When algebra met proof theory
Researchers decided it was not sufficient to compare the results of the two different approaches to establish the connection between algebra and proof theory. Therefore, it was necessary to apply methods and techniques from each field to obtain proofs in the other. The new discipline that emerged from the 'Proof-theoretic methods in algebra' (PROALG) project was named algebraic proof theory. Under the auspices of the PROALG project, mathematicians from across the world pooled their efforts. Some provided expertise in proof theory, where proofs are represented as mathematical formulas with logical structure that can be analysed. Others contributed with their extensive experience in algebraic structures, consisting of elements bound by operators which satisfy certain of axioms. The question that they all faced was what proofs will a given formula have, if it is provable? During the course of the PROALG project, scientists searched for and found proofs for algebraic structures such as partially ordered groups. They also used them to obtain proofs for fundamental theorems of each class of algebraic structures. The relationship found between algebra and proof theory was then used for a thorough mathematical analysis of fuzzy logic rules. The structures in fuzzy logic attempt to capture the fuzziness' or imprecision of the real world, something that classical mathematics touches on through probability theory. The PROALG scientists were successful in defining new calculus proofs to capture uncertainty and necessity. The development of analytic calculus proofs for fuzzy logic and other kinds of non-classical logic, was not only a theoretical task for the PROALG project. Besides establishing their fundamental properties, including consistency and computational complexity, it is key to their applications. The calculus proofs proceed by breaking down formulas step by step. This methodology is a prerequisite for the automatisation of proof search. Through the close connections with computer science, algebraic proof theory is expected to contribute to research areas outside traditional mathematics. Among others, the new theory can contribute to the verification of computer programs. The key findings of the PROALG project were shared with the research community in eight journal papers and conference presentations.
