Unit:
Διαπανεπιστημιακό ΠΜΣ Διδακτική και Μεθοδολογία των ΜαθηματικώνLibrary of the School of Science
Author:
Γαλιουδάκης Εμμανουήλ
Supervisors info:
Στυλιανός Νεγρεπόντης Ομ. Καθηγ. (Eπιβλέπων), Βασιλική Φαρμάκη Καθηγ., Νικόλαος Παπαναστασίου Αναπλ. Καθηγ.
Original Title:
Η αξιωματική θεμελίωση της Ευκλείδειας γεωμετρίας κατά Hilbert στο πνεύμα των Στοιχείων του Ευκλείδη
Translated title:
The axiomatic foundation of Euclidean geometry by Hilbert based on Euclid's Elements
Summary:
Euclid’s Elements is the first known systematic effort for the axiomatic
foundation of Mathematics. Elements consist of thirteen books, includes
Definitions, Postulates, Common notions and Propositions and covers mainly
geometry, proportion theory and number theory.
In proofs of Elements’ propositions, some weaknesses have been observed.
Notions like the interior point of an angle, the intersection of two circles
are based on intuition and on figures instead of the five Postulates or the
common notions in Elements. The insufficiency of the five Postulates and common
notions lead to a different axiomatic foundation by Hilbert.
In this thesis we prove the Propositions from Books I, II, V and some from VI
of Elements based on Hilbert’s axiomatic system. Our aim is to stay as close as
possible to Elements thus we don’t introduce Hilbert’s axiomatic foundation
collectively but gradually and according to Elements’ order and needs.
During our attempt not to depart from the Elements, we developed the line
segments proportion theory following the Elements’ Book V, regarding line
segments. The theory developed up to this point allows us to do so without the
necessity of introducing algebraic operations between line segments. This
constitutes the main deviation from Hilbert’s axiomatic foundation.
Keywords:
Euclid, Axiomatic foundation, Elements, Point, Hilbert
