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The Social Life of Numbers: A Quechua Ontology of Numbers and Philosophy of Arithmetic

2000; Duke University Press; Volume: 80; Issue: 3 Linguagem: Inglês

10.1215/00182168-80-3-578

1527-1900

Enrique Mayer,

Indigenous Cultures and Socio-Education

Ñawpaq (first): This is an extraordinary book. It is easily readable even for the non-mathematically inclined and non-Andeanists. It deals with issues of why one counts, what is counted, and how arithmetic operations are used in social life. Drawings, diagrams, and tables allow for the easy perusal of the material, and the profusion of examples facilitate understanding. The material comes from wonderful ethnographic fieldwork in southern Bolivia, conducted with exquisite linguistic skills by anthropologist Gary Urton paired up with Primitivo Nina Llanos, a professor of Quechua at the University of Sucre.Iskayñaken (second): It is also a deeply philosophical work, delving into the underlying logic and principles involved in Quechua ways with numbers. It argues that Andean civilization scores high on numeracy, but low on literacy (lacking a writing system). Discussing ordinal numbers, Urton introduces the prototypical model of an ordinal sequence, derived from the social relations of a mother and her offspring, mother being first, and the children succeeding, one by one, in a hierarchical relationship of seniority up to and usually including five. Ordinal numbers in the Quechua world should form pairs, and therefore odd numbers are complicated: they are “bad” omens in prognostication, and they socially stand alone as incomplete. Pairs, however, are conjunctions of two opposite but complementary things intimately bound together as something that can be one. Thus, for example, yanantin is a husband-wife team, a unit that expresses part-to-whole relations rather than shared attributes that put things in a class. A member of a pair that is torn asunder is said to be single, alone, incomplete, and odd. Pairing elements are therefore fundamental elements of ordination, and an aspect of the dual organization that is so fundamental to the Andes.Kimsañaken (third): Urton shows that enumerating, that is, ranking and ordering are more important in Andean social life than counting. Thus, first and last, junior and senior, progenitor and offspring place individuals in an ordered world, and give meaning to cardinal numbers, which, by themselves are not considered to have an independent existence. There are also things one should not count, notably animals in a herd, because numbers tend to separate things, to isolate one unit from another. And this is not a good thing to do when applied to a reproductive group such as a herd. Animals are enumerated by checking off the individual names of each from a mental list. Enumeration is dependent on the social categories in which things are put into. In the decimal system of the Quechua, a unit of ten is made up of a set of two fives, the first five (odd) united or paired with its even unit of five (six to ten). Enumeration is therefore also always concerned with a sense of completeness, balance and symmetry that is an ideal to be achieved.Tawañaquen (fourth): Arithmetic operations of summation, subtraction, multiplication and division serve social principles of “rectification,” a term Urton introduces and discusses at length. When circumstances produce imbalances, incomplete sets or disharmony, action must be taken to restore balance, and the arithmetic procedures of subtracting from one set, adding them to the other, re-dividing units, and so on, are aimed at “rectifying” the imbalance. They are, apart from the mental operations that manipulate the numbers, a set of social and political processes that require authority to carry out. For example, in Inca times, excess men from one decimal unit (pachaca) would be re-affiliated with a unit that was deficient. The change had to be recorded in someone’s list. And this is what the khipu, the knotted strings, used in Inca administration, do so well. Based on the pioneering work of Marcia and Robert Asher, Urton’s analysis explicitly points to the underlying social principles of khipu-keeping: the enumeration of ranked items in paired categories that can be changed as required by untying and retying knots on a string, such that there is a column of items in an ordered sequence, with the earliest ones being tied furthest away from the place where it joins the main chord. All the members recorded in a khipu are placed in it, each one in its place and category. That one can arrive at a total summation is incidental to its real purpose.Qhipañaquen (last): Rectification is modeled as yapa, a word that comes from the world of trading. Yapa is the little gift added on to a quantity already agreed upon in bartering or sale, meant to even out any potential imbalance in the exchange. This is the only term from the world of trade that Urton discuses. Regrettably, the social life of numbers in the market place is given insufficient treatment. He does so explicitly, because, when he discusses European logic of numbers introduced by the Spanish, Urton sees the two worlds as separate and non-interfering. Which is not quite the case, as when he gives an example of expert weavers (skilled in the art of counting) who are seemingly unable to count money. In the world of exchange things transferred from one person to another need to be measured. This adds a whole domain to counting, accounting, enumerating and arithmetic operations not adequately discussed in this book. Although contemporary Quechua use Spanish units for weight, volume, surface area and coinage, the way these units are used, often respond to an Andean logic that should have been brought under consideration in this book. A fanega, for example, is a bushel of maize but it is also the amount of land that can be seeded with a fanega of seed which is indeterminate depending on slope, soil, and seeding density. This is, as I see it, an application of the art of rectification; for example, barter, where units to be exchanged are often paired. Economic valuation is also an aspect of ordination, as when a clay pot to be bartered is considered equivalent in value to its volume in fine grains, but two volumes for lumpy items such as potatoes. Urton follows too closely the teachings of our respective mentors, R. T. Zuidema and John V. Murra, who tended to de-emphasize the role of markets and exchange of goods in the Andes, in prehispanic, colonial, and contemporary contexts. It would have been interesting to have Urton’s excellent talents also applied to the social life of numbers in this sphere, a rectification, I hope he, myself, and other students of the Andes will soon bring about.