`Harmonizing Euclidean geometry': Hilbert and the theory of proportion

GIOVANNINI, EDUARDO N.

Salvador de Bahía

Congreso; 4th International Meeting of the Association for the Philosophy of Mathematical Practice; 2017

Universidad Federal de Bahía/ Association for the Philosophy of Mathematical Practice

The aim of this talk is to provide a historically sensitive discussion of Hilbert?s reconstruction of the theory of proportion in his groundbreaking monograph Foundations of geometry (1899). It will be argued that Hilbert bestowed to this reconstruction a crucial epistemological and methodological significance. On the one hand, the theory of proportion was for Hilbert one of the central parts of elementary geometry that called more urgently for a new solid foundation. On the other hand, an adequate ?purely geometrical? grounding for the notion of proportionality was essential for his main aim of providing an independent basis for geometry, since this notion was necessary for the reconstruction of other important parts of elementary geometry, such as the theories of similar triangles and plane area. The presentation will be structured in two main parts. In the first part, I will present and analyze some critical comments formulated by Hilbert to Euclid?s theory of proportion developed in Book V of the Elements. These critical remarks consist in pointing out that the Euclidean theory does not have a ?purely geometrical? character, since Euclid never explains what it means geometrically for two pairs of geometrical elements ? e.g., line segments ? to be proportional. Moreover, Hilbert observes that the Euclidean theory of proportion is grounded on an arithmetical basis, which can be noted in the fact that the definition of proportionality provided by Euclid requires the validity of a continuity principle such as the axiom of Archimedes. Then, I will argue that Hilbert?s objections to Euclid?s theory of proportion and similar triangles consisted not only in pointing out the existence of implicit assumptions, but also in raising very explicit purity concerns. In the second part of the presentation, I will expound briefly the technical content of Hilbert?s theory of proportion, that is, his definition of proportionality on the basis of the arithmetic of segments [Streckenrechnung]. Hilbert showed that, once the operations of sum and product of line segments have been defined in an adequate and purely geometrical way, it is possible to use the classical theorems of Desargues and Pascal to prove that these operations satisfy all the properties of an ordered field. This purely geometrical construction of a set of segments, which satisfies all the properties of an ordered field, allowed him to reconstruct the classical Euclidean theory of proportions and similar triangles, to which he finally resorted to perform an internal arithmetization of geometry. Hence, Hilbert produced a unification of two theories, which before were grounded on different foundations, giving at the same time a new answer to the problem of the role of numbers in geometry.