By Marcia Ascher. Pacific Grove, CA: Brooks/Cole, 1991. ix + 203 pages.
Years ago Bob Seger sang the song “Feel Like a Number”; credit card numbers, Social Security Numbers, and PINS are now a part of daily life for most in the United States. The census, opinion polls, and aggregate demographic data used for marketing purposes further underscore the importance of numbers and their manipulation in modern society. Debates about “quantitative” versus “qualitative” judgments influence topics as diverse as pedagogy and the physical sciences. Digital computers processing strings of 1s and 0s might as well serve as a symbol of the age.
The ontological status of numbers, however, has been in flux for at least a hundred years—are they a sort of Platonic ideal? are they a human psychological universal? are they merely handy logical constructs? Kant claimed that mathematics was synthetic and thus led to new knowledge, yet at the same time was a priori. Having taught math for a number of years, Kant was no mathematical neophyte, yet what Kant understood as mathematical was limited by the Western mathematical tradition to which he was exposed. Indeed, many histories of mathematics treat the history of mathematics as a tradition that has progressed over the ages, building pieces of knowledge upon pieces of knowledge. Such a project contains contributions not only by “modern” cultures, but also by “classical” cultures that left us written records, be they Greek, Indian, or Arab.
Two questions then arise: do all cultures formulate mathematical ideas in the same (or similar) ways, and what can we know about how mathematical ideas are used and expressed in cultures that don’t leave behind written records?
A slender, handsome, and topically illustrated book, Marcia Ascher’s Ethnomathematics deals with these questions and touches upon a variety of mathematical topics, providing varying degrees of insight into each. Ascher’s book deals with six primary areas of inquiry and concludes with a more theoretically-oriented discussion of the present and future of ethnomathematics. The author draws most of her empirical data from a handful of North/South American, African, and Oceanic indigenous cultures, supplying sufficient ethnographic data so as to contextualize her mathematical arguments.
The first chapter, dealing the the concept of numbers and their representations, is the broadest in its treatment of a “multicultural” consideration of mathematical ideas; the approach is comparative, yet also provides an in-depth “case study” dealing with Incan quipu, or colored, knotted cotton cords used for encoding numerical data. Subsequent chapters read more like detached and narrowly-focused essays; in each the emphasis is on one or two case studies (or data sets) and a specific mathematical idea: (2) graphs, (3) logical reasoning, (4) probability and counting, (5) spacial modeling, and (6) symmetry.
Contributions to mathematical knowledge by Indian and Arab cultures have been dealt with in other books and articles; Ascher does not attempt to pursue a similar goal for less well-known cultures. Indeed, she implies that such a project would be pointless; mathematics as we know it is an institutionalized form of knowledge, a specific discourse practiced by mathematicians. Instead, the author looks for traces of “mathematical ideas,” thus the topics of the individual chapters, which demonstrate how certain cultural practices (be they ritual, artistic, pragmatic, or recreational endeavors) embody certain types of mathematical ideas.
Individual examples and pieces of trivia—such as the wide-spread occurrence of a specific logic puzzle across continents and cultures—prove to be the most interesting aspects of the book. The author’s exhaustive treatment of the group theory of kinship relations, the possible results in a game of strategy, and the algebra of symmetry in certain geometric designs are admirable examples of analysis, yet come across as dry and pedantic after a while—they seem too focused toward specialists interested in the data under consideration.
Ethnomathematics (implicitly) argues for a relativity of mathematical expression yet an universalism of mathematical ideas; only in the discussion of the conceptualization of space and time by different cultures (the chapter most devoted to how people think rather than to cultural artifacts) does the book provide a negative answer to the question do all cultures formulate mathematical ideas in the same (or similar) ways? In the concluding chapter Ascher discusses how her project drew upon anthropology, ethnology, archaeology, and linguistics as much as mathematics, and perhaps it is for this reason that many of the “mathematical ideas” under consideration don’t feel strictly mathematical in nature. This is not necessarily a weakness, for it does provide an interesting mix of detailed and general discussion, making the book accessible to a wide variety of readers; at the same time, one must wonder why certain other types of “mathematical” knowledge were not considered, specifically aspects of music across cultures.
Furthermore, the tie-ins with linguistics and cognitive science seem to merit a more detailed discussion of linguistic relativism (i.e. the Sapir-Whorf hypothesis, etc.); likewise a greater comparative emphasis rather than one so focused on “case studies” would lead to a work that seemed less like a compendium or collection of articles and more like a book.
Ascher’s Ethnomathematics is a valuable contribution both to the history and philosophy of mathematics as well as to cultural studies in general. The author refrains from excessive claims, engages in meticulous analysis, and supplies wonderful examples and documentation. Its shortcomings leave the reader unfulfilled, yet at the same time point toward possibilities for further research and future books.