Preface

Light travels at a speed of approximately 186,000 miles per second: 186,000 miles is over seven times the circumference of the earth. On a clear, moonless night a faint patch of light can be seen in the constellation Andromeda. This is the Andromeda nebula. It is a galaxy of about a hundred thousand million stars, each of them a sun like our own. Its light takes some two million years to reach us. It is the farthest object visible to the naked eye. Yet by comparison with other galaxies it is a close neighbour.

Who can fail to be moved by the sheer scale of it all? Who can deny the humbling and awesome effect of contemplating those vast, silent reaches?

It is the same when we think about the past. Imagine that the five-thousand-million-year history of the earth were condensed into a decade. Then dinosaurs died out between two and three months ago; about a fortnight ago proconsul apes appeared; nine hours ago humans were beginning to make tools; approximately two minutes ago Jesus Christ was born; and three seconds ago the atom bomb was exploded.

Yet we are still unmistakably in the realms of the finite. The two million light-years separating us from the Andromeda nebula, and the one hundred and fifty million years separating us from the dinosaurs: these are finite ‘bits’ of space and time. The infinite seems, not bigger, or at any rate not just bigger, but of a different kind.

This is a book about the infinite. It is also a book about the finite—about our own finitude. A sense of our own finitude is what underlies our sense of the infinite. We know that we are finite. This is not just a matter of our being tiny, and ephemeral. There is something more fundamental than that. There is the fact that we find ourselves cast into a world that is not of our own making, the fact that we find ourselves confronted with what is other than us. And if scientific investigation should reveal that all the suns, and planets, and meteors were contained in a finite region of space, and that they were all debris from some cosmic explosion that took place finitely many years ago—nay, that space and time themselves were finite—still we should have a contrasting sense of the infinitude that surrounds us:

-xx-

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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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