PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2015 | 7 | 41--54
Tytuł artykułu

Parthood and Convexity as the Basic Notions of a Theory of Space

Autorzy
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A deductive system of geometry is presented which is based on atomistic mereology ("mereology with points") and the notion of convexity. The system is formulated in a liberal manysorted logic which makes use of class-theoretic notions without however adopting any comprehension axioms. The geometry developed within this framework roughly corresponds to the "line spaces" known from the literature; cf. [1, p. 155]. The basic ideas of the system are presented in the article's Introduction within a historical context. After a brief presentation of the logical and mereological framework adopted, a "pregeometry" is described in which only the notion of convexity but no further axiom is added to that background framework. Pregeometry is extended to the full system in three steps. First the notion of a line segment is explained as the convex hull of the mereological sum of two points. In a second step two axioms are added which describe what it means for a thus determined line segment to be "straight". In the final step we deal with the order of points on a line segment and define the notion of a line. The presentation of the geometric system is concluded with a brief consideration of the geometrical principles known by the names of Peano and Pasch. Two additional topics are treated in short sections at the end of the article: (1) the introduction of coordinates and (2) the idea of a "geometrical algebra".(original abstract)
Słowa kluczowe
PL
EN
Rocznik
Tom
7
Strony
41--54
Opis fizyczny
Twórcy
  • University of Southern Denmark
Bibliografia
  • M. L. J. van der Vel, Theory of Convex Structures. Amsterdam: North- Holland Publishing Company, 1993.
  • Euclid, The Thirteen Books of the Elements. 3 vols., 2nd ed. New York: Dover, 2000.
  • H. Zeuthen, Geschichte der Mathematik im Altertum und Mittelalter. Copenhagen: Höst, 1896.
  • G. W. Leibniz, Mathematische Schriften. Ed. C. J. Gerhardt. Hildesheim and New York: Olms, 1962, vol. 5, ch. De analysi situ, pp. 178-183, reprint of the original edition from 1858.
  • V. Debuiche, "Perspective in Leibniz's invention of Characteristica geometrica: the problem of Desargues' influence," Historia Mathematica, vol. 40, pp. 359-385, 2013. doi: 10.1016/j.hm.2013.08.001
  • S. C. J. and P. Schreiber, 5000 Jahre Geometrie. Geschichte, Kulturen, Menschen. Berlin etc.: Springer, 2001.
  • H. Grassmann, Gesammelte mathematische und physikalische Werke. Vol. I. Part 1. Leipzig: Teubner, 1894, reprinted New York: Chelsea Pub. Co., 1969.
  • W. Prenowitz, "Projective geometries as multigroups," American Journal of Mathematics, vol. 65, pp. 235-256, 1943.
  • W. Prenowitz and J. Jantosciak, Join Geometries. A Theory of Convex Sets and Linear Geometry. Berlin etc.: Springer, 1979.
  • W. V. O. Quine, "New foundations for mathematical logic," American Mathematical Monthly, vol. 44, pp. 70-80, 1937, reprinted as Chapter V in W. V. O. Quine: From a Logical Point of View. Cambridge MA: Harvard University Press, 1961.
  • A. Schmidt, "Über deduktive Theorien mit mehreren Sorten von Grunddingen," Mathematische Annalen, vol. 115, pp. 485-506, 1938.
  • D. Hilbert, Grundlagen der Geometrie. Leipzig: Teubner, 1899, 14th edition. Edited and supplied with appendices by Michael Toepell. Stuttgart etc.: Teubner. - English translation: Foundations of Geometry. LaSalle IL: Open Court 1971. 10th printing 1999.
  • J.-M. Glubrecht, A. Oberschelp, and G. Todt, Klassenlogik. Mannheim: BI Wissenschaftsverlag, 1983.
  • P. M. Gruber and W. J. M., Handbook of Convex Geometry. 2 vols. Amsterdam: North-Holland Pub. Co., 1993.
  • J. O'Rourke, Computational Geometry, 2nd ed. Cambridge GB: Cambridge University Press, 1998.
  • P. Gärdenfors, Conceptual Spaces. The Geometry of Thought. Cambridge MA: MIT Press, 2000.
  • D. A. Randell, Z. Cui, and A. G. Cohn, "A spatial logic based on regions and connections," in Principles of Knowledge Representation and Reasoning. Proceedings of the 3rd International Conference, B. Nebel, C. Rich, and W. Swartout, Eds. Los Altos CA: Morgan Kaufmann, 1992, pp. 165-176.
  • I. Pratt, "First-order qualitative spatial representation languages with convexity," Spatial Cognition and Computation, vol. 1, pp. 181-204, 1999.
  • A. Trybus, "An axiom system for a spatial logic with convexity," Ph.D. dissertation, University of Manchester, Manchester, 2011.
  • D. Lewis, Parts of Classes. Oxford: Blackwell, 1991.
  • R. Urbaniak, Le´sniewski's Systems of Logic and Foundations of Mathematics. Cham: Springer, 2014.
  • R. E. Clay, "Relation of Le´sniewski's mereology to Boolean algebra," The Journal of Symbolic Logic, vol. 39, pp. 638-648, 1974.
