Regions-based two dimensional continua: The Euclidean case

Geoffrey Hellman, Stewart Shapiro

DOI: http://dx.doi.org/10.12775/LLP.2015.011

Abstract


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.

Keywords


mereology, point-free geometry, Euclidean geometry, Tarski, gunk, Archimedean property, points

Full Text:

PDF

References


Aristotle, The basic works of Aristotle, edited by R. McKeon, Random House, 1941.

Bennett, B., “A categorical axiomatization of region-based geometry”, Fundamenta Informaticae, 46 (2001): 145–158.

Biacino, L., and G. Gerla, “Connection structures: Grzegorczyk’s and Whitehead’s definitions of point”, Notre Dame Journal of Formal Logic, 37 (1996): 431–439. DOI: 10.1305/ndjfl/1039886519

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.








ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism