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. (shrink)

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 innature 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. (shrink)

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 innature 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. (shrink)