From abstraction and indiscernibility to classification and types:

Abstract

At the end of the 19th century, the Peano School elaborated its famous theory of “definitions by abstraction”. Two decades later, Hermann Weyl elaborated a generalization of the former, termed “creative definitions”, capable of covering various cases of ideal elements (Peano’s abstracta being among them). If the Peano School proposal eventually appeared to be based on the nowadays standard classificatory process of quotienting a set by an equivalence, Weyl’s proposal still lacks a set-theoretical, classificatory interpretation. In this paper, we define and investigate the notion of relational indiscernibility (upon which Weyl’s creative definitions are based) and show that a bridge from the concept of indiscernibility to the notion of type (sets closed by bi-orthogonal) may be built from the observation that individuals are indiscernible exactly when they belong to exactly the same types. In the last part, we investigate some philosophical consequences of those observations concerning the theory of abstraction.