Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

By Paolo Mancosu | Go to book overview

5

Paradoxes of the Infinite

In the introduction to Paradoxien des Unendlichen Bernard Bolzano remarked that most paradoxical results found in mathematics rest on the concept of the infinite. 1 The seventeenth century provided many of the paradoxes of the infinite that constitute the topic of Bolzano's treatise. If one restricts attention only to those paradoxes that generated foundational discussions, two classes emerge from the plethora of surprising results provided by seventeenth-century mathematicians and philosophers. First are the paradoxes having to do with what Bolzano would have called the general theory of magnitudes, especially the composition of continuous quantities. These will be analyzed with reference to the theory of indivisibles ( Cavalieri, Galileo, Torricelli, Tacquet, and Leibniz). Second are the paradoxes relating to the theory of space. These will be investigated with reference to Torricelli's cubature of an infinitely long solid and the varied philosophical reactions generated by this result, or plane versions of it. The use of infinity in the Leibnizian calculus is the topic of chapter 6.

The term paradox will be taken in its original meaning -- a statement that contradicts a previous belief -- and in the meaning of an outright logical contradiction. Particularly useful to us, paradoxes often reveal the implicit assumptions and tacit conventions that form the basis of a determinate mathematical or philosophical viewpoint. Their study allows us to appreciate the mental obstacles our predecessors had to overcome before they could make further progress in the development of philosophy and mathematics.

The goal of the chapter is to highlight both the mathematical and the philosophical relevance of the topic of infinity for seventeenth-century philosophy of mathematics. In section 5.1 the analysis of the technical objections to the geometry of indivisibles will show that many mathematicians and philosophers endeavored to justify Cavalieri's theory in terms of homogeneous quantities, also in response to the threat of paradoxes. At the same time, the early exchange between Galileo and Cavalieri on the possibility of a geometry of indivisibles reveals how problematic was the idea of a theory of infinite quantities in the early seventeenth century. Section 5.2 moves on to the topic of the infinitely large, a topic that has been largely ignored in the literature in favor of the debates on the infinitely small. But it is in connection with the infinitely large that I believe the most important foundational reflections are

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Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century
Table of contents

Table of contents

  • Title Page iii
  • Preface vii
  • Contents *
  • Introduction 3
  • 1 - Philosophy of Mathematics and Mathematical Practice in the Early Seventeenth Century 8
  • 2 - Cavalieri's Geometry of Indivisibles and Guldin's Centers of Gravity 34
  • 3 - Descartes' Géométrie 65
  • 4 - The Problem of Continuity 92
  • 5 - Paradoxes of the Infinite 118
  • 6 - Leibniz's Differential Calculus and Its Opponents 150
  • Appendix 178
  • Notes 213
  • References 249
  • Index 267
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