Investigations into the Applicability of Geometry

Philosophical reflection about the sciences has persistently given rise to worries that mathematics, while true of its own special objects, is inapplicable to nature or to the physical world. Focusing on the case of geometry, and drawing on the histories of philosophy and science, I articulate a series of challenges to the applicability of geometry based on the general idea that geometry fails to fit nature. This series of challenges then plays two major roles in the dissertation: it clarifies the ways in which the applicability of geometry poses a problem for two major early modern natural philosophers, viz., Galileo and Leibniz, and it allows for the investigation of the relationship between geometric structures and nature by means of an investigation of the applicability of geometry. I begin with the challenge pressed by some thinkers in the Aristotelian tradition that the results which geometry proves about its objects are false when interpreted as assertions about objects in nature. Despite the durable influence of this challenge and the Aristotelian theory of science which inspires it, I argue that Aristotle himself did not oppose the use of geometry in empirical inquiry, but rather offered an account of it. I then examine how Galileo takes on the objection that geometric results are false if understood as claims about nature in his _Dialogue Concerning the Two Chief World Systems_. On my interpretation, Galileo argues the objection should be recast as the claim that there are no geometric points, lines, or surfaces in nature. This is an objection both Galileo and Leibniz take seriously in developing their new mathematical physics, although I argue that Galileo and Leibniz react to the objection very differently: Galileo rejects the objection as false and grounded on a misconception of the relationship between geometry and nature, whereas Leibniz grants the truth of the objection and tries to show that it is not damaging for the project of mathematical physics. In defending the applicability of geometry, both Galileo and Leibniz help to develop and employ notions of approximation in the sciences. Their work highlights an important presupposition of approximations: that there must be determinate discrepancies between an object being approximated and its approximation. I conclude the dissertation with an argument that actual applications of geometry in empirical science require that there be determinate discrepancies between geometric structures and nature.