Whitehead's point-free geometry

In mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as connection theory.

Motivation

Point-free geometry was first formulated in Whitehead (1919, 1920), not as a theory of geometry or of spacetime but of "events" and of an "extension relation" between events. Whitehead's purposes were as much philosophical as scientific and mathematical.

Whitehead did not set out the theories discussed in this entry in a manner that would satisfy present-day canons of formality. This entry consists of Whitehead's theories reformulated by others as first order theories. The domain for both theories consists of "regions." Inclusion, denoted by infix "≤", is a fundamental binary relation. 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 (x<y) ↔ (x≤y ∧ x≠y).

All unquantified variables in this entry should be taken as tacitly universally quantified; hence all axioms should be taken as universal closures. No axiom requires more than three quantified variables. Both sets of axioms have four existential quantifiers.

Point-free geometry

The following axiomatic system is, except for numbering, Def. 2.1 in Gerla and Miranda (2008). Inclusion is primitive. The axioms are:

• Inclusion partially orders the domain.

G1. $x\leq x.$ (reflexive)

G2. $(x\leq z\land z\leq y)\rightarrow x\leq y.$ (transitive) WP4.

G3. $(x\leq y\land y\leq x)\rightarrow x=y.$ (anti-symmetric)

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

G4. $\exists z[x\leq z\land y\leq z].$

• Proper Part densely orders the domain. WP5.

G5. $x<y\rightarrow \exists z[x<z<y].$

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

G6. $\exists yz[y<x\land x<z].$

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

G7. $\forall z[z<x\rightarrow z<y]\rightarrow x\leq y.$

A model of G1-G7 is an inclusion space.

Definition (Gerla and Miranda 2008: Def. 5.1). Given some inclusion space, an abstractive class is a class G of regions such that G is totally ordered. 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.

Point-free geometry is essentially a fragment of Simons's (1987: 83) system W, except that W is cast solely in terms of Proper Part, a strict partial order. The verbal description of each axiom ends with the identifier of the corresponding axiom in Simons, having the form "WPn." W, in turn, is an adaptation of the system of Whitehead (1919), featuring a primitive binary relation K, such that xKy ↔ y<x. Hence K is the converse of Proper Part. WP1 asserts that Proper Part is irreflexive and so corresponds to G1. G3 establishes that inclusion, unlike Proper Part, is anti-symmetric.

Because point-free geometry can be formulated using a strict partial order, namely Proper Part, that geometry is closely related to the first order theory of dense sets D, consisting of a primitive binary relation "<", assumed to be a strict total order, and the axioms G5 and G6. Hence point-free geometry would be a proper extension of D (namely D∪{G4, G7}), were it not that D assumes totality.

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. Theories combining inclusion with primitive topological notions are called meretopologies. Connection theory distills the first 12 of the 31 assumptions in chpt. 2 of Process and Reality into the 6 axioms of the theory C. Clarke (1981) noted that these axioms define not a point-free geometry but rather a mereology, an example of which is point-free geometry described above.

Unlike the case with inclusion spaces, connection theory enables defining "non-tangential" inclusion, a total order adequate for the construction of abstractive classes. Gerla and Miranda (2008) argue that only thus can mereotopology unambiguously define a point.

The axiomatic system C set out below is Def. 3.1 of Gerla and Miranda (2008) except for numbering. C is a first order theory with one primitive binary relation, "connection," denoted by the prefixed predicate letter C. That x is included in y can now be defined as x≤y ↔ ∀z[Czx→Czy]. The axioms of C are:

• C is reflexive. C.1.

C1. $\ Cxx.$

• C is symmetric. C.2.

C2. $Cxy\rightarrow Cyx.$

• C is extensional. C.11.

C3. $\forall z[Czx\leftrightarrow Czy]\rightarrow x=y.$

• All regions have proper parts, so that C is an atomless theory. P.9.

C4. $\exists y[y<x].$

• Given any two regions, there is a region connected to both of them.

C5. $\exists z[Czx\land Czy].$

• All regions have at least two unconnected parts. C.14.

C6. $\exists yz[(y\leq x)\land (z\leq x)\land \neg Cyz].$

A model of C is a connection space.

The verbal description of each axiom ends with 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). (Grzegorczyk (1960) proposed a similar theory, whose motivation was primarily topological.) Any mereotopology can be made atomless by invoking C4, without risking paradox or triviality. Hence C is a proper extension of the atomless variant of SMT. The extension consists of the axioms C5 and C6, suggested by chpt. 2 of Process and Reality. For an advanced discussion of systems related to C, see Roeper (1997).

Biacino and Gerla (1991) criticize Clarke's approach by showing that its connection relation coincides with the overlapping relation of mereology, so that Boolean algebra is a model of Clarke's axioms. It is doubtful that Whitehead intended this.

See also

• Mereology
• Mereotopology
• Pointless topology

References

• Biacino L, Gerla G., 1991, "Connection Structures," Notre Dame Journal of Formal Logic 32: 242-47.
• Casati, R., and Varzi, A. C., 1999. Parts and places: the structures of spatial representation. MIT Press.
• Clarke, Bowman L., 1981, "A calculus of individuals based on ‘connection’," Notre Dame J. Formal Logic 22: 204–218.
• ------, 1985, "Individuals and Points," Notre Dame Journal of Formal Logic 26: 61-75.
• De Laguna, Theodore, 1922, "Point, line and surface as sets of solids," The Journal of Philosophy 19: 449-461.
• Gerla, Giangiacomo, 1995, "Pointless Geometries" in Buekenhout, F., Kantor, W. eds., Handbook of incidence geometry: buildings and foundations. North-Holland: 1015-31.
• --------, and Miranda, A. (2008), "Inclusion and Connection in Whitehead's Point-Free Geometry."
• --------, and Miranda, A., "From the inclusion-based to the connection-based point-free geometry," to appear in Handbook of Whiteheadian Process Thought.
• Grzegorczyk, A., 1960, "Axiomatizability of geometry without points," Synthese 12: 228-235.
• Kneebone, Geoffrey, 1963. Mathematical Logic and the Foundation of Mathematics. Dover reprint, 2001.
• Lucas, J. R., 2000. Conceptual Roots of Mathematics. Routledge. Chpt. 10, on "prototopology," discusses Whitehead's systems and is strongly influenced by the unpublished writings of David Bostock.
• Roeper, Peter, 1997, "Region-Based Topology," Journal of Philosophical Logic 26: 251-309.
• Simons, Peter, 1987. Parts: A Study in Ontology. Oxford Univ. Press.
• Whitehead, A.N., 1919. An Enquiry Concerning the Principles of Natural Knowledge. Cambridge Univ. Press. 2nd ed., 1925.
• –––---, 1920. The Concept of Nature. Cambridge Univ. Press. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at Trinity College.
• ------, 1979 (1929). Process and Reality. Free Press.