Paulus Gerdes

MEP Publications, Minneapolis

Copyright © 2003 by Paulus Gerdes, All rights reserved.

With Foreword by Dirk J. Struik

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?
#### 4. Organizing spatial experiences

#### 5. Who is able to perceive geometric forms and relationships?

Howard Eves, in his paper “The History of Geometry,”
answers the question, “Which human beings are able to perceive geometric forms and relationships?” in the same way as
Coolidge: “All.” However, he presents other reasons: “The first
geometrical considerations of Man . . . seem to have had their
origin in simple observations stemming from human ability to
recognize physical form and to compare shapes and sizes” (1969,
165). Here he presupposes the ability to recognize and compare
forms as a natural, a once-and-for-all given quality of human
beings. Consequently, it turns out to be relatively easy to explain
the origin of early geometrical concepts. For instance, the outline
of the sun and the moon, the shape of the rainbow, and the seed
heads of many flowers, etc. led to the conception of circles. A
thrown stone describes a parabola; an unstretched cord hangs in
a catenary curve; a wound-up cord lies in a spiral; spider webs
illustrate regular polygons, etc. (168).

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).