| There is a fundamental tension in Greek philosophies of mathematics arising from the descriptive origins of mathematics (its natural ontology) and the deductive power of formal systems (its natural epistemology). Sophists rejected the epistemology; Platonists the ontology. Aristotle tries to preserve both by adopting a mathematical realism and grounding epistemology in experience. Hence there are three central questions in Aristotle's philosophy of mathematics: what are the objects of mathematics, how can we have knowledge of them, and how can we get knowledge of the physical world from knowledge of mathematical entities? My dissertation is in two parts: ontology and epistemology.; Since Aristotle conceives of a substance as a subject for other entities which are in turn subjects for other entities, etc., mathematical understanding involves treating substances qua triangle, i.e. treating the triangle which belongs to substance as if it were substance. I call this view epistemological nominalism. By making the objects of mathematical material potential divisions, Aristotle can claim that geometry is true. However, in lower sciences such as meteorology idealizations become necessary. I examine Aristotle's treatments of different types of quantities. Since Aristotle's views on mathematics depend on his views on substance, I emphasize developmental features. Although I am sceptical about Aristotle endorsing universal mathematics, the importance of analogy in his middle period may arise from the Eudoxan theory of proportions.; In Part II, Aristotle's treatment of mathematical elements is seen in a broad context. I compare treatments of first principles, distinctions between theorem and problem, and views on the basic objects of geometry in Aristotle, Archimedes, Euclid, and others. The Aristotelian elements are inadequate for doing geometry, but do not constitute all his geometrical hypotheses. Aristotle also thinks that mathematical demonstrations are syllogisms. The solution to both anomalies is that whereas demonstration constitutes the explanatory core of mathematical argument, induction provides the premises for deduction. Since induction requires imagination and hence perception, the diagram as image becomes the source of actual knowledge, knowledge of the universal in the particular. The epistemology is fully grounded in perception. However problems in Aristotle's account show that the tension is not resolved successfully. |
