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All human knowledge thus begins with intuitions, proceeds thence to concepts, and ends with ideas.
Quoted in Hilbert's Foundations of Geometry.
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Science and Mathematics in Ancient Greek Culture
C.J. Tuplin and T.E Rhill (eds.), Science and Mathematics in Ancient Greek Culture. New York: Oxford University Press, 2002, xvi + 379 pages + illustrations. $ 85.00. ISBN 0198152485,800-451-7556,http://www.oup.com/us .
This book is a collection of essays on Greek science and mathematics; however, only a few essays consider mathematics. The initial essay in this collection offers the interested student a perspective for viewing much of secondary mathematics and the thinking underlying it. “Introduction: Greek Science in Context” by T.E. Rhill, provides a common perspective for the remaining essays. The author reminds us, “For the Greeks, a philosopher was a person who loved wisdom, and wisdom comes in many forms. But whatever they did or believe, they were in broad agreement that reasoned argument and rational debate were the tools by which one could discover or create knowledge.” For the historically curious, every chapter is thoughtful if not provocative. All discussions provide useful information for teachers and students alike.
Anyone studying/teaching spherical geometry will be delighted to learn, in J.L. Berggren’s essay “Ptolemy’s Map as an Introduction to Ancient Science,” that an ancient map can offer “an entry-point onto some of the most common and scientific ideas of the Greeks and Romans,” for instance, “the Greek geometrical model of the cosmos and its companion theory of spherics , the measurement of the circumference of the earth (see a Lesson Plan for this in Convergence), the table of chords in a circle and some of its uses, the solution of right triangles.” All of these concepts may be found in the study of ancient maps.
R. Netz’s contribution, “Greek Mathematicians: A Group Picture,” uses the tools of statistical analysis to create a plausible picture of who was active in Greek mathematics, male or female, provided they lived from classical times to 500 A.D. Four mathematical instruments, the sundial, equinoctial rings, the dioptra, and the catapult, are discussed in three sequential essays by R. Hanna, L. Taub, and S. Cuomo. C.M. Taisbak gives the best explanation of the origin of prime numbers that I have ever read, as he discourses on “The Fundamental Theorem of Arithmetic,” not to ignore the jewels he reveals in his discussion of the multiplication table!
Aristotle is not commonly thought of as a mathematician. As we understand the word “mathematician” today, it is a product of the late Middle Ages, not of ancient Greece. Even Euclid, in his time, was called merely a ‘”scholar.” Aristotle used mathematics in his study of natural science and the processes and substances of nature. He envisioned the fundamental principles of arithmetic and geometry as real limits “on what the world can be like.” E. Hussey, in his essay, “Aristotle and Mathematics,” urges a cautious reading of Aristotle's works.
Members of Math Clubs and better students in high school math classes will profit from all that has been described above. Unfortunately, the price of the book probably limits its purchase to school libraries.
Barnabas Hughes, O.F.M., Professor Emeritus, California State University, Northridge