  • A. Tarski, "Les fondements de la géométrie des corps," Księga Pamiątkowa Pierwszego Polskiego Zjazdu Matematycznego (Supplement to Annales de la Société Polonaise de Mathématique), pp. 29-33, 1929, English translation in: Tarski, Alfred (1983): Logic, Semantics, Metamathematics. 2nd edition ed. by John Corcoran. Indianapolis IN: Hackett. 24-29.
  • A. Grzegorczyk, "Axiomatizability of geometry without points," Synthese, vol. 12, pp. 228-235, 1960.
  • B. L. Clarke, "Individuals and points," Notre Dame Journal of Formal Logic, vol. 26, no. 61-75, 1985. doi: 10.1305/ndjfl/1093870761
  • R. Gruszczy´nski and A. Pietruszczcak, "Full development of Tarski's geometry of solids," The Bulletin of Symbolic Logic, vol. 14, pp. 481-540, 2008. doi: 10.2178/bsl/1231081462
  • T. Hahmann, M. Winter, and M. Gruninger, "Stonian portholattices: A new approach to the mereotopology RT0," Artificial Intelligence, vol. 173, pp. 1424-1440, 2009. doi: 10.1016/j.artint.2009.07.001
  • R. Casati and A. C. Varzi, Parts and Places. Cambridge MA: MIT Press, 1999.
  • A. N. Whitehead, An Enquiry Concerning the Principles of Natural Knowledge. Cambridge GB: Cambridge University Press, 1919.
  • The Concept of Nature. The Tarner Lectures Delivered in Trinity College 1919. Cambridge GB: Cambridge University Press, 1920.
  • Process and Reality. An Essay in Modern Cosmology. New York: McMillan, 1929, paperback edition 1969.
  • Aristotle, The Metaphysics, 2nd ed. London: Penguin, 2004.
  • Proclus, A Commentary on the First Book of Euclid's Elements. Princeton NJ: Princeton University Press, 1992.
  • R. Strohal, Die Grundbegriffe der reinen Geometrie in ihrem Verhältnis zur Anschauung. Leipzig: Teubner, 1925.
  • T. d. Laguna, "Point, line, and surface, as sets of solids," Journal of Philosophy, vol. 19, pp. 449-461, 1922. doi: 10.2307/2939504
  • S. Le´sniewski, S. Leśniewski's Lecture Notes in Logic. Dordrecht: Kluwer, 1988, ch. 6. Whitehead's theory of events, pp. 171-178.
  • E. A. Marchisotto and J. T. Smith, The Legacy of Mario Pieri in Geometry and Arithmetic. Boston etc.: Birkhäuser, 2007.
  • J. Renz, Qualitative Spatial Reasoning with Topological Information. Berlin etc.: Springer, 1998.
  • C. Eschenbach, "A mereotopological definition of 'point'," in Topological Foundations of Cognitive Science. Papers from the Workshop at the FISI-CS, Buffalo, NY, July 9-10, 1994.
  • Hamburg: Graduierenkolleg Kognitionswissenschaft, 1994, pp. 63-80.
  • G. Gerla, "Pointless geometries," in Handbook of Incidence Geometry, F. Buekenhout, Ed. Amsterdam: Elsevier Science, 1995, pp. 1015-1031.
  • K. Menger, "Topology without points," Rice Institute Pamphlets 27, pp. 80-107, 1940.
  • G. Dimov and D. Vakarelov, "Contact algebras and region-based theory of space i, ii," Fundamenta Informaticae, vol. 74, pp. 209-249, 251-282, 2006.
  • D. Vakarelov, "Region-based space: algebra of regions, representation theory, and logics," in Mathematical Problems from Applied Logic II. Logics for the XXIst Century, D. Gabbay, Goncharov, S. S., and M. Zakharyaschev, Eds. New York: Springer, 2007, pp. 267-347.
  • T. Hahmann, M. Winter, and M. Gruninger, "On the algebra of regular sets: Properties of representable Stonian p-ortholattices," Annals of Mathematics and Artificial Intelligence, vol. 65, pp. 35-60, 2012. doi: 10.1007/s10472-012-9301-2
  • C. Coppola and G. Gerla, "Multi-valued logic for a point-free foundation of geometry," in Mereology and the Sciences. Parts and Wholes in the Contemporary Scientific Context, C. Calosi and P. Graziani, Eds. Springer, 2012, pp. 105-122.
  • P. Suppes, Representation and Invariance of Scientific Structures. Stanford CA: Center for the Study of Language and Information, 2002.
  • E. Rubin, Synsoplevede Figurer. Studier i psykologisk Analyse. Første Del. Copenhagen: Gyldendal, 1915.
  • O. Selz, "Die Struktur der Steigerungsreihen und die Theorie von Raum, Zeit und Gestalt," in Bericht über den XI. Kongreß für experimentelle Psychologie in Wien vom 9.-13. April 1929, H. Volkelt, Ed. Jena: Fischer, 1930, pp. 27-45.
  • "Appendix E. An alternative system for P and T," In: J. H. Woodger: The Axiomatic Method in Biology. Cambridge GB: Cambidge University Press, 1937. 161-172.
  • W. Schwabhäuser, Wolfram; Szmielew and A. Tarski, Metamathematische Methoden in der Geometrie. Berlin etc.: Springer, 1983.
  • M. Pasch, Vorlesungen über neuere Geoemtrie. Leipzig: Teubner, 1882, 2nd edition with an appendix by Max Dehn. Berlin: Springer 1926. Reprinted 1976.
  • "Die Begründung der Mathematik und die implizite Definition. Ein Zusammenhang mit der Lehre vom Als-Ob," Annalen der Philosophie, vol. 2, pp. 145-162, 1921.
  • W. A. Coppel, Foundations of Convex Geometry. Cambridge GB: Cambridge University Press, 1998.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.ekon-element-000171419070

Zgłoszenie zostało wysłane

Zgłoszenie zostało wysłane

Musisz być zalogowany aby pisać komentarze.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.