The Infinite

This historical study of the infinite covers all its aspects from the mathematical to the mystical. Anyone who has ever pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of the subject. Beginning with an entertaining account of the main paradoxes of the infinite, including those of Zeno, A.W. Moore traces the history of the topic from Aristotle to Kant, Hegel, Cantor, and Wittgenstein.

Preface to the Second Edition

xi

Preface

xx

Introduction: Paradoxes of the Infinite

1

(2)

Paradoxes of the infinitely small

3

(2)

Paradoxes of the infinitely big

5

(4)

Paradoxes of the one and the many

9

(2)

Paradoxes of thought about the infinite

11

(6)

PART ONE: THE HISTORY

Early Greek Thought

17

(17)

Anaximander and to apeiron

17

(2)

The Pythagoreans

19

(4)

The Eleatics

23

(3)

Plato

26

(2)

Early Greek mathematics

28

(6)

Aristotle

34

(11)

Preliminaries

34

(2)

The problem

36

(3)

The solution: the potential infinite and the actual infinite

39

(1)

Application of the solution

40

(4)

A remaining difficulty

44

(1)

Medieval and Renaissance Thought

45

(12)

The Greek legacy: reactions and developments

45

(3)

Aquinas

48

(2)

Later developments: the mathematically infinite

50

(5)

Nicholas of Cusa. The end of the Renaissance

55

(2)

The Calculus

57

(18)

The fundamental principles of the calculus

57

(6)

A brief history of the calculus

63

(7)

Taking stock

70

(5)

The Rationalists and the Empiricists

75

(9)

The rationalists

75

(5)

The empiricists

80

(4)

Kant

84

(12)

The background: an outline of Kant's philosophy

84

(2)

The metaphysically infinite and the mathematically infinite

86

(1)

The infinitude of the world. The antinomies

87

(6)

The infinitude of reason

93

(3)

Post-Kantian Metaphysics of the Infinite

96

(14)

Hegel

96

(4)

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

(2)

Currents of thought in post-Hegelian metaphysics of the infinite III: the existentialists

105

(3)

Nietzsche

108

(2)

The Mathematics of the Infinite, and the Impact of Cantor

110

(21)

Bolzano

112

(1)

Turn-of-the-century work on the foundations of mathematics

113

(5)

The main elements of Cantor's theory. Its early reception

118

(5)

The theory of ordinals. The Burali-Forti paradox

123

(4)

Cantor's attitude to the paradoxes

127

(1)

Later development: axiomatization

128

(3)

Reactions

131

(16)

Intuitionism

131

(2)

Finitism

133

(4)

Wittgenstein

137

(4)

Current thought

141

(6)

PART TWO: INFINITY ASSESSED

Transfinite Mathematics

147

(12)

The iterative conception of a set. The paradox of the Set of all Sets

147

(4)

Ordinals as sets

151

(1)

Cardinals. Measuring infinite sets

152

(2)

The continuum hypothesis

154

(2)

Further thoughts on the infinite by addition and the infinite by division

156

(3)

The Lowenheim-Skolem Theorem

159

(13)

An introduction to the Lowenheim-Skolem theorem. Reactions and counter-reactions

159

(4)

The solution to Skolem's paradox. Scepticism and relativism

163

(2)

Scepticism and relativism rebutted

165

(2)

Meaning and understanding. The Lowenheim-Skolem theorem finally defused

167

(2)

A lingering paradox

169

(3)

Godel's Theorem

172

(14)

Introduction: the Euclidean paradigm

172

(2)

A sketch of the proof of Godel's theorem

174

(4)

Hilbert's programme

178

(2)

The human mind and computers

180

(1)

Self-consciousness

181

(1)

Meaning and understanding

182

(4)

Saying and Showing

186

(15)

The saying/showing distincton in the Tractatus

187

(3)

The very idea of a saying/showing distinction

190

(2)

Wittgenstein's early views on the infinite

192

(5)

The infinite and the ineffable

197

(4)

Infinity Assessed. The History Reassessed

201

(17)

The infinite and the ineffable: early Greek thought, medieval and Renaissance thought, post-Kantian thought

202

(1)

Aristotle and Kant: an unsuccessful compromise?

203

(2)

The empiricists: an uncompromising success?

205

(1)

The Wittgensteinian critique. Aristotle and Kant vindicated?

206

(2)

The impossibility of an infinite co-incidence, and the law of the excluded middle

208

(3)

A problem for intuitionism

211

(7)

Human Finitude

218

(16)

The nature of human finitude

218

(3)

Time

221

(1)

The infinite as an Idea of reason. The saying/showing distinction revisited

222

(4)

The poignancy of human finitude. Death

226

(4)

Being finite

230

(4)

Glossary

234

(2)

Notes

236

(14)

Bibliography

250

(11)

Index

261

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

Summary

Table of Contents

Supplemental Materials