Descartes held that practicing mathematics was important for developing the mental faculties necessary for science and a virtuous life. Otherwise, he maintained that the proper uses of mathematics were extremely limited. This article discusses his reasons which include a theory of education, the metaphysics of matter, and a psychologistic theory of deductive reasoning. It is argued that these reasons cohere with his system of philosophy.

How does Newton approach the challenge of mechanizing gravity and, more broadly, natural philosophy? By adopting the simple machine tradition’s mathematical approach to a system’s co-varying parameters of change, he retains natural philosophy’s traditional goal while specifying it in a novel way as the search for impressed forces. He accordingly understands the physical world as a divinely created machine possessing intrinsically mathematical features, and mathematical methods as capable of identifying its real features. The gravitational force’s physical cause remains an outstanding (...) problem, however, as evidenced by Newton’s onetime reference to active principles as the “genuine principles of the mechanical philosophy”. (shrink)

We argue that a conflict between two conceptions of “quantity of matter” employed in a corollary to proposition 6 of Book III of the Principia illustrates a deeper conflict between Newton’s view of the nature of extended bodies and the concept of mass appropriate for the theoretical framework of the Principia. We trace Newton’s failure to recognize the conflict to the fact that he allowed for the justification of natural philosophical claims by two types of a posteriori, empiricist methodologies. Newton's (...) thoughts on these methodologies demonstrate that his natural philosophy continued to develop after the publication of the first edition of Principia and that De Grav should be understood as an early, and not necessarily representative, text. (shrink)

In the history of the scientific revolution, Descartes is often considered as the mechanical philosopher par excellence, and opposed as such to the founder of mechanical science, that is to say, Galileo: this cliché is not without foundation, but it must not make us forget that Descartes was himself a practitioner of mechanical science. In the article "Cartesian Mechanics" I detail the meaning and reach of "mechanics" in the Cartesian corpus, and do so in three steps. 1. I begin by (...) explaining the genesis of the thesis which states that there is no difference between the physical and the mechanical; this thesis is so famous that it is often imagined that it is constituted in Descartes's first writings. But we can in fact show that the first works of Descartes and Beeckman arise from a "physico-mathematical" practice which does not necessarily imply a complete reworking of traditional physics, even if it might have favored the Cartesian ambition for a physics as certain as geometry; it was only in the late 1630s that Descartes began to systematically affirm the identity of physics and mechanics, or of "rules of motion", of "laws of Nature" and of "laws of mechanics". I then analyze the consequences of this affirmation. 2. In the second part of this article I provide a step-by-step commentary of the response proposed by Descartes in a letter to Mersenne in July 1638 to the question raised by Beaugrand's Geostatice (...) dissertatio mathematica (1636), that is to say the question of whether a body weighs more when it is farther from the center of Earth than when it is close. Compared with other texts of the Cartesian corpus, this sample of mechanics has been little analyzed (to my knowledge, the only analyses are those of P. Duhem, P. Costabel, A. Gabbey and D. Garber). This text is nonetheless extremely interesting: Descartes responds to the geostatic question in turn in terms of physics, then mathematics; even more interesting, the examination of the details of his procedures allows us to confront Cartesian statics to other writings, whether those of Guidobaldo del Monte, Galileo, Stevin, Mersenne or Roberval. 3. In the third and final section, I analyze the reasons that lead Descartes to exclude velocity from his statics, I examine the difficulties that his explanation of gravity created for statics, and conclude with the confrontation between mechanical philosophy and mechanical science in the case of Descartes. (shrink)