Philosophy of Mathematics

This course explores the role that mathematics plays in Plato's philosophy with a special focus on the concept of incommensurability. We will be reading Platonic dialogues in which mathematical practice figures prominently and our goal will be to inquire into the ways that mathematical practice is similar to philosophical practice and the ways it can serve as a useful exemplar. We will also inquire into the ways that mathematics falls short of philosophy, which will give us a better sense of what the philosophical goals are. Finally, we will consider the challenges presented by mathematical incommensurability and we will investigate the ways that this concept is appropriated by Plato for philosophical purposes.

Texts will include: Meno, Republic 5-7, Timaeus, Theaetetus, Statesman. We will read some secondary literature on Plato (e.g. S. Menn, H. Benson, T. Echterling) and on the mathematics of the time (W. Knorr, J. Klein) but not every time. (B)

This course surveys recent work on explanation across philosophical disciplines. Beginning with classic accounts of scientific explanation we will proceed to consider recent work on mechanical explanation, mathematical explanation, causal explanation (particularly in the physical and social sciences), the relation between explanation and understanding, and metaphysical explanation (particularly the idea of explanation as ground). (II)

In this course we will read Whitehead with the aim of understanding how he arrived at his mature views, i.e., the "philosophy of organism" expressed in Process and Reality (1929). The development of Whitehead's philosophy can be traced back to a planned fourth volume of Principia Mathematica (never completed) on space and time. This course will examine how these concerns with natural philosophy led Whitehead to develop his philosophy of organism. Beginning in the late 1910s, we will read over 10 years of published work by Whitehead, supplemented by recently discovered notes from his Harvard seminars 1924/25 and selected commentaries. (II)

This course will broadly be about the concept of mathematical proof, focusing on the case of geometry, and more specifically, focusing on the works of Euclid. While many mathematicians think of Euclid as the pioneer of the modern axiomatic method, this way of thinking seems somewhat anachronistic. How then should we think of Euclidean proofs? What does a Euclidean proof accomplish, how does it accomplish it, and what does this tell us about the nature of mathematical proof more generally? This course will look both at ancient sources and modern sources as a way of tackling these questions. (II)