The seventeenth century saw dramatic advances in mathematical theory and practice than any era before or since. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, analytic geometry, the geometry of indivisibles, the
arithmetic of infinites, and the calculus had been developed. Although many technical studies have been devoted to these innovations, Paolo Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics
of the period. Beginning 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 relationship between empiricist epistemology and infinitistic theorems in geometry, and the debates concerning the foundations of the Leibnizian calculus In the process Mancosu draws a sophisticated picture of the subtle dependencies between
technical development and philosophical reflection in seventeenth century mathematics.
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'Hôpital'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
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