CHAPTER 3

Medieval and Renaissance Thought 1

A finite intellect…cannot by means of comparison reach the absolute truth of things. Being by nature indivisible, truth excludes ‘more’ or ‘less’, so that nothing but truth itself can be the exact measure of truth …. In consequence, our intellect, which is not the truth, never grasps the truth with such precision that it could not be comprehended with infinitely greater precision. (Nicholas of Cusa)


1 The Greek legacy: reactions and developments

Two things dominated medieval and Renaissance philosophy in general, and philosophical thought about the infinite in particular: the legacy of the Greeks; and religion. Religion in this context virtually meant Christianity, but not exclusively so. And the legacy of the Greeks was eventually to become, more than anything else, the legacy of Aristotle—albeit tempered and informed by strands of Platonism.

Before that, however, there was something of a reaction against Aristotle and a reversion to the ideas of Plato—albeit tempered and informed by strands of Aristotelianism. Plotinus (205-270) played a major role here. He was probably born in Upper Egypt and may have been a Hellenized Egyptian. He was not a medieval thinker. He stood, rather, at the end of antiquity. But this is an apt point at which to consider him.

He drew heavily on the ideas of Plato, founding what became known as Neoplatonism. This was to have a profound influence on Christian thought. First and foremost he wanted to resurrect (something like) the very radical appearance/reality distinction that had been integral to Plato, and before him Parmenides and the Eleatics. He believed in an utterly transcendent realm of being that underlies and sustains, yet is quite separate from, all that we directly experience. It was hostility to this very belief that had been at the roots of Aristotle’s naturalistic espousal of the mathematically infinite, and his repudiation of the metaphysically infinite. By readopting the belief, Plotinus was in a position to upturn much of what Aristotle had argued for and to rehabilitate the metaphysically infinite. He

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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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