Awakening of Geometrical Thought in Early Culture
MEP Publications, Minneapolis
Copyright © 2003 by Paulus Gerdes, All rights reserved.
With Foreword by Dirk J. Struik
Chapter 1. Mathematicians on the origin of elementary geometrical concepts
Please review the Forward, Preface and First Chapter of the book here, or get the complete book at Biblio: Click Here!
- 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
Chapter 2. How did people learn to geometrize?
- 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
Chapter 3. Early geometrical concepts and relationships in societal activities
- 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
Chapter 4. Societal activity in the formation of ancient geometry
- 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?
Chapter 5. Conclusion: Awakening of geometrical thought
- 5.1 Methodology
- 5.2 Activity and the awakening of geometrical thought
- 5.3 New hypotheses on the history of ancient geometry
Foreword by Dirk J. Struik [1894-2000]
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.