On the Cartesian significance of David Hilbert's Grundlagen der Geometrie

GIOVANNINI, EDUARDO N.

Zürich

Congreso; 5th International Meeting of the Association for the Philosophy of Mathematical Practice; 2020

Eidgenössische Technische Hochschule Zürich (ETH Zürich)

The aim of this talk is to explore the historical and philosophical significance of Hilbert? axiomatic investigations into geometric construction problems. More specifically, we will analyze the meaning of these investigations by considering them from the perspective of the central ?Cartesian? program in early modern geometry, which aimed at the classification of geometrical problems according to the simplest means for their solution. As is well known, throughout La géemétrie (1637), Descartes sketched a hierarchy of problems by sorting them out into classes according to the degree of their associated equations. In other words, he showed how the degree of the equation associated with a problem contained information about the constructability of the geometric problem itself, which could be also used to establish an algebra-based classification of problems.On the one hand, we will argue that Hilbert?s axiomatic investigations can be taken, from a conceptual perspective, as the accomplishment of Descartes? original geometrical program laid out in La géométrie (1637). In particular, we will claim that Hilbert?s main contribution to this programconsisted in providing rigorous proofs of the impossibility of solving geometric construction problems with certain restricted means. On the other hand, we will focus on the programmatic character of these investigations, for they prompted the development of a novel and fruitful mathematical theory, namely plane construction field theory.