This course is a first introduction to philosophy of mathematics, via one of our most fascinating and perplexing concepts: the infinite. We encounter the concept of infinity in myriad ways. In Zeno’s paradoxes of time, space, and motion, the idea of infinite division is used to argue in favour of a radical monism. The ancient atomists Leucippus and Democritus claimed that the universe consisted of an infinity of atoms moving in an infinite void, and contemporary cosmology still considers the issue of whether the universe is infinite to be an open question.

But what does it mean for something to be infinite? It is mathematics that offers us the precise definitions that let us begin to answer this question, and thus in mathematics that many of the most important questions concerning the infinite arise. Do the infinite structures that we talk about in mathematics really exist? If so, how can we have knowledge of them? Is it even coherent to talk about the truly infinite, or does it fall victim to paradox? This course will investigate these and other questions by engaging with the ideas of philosophers and mathematicians from across history, with a focus on the reception of Georg Cantor’s theory of sets, and the crisis in the foundations of mathematics that it precipitated.