On Hilbert's foundations of the theory of plane area

GIOVANNINI, EDUARDO N.

Bologna

Congreso; Triennial International Conference of the Italian Society for Logic and Philosophy of Science; 2017

Italian Society for Logic and Philosophy of Science

The importance that Hilbert bestowed to ?purity? in his early axiomatic investigations is well known, and it has been recently analyzed in two important works (Hallett, 2008; Arana & Mancosu, 2012). In this context, ?purity? is tied to the requirement of the ?purity of methods of proof?, according to which theorems must be proved, if possible, using means that are suggested by their content (Cf. Majer & Hallett , 2004, pp. 315?316). A prominent example of this kind of purity inquire is the (im)possibility of finding a purely projective proof of Desargues? theorem in the plane, avoiding any kind of spatial assumptions. Now, it can be argued that purity demands were also operating more generally in Hilbert?s axiomatic construction of Euclidean geometry. To be more precise, a central concern that motivated Hilbert?s axiomatic investigations from very early on was the aim of providing an independent basis for geometry. By proving that one is not required to resort to any kind of numerical assumptions in the construction of a major part of elementary geometry, Hilbert was pursuing the central epistemological goal of showing that geometry should be considered, regarding its foundations, a self?sufficient or autonomous science. Then, a main goal of Hilbert?s axiomatization was not only to show that geometry should be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the constructionof such an axiomatic system one could proceed purely geometrically, avoiding concepts borrowed from other mathematical disciplines like arithmetic or analysis.The aim of this presentation is to analyze the relationship between these purity demands and Hilbert?s reconstruction of the theory of plane area in Foundations of Geometry. On the one hand, I will argued that the construction of this central part of elementary geometry presented a serious and appealing challenge to Hilbert?s general aim of providing a purely synthetic axiomatization of this geometrical theory; in other words, to his epistemological and methodological concerns of constructing elementary geometry without resorting to anykind of numerical assumption. On the other hand, I will claim that purity concerns were also behind Hilbert?s search for an ?elementary proof? of the axiom of De Zolt, i.e. a geometrical proposition whose validity is indispensable for the construction of the theory of plane area.Finally, I will conclude with a more general discussion on the role played by ?purity?, as a methodological and epistemological guiding principle, in Hilbert?s axiomatic construction of Euclidean geometry.