Space, points and mereology. On foundations of point-free Euclidean geometry

@article{Gruszczynski2009SpacePA,
  title={Space, points and mereology. On foundations of point-free Euclidean geometry},
  author={Rafal Gruszczynski and A. Pietruszczak},
  journal={Logic and Logical Philosophy},
  year={2009},
  volume={18},
  pages={145-188}
}
This article is devoted to the problem of ontological foundations of three-dimensional Euclidean geometry. Starting from Bertrand Russell’s intuitions concerning the sensual world we try to show that it is possible to build a foundation for pure geometry by means of the so called regions of space. It is not our intention to present mathematically developed theory, but rather demonstrate basic assumptions, tools and techniques that are used in construction of systems of point-free geometry and… Expand
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References

SHOWING 1-10 OF 14 REFERENCES
Full Development of Tarski's Geometry of Solids
TLDR
It is shown that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. Expand
A Pointless Theory of Space Based on Strong Connection and Congruence
We present a logical theory of space where only tridimensional regions are assumed in the domain. Three distinct primitives are used to describe their mereological, topological and morphologicalExpand
Pointless Metric Spaces
TLDR
This paper proposes and examines a system of axioms for the pointless space theory in which “regions”, “inclusion”. Expand
Pieces of mereology
In this paper we will treat mereology as a theory of some structures that are not axiomatizable in an elementary language (one of the axioms will contain the predicate ‘belong’ (‘∈’) and we will useExpand
Axioms, Algebras and Topology
TLDR
Axiomatic theories provide a very general means for specifying the logical properties of formal concepts by interpreting logical operations as functions of these formal symbols corresponding to the formal symbols of the theory. Expand
Pieri's Structures
We present basic notions of Pier's system of geometry in an intuitive way and from a heuristic point of view, trying to focus on the "essence" of Italian mathematician's constructions. However, it isExpand
Connection Structures: Grzegorczyk's and Whitehead's Definitions of Point
TLDR
Two definitions of point in a system in which the inclusion relation and the relation of being separated were assumed as prim- itive are compared. Expand
Axiomatizability of geometry without points
The aim of this paper is to make more precise the well-known conviction that geometry may be built without speaking about points. In the first section we prepare some general syntactical theoremsExpand
Region-Based Topology
  • P. Roeper
  • Mathematics, Computer Science
  • J. Philos. Log.
  • 1997
A topological description of space is given, based on the relation of connection among regions and the property of being limited. A minimal set of 10 constraints is shown to permit definitions ofExpand
The Calculus of Individuals and Its Uses
TLDR
An unfortunate dependence of logical formulation upon the discovery and adoption of a special physical theory, or the presumption that such a suitable theory could in every case be discovered in the course of time, indicates serious deficiencies in the ordinary logistic. Expand
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