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

Language as Mathematical Structure: Examining Semantic Field Theory Against Language Games Dimitris Vartziotis1,2 1TWT Science & Innovation, Stuttgart, Germany 2NIKI - Digital Engineering, Ioannina, Greece Abstract 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 formal- ize 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 regu- larities 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 mathe- matical 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 clar- ifies the scope and limits of purely statistical models of language and motivates new directions for theoretically informed AI architectures. The emergence of large language models (LLMs) achieving near-human linguistic performance through purely mathematical operations1,2 poses a fundamental challenge to dominant theories of meaning. Social constructivist accounts, following Wittgenstein’s later philosophy3, insist that language cannot be reduced to formal structures. Yet transformer architectures discover 1 arXiv:2601.00448v1 [cs.CL] 1 Jan 2026 systematic semantic relationships without social grounding4, suggesting language may possess inherent mathematical structure. The author14–16, anticipated this development with remarkable prescience. Writing in 2012—years before the transformer revolution—his commentary on Wittgenstein’s collected aphorisms5,6 pro- posed that words create ‘semantic fields’ (Lexfelder) that interact according to mathematical laws, producing composite ‘linguistic fields’ (Lingofelder). This framework offers a radical alternative: meaning as mathematical discovery rather than social construction. The genesis of semantic field theory appears in Vartziotis’s response to Wittgenstein’s observation: Wittgenstein: “Die Sprache ist nicht gebunden, doch der eine Teil ist mit dem anderen verknüpft.” (Language is not bound, yet one part is connected to another.) Vartziotis: “Sie ist ein schwebendes Netz. Eine Mannigfaltigkeit. Jedes Wort hat sein eigenes ‘Gravitationsfeld’. Wir können es ja ‘Lexfeld’ nennen.” (It is a floating net. A manifold. Each word has its own ‘gravitational field’. We can call it a ‘lexical field’.)6 Semantic field theory formalized From usage to fields The crucial divergence between Wittgenstein and the author emerges in their treatment of linguistic meaning. Consider their exchange on what gives life to signs: Wittgenstein: “Jedes Zeichen scheint allein tot. Was gibt ihm Leben? - Im Gebrauch lebt es. Hat es da den lebenden Atem in sich? - Oder ist der Gebrauch sein Atem?” (Every sign by itself seems dead. What gives it life? - In use it lives. Does it have living breath in itself? - Or is use its breath?) Vartziotis: “Akzeptieren wir kurz, dass das Wort (Zeichen) eine Art ‘Lexfeld’ hat. Die Wörter bilden ein komplexes Feld (Lingofeld), den Feldern der Physik entsprechend, welches definiert werden muss. Dann lässt sich manches erklären! Selbst gebogene und 2 verdrehte Bedeutungen.” (Let us briefly accept that the word (sign) has a kind of ‘lexical field’. Words form a complex field (linguistic field), corresponding to the fields of physics, which must be defined. Then much can be explained! Even bent and twisted meanings.)6 This insight led the author to identify what he called the “Dreiwörterproblem” (three-word problem), analogous to the three-body problem in physics—suggesting that linguistic complexity emerges from nonlinear field interactions. Definition 1 (Lexical Field). Let S = Rn be the semantic space, where each dimension corresponds to a latent semantic feature. For any point q ∈S: Lw(q) = Sw · G(∥q −qw∥; σw) (1) measures the semantic field strength of word w at position q,where qw ∈Rn represents the word’s position in n-dimensional semantic space, Sw its inherent semantic strength, and G a monotonically decreasing kernel function with characteristic width σw. Definition 2 (Linguistic Field). Let W = {w1, w2, ..., wm} be an ordered sequence of words forming a phrase. The composite linguistic field ΦW : S →R at any point q ∈S is defined by: ΦW(q) =

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