David Hilbert and the foundations of the theory of plane area

Abstract

This paper provides a detailed study of David Hilbert’s axiomatization of the theory of plane area, in the classical monograph Foundation of Geometry (1899). On the one hand, we offer a precise contextualization of this theory by considering it against its nineteenth-century geometrical background. Specifically, we examine some crucial steps in the emergence of the modern theory of geometrical equivalence. On the other hand, we analyze from a more conceptual perspective the significance of Hilbert’s theory of area for the foundational program pursued in Foundations. We argue that this theory played a fundamental role in the general attempt to provide a new independent basis for Euclidean geometry. Furthermore, we contend that our examination proves relevant for understanding the requirement of “purity of the method” in the tradition of modern synthetic geometry.

From Magnitudes to Geometry and Back: De Zolt's Postulate

“…Hilbert, 1898/1899 ). For a historical discussion of these developments and a detailed analysis of Hilbert's proof, see Giovannini ( 2021 ).…”