# Proof Theory Research Area

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A Proof Theory Research Area is a Mathematical Logic Research Area whose mathematical object of analysis are mathematical proofs.

**Context:**- It can propose of Formal Logic System.
- It can be related to a Programming Language Theory Research Area (e.g. by a Curry-Howard Isomorphism).
- It can (typically) present Mathematical Proofs as Inductively-defined Data Structures.
- …

**Counter-Example(s):****See:**Axiom, Rule of Inference, Syntax (Logic), Model Theory, Formal Semantics (Logic), Axiomatic Set Theory, Recursion Theory, Philosophical Logic.

## References

### 2014

- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/proof_theory Retrieved:2014-8-9.
**Proof theory**is a branch of mathematical logic that 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 syntactic in nature, in contrast to model theory, which is 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.^{[1]}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.

- ↑ E.g., Wang (1981), pp. 3–4, and Barwise (1978).