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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Notes
Lemmermeyer (2000) is a thorough and useful entry point to this literature.
A handful of noteworthy examples, in chronological order: Steiner (1978), Resnik and Kushner (1987), Hafner and Mancosu (2005), Tappenden (2005), Mancosu (2008), Lange (2009), Frans and Weber (2014), Lange (2014), Pincock (2015), Inglis and Mejía-Ramos (2019) and D’Alessandro (2020). Some of these and other works on mathematical explanation are discussed below, especially in Sects. 3 to 5.
A recent exception is Colyvan et al. (2018), which discusses a disagreement about how best to explain the free group theorem. Colyvan et al. suggest that there are at least two fundamentally different ways for a proof to be explanatory; they call the relevant properties “abstractness” and “constructiveness”, although neither these qualities nor their connection to explanation are analyzed in much detail.
Legendre (1830), p. 230.
A referee worries that these questions are predicated on a doubtful assumption about mathematical explanation: namely, that every mathematical fact has a unique correct explanation. As the referee points out, this claim seems false. Some mathematical facts seem not to have any explanation at all. And others seem to have more than one good explanation.
One could take this worry in two ways. The first interpretation is that I myself appear to be committed to the above assumptions. In fact, I’m not: I reject both the existence claim and the uniqueness claim. In particular, I don’t assume at the outset that we have explained or can explain QR, or that QR must have a single correct explanation if it has any. As far as I can tell, I haven’t said anything that presupposes or appears to endorse either claim.
The second interpretation is that the doubtful assumptions are made by the mathematicians whose views on QR are at issue. Perhaps the worry is that, if the disagreement about QR is based on mistaken endorsement of a false theory of explanation, then the disagreement ceases to look very interesting: rather than bothering ourselves about the details of these mathematicians’ views, we should just note their incorrect starting point and pay the controversy no further mind. I have two replies. First, it’s unclear that the QR debate would be uninteresting even if it partly depended on false assumptions. Second and more importantly, I see no reason to think that these mathematicians actually do endorse either the existence or uniqueness claim in general. As I show below, some of them do accept one kind of proof as explanatory and reject another kind as unexplanatory. But this isn’t inherently problematic. The claim that one has an explanation for QR doesn’t imply that every mathematical fact has an explanation. The claim that a certain purported explanation is unacceptable doesn’t imply that QR can have only one correct explanation. And even the claim that QR does have only one correct explanation doesn’t imply that this is true of every mathematical fact. So I don’t think the disagreement about QR is based on an obvious mistake of the above sort. Neither version of the worry, then, appears to be a real problem.
To be clear, the meaning I’m giving to ‘transparent’ should be understood as somewhat stipulative. Although it’s compatible with (and inspired by) the way many mathematicians use the term, I’m not claiming that my definition exactly captures what they all have in mind. Indeed, there’s probably some variation in different authors’ understanding of transparency.
An infamous example is Don Zagier’s “one-sentence proof” that every prime congruent to 1 modulo 4 is a sum of two squares (Zagier 1990).
In a similar vein, I’ve argued in D’Alessandro (2020) that some explanations aren’t grounded in objective dependence relations, and that some type of epistemic or cognitive account of explanation may be the best alternative.
Thanks in part to a 2015 special issue of Philosophia Mathematica on mathematical depth (Vol. 23, No. 2).
See D’Alessandro (2018) for a sustained argument that the reduction of arithmetic to set theory isn’t explanatory.
“The choice of this lake—the largest and deepest freshwater reservoir in the world—was determined by the high transparency of its water, its depth, and the ice cover that allows the installation of deep-water equipment during two months in winter” (Domogatsky 2015, p. 23).
The argument was Gauss’s third published proof, but the fifth proof he discovered. The conventional numbering of Gauss’s proofs is based on the order of publication.
