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Proof theory, studied as a branch of mathematical logic, represents 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 rules of inference of the logical system. As such, proof theory is closer to syntax, while model theory is more purely semantical. 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.
That represents the position as of about 1940 onwards. The subject of proof theory has a significant if somewhat opaque prehistory as metamathematics, the proposed theory under development from the start of the twentieth century, for a generation, under the influence of David HilbertDavid Hilbert ( January 23, 1862 February 14, 1943) was a German mathematician born in Wehlau, near Konigsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th cen. The aim, of a convincing consistency proofProof theory In mathematical logic, a formal system is said to be consistent if it doesn't contain a contradiction, or, more precisely, for no proposition are both and provable. A consistency proof is a formal proof that a formal system is consistent. for mathematics, was not to be realised, for reasons later understood: proof theory can only sweep the metaphysical dust into tidier heaps under carpets with more attractive patterns. Hilbert's ideas seem to have been based on an analogy, in fact false, with the elimination theoryIn algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables. The linear case would now routinely be handled by Gauss-Jordan elimination, rather than the theoretical sol of algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations . When there is more than o familiar to him from his early work in algebra. The real insights of proof theory, such as cut-elimination and the isolation of the structural rules, were not to come from this direction.
Proof theory can also be considered a branch of philosophical logicPhilosophical logic is the study of the more specifically philosophical aspects of logic: the term contrasts with mathematical logic''. It is concerned with characterising notions like inference, rational thought, truth, and contents of thoughts, in the m, where the primary interest is in the idea of a proof-theoretic semanticsProof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or, an idea which depends upon technical ideas in structural proof theory to be feasible.