用数学结构解析语言，揭示大模型为何懂语义却难懂语境。
Language as Mathematical Structure: Examining Semantic Field Theory Against Language Games
- 将词汇场与语言场建模为连续语义空间中的交互结构。
- 发现大模型的注意力机制等特性契合数学化语义规律。
- 适合关注语言本质与AI理论根基的研究者阅读。
大型语言模型（LLMs）为检验长期存在的语言意义理论提供了新的实证场景。本文对比了两种范式：与语言游戏相关的社会建构主义观点，以及我们称之为语义场理论（Semantic Field Theory）的数学导向框架。基于作者前期工作，我们将词汇场（Lexfelder）和语言场（Lingofelder）形式化为连续语义空间中相互作用的结构。接着分析了变压器架构的核心特性——如分布式表示、注意力机制及嵌入空间的几何规律——如何与这些概念关联。我们主张，大模型在捕捉语义规律上的成功支持语言具有潜在数学结构的观点；而其在语用推理和语境敏感性方面的持续局限，则与哲学中强调社会根基的语言使用观一致。因此，数学结构与语言游戏可视为互补而非对立视角。该框架厘清了纯统计语言模型的边界，并为理论驱动的AI架构指明新方向。
原文摘要 · Abstract (English)
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.
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