"The whole is greater than the part": Mereology in Euclid's elements

@article{Robering2016TheWI,
  title={"The whole is greater than the part": Mereology in Euclid's elements},
  author={Klaus Robering},
  journal={Logic and Logical Philosophy},
  year={2016},
  volume={25},
  pages={371-409}
}
  • K. Robering
  • Published 27 May 2016
  • Philosophy
  • Logic and Logical Philosophy
The present article provides a mereological analysis of Euclid’s planar geometry as presented in the first two books of his Elements . As a standard of comparison, a brief survey of the basic concepts of planar geometry formulated in a set-theoretic framework is given in Section 2. Section 3.2, then, develops the theories of incidence and order (of points on a line) using a blend of mereology and convex geometry. Section 3.3 explains Euclid’s “megethology”, i.e., his theory of magnitudes. In… 
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References

SHOWING 1-10 OF 23 REFERENCES
Regions-based two dimensional continua: The Euclidean case
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
Geometry: Euclid and Beyond
1. Euclid's Geometry.- 2. Hilbert's Axioms.- 3. Geometry over Fields.- 4. Segment Arithmetic.- 5. Area.- 6. Construction Problems and Field Extensions.- 7. Non-Euclidean Geometry.- 8. Polyhedra.-
Elementary Geometry
YOUR correspondent, “A Father,” has in view a very desirable object—to teach a young child geometry—but I fear that he is likely to miss altogether the path by which it may be reached. His principle,
Ontologies for Plane, Polygonal Mereotopology
TLDR
It is concluded that useful alternative ontologies of the plane are, if anything, less parsimonious than the one which they are supposed to replace.
Menger and Nöbeling on Pointless Topology
This paper looks at how the idea of pointless topology itself evolved during its pre-localic phase by analyzing the definitions of the concept of topological space of Menger and Nobeling. Menger put
The Aristotelian Continuum. A Formal Characterization
  • P. Roeper
  • Philosophy
    Notre Dame J. Formal Log.
  • 2006
TLDR
A formal account of Aristotle's conception of the linear continuum as continuous and infinitely divisible, without ultimate parts can be given employing a theory of quantification for nonatomic domains and a theories of region-based topology.
A Spatial Logic based on Regions and Connection
TLDR
An interval logic for reasoning about space is described, which supports a simpler ontology, has fewer functions and relations, yet does not su er in terms of its useful expressiveness.
The Complete Works of Aristotle the Revised Oxford Translation
The Oxford Translation of Aristotle was originally published in 12 volumes between 1912 and 1954. It is universally recognized as the standard English version of Aristotle. This revised edition
A more expressive formulation of many sorted logic
  • A. Cohn
  • Computer Science
    Journal of Automated Reasoning
  • 2004
TLDR
The many sorting logic described here has several unusual features which not only increase expressiveness but also can reduce the search space even more than a conventional many sorted logic.
...
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3
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