Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

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Oxford University Press, Jan 18, 1996 - Mathematics - 288 pages
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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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Contents

1 Philosophy of Mathematics and Mathematical Practice in the Early Seventeenth Century
8
2 Cavalieris Geometry of Indivisibles and Guldins Centers of Gravity
34
3 Descartes Géométrie
65
4 The Problem of Continuity
92
5 Paradoxes of the Infinite
118
6 Leibnizs Differential Calculus and Its Opponents
150
Giuseppe Biancanis De Mathematicarum Natura
178
Notes
213
References
249
Index
267
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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 12 - Of all the figures the most scientific is the first. Thus, it is the vehicle of the demonstrations of all the mathematical sciences, such as arithmetic, geometry, and optics, and practically of all sciences that investigate causes : for...
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...

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