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Why Study Quechua Numbers and Mathematics?
I expect that some (if not many) who pick up this book will ask themselves the question posed in the heading to this section. The question is certainly not without merit, as I asked it of myself many times during the early days and months of research and writing on this book, which concerns the numerical knowledge and arithmetic practices of the Quechua-speaking peoples of the Bolivian and Peruvian Andes. The perceptive reader will of course recognize a confession buried in the previous sentence. That is, I must admit that I did not take up this study from a profound, single-minded interest in numbers and mathematics. Behind this confession lies another, decidedly more embarrassing one, which is that while I have always had a great interest in science and mathematics, I am not a math whiz--the type of person who seeks out mathematical problems and puzzles against which to test their mathematical acumen. So, the question in the section heading now becomes both more urgent and personal: Why would someone who is not an expert in the art of mathematics take up the study of numbers, arithmetic, and mathematics? There are two answers to this question, which, I am quick to point out, have undergone a complete reversal in terms of their priority in motivating my work on this topic. The first has a historical focus, the second an ethnographic one.
Initially, I came to the study of contemporary Quechua numerical knowledge and arithmetic and mathematical practice as what I saw early on to be an essential first step in pursuing my primary interest at the time, which was the analysis of the Inka khipus. Khipus are the knottedstring devices that were used by the Inkas to record both quantitative data (such as census accounts and tribute records) and information that was said to have been used in the recording and retrieval--or "reading"--of Inka histories, genealogies, and myths. We know with some certainty that many of the khipus preserved today in museum and private collections contain quantitative data (see Ascher and Ascher 1981; Locke 1923; Radicati di Primeglio 1979). Since the beginning of this century, scholars have succeeded in "cracking" the code, or organization, of the numerical information in the khipus, thus allowing us a glimpse into the numerical knowledge and--indirectly at least--the mathematical practices of the Inkas.
There is, however, a fundamental methodological problem in the way that all previous studies of the numerical information recorded on the khipus have been carried out. That is, the numbers have in all cases been translated using Hindu-Arabic numerals or the language of numbers of the investigator, especially English and Spanish. Such approaches to the khipu numbers inevitably mask, and eliminate from analysis, any values and meanings that may have been attached to these numbers by the Quechua-speaking bureaucrats of the Inka empire who recorded the information. Such symbolic, metaphorical, and other meanings that may have been associated with these numbers (or number groupings, patterns, etc.) could be of great importance in our continuing efforts to understand Inka numerical and mathematical concepts and practices. But, in addition, we might find that such values could provide us with a basis for approaching the task of "deciphering" the narratives recorded on the khipus--a challenge which we have not addressed in a serious way to the present day (Urton, n.d.).
It was this set of interests and objectives that most directly and profoundly motivated my study of contemporary Quechua numerical knowledge and mathematical practices initially; that is, I was seeking a body of knowledge and practice in the present that could help direct and inform my investigations of the past. In this regard, I should note that such historically motivated and, frankly, superficially "romantic" investigations of contemporary Quechua society and culture as that outlined here are often viewed with uneasiness, if not with outright disdain, especially by some who insist on seeing Andean peoples today solely as the victims of poverty and as either the perpetrators or the victims of violence (see, for example, Starn 1991). According to this view, Andean peoples have nothing to add to the human record other than the misfortunes of their material conditions of life. However, I (and others) continue to insist on seeing Andean peoples as bearers and manipulators of bodies of knowledge and practice--from animal husbandry and agriculture (see Van der Ploeg 1993) to astronomy (Urton 1988)that are profoundly important for us to understand in historical and comparative terms. To ignore, and thereby undervalue, the complex traditions and systems of knowledge that have been maintained and continually rethought and reformulated by Andean peoples through history strikes me not only as pernicious, but, more importantly, as performing a disservice to people who have been regarded over the past five hundred years as too culturally and intellectually debilitated to offer anything of interest to the record of human accomplishments.
The second reason that I embarked on a study of Quechua numerical knowledge and arithmetic and mathematical practice was because this is an almost totally unstudied domain of knowledge and practice in Andean studies. Having earlier investigated the astronomical knowledge and cosmological beliefs of people in a community near Cusco, Peru (Urton 1988), and having found there beliefs, ideas, and traditions that are of considerable complexity and (I think) of great comparative interest and importance, it seemed to me that the same might prove to be true of numbers and mathematics. I admit here that this second motivation for undertaking the present study was initially far less compelling than the first--my interest in the Inka khipus. Therefore, I was quite unprepared for, if not for some time actually resistant to, what I would find in this study. For what I found as my research proceeded, especially during a year of linguistic study and ethnographic research carried out in and around Sucre, Bolivia, in 1993-1994, was a complex and well-articulated set of ideas and practices that represent a new (to us) and arguably unique ontology of numbers and philosophy of arithmetic and mathematics. I hope to show that the ontology and philosophy to be elaborated herein provide us with a new perspective--perhaps a new paradigm--with which to reexamine and rethink well-known bodies of ethnographic and historical data pertaining to Andean societies and cultures, past and present.
This is a broad and immodest claim for such a slim book. It is therefore imperative that I inform the reader from the beginning about the basic arguments to be presented in the following chapters. However, in order for this summary to be meaningful, I must first make clear exactly what the intended topic of study is here and who the people are whom I have been referring to as the "Quechua-speaking people of the Andes."
The Topic of Study: Is It Ethnomathematics or Ethnoarithmetic?
