The application of mereology to grounding of elementary geometry

Glibowski, Edmund

doi:10.1007/bf02134305

Cited by 3 publications

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“…The geometries thus developed are unsatisfactory in our view since they make a direct use of points (as a different sort) or regard some of the individuals as point-like entities. The last case is exemplified by [Gli69], which uses the notion of (finite) segment and congruence, and [GV85] based on the notions of solid (among which the so-called ε-points), parthood (here called inclusion) and distance. Among the first type of systems instead we cite [Pam04] that uses as primitives points and triangles (or squares) and the relation of point-triangle (or pointsquare), and [Pra83, Pam00] based on points, circles and the relation of point-circle incidence.…”

Section: Literaturementioning

confidence: 99%

Spheres, cubes and simple

Borgo

2013

LLP

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Abstract. In 1929 Tarski showed how to construct points in a region-based first-order logic for space representation. The resulting system, called the geometry of solids, is a cornerstone for region-based geometry and for the comparison of point-based and region-based geometries. We expand this study of the construction of points in region-based systems using different primitives, namely hyper-cubes and regular simplexes, and show that these primitives lead to equivalent systems in dimension n ≥ 2. The result is achieved by adopting a single set of definitions that works for both these classes of figures. The analysis of our logics shows that Tarski's choice to take sphere as the geometrical primitive might be intuitively justified but is not optimal from a technical viewpoint.

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Section: Literaturementioning

confidence: 99%