The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development

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Springer Science & Business Media, Dec 6, 2012 - Science - 250 pages
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 Gödel 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.
 

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Contents

THE MATHEMATICS OF ANCIENT GREECE
10
viii
19
THE DEVELOPMENT OF THE NUMBER CONCEPT
29
THE EVOLUTION OF ALGEBRA I
53
THE EVOLUTION OF ALGEBRA II
72
THE EVOLUTION OF ALGEBRA III
89
THE DEVELOPMENT OF GEOMETRY I 1 11
111
Higher Dimensional Spaces
120
THE DEVELOPMENT OF GEOMETRY II
130
THE CALCULUS AND MATHEMATICAL ANALYSIS
153
THE CONTINUOUS AND THE DISCRETE
174
THE PHILOSOPHY OF MATHEMATICS
193
THE INSOLUBILITY OF SOME GEOMETRIC
210
THE CALCULUS IN SMOOTH INFINITESIMAL
224
INDEX OF NAMES
239
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