Elsevier

Historia Mathematica

Volume 44, Issue 1, February 2017, Pages 31-69
Historia Mathematica

Strategical use(s) of arithmetic in Richard Dedekind and Heinrich Weber's Theorie der algebraischen Funktionen einer Veränderlichen

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Abstract

In this paper, I study Richard Dedekind and Heinrich Weber's 1882 Theorie der algebraischen Funktionen einer Veränderlichen, with a focus on the inherently arithmetical aspects of their work. I show that their paper provides an arithmetical rewriting of Riemannian function theory, i.e. a rewriting built on elementary arithmetical notions such as divisibility. I start with contextual elements concerning what is “arithmetical”, to put Dedekind and Weber's works into perspective from that viewpoint. Then, through a detailed analysis of the 1882 paper and using elements of their correspondence, I suggest that Dedekind and Weber deploy a strategy of rewriting parts of mathematics using arithmetic, and that this strategy is essentially related to Dedekind's specific conception of numbers and arithmetic as intrinsically linked to the human mind.

MSC

01A55
14-03
11-03

Keywords

Algebraic functions
19th century
Arithmetic
Dedekind
Weber
Riemann

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