Mereological foundations of point-free geometry via multi-valued logic
DOI:
https://doi.org/10.12775/LLP.2015.019Keywords
point-free geometry, multi-valued logic, fuzzy logic, continuous logic, metric geometry, mereology, naïve scienceAbstract
We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. ‘close’, ‘small’, ‘contained’. Accordingly, some first-order multi-valued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking into account that interesting metrical approaches to geometry exist, this looks to be promising for a point-free foundation of the notion of space. We hope also that this way to face point-free geometry provides a tool to illustrate the passage from a naïve and ‘qualitative’ approach to geometry to the ‘quantitative’ approach of advanced science.References
Banaschewski, B., and A. Pultr, “A new look at pointfree metrization theorems”, Comment. Math. Univ. Carolinae, 39 (1998): 167–175.
Blumenthal, L.M., Theory and Applications of Distance Geometry, Chelsea Publishing Company, N.Y., 1970.
Bozzi, P., Fisica ingenua, Garzanti, 1990.
Casati, R., and A. Varzi, “Spatial entities”, pages 73–96 in Spatial and Temporal Reasoning, O. Stock (ed.), Dordrecht, Kluwer, 1997. DOI: 10.1007/978-0-585-28322-7_3
Casati, R., and A. Varzi, “Un altro mondo?”, Rivista di Estetica, 42, 1 (2002): 131–150.
Chang, C.C., and H.J. Keisler, Continuous Model Theory, Princeton University Press, 1966.
Coppola, C., and G. Gerla, “Special issue on point-free geometry and topology: An introduction”, Logic and Logical Philosophy, 22 (2013): 139–143. DOI: 10.12775/LLP.2013.008
Coppola, C., and G. Gerla, “Multi-valued logic for a point-free foundation of geometry”, pages 105–122 in: Mereology and the Sciences. Parts and Wholes in the Contemporary Scientific Context., C. Calosi and P. Graziani (eds.), Springer Synthese Library, vol. 372, 2014. DOI: 10.1007/978-3-319-05356-1_5
Coppola, C., G. Gerla, and A. Miranda, “Point-free foundation of geometry and multi-valued logic”, Notre Dame Journal of Formal Logic, 51, 3 (2010): 383–405. DOI: 10.1215/00294527-2010-024
Di Concilio, A., and G. Gerla, “Quasi-metric spaces and point-free geometry”, Mathematical Structures in Computer Science, 16, 1 (2006): 115–137. DOI: 10.1017/S0960129506005111
Gerla, G., “Pointless metric spaces”, Journal of Symbolic Logic, 55 (1990): 207–219. DOI: 10.2307/2274963
Gerla, G., “Pointless geometries”, pages 1015–1031 in Handbook of Incidence Geometry, F. Buekenhout (ed.), Elsevier Science, 1994. DOI: 10.1016/B978-044488355-1/50020-7
Gerla, G., “Approximates similarities and Poincaré paradox”, Notre Dame Journal of Formal Logic, 49, 2 (2008): 203–226. DOI: 10.1215/00294527-2008-008
Gerla, G., and A. Miranda, “Graded inclusion and point-free geometry”, International Journal of Pure and Applied Mathematics, 11 (2004): 63–81.
Gerla, G., and A. Miranda, “Mathematical features of Whitehead’s point-free geometry”, pages 507–533 in Handbook of Whiteheadian Process Thought, M. Weber and W. Desmond (eds.), Ontos Verlag, 2008.
Gerla, G., and B. Paolillo, “Whitehead’s point-free geometry and diametric posets”, Logic and Logical Philosophy, 19 (2010): 289–308. DOI:10.12775/LLP.2010.010
Gerla, G., and R. Volpe, “Geometry without points”, The American Mathematical Monthly, 92 (1985): 707–711. DOI: 10.2307/2323221
Gruszczynski, R., and A. Pietruszczak, “Full development of Tarski’s geometry of solids”, Bulletin of Symbolic Logic, 14, 4 (2008): 481–540. DOI:10.2178/bsl/1231081462
Hájek, P., Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.
Hayes, P.J., The Naive Physics Manifesto, in Expert Systems in the Microelectronic Age, D. Michie (ed.), Edimburg, Edimburgh University Press, 1979.
Pavelka, J., “On fuzzy logic I. Many-valued rules of inference”, Zeitschr. f. math. Logik und Grundlagen d. Math., 25 (1979): 45–52.
Pultr, A., “Pointless uniformities II: (Dia)metrization”, Comment. Math. Univ. Carolinae, 25 (1984): 104–120.
Pultr, A., “Diameters in locales: How bad they can be?”, Comment. Math. Univ. Carolinae, 4 (1998): 731–742.
Smith, B., and R. Casati, “Naïve physics: An essay in ontology”, Philosophical Psychology 7 (1994): 225–244. DOI: 10.1080/09515089408573121
Tarski, A., “Foundations of the geometry of solids”, pages 24–29 in Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Clarendon Press, Oxford, 1956.
Varzi, A.C., “Boundaries, continuity, and contact”, Nous, 31 (1997): 26–58. DOI: 10.1111/0029-4624.00034
Whitehead, A.N., An Inquiry Concerning the Principles of Natural Knowledge, Univ. Press, Cambrige, 1919.
Whitehead, A.N., The Concept of Nature, Univ. Press Cambrige, 1920.
Whitehead, A.N., Process and Reality, Macmillan, N.Y., 1929.
Yaacov, I.B., and A. Usvyatsov, “Continuous first-order logic and local stability”, Trans. Amer. Math. Soc., 362 (2010), 5213–5259.
Downloads
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 473
Number of citations: 1