As will soon become apparent, my principal concern in this study is with understanding the cultural significance of numbers and arithmetic among Quechua-speaking peoples in the Andes. Given that the primary focus here will be on the positive integers (1, 2 , 3, 4 , 5, . . . ) and the arithmetic procedures for manipulating relations among them (i.e., addition, subtraction, multiplication, and division), this study falls comfortably under the heading of the "anthropology of numbers" (Crump 1990) or the more common, but somewhat more problematic, label "ethnomathematics" (Ascher 1991). Identified either way, the field of studies we are concerned with is new and little developed to date; thus, there remain several problems in the definition of terms, as well as in the conceptualization of topics to be addressed, assumptions made, and approaches used, by researchers in the field. To get straight to some of these problems, we may begin by defining the two principal fields of study in Western academic practice most closely connected with our topic of study--arithmetic and mathematics--and then move on to discuss the meaning and significance of attaching the prefix "ethno-" to one or the other (or both) of these terms.
In James and James's Mathematics Dictionary (1976), "arithmetic" is defined as: "The study of the positive integers, 1, 2, 3, 4, 5.... under the operations of addition, subtraction, multiplication, and division, and the use of the results of these studies in everyday life." This definition encompasses the great majority of topics and practices that I will be concerned with in this work. However, there are certain equally important topics discussed here that fall outside the boundaries of the above definition of arithmetic. These include such questions as the ontology of numbers generally; the logical foundation, motivation, and symbolic and metaphorical meanings of the cardinal and ordinal numeral sequences in Quechua; and the relationship between shape and quantity (as in the geometrical designs of Andean weavings). Certain of these matters fall more naturally within the domain of what we know in the West as mathematics.
As many authors have noted, it is virtually impossible to define mathematics in any concise way, thus producing a notable resistance on the part of scholars who have written on comparative and historical mathematics to provide anything but the most general characterization of this subject (e.g., Ascher 1991: 2-3, and Joseph 1991: 3). Nonetheless, we have to begin somewhere, especially if we propose later to attach to this undefinable--something the even more ambiguous "ethno-" prefix! Citing James and James's work again, mathematics can be defined as:
The logical study of shape, arrangement, quantity and many related concepts. Mathematics often is divided into three fields: algebra, analysis, and geometry. However, no clear divisions can be made, since these branches have become thoroughly intermingled. Roughly, algebra involves numbers and their abstractions, analysis involves continuity and limits, and geometry is concerned with space and related concepts. (1976:239)
It is also relevant (especially in relation to topics discussed later in this introduction) to take note of James and James's definition of pure mathematics as "the study and development of the principles of mathematics for their own sake and possible future usefulness, rather than for their immediate usefulness in other fields of science or knowledge. The study of mathematics independently of experience in other scholarly disciplines" (1976: 239).
Before taking stock of the relevance of the terms arithmetic and/or mathematics for our interests here, we should consider the significance of the prefix "ethno-." Ascher argues that the appropriate object of study in "ethnomathematics" is mathematical ideas not "mathematics" per se, the latter of which, she argues, is a Western category and thus is not even to be found in traditional cultures. Among the mathematical ideas that Ascher proposes as appropriate for, and accessible to, study are "those involving number, logic, spatial configuration, and, even more significant, the combination or organization of these into systems or structures" (1991: 2).
"Ethno-" is, of course, used today quite liberally to indicate that the investigation of a particular field of study (such as biology or astronomy) is being discussed from the perspective, and on the basis of the knowledge, of the people of some non-Western, "traditional" society. In such contexts, "traditional" is used to denote a society that, at least as characterized in the "ethnographic present" of the published anthropological literature, is largely nonliterate, nonindustrialized, and overwhelmingly rural and agriculturally based. Now, as has been recognized in the anthropological literature for some time, the designation of any particular society as "traditional" is highly problematic, especially when this category is considered to stand in some meaningful opposition to a "modern" society. For our purposes here, the point that I want to stress regarding this dual classification is that "modern" societies are generally assumed to be so partially in relation to their possession of science, whereas "traditional" societies are considered not to have science. The questions for us to consider in this regard are: How does the science/ nonscience dualism characterizing the split between modern and traditional societies condition the pursuit of the investigation of numbers, arithmetic, and mathematics in a particular non-Western society? And, more to the point, why has the field of studies we are engaged in here come to be known as "ethnomathematics" and not, for instance, "ethnoarithmetic?"
There are two observations that will direct us on our way to finding answers to the questions raised above. The first observation, which has not been addressed specifically in any previous ethnomathematical studies that I am aware of, is this: In the classification of societies according to the traditional/modern dualism, there has been at least an implicit recognition that traditional societies possess, to varying degrees, numbers and arithmetic, while modern societies possess numbers (to the nth degree), arithmetic, and mathematics. It is important to note that the collection of knowledge and practices referred to as mathematics is often accorded to a handful of earlier, literate societies, such as China, India, Greece, and Rome, as well as to the schools of Islamic scholars who were directly responsible for preserving, and innovating on, both Greco-Roman and Eastern mathematical traditions (see Joseph 1991).
