The Infinite / Edition 2

The Infinite / Edition 2

by A.W. Moore
     
 

ISBN-10: 0415252857

ISBN-13: 9780415252850

Pub. Date: 03/09/2001

Publisher: Taylor & Francis

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.  See more details below

Overview

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

Product Details

ISBN-13:
9780415252850
Publisher:
Taylor & Francis
Publication date:
03/09/2001
Series:
Problems of Philosophy Series
Edition description:
Revised
Pages:
296
Product dimensions:
5.50(w) x 8.50(h) x 0.67(d)

Table of Contents

Preface to the Second Edition
Preface
Introduction: Paradoxes of the Infinite1
1Paradoxes of the infinitely small3
2Paradoxes of the infinitely big5
3Paradoxes of the one and the many9
4Paradoxes of thought about the infinite11
Pt. 1The History
1Early Greek Thought17
1Anaximander and to apeiron17
2The Pythagoreans19
3The Eleatics23
4Plato26
5Early Greek mathematics28
2Aristotle34
1Preliminaries34
2The problem36
3The solution: the potential infinite and the actual infinite39
4Application of the solution40
5A remaining difficulty44
3Medieval and Renaissance Thought45
1The Greek legacy: reactions and developments45
2Aquinas48
3Later developments: the mathematically infinite50
4Nicholas of Cusa. The end of the Renaissance55
4The Calculus57
1The fundamental principles of the calculus57
2A brief history of the calculus63
3Taking stock70
5The Rationalists and the Empiricists75
1The rationalists75
2The empiricists80
6Kant84
1The background: an outline of Kant's philosophy84
2The metaphysically infinite and the mathematically infinite86
3The infinitude of the world. The antinomies87
4The infinitude of reason93
7Post-Kantian Metaphysics of the Infinite96
1Hegel96
2Currents of thought in post-Hegelian metaphysics of the infinite I: the 'metaphysically big'100
3Currents of thought in post-Hegelian metaphysics of the infinite II: the 'metaphysically small'103
4Currents of thought in post-Hegelian metaphysics of the infinite III: the existentialists105
5Nietzsche108
8The Mathematics of the Infinite, and the Impact of Cantor110
1Bolzano112
2Turn-of-the-century work on the foundations of mathematics113
3The main elements of Cantor's theory. Its early reception118
4The theory of ordinals: the Burali-Forti paradox123
5Cantor's attitude to the paradoxes127
6Later development: axiomatization128
9Reactions131
1Intuitionsim131
2Finitism133
3Wittgenstein137
4Current thought141
Pt. 2Infinity Assessed
10Transfinite Mathematics147
1The iterative conception of a set. The paradox of the Set of all Sets147
2Ordinals as sets151
3Cardinals. Measuring infinite sets152
4The continuum hypothesis154
5Further thoughts on the infinite by addition and the infinite by division156
11The Lowenheim-Skolem Theorem159
1An introduction to the Lowenheim-Skolem theorem. Reactions and counter-reactions159
2The solution to Skolem's paradox, Scepticism and relativism163
3Scepticism and relativism rebutted165
4Meaning and understanding. The Lowenheim-Skolem theorem finally defused167
5A lingering paradox169
12Godel's Theorem172
1Introduction: the Euclidean paradigm172
2A sketch of the proof of Godel's theorem174
3Hilbert's programme178
4The human mind and computers180
5Self-consciousness181
6Meaning and understanding182
13Saying and Showing186
1The saying/showing distinction in the Tractatus187
2The very idea of a saying/showing distinction190
3Wittgenstein's early views on the infinite192
4The infinite and the ineffable197
14Infinity Assessed. The History Reassessed201
1The infinite and the ineffable: early Greek thought, medieval and Renaissance thought, post-Kantian thought202
2Aristotle and Kant: an unsuccessful compromise?203
3The empiricists: an uncompromising success?205
4The Wittgensteinian critique. Aristotle and Kant vindicated?206
5The impossibility of an infinite co-incidence, and the law of the excluded middle208
6A problem for intuitionism211
15Human Finitude218
1The nature of human finitude218
2Time221
3The infinite as an Idea of reason. The saying/showing distinction revisited222
4The poignancy of human finitude. Death226
5Being finite230
Glossary234
Notes236
Bibliography250
Index261

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