Giuseppe Veronese e i fondamenti della geometria [Giuseppe Veronese and the Foundations of Geometry]

Paola Cantu (1)

1 Dipartimento di Filosofia

The book revises certain errors in the history of philosophy and mathematics concerning the role and the meaning of Giuseppe Veronese in the foundations of geometry, showing the originality and the epistemological interest of his works. At the end of the 19th century Veronese suggested a radically new solution to the problem of continuity: he introduced a system of non-Archimedean magnitudes, infinitely great and small, and advanced an original solution to the question of the demarcation between geometry and arithmetic. The study shows why the priority of the discover of a non-Archimedean geometry should go to Veronese rather than to Hilbert, and not only for chronological reasons: Veronese’s philosophical and epistemological insight were deeper, as it was shown by the successive algebraic developments, which proved that Veronese’s system of numbers was an example of a non-Archimedean ordered structure that could include real numbers as a particular case. The author investigates Veronese’s works in the light of the relation between philosophical ideas and mathematical results: e.g. between the synthetic understanding of continuity and the construction of the non-Archimedean geometry, between the property of thinking and the development of hyper-spaces, between the role given to intuition and the definition of geometry as a mixed science.

Mots clés: geometry; foundations of mathematics; continuum; non-Archimedan; hyper-spaces; real numbers; magnitudes

Domaines: Philosophie

Paola Cantu. Giuseppe Veronese e i fondamenti della geometria [Giuseppe Veronese and the Foundations of Geometry]. Unicopli, 1999, 9788840005898. ⟨hal-01194312⟩