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Logic and Logical Philosophy

Whitehead's pointfree geometry and diametric posets
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Whitehead's pointfree geometry and diametric posets

Authors

  • Giangiacomo Gerla Università di Salerno
  • Bonaventura Paolillo Università di Salerno

DOI:

https://doi.org/10.12775/LLP.2010.010

Keywords

Whitehead, pointfree geometry, lattices, diameters

Abstract

This note is motivated by Whitehead’s researches in inclusion-based point-free geometry as exposed in An Inquiry Concerning the Principles of Natural Knowledge and in The concept of Nature. More precisely, we observe that Whitehead’s definition of point, based on the notions of abstractive class and covering, is not adequate. Indeed, if we admit such a definition it is also questionable that a point exists. On the contrary our approach, in which the diameter is a further primitive, enables us to avoid such a drawback. Moreover, since such a notion enables us to define a metric in the set of points, our proposal looks to be a good starting point for a foundation of the geometry metrical in nature (as proposed, for example, by L.M. Blumenthal).

Author Biographies

Giangiacomo Gerla, Università di Salerno

Dipartimento di Matematica

Bonaventura Paolillo, Università di Salerno

Dipartimento di Matematica

References

Banaschewski, B., and A. Pultr, “Cauchy points of metric locales”, Canad. J. Math. 41, 5 (1989): 830–854.

Banaschewski, B., and A. Pultr, “A new look at pointfree metrization theorems”, Comment. Math. Univ. Carolinae 39, 1 (1998): 167–175.

Blumenthal, L.M., Theory and Applications of Distance geometry, N.Y., Chelsea Publishing Company, 1970.

Coppola, C., and T. Pacelli, “Approximate distances, pointless geometry and incomplete information”, Fuzzy Sets and Systems 157 (2006): 2371–2383.

Gerla, G., “Pointless metric spaces”, The Journal of Symbolic Logic 55 (1990): 207–219.

Gerla, G., “Pointless Geometries”, pp. 1015–1031 in: Handbook of Incidence Geometry: Buildings and Foundations, F. Buekenhout and W. Kantor (eds.), Amsterdam: North-Holland, 1995.

Gerla, G., and A. Miranda, “Inclusion and connection in Whitehead’s pointfree geometry, in: Handbook of Whiteheadian Process Thought, M. Weber (ed.), 2006.

Gerla, G., and R. Volpe, “Geometry without points”, The American Math. Monthly 92 (1985): 707–711.

Previale, F., “Reticoli metrici”, Boll. Un. Mat. Ital. 21 (1966): 243–250.

Previale, F., “Su una caratterizzazione reticolare del concetto di spazio metrico”, Atti d. Accad. d. Sc. di Torino,Cl. Sc. Fis. 100 (1966): 766–779.

Pultr, A., “Diameters in locales: How bad they can be?” Comm. Math. Universitatis Carolinae 4 (1988): 731–742.

Pultr, A., “Categories of diametric frames”, Math. Proc. Camb. Phil. Soc. 105 (1989): 285–297.

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

Whitehead, A.N., The Concept of Nature, Univ. Press. Cambridge, 1920.

Whitehead, A.N., Process and Reality, N.Y, Macmillan, 1929.

Logic and Logical Philosophy

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Published

2010-12-30

How to Cite

1.
GERLA, Giangiacomo and PAOLILLO, Bonaventura. Whitehead’s pointfree geometry and diametric posets. Logic and Logical Philosophy. Online. 30 December 2010. Vol. 19, no. 4, pp. 289-308. [Accessed 1 July 2025]. DOI 10.12775/LLP.2010.010.
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Vol. 19 No. 4 (2010)

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