Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century
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.
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2 Cavalieris Geometry of Indivisibles and Guldins Centers of Gravity
3 Descartes Géométrie
4 The Problem of Continuity
5 Paradoxes of the Infinite
6 Leibnizs Differential Calculus and Its Opponents
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algebraic analysis Archimedes Aristotelian Aristotle arithmetic Arnauld Barrow Bernoulli Biancani Bolzano causal Cavalieri center of gravity chapter circle circumference claim Clavius Clüver concept conclude cone considered construction curves cylinder debate definition Descartes differential direct proofs discussion equal equation Euclid Euclid's Elements example finite follows Fontenelle formal cause foundational Gassendi Géométrie geometry given Gottignies Guldin Hobbes hyperbolic solid infinitely long solid infinitely small infinitesimal infinity issue Leibniz Leibnizian calculus magnitudes material cause mathematical demonstrations mathematical practice mathematicians Mersenne method of indivisibles motion nature Nieuwentijt objections Pappus paradox Pereyra perfect demonstration philosophy of mathematics Piccolomini plane figures Posterior Analytics principles problem proceeds Proclus proof by exhaustion proofs by contradiction proposition proved qu'il quadratrix quadrature Quaestio quantities ratio reason reductio reductio ad absurdum right angles Rolle segments seventeenth century square straight line superposition tangent theorem theory things Torricelli Torricelli's result triangle truth Varignon Wallis
Page 11 - We suppose ourselves to possess unqualified scientific knowledge of a thing, as opposed to knowing it in the accidental way in which the sophist knows, when we think that we know the cause on which the fact depends, as the cause of that fact and of no other, and, further, that the fact could not be other than it is.
Page 11 - I now assert is that at all events we do know by demonstration. By demonstration I mean a syllogism productive of scientific knowledge, a syllogism, that is, the grasp of which is eo ipso such knowledge. Assuming then that my thesis as to the nature of scientific knowing is correct, the...