Point-free geometry is based on the idea that in geometry it is not necessary to assume as a primitive the notion of point. Instead, we can start from the notion of a "region" in the space. The points are defined by suitable abstraction processes, i.e. order-reversing sequences of regions. Firstly, such a question was analized by A. N. Whitehead in the books "An Inquiry Concerning the Principles of Natural Knowledge" and "The concept of Nature". In these books the inclusion relation is the only primitive. Successively in "Process and Reality" Whitehead proposed a new approach in which the connection relation is considered. Whitehead's analysis was philosophical in nature. Successively several autors translated this analysis into systems of axioms for a mathematical treatment of point-free geometry (for further information see [1]). A totally different approach to point-free geometry is proposed in a very interesting book by H. J. Schmidt.

References

1. G. Gerla, Pointless geometries, in Handbook of Incidence Geometry, F. Buekenhout and W. Kantor (eds) 1994 North-Holland.

2. H. J. Schmidt, Axiomatic Characterization of Physical Geometry, Lecture Notes in Physics, Springer-Verlag, Berlin Heidelberg 1979.

3. A. N. Whitehead, An Inquiry Concerning the Principles of Natural Knowledge, Camb. Univ. Press, Cambridge 1919.

4. A. N. Whitehead, The concept of Nature, Camb. Univ. Press, Cambridge 1920.

5. A. N. Whitehead, Process and Reality, The Macmillan Co., New York 1929.