Character and Object

The concept of function permeates most branches of modern mathematics: from analysis, algebra and logic to probability and mathematical physics. However, the set-theoretic conception of function as a particular mathematical object (specifically, as a set of ordered pairs satisfying conditions of totality and single valuedness) was a long time in development. Indeed, historians generally cite the eighteenth century as the time during which the concept of function first made an explicit appearance. Moreover, from the time it was first introduced to the present day, the concept of function underwent significant development and changed considerably (see e.g. [Mon72], [You76], [Kle89], [Luz98a], and [Luz98b]). Most of the historical literature focuses primarily on the development of the function concept within mathematical analysis, but the concept also underwent significant changes within number theory. In this thesis, we will explore a facet of the development of the function concept in number theory by looking at the evolution of particular types of functions, and the associated philosophical consequences. In particular, we shall examine the history of certain functions called Dirichlet characters and Dirichlet L-functions used in the proof of a result first established by Dirichlet in 1837 [Dir37a]. This proof was (re)presented and discussed by various other mathematicians in the nineteenth century, including Dedekind [DD99], de la Vallée Poussin [dlVP96], [dlVP97], Hadamard [Had96] and Landau [Lan09], [Lan27] and appears in many modern textbooks (for example Everest and Ward [EW05]). An analysis of these various presentations reveals a distinct change in the way in which the characters were conceived, and in particular, we will see that mathematicians gradually became more willing to treat the characters as mathematical objects in their own right. We shall refer to this as the process of reification of characters and will explain how it is to be understood more precisely throughout the rest of the present paper. Moreover, we will see that there are important consequences associated with the different methods of working with and