Gauss’s lemma is the following statement, which is used in quite a few proofs of quadratic reciprocity. Let p be an odd prime and let a be an integer coprime to p. Consider the residues a (mod p), 2a (mod p), 3a (mod p), \(\ldots \) , \(\left( \frac{p-1}{2}\right) a\) (mod p). (These are all distinct.) Suppose that n of these residues are greater than p/2. Then, writing \(\left( \frac{a}{p}\right) \) for the Legendre symbol, we have \(\left( \frac{a}{p}\right) =\left( -1\right) ^{n}\).
I.e., so that the sum of an even and an odd parity is odd, and the sum of two odd or even parities is even.
Artin reciprocity says that two algebraic objects—the abelianization of a certain kind of Galois group, and a certain kind of idele class group—are isomorphic. Defining these groups in a precise way would take quite a few layers of definitions.
For the record, Steiner’s theory claims that a proof is explanatory when it’s “deformable” in a certain respect, while Kitcher’s account is a mathematical application of his well-known unificationist theory of scientific explanation. Some philosophers remain sympathetic to modified or restricted versions of these views. For instance, Weber and Verhoeven (2002) develop an amended Steinerian account that’s only intended to handle certain kinds of explanatory proof. And Tappenden (2005) is a sympathetic exploration of the idea that unification promotes explanation in mathematics. See D’Alessandro (2019) for more on Steiner’s and Kitcher’s views and their respective receptions.
References
Aigner, M., & Zeigler, G. M. (2010). Proofs from the book (4th ed.). Berlin: Springer.
Arana, A. (2015). On the depth of Szemerédi’s theorem. Philosophia Mathematica, 23, 163–176.
Baumgart, O. (2015). The quadratic reciprocity law: A collection of classical proofs (F. Lemmermeyer, Ed. and Trans.). Heidelberg: Birkhäuser.
Booß-Bavnbek, B., & Wojciechowski, K. P. (1993). Elliptic boundary problems for dirac operators. New York: Springer.
Cassels, J. W. S. (1950). The rational solutions of the diophantine equation. Acta Mathematica, 82, 243–273.
Colyvan, M. (2012). An introduction to the philosophy of mathematics. Cambridge: Cambridge University Press.
Colyvan, M., Cusbert, J., & McQueen, K. J. (2018). Two flavours of mathematical explanation. In A. Reutlinger & J. Saatsi (Eds.), Explanation beyond causation: Philosophical perspectives on non-causal explanations (pp. 231–249). New York: Oxford University Press.
Cox, D. A. (2013). Primes of the form\(x^{2}+ny^{2}\): Fermat, class field theory, and complex multiplication (2nd ed.). New York: Wiley.
D’Alessandro, W. (2018). Arithmetic, set theory, reduction and explanation. Synthese, 195, 5059–5089.
D’Alessandro, W. (2019). Explanation in mathematics: Proofs and practice. Philosophy Compass,. https://doi.org/10.1111/phc3.12629.
D’Alessandro, W. (2020). Viewing-as explanations and ontic dependence. Philosophical Studies, 177, 769–792.
Di Bucchianico, A., & Loeb, D. E. (1998). Natural exponential families and umbral calculus. In B. E. Sagan & R. P. Stanley (Eds.), Mathematical essays in honor of Gian-Carlo Rota (pp. 195–212). Boston: Birkhäuser.
Domogatsky, G. (2015). A new neutrino telescope for Lake Baikal. CERN Courier, 55, 23–24.
Dudley, U. (2009). A guide to elementary number theory. Washington, D.C: Mathematical Association of America.
Edwards, H. M. (1977). Fermat’s last theorem: A genetic introduction to algebraic number theory. New York: Springer.
Frans, J., & Weber, E. (2014). Mechanistic explanation and explanatory proofs in mathematics. Philosophia Mathematica, 22, 231–248.
Frei, G. (1994). The reciprocity law from Euler to Eisenstein. In S. Chikara, S. Mitsuo, & J. W. Dauben (Eds.), The intersection of history and mathematics (pp. 67–90). Basel: Birkhäuser Verlag.
