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Logic and Logical Philosophy

“The whole is greater than the part.” Mereology in Euclid's Elements
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“The whole is greater than the part.” Mereology in Euclid's Elements

Authors

  • Klaus Robering University of Southern Denmark

DOI:

https://doi.org/10.12775/LLP.2016.011

Keywords

atomistic mereology, convex geometry, Euclidean plane, polygons, points, continuum, measure theory

Abstract

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 Euclid’s system of geometry, megethology takes over the role played by the theory of congruence in modern accounts of geometry. Mereology and megethology are connected by Euclid’s Axiom 5: “The whole is greater than the part.” Section 4 compares Euclid’s theory of polygonal area, based on his “Whole-Greater-Than-Part” principle, to the account provided by Hilbert in his Grundlagen der Geometrie. An hypothesis is set forth why modern treatments of geometry abandon Euclid’s Axiom 5. Finally, in Section 5, the adequacy of atomistic mereology as a framework for a formal reconstruction of Euclid’s system of geometry is discussed.

Author Biography

Klaus Robering, University of Southern Denmark

Department of Communciation and Design

References

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Logic and Logical Philosophy

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Published

2016-05-27

How to Cite

1.
ROBERING, Klaus. “The whole is greater than the part.” Mereology in Euclid’s Elements. Logic and Logical Philosophy. Online. 27 May 2016. Vol. 25, no. 3, pp. 371-409. [Accessed 4 July 2025]. DOI 10.12775/LLP.2016.011.
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