Yvon Gauthier

Dans cet article, je compare les vues de Lautman et Herbrand sur la théorie des nombres et la philosophie de l’arithmétique. Je montre que, bien que Lautman eût avoué avoir été marqué par l’influence de Herbrand, les postures fondationnelles des deux amis divergent considérablement. Alors que Lautman versait dans un réalisme platonicien, Herbrand est resté fidèle au finitisme hilbertien. Il est vrai que Lautman était philosophe et que Herbrand était avant tout arithméticien et logicien, mais il demeure que l’oeuvre de Herbrand a une portée philosophique mieux accordée à la logique et aux mathématiques contemporaines.In this paper, I am contrasting Lautman’s and Herbrand’s views on number theory and philosophy of arithmetic. It is argued that despite the fact that Lautman had acknowledged Herbrand’s major influence on his own work, their foundational stances diverge profoundly. Lautman defended a variety of Platonism and Herbrand advocated a personal version of Hilbertian finitism. Of course, Lautman was a philosopher while Herbrand dealt mainly with number theory and logic. It remains though that Herbrand’s work is more in tune with contemporary logic and mathematics from a philosophical perspective

ABSTRACT: It is argued that the equivalence, which is usually postulated to hold between infinite descent and transfinite induction in the foundations of arithmetic uses the law of excluded middle through the use of a double negation on the infinite set of natural numbers and therefore cannot be admitted in intuitionistic logic and mathematics, and a fortiori in more radical constructivist foundational schemes. Moreover it is shown that the infinite descent used in Dedekind-Peano arithmetic does not correspond to the infinite descent of classical Fermatian arithmetic or number theory. However, from the point of view of classical logic, the principles of complete induction and transfinite induction, the least number principle and infinite descent are all equivalent. We find here a focal point for a foundational critique that aims for a a clarification of philosophical options in the foundations of logic and mathematics