Current fears of a "reproducibility crisis" have led researchers, sources of scientific funding, and the public to question both the efficacy and trustworthiness of science (1, 2). Suggested policy changes have been focused on statistical problems, such as p-hacking, and issues of experimental design and execution (3, 4). However, "reproducibility" is a broad concept that includes a number of issues (5) (see also www.pnas. org/improving_reproducibility). Furthermore, reproducibility failures occur even in fields such as mathematics or computer science that do not have statistical problems or issues with experimental design. Most importantly, these proposed policy changes ignore a core feature of the process of scientific inquiry that

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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. We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet's proof in the nineteenth and early twentieth centuries, with an eye towards understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method. * This essay draws extensively on the second author's Carnegie Mellon MS thesis [64]. We are grateful to Michael Detlefsen and the participants in his Ideals of Proof workshop, which provided feedback on portions of this material in July

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.

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