On a Geometric Foundation of Mathematics (Su una Fondazione Geometrica della Matematica)

Abstract

Frege with Grundlagen der Arithmetik and Hilbert with Grundlagen der Geometrie are two outstanding figures that are attributed to a fundamental role in the arithmetization of mathematics. However, the latest writings of Frege, released posthumously, testify to his reflection on the nature of mathematics. In them Frege argues that mathematics is all about geometry and begins a theory that aims to define complex numbers geometrically. For this purpose he introduced a notion of identical relationships that tends to set up a geometric aspect ratio. In addition, Grundlagen der Geometrie can be given a radically different reading from that which emphasizes Hilbert's exclusive intention to found geometry on a purely formal axiomatic system. Several authors argue that by his work, and in particular through the arithmetic of the segments introduced in it, Hilbert wanted to emancipate the geometry from instruments outside her, such as numbers, finding them within a substantially synthetic geometry.