Space, points and mereology. On foundations of point-free Euclidean geometry

Rafał Gruszczyński; Andrzej Pietruszczak

doi:10.12775/LLP.2009.009

Authors

Rafał Gruszczyński Nicolaus Copernicus University, Toruń
Andrzej Pietruszczak Nicolaus Copernicus University, Toruń http://orcid.org/0000-0001-9133-5081

DOI:

https://doi.org/10.12775/LLP.2009.009

Keywords

space, points, mereology, pointless geometry, point-free geometry, geometry of solids, foundations of geometry, point-free topology

Abstract

This article is devoted to the problem of ontological foundations of three-dimensional Euclidean geometry. Starting from Bertrand Russell’s intuitions concerning the sensual world we try to show that it is possible to build a foundation for pure geometry by means of the so called regions of space.

It is not our intention to present mathematically developed theory, but rather demonstrate basic assumptions, tools and techniques that are used in construction of systems of point-free geometry and topology by means of mereology (resp. Boolean algebras) and Whitehead-like connection structures. We list and briefly analyze axioms for mereological structures, as well as those for connection structures. We argue that mereology is a good tool to model so called spatial relations. We also try to justify our choice of axioms for connection relation.

Finally, we briefly discuss two theories: Grzegorczyk’s point-free topology and Tarski’s geometry of solids.

Author Biographies

Rafał Gruszczyński, Nicolaus Copernicus University, Toruń

Department of Logic

Andrzej Pietruszczak, Nicolaus Copernicus University, Toruń

Department of Logic

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