CHAPTER 4

The Calculus 1

And what are…fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? (George Berkeley)


1 The fundamental principles of the calculus

There is a cluster of mathematical problems which at first sight seem quite disparate but which come together in a remarkable way. They have a history going back at least as far as Eudoxus and Archimedes, with their method of exhaustion, and continuing to the present day. This history is a vital component strand within the broader history of the infinite. I turn to it now, because some of the most significant breakthroughs that go to make it up were made in the seventeenth century.

What are the problems to which I refer? They fall into two groups. Those in the first group all concern curves. They include: how to determine the area of a curved figure (a figure bounded by curves); how to determine the slope of the tangent to a curve at a point; and suchlike. The problems in the second group concern the idea of the continuous variation of one quantity with respect to another. (For example, if an object moves straight from A to B, accelerating all the time, then, during that period, both its distance from A and its speed increase continuously with respect to time. ) These problems include: how to analyze such continuous variation; how to determine its rate; and suchlike.

All of these problems come together in that branch of mathematics known as the calculus. The first step towards a grasp of the calculus is to understand what is meant by a graph. Suppose two lines, or axes, at right angles to each other, represent the different possible values of two quantities. Then a point on the same plane as them can represent a pair of these values, as determined by its distance, in the relevant direction, from each axis (see Figure 4.1). The two values are known as the point’s co-ordinates. A graph can be thought of as a set of such points. (For

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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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