A Working Mathematician between Philosophers: On the Logical Analysis of Magnitudes

GIOVANNINI, EDUARDO N.; LASSALLE CASANAVE, ABEL

A Question is More Illuminating than an Answer. A Festscrift for Paulo A. S. Veloso

College Publicaitons

Lugar: London; Año: 2021; p. 120 - 135

Much has been written about Hilbert?s strive for ?purity? in Foundations of geometry (1899), especially in relation to the exclusion of arithmetical concepts in the treatment of geometrical problems. But little has been written about another, no less fundamental, aspect of his program, namely the exclusion of the general concept of magnitude, a thesis which bears many resemblances to Dedekind?s program for the foundations of arithmetic. In this paper, we will illustrate this aspect of Hilbert?s geometrical enterprise by examining the elementary case of straight line segments, their operations, and relations. In particular, we will analyze how Hilbert explicitly avoided the use of principles for general magnitudes, while he proved their validity for the relevant geometrical objects from his geometrical axioms.