Whitehead's point-free geometry

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In

The axioms are:[b]

• Inclusion
  partially orders the domain
  .

G1. ${\displaystyle x\leq x.}$ (reflexive)

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

G3. ${\displaystyle (x\leq y\land y\leq x)\rightarrow x=y.}$ (antisymmetric)

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

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

• Proper Part densely orders the domain. WP5.

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

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

A model of G1–G7 is an inclusion space.

Definition (Gerla and Miranda 2008: Def. 4.1). Given some inclusion space S, an abstractive class is a class G of regions such that S\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

Point-free geometry is closely related to a dense linear order D, whose axioms are G1-3, G5, and the totality axiom ${\displaystyle x\leq y\lor y\leq x.}$^[2] 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.

Connection theory (mereotopology)

A different approach was proposed in Whitehead (1929), one inspired by De Laguna (1922). Whitehead took as primitive the

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[Czx→Czy]. Unlike the case with inclusion spaces, connection theory enables defining "non-tangential" inclusion,^[c] 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. ${\displaystyle \ Cxx.}$

• C is
  symmetric
  . C.2.

C2. ${\displaystyle Cxy\rightarrow Cyx.}$

• C is extensional. C.11.

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

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

C4. ${\displaystyle \exists y[y<x].}$

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

C5. ${\displaystyle \exists z[Czx\land Czy].}$

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

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

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).^[d] 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 chapter 2 of part 4 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.

See also

• Mereology
• Mereotopology
• Pointless topology

Notes

References

Bibliography

• Biacino L., and 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, 1981, "A calculus of individuals based on 'connection'," Notre Dame Journal of Formal Logic 22: 204-18.
• ------, 1985, "Individuals and Points," Notre Dame Journal of Formal Logic 26: 61-75.
• De Laguna, T., 1922, "Point, line and surface as sets of solids," The Journal of Philosophy 19: 449-61.
• Gerla, G., 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," in Michel Weber and Will Desmond, (eds.), Handbook of Whiteheadian Process Thought, Frankfurt / Lancaster, ontos verlag, Process Thought X1 & X2.
• Gruszczynski R., and Pietruszczak A., 2008, "Full development of Tarski's geometry of solids," Bulletin of Symbolic Logic 14:481-540. The paper contains presentation of point-free system of geometry originating from Whitehead's ideas and based on Lesniewski's mereology. It also briefly discusses the relation between point-free and point-based systems of geometry. Basic properties of mereological structures are given as well.
• Grzegorczyk, A., 1960, "Axiomatizability of geometry without points," Synthese 12: 228-235.
• Kneebone, G., 1963. Mathematical Logic and the Foundation of Mathematics. Dover reprint, 2001.
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• Simons, P., 1987. Parts: A Study in Ontology. Oxford Univ. Press.
• Whitehead, A.N.
  , 1916, "La Theorie Relationiste de l'Espace," Revue de Metaphysique et de Morale 23: 423-454. Translated as Hurley, P.J., 1979, "The relational theory of space," Philosophy Research Archives 5: 712-741.
• --------, 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.