Operational Arithmetic as Numerogrammar: A Semiotic Foundation for Arithmetic Learning

This paper develops a semiotic account of arithmetic meaning grounded in mathematical practice rather than representation or skill. Building on Role-Binding Hypostasis (Part I), which introduces a general mechanism by which signs acquire determinate meaning through their position in a culturally inherited grammar, the present paper applies that mechanism to arithmetic. It argues that arithmetic constitutes a distinct model-semiotic whose criterion of meaning is being calculable rather than being referential. Arithmetic meaning is shown to reside in grammatical position within structured configurations, not in numerals taken as names of quantities. The fundamental unit of arithmetic meaning is the synset—an ordered triple specifying a well-formed configuration under additive or multiplicative grammar. Natural-language formulations of quantitative relations map to synsets via structure-preserving role re-binding, not translation. This framework explains the systematic persistence of the cardinal fallacy, in which relational numerals (differences, quotients) are misinterpreted as entity terms, and clarifies why operational arithmetic functions as a necessary intermediate practice between natural language and algebraic formalism. The paper concludes by outlining a grammar-preserving semantic compiler that formalizes these transformations and makes explicit the role-structure underlying arithmetic reasoning.