Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth

Synthese (9):1-44 (2020)
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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 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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William D'Alessandro
Ludwig Maximilians Universität, München

References found in this work

Explanatory Unification and the Causal Structure of the World.Philip Kitcher - 1989 - In Philip Kitcher & Wesley Salmon (eds.), Scientific Explanation. Minneapolis: University of Minnesota Press. pp. 410-505.
The Philosophy of Mathematical Practice.Paolo Mancosu (ed.) - 2008 - Oxford, England: Oxford University Press.
Scientific Explanation and the Sense of Understanding.J. D. Trout - 2002 - Philosophy of Science 69 (2):212-233.
Viewing-as Explanations and Ontic Dependence.William D’Alessandro - 2020 - Philosophical Studies 177 (3):769-792.

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