Gourt > Science > Math > Logic and Foundations > Proof Theory (1)
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.
Proof theory can also be considered a branch of 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.
History
Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system; then his incompleteness theorems showed that this is unattainable. All of this work was carried out with the proof calculi called the Hilbert systems. More on [ Proof theory ]
Automated Reasoning :: Computational Logic
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Proof Theory on the Eve of Year 2000 - A survey consisting of 10 questions asked by Solomon Feferman and 29 responses.
Frogs Discussion Group - Internet forum focused on the calculus of structures.
G. Japaridze's Papers - Full list of publications by this author, with about 30 items in the area of proof theory. Many of the papers are downloadable.
Linear Network - Linear Logic in Computer Science is lead by the Logic of Programming research team. Its thematic is focused on developing the theory and the applications of Linear Logic. It is formed by seven sites located in Marseille, Bologna, Cambridge, Edinburgh, Lisboa, Paris and Roma, and a few subsites.
Logic of programming - The Logic of Programming research team is interested in proof theory and its relations with theoretical computer science. The main topic is mathematical interpretation of proofs : nets (proof = graph), denotational semantics (proof = function), and game semantics (proof = strategy). Two realisations of this working programm are Linear Logic and Ludics.
Mathematical Reasoning Group - Research group based in Edinburgh, it is running on the interaction between logic, mathematics and informatics. Links to publications, homepages, reports.
Meta Description: [ School of Informatics, The University of Edinburgh, Scotland, UK. Research, undergraduate and postgraduate courses. BSc, BEng, MSc, PhD degrees in Informatics, Artificial Intelligence, Computer Science, Computational Linguistics, Software Engineering and Cognitive Science. ]
Proof Theory - Open Encyclopedia entry. Hierarchically organized by subtopics.
Meta Description: [ Science: Mathematics: Logic: Proof Theory - Open Site. ]
Proof Theory as an Alternative to Model Theory - Short article by Dale Miller, arguing that logic programming languages should base their semantics on proof theory, not model theory.
ProofTheory.ORG - Basic material on proof theory and the home page of the only mailing list devoted to proof theory, with hundreds of experts.
The Calculus of Structures - The calculus of structures is a new proof theoretical formalism. It exploits a top-down symmetry of derivations made possible by deep inference.
The Epsilon Calculus - Discussion of David Hilbert's development of this type of logical formalism with emphasis on proof-theoretic methods.
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Topics in Logic and Proof Theory - Brief introductions to combinatory logic, the incompleteness theorems and independence results, by Andrew D Burbanks.
Meta Description: [ Andrew Burbanks Staff Home Page ]
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