 The Two Sides of Modern Axiomatics: Dedekind and Peano, Hilbert and BourbakiJosé Ferreirós ()
Chapter Chapter 6 in Axiomatic Thinking I, 2022, pp 83-103 from Springer Abstract: 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. Date: 2022 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-77657-2_6 Ordering information: This item can be ordered from DOI: 10.1007/978-3-030-77657-2_6 Access Statistics for this chapter More chapters in Springer Books from Springer |
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