The varieties of mathematical experience: ethnomathematics is a powerful tool for understanding other cultures

Mathematics Elsewhere: An Exploration of Ideas across Cultures by Marcia Ascher Princeton University Press, 2002; $24.95

The Incan quipu is an unusual object, an assemblage of slender, knotted cords tied along a thicker, main cord. The cords are dyed a variety of colors: when it's bundled up, a quipu looks like a multicolored mop; when it's spread out, it resembles a long rope necklace or a grass skirt. The quipus of the ancient Incas of Peru encoded a wide range of data about people, land, and crops for the government bureaucracy. The code was efficient and compact: the color, number, and relative spacing of the cords, and the number and type of knots tied into each cord, all held significance. A quipu might include as many as 2,000 cords, in some fifty or sixty different colors. I won't venture to estimate the storage capacity of a quipu in bits or bytes, but the system was, in its unique way, a pre-Columbian database for the Andes--an artifact of a mathematical tradition that developed entirely outside Western models.

Marcia Ascher, emerita professor of mathematics at Ithaca College in New York, and her husband Robert Ascher were instrumental in deciphering the code of the quipu (their book Code of the Quipu: A Study in Media, Mathematics, and Culture was published in 1981). Since then Marcia Ascher has focused her considerable analytic skills on a whole range of similar mathematical artifacts and concepts outside mainstream Western culture. Her latest offering, Mathematics Elsewhere: An Exploration of Ideas across Cultures, is a collection of essays on mathematical concepts in use by small-scale, traditional societies: a series of reports from an explorer "in the field" Ascher both examines the nature of the mathematics put into practice by individual societies and considers how those non-Western mathematical concepts fit into and express the ethos of the cultures that gave rise to them.

A scher's book is at once a scholarly progress report and an introduction for the curious general reader to a relatively new area of study known as ethnomathematics. The field, which has emerged in the past two decades, lies at the intersection of anthropology, education, and mathematics. For the ethnomathematician, all signs of counting, measuring, designing, patterning, modeling, sorting, or reasoning are evidence for the existence of mathematical ideas. Such ideas, whether implicit or explicit, past or present, and no matter what the cultural setting, are grist for the ethnomathematician.

Among the Iqwaye people of Papua New Guinea, for instance, fingers, toes, and the spaces between toes are tools for counting to numbers much higher than 10 or 20 or 28; instead, they form the basis of a sophisticated numbering system that can count to numbers of indefinitely large size. Among the Cayuga of New York state, the rules of a game of chance called dish, which were documented in the late nineteenth century, clearly demonstrate that the players understood the laws of probability. The assigned point values for each possible outcome of the game closely corresponded to their associated probabilities--at least as clearly as the rule in poker that four-of-a-kind beats a full house.

In the brief history of ethnomathematics, two international conferences on the topic have already been convened, the first in Granada, Spain, in 1998, and the second one last year, in Ouro Preto, Brazil. The International Study Group on Ethnomathematics claims membership from around the world. The Brazilian mathematician Ubiratan D'Ambrosio, emeritus professor of mathematics at Brazil's State University of Campinas, who is generally credited with defining the field, has called it a "research program in the historical and epistemological foundations of mathematics with pedagogical implications." That entails, in part, charting the diversity among groups of people in the realm of mathematics: the ways numbers are understood and conceived, the methods of reasoning, and the systems people adopt to model and find patterns in their own social and natural environments. D'Ambrosio's program aims at compiling a universal history of mathematics that includes contributions from every culture on the planet.

Mathematics Elsewhere fits squarely into D'Ambrosio's program, and she organizes the bounty of cases she cites around the themes of divination, time, maps, relationships, and art. From Madagascar, for instance, she describes a divination practice that has endured for four centuries, in essentially the same form, among members of the island's diverse ethnic and sociopolitical groupings. Madagascans seeking advice and guidance consult an expert, known as an ombiasy, in a divination system called sikidy. The diviner grabs a fistful of the seeds from a local tree out of a bag and makes a column of four random piles. He then removes the seeds from the piles two at a time, until each pile is reduced to either one or two seeds. He then repeats the process three more times, each time placing the new column of piles to the left of the preceding column. In the end he has sixteen piles of seeds before him, each containing either one or two seeds, arranged in four columns.