  Proof theory began in the 1920's as a part of Hilbert's program, which 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. Today such a viewpoint has
applications in mathematics, computer science, and the philosophy of
mathematics.