According to the above characterization, numbers and arithmetic are not only considered to be knowable and doable by people in all societies, but there is also the presumption (suggested by the fact that there is no talk in this body of literature about either "ethnonumbers" or "ethnoarithmetic") that the concept of numbers and the procedures of arithmetic are unaffected by cultural differences. Before I comment on this point, let me raise a second related point, which is that, while it is common to discuss the philosophy of mathematics, there is not a comparable body of literature dealing with the "philosophy of arithmetic." The conclusion that I draw from these two observations is that when scholars talk about "ethnomathematics," they generally do not by the use of that label mean to suggest that culture can affect the nature and meaning of numbers, nor the procedures of arithmetic or mathematics. That is, no one seems to be suggesting that, in some other culture, two plus two might equal anything other than four, or that transfinite numbers might be imagined in some cultural tradition in such a way as to possess characteristics other than those assigned to them in Western mathematics. Rather, it is clear that by the use of the label ethnomathematics, what is considered to be susceptible to cultural influence are the conceptions of numbers and the philosophy of mathematics. In other words, to my mind at least, ethnomathematics is actually concerned with ethnophilosophy. The elision of "philosophy" in the word ethnomathematics masks the idea, firmly held by most practitioners of ethnomathematics, that while philosophy may be influenced by culture, mathematics is not.
Now, I think it has been clearly demonstrated that the philosophy of mathematics is indeed subject to different cultural, linguistic, and perhaps even national formulations. This is evident from any survey of the Western history and philosophy of mathematics, in which we find a record of a few significantly different traditions of mathematical philosophy, such as Platonism, Intuitionism, and Formalism. The question for us to consider here, however, is: Where does this leave us in defining what it is we are (or ought to be) concerned with in an anthropological study of numbers, arithmetic, and mathematics? While I will have more to say later in this chapter on the question of the intersection of anthropology and mathematics, the position that I will adopt here is to place everything on the table--numbers, arithmetic, philosophy, and mathematics--not privileging any particular form of human knowledge and practice as being above either language or culture. That is, I assume that to some degree and in varying ways (linguistic, logical, metaphorical, etc.), all of these forms of knowledge and practice are susceptible to formulations reflecting the differing customs, values, and ways of constructing and pursuing logical arguments among cultures worldwide. Whether or not that susceptibility is realized, and what its consequences might be, are matters to be determined on the basis of intensive ethnographic and linguistic research on number theory and ethnomathematics in different cultures. For instance, while we find that the Hindu-Arabic numerals 1, 2, 3.... are used to sign the same quantities in all cultures that have adopted these symbols, nonetheless, we will find in this study that the relations between any two adjacent numbers are conceptualized quite differently in Quechua numerical ontology than they are in Western number theory. At this level, then, I will argue that there are "ethnonumbers" and that there can be (and is) such a thing as "ethnoarithmetic." As far as I am aware from the published literature, we have barely even begun comparative investigation of numbers at the level outlined above (one of the few such studies is that carried out by Mimica ).
In summary, while I will not be overly concerned with using a particular label (such as "ethnomathematics") for what it is I am investigating here, I will try to label the different topics that I address below (for example, the ontology of numbers, the philosophy of arithmetic and/or mathematics) consistently so that the reader will be clear about what concept or practice is at issue, at least as it is to be understood on the basis of the normal English usage of the terms involved. Before turning to discuss more general problems in the anthropological study of numbers, arithmetic, and mathematics, I will clarify the particular language and culture that we will focus on in this study.
Defining the Language of Study
"Quechua" refers to a widespread language family, the numerous variants of which are spoken by some six to ten million people (Crystal 1987: 442) primarily in the Andean nations of Peru, Bolivia, Ecuador, Colombia, and Argentina. The language family has its present, widespread distribution as a result of a long and complex history of language
dispersal, contact, and attempts at cultural unification that occurred both before and after the Spanish conquest of the Andes, beginning in 1532. The Quechua language is thought to have had its origins in central or coastal Peru. From its origins, there developed two main branches: Central Quechua and Peripheral Quechua, the latter of which was divided primarily between two sub-branches, one to the north and the other to the south of Central Quechua (Parker 1963; Torero 1964, 1974; Mannheim 1991). We will be concerned in this study with two varieties of the southern sub-branch of Peripheral Quechua: Southern Peruvian and Bolivian Quechua.
The present distribution of Southern Peruvian and Bolivian Quechua is intimately related to the history of the Inka empire and its conquest and colonization by Europeans. Beginning several centuries before the European invasion, populations from central Ecuador southward to central Chile were under at least the nominal, if not actual, control of the Inkas, whose empire was centered in Cusco, located in what is today south-central Peru. While there were many different languages spoken throughout the territory of Tawantinsuyu ("the four united quarters/ parts")--the name that was used by the Inkas to refer to their empire--the Southern Peruvian variety of Quechua spoken in the heartland of the empire, the southern highlands of Peru, had the status of a lingua franca, partially because this was the language of administration in the Inka empire. Provincial nobility throughout the empire were required to send their heirs (their sons and/or brothers) to Cusco for training in Quechua. As Damián de la Bandera noted in ca. 1557, "All the caciques and chief persons of the whole kingdom who held any office or position in the state were obliged to know the general language [Quechua] in order to be able to give whatever information was necessary to their superiors" (cited in Rowe 1982: 96). However, the Inkas seem not to have been particularly disturbed by the myriad other languages that were spoken within the empire. In addition, through such institutions as that of the mitmaq--the relocation of populations within the empire for purposes of pacification, or the resettling of people where their particular skills were needed--the discontinuities in the distribution of languages became, if anything, more pronounced during imperial times. The relocation of Southern Peruvian Quechua-speaking mitmaq to the Cochabamba valley and elsewhere in central Bolivia (see, for example, Wachtel 1982) may at least partially account for the development of Bolivian Quechua, a variety of southern Peripheral Quechua that is mutually intelligible with the Southern Peruvian, the Inka lingua franca.
