Author:
AVIGAD JEREMY,MORRIS REBECCA
Abstract
AbstractIn 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly of higher-order, in that they involve quantifying over and summing overDirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation.In this essay, we study the history of Dirichlet’s theorem with an eye towards understanding the methodological pressures that influenced some of the ontological shifts that occurred in nineteenth century mathematics. In particular, we use the history to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment.
Publisher
Cambridge University Press (CUP)
Subject
Logic,Philosophy,Mathematics (miscellaneous)
Reference37 articles.
1. Pioneers of Representation Theory: Frobenius,
Burnside, Schur, and Brauer
2. Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält;Dirichlet;Abhandlungen der königlich Preussischen Akademie der Wissenschaften,1837b
3. Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939
4. Untersuchungen über die Theorie der complexen Zahlen;Dirichlet;Journal für die reine und angewandte Mathematik,1841
5. Sur la distribution des zéros de la fonction $\zeta(s)$ et ses conséquences arithmétiques
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