Regions-based two dimensional continua: The Euclidean case

• Authors: Hellman Geoffrey , Shapiro Stewart
• 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.

Keywords

EN

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

• Publisher: Uniwersytet Mikołaja Kopernika (Nicolaus Copernicus University in Toruń)
• Journal: Logic and Logical Philosophy
• Year: 2015
• Volume: 24
• Issue: 4 Mereology and Beyond
• Pages: 499-534

Dates

online

2015-05-15

Contributors

author

Hellman Geoffrey

• Department of Philosophy, University of Minnesota, Minneapolis, MN, USA

author

Shapiro Stewart

• Department of Philosophy, Ohio State University, Columbus, OH, USA

References

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Identifiers

DOI

10.12775/LLP.2015.011