Abstract

Available from UMI in association with The British Library. ;Plato's philosophy of mathematics must be a philosophy of 4th century B.C. Greek mathematics, and cannot be understood if one is not aware that the notions involved in this mathematics differ radically from our own notions; particularly, the notion of arithmos is quite different from our notion of number. The development of the post-Renaissance notion of number brought with it a different conception of what mathematics is, and we must be able to see the differences before we can begin to appreciate Plato's views, which must never be seen as applying to a revolutionary notion of mathematics . For this reason the first part of this dissertation must comprise a detailed analysis of these differences, along with some criticisms of those of Plato's modern interpreters who have not taken this point. Much of what will be found here can be found in Jacob Klein's Greek Mathematics and the Origin of Algebra, which will be extensively quoted and interpreted. This book is often quoted, but very seldom understood; or at least, those lessons which should be taken from it are rarely applied to Platonic exegesis. In the second part I attempt a better explanation of Plato's mathematical ontology, beginning with the most extensive passage in the dialogues, the triple image of Sun, Line and Cave in Republic. This reveals a relation between sensibles and forms which is at odds with the traditional view that mathematical objects are never precisely accurate instances of their forms--but this view can be refuted. Plato's view of the nature of mathematical objects will be seen to be a metaphorical way of saying what Aristotle says logically, and is entirely appropriate to the aims and methods of his contemporary mathematicians. Aristotle's criticism of Plato's views is examined and found to make sense and to be appropriate to views which Plato could consistently have held