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

Rafał Gruszczyński; Andrzej Pietruszczak

doi:10.12775/LLP.2009.009

Autor

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

DOI:

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

Słowa kluczowe

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

Abstrakt

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.

Biogramy autorów

Rafał Gruszczyński - Department of Logic, Nicolaus Copernicus University

Department of Logic

Andrzej Pietruszczak - Nicolaus Copernicus University, Toruń

Department of Logic

Bibliografia

Bennett, B., A.G. Cohn, P. Torrini and S.M. Hazarika, “Region-Based Qualitative Geometry”, University of Leeds, School of Computer Studies, Research Report Series, Report 2000.07.

Bennet, B., and I. Düntsch, “Axioms, algebras and topology”, pp. 99–159 in: Handbook of Spatial Logics, M. Aiello, I. Pratt-Hartmann, and J. Van Benthem (eds.), Springer, 2007.

Biacino, L., and G. Gerla, “Connection structures: Grzegorczyk’s and Whitehead’s definitions of point”, Notre Dame Journal of Formal Logic 37, 3 (1996): 431–439.

Borgo, S., N. Guarino, and C. Masolo, “A pointless theory of space based on strong connection and congruence”, in: Principles of Knowledge Representation and Reasoning, Proceedings of the 5th International Conference KR96, L.C. Aiello and J. Doyle (eds.), Morgan Kaufmann, 1996.

Borsuk, K., and W. Szmielew, Foundations of Geometry: Euclidean and Bolyai-Lobachevskian Geometry, Projective Geometry, North Holand Publishing Company, Amsterdam, 1960.

Euclid, Project Gutenberg’s first six books of the Elements of Euclid, Link, 2007.

Gerla, G., “Pointless metric spaces”, Journal of Symbolic Logic 55, 1 (1990): 207–219.

Gruszczyński, R., and A. Pietruszczak, “Pieri’s structures”, Fundamenta Informaticae 81, 1–3 (2007): 139–154. PDF (page 1)

Gruszczyński, R., and A. Pietruszczak, “Full development of Tarski’s geometry of solids”, The Bulletin of Symbolic Logic 14, 4 (2008): 481–540. DOI: 10.2178/bsl/1231081462

Grzegorczyk, A., “Axiomatizability of geometry without points”, Synthese 12, 2–3 (1960): 228–235.

Hilbert, D., Foundations of Geometry, The Open Court Publishing Company, La Salle, Illinois, 1950.

Leśniewski, S., “O podstawach matematyki I”, Przegląd Filozoficzny, XXX (1927): 164–206.

Leśniewski, S., “O podstawach matematyki V”, Przegląd Filozoficzny XXXIV (1931): 142–170.

Leśniewski, S., “On the foundations of mathematics”, pp. 174–382 in: Collected Works, vol. I, S.J. Surma et al. (eds.), Nijhoff International Philosophy Series, no. 44, Kluwer Academic Publishers, Dordrecht, 1991.

Leonard, H.S., and N. Goodman, “The calculus of individuals and its uses”, Journal of Symbolic Logic 5 (1940): 45–55.

Marchisotto, E.A., and J.T. Smith (eds.), The Legacy of Mario Pieri in Geometry and Arithmetic, Birkhäuser, Boston-Basel-Berlin, 2007.

Pieri, P., “La geometria elementare istituita sulle nozioni ‘punto’ é ‘sfera’”, Matematica e di Fisica della Società Italiana delle Scienze 15 (1908): 345–450.

Pietruszczak, A., Metamereologia, Wydawnictwo Uniwersytetu Mikołaja Kopernika, Toruń, 2000.

Pietruszczak, A., “Pieces of mereology”, Logic and Logical Philosophy 14, 2 (2005): 211–234. MathSciNet DOI: 10.12775/LLP.2005.014

Roeper, P., “Region-based topology”, Journal of Philosophical Logic 26, 3 (1997): 251–309.

Russell, B., Our Knowledge of the External World, George Allen and Unwin LTD, London, 1914. http://www.archive.org/details/ourknowledgeofth005200mbp

Simons, P., Parts. A Study in Ontology, Clarendon Press, Oxford, 1987.

Steen, L.A., and J.A. Seebach, Counterexamples in Topology, Dover Publications, Inc., New York, 1995.

Tarski, A., “Foundations of the geometry of solids”, pp. 24–29 in: Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Oxford, 1956.

Varzi A.C., “Spatial reasoning and ontology: parts, wholes and locations”, pp. 945–1038 in: Handbook of Spatial Logics, M. Aiello, I. Pratt-Hartmann, and J. Van Benthem (eds.), Springer, 2007.