Following the Spanish conquest of the Andes, the European colonizers were faced with an exceptionally complex linguistic picture. This included the myriad minor languages spoken in different locales; a few languages--such as Aymara, Puquina, and Yunga--that had a fairly widespread distribution and which were (initially at least) encouraged over local languages by the Spaniards; and the Inka lingua franca--Southern Peruvian Quechua. As a part of their strategy of administering to the native population in both civil and ecclesiastical terms, the Spaniards increasingly promoted the use of Quechua over other languages. In addition, from early colonial times, the Spaniards were responsible for the publication in Quechua of certain works, especially ecclesiastical items such as catechisms and confessional manuals (see Barnes 1992; Mannheim 1991: 34). Thus, as has been made clear by a number of students of Quechua over the past few decades, the present widespread distribution of the Quechua language throughout the Andes is primarily a result of Spanish initiatives at language standardization, rather than a reflection of an attempt by the Inkas to eliminate linguistic diversity within the empire (Mannheim 1991; Torero 1974).
What is important for our purposes about the above observations is, first, that the people who speak Quechua today throughout the Andes do so as a consequence not of descent, in some pure manner, from the Inka nobility in Cusco or from their provincial administrators throughout the empire, but rather as a result of the project of linguistic homogenization that was carried out by the Spaniards during the colonial era. Nonetheless, and this is the second point of interest, the Southern Peruvian and Bolivian varieties of Quechua spoken today in southern Peru and central Bolivia are the historical descendants of the language spoken by the Inkas in Cusco (Mannheim 1991: 11). While many changes in phonetics, vocabulary, and, to a lesser degree, grammar have occurred in Southern Peruvian and Bolivian Quechua since the European invasion (see Mannheim 1991: 113 ff), it is clear--for example, from dictionaries and other early colonial sources in Quechua--that there are far more similarities than differences between the Quechua spoken today and that spoken in late pre-Hispanic and early colonial times throughout southern Peru and central Bolivia. In other words, there would be mutual comprehension between and among the speakers of these two varieties of Quechua at the two end points of the historical continuum from late pre-Hispanic times to the present day.
The question that is raised for us by these observations is: What is the status, in terms of continuities from colonial times, in the lexicon, syntax, and semantics of the language of numbers and arithmetic operations (the terms and phrases used for addition, subtraction, etc.) in the varieties of Quechua spoken today in southern Peru and central Bolivia? While I will address different aspects of this question in each of the following chapters, the short answer to this question is that the language of numbers, arithmetic, and mathematics does not appear to have undergone significant changes, either in terms of the vocabularies involved or in the ways terms are strung together syntactically in phrases denoting compound numbers and arithmetic operations.
One thing that may account for the persistence of Quechua numbers to the present day is that, like the numbering system of the conquering Europeans, Quechua utilizes a base-10, or decimal, system of numeration. Thus, there are no fundamental structural discontinuities in the two numbering systems as we might expect would arise, for example, were the number system of the conquering society to have been based on a binary, quinary, or vigesimal principle. This means that values expressed in one of the two languages (Quechua or Spanish) can be easily translated using the number names and grammatical constructions of the other. This is not to say, however, that even in predominately Quechua-speaking villages Quechua numbers are utilized on all occasions, for not only do Spanish number words predominate in certain settings (such as marketing and telling time), but also Quechua and Spanish number names and constructions are freely mixed by certain types of people in certain settings. This will be clear, for instance, to anyone who becomes attuned to the number words and phrases used in urban and provincial markets throughout the Andes today. I will return in Chapter 6 to the discussion of a number of issues concerning uses of, and conflicts over, Quechua and Spanish numbers in the Andes during the colonial era.
To return to the question of linguistic continuities in Southern Peruvian and Bolivian Quechua, wherever and whenever Quechua number names and phrases are used today, there are few differences from colonial times in such features as, for instance, the grammatical entailments of combining "minor" units (1-9) with "major" units (10, 100, and 1,000) to form compound numbers--those combining two or more primary lexemes in the decimal system of numeration, as English "seventeen" (7 10). Therefore, generally speaking, we can use the material on numbers and arithmetic from one era (for example, the colonial dictionaries of Quechua) to enlarge, enrich, and in some cases contextualize data collected in another era (such as contemporary ethnographic data from Quechua-speaking communities in southern Peru and Bolivia).
With this brief overview of the Quechua language in the Andes as background, we now turn to a summary of the principal arguments to be made in the following chapters concerning the ontology of numbers and the philosophy of arithmetic and mathematics among contemporary Quechua-speakers of southern Peru and Bolivia. This summary will also inform the discussion, to be taken up later in this introduction, of the question: What contribution can anthropology make to the science of mathematics?
Outline of a Quechua Ontology of Numbers and Philosophy of Mathematics
When I speak of an "ontology" of numbers, I am referring to ideas about the origin and nature of numbers. That is: Where do numbers come from? What kind(s) of thing(s) are they? And how is one number related to another, or the next, number? These are questions that have been of central concern to philosophers of mathematics in the West for several centuries (Boyer 1968; Dummett 1991; Wittgenstein 1978). In this work, I will show that the characteristics and identities of numbers as conceived of and formulated in Quechua language and culture are predicated in terms of relations and identities constituting and governing social life in Quechua communities. More concretely, family relations and kinship roles and statuses represent the principal types of relations--such as hierarchy, descent, succession--in terms of which numbers are conceptualized, organized, and talked about. Any given sequence of natural numbers (as four, five, six.... ) or ordinal numerals (as fourth, fifth, sixth.... ) is motivated, and finds its rationale in, the social and biological relations uniting and organizing a descent group. The rules, or organizing principles, underlying succession, which is one of the central processes that must be accounted for in any ontology of numbers, are identified on two levels in Quechua ideology; the first is that between a mother and her offspring, with the former as the prior number (usually "one") and the latter as her successor(s); the second is that of age-grading among siblings, with an older sibling as a prior number and a younger sibling as its successor. Thus, kinship and social relations will be shown to form the matrix for thinking about, and talking with, numbers.
