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2015 | 24 | 4 Mereology and Beyond | 499-534
Article title

Regions-based two dimensional continua: The Euclidean case

Title variants
Languages of publication
EN
Abstracts
EN
We extend the work presented in [7, 8] to a regions-based, two-dimensional, Euclidean theory. The goal is to recover the classical continuum on a point-free basis. We first derive the Archimedean property for a class of readily postulated orientations of certain special regions, “generalized quadrilaterals” (intended as parallelograms), by which we cover the entire space. Then we generalize this to arbitrary orientations, and then establishing an isomorphism between the space and the usual point-based R × R. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause” (to the effect that “these are the only ways of generating regions”), and we have no axiom of induction other than ordinary numerical (mathematical) induction. Finally, having explicitly defined ‘point’ and ‘line’, we will derive the characteristic Parallel’s Postulate (Playfair axiom) from regions-based axioms, and point the way toward deriving key Euclidean metrical properties.
Year
Volume
24
Pages
499-534
Physical description
Dates
online
2015-05-15
Contributors
  • Department of Philosophy, University of Minnesota, Minneapolis, MN, USA
  • Department of Philosophy, Ohio State University, Columbus, OH, USA
References
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  • de Laguna, T., “Point, line, and surface, as sets of solids”, Journal of Philosophy, 19 (1922): 449–461.
  • Grzegorczyk, A. “Axiomatizability of geometry without points”, Synthese, 12 (1960): 228–235. DOI: 10.1007/BF00485101
  • Gruszczyński, R., and A. Pietruszczak, “Full development of Tarski’s geometry of solids”, The Bulletin of Symbolic Logic, 14 (2008): 481–540. DOI: 10.2178/bsl/1231081462
  • Hellman, G., and S. Shapiro, “Towards a point-free account of the continuous”, Iyyun: The Jerusalem Philosophical Quarterly, 61 (2012): 263–287.
  • Hellman, G., and S. Shapiro, “The classical continuum without points”, Review of Symbolic Logic, 6 (2013): 488–512. DOI: 10.1017/S1755020313000075
  • Nagel, E., “The meaning of reduction in the natural sciences”, pp. 288–312 in Philosophy of Science, A. Danto and S. Morgenbesser (eds.) Cleveland: Meridian Books, 1960.
  • Nagel, E., The Structure of science, New York: Harcourt, Brace, and World, 1961.
  • Pieri, M. “La geometria elementare instituita sulle nozione di ‘punto’ e ‘sfera’”, Memorie di Matematica e di Fisica della Società Italiana delle Scienze, Serie Terza, 15 (1908): 345–450.
  • Tarski, A. “Foundations of the geometry of solids”, pp. 24–29 in Logic, Semantics, and Metamathematics: Papers from 1923 to 1938, Oxford, 1956.
  • Whitehead, A.N., Process and Reality, New York, The MacMillan Company, 1929.
Document Type
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.desklight-74d4ae3a-db4f-4f99-b6cb-f9b22fec0bbb
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