Was Plato a mathematical Platonist?: An inquiry into the nature of the intermediates, their effects on Plato's metaphysics, and Plato's resulting mathematical ontology

Most discussions of Plato's ontology refer to his dualist ontology of Forms and the physical objects that exemplify them. Aristotle suggests that in Mathematics, Plato countenanced a third type of entity. Specifically, in Metaphysics Mu and Nu, one of the many objects of Aristotle's attacks is Plato's purported belief in the existence of intermediate mathematical entities, which serve as the objects of Mathematics. Assuming that Plato in fact thought that these mathematical objects exist, the general aim of this dissertation is to find out what they are and how they relate to the other objects in Plato's ontology. More specifically, the aim of this dissertation is to uncover the epistemological status of the Intermediates and whether Plato would have been justified in positing the Intermediates given the metaphysical requirements of Mathematics.;My dissertation starts with a detailed account of how Plato's metaphysics would look if he actually did posit the existence of these mathematical objects. This analysis includes an argument for how the existence of intermediate entities would demand that the natures of the two other categories in Plato's ontology become more precise. Specifically, I argue that if Plato posited intermediate mathematical objects, he could sidestep the Third Man regress. Unless we want to accuse Plato of a redundancy, Forms could not be self-predicating if the Intermediates perfectly exemplify them.;I next turn to the issue of why Plato may have wanted to posit intermediate mathematical objects. I offer several justifications for why Plato may have needed both the arithmetic and geometric Intermediates given the prevailing views on Arithmetic and Geometry when Plato was writing. However, I also show that Aristotle overlooked a significant problem with the nature of the arithmetic intermediates.;Finally, I turn to the epistemological issues surrounding the mathematical Intermediates. I present an argument to show that these mathematical intermediate objects could be the objects of knowledge, like the Forms. Contrary to this argument, I consider evidence from the Republic's passages on the Divided Line to show that the Intermediates could only be the objects of thought, not of knowledge.