Proof theory refers to a line of mathematical reasoning that symbolizes proofs as formal mathematical items. It facilitates the analysis of these objects/items through a series of mathematical techniques. Generally, it presents these proofs as inductively defined data structures like plain lists, trees or boxed lists that are based on the axioms and the inference roles of the logical system. Consequently, proof theory which is basically syntactic compared to the semantic model theory. Proof theory, alongside model theory, recursion theory, axiomatic set theory, can be termed as one of the four pillars mathematics’ foundation.

** History**

While formalization of logicis associated much with proponents such as Giueseppe Peano, Gottlob Frege, Richard Dedekind and Bertrand Russell, modern proof theory is much associated with David Hilbert, who developed “the Hilbert program” in mathematics. In his Research in proof theory, Kurt Gödel’s first advanced and then refuted this program. He developed his completeness theorem which initially appeared to integrate well with Hilbert’s aim of cutting down all mathematics into a finitist formal system although this argument was found to be unattainable due to the incompleteness. All of his work was performed within the principles of proof calculi referred to as the Hilbert systems.

In deed, Proof theory is paramount in philosophical logic, in which the principal interest is based in the argument of proof-theoretic semantics. Note that the argument of the proof theoritic semantics, for it to be feasible, depends upon the technical concepts in structural proof theory.

Proof theory aims at establishing an understanding a structures in mathematical proofs and has gone through various phases: consequently, it has been general, reductive and, structural. Particularly thanks to subsequent calculus formalizations as deep results were realized where pure logics proofs and arithmetic are involved.

As it is closely related to the science of computers, proof theory has lead to the discovery of new fields of research that are outside traditional mathematics. These include verification computer programs’ accuracy. Natural deduction has resulted into the Curry-Howard correspondence as well as the connections to the functional programing as well as the sequent culculas which is usually applied in the automatic proof search systems as wel as as well as in the logic programing. Recently, there has been a development in the proof theoric semanticsfrom the general proof theory to substitute the typical denotational truth-condition semantics.

## Formal and informal proofs

The informal proofs of the proofs theory and which are found each day’s mathematical practices are different from the formal ones. They resemble high-level blueprints that would enable an expert to rebuild a formal proof not less than the principle if he is patient and is given enough time. In fact, most mathematicians find it quite pedantic and also long-winding to write a full formal proof for it to be in common application. Formal proofs are designed using computers in an interactive theorem proving.

As earlier stated, Hilbert’s program has inspired most of the mathematical research of proofs in the formal theories. The key argument in this program is that; if we can come up with a finitialy proof in cnsistencey of all the complex formal theories required by mathematicians, then we can explain these theories in terms of metamathematical arguments. This shows that all these purely universal assertions are indeed more technically workable and also finitarily true and that by grounding them, there can’t be any need to care about the non-finitary meanings of the existential theorems and pertaining to the pseudo-meaningful conditions on the existence of ideal the entities.

## Types of proof calculi

### There are three well-known types of proof calculi:

- The Hilbert calculi
- The natural deduction calculi
- The sequent calculi

Note that, each one of these can result into a complete, axiomatic formalization of propositional/predicate logic in both the traditional or intuitionistic style and almost in all of the modal logics and many sub-structural logics, like the relevance logic or the linear logic. In fact it is unconventional to find a logic which resists being displayed in one of these calculi.

## Structural proof theory

Structural proof theory is a sub-discipline of proof theory. It studies proof calculi which supports the arguments of analytic proof. This notion of analytic the proof was launched by Gentzen. Analytic proofs refer to the proofs that that are cut-free. In his natural deduction, calculus supports the idea of analytic proof, as it was shown by Dag Prawitz. Who defined analytic proofs as the normal forms related to the idea of normal-form in terms rewriting. In addition, Structural proof theory is related to the type theory by way of the Curry-Howard correspondence.

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