Mathematical Linguistics: Syntax, Pragmatics, and Structural Generativity in Mathematical Practice

This paper proposes a theoretical framework called mathematical linguistics: a study of mathematics as a language whose productivity is not exhausted by formal syntax, nor by static semantics alone, but is generated through a pragmatic relation between structural cognition and symbolic form. The central thesis is that mathematical practice is driven by structure-sensitive pragmatic operations—operations by which mathematicians identify, compress, repackage, and recursively extend structural relations under syntactic constraints. In this view, formal axioms and inferential rules do not merely restrict mathematical expression; they also enable a space of pragmatic transformations through which new theories emerge. We develop a tripartite model distinguishing syntax, pragmatics, and theory, and we formalize pragmatic generation as an iterative closure process over derivations and structural representations. The framework clarifies several features of mathematical work: the standardization of proof templates, the role of intuition in discovery, the reuse and encapsulation of old theories in newer formalisms, and the coexistence of multiple foundational languages. We also introduce notions of pragmatic validity, pragmatic compression, and pragmatic complexity, and discuss how the framework can illuminate the relation between mathematical creativity, proof theory, and cognitive structure. The paper is exploratory and conceptual, but the formal model is precise enough to support future mathematical and empirical development.