“A Picture of Geometrical Reality”: Hilbert’s Axiomatic Conception of Geometry in the Light of Heinrich Hertz’s Bildtheorie

Abstract

The paper outlines an interpretation of David Hilbert’s early axiomatic approach to geometry, i.e., the one developed between 1891 and 1905. It is argued that several aspects of this approach, which could be initially seen as problematic, can be better understood when Hilbert’s view is contrasted with one of its most important influences in this period: Heinrich Hertz’s “picture theory” [Bildtheorie]. Especially, it will be claimed that an examination of Hilbert’s axiomatic conception in the light of the Hertzian view of mechanical theories can be used to relieve the former from certain manifest tensions; more precisely, the tensions that arise when maintaining, at the same time, an abstract or formal axiomatic position and an empiricist view of geometry, which regards it as a kind of natural science.