Towards a history of the geometric foundations of mathematics

Many XIX^th century «geometers»—such as Bernhard Riemann, Hermann von Helmholtz, Felix Klein, Riccardo De Paolis, Mario Pieri, Henri Poincaré, Federigo Enriques, and others—played an important role in the discussion about the foundations of mathematics. But in contrast to Euclid's ideas, they did not simply identify “physical space» with the «space of the senses». On the basis of our experience in space, they intended to determine the main properties of space and put them at the very foundation of geometry. The axioms of geometry were hence based on active knowledge of space and were not aa priori, as in the case according to kantian philosophy. Moreover, in the last decade of the century some Italian mathematicians—De Paolis, Gino Fano, Pieri, and others—founded the concept of number itself on geometry, by using results of projective geometry. Arithmetic, was then founded on geometry and not reversely, as David Hilbert tried—without success—to do some years later.