Following from these observations, the prototype of a sequence of cardinal numbers or ordinal numerals combines the two principles of succession described above in the group formed by a mother and her several offspring born successively over time (see Table 1.1). I will explain later (Chapters 3 and 4) why the prototype, or model, of number/numeral sequences contains five elements.
Thus, numbers are not conceived of in Quechua ideology as abstractions whose nature and relations to each other rely on the predications of pure logic, as in the West. Rather, numbers are conceptualized in terms of social-especially family and kinship-roles and relations. One consequence of their participation in social life is that all of the linguistic formulations and grammatical constructions that are used to talk about inheritance, succession, dependence, and interdependence within a kin group (such as the Andean ayllu) can be applied as well to number identities and numerical relations. I would note that the metaphorical links between kinship and numerical relations outlined here are consistent with what Turner (1987) has so cogently identified as the central role of kinship relations in the construction of metaphors more generally.
With this stark outline of the ontology of numbers as background, we now turn to a summary of the philosophy of arithmetic and mathematics to be encountered herein (see especially Chapter 5). By a "philosophy" of arithmetic, I am referring to ideas, discourse, and practices concerning when and why, or under what conditions, certain manipulations of numbers, or mathematical operations, are both called for and are considered to be appropriate. The operations that we will examine include the arithmetic operations of addition, subtraction, multiplication, and division. The fundamental notion on which the performance of any one of these operations is premised is the idea that the desired and proper state of affairs in the world is one in which resources, labor, behavior, and all other objects, relations, and attitudes are in a state of balance, equilibrium, and harmony. If and when a state of imbalance or disequilibrium emerges (as through an inappropriate distribution of resources, or inappropriate behavior), then the imbalance and disequilibrium must be rectified. The arithmetic operations of addition, subtraction, multiplication, and division represent several of the principal forms of rectification that may be carried out. That is, for example, if resources are distributed inappropriately (note: "inappropriate" is not necessarily synonymous with "unequal"), then the situation must be rectified, perhaps by taking something from one part (subtraction) and applying it to another (addition).
In Chapter 5, I will refer to the philosophical principles outlined above as constituting an arithmetic and a mathematics of "rectification." It will be seen that these notions of the nature, logic, and purpose of arithmetic practice are radically different from those encountered in the West, while potentially yielding the same results, at least insofar as numbers are manipulated in Quechua practice. That is, in Western mathematics, one need have no purpose or justification for manipulating numbers, nor are arithmetic and mathematics to be justified solely on the grounds of their "utility." In particular, pure mathematics undertakes its manipulations of numbers for the purpose of exploring the properties of numbers, shapes, spatial relations, etc. If the properties that are discovered result in having some use value, so much the better; however, the search for such values (that is, "applications") is not the driving force behind mathematical explorations in their purest form (see Hardy 1993; see also above, in this chapter).
When we consider the Quechua ontology of numbers and philosophy of mathematics summarized above, we can appreciate the uniqueness of these concepts as well as the potential value to Andean studies, conceived of broadly, of the investigation of these numerical, arithmetic, and mathematical forms of reasoning. As I noted, these issues have received virtually no attention to date. These observations also provide an appropriate point of departure for a consideration of the broader question of the potential contribution that anthropology can make to the study of number theory, arithmetic, and mathematics. Before taking up the latter question, let me clarify the ethnographic status of the two terms used above (and in the subtitle to this work) to refer to the general topics of inquiry here--"ontology" and "philosophy."
Except in the case of my principal teacher and informant, Primitivo Nina Llanos, the Quechua-speaking people with whom I discussed numbers, arithmetic, and related matters (such as "sets" of objects; correlations between one quantity and another) in villages in Peru and Bolivia did not use either one of these metalinguistic terms (ontology and philosophy) to describe, in an abstract sense, what general category of knowledge their comments referred to. Primitivo Nina, who is a literate, highly analytical professor of Quechua, easily makes the transition and connections between individual acts and such abstract, analytical categories of knowledge as "ontology." My other informants also were entirely capable of using abstract, analytical terms for types of people, forms of behavior, and categories of kinship and other social relations. Thus, for example, people in Candelaria would daily use a term such as mama to refer equally to "mother," the "thumb," the first "ear of corn" sprouting on a cornstalk, and the first color to appear in a rainbow (see Chapter 3). However, the people in this community (or in any other that I am familiar with) have no interest in, or need for, a set of analytical terms for talking about their thinking and speech (see below, Chapter 4). Out of such general uses of terms and correlations of phenomena and actions as those given above for mama, I have grouped together speech acts, objects of reference, etc., as indicative of the subject matter of a Quechua "ontology" of numbers. I have done the same for what I call herein the "philosophy" of arithmetic (in a manner similar to Isbell and Roncalla Fernandez's  discussion of the "ontogenesis" of metaphor in Quechua). My hope is that by the end of this study, I will have successfully explained and interpreted the linguistic and ethnographic materials to the point that I will have justified my use of the particular metalinguistic labels "ontology" (of numbers) and "philosophy" (of arithmetic).
