Oleg Prosorov

We study the process of interpretation of a text written in some unspecified natural language, say in English, considered as a means of communication. Our analysis concerns the only texts written “with good grace” and intended for human understanding; we call them 'admissible'. Whether a part of an admissible text is meaningful or not depends on some accepted 'criterion of meaningfulness'. We argue that the criterion of meaningfulness conveying an idealized reader's linguistic competence meant as ability to grasp a communicative content gives rise to a genuine topology on an admissible text, which we call 'phonocentric'. We argue that the following properties of phonocentric topology are 'linguistic universals' of topological nature: (i) T0-separability, (ii) topological connectedness, (iii) acyclicity of corresponding Hasse diagram. This way, we interpret linguistic notions in terms of topology and specialization order; their geometric studies is a kind of 'Topological Formal Syntax'. For an admissible text X, we define the Schleiermacher category Schl(X) of sheaves of fragmentary meanings, and we define the category Context(X) of étale bundles of contextual meanings. Their geometric studies is a kind of 'Sheaf-Theoretic Formal Semantics' that allows us to generalize Frege's compositionality and contextuality principles related by 'Frege Duality' defined in categorical terms. It reveals that the acceptance of one of these principles implies the acceptance of the other, and it gives rise to a functional representation of fragmentary meanings that allows us to develop a kind of 'Dynamic Semantics' describing how the interpretation of a given text proceeds by induction over the discrete time as a successive extension of a continuous function, which represents a meaning, to the whole finite topological space naturally attached to interpreted text.