Abstract

This dissertation argues that Aristotle intended his so-called theory of abstraction to serve primarily as the resolution of a special problem in the philosophy of mathematics; i.e., the ontological status of mathematical objects. My general approach is dictated by the view that Aristotle's 'theories' must be understood in terms of the particular problems that he is trying to resolve. Thus, most of my dissertation is devoted to examining his treatment of the problem which I show to be relevant to the theory of abstraction. In fact, it contains a methodological study of Aristotle's characteristic manner of treating a philosophical problem. ;My first chapter shows: how a problem must arise for Aristotle about the mode of being of mathematical objects; why this is a peculiarly difficult problem for him to resolve; and that he recognizes it as an appropiate problem for First Philosophy. The second chapter selects two relevant puzzles) from among those listed in Metaphysics Beta and examines Aristotle's review of the difficulties associated with each puzzle. This even-handed review is distinguished from his elenctic treatment of the same general problem in Mu 1-2, where his purpose is to refute the Platonists and to prepare the way for his own solution in Mu 3. ;My final chapter elucidates this positive solution and establishes its connection with the terminology of abstraction. I show that the crucial issue for Aristotle is the precise mode of being of mathematical objects, given that they are the products of a dialectical method of subtraction ). Thus, I argue that he makes these entities doubly dependent, in contrast with the independent substances posited by the Platonists. In one respect mathematical quantities are potentially in sensible substances but, precisely as they are defined, they are actualized only through the abstractive activity of the mathematician. To support my interpretation, I show that such a complex solution is paralleled in Aristotle's account of the mode of being of the infinite. Furthermore, I show that this solution "saves the phenomena", in the sense that it resolves the outstanding difficulties associated with the problem