Giuseppe Veronese: The Fascination of Infinity

Paolo Busotti (San Marino in Storia della Scienza) gives a talk at the MCMP Colloquium (7 May, 2015) titled "Giuseppe Veronese: The Fascination of Infinity". Abstract: Giuseppe Veronese (1854-1917) is one of the most interesting mathematicians lived between the end of the 19th century and the beginning of the 20th. He gave important contributions to geometry, in particular he developed the non-Archimedean geometries and David Hilbert (1862-1943) mentioned some of Veronese’s results in his Grundlagen der Geometrie. In connection to his geometrical researches, Veronese developed a theory of infinite numbers. In his huge (more than 600 pages) essay Fondamenti di geometria, 1891 (Foundations of geometry), Veronese premised an introduction which is a very treatise (about 200 pages) in which he developed a theory of the continuum and of the infinite numbers which was completely different from Cantor’s (1845-1918) and which, in the mind of his author, had to represent an alternative to Cantorian set theory. The great difference, in comparison to Cantor, was that Veronese admitted the existence of infinitesimal actual numbers, while Cantor always denied this possibility. Basing on his actual infinite and infinitesimal numbers Veronese constructed the continuum in a manner which is different from Cantor’s and Dedekind’s (1831-1916). Other mathematicians, as Paul Dubois-Reymond (1831-1889) and Otto Stolz (1842-1905) faced the problem of the infinite actual magnitudes in an original way, but they did not develop an entire theory, while Veronese did. From a mathematical point of view Veronese’s theory is problematic, because there are some serious inaccuracies and it is not developed in every detail. Nevertheless, the situation is very interesting from an epistemological and logical standpoint because many of the ideas carried out by Veronese were resumed by Abraham Robinson (1918-1995) in his famous book Non standard Analysis (1966), where a coherent theory of non-archimedean numbers is explained. Many of Robinson’s idea had already been expounded by Veronese, though in nuce. In my talk, I am going to explain Veronese’s theory of infinite numbers in comparison to Cantor’s as well as Veronese’s conception of the continuum.