CHAPTER 1

Early Greek Thought

It is incumbent upon the person who treats of nature to discuss the infinite and to enquire whether there is such a thing or not, and, if there is, what it is…[And] all who have touched on this kind of science in a way worth considering have formulated views about the infinite.

(Aristotle)


1

Anaximander and to apeiron1

The Greek word ‘peras’ is usually translated as ‘limit’ or ‘bound’. To apeiron’ denotes that which has no peras, the unlimited or unbounded: the infinite.

To apeiron made its first significant appearance in early Greek thought with Anaximander of Miletus (c. 610 BC to shortly after 546 BC). Its role was very different from that which it tends to play in modern thought. It was introduced in response to what was then (and has remained) a basic intellectual challenge: to identify the stuff of which all things are made. What, as the Greeks would have put it, is the ‘principle’ of all things? Thales had earlier proposed that it is water. Perhaps he had been impressed by the natural processes whereby the sea evaporates under the influence of the sun, then forms clouds, dissolves in the form of rain, and soaks into the earth, moistening the food by which living things are nourished. Still, why single out water in this way as anything more than just one of the many forms that basic stuff could take? There was something arbitrary about this. So Anaximander’s proposal was that the primal substance of which all things are made is to apeiron. This he conceived as something neutral, the boundless, imperishable, ultimate source of all that is.

But it was not just that. It was also something divine, something with a deeper significance. Given the processes whereby substances change into one another, the losses and compensating gains, it made good metaphysical sense to suppose that there was an underlying changeless substratum. But for Anaximander it made as much ethical sense. He saw, in

-17-

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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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