# Arithmetic, Set Theory, Reduction and Explanation

*Synthese*195 (11):5059-5089 (2018)

**Abstract**

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 set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences.

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**Reprint years**

2018

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**PhilPapers/Archive ID**

DALAST-2

**Revision history**

Archival date: 2017-05-21

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References found in this work BETA

Word and Object.Quine, Willard Van Orman

Word and Object.Quine, Willard Van Orman

No Understanding Without Explanation.Strevens, Michael

Realism in Mathematics.MADDY, Penelope

The Structure of Science.Nagel, Ernest

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Citations of this work BETA

Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth.D'Alessandro, William

Explanation in Mathematics: Proofs and Practice.D'Alessandro, William

Viewing-as Explanations and Ontic Dependence.D’Alessandro, William

In Defense of Benacerraf’s Multiple-Reductions Argument.Ginammi, Michele

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