122. Philosophy of Mathematics

Philosophical conundrums pervade mathematics, from fundamental questions of mathematical ontology to deep questions of epistemology. What are numbers? What is the nature of infinity? How do or can we come to mathematical knowledge? What are the relations between truth, proof, and meaning? Does every mathematical truth admit of proof? What role do figures play in geometric argument? Do mathematical objects exist that we cannot construct? Can every mathematical question be solved in principle by computation? By what criteria are we to accept or reject mathematical axioms? These are merely a few of the questions we shall consider while exploring various philosophical positions, including platonism, realism, logicism, structuralism, formalism, constructivism, and many others. No specific mathematical knowledge is required for study in this topic, but a stronger mathematical background may enable a deeper understanding; it will be helpful to have studied mathematics at A-levels or similar as well as logic in the Prelims/Mods.