Paulus Gerdes

MEP Publications, Minneapolis

Copyright © 2003 by Paulus Gerdes, All rights reserved.

With Foreword by Dirk J. Struik

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- 1.1 Did Geometry Have a Beginning
- 1.2 Does geometry equal deductive geometry?
- 1.3 Still in the dark: What is geometry?
- 1.4 Organizing spatial experiences

- 2.1 The birth of geometry as a science
- 2.2 An example of the influence of labor on the emergence of early geometrical notions
- 2.3 An unexplored field: Geometrical concepts in weaving

- 3.1 The concept of a right angle
- 3.2 Where did the idea of a regular hexagon arise?
- 3.3 How can one braid strands together?
- 3.4 How can one weave a button?
- 3.5 The concept of a circle
- 3.6 The idea of a regular pentagon
- 3.7 How can one weave baskets with a flat bottom?
- 3.8 The origin of some plaiting patterns and a unit for the measurement of volume

- 4.1 Did geometry have a ritual origin?
- 4.2 The possible formation of pyramid concepts
- 4.3 The “pinnacle of achievement” of mathematics in ancient Egypt
- 4.4 How could the Theorem of Pythagoras be discovered thousands of years before Pythagoras?
- 4.5 How did ancient Mesopotamians and Egyptians determine the area of a circle?

- 5.1 Methodology
- 5.2 Activity and the awakening of geometrical thought
- 5.3 New hypotheses on the history of ancient geometry

How did our mathematical concepts originate? And how did our science of mathematics come into being?

For many mathematicians the answer to the second question has been easy. Mathematics is a deductive science, and therefore originated with the Greeks, beginning with Thales and Pythagoras about 500 B.C. From them came many of our terms, even the term mathematics itself. The term geometry shows that the Greeks took many of their ideas from the Egyptians, because it referred to the annual surveying of the land after the floods of the Nile. Thus, according to this theory, Egyptians, as well as Babylonians, had mathematics, but mostly in an empirical way. The same held for China.

When, with the publication of such work as that of Neugebauer on Babylonia and Needham on China in the 1930s and later, it became clear that the mathematics of the Bronze Age empires was far more sophisticated than was believed, many mathematicians were willing to admit that the origin of mathematics as a science had to be traced back from the fourth century B.C. to the Sumerians and perhaps the Egyptians and Chinese as well.

This meant that mathematics began in the period when scribes of the Bronze Age states began to use symbols and special terms for mathematical concepts. But where did these concepts, and some of the terms already in existence, come from?

In years long past, there was a simple answer. God had bestowed on Adam in Paradise a lot of mathematical knowledge, which after his expulsion he bequeathed to his son Seth, the father of Enos. Enos, having a foreboding of the Flood, had his knowledge inscribed on two pillars, which survived the Flood. In the course of time they were seen and studied by many a traveler, among them the patriarch Abraham, who brought his knowledge to Egypt. And the Egyptians taught the Greeks.

We find such a story in Josephus, in the writings of the seventeenth-century mathematician Tacquet, and in other places. We present this story to our friends the Creationists, but prefer to search for the origin of mathematical concepts elsewhere.

We shall have to watch the gradual evolution of Homo sapiens all through the millennia of the prehistoric period for the earliest stages of tool-making, of fishing and hunting to agriculture, cattle raising, and trade—all through the Stone Age.

There has been much speculation on how the process of acquiring knowledge of mathematical concepts, of forms and number, has actually occurred.

One approach can be found in the words of one historian that “the first geometrical [and arithmetical] considerations of man . . . seem to have had their origin in simple observation, stemming from human ability to recognize physical form [and quantity], and compare shapes and sizes.” For instance: the form of sun, moon, and certain flower heads led to the concept of a circle, the shape of ropes to line and curves, further spider webs and honeycombs to more intricate forms, to triangles, spirals, solids. Comparing heaps of objects to each other led to counting, first only one, two, many, etc. This approach stresses onlooking, reflection. It is a static point of view. We can call this the attitude of homo observans.

