The Formalisation of Logic in the late XIX and early XX centuries

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Peano’s structuralism and the birth of formal languages

Publication . Bertran-San-Millán, Joan

Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that compose these theories. Second, I shall claim that, in the context of independence arguments, Peano developed a schematic understanding of the axioms which, despite diverging in some respects from Dedekind’s construction of arithmetic, should be considered structuralist. From this stance I shall argue that this schematic understanding of the axioms anticipates the basic components of a formal language.

2022Journal article

Open access

Peano’s Geometry: From Empirical Foundations to Abstract Development

Publication . Bertran San-Millán, Joan

In Principii di Geometria (1889b) and ‘Sui fondamenti della Geometria’ (1894) Peano offers axiomatic presentations of projective geometry. There seems to be a tension in Peano's construction of geometry in these two works: on the one hand, Peano insists that the basic components of geometry must be founded on intuition, and, on the other, he advocates the axiomatic method and an abstract understanding of the axioms. By studying Peano’s empiricist remarks and his conception of the notion of mathematical proof, and by discussing his critique of Segre’s foundation of hyperspace geometry, I will argue that the tension can be dissolved if these two seemingly contradictory positions are understood as compatible stages of a single process of construction rather than conflicting options.

2023Book part

Open access