Synopses & Reviews
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.
Review
"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
Description
Includes bibliographical references (p. 249-265) and index.
Table of Contents
1. Philosophy of Mathematics and Mathematical Practice in the Early Seventeenth Century 1.1. The Quaestio de Certitudine Mathematicarum
1.2. The Quaestio in the Seventeenth Century
1.3. The Quaestio and Mathematical Practice
2. Cavalieri's Geometry of Indivisibles and Guldin's Centers of Gravity
2.1. Magnitudes, Ratios, and the Method of Exhaustion
2.2. Cavalieri's Two Methods of Indivisibles
2.3. Guldin's Objections to Cavalieri's Geometry of Indivisibles
2.4. Guldin's Centrobaryca and Cavalieri's Objections
3. Descartes' Géométrie
3.1. Descartes' Géométrie
3.2. The Algebraization of Mathematics
4. The Problem of Continuity
4.1. Motion and Genetic Definitions
4.2. The "Casual" Theories in Arnauld and Bolzano
4.3. Proofs by Contradiction from Kant to the Present
5. Paradoxes of the Infinite
5.1. Indivisibles and Infinitely Small Quantities
5.2. The Infinitely Large
6. Leibniz's Differential Calculus and Its Opponents
6.1. Leibniz's Nova Methodus and L'Hopital's Alalyse des Infiniment Petits
6.2. Early Debates with Clüver and Nieuwentijt
6.3. The Foundational Debate in the Paris Academy of Sciences
Appendix: Giuseppe Biancani's De Mathematicarum Natura, Translated by Gyula Klima
Notes
References
Index