Point-free geometry

Point-free geometry is based on the idea that we can consider the notion of point not as a primitive since we can define it by the more concrete notion of region. The point are defined by suitable sequences of regions. The basic ideas of point-free geometry were formulated by Alfred North Whitehead in An Inquiry Concerning the Principles of Natural Knowledge and in The Concept of Nature where the events and the extension relation between events are proposed as primitives. While the analysis of Whitehead is qualitative in nature, there is no difficulty to translate it into a system of axioms for a well-based theory. Indeed, it is easy to prove that first properties proposed by Whitehead are equivalent to the following system of axioms:

i) $\forall x$(x ≤ x) (reflexive)

ii) $\forall x\forall y\forall z$(x≤z $\land$ z≤y) $\rightarrow$ (x≤y) (transitive)

iii) $\forall x\forall y$(x≤y $\land$ y≤x $\rightarrow$ x = y) (anti-symmetric)

iv) $\forall z$$\exists x$(x<z) (there is no minimal region)

v) $\forall x\forall y$(x<y $\exists$z$\rightarrow$z (x<z<y) (dense)

vi) $\forall x\forall y$($\forall$x'(x’<x $\rightarrow$ x’<y) $\rightarrow$ x≤y)

vii) $\forall x\forall y$$\exists$z(x≤z$\land$ y≤z) (upward-directed)

viii) $\forall z$$\exists x$(z<x) (there is no maximal region).

We call inclusion space any model of i)-viii). The points, the lines and all the “abstract” geometrical entities are defined by Whitehead through the basic notion of “abstractive class”. i.e. a totally ordered family P of regions such that no region exists which is contained in all the regions in C. As a matter of fact, as observed in Casati and Varzi 1997, such a definition gives a basis for a "mereology" (i.e. an investigation about the part-whole relation) rather than for a point-free geometry. So, it is not surprising the fact that, later, in Process and Reality, Whitehead proposed a different approach, inspired to De Laguna 1922, in which the topological notion of “contact” between two regions is a primitive and the inclusion is defined. While in this book a very long list of “assumptions” is proposed, we can translate the main nucleus of Whitehead ideas into the following system of axioms. C denotes the connection relation and x≤y denotes the formula $\forall$z(zCx→zCy). We call connection space every model of the following system of axioms:

C1 $\forall$xy(xCy$\rightarrow$yCx) (symmetry)

C2 $\forall$z$\exists$xy((x≤z)$\land$ (y≤z)$\land$ $\neg$(xCy)).

C3 $\forall x\forall y$$\exists$z(zCx$\land$ zCy)

C4 $\forall$x(xCx)

C5 ($\forall$z(zCx $\leftrightarrow$ zCy)) $\rightarrow$ x = y

C6 $\forall$x$\exists$y(y<x).

The investigation about theories of such a kind in which apart the inclusion relation is involved also a topological notion is called mereotopology. The points are defined by a notion of abstractive class where inclusion is substituted with non-tangential inclusion.

Notice that an analogous approach was proposed independently in Grzegorczyk 1960 as a first order theory. Also, a point-free geometry is considered by J. von Neumann in connection with quantum theory. Indeed, due to Heisenberg’s uncertainty relation, points become meaningless so that one has to replace points by so called “Planck cells”. Von Neumann called the resulting geometry “pointless geometry” which was later on labeled as “noncommutative geometry”.

References

• Casati R., Varzi A., 1997. ‘’Spatial Entities’’, Spatial and Temporal Reasoning, Oliviero Stock (ed), Dordrecht: Kluwer , 73-96.
• Clarke B. L., 1981, A calculus of individuals based on ‘connection’, Notre Dame J. Formal Logic 22 204 –218.
• De Laguna T., Point, line and surface as sets of solids, ‘’The Journal of Philosophy’’, 19, 1922 449-461.
• Gerla G. 1994. ‘’Pointless geometries’’, Handbook of Incidence Geometry, F.Buekenhout ed., Elsevier Science, 1015-1031.
• Gerla G., 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.
• Neumann J.V., 1998, Continuous geometry, Princeton University Press.
• Pratt I. and Lemon O., Expressivity in polygonal, plane mereotopology, ‘’Journal of Symbolic Logic, 65, (2000) 822-838.
• Whitehead, A. N. 1919. An Inquiry Concerning the Principles of Natural Knowledge. Univ. Press. Cambridge.
• Whitehead, A. N. 1920. The Concept of Nature. Univ. Press. Cambridge.
• Whitehead, A. N. 1929. Process and Reality. Macmillan, N.Y.