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The Two Sides of Modern Axiomatics: Dedekind and Peano, Hilbert and Bourbaki

In: Axiomatic Thinking I

Author

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  • José Ferreirós

    (Universidad de Sevilla)

Abstract

This chapter focuses on two different facets of axiomatics: 1. the formal-logical side, linked to careful, rigorous establishing of the inferential structure of a theory, and 2. the conceptual-mathematical side, often linked to the establishment of new interconnections in mathematics, or ‘deeper’ ways of grounding some results. We explore this question, first by offering some classic examples in papers due to Hilbert and Bourbaki (and Hausdorff), and then going on to a simple example: the treatment of arithmetic around 1889 in the hands of Dedekind and Peano. In this thumbnail example, one can already find the above-mentioned duality. It is usual to insist on the equivalence of the works of Peano and Dedekind, but we shall argue that they had different aims—Peano focused on elementary arithmetic and its precise formulation in a new artificial language, while Dedekind aimed to systematize and “deepen the foundations” of number theory (elementary or not). We offer arguments for these claims, including a discussion of “modern” number theory in the 19th century, and we close with some philosophical remarks.

Suggested Citation

  • José Ferreirós, 2022. "The Two Sides of Modern Axiomatics: Dedekind and Peano, Hilbert and Bourbaki," Springer Books, in: Fernando Ferreira & Reinhard Kahle & Giovanni Sommaruga (ed.), Axiomatic Thinking I, chapter 0, pages 83-103, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-77657-2_6
    DOI: 10.1007/978-3-030-77657-2_6
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