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  1. The Structuralist Approach to Underdetermination.Chanwoo Lee - 2022 - Synthese 200 (2):1-25.
    This paper provides an exposition of the structuralist approach to underdetermination, which aims to resolve the underdetermination of theories by identifying their common theoretical structure. Applications of the structuralist approach can be found in many areas of philosophy. I present a schema of the structuralist approach, which conceptually unifies such applications in different subject matters. It is argued that two classic arguments in the literature, Paul Benacerraf’s argument on natural numbers and W. V. O. Quine’s argument for the indeterminacy of (...)
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  • In Defense of Benacerraf’s Multiple-Reductions Argument.Michele Ginammi - 2019 - Philosophia Mathematica 27 (2):276-288.
    I discuss Steinhart’s argument against Benacerraf’s famous multiple-reductions argument to the effect that numbers cannot be sets. Steinhart offers a mathematical argument according to which there is only one series of sets to which the natural numbers can be reduced, and thus attacks Benacerraf’s assumption that there are multiple reductions of numbers to sets. I will argue that Steinhart’s argument is problematic and should not be accepted.
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  • On Number-Set Identity: A Study.Sean C. Ebels-Duggan - forthcoming - Philosophia Mathematica.
    Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.
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  • Viewing-as Explanations and Ontic Dependence.William D’Alessandro - 2020 - Philosophical Studies 177 (3):769-792.
    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 part of the paper defends this claim (...)
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  • Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth.William D’Alessandro - 2020 - Synthese (9):1-44.
    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 geometric interpretation, due to Gauss (...)
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  • Explanation in Mathematics: Proofs and Practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11).
    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 explanatory proofs, and if so, how do (...)
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  • Breaking the Tie: Benacerraf’s Identification Argument Revisited.Arnon Avron & Balthasar Grabmayr - forthcoming - Philosophia Mathematica.
    ABSTRACT Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions (...)
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  • Abstraction and Idealization: Mutually Exclusive or Not?Mustafa Efe Ateş - 2021 - The Reasoner 15 (3):19-20.
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