CHAPTER 6

Kant 1

When I look up at Thy heavens, the work of Thy fingers, the moon and the stars set in their place by Thee, what is man, that Thou shouldst remember him, mortal man that Thou shouldst care for him? Yet Thou hast made him little less than a god, crowning him with glory and honour. (Psalm VIII)


1

The background: an outline of Kant’s philosophy2

It is often said that the three greatest philosophers of all time were Plato, Aristotle, and Kant. The third of these now makes his mark on our enquiry.

Immanuel Kant (1724-1804) was born in Königsberg in Prussia and lived there all his life. Much of his philosophy was devoted to taking rival systems of thought and rooting out the inveterate assumptions common to them. On the one hand this enabled him to show that some of the fundamental points of controversy between them were ill-conceived. On the other hand it enabled him to salvage and to reconcile some of their apparently irreconcilable insights. The latter was something that he sought to do above all in the case of the conflict between traditional Christian morality and Newton’s (by now) well-established mechanics. Christian morality seemed to make no sense without human freedom, but there was no room for human freedom, it seemed, in Newton’s world of inexorable mechanical laws. In attempting a reconciliation here, Kant developed a philosophical system of breathtaking depth and power. Indeed it enabled him at the same time and in much the same way to arbitrate between the rationalists and the empiricists. Since they have just been the focus of our attention, let us broach Kant’s system in those terms.

Kant wanted to accept, with the rationalists, that we had substantial a priori knowledge. Yet, in line with the empiricists, he did not see how we could know anything substantial about what was out there, independent of us, without letting it impinge on us through experience. He resolved the apparent conflict here by arguing that the a priori knowledge in question was not after all knowledge about what was out there, independent of us.

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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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