Notions about the nature of Geometry

TL;DR

This paper explores the historical and philosophical evolution of Euclidean geometry's postulates, discusses the unification of geometries through Klein's Erlangen Program, and presents specific Cayley-Klein geometries with educational insights.

Contribution

It provides a historical and philosophical analysis of geometric postulates, reviews the unification of geometries via Klein's Erlangen Program, and introduces specific Cayley-Klein geometries with didactic perspectives.

Findings

Analysis of the evolution of Euclidean postulates

Overview of Klein's Erlangen Program and geometry unification

Presentation of three Cayley-Klein geometries

Abstract

In this paper we discuss, from a historical and philosophical point of view, a variation of the meaning of the five postulates in Euclidean Geometry and we make a short reference to D. Hilberts formalism. We examine, throughout the ages, the question what is Geometry by studying the segmentation and the unification of various geometries, been introduced by F. Kleins Erlangen Program. We concisely present three out of nine Cayley-Klein geometries and then we conclude with some didactic perspectives.