Independent Scholar
  •  8
    In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice …Read more
  •  13
    Prominent mathematician William Thurston was praised by other mathematicians for his intellectual generosity. But what does it mean to say Thurston was intellectually generous? And is being intellectually generous beneficial? To answer these questions I turn to virtue epistemology and, in particular, Roberts and Wood's (2007) analysis of intellectual generosity. By appealing to Thurston's own writings and interviewing mathematicians who knew and worked with him, I argue that Roberts and Wood's a…Read more
  •  1
    Plans and planning in mathematical proofs
    Review of Symbolic Logic 1-40. forthcoming.
    In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The sta…Read more
  •  22
    Motivated proofs: What they are, why they matter and how to write them
    Review of Symbolic Logic 13 (1): 23-46. 2020.
    Mathematicians judge proofs to possess, or lack, a variety of different qualities, including, for example, explanatory power, depth, purity, beauty and fit. Philosophers of mathematical practice have begun to investigate the nature of such qualities. However, mathematicians frequently draw attention to another desirable proof quality: being motivated. Intuitively, motivated proofs contain no "puzzling" steps, but they have received little further analysis. In this paper, I begin a philosoph…Read more
  •  24
    Scientific explanations are widely recognized to have instrumental value by helping scientists make predictions and control their environment. In this paper I raise, and provide a first analysis of, the question whether explanatory proofs in mathematics have analogous instrumental value. I first identify an important goal in mathematical practice: reusing resources from existing proofs to solve new problems. I then consider the more specific question: do explanatory proofs have instrumental valu…Read more
  •  20
    Opinion: Reproducibility failures are essential to scientific inquiry
    with A. David Redish, Erich Kummerfeld, and Alan Love
    Proceedings of the National Academy of Sciences 115 (20): 5042-5046. 2018.
    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. Suggested policy changes have been focused on statistical problems, such as p-hacking, and issues of experimental design and execution. However, “reproducibility” is a broad concept that includes a number of issues. Furthermore, reproducibility failures occur even in fields such as mathematics or computer science that do not…Read more
  •  22
    Dedekind’s structuralism is a crucial source for the structuralism of mathematical practice—with its focus on abstract concepts like groups and fields. It plays an equally central role for the structuralism of philosophical analysis—with its focus on particular mathematical objects like natural and real numbers. Tensions between these structuralisms are palpable in Dedekind’s work, but are resolved in his essay Was sind und was sollen die Zahlen? In a radical shift, Dedekind extends his mathemat…Read more
  •  20
    Character and object
    Review of Symbolic Logic 9 (3): 480-510. 2016.
    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 thi…Read more
  •  12
    The concept of “character” in Dirichlet’s theorem on primes in an arithmetic progression
    Archive for History of Exact Sciences 68 (3): 265-326. 2014.
    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 ofDirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method