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

• Authors: Robering Klaus
• Languages of publication: EN

Abstracts

EN

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.

Keywords

EN

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

• Publisher: Uniwersytet Mikołaja Kopernika (Nicolaus Copernicus University in Toruń)
• Journal: Logic and Logical Philosophy
• Year: 2016
• Volume: 25
• Issue: 3 Mereology and Beyond (II)
• Pages: 371-409

Dates

online

2016-05-27

Contributors

author

Robering Klaus

• Department of Communciation and Design, University of Southern Denmark, Kolding, Denmark

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Identifiers

DOI

10.12775/LLP.2016.011