Whitehead's Point-free Geometry - Inclusion-based Point-free Geometry

Inclusion-based Point-free Geometry

The axioms G1-G7 are, but for numbering, those of Def. 2.1 in Gerla and Miranda (2008). The identifiers of the form WPn, included in the verbal description of each axiom, refer to the corresponding axiom in Simons (1987: 83).

The fundamental primitive binary relation is Inclusion, denoted by infix "≤". (Inclusion corresponds to the binary Parthood relation that is a standard feature of all mereological theories.) The intuitive meaning of x≤y is "x is part of y." Assuming that identity, denoted by infix "=", is part of the background logic, the binary relation Proper Part, denoted by infix "<", is defined as:

The axioms are:

• Inclusion partially orders the domain.

G1. (reflexive)

G2. (transitive) WP4.

G3. (anti-symmetric)

• Given any two regions, there exists a region that includes both of them. WP6.

G4.

• Proper Part densely orders the domain. WP5.

G5.

• Both atomic regions and a universal region do not exist. Hence the domain has neither an upper nor a lower bound. WP2.

G6.

• Proper Parts Principle. If all the proper parts of x are proper parts of y, then x is included in y. WP3.

G7.

A model of G1–G7 is an inclusion space.

Definition (Gerla and Miranda 2008: Def. 4.1). Given some inclusion space, an abstractive class is a class G of regions such that G is totally ordered by Inclusion. Moreover, there does not exist a region included in all of the regions included in G.

Intuitively, an abstractive class defines a geometrical entity whose dimensionality is less than that of the inclusion space. For example, if the inclusion space is the Euclidean plane, then the corresponding abstractive classes are points and lines.

Inclusion-based point-free geometry (henceforth "point-free geometry") is essentially an axiomatization of Simons's (1987: 83) system W. In turn, W formalizes a theory in Whitehead (1919) whose axioms are not made explicit. Point-free geometry is W with this defect repaired. Simons (1987) did not repair this defect, instead proposing in a footnote that the reader do so as an exercise. The primitive relation of W is Proper Part, a strict partial order. The theory of Whitehead (1919) has a single primitive binary relation K defined as xKy ↔ y<x. Hence K is the converse of Proper Part. Simons's WP1 asserts that Proper Part is irreflexive and so corresponds to G1. G3 establishes that inclusion, unlike Proper Part, is anti-symmetric.

Point-free geometry is closely related to a dense linear order D, whose axioms are G1-3, G5, and the totality axiom Hence inclusion-based point-free geometry would be a proper extension of D (namely D∪{G4, G6, G7}), were it not that the D relation "≤" is a total order.