Peano and Gödel

Résumé En

This paper will take into account several passages from Gödel’s unpublished philosophical manuscripts, the Max Phil, and compare them with relevant passages from the Formulary and from Russell’s Principia Mathematica on definite descriptions, definitions and functions. Gödel’s remarks together with a reconstruction of Peano’s general notion of function will be used to prove how intensively Gödel had worked on Peano’s writings and to oppose, or at least to restrict, the conceptual continuity between Peano and Russell outlined in recent literature. In particular, I will claim that a definite description that fails to satisfy the conditions of existence and uniqueness is meaningful in Peano, although it can not play the role of a definition. This is possible because 1) the function that characterizes definite descriptions is introduced as the inverse of the iota function, and 2) functions can be intended in a general sense, without a determined domain, or in a strict sense, and 3) only in the second case the conditions of existence and uniqueness need to be proved in order to use the definite description as a definition. I will conclude by suggesting some reasons that might explain Gödel’s interest in Peano’s approach.