Investigações sobre sistemas axiomáticos na geometria euclidiana - Rodrigues, Douglas Alexandre [UNESP] (2014)
The objective of this research is to analyze the historical development of the classical work of geometry named The Elements and written by Euclid and the foundations of geometry Grundlangen der Geometrie (Foundations of Geometry) written by David Hilbert by studying the axiomatic structure of geometry dealt with by each author. The deductive rigor used by Euclid, which is based on the classical logic of Aristotle, has received several criticisms from modern mathematicians with regard to the gaps in its mathematical deductive system. The various uncertainties regarding the axiomatic system threatened its logical development and in the specific case of geometry, many discussions arose on the acceptance of the Euclid's fifth postulate. Only in the late nineteenth century, axiomatic systems reached deeper levels in the foundations of geometry and, in an attempt to complete the axiomatic geometry, Hilbert publishes “Grundlangen der Geometrie”, which is the axiomatic approach more widely adopted in the Euclidean geometry. In this context, we discuss the different concepts of classical and modern axiomatic systems , studying their logical meanings and its relations with the objects of geometry . As part of the final thoughts , this paper highlights some considerations on the concept of motion in geometry and a possible axiomatic approach to it