Philosophy Lexicon of Arguments

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Euclid: Euclides of Megara, mathematician, 5th century BC. He was younger than Plato but older than Aristotle. His main works are the elements that develop his axioms. (See Der kleine Pauly, Lexikon der Antike, Munich 1979).

Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

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A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967

Axioms/Euclid/d'Abro: We know today that the explicit assumptions of Euclid are not sufficient as a basis for the Euclidean geometry. It has unconsciously made additional assumptions.
For example, in the equality of two triangles, he assumes congruence without having developed it. He had unconsciously assumed that the triangles can be moved.
Terminology: Euclid has distinguished between axioms and postulates. Today this distinction is no longer considered.
Hilbert's postulate system consists of 21 postulates that should define relationships between points, lines, and planes.
E.g. Continuity had been assumed tacitly by Euclid, and was explicitly demanded by Hilbert. ("Archimedean Postulate") Euclid was unconsciously guided by the idea of solid bodies.

Definition "Archimedian Postulate"/Hilbert: Assumption of Continuity (is assumed tacitly in Euclid).

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
d’Abro, A.

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Ed. Martin Schulz, access date 2018-04-23