Paulus Gerdes

MEP Publications, Minneapolis

Copyright © 2003 by Paulus Gerdes, All rights reserved.

With Foreword by Dirk J. Struik

Then there are the famed cave paintings in Spain and *Struik refers to a bone found at Ishango (Congo). Dating estimates of this bone now range from 8,000 to 20,000 B.C. A still older bone with twenty-nine clearly marked notches was found iSn a cave in the Lebombo Mountains on the border between South Africa and Swaziland. This bone has been dated at approximately 35,000 B.C. (see Gerdes 1994).

France, also very ancient, which show mathematical traces, if only by the fact that they are two-dimensional projections of solid bodies, hence exercises in mapping. We can also study arrowheads and other artifacts.

Much richer information can be obtained by studying the culture of present-day indigenous peoples still living in Stone Age conditions or at any rate retaining customs and memories of older times before Western influence set in. Their culture may contain many strains millennia old. Though we have some accounts of mathematical lore by travelers or missionaries, such as some reports on the counting of American Indians or the games of Polynesians dating to the nineteenth and early twentieth century, a systematic study of these cultures from a mathematical point of view only took place in the years after World War II, and has led to a novel field called ethnomathematics. This term was proposed by Professor Ubiratan D’Ambrosio of Brazil, who has studied, among other things, Latin American indigenous cultures.

One of the reasons for this interest has been political— anticolonialism. Starting with the impetus given by the Russian Revolution, the struggle against colonialism has led after the Second World War to the dissolution of the old colonial empires. The new politically independent states had to cope with the devastating influence of the colonial regime on the old native cultures, especially in Africa, Polynesia, and Micronesia. It has been a struggle to recoup native identities, if possible. The search for mathematical concepts inherent in these native cultures is part of this search for identity.

Pioneering here has been the work of Claudia Zaslavsky; in her book Africa Counts (1973), she surveys the mathematical (or “protomathematical,” if you prefer) ideas in the cultures of peoples living south of the Sahara. She finds them in their counting, architecture, ornamentation, games, riddles, taboos, concepts of time, weights and measures, even magic squares.

Since the appearance of her book, many studies in this field have been published. We mention only Marcia Ascher’s book Ethnomathematics (1991), which gives examples from many parts of the third world, including even kinship relations. As to Africa, here the main investigations have been led by Paulus

Gerdes and his collaborators. In this book he deals with the geometrical and ornamental aspect of native mathematics. We learn in this book how mathematical concepts were involved in the construction of baskets, mats, bags, from reeds, leaves, and other parts of plants, as well as in the construction of homes and pyramids. In the course of the centuries, the artifacts and the methods of construction were improved, and so the concepts of triangle, hexagon, circle, and rectangle could be developed until they led to the abstractions of the science of mathematics.

Gerdes shows how, in the course of time, properties of these geometrical figures could be discovered, including the Theorem of Pythagoras. It has always been a mystery how knowledge of this theorem appears in Babylonia around 2000 B.C.—where did it come from? This look at the construction, use, and improvement of artifacts can also lead to other properties. Is it possible that Greek knowledge of the volume of a pyramid was developed out of the way fruit (say apples) is piled up in the markets and could this also have led to Pascal’s triangle? Gerdes believes that the knowledge of the volume of the truncated pyramid could also have been the result of sophisticated methods born out of practices.

There is still another side of ethnomathematical study. It is its importance for education. If pupils from the villages (and ghettos) come to school and enter modern classrooms, will not the indigenous mathematics in their upbringing facilitate their acquisition of the modern mathematics of the classroom? This use of the “intuitive” native mathematics may well be of help in easing the mathematical angst we hear so much about.

This brings ethnomathematics in as a factor in the widespread discussion on the improvement of mathematical instruction in our schools. His ideas can have wide application. To the literature and the discussion of this subject, other writings of Professor Gerdes have also made their contribution. Dirk J. Struik

Belmont, Massachusetts March 1998Most standard histories of mathematics ignore completely or pay little attention to the existence of mathematical traditions outside the so-called West. Geometry is presented as something very special, born among the “rational Greeks.” Before them, at most some practical rules would have been known. Most standard textbooks ignore geometrical thinking in daily life, in particular in the daily life of the peoples of the “third world,” of the “South.”

Strong protests have arisen in recent decades against the ignorance of mathematics outside the “West” and “North,” especially from the ethnomathematical movement. Claudia Zaslavsky’s Africa Counts: Number and Pattern in African Culture (1999, first edition 1973), Ubiratan D’Ambrosio’s Sociocultural Bases for Mathematics Education (1985) and Etnomatemática (1990), Alan Bishop’s Mathematical Enculturation (1988), Marcia and Robert Ascher’s The Code of the Quipu: A Study in Media, Mathematics and Culture (1981), Marcia Ascher’s Ethnomathematics: A Multicultural View of Mathematical Ideas (1991), Michael Closs’s Native American Mathematics (1986), George Gheverghese Joseph’s The Crest of the Peacock: Non-European Roots of Mathematics (1991), and Arthur B. Powell and Marilyn Frankenstein’s Ethnomathematics: Challenging Eurocentrism in Mathematics Education (1997) are extremely important in demystifying the dominant views about mathematics and in contributing to an alternative picture of mathematics as a panhuman activity.

In this perspective, Awakening of Geometrical Thought in Early Culture considers early geometrical thinking, both as embedded in various social activities surviving colonization in the life of the peoples of the “South,” and in early history. Chapter 1 discusses briefly some standard views of the origin of geometrical concepts. Chapter 2 analyzes alternative views of geometry stimulated by the philosophical reflections of Frederick Engels and presents a wholly unexplored field of research: geometrical thinking as embedded in mat- and basket-weaving. Chapter 3, constituting the principal part of the book, analyzes the emergence of a series of early geometrical concepts and relationships in socially important activities. Questions such as these are considered: Where could the concept of a right angle have come from? Where did the idea of a regular hexagon arise? How is it possible to determine the rectangular base of a building? Chapter 4 presents, on the basis of the ideas and the methodology developed in the previous chapter, a series of hypotheses on the possible role of social activity in the development of geometry in ancient Mesopotamia and Egypt. The last chapter offers some general ideas on the awakening of geometrical thought based on the analysis in this book.