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There is a tradition of opposition between adherents of induction and of deduction. In my view it would be just as sensible for the two ends of a worm to quarrel.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
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Awakening of Geometrical Thought in Early Culture
Awakening of Geometrical Thought in Early Culture, Paulus Gerdes, 2003. xvi + 171 pp., $49.50 hardbound. ISBN 0-939656-75-X. MEP Publications, University of Minnesota, 116 Church Street SE, Minneapolis MN 55455-0112. http://umn.edu/home/marqu002
For more than twenty years, Paulus Gerdes has been investigating activities of everyday life in non-Western cultures, looking for mathematical thought and seeking ways that traditional work can lead to mathematical concepts. He has published his work in scattered publications in several languages in Europe and Africa. This book represents a consolidation of several past works with new material, published in a more accessible English-language form.
His central argument is that geometrical ideas are not discovered as mathematics in the Platonic sense; nor are they the product of passive observation of the geometric shapes in nature. Rather, they come from active design and the making of tools and other everyday products. Often the material structure used in these activities and the necessity for maximum strength or volume or other properties dictates the shapes and geometrical relationships that emerge. From these physical requirements, he suggests early people came to recognize not only circles and straight lines, but also such specialized shapes as hexagons and properties such as the Pythagorean relationship.
The biggest part of the book demonstrates the technical details of how the geometrical notions emerge from the activities. In the process, he teaches the reader specifics about basket-weaving, pottery-making, and other traditional activities. For teachers, many of the examples could be developed class activities; they are not written as activities, but are well described. This can offer students more demonstrations of the contributions of world historical cultures to mathematical thinking.
Lawrence Shirley, Professor of Mathematics, Towson State University