Anthropology's Potential Contribution to Mathematics
What can anthropology contribute to the study of numerical and mathematical systems? While the literature relevant for addressing this question is not extensive, it is not my intention to summarize or review that material here. Two very good summaries or overviews of mathematical and numerical traditions in non-Western societies have appeared in recent years (Ascher 1991; Crump 1990). The reader is referred to those works, and others (e.g., Gillings 1978; Hallpike 1979; Joseph 1991; Menninger 1969; Mimica 1992; Nakayama 1978; Needham 1959; van der Waerden 1978; Zaslavsky 1990) for comparative descriptions and analyses of the nature, typologies, and evolutionary significance of the varieties of arithmetic, mathematical, numbering, and counting systems encountered worldwide.
In arriving at a statement of what I see as perhaps the most valuable contribution that anthropology can offer to mathematics, I will begin by questioning the degree to which our comparative studies of mathematical knowledge and practice in non-Western cultures have added anything fundamentally new or challenging to the prevailing, Westernoriented view of the nature and meaning of mathematics. What I mean by this is the following. The dominant view among Western philosophers and mathematicians concerning the nature of mathematical propositions and "truths" is embodied in the theory referred to as "Platonism." This theory posits that mathematics "discovers" and validates logical truths which exist in a timeless, ideal state. As the case for Platonism (as opposed to Intuitionism and Formalism) was stated by G. H. Hardy: "I believe that mathematical reality lies outside us, and that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply our notes of our observations" (cited in Barrow 1992: 261).
Since, in the Platonist view, mathematicians observe and describe a reality that exists beyond the human realm--that is, beyond the realm of experience, change, and language--this reality ought to be describable in the same terms regardless of cultural and/or linguistic differences that distinguish one society from another. In fact, the view that mathematical truths do not--cannot--differ cross-culturally is a central tenet of the Platonist vision of the reality of numbers, shapes, sets, etc. (Barrow 1992: 263; Rotman 1988: 5). It is important to stress that for Frege, whose work is the principal source of twentieth-century Platonism, numbers are a part of this non-sensible, objective reality; that is, number is neither spatial nor physical, nor (unlike ideas) subjective. Dummett (1991: 81) characterizes Frege's notion of "objectivity" as "independence from our experience, intuition and imagination and from the delineation of inner images from the memory of earlier experiences."
Now, Frege himself pointedly stressed that, in investigating the fundamental principles of mathematics, it may be of some use to examine ideas and changes of ideas that occur during the practice of mathematics. Nonetheless, he notes, "psychology should not imagine that it can contribute anything to the foundation of arithmetic" (cited in Dummett 1991: 18). The question for us at this point, obviously, is: Should anthropology imagine that it can make such a contribution? As a prelude to answering this question and, in the process, of returning to explain why I have questioned whether or not previous ethnomathematical studies have added anything fundamentally new to the corpus of mathematical truths or have challenged the prevailing Western (i.e., Platonist) view on the nature of mathematical truths, we first have to answer the question of what type(s) of contributions anthropological data might possibly make to the comparative study of mathematics.
I would suggest that there are basically two types of data that would have one or the other of the effects noted above. The first would be at the level of the mathematical truths that might be apprehended by the people of some "other," non-Western tradition. That is, if we were to find, in the mathematics of society X, a recognition of some fundamental property of numbers and their relations that had not been recognized in the West, such knowledge could indeed represent a unique contribution to world mathematics. Now, in my reading of the anthropologically oriented studies that have been published to date, it is primarily at this first level that they seek to make a contribution to the mathematical literature. While these studies have presented a host of unusual, challenging, and novel mathematical formulations from cultures around the world (such as puzzles and unusual number matrices incorporated in weavings, metalwork, architecture, and games), none has identified mathematical truths or properties and relations of numbers that have not been recognized, or are not entirely consistent with, the properties of numbers identified in Western mathematics. This is the case, I would argue, despite the fact that many of the properties and relations that have been identified in such contexts are of a quite sophisticated nature (see, for example, Ascher's analysis [1991: 74 f.] of Walpiri kinship terms and relations, which, she shows, are organized according to what mathematicians call "the dihedral group of order 8"). Nor have these comparative data challenged the ideas about the nature and location of mathematical truths as conceived of in Western Platonist philosophy.
The second way that anthropology can potentially contribute to the mathematical literature is at--what I see as--the more fundamental level of a challenge to the philosophical and logical foundations of Platonism. In the characterization of some of the principal tenets of Platonism given above, we learned that mathematical truths are conceived of as existing in a reality "out there," apart from human ideas, actions, language, and experiences. What humans may do is attempt to discover, examine, and formulate theorems identifying and validating these objective truths. But just who says--that is, how and from whom do we know--that this notion of the location of mathematical truths is, in fact, true? If we find, for instance, that the Quechua conceive of numbers as grounded in family relations and kinship statuses and numerical relations as premised on social relations, both of which are not only apprehended but experienced and talked about (in Quechua) daily, how can we deny the truth of these views providing that they do not produce numerical values and mathematical formulations that are at odds with those recognized in other mathematical traditions?
