In this talk, I shall claim that, on the one hand, Peano’s understanding of the axioms and the primitive terms which occur in them, and, on the other, the methodology adopted in his independence arguments amount to viewing the primitive terms as uninterpreted non-logical constants. The understanding of mathematical symbols in the context of independence arguments is a crucial step towards the formalisation of mathematical theories and, moreover, of the calculus of classes. However, I shall also claim that Peano’s incomplete separation between logical and interpreted language (the language of the calculus of classes), and the absence of a global programme regarding meta- mathematical investigations, makes him fall short of developing a genuine formal language.