Vectors, viewpoint and viewpoint shift

Cognitive Linguistics has developed many insights into language but no systematic formalism. This article proposes that the discipline can benefit from standard geometrical formalisms. It outlines an approach called Discourse Space Theory (DST) that uses coordinate geometry and vector spaces. It describes how vectors have been used by several scholars to characterise the semantics of spatial prepositions, extending this approach to an abstract, deictically centred discourse space. Vector geometry comes with standard properties and operations that can serve to describe many linguistic and discourse phenomena. One of these is transformation of axes, which can be used to represent the phenomenon of viewpoint shift in discourse. The paper illustrates how DST can systematically describe viewpoint shift in foreground-background shifts, deictic verbs, tense, complement clauses and conditional constructions, especially counterfactuals.