E-Book Overview
A compact survey, at the elementary level, of some of the most important concepts of mathematics. Attention is paid to their technical features, historical development and broader philosophical significance. Each of the various branches of mathematics is discussed separately, but their interdependence is emphasised throughout. Certain topics -- such as Greek mathematics, abstract algebra, set theory, geometry and the philosophy of mathematics -- are discussed in detail. Appendices outline from scratch the proofs of two of the most celebrated limitative results of mathematics: the insolubility of the problem of doubling the cube and trisecting an arbitrary angle, and the Godel incompleteness theorems. Additional appendices contain brief accounts of smooth infinitesimal analysis -- a new approach to the use of infinitesimals in the calculus -- and of the philosophical thought of the great 20th century mathematician Hermann Weyl. Readership: Students and teachers of mathematics, science and philosophy. The greater part of the book can be read and enjoyed by anyone possessing a good high school mathematics background.
E-Book Content
JOHN L. BELL Department of Philosophy, University of Western Ontario
THE ART OF THE INTELLIGIBLE An Elementary Survey of Mathematics in its Conceptual Development
To my dear wife Mimi
The purpose of geometry is to draw us away from the sensible and the perishable to the intelligible and eternal. Plutarch
TABLE OF CONTENTS FOREWORD
page xi
ACKNOWLEDGEMENTS
xiii
CHAPTER 1
NUMERALS AND NOTATION
1
CHAPTER 2
THE MATHEMATICS OF ANCIENT GREECE
9
CHAPTER 3
THE DEVELOPMENT OF THE NUMBER CONCEPT
28
THE THEORY OF NUMBERS Perfect Numbers. Prime Numbers. Sums of Powers. Fermat’s Last Theorem. The Number π . WHAT