Now, the skeptical, cynical, or plain anthropologically disinterested reader might give the tag-on at the end of the last sentence the following significance: If the Quechua view does not produce a mathematics contrary to generally accepted mathematical truths based on Platonism, then why should we not disregard Quechua ontology and philosophy, since it comes loaded down with a lot of cultural baggage, such as kinship and social organization, that can potentially complicate matters, as well as vary significantly from one society to the next? However, the anthropologically interested interpretation of the above tag-on would argue that, if we do identify a philosophy that does not disturb mathematical truths as formulated in our (Western) theorems but does allow us to link mathematics, cultural values, and social organization, then why not pursue that philosophy to its ends in order to discover what we can learn. For instance, we may find that the Cartesian split between the real and the ideal, which underlies the modern philosophical formulation of Western Platonism, represents an unnecessary sacrifice in our pursuit of the truth of mathematics. That is, might we find in this way that it is, in fact, possible to effect a convergence of the interests and objectives of anthropology and those of mathematics, at least at a certain level of analysis? The level of analysis that I am referring to is that of the investigation and articulation of the ontological and philosophical foundations of the description and analysis of objects, sets, collections, and the principles and forces organizing relations among them. If mathematical philosophies can be shown to vary in a fundamental way crossculturally, then does anthropology not hold out to mathematics the offer of expanding and clarifying its philosophical grounding, thereby enriching both mathematics and anthropology?
It is in terms of the ideas laid out in the previous paragraph that I think anthropology can make a unique and significant contribution to mathematics. It is in these terms that I hope the present work will be of interest, both to anthropologists and philosophers. It is important to discuss here two works that have already begun, in different but complementary ways, the task of bringing together anthropology and mathematics. One work, Jadran Mimica's Intimations of Infinity (1992), was produced by an anthropologist; the other, Brian Rotman's "Toward a Semiotics of Mathematics" (1988), by a mathematician.
Mimica's book is a study of the numbering system and the ontology of numbers among the Iqwaye, of Papua New Guinea. The Iqwaye formulate their ideas about the nature and organization of numbers on the basis of complex symbols and metaphors relating fingers, toes, and genitalia to basic units, sets, and numerical relations (such as even and odd) of their numbering and counting systems. In addition, the organization of numbers, especially in terms of principles of hierarchy and succession found in sequences of cardinal numbers and ordinal numerals, is also formulated in terms of kinship relations and age-grading. For example, in Igwaye numerical ontology,
Humans are permanently metaphorised as fingers. But they are not metaphorical fingers in some indefinite sense. Birth order is the order in the realm of substantial consanguineal relatedness. Children emerge from the genetrix's womb as a consequence o f the procreative process engendered by genitor's semen and genetrix's blood and milk. Therefore, the metaphorical significance of the birth order suffixes [which are also assigned to fingers] is not simply--humans as fingers--but brothers and sisters as fingers which, as such, represent the hand(s) of a higher whole, their male genitor, father. (Mimica 1992: 61)
In short, Mimica's presentation and analysis of the Iqwaye numbering system and ontology of numbers inform us of an ideology of numbers and mathematics that challenges the Platonist philosophy of a separation between the reality of numerical and mathematical truths, on the one hand, and the human body, culture, and the experience of social life, on the other (for another study showing the merging of these domains of experience in Papua New Guinea, see Biersack 1982). More specifically, Mimica (1992: 107-120) uses his analysis of Iqwaye concepts of number as the basis for, and as an entry into, an analysis of the philosophical principles motivating Cantor's formulation of the concept of transfinite numbers. This discussion leads, in turn, to Mimica's critique of Western philosophical, psychological, and mathematical conceptions of number and the concept of infinity.
An especially important and informative part of Mimica's study is his decidedly scathing evaluation of certain earlier works, both by anthropologists (such as Hallpike and Lévi-Strauss) and psychologists (Piaget), who have articulated--not always on the basis of compelling ethnographic materials, such as those offered by Mimica for the Iqwaye--unfavorable evaluations and over-generalized commentaries on the evolutionary significance of "primitive" numbering systems and precise knowledge in general. For instance, in making a transition between Piaget's notion of "primitive" counting and conceptions of number as indicative of "pre-operatory" thought and the use to which Hallpike put such a notion in his evaluation of the foundations of primitive thought, Mimica notes that
there is no need to assume that the structures constitutive of the category of number in its everyday and primordial sense of 'the number of things' or simply as 'number qua counting' are somehow less real or true than those formulated in logic and mathematics. One has to understand that the latter developed and exists in relation to the former and not the other way around.... Piaget's emphasis on the formal operatory stage in human development reflects the biases of a specific trend in Western metaphysics which, having become a historically and culturally developed mode of self-understanding, is projected upon humanity as a whole, and is usable for a wholesale self-aggrandisement. (1992: 151-152; my emphasis)
I will return in Chapter 4 to a discussion of certain issues raised by Piaget's (and later Hallpike's) views on pre-operatory and analytical conceptions of numbers.
Brian Rotman's study of the semiotics of mathematics is more difficult to summarize, both because of its brevity and because of the large task it sets for itself, which is essentially that of "unmasking" the elaborate fiction propping up Platonist philosophy and language concerning the nature of mathematical truths. Rotman argues that the development of mathematics in the West (as elsewhere) must ultimately be understood as the product of historical, cultural, and linguistic forces and processes (see also Kitcher 1984; Hurford 1987: viii, 69-71, 184-185). This may seem to be obvious to some readers, but if so, such an understanding will most likely be based on philosophical or political inclinations. This is because mathematicians have consistently and purposely written themselves as individuals--as subjects and agents of history and culture--out of the mathematical operations they perform. Therefore, since mathematical manipulations are formulated discursively as though they were objective and independent of the hand that scribbles them, the only way to confidently assert that mathematics is indeed a historical and cultural activity is to expose the nature of, and any weaknesses in, the linguistic, semantic, and ideological--that is, the semiological--constructions that prop up and perpetuate the proposition that mathematical truths are objective and timeless. This is Rotman's objective in the article cited.
