Language as Mathematical Structure: Examining Semantic Field Theory Against Language Games

TL;DR

This paper compares social constructivist and mathematical approaches to linguistic meaning using large language models, proposing a formal framework that highlights their complementary roles in understanding language.

Contribution

It formalizes lexical and linguistic fields within Semantic Field Theory and relates them to transformer architectures, bridging social and mathematical perspectives on language.

Findings

LLMs capture semantic regularities supporting mathematical structure of language.

Limitations in pragmatic reasoning align with social grounding theories.

Framework clarifies scope of statistical models and guides new AI architectures.

Abstract

Large language models (LLMs) offer a new empirical setting in which long-standing theories of linguistic meaning can be examined. This paper contrasts two broad approaches: social constructivist accounts associated with language games, and a mathematically oriented framework we call Semantic Field Theory. Building on earlier work by the author, we formalize the notions of lexical fields (Lexfelder) and linguistic fields (Lingofelder) as interacting structures in a continuous semantic space. We then analyze how core properties of transformer architectures-such as distributed representations, attention mechanisms, and geometric regularities in embedding spaces-relate to these concepts. We argue that the success of LLMs in capturing semantic regularities supports the view that language exhibits an underlying mathematical structure, while their persistent limitations in pragmatic reasoning and context sensitivity are consistent with the importance of social grounding emphasized in philosophical accounts of language use. On this basis, we suggest that mathematical structure and language games can be understood as complementary rather than competing perspectives. The resulting framework clarifies the scope and limits of purely statistical models of language and motivates new directions for theoretically informed AI architectures.