Inclusion and Connection in Whitehead ’ s point-free geometry

Whitehead's considerations on geometry exposed in PNK, CN and, successively , in PR are a very promising basis for a modern point-free foundation of geometry. The aim of this paper is to emphasize this by translating Whitehead's ideas into suitable first order theories and by examining them from a mathematical point of view. The possibility of involving multi-valued logic is also proposed. 1. Introduction Recently the researches in point-free geometry received an increasing interest in different areas. As an example, we can quote computability theory, lattice theory, computer science. Now, the basic ideas of point-free geometry were firstly formulated by A. N. Whitehead in PNK and CN where White-head proposed as primitives the events and the extension relation between events. The points, the lines and all the " abstract " geometrical entities are defined in a suitable way. As a matter of fact, as observed in Casati and Varzi 1997, the approach proposed in these books is a basis for a "mereology" (i.e. an investigation about the part-whole relation) rather than for a point-free geometry. Indeed, the inclusion relation is set-theoretical and not topologi-cal in nature. This generates several difficulties. As an example it is a very hard enterprise to give a suitable definition point (see Section 4). So, it is not surprising that some years later the publication of PNK and CN, Whitehead in PR proposed a different idea in which the primitive notion is the one of connection relation. This idea was suggested in de Laguna 1922. The aim of this paper is not to give a precise account of Whitehead's ideas but only to emphasize their mathematical potentialities. So, we translate the analysis of Whitehead into suitable first order theories and we examine these