Unification in Descartes' Philosophy of Mathematics

Liu, Cathay

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March 22, 2019

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Liu, Cathay
- Affiliation: College of Arts and Sciences, Department of Philosophy

Abstract

I argue that Descartes unified mathematics in a rather striking and strong way. He held that there was a single shared subject matter of all mathematics, that the objects of mathematics are ontologically identical to extended substance. The most entrenched conception that I argue against in my dissertation is the view that for Descartes mathematical objects--particularly numbers--have some sort of ontological status independent of material extension. It is easier to see that geometrical objects such as triangles depend on geometrical extension or reduce to extension but similar claims about the dependence on extension for numbers or other abstract, arithmetic, or algebraic objects doesn't not seem as immediately plausible. I argue that the best way to understand Descartes' views about mathematics is to see that Descartes viewed numbers as epistemically and metaphysically dependent on extension. I give three arguments for this view. The first argument I offer is based on ontological problems with mathematical realism as an interpretation of Descartes. Many commentators have thought Descartes is best understood as having some form of realism about the universals in our mathematical cognitions. Problems for these realist readings arise when the need to ontologically locate the universals conflicts with central Cartesian metaphysical doctrines. The alternative is a nominalism about mathematical objects. The second argument I offer is found in Descartes' concept of numbers. There I show that in order for numbers to be conceived at all, a prior idea of extension (specifically its divisibility) must be contained in the idea. Without the idea of the nature of extension, there can be no idea numbers. Knowledge of extension is epistemically prior to and necessary for mathematical knowledge. The final argument concerns Descartes' unification of algebra and geometry. I show that the best account of Descartes' mathematics is given using this way of distinguishing the metaphysical and epistemic unity and priority between algebra and geometry. Descartes' development of analytical geometry exhibits the dependence of number on extension through the dependence of algebra on geometry and the priority of geometrical magnitudes to discrete algebraic multitudes.

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