Hilbert's \textit{Grundlagen der Geometrie} and their status in the history of the foundational debate (Q2443850)

The paper is devoted to the discussion of the meaning and influence of Hilbert's \textit{Grundlagen der Geometrie} (1899). This work is considered in a wide historical perspective taking into account ways of understanding mathematical theories and in particular geometry. The starting point is the conception of geometry by the Pythagoreans. Next, Euclid's axiomatic approach is considered and contributions of persons such as Aristotle, Heron, Descartes, Pascal, Hobbes, Tschirnhaus et al. are discussed. Further, the formal approach to geometry (Pasch, Wiener et al.) is considered. In the next sections the work \textit{Grundlagen der Geometrie} itself is presented and discussed. The author stresses the completeness axiom and the structuralistic elements in Hilbert's approach. The paper ends with considerations devoted to the evolution of the idea of relations between geometry as a mathematical theory and the real world.