La concezione dell'infinito in Federigo Enriques

Bussotti, Paolo

Matematica, cultura e società, Série 1, Tome 1 (2016) no. 1, pp. 65-86

Résumé

In this paper, I deal with Enriques' concept of mathematical infinity and infinitesimal. As to infinity and infinitesimal in their potential form, Enriques' considerations are remarkable. Nevertheless, his conceptions are particularly original as far as the actual infinity and infinitesimal are concerned. Enriques accepts entities as Cantor's transfinite ordinal numbers, cardinalities as those of the denumerable and the continuum, Veronese's infinite and infinitesimal numbers, but refuses concepts as Cantor's $\aleph_1$, or propositions as Zermelo's axiom of choice. This might appear strange, but, in fact, I will try to show the refusal of a certain use of actual infinity is one of the keys to fully understand Enriques' conception of mathematics. Not only: the steps of this argumentation by Enriques will enable us to enter his thought, till reaching a full comprehension of the reason why mathematics, gnoseology, history of philosophy and science were strictly interconnected in his way of thinking. Thence, the theme of the actual infinity and infinitesimal is a good perspective lens to face several topoi of Enriques' speculation. Almost the whole paper is, thus, dedicated to this problem. However, in the conclusive part, I have added a section on potential infinity and infinitesimal because an explanation of how Enriques interpreted the birth of geometry and of philosophy itself can be offered by addressing these concepts.