Aristotle on the metaphysical status of mathematical entities
The purpose of this dissertation is to provide an account of the metaphysical status of mathematical entities in Aristotle. Aristotle endorses a form of realism about mathematical entities: for him as well as for Platonists, anti–realism, the view that mathematical objects do not exist, is not a viable option. The thesis consists of two main parts: a part dedicated to the objects of geometry, and a part dedicated to numbers. Furthermore, I have included an introductory chapter about a passage in the second chapter of Book B of the Physics (193b31- 194a7) where Aristotle endorses a form of naïve realism with regard to mathematical entities. Many of the passages that give us an insight into Aristotle’s philosophy of mathematics are to be found in the third chapter of Book M of the Metaphysics. Aristotle’s primary concern there, however, is not so much to present his own positive account as to provide answers to a series of (not so obvious) Platonic arguments. In the second chapter of my thesis, I discuss some of those arguments and highlight their role in Aristotle’s own position about the metaphysical status of geometrical entities. In a passage that is of crucial importance to understand Aristotle’s views regarding the mode of existence of the objects of mathematics (Meta. M.3, 1078a25-31), Aristotle allows for the potential existence of them. I argue that Aristotle’s sketchy remarks in Meta. M.3 point towards a geometry based on the commonsensical notion of the solid. This account can be further developed if we take into consideration the purpose of the preceding chapter M.2: to refute Platonic arguments that attribute greater metaphysical status to ‘limit entities’ (entities bounding and within a physical body), that is, to points, lines, and surfaces. According to Aristotle, such ‘limit entities’ have only a potential existence–what does this claim amount to? To answer this question, I will explore a more traditional reading of this claim and I will also put forward a more radical one: from a contemporary perspective, this reading makes Aristotelian geometry a distant cousin of modern Whiteheadian or Tarskian geometries. Providing an account of the metaphysical status of number in Aristotle poses quite a few challenges. On the one hand, the scarcity of the evidence forces commentators to rely on a few scattered remarks (primarily from the Physics) and to extract Aristotle’s own views from heavily polemical contexts (such as the convoluted arguments that occupy much of books M and N of the Metaphysics). On the other hand, the Fregean tradition casts a great shadow upon the majority of the interpretations; indeed, a great amount of the relevant scholarship is dominated by Fregean tendencies: it is, for example, widely held that numbers for Aristotle are not supposed to be properties of objects, much like colour, say, or shape, but second-order properties (properties–of–properties) of objects. The scope of the third chapter is to critically examine some of the Fregean-inspired arguments that have led to a thoroughly Fregean depiction of Aristotle, and to lay the foundations for an alternative reading of the crucial texts.
Aristotle, philosophy of mathematics, realism
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