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

Abstract: 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.