Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

By Paolo Mancosu | Go to book overview

2

Cavalieri's Geometry of Indivisibles and
Guldin's Centers of Gravity

The passage from classical Greek mathematics to seventeenth-century mathematics is marked by two dramatic events: the widespread use of infinitary techniques in geometry and the algebraization of geometry. At first independent, they show a close interaction later in the century. In this chapter I shall begin the analysis of the relevance of the use of infinitistic techniques for the philosophy of mathematics, and in later chapters I shall deal with the issue of analytic techniques.

It is often said that the Greeks avoided recourse to the infinite in their mathematics. However, one should be careful here. The remark is correct only if referred to the formal expositions found in Greek mathematical texts. Indeed, it is quite well known that heuristically the Greeks appealed to infinitistic techniques, as witnessed by Archimedes' Method, a short treatise rediscovered in 1906 which contains, in the form of a letter from Archimedes to Eratosthenes, an explanation of Archimedes' heuristic strategies for the discovery of theorems in geometry concerning areas, volumes, and centers of gravity. Archimedes explains in his Method how to exploit in geometrical investigations mechanical considerations and indivisibilist techniques, where the latter denomination refers to the idea of considering a plane figure (or a solid) as being "constituted" by lines (or planes).

Starting from the Method, W. Knorr in his "Before and after Cavalieri: the method of indivisibles in ancient geometry" has given intriguing evidence to support the claim that the use of indivisibilist techniques in ancient times must have been a widespread heuristic tool for the geometers.

The indivisibilist technique in geometry was not an isolated and idiosyncratic whim of Archimedes, published in only a single work the Method; but rather . . . it was a technique already familiar to Archimedes from his own technical sources, whose tradition could still be drawn upon by later commentators like Hero and Theon. 1

Thus, in which sense is the widespread use of infinitesimalist techniques such a novelty in the seventeenth century? There are three orders of reason that

-34-

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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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