Some Words on a Proof in Mathematics

Abstract

The aim of the paper is to study the role and features of proofs in mathematics. Formal and informal proofs are distinguished. It is stressed that the main roles played by proofs in mathematical research are verification and explanation. The problem of the methods acceptable in informal proofs, in particular of the usage of computers, is considered with regard to the proof of the Four-Color Theorem. The features of in-formal and formal proofs are compared and contrasted. It is stressed that the concept of an informal proof is not precisely defined, it is simply practised and any attempts to define it fail. It is — so to speak — a practical notion, psychological, sociological and cultural in character. The second one is precisely defined in terms of logical con-cepts. Hence it is a logical concept which is rather theoretical than practical in char-acter. The first one is — in part at least — semantical in nature, the second is entirely syntactical. A proof-theoretical thesis, similar to the Turing-Church Thesis in the re-cursion theory, is formulated. It says that both concepts of a proof in mathematics are equivalent. Arguments for and against it are formulated.