cooked manioc). All these morphemes have some common meaning which ranges, it seems to me, from 'to miss some pre-existent or characteristic', to 'to be in a non-central position'. An etymological speculation could lead us to propose 'uneven' as an original meaning of the form for 'three'.
It is likely that the form manáintyu arrived in Shuar in relatively recent times, to replace an older form similar to that used at present in Achuar and in Aguaruna. Karsten (1935) stated that "among the Upano7 tribes was used for 'three' menéindu or kámbatama." At the time he collected his data some variation between the two forms should have existed in Shuar.
I am unable to associate the Achuar and Aguaruna form to other morphemes of the two languages or to any morpheme of the Shuar. We should note that the difference in the semifinal vowels can be explained as a systematic phonological correspondence between the two languages.
The Shuar form for 'four' is probably related to the morphemes aínik, aíniu, ániu, 'similar, equal, even', ain-kia-, 'to do the same', aint-ra-, 'to go together, to follow'. We could associate the root ain(t)- with a meaning 'equal, together'. Both the Shuar and the Achuar forms are reduplicated. The semantics of reduplication in Shuar is not easy to catch because it is a widely used device in the language. I would say that it has in general an intensive meaning, so that the Shuar form for 'four' could be interpreted as 'the very even' or something very similar. The Aguaruna form for 'four' is very interesting. The Jivaros used to count by putting down fingers, starting from the minor one, so that the index is the fourth finger. The index is used to paint the face with red pigment from the Bixa Orellana. The meaning of the Aguaruna name is 'for painting'. In Shuar the name of the finger is ipyáksuntai, 'for painting with Bixa Orellana (ipyák) pigment'. A related form -(u)sumtai- is used in Shuar to refer to 'nine'.
The fact that the Aguaruna name of the number is indeed the name of the fourth finger constitues, in some sense, the actual link between the verbal names of the numbers and the gestual activity of reckoning on the fingers. In the short numerical distance from 'one' to 'four' we find a great linguistic distance from the absolute etymological opaqueness of the first two numerals to the relative transparency of some of the five names for 'three' and 'four'. In some sense there is also a difference in degree of arbitrariness, because a form which is more transparent and more easy to relate to other forms of the language is in relative terms less arbitrary than a form which is not. As mentioned earlier, the Aguaruna form for 'four' is particularly interesting not only because it is transparent in its meaning but because it relates the names of numbers to the 'concrete' action of reckoning.
The expression used for 'five' in Shuar is ewéh amus, 'the hand is complete'. It is a descriptive sentence.
For numbers beyond 'five' we observe a good degree of variation in the expressions which are used together with the gestures. Pellizarro (1969, p.23) gives a set of expressions which are different from those given by Karsten and Ghinassi. The numerals 'one', 'two', 'three', and 'four' are followed by the verbal expression íraku, 'added'. Thus, one has chikíchik íraku, 'one added', Jimerá íraku, 'two added', etc., for 'six', 'seven', and so on.
The differences between the authors become still greater when we go beyond 'ten'. For example, for 'eleven', Pellizarro gives the expression chikíchik nawén íraku, 'one of the foot added'. However, he also states that "when they have more than four or five things to count they use to say untsurí 'many'". Both the divergence between Karsten's and Pellizarro's accounts, and the alternatives that Pellizarro gives, are important in the perspective I am presenting here. They confirm my field observations of the variation which is found after the first few established numerals. The variation is a central characteristic in the dynamic view of language and culture. These data bring insight into the growth of numerical terminology and demonstrate that the establishing of a numerical system can be a slow process in the history of a language. In this sense we can expect that some number names are connected with other roots of the language.
Recently a Shuar mathematical terminology was proposed as a part of a process of language standardization for a bilingual and bicultural education-by-radio program.8 A group of Shuar teachers 'adapted' lexical items of the language to express elementary mathematical concepts. I already mentioned the root nekap-, used to express the meanings of 'number' and 'to count'. The form eweh, 'hand', was used for 'five'. Numerals for 'six', 'seven', and 'eight' were proposed. As for 'nine' the form usúmtai, which I have already mentioned, was introduced. For 'ten' the form náwe, 'foot', was proposed, to avoid expressions such as mái ewéh amúkhai, 'I completed both my hands'. For 'hundred' was proposed the form wášim and for 'thousand' the form nupantí, both being terms taken from the language. The entire numeral system is built on the base of these numerals and two basic syntactic principles of the language: the numeral which precedes the forms for 'ten', 'hundred', and 'thousand' has the function of a multiplier, the numeral which follows represents an added value. Thus, for example, himyára náwe čikíčik, 'two (times) ten (and) one', signifies 21 and menáintyu wášim áintiuk náwe ewéh, 'three (times) hundred (and) four (times) ten (and) five', signifies 345.
In addition to the numerals, approximately 45 terms were proposed for other concepts of elementary mathematics. I will mention some of these, to show both the linguistic problems and the internal possibilities that can be found in a native language. In the Shuar language there are various ways to express the disjunction. In mathematical terminology, the expressions niniák páčitsuk, 'not naming that', and tumátskeša or turútskeša, 'not doing this way', have been used. Such expressions are also found in the everyday language.
To express the concept of comparative order a current Shuar form was used. This is the plain adjective učič, 'small', for 'less than', and uúnt, 'big', for 'greater than'. To state, as many linguists would, that the Shuar language does not have comparative forms would mean that some standard model of comparative construction is taken as reference.
To express concepts such as 'less than or equal to', in which both the disjunction and the comparative are present, the disjunction was expressed in a way different from the translations already mentioned. In these cases it was simply omitted with učič', metéketai signifying 'is small, equal to', and uúnt, méteketai signifying 'is big, equal to'.
To conclude, I would like to emphasize once again the basic hypothesis of this paper. We cannot expect to find a specific mathematical activity in most cultures without a tradition of writing, but we should be able to identify basic capacities upon which a mathematical activity can be built. In cultures without a written tradition, the whole perspective on knowledge and on the capacities of reflection is deeply different from our own. The process of accumulation of knowledge and of reflection on nature exists in a dimension very different from the one we know. Nevertheless, the language may "register" some aspect of such a process.
NOTES