Garrett, P. (2005). Cryptographic primitives. In P. Garrett & D. Lieman (Eds.), Public-key cryptography (pp. 1–62). Providence, RI: American Mathematical Society.
Gauss, C. F. (1863). Werke (Vol. II). Göttingen: Königlichen Gesellschaft der Wissenschaften.
Gauss, C. F. (1966). Disquisitiones arithmeticae (English edition, A. A. Clarke, Trans). New York: Springer.
Givental, A. (2006). The Pythagorean theorem: What is it about? American Mathematical Monthly, 113, 261–265.
Gouvêa, F. Q. (2015). Review of The quadratic reciprocity law: A collection of classical proofs, by Oswald Baumgart (F. Lemmermeyer, Ed. and Trans.). MAA Reviews, June 30, 2015, online at https://www.maa.org/press/maa-reviews/the-quadratic-reciprocity-law-a-collection-of-classical-proofs.
Gowers, T. (2000). The two cultures of mathematics. In V. Arnold, M. Atiyah, P. Lax, & B. Mazur (Eds.), Mathematics: Frontiers and perspectives (pp. 65–78). Providence, RI: American Mathematical Society.
Gowers, T. (2008). Mathematics, memory and mental arithmetic. In M. Leng, A. Paseau, & M. Potter (Eds.), Mathematical knowledge (pp. 33–58). Oxford: Oxford University Press.
Gray, J. (2015). Depth—A Gaussian tradition in mathematics. Philosophia Mathematica, 23, 177–195.
Gray, J. (2018). A history of abstract algebra: From algebraic equations to modern algebra. Cham: Springer.
Grimm, S. R. (2010). The goal of explanation. Studies in the History and Philosophy of Science, 41, 337–344.
Hafner, J., & Paolo, M. (2005). The varieties of mathematical explanation. In P. Mancosu, K. Frovin Jørgensen, & S. A. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 215–250). Berlin: Springer.
Hanna, G. (2018). Reflections on proof as explanation. In A. G. Stylianides & G. Harel (Eds.), Advances in mathematics education research on proof and proving: An international perspective (pp. 3–18). Cham: Springer.
Hardy, G. H. (2012). A mathematician’s apology. New York: Cambridge University Press.
Harriss, E. O. (2006). Review of “The Pythagorean theorem: What is it about?”, by Alexander Givental. Mathematical Reviews, MathSciNet, MR2204490.
Hecke, E. (1981). Lectures on the theory of algebraic numbers (G. U. Brauer & J. R. Goldman, Trans.). New York: Springer.
Hilbert, D. (1998). The theory of algebraic number fields (I. T. Adamson, Trans.). Berlin: Springer.
Inglis, M., & Mejía-Ramos, J. P. (2019). Functional explanation in mathematics. Synthese,. https://doi.org/10.1007/s11229-019-02234-5.
Ioffe, A. D. (2000). Review of A. A. Milyutin and N. P. Osmolovskii, Calculus of variations and optimal control. In D. G. Babbitt & J. E. Kister (Eds.), Featured reviews in mathematical reviews, 1997–1999 (pp. E63–E64). Providence, RI: American Mathematical Society.
Ireland, K., & Rosen, M. (1990). A classical introduction to modern number theory (2nd ed.). New York: Springer.
Janusz, G. J. (1996). Algebraic number fields (2nd ed.). Providence, RI: American Mathematical Society.
Kazdan, J. L. (1981). Another proof of Bianchi’s identity in Riemannian geometry. Proceedings of the American Mathematical Society, 81, 341–342.
Kedlaya, K. S. (2008). From quadratic reciprocity to class field theory. In T. Gowers, J. Barrow-Green, & I. Leader (Eds.), The princeton companion to mathematics (pp. 39–42). Princeton: Princeton University Press.
Kelp, C. (2016). Towards a knowledge-based account of understanding. In S. R. Grimm, C. Baumberger, & S. Ammon (