For instance, why, Rotman asks, should one not believe that mathematics is about some timeless, ideal world full of unchanging objects--the mathematical truths--that are independent of human language and consciousness, and that theorems express what is eternally true about these objects? One response, Rotman suggests, is to question the semiotic coherence of the notion of pre-linguistic referents that is required in such a formulation. If the relationship between signs and signifiers were what it is purported to be in the Platonist view, then language--as cultural mediation--would be inextricable from the process of referring.
This will mean that the supposedly distinct and opposing categories of reference and sense interpenetrate each other, and that the object referred to can neither be separated from nor antedate the descriptions given of it. Such a referent will be a social historical construct; ... it will be no more timeless, spaceless or subjectless than any other social artefact. (Rotman 1988: 25-26)
In addition to raising questions about the status of the signs and referents of mathematical theorems, Rotman questions the various identities and epistemological statuses engaged in the performance of a mathematical theorem. These include: the Mathematician, who imagines the imperatives of the theorem ("consider all . . . "; "let x = y. . . "); the Agent, who executes the actions within the fabricated world of the theorem; and the Person, the one whom the mathematician imagines would become convinced of the mathematician's proofs were the Agent to perform the operations demanded of him by the Mathematician. Platonism, in fact, occludes the identity and role of the Mathematician by flattening this trichotomy into an opposition between the subjective, changeable, mortal Person and the idealized, infinitary Agent, the supposed source of objective, eternal "thoughts." As Rotman notes: "it is precisely about the middle term [the Mathematician], which provides the epistemological link between the two [i.e., the Agent and the Person], that platonism is silent" (Rotman 1988: 29). This all-too-common obfuscation of the identity and activity of the Mathematician is as well one of the foci of Kitcher's critique of mathematical thinking and practice, a critique that guides his attempt to "dissolve the mysteries" which Platonism spawns by "viewing Platonism as a convenient façon de parler, a position which errs by adopting a picture of mathematical reality without recognizing the route through which the picture emerged" (cited in Hurford 1987: 184).
I do not know whether Frege was right when he suggested that psychology has nothing to contribute to mathematics; I do know, however, that anthropology ought to have something to say about practitioners in a society who construct and elide their various identities in a cultural production (like mathematics) in the manner Rotman argues is done by Western mathematicians.
Finally, we should take note of Rotman's comments concerning the nature of numbers. He argues that the mathematician's conviction to the effect that the integers are not social, cultural, historical artifacts, but natural objects, is to be explained partially by the same processes of social alienation that Marx identified whereby, in order to be bought and sold, commodities must be "fetishized." That is:
human products [e.g., commodities or numbers] frequently appear to their producers as strange, unfamiliar, and surprising; that what is created may bear no obvious or transparent markers of its human (social, cultural, historical, psychological) agency, but on the contrary can, and for the most part does, present itself as alien and prior to its creator. (Rotman 1988: 30)
This brief summary does not do justice to Rotman's masterful probing of the philosophical, linguistic, and ideological foundations of mathematical Platonism. It does, however, help us understand and appreciate (intellectually and historically) mathematical philosophies, like those of the Quechua and Iqwaye, in which the producers of the artifacts we call "numbers" do not systematically alienate those products from the social and cultural environments and processes in which they are produced. This brings up a final point regarding Rotman's work.
In his recent book Ad Infinitum (1993), Rotman further develops his critique of Platonism in the context of a larger questioning of the semiological and philosophical issues raised by the mathematical sign ( ... ), the sign used for the injunction to "carry on counting to infinity" (e.g., 1, 2, 3, . . .). Rotman's goal in this work, aside from an analysis of the relationship between Platonism and the ad infinitum principle as used in mathematics, is to establish the logic of a corporeal, non-Euclidean science of numbers (roughly: a science of large, but finite, numbers). While I find Rotman's book extremely thought-provoking, I do not think he entirely succeeds in the challenge he sets for himself, particularly with respect to non-Euclidean numbers. The reason for this, in my view, is because Rotman seems not to recognize that a corporeal science of numbers is, in fact, an "anthropology of numbers." What is called for, then, is the development of a theory of numbers and infinity informed by anthropological data, theories, and insights. Mimica, in his book Intimations of Infinity (1992), has begun such a project with his analyses of Iqwaye numbers and the perspective that we gain from these data on such Western philosophical mathematical topics as Cantor's transfinite numbers (see esp. Mimica 1992: 107-125). The more general goal of this study of Quechua numbers and arithmetic is to contribute in some small way to this larger project initiated by Mimica and Rotman.
The main point that I want to leave the reader with at this point is similar to that which Hurford articulates at the beginning of his masterful and stimulating work, Language and Number (1987: 185): "it seems in most cases unobjectionable to treat numbers (as opposed to collections) as real, but abstract, objects created through an interaction of people, language, and the world." It will be the task of this book as a whole to attempt to elucidate how the Quechua ontology of numbers and philosophy of mathematics construct--grammatically and through symbols and metaphors--a concept of numbers and procedures for their manipulation (addition, subtraction, etc.) premised on values, institutions, and practices of Quechua society and culture more generally.