Teoremi configurazionali e coordinatizzazione dei piani affini

The non-Euclidean revolution has imposed the search for a foundation of mathematics. Therefore the language of mathematics has become more abstract and formal. As a consequence, in the last decades the university teaching of mathematics has absorbed these features and eventually lost all connections with the traditional language and approach.This has gradually turned the culture and training of the new generations of mathematicians much poorer and fragile. In order to contrast this fatal trend that strongly conditions the quality of mathematics teaching in all school levels, we started a study aimed at highlighting the links between the traditional language of geometry and the current one, so as to allow us to appreciate the advantages and qualities of both approaches.In the paper, in order to illustrate an important theorem of the foundation of Geometry, we face the case of the affine plane and we show the role of theorems by Desargues and Pappus in the coordinatization of the plane. This exposition highlights the connection between the geometrical properties assumed as axioms of the algebraic properties.