Introduction: Paradoxes of the Infinite

The infinite has always stirred the emotions of mankind more deeply than any other question; the infinite has stimulated and fertilized reason as few other ideas have; but also the infinite, more than any other notion, is in need of clarification. (David Hilbert)

The aim of this book is to arrive at an understanding of the infinite—via an understanding of how it has been understood by other thinkers in the west over the past two and a half millennia.

It would be inappropriate to try to begin with a crisp, substantive, uncontroversial definition of the infinite. There are two special reasons for this. First, one of the central issues concerning the infinite is whether it can be defined. Many have felt that it cannot; for if we try to define the infinite as that which is thus and so, we fall foul of the fact that being thus and so is already a way of being limited or conditioned. (It is as if the infinite cannot, by definition, be defined. This is one of the paradoxes that we shall be looking at later in this introduction. ) Despite this, there have been many attempts throughout the history of thought about the infinite to define it, or at least to explain why it cannot be defined by those persuaded that it cannot. And these supply the second reason why it would be inappropriate, in a book where historical impartiality at the outset is crucial, to try to begin with a preferred definition: these attempts have revealed a striking lack of consensus. It is not just that different thinkers have focused on different aspects of the infinite. Again and again we find new accounts of the infinite being presented in the firm conviction that what had been handed down as orthodoxy was just wrong.

Two clusters of concepts nevertheless dominate, and much of the dialectic in the history of the topic has taken the form of oscillation between them. Within the first cluster we find: boundlessness; endlessness; unlimitedness; immeasurability; eternity; that which is such that, given any determinate part of it, there is always more to come; that which is greater than any assignable quantity. Within the second cluster we find: completeness; wholeness; unity; universality; absoluteness; perfection; self-sufficiency;

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The Infinite
Table of contents

Table of contents

  • Title Page iii
  • Contents vii
  • Preface to the Second Edition xi
  • Preface xx
  • Introduction: Paradoxes of the Infinite 1
  • Part One - The History 15
  • Chapter 1 - Early Greek Thought 17
  • Chapter 2 - Aristotle 34
  • Chapter 3 - Medieval and Renaissance Thought 45
  • Chapter 4 - The Calculus 57
  • Chapter 5 - The Rationalists and the Empiricists 75
  • Chapter 6 - Kant 84
  • Chapter 7 - Post-Kantian Metaphysics of the Infinite 96
  • Chapter 8 - The Mathematics of the Infinite, and the Impact of Cantor 110
  • Chapter 9 - Reactions 131
  • Part Two - Infinity Assessed 145
  • Chapter 10 - Transfinite Mathematics 147
  • Chapter 11 - The Löwenheim-Skolem Theorem 159
  • Chapter 12 - Gödel's Theorem 172
  • Chapter 13 - Saying and Showing 186
  • Chapter 14 - Infinity Assessed. the History Reassessed 201
  • Chapter 15 - Human Finitude 218
  • Glossary 234
  • Bibliography 250
  • Index 261
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