Continuity and Mathematical Ontology in Aristotle

In this paper I argue that Aristotle's understanding of mathematical continuity constrains the mathematical ontology he can consistently hold. On my reading, Aristotle can only be a mathematical abstractionist of a certain sort. To show this, I first present an analysis of Aristotle's notion of continuity by bringing together texts from his Metaphysica and Physica, to show that continuity is, for Aristotle, a certain kind of per se unity, and that upon this rests his distinction between continuity and contiguity. Next I argue briefly that Aristotle intends for his discussion of continuity to apply to pure mathematical objects such as lines and figures, as well as to extended bodies. I show that this leads him to a difficulty, for it does not at first appear that the distinction between continuity and contiguity can be preserved for abstract mathematicals. Finally, I present a solution according to which Aristotle's understanding of continuity can only be saved if he holds a certain kind of mathematical ontology.

The Cartesians

This chapter examines the crisis of perception as it figures in the work of four of Descartes’s immediate successors: Louis de la Forge, Robert Desgabets, Pierre-Sylvain Régis, and Antoine Arnauld. La Forge opts for a version of Descartes’s last view, which has no place for natural geometry. Desgabets defends a version of Descartes’s earliest view, which requires the mind to turn to the brain image. Régis thinks we sense colors and sounds and the rest and then use these to imagine extension. Arnauld’s case is especially problematic, since he rejects the mind-independent existence of sensible qualities but seems committed to some version of direct realism. He is then left with the question how the mind projects these illusory states on to extended bodies, a question for which he has no answer.

Body and Corporeal Substance

This chapter discusses Leibniz’s conception of body and the closely related concept of corporeal substance. Leibniz saw problems with Descartes’s and Hobbes’s view and introduced a new conception of body based on a metaphysical argument that multiplicity presupposes unities, leading Leibniz to the view that extended bodies are made up of corporeal substances, genuine unities on the model of living animals, and a physical argument that the proper laws of nature require forces, active and passive in bodies; thus, that extended bodies are to be understood in terms of corporeal substances considered as unities of form (active force) and matter (passive force). The chapter traces the development of this view of body as Leibniz introduces monads as metaphysically more fundamental than corporeal substances and struggles to integrate them into the world of nonextended monads.