Proof theory began in the 1920’s as a part of Hilbert’s program. That program aimed to secure the foundations of mathematics by modeling infinitary mathematics with formal axiomatic systems, and proving those systems consistent using restricted, “finitary” means. The program thus viewed mathematics as a system of reasoning with precise linguistic norms, governed by rules that can be described and studied in concrete terms. Such a viewpoint, today, has applications in mathematics, computer science, and the philosophy of mathematics.