De la Práctica Euclidiana a la Práctica Hilbertiana: las Teorías del Área Plana

Abstract

This paper analyzes the theory of area developed by Euclid in the Elements and its modern reinterpretation in Hilbert?s influential monograph Foundations of Geometry. Particular attention is 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 argue 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 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, in an appendix logical relations among equivalence, comparison and addition of magnitudes are examined schematically in an abstract setting.