The Infinite / Edition 2

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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.

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Editorial Reviews

From the Publisher
'Moore's book points to deep and unresolved issues in the philosophy of mathematics, and even deeper issues in general philosophy ... It deserves serious study by both mathematicians and philosophers.' - Thomas Tymoczko, Philosophia Mathematica

'[Moore's treatment of] the problems with which the history of thought about the infinite confronts us today ... shows that questions concerning the nature and existence of the infinte are still very much alive ... The importance of [his] book lies ... in its highly stimulating account of the nature of infinity and its bold defence of finitism.' - W.L.Craig, International Philosophical Quarterly

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Product Details

  • ISBN-13: 9780415252850
  • Publisher: Taylor & Francis
  • Publication date: 3/9/2001
  • Series: Problems of Philosophy Series
  • Edition description: Revised
  • Edition number: 2
  • Pages: 296
  • Product dimensions: 5.50 (w) x 8.50 (h) x 0.67 (d)

Table of Contents

Preface to the Second Edition
Introduction: Paradoxes of the Infinite 1
1 Paradoxes of the infinitely small 3
2 Paradoxes of the infinitely big 5
3 Paradoxes of the one and the many 9
4 Paradoxes of thought about the infinite 11
Pt. 1 The History
1 Early Greek Thought 17
1 Anaximander and to apeiron 17
2 The Pythagoreans 19
3 The Eleatics 23
4 Plato 26
5 Early Greek mathematics 28
2 Aristotle 34
1 Preliminaries 34
2 The problem 36
3 The solution: the potential infinite and the actual infinite 39
4 Application of the solution 40
5 A remaining difficulty 44
3 Medieval and Renaissance Thought 45
1 The Greek legacy: reactions and developments 45
2 Aquinas 48
3 Later developments: the mathematically infinite 50
4 Nicholas of Cusa. The end of the Renaissance 55
4 The Calculus 57
1 The fundamental principles of the calculus 57
2 A brief history of the calculus 63
3 Taking stock 70
5 The Rationalists and the Empiricists 75
1 The rationalists 75
2 The empiricists 80
6 Kant 84
1 The background: an outline of Kant's philosophy 84
2 The metaphysically infinite and the mathematically infinite 86
3 The infinitude of the world. The antinomies 87
4 The infinitude of reason 93
7 Post-Kantian Metaphysics of the Infinite 96
1 Hegel 96
2 Currents of thought in post-Hegelian metaphysics of the infinite I: the 'metaphysically big' 100
3 Currents of thought in post-Hegelian metaphysics of the infinite II: the 'metaphysically small' 103
4 Currents of thought in post-Hegelian metaphysics of the infinite III: the existentialists 105
5 Nietzsche 108
8 The Mathematics of the Infinite, and the Impact of Cantor 110
1 Bolzano 112
2 Turn-of-the-century work on the foundations of mathematics 113
3 The main elements of Cantor's theory. Its early reception 118
4 The theory of ordinals: the Burali-Forti paradox 123
5 Cantor's attitude to the paradoxes 127
6 Later development: axiomatization 128
9 Reactions 131
1 Intuitionsim 131
2 Finitism 133
3 Wittgenstein 137
4 Current thought 141
Pt. 2 Infinity Assessed
10 Transfinite Mathematics 147
1 The iterative conception of a set. The paradox of the Set of all Sets 147
2 Ordinals as sets 151
3 Cardinals. Measuring infinite sets 152
4 The continuum hypothesis 154
5 Further thoughts on the infinite by addition and the infinite by division 156
11 The Lowenheim-Skolem Theorem 159
1 An introduction to the Lowenheim-Skolem theorem. Reactions and counter-reactions 159
2 The solution to Skolem's paradox, Scepticism and relativism 163
3 Scepticism and relativism rebutted 165
4 Meaning and understanding. The Lowenheim-Skolem theorem finally defused 167
5 A lingering paradox 169
12 Godel's Theorem 172
1 Introduction: the Euclidean paradigm 172
2 A sketch of the proof of Godel's theorem 174
3 Hilbert's programme 178
4 The human mind and computers 180
5 Self-consciousness 181
6 Meaning and understanding 182
13 Saying and Showing 186
1 The saying/showing distinction in the Tractatus 187
2 The very idea of a saying/showing distinction 190
3 Wittgenstein's early views on the infinite 192
4 The infinite and the ineffable 197
14 Infinity Assessed. The History Reassessed 201
1 The infinite and the ineffable: early Greek thought, medieval and Renaissance thought, post-Kantian thought 202
2 Aristotle and Kant: an unsuccessful compromise? 203
3 The empiricists: an uncompromising success? 205
4 The Wittgensteinian critique. Aristotle and Kant vindicated? 206
5 The impossibility of an infinite co-incidence, and the law of the excluded middle 208
6 A problem for intuitionism 211
15 Human Finitude 218
1 The nature of human finitude 218
2 Time 221
3 The infinite as an Idea of reason. The saying/showing distinction revisited 222
4 The poignancy of human finitude. Death 226
5 Being finite 230
Glossary 234
Notes 236
Bibliography 250
Index 261
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