On the problem of describing semantic structures and semantic activity in formal mathematics and logic

Abstract

The text considers the impossibility of abstracting away from the sense of formal con­structions in logical and mathematical researches. The validity of the application of the “formal methodology” is allowed only after some system of conventional notations and agreements has been accepted. The context determined by such agreements is called formal. A correlation of constructions and results obtained by formal methods within sev­eral formal contexts is impossible without a consideration of the various semantic aspects of the correlated formal constructions. The author calls such correlations intercontextual. The paper examines two examples of such intercontextual comparisons to demonstrate the necessity of taking into account different semantic components of the compared for­mal constructions. In the context of these conclusions, the author raises the question of the structure and origin of some senses of the “symbols” used in formal constructions and of the “sequences of symbols” constructed from them. The author identifies three main sources of the semantic load carried by formal constructions. Firstly, these are the various aspects of semiotic usage: first of all, the general cultural and general profes­sional semiotic skills of the “interpreter”. Secondly, it is the sense given to formal con­structions by verbal comments, descriptions of the construction process and the associ­ated knowledge of the “interpreter”. Thirdly, these are the senses set by the formal con­structions themselves: at the stage of defining a formal language and at the stage of con­structing a formal deductive or semantic system. The author also considers the fallacy of the assumption of the existence of some universal “global intuition” associated with the very possibility of formal methodology.