Geometry without Numbers: Hilbert’s Axiomatic Program Reconsidered

GIOVANNINI, EDUARDO N.

Orange, California

Conferencia; OCIE Seminar in the History and Philosophy of Mathematics and Logic; 2022

Chapman University

The aim of this talk is to discuss the geometrical program carried out by Hilbert in his influential book Foundations of geometry (1899). Hilbert´s axiomatization is often considered the most elaborated and systematic instance of the modern synthetic reconstruction of Euclidean geometry. The ´synthetic´ character of this axiomatization usually refers to the development of geometry independently of the concept of real number and, therefore, of continuity assumptions. First, I will offer a historical sensitive discussion of the shaping of modern synthetic geometry in the second half of the nineteenth century. I will argue that the modern idea of a synthetic approach to geometry also emerged in connection to other foundational programs in nineteenth-century mathematics outside the domain of geometry, such as the movement known as the arithmetization of analysis. Second, I will examine the key technical result in Hilbert´s program, namely the construction of a "geometrical theory of numbers". I will contend that the strategy of building a new independent foundation based on a segment arithmetic raises important questions concerning the proclaimed strictly synthetic nature of Hilbert´s axiomatization.