Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

by Paolo Mancosu

Overview

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.

Product Details

ISBN-13:	9780195084634
Publisher:	Oxford University Press
Publication date:	01/18/1996
Pages:	288
Product dimensions:	9.30(w) x 6.30(h) x 1.00(d)
Lexile:	1500L (what's this?)

About the Author

University of California at Berkeley

Table of Contents

1.	Philosophy of Mathematics and Mathematical Practice in the Early Seventeenth Century	8
1.1	The Quaestio de Certitudine Mathematicarum	10
1.2	The Quaestio in the Seventeenth Century	15
1.3	The Quaestio and Mathematical Practice	24
2.	Cavalieri's Geometry of Indivisibles and Guldin's Centers of Gravity	34
2.1	Magnitudes, Ratios, and the Method of Exhaustion	35
2.2	Cavalieri's Two Methods of Indivisibles	38
2.3	Guldin's Objections to Cavalieri's Geometry of Indivisibles	50
2.4	Guldin's Centrobaryca and Cavalieri's Objections	56
3.	Descartes' Geometrie	65
3.1	Descartes' Geometrie	65
3.2	The Algebraization of Mathematics	84
4.	The Problem of Continuity	92
4.1	Motion and Genetic Definitions	94
4.2	The "Causal" Theories in Arnauld and Bolzano	100
4.3	Proofs by Contradiction from Kant to the Present	105
5.	Paradoxes of the Infinite	118
5.1	Indivisibles and Infinitely Small Quantities	119
5.2	The Infinitely Large	129
6.	Leibniz's Differential Calculus and Its Opponents	150
6.1	Leibniz's Nova Methodus and L'Hopital's Analyse des Infiniment Petits	151
6.2	Early Debates with Cluver and Nieuwentijt	156
6.3	The Foundational Debate in the Paris Academy of Sciences	165
Appendix	Giuseppe Biancani's De Mathematicarum Natura	178
	Notes	213
	References	249
	Index	267

Editorial Reviews

"Students of the history of mathematics and philosophers of mathematics will find this a valuable addition to the literature."—Choice

"Mancosu's book shows philosophical acumen as well as high technical competence—and it makes good reading even as it explores abstruse notions or involved technicalities. For historians of early modern mathematics, it is essential reading."—Isis

"Mancosu tells the story well and is good at bringing out significant points"—International Philosophical Quarterly

"Mancosu's book shows philosophical acumen as well as high technical competence—and it makes good reading even as it explores abstruse notions or involved technicalities. For historians of early modern mathematics, it is essential reading."—Isis

"This is a very carefully researched and documented analysis of the rich relationship between philosophy of mathematics and mathematical practice during the 17th century."—Mathematical Reviews

From the Publisher