Another approach, presented by Seidenberg, looks at religious impulses like the building of altars. As explained in this book, this is not very satisfactory. What Gerdes stresses goes beyond this and also beyond observation, and is the approach through the effects of labor. Ever since the hominoids began to walk erectly, their hands became free to make tools in the production of their livelihood—first very primitive, but gradually evolving into well-constructed artifacts. Man discovers, improves, constructs, uses all kinds of forms. The number concept grows. Man builds tents, houses; makes baskets, bags, nets, pottery, and weapons.

Through the millennia, first very slowly, then more rapidly, a great amount of knowledge of a mathematical art is obtained. This is a dynamic approach, the approach of homo laborans. It is implicit in the Marxian point of view, and we find it, for instance, in a note by Frederick Engels (1885), where he points out that the basic ideas of line, surface, angle, and number are all borrowed from reality in the interplay of Man and Nature. The objects seen in nature and in tools, in the villages and in the fields, are never exact straight lines, circles, triangles, squares. Only by activity throughout the centuries could Man be led from these forms to the abstract concepts of mathematics.

Man, changing Nature, changes himself.

We do not entirely deny the value of the other approaches; they stand in a dialectical relation to each other and the dynamic point of view. There are still other factors to take into account, for instance that of playing man, the man of games with a mathematical strand. The homo ludens.

During the many centuries the tools improved. For instance, arrowheads and hand axes become more efficient, well made; the same holds for baskets, pottery, nets. The tools became more symmetrical because of increased efficiency; and so we find, for instance, baskets taking the form of cylinders or prisms.

Incidentally, the symmetry and harmony of forms that turn out to be most efficient (many examples appear in this book) also strike us as more agreeable, beautiful. A source of the birth of aesthetics? We can refer to the book.

In order to obtain more factual information on Stone Age development, we can search for remnants of this age. There are some rods, wood or bone, found in Africa, perhaps 10,000 years old,* with carvings of parallel lines, perhaps the tally of hunting results. 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.

In other work, I have tried to build upon ideas developed in Awakening of Geometrical Thought in Early Culture and, in particular, to give concrete examples of how (reconstructed) geometrical traditions may be incorporated into mathematics education. One of the objectives of ethnomathematical research is improving the teaching of mathematics by embedding it into the cultural context of students and teachers. Such mathematics education can heighten the appreciation of the scientific knowledge inherent in culture by using this knowledge to lay the foundations for providing quicker and better access to the scientific heritage of the whole of humanity.

Awakening of Geometrical Thought in Early Culture is a briefer version in English of a book originally written in 1985 in German and Portuguese. A German-language version was published in 1990 under the title Ethnogeometrie: Kulturanthropologische Beiträge zur Genese und Didaktik der Geometrie (Bad Salzdethfurth: Verlag Franzbecker), with a preface by Professor Peter Damerow (now at the Max Planck Institute for the History of Science, Berlin), and including chapters on the didactics of geometry in the context of an African country. Shorter Portuguese-language editions have been published by Universidade Pedagógica in Mozambique under the title Cultura e o despertar do pensamento geométrico (1991) and by the Universidade Federal do Paraná (Curitiba, Brazil, 1992) under the title Sobre o despertar do pensamento geométrico, with a preface by Professor Ubiratan D’Ambrosio (Universidade Estadual de Campinas). These three editions include a chapter on the artistic elaboration of symmetry ideas emerging from social activity that has not been included in the English version. Neither the Portuguese-language editions nor the English-language edition include the original introduction on mathematical underdevelopment. The English edition includes a section on ancient Mesopotamian and Egyptian methods for the determination of the area of a circle that does not appear in the Portugueselanguage editions. The German-language edition may be consulted for more notes and an extended bibliography, Acknowledgments

I feel very honored that the late Dirk J. Struik, the “Nestor of the historians of mathematics” (and professor emeritus at the Massachusetts Institute of Technology), had been kind enough to provide the foreword to this book. His century-long work and active life have stimulated several generations of mathematicians and mathematics educators to reflect on the material and sociocultural roots of mathematics, and to deepen understanding of the philosophy and history of mathematics. His letters to me for over two decades and our more recent conversations have encouraged and challenged me to pursue my research.

