Any natural language can be considered as a tool for producing large
databases (consisting of texts, written, or discursive). This tool for its
description in turn requires other large databases (dictionaries, grammars
etc.). Nowadays, the notion of database is associated with computer processing
and computer memory. However, a natural language resides also in human brains
and functions in human communication, from interpersonal to intergenerational
one. We discuss in this survey/research paper mathematical, in particular
geometric, constructions, which help to bridge these two worlds. In particular,
in this paper we consider the Vector Space Model of semantics based on
frequency matrices, as used in Natural Language Processing. We investigate
underlying geometries, formulated in terms of Grassmannians, projective spaces,
and flag varieties. We formulate the relation between vector space models and
semantic spaces based on semic axes in terms of projectability of subvarieties
in Grassmannians and projective spaces. We interpret Latent Semantics as a
geometric flow on Grassmannians. We also discuss how to formulate G\"ardenfors'
notion of "meeting of minds" in our geometric setting.Comment: 32 pages, TeX, 1 eps figur
