Proof theory is a branch of mathematical logic that represents Mathematical proof|proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rule of inference|rules of inference of the logical system. As such, proof theory is syntax (logic)|syntactic in nature, in contrast to model theory, which is Formal semantics (logic)|semantic in nature. Together with model theory, axiomatic set theory, and recursion theory, proof theory is one of the so-called four pillars of the foundations of mathematics.
Proof theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.