University of Cambridge > > Departmental Seminars in History and Philosophy of Science > Picturability and the mathematical ideals of knowledge: Leibniz versus Newton

Picturability and the mathematical ideals of knowledge: Leibniz versus Newton

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There had been a widely-held view in the seventeenth century that the Fall had dulled Adam’s senses, and that the use of artificial aids might effect the restoration of their pre-lapsarian acuity. But while there had also been a widespread view that reason too had been impaired in the Fall, this was generally diagnosed in terms of the passions triumphing over reason, the remedy being to control the passions. I offer an interpretation of Leibniz’s account of the calculus as holding that artificial aids can correct reason itself.

Newton developed a version of infinitesimal calculus in the early 1670s but abandoned it on the grounds that it used procedures that could not be justified. They were black boxes: one put in the premisses and generated the right results, but had no grasp on what was going on in the middle. In fact, both Newton and Leibniz agreed that infinitesimal calculus required justification in terms of limit procedures, which were geometrical and open to inspection at every stage. The difference was that Newton believed that this meant that any procedure using infinitesimal calculus had to be translated into geometrical limit procedures, whereas Leibniz believed that it was only the general technique that had to be justified in terms of limit procedures, and that, once this was done, it was not required that one justify each and every operation employing infinitesimals in this way. Leibniz’s approach is not driven by pragmatic concerns, however, but rather by a view that the calculus extends human capacities in new ways into new areas: it goes beyond our natural faculties and hence we cannot expect our natural faculties to be able to legitimate it. This raises the general question of whether we can employ procedures of enquiry whose workings transcend our faculties.

This talk is part of the Departmental Seminars in History and Philosophy of Science series.

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