Andrew Arana (University of Paris 1), Non-Euclidean geometry and geometrical content

In 1868, Beltrami described himself as having identified a "real substrate” for hyperbolic geometry, "rather than admit the necessity for a new order of entities and concepts." He sought to show that hyperbolic geometry described the geometry on curved surfaces in "real", i.e. Euclidean, space. In 1873, Klein wrote that such investigations are "by no means intended to decide the validity of the parallel axiom, but only whether the parallel axiom is a mathematical consequence of the remaining axioms of Euclid". But relative consistency is a logical relationship between statements, whereas Beltrami's reduction of hyperbolic geometry to the geometry of the surfaces of Euclidean objects is a geometric relationship between objects. The logical relationship identified by Klein, and soon Poincaré and Hilbert, is the heart of what we now consider at issue in the legitimization of non-euclidean geometry. Yet the move from Beltrami's view to this "modern" one involves a considerable conceptual leap, in which geometers came to accept that they could "reinterpret" the geometrical terms of geometric statements. In this talk I want to consider the nature of this conceptual leap, toward better understanding this "modern" understanding of the content of geometrical statements.