Character and object
In 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 higher-order, in that they involve quantifying over and summing over Dirichlet 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 describe an approach to the philosophy of mathematics in which it is an important task to understand the roles of our ontological posits and assess the extent to which they enable us to achieve our mathematical goals. We use the history of Dirichlet’s theorem 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. We also use these considerations to illuminate the formal treatment of functions and objects in Frege’s logical foundation, and we argue that his philosophical and logical decisions were influenced by many of the same factors.