David Hilbert and the foundations of the theory of plane area

GIOVANNINI, EDUARDO N.

ARCHIVE FOR HISTORY OF EXACT SCIENCES

SPRINGER

Lugar: Berlin; Año: 2021 vol. 75 p. 649 - 698

0003-9519

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.