Douglas Bertrand Marshall

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.

In her sole philosophical treatise, The Principles of the Most Ancient and Modern Philosophy, Anne Conway (1631-1679) offers a demonstration of the proposition that, in addition to God and creatures, there is a being whose essence is the medium between God’s essence and creatures’ essence. We offer an interpretation of Conway’s demonstration that reveals its dependence on a rational principle ('PME'): if beings with extreme natures are united, then they are united by means of a being whose nature is the medium between the extremes. We also assess the extent to which Conway offers a justification for her metaphysics by demonstrating her claims from principles known by the understanding. Conway’s philosophical demonstrations are suggestive of a rationalist position on which her metaphysics may be proved from a small number of propositions established independent of experience. However, we ultimately argue that Conway’s philosophical methodology is not rationalist in this sense: for Conway is willing to take our daily experience as the evidence for a metaphysical principle, and also to argue for her philosophy on the basis of a consensus among authoritative texts.

Leibniz holds that nothing in nature strictly corresponds to any geometric curve or surface. Yet on Leibniz’s view, physicists are usually able to ignore any such lack of correspondence and to investigate nature using geometric representations. The primary goal of this essay is to elucidate Leibniz’s explanation of how physicists are able to investigate nature geometrically, focussing on two of his claims: (i) there can be things in nature which approximate geometric objects to within any given margin of error; (ii) the truths of geometry state laws by which the phenomena of nature are governed. A corollary of Leibniz’s explanation is that physical bodies do have boundaries with which geometric surfaces can be compared to very high levels of precision. I argue that the existence of these physical boundaries is mind-independent to such an extent as to pose a significant challenge to idealist interpretations of Leibniz.

Just as mathematics helps us to represent and reason about the natural world, in its internal applications one branch of mathematics helps us to represent and reason about the subject matter of another. Recognition of the close analogy between internal and external applications of mathematics can help resolve two persistent philosophical puzzles concerning its applicability: a platonist puzzle arising from the abstractness of mathematical objects; and an empiricist puzzle arising from mathematical propositions’ lack of empirical factual content. In order to see how this is the case, we will examine what it is to apply mathematics internally and describe examples.