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1

Regions-based two dimensional continua: The Euclidean case

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Hellman G., Shapiro S.

Logic and Logical Philosophy

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2015

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vol. 24

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issue 4 Mereology and Beyond

499-534

EN

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 property for a class of readily postulated orientations of certain special regions, “generalized quadrilaterals” (intended as parallelograms), by which we cover the entire space. Then we generalize this to arbitrary orientations, and then establishing an isomorphism between the space and the usual point-based R × R. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause” (to the effect that “these are the only ways of generating regions”), and we have no axiom of induction other than ordinary numerical (mathematical) induction. Finally, having explicitly defined ‘point’ and ‘line’, we will derive the characteristic Parallel’s Postulate (Playfair axiom) from regions-based axioms, and point the way toward deriving key Euclidean metrical properties.

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

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Robering K.

Logic and Logical Philosophy

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2016

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vol. 25

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issue 3 Mereology and Beyond (II)

371-409

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.

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Typological, chronological and cultural verification of Pleistocene and early Holocene bone and antler harpoons and points from the southern Baltic zone

67%

Galiński T.

Przegląd Archeologiczny

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2013

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vol. 61

93-144

EN

The study catalogues all currently known finds of bone and antler harpoons and points associated with Paleolithic, Mesolithic and Protoneolithic culture in the southern Baltic zone, between the mouths of the Oder and Niemen rivers. It undertakes an analysis of the category in typological, chronological and cultural terms, taking into consideration results of recent paleogeographic investigations and research on the Stone Age in this region. An important element of this study are drawn plates of nearly all of the discussed objects as well as distribution maps. The author gives a critical analysis of the classic harpoon and point typology presented by J.G.D. Clark (1936) in the context of a broader source base, encompassing finds from the entire Baltic zone. A detailed morphological classification of harpoon barbs is one of the most important tools essential to this end.

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2 Logic and Logical Philosophy

1 Przegląd Archeologiczny

1 2016

1 2015

1 2013

1 Galiński T.

1 Hellman G.

1 Robering K.

1 Shapiro S.

Regions-based two dimensional continua: The Euclidean case

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

Typological, chronological and cultural verification of Pleistocene and early Holocene bone and antler harpoons and points from the southern Baltic zone