•  706
    Explicitism about Truth in Fiction
    British Journal of Aesthetics 56 (1): 53-65. 2016.
    The problem of truth in fiction concerns how to tell whether a given proposition is true in a given fiction. Thus far, the nearly universal consensus has been that some propositions are ‘implicitly true’ in some fictions: such propositions are not expressed by any explicit statements in the relevant work, but are nevertheless held to be true in those works on the basis of some other set of criteria. I call this family of views ‘implicitism’. I argue that implicitism faces serious problems, where…Read more
  •  404
    Arithmetic, Set Theory, Reduction and Explanation
    Synthese 195 (11): 5059-5089. 2018.
    Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to …Read more
  •  249
    Explanation in mathematics: Proofs and practice
    Philosophy Compass 14 (11). 2019.
    Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanat…Read more
  •  192
    Viewing-as explanations and ontic dependence
    Philosophical Studies 177 (3): 769-792. 2020.
    According to a widespread view in metaphysics and philosophy of science, all explanations involve relations of ontic dependence between the items appearing in the explanandum and the items appearing in the explanans. I argue that a family of mathematical cases, which I call “viewing-as explanations”, are incompatible with the Dependence Thesis. These cases, I claim, feature genuine explanations that aren’t supported by ontic dependence relations. Hence the thesis isn’t true in general. The first…Read more
  •  111
    Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive…Read more
  •  92
    This is a teaching and learning guide to accompany "Explanation in Mathematics: Proofs and Practice".
  •  66
    Mathematical Explanation beyond Explanatory Proof
    British Journal for the Philosophy of Science. 2017.
    Much recent work on mathematical explanation has presupposed that the phenomenon involves explanatory proofs in an essential way. I argue that this view, ‘proof chauvinism’, is false. I then look in some detail at the explanation of the solvability of polynomial equations provided by Galois theory, which has often been thought to revolve around an explanatory proof. The paper concludes with some general worries about the effects of chauvinism on the theory of mathematical explanation.
  •  14
    Mathematical Explanation beyond Explanatory Proof
    British Journal for the Philosophy of Science 71 (2): 581-603. 2020.
    Much recent work on mathematical explanation has presupposed that the phenomenon involves explanatory proofs in an essential way. I argue that this view, ‘proof chauvinism’, is false. I then look in some detail at the explanation of the solvability of polynomial equations provided by Galois theory, which has often been thought to revolve around an explanatory proof. The article concludes with some general worries about the effects of chauvinism on the theory of mathematical explanation. 1Introdu…Read more