Euclid and Hilbert on the theory of plane area

GIOVANNINI, EDUARDO N.

Santa Fe

Workshop; Workshop Internacional sobre pruebas y diagramas en lógica y matemática.; 2018

Facultad de Humanidades y Ciencias, UNL

The aim of this talk is to analyze the theory of area developed by Euclid in the Elements and its modern reinterpretation in Hilbert?s influential monograph Foundations of Geometry. Particular attention will be bestowed upon the role that two specific principles play in these theories, namely the famous common notion 5 and the geometrical proposition known as De Zolt's postulate. On the one hand, we will be argued that an adequate elucidation of how these two principles are conceptually related in the theories of Euclid and Hilbert is highly relevant for a better understanding of the respective geometrical practices. On the other hand, we will claim that these conceptual relations unveil interesting issues between the two main contemporary approaches to the study of area of plane rectilinear figures, i.e., the geometrical approach consisting in the geometrical theory of equivalence and the metrical approach based on the notion of measure of area. Finally, we will analyze how the different strategies commonly used to provide a proof of De Zolt's postulate raise interesting issues for the current discussions on the purity of method.