I thank Erwin and Doris Marquit for their hospitality when they received me in Minneapolis, and I am grateful for their able editing of my draft translation of the book into English.

MEP Publications released in 1985 Beatrice Lumpkin’s translation of my book on the mathematical writings of Marx, Karl Marx: Arrancar o véu misterioso à matemática (Eduardo Mondlane University, Maputo, 1983) under the title Marx Demystifies Calculus.

I am pleased that MEP Publications has once again been interested in publishing one of my books, making it available also to the North American public, and contributing in this way to the debate on the questions raised in this book. Paulus Gerdes

Universidade Pedagógica Maputo, MozambiqueIn this chapter, I shall consider some widely held opinions of mathematicians about the origin and the early development of geometry; in chapters 4 and 5, I shall discuss other common ideas about early geometrical thought.

“Did geometry once have a beginning?” is a question that Julian Coolidge implicitly raises when he writes in his History of Geometrical Methods (1963), “Whatever be our definition of the Homo sapiens, he must be accorded some geometrical ideas; in fact, there would have been geometry if there had been no Homines sapientes at all” (1). Geometrical forms appear both in inanimate nature and also in organic life, and this phenomenon may be explained as a consequence of mechanical and physiological causes. Apart from this mechanical necessity—so asks Coolidge—what is the earliest example of an intentional geometrical construction? Maybe the making of a cell structure of the honey bee, “if we avoid metaphysical difficulties over the problem of the freedom of the will”? (1). No, the honeybee only optimizes, but “the ablest geometer among the animals is surely the spider” that weaves such beautiful (!) webs (2). According to Coolidge, geometry exists outside humans and their activities. Geometry is eternal. Coolidge’s history of (human?) geometrical methods begins completely arbitrarily in Mesopotamia, [i] as he is lacking any criterion to find out when or which human beings became able to observe or perceive geometrical forms in nature. [ii]

Quite often it is said that geometry started in ancient Egypt. [iii] Problems of field measurement led to a series of mostly only approximate formulas, but as Leonard Blumenthal asserts in his Modern View of Geometry, “the Egyptian surveyors were no more geometers than Adam was a zoologist when he gave names to the beasts of the field” (1961, 1). In his view, geometry emerged as a science as soon as it became deductive in ancient Greece. Even if one agrees to identify geometry with deductive geometry, another doubt arises: were not pre-Greek observations of, and reflections about, space rarely or never deductive? And does an induction not presuppose a deduction?

Also Herbert Meschkowski begins his well-known book Evolution of Mathematical Thought (1965) with Euclid’s Elements. He argues that the first childish steps were surpassed with the development of a rigorous system of mathematical proofs. Although it might be true that the ancient Egyptians and Babylonians had discovered quite a lot of theorems, nevertheless “these insights were acquired by intuition or by direct observation” (emphasis added). The transition from intuition and direct observation to the rigorous system of mathematical proofs remains without explanation and appears therefore absolute. And should not in particular this transition—if it had taken place in reality—have been one of the most important transformations in the evolution of mathematical thought?

Now this transition seems to be a (nondialectical) leap. On the other hand, would, for example, the so-called Theorem of Pythagoras have been found through mere intuition? Or would it have been the result of pure direct observation?

Raymond Wilder, the late chairman of the American Mathematical Society (1955–1956) and of the Mathematical Association of America (1965–1966), stresses in the chapter on geometry in his book Evolution of Mathematical Concepts that “instead of looking for miracles or gods or superhuman individuals” in order to understand the level of Greek geometry, one should try to find the continuous line that leads from Egyptian and Babylonian geometry to Greek geometry (1968, 88). If one agrees, then one may still raise the question whether this line started in the ancient Orient or still earlier elsewhere. Wilder’s answer remains in the dark: “There was a time” [where and when?] “when mathematics included nothing that one would place in a separate category and label geometry. . . . For at that time mathematics consisted solely of an arithmetic of whole numbers and fractions, together with an embryonic (albeit quite remarkable) algebra” (1968, 88). Would fractions have emerged earlier as the first geometrical concepts? If so, what then is geometry?

Contrary to Blumenthal and Meschkowski, the well-known geometer and didactician of mathematics Han Freudenthal evaluates in a completely different way the significance of the Greek deductive method when he notes forcefully: “Rather than as a positive element, I am inclined to view the Greek efforts to formulate and prove knowledge . . . by means of clumsy methods and governed by strict conventions, as a symptom of a terrifying dogmatism” that until today has retarded and sometimes endangered the spread and dissemination of mathematical knowledge (1982, 444). In Freudenthal’s view, geometry did not begin late in history with the formulation of definitions and theorems, but as early as the organization of the spatial experiences that led to these definitions and theorems (1978, 278).

Why, when, and where did this organizing of spatial experiences begin? Or, which human beings are able to perceive geometric forms and relationships?

So far, Eves’s position may seem empiricist: the properties that are common to different objects are of an immediately visible and perceivable character. This perception remains mostly passive. Nevertheless he notes, “Physical forms that possess an ordered character, contrasting as they do with the haphazard and unorganized shapes of most bodies, necessarily attract the attention of a reflective mind—and some elementary geometric concepts are thereby brought to light,” leading to a “subconscious geometry” [iv] (166; emphasis added). But how do people know which forms possess an ordered character? Or better still, why and how did humans necessarily learn to discover order in nature? Why does the “subconscious geometry” transform itself in ancient Egypt and Mesopotamia, as Eves asserts (167), into “scientific geometry”? [v]

These questions indicate already how Eves’s position may be dialectically sublated (aufgehoben): in order to geometrize, not only are geometrizable objects necessary, but also, to consider and perceive these objects, the ability to abstract all their other properties apart from their shape is also needed. This ability is the result of a long historical development based on experience, to paraphrase Frederick Engels. [vi]

1. Would it be by chance that for the same Coolidge the choice of geometrical axioms is completely arbitrary? See Coolidge, 1963, 423.

2. Cf. Simon: “Never and nowhere mathematics was invented. . . .
Mathematical ideas are not at all restricted to Man. . . . When the spider produces its web, it uses its particularly built foot as a compass; the bees have
solved a difficult maximum problem when they construct their hexagonal cells”
(1973, xiii).

3. Cf. Ball: “Geometry is supposed to have had its origin in land surveying. . . . [S]ome methods of land-surveying must have been practiced from very
early times, but the universal tradition of antiquity asserted that the origin of
geometry was to be sought in Egypt” (1960, 5).

4. In what sense “subconscious”? “For want of a better name,” Eves calls
this knowledge of elementary geometric concepts “subconscious geometry.” He
notes, “This subconscious geometry was employed by very early man in the
making of decorative ornaments and patterns, and it is probably quite correct to
say that early art did much to prepare the way for later geometric development.
The evolution of subconscious geometry in little children is well known and
easily observed” (166).

5 Cf. Cantor: “Also geometrical concepts . . . must have emerged early in
history. Objects and figures limited by straight lines and curves must have
attracted the eye of Man, as soon as he started not only to see, but to look around
himself” (1922, 1:15). What, however, could have caused this changeover from
“seeing” to “looking around himself”?

6. “Counting requires not only objects that can be counted, but also the ability to exclude all properties of the objects considered except their number—and
this ability is the product of a long historical development based on experience”
(Engels 1987a, 36–37).