Peano's Philosophical Views between Structuralism and Logicism

Overview & Synopsis

The paper investigates Peano’s epistemology through a detailed analysis of several mathematical practices that can be considered as markers of logicism: the primacy of functions on relations or viceversa, the presence or absence of metatheoretical investigations, a syntactic or model-theoretic notion of interpretation, the role of definitions by abstraction, and the foundational or non-foundational role of axiomatics. Peano’s epistemology will be characterized as a form of structuralist algebraism, that differs both from the algebra of logic tradition that used mathematical symbols to express logical calculi, and from logical investigations centered on the effort to understand the functional nature of predication. Logic is the invariant that emerges by algebraic syntactic substitutions, and that precedes mathematics because of its abstract generality that makes it a structuring element of mathematics.