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

by Gary Urton

Unraveling all the mysteries of the khipu ―the knotted string device used by the Inka to record both statistical data and narrative accounts of myths, histories, and genealogies―will require an understanding of how number values and relations may have been used to encode information on social, familial, and political relationships and structures. This is the problem Gary Urton tackles in his pathfinding study of the origin, meaning, and significance of numbers and the philosophical principles underlying the practice of arithmetic among Quechua-speaking peoples of the Andes. Based on fieldwork in communities around Sucre, in south-central Bolivia, Urton argues that the origin and meaning of numbers were and are conceived of by Quechua-speaking peoples in ways similar to their ideas about, and formulations of, gender, age, and social relations. He also demonstrates that their practice of arithmetic is based on a well-articulated body of philosophical principles and values that reflects a continuous attempt to maintain balance, harmony, and equilibrium in the material, social, and moral spheres of community life.

Genres: History, Language, Mathematics

267 pages, Paperback

First published January 1, 1997

About the author

Gary Urton is Dumbarton Oaks Professor of Pre-Columbian Studies in the Department of Anthropology at Harvard University.

Reviews

[Tuuli Ahlholm · Rating 4 out of 5]

A fascinating, and in many ways, successful "ethnomathematical" (though Urton has some qualms about the term) survey into both contemporary and ancient number-culture of the Quechua-speaking peoples of the Andes. Urton puts to good use both his vast expertise in Incan history (quipu-accounting in particular), and months of fieldwork in Andean farming and weaving communities, where he has gained intimate knowledge of Quechuan vocabulary and social practices surrounding counting. "The Social Life of Numbers" is at its best when Urton uses this unique knowledge combo to reveal us the long scope, such as when he shows the perseverance of a distinct, local "philosophy" of numbers which the Andeans have preserved as remarkably constant for over 500 years, through the turmoils of colonial invasions and governments.

Urton's prose is smooth, conversational even, and especially the last chapters with their colourful examples from both Inca history and modern Quechua communities make for engaging reading, whereas the first chapters (after the introduction) with their detailed discussion of grammar and individual number-words etc. will probably be read in detail only by hardcore enthusiasts of linguistics etc..

I unfortunately doubt that Urton succeeds in the most ambitious goal that he sets down for himself in his Introduction, that is, in convincing hardcore mathematicians and philosophers that anthropological surveys of non-Western number-cultures can contribute to philosophy of mathematics. Partly because the book is very light on mathematics (which Urton recognises), partly because it does not engage deeply enough with any whatever contemporary theory his research could influence (Platonism?), and he proves his arguments by illustration: the style of the book will appeal to anthropologists and humanists, not so much scientists. But, also because his main findings do not, in my opinion, pose any threat or otherwise illuminate modern Platonist theories of numbers?

It is true that Urton convincingly shows that Quechua culture appears to have a distinct (what Urton calls) "ontology and philosophy" of numbers: whereas numbers and their manipulation are thought as abstract, objective and value-free in the West, the Quechua vocabulary and metaphor for numbering is intimately linked with e.g. social relationship (kinship, coupling, hierarchy...) and moral values. E.g. whereas Western mathematicians use numbers as a "neutral" way to represent reality, the Quetchua counters tend to see counting as a "moral and ethical" task of "rectifying" balances in the universe. Some things that belong together should never be separated into units from their "natural group-state" through counting, such as herd animals; and pre-Columbian Inca tribute lists betray the desire to record and organise society so that people and resources exist/manifest only as "ideal", decimal (10, 20, 50 etc.) number groups.

However, is this culture-specific "ontology and philosophy" of numbers that Urton describes, not best titled only as (as Urton sometimes does) "experience of numbers" or "social life of numbers"? Philosophers of mathematics tend to be concerned primarily with questions of what numbers metaphysically *are*, what do they *refer to*; Platonists, who continue to represent the most popular school of thought, would say that to abstract objects that exist independently of us. Regardless of the semiotic sign (numeral, number-word, quipu-knot, five fingers...) that a human in any culture could use to communicate the number "5", it always refers to this same Platonic "5" - which is why any culture-specific mathematical system is ultimately always translatable to all other languages/systems. The Quechua "5" might carry tones and associations that the English "5" does not (of perfection, completeness, "younger descendants" etc. according to Urton), just like all words in different languages tend to have different social histories and thus slightly different messages - but that does change that their common nominator is the same reference to to the same Platonic "5"? Why should it be of concern to Platonists that not all human cultures treat mathematics as universal and value-free?

[Robert Murphy · Rating 3 out of 5]

While it was pleasant to read of someone trying to break free of Platonism and the dehistoricized concept of true intrinsic to Western mathematics, this book was actually just the warm up for Urton's next book on quipus, the knot work of the Inca civilization. These artifacts are pre-literate creations that were something between family trees and full-blown narratives. Navigating this ethno-mathematical semantic domain is the actual goal of this book. It was very interesting, but not as philosophically precise as I would have wished.

[Maddie Walker · Rating 2 out of 5]

This book was alright and I appreciated its focus on the more historical and cultural sides but I had a hard time getting through it for math class.