CHAPTER 2

Aristotle 1

In general, the infinite exists through one thing being taken after another, what is taken being always finite, but ever other and other.

(Aristotle)


1 Preliminaries

Aristotle (384 BC-322 BC) is a touch-stone for this whole enquiry. Born in Stagira, he lived most of his life in Athens, where he studied under Plato in the Academy that he (Plato) had founded. He was a remarkable polymath. He made major contributions to logic, metaphysics, the natural sciences (above all biology), psychology, ethics, politics, and literary criticism; and some of these disciplines he can even be said to have founded. His entry into our particular drama in many respects marks the end of the prologue and serves to inaugurate the action proper. Many of the concepts that have shaped and informed subsequent discussion, indeed much of what has actually been discussed, originated with Aristotle.

He himself began with the views of his predecessors. He noted in particular one recurring and dominant theme in what they had been saying: whatever is infinite is ipso facto a ‘principle’, that is to say something fundamental from which other things are derived or in terms of which other things are to be explained. Otherwise, it would be derivative and thus limited. This is why, despite profound differences in their views, earlier thinkers had all held the infinite to be ungenerable, indestructible, and eternal. It is also why they had tended to refrain from attributing any particular or determinate qualities to it. These too would have counted as limiting it, leaving it open to explanation in more fundamental terms.

Such remarks apply both to those thinkers who regarded the infinite as an entity in its own right (paradigmatically Anaximander, for whom it was actually a substance) and to those who thought of infinitude in a more modern vein as a property that other entities possessed. Among the latter were some philosophers not discussed above. Anaxagoras, for example, had held that all the different substances in the world were originally

-34-

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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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