Aristotle on mathematical objects

Abstract

My thesis is an exposition and defence of Aristotle’s philosophy of mathematics. The first part of my thesis is an exposition of Aristotle’s cryptic and challenging view on mathematics and is based on remarks scattered all over the corpus aristotelicum. The thesis’ central focus is on Aristotle’s view on numbers rather than on geometrical figures. In particular, number is understood as a countable plurality and is always a number of something. I show that as a consequence the related concept of counting is based on units. In the second part of my thesis, I verify Aristotle’s view on number by applying it to his account of time. Time presents itself as a perfect test case for this project because Aristotle defines time as a kind of number but also considers it as a continuum. Since numbers and continuous things are mutually exclusive this observation seems to lead to an apparent contradiction. I show why a contradiction does not arise when we understand Aristotle properly. In the third part, I argue that the ontological status of mathematical objects, dubbed as materially [hulekos, ÍlekÀc] by Aristotle, can only be defended as an alternative to Platonism if mathematical objects exist potentially enmattered in physical objects. In the fourth part, I compare Aristotle’s and Plato’s views on how we obtain knowledge of mathematical objects. The fifth part is an extension of my comparison between Aristotle’s and Plato’s epistemological views to their respective ontological views regarding mathematics. In the last part of my thesis I bring Frege’s view on numbers into play and engage with Plato, Aristotle and Frege equally while exploring their ontological commitments to mathematical objects. Specifically, I argue that Frege should not be mistaken for a historical Platonist and that we find surprisingly many similarities between Frege and Aristotle. After having acknowledged commonalities between Aristotle and Frege, I turn to the most significant differences in their views. Finally, I defend Aristotle’s abstractionism in mathematics against Frege’s counting block argument. This whole project sheds more light on Aristotle’s view on mathematical objects and explains why it remains an attractive view in the philosophy of mathematics.