Arithmetizing the geometry from inside: David Hilbert's segment calculus

Abstract

On the basis of a set of unpublished notes for lecture courses on geometry, the paper seeks to contextualize and analyze one of the most important and original contributions of David Hilbert's celebrated monograph Foundations of Geometry (1899), namely its arithmetic of line segments (Streckenrechnungen). It is argued that Hilbert attributed to his arithmetic of segments an important epistemological and methodological meaning, in addition to its relevance as an original mathematical result. In particular, it is claimed that for Hilbert his arithmetic of segments represented a clear example of one of the most fruitful and attractive traits of his new formal axiomatic method, i.e., the power to discover and exhibit inner or structural connections among different mathematical theories.

Hilbert; Axiomatic method; Unity of mathematics; Euclidean geometry; Arithmetization