The paper focuses on the history of the “Archimedes axiom”, a name that was first given by Stolz in 1882 to a mathematical sentence occurring in Euclid’s Book V to express an intuitive content concerning quantity measurement in proportion theory. The name was later associated to the analysis of continuity in Hilbert’s Grundlagen, where it first expressed continuity tout court, and subsequently...
My talk will be divided into two parts: in the first and briefer one, I will explain the non-Archimedean numerical system of infinite and infinitesimal numbers devised by Giuseppe Veronese (1854-1917) in the long Introduction (more than 200 pages) to his monumental Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee (1891). Relying on his non-Archimedean numbers...
If the world were Archimedean, then statements such as the Euclidean parallel postulate, the existence of rectangles, the fact that perpendiculars to the sides of a right triangle intersect, would all be equivalent. Aristotle's axiom, that the distances between two legs of an angle grow without bounds, would be simply true in all models of absolute geometry if the Archimedean axiom were to...
Saša Popović, Revisiting the reception history of Veronese’s non-Archimedean geometry: Fondamenti di geometria between modernism and counter-modernism
Abstract. Following his important contributions to the projective geometry of hyperspaces from the 1870s and 80s, Giuseppe Veronese (1854–1917) introduced non-Archimedean geometry between 1889 and 1891, and elaborated it in a series of seminal...
