Connection Theory
In his 1929 Process and Reality, A. N. Whitehead proposed a different approach, one inspired by De Laguna (1922). Whitehead took as primitive the topological notion of "contact" between two regions, resulting in a primitive "connection relation" between events. Connection theory C is a first order theory that distills the first 12 of the 31 assumptions in chpt. 2 of Process and Reality into 6 axioms, C1-C6. C is a proper fragment of the theories proposed in Clarke (1981), who noted their mereological character. Theories that, like C, feature both inclusion and topological primitives, are called mereotopologies.
C has one primitive relation, binary "connection," denoted by the prefixed predicate letter C. That x is included in y can now be defined as x≤y ↔ ∀z. Unlike the case with inclusion spaces, connection theory enables defining "non-tangential" inclusion, a total order that enables the construction of abstractive classes. Gerla and Miranda (2008) argue that only thus can mereotopology unambiguously define a point.
The axioms C1-C6 below are, but for numbering, those of Def. 3.1 in Gerla and Miranda (2008).
- C is reflexive. C.1.
- C1.
- C is symmetric. C.2.
- C2.
- C is extensional. C.11.
- C3.
- All regions have proper parts, so that C is an atomless theory. P.9.
- C4.
- Given any two regions, there is a region connected to both of them.
- C5.
- All regions have at least two unconnected parts. C.14.
- C6.
A model of C is a connection space.
Following the verbal description of each axiom is the identifier of the corresponding axiom in Casati and Varzi (1999). Their system SMT (strong mereotopology) consists of C1-C3, and is essentially due to Clarke (1981). Any mereotopology can be made atomless by invoking C4, without risking paradox or triviality. Hence C extends the atomless variant of SMT by means of the axioms C5 and C6, suggested by chpt. 2 of Process and Reality. For an advanced and detailed discussion of systems related to C, see Roeper (1997).
Biacino and Gerla (1991) showed that every model of Clarke's theory is a Boolean algebra, and models of such algebras cannot distinguish connection from overlap. It is doubtful whether either fact is faithful to Whitehead's intent.
Read more about this topic: Whitehead's Point-free Geometry
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