Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 5 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, https://mitpress.mit.edu/books/lectures-philosophy-mathematics Lecture 5. Proof What is proof? What is the relation between proof and truth? Is every mathematical truth true for a reason? After clarifying the distinction between syntax and semantics and discussing various views on the nature of proof, including proof-as-dialogue, we shall consider the nature of formal proof. We shall highlight the importance of soundness, completeness, and verifiability in any formal proof system, outlining the central ideas used in proving the completeness theorem. The compactness property distills the finiteness of proofs into an independent, purely semantic consequence. Computer-verified proof promises increasing significance; its role is well illustrated by the history of the four-color theorem. Nonclassical logics, such as intuitionistic logic, arise naturally from formal systems by weakening the logical rules.