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Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth

D'Alessandro, William (2020) Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth. Synthese. ISSN 1573-0964

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Abstract

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 eighteenth 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 and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency— which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation.


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Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
D'Alessandro, Williamwdales2@uic.edu0000-0002-5451-079X
Keywords: Quadratic reciprocity, number theory, Gauss, mathematical explanation, explanation, explanation in mathematics, explanatory proof, philosophy of mathematics, mathematical practice, history of mathematics
Subjects: Specific Sciences > Mathematics > Epistemology
Specific Sciences > Mathematics > Explanation
Specific Sciences > Mathematics > History
Specific Sciences > Mathematics > Methodology
Specific Sciences > Mathematics > Practice
Specific Sciences > Mathematics > Proof
Specific Sciences > Mathematics > Values
Specific Sciences > Mathematics
Depositing User: Dr. William D'Alessandro
Date Deposited: 02 Mar 2020 16:13
Last Modified: 02 Mar 2020 16:13
Item ID: 16959
Journal or Publication Title: Synthese
Publisher: Springer (Springer Science+Business Media B.V.)
Official URL: https://link.springer.com/article/10.1007%2Fs11229...
DOI or Unique Handle: https://doi.org/10.1007/s11229-020-02591-6
Subjects: Specific Sciences > Mathematics > Epistemology
Specific Sciences > Mathematics > Explanation
Specific Sciences > Mathematics > History
Specific Sciences > Mathematics > Methodology
Specific Sciences > Mathematics > Practice
Specific Sciences > Mathematics > Proof
Specific Sciences > Mathematics > Values
Specific Sciences > Mathematics
Date: 1 March 2020
ISSN: 1573-0964
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/16959

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