A Study on ‘Point-Free Foundation of Geometry’

Abstract

The name “point-free geometry” denotes a series of researches on foundation of geometry in which the main primitive notion is the one of “solid body” (or, better, “region”), while “points”, “lines”, “planes” are defined. The main motivation of these researches is ontological in nature since the existence of solid bodies in the space looks more acceptable than the existence of points, lines, planes. In particular, an entity without extension as a point is not conceivable and it is inconceivable that three-dimensional entities are formed by entities without extension. Regardless of the validity of these motivations, the researches on point-free geometry show that the historically dominant choice of assuming points as primitive entities is only one of the possible choices (maybe the best one) and that alternative are possible. In our opinion this fact alone is sufficient to show the importance of point-free geometry.