ABSTRACT
This chapter investigates Peano’s philosophical views through a detailed analysis of several mathematical practices that can be considered markers of logicism: the link between functions and relations, the role of metatheoretical investigations, the kind of semantics, the use of definitions by abstraction, and the foundational or non-foundational value of axiomatics. Peano’s view is characterized as a form of structural algebraism, which differs from both the algebra of logic tradition using mathematical symbols to express logical calculi and from Frege’s logical investigation centered on the effort to understand the functional nature of predication. The influence of Leibniz is present both in the development of a kind of semantics that is neither representational nor conceptualist (symbols mean linguistic items of ordinary language) and in the consideration of symbolic signs as mirroring what they stand for. The symbolic logic developed in Peano’s Formulario is the root of mathematics but not its foundation: like linguistic roots in ancient languages, logical symbols are a tool to grasp the invariant features of mathematical functions and operations.
