Unit:
Κατεύθυνση Διδακτική και Μεθοδολογία των ΜαθηματικώνLibrary of the School of Science
Author:
Malesiou Konstantina
Supervisors info:
Δ. Λάππας Αναπλ. Καθηγητή Τμήμα Μαθηματικών
Ε. Ράπτης Καθηγητή Τμήμα Μαθηματικών
Π. Σπύρου τ. Αναπλ. Καθηγητή Τμήμα Μαθηματικών
Original Title:
Αξιώματα Αρχιμήδους – Ευδόξου. Από τα «Στοιχεία» του Ευκλείδη στις Μη Αρχιμήδειες Γεωμετρίες: Μια ιστορική διαδρομή με διδακτικές προεκτάσεις.
Translated title:
Archimedes' – Eudoxus' Axioms. From Euclid's "Elements" to Non-Archimedean Geometries: A historical approach with teaching implications.
Summary:
The purpose of this thesis is to present the division form of the Axiom about magnitudes measurement, referred as “Archimedes’ – Eudoxus’ Axiom” and to highlight its role, as it used in Ancient Greek Mathematics (: in Euclid’s “Elements” and in Archimedean Corpus), as well as in Modern Mathematics (: in Dedekind’s theory of Real numbers and Hilbert’s Axiomatic Foundation of Elementary Geometry).
The magnitudes which are usually studied (such as angles, segments, area) are subject to the “Archimedes’ – Eudoxus’ Axiom”. But there also exist geometric objects (such as the horned angles) for which the Axiom is not valid and they are referred as Non- Archimedean. We analyze how the Axiom is useful in demonstrating results through the “Method of Exhaustion”. It is also underlying the difference between “Archimedes’ Axiom” and “Eudoxus’ Axiom”.
The role of this Axiom was also present in early results of “Non-Euclidean Geometry”, as Saccheri and Legendre used it in their study of the sum of angles of a triangle in the so-called Neutral Geometry.
In a modern Analytic approach, there are developed Non-Archimedean ordered fields and Algebraic models for Hilbert’s Axiomatic Foundation of Elementary Geometry are constructed.
Finally, we conclude this thesis, by presenting a teaching proposal concerning the construction of a “Non-Archimedean Geometry” based on an abstract ordered field.
Main subject category:
Science
Keywords:
Eudoxus, Archimedes, Axiom, Hilbert, Non-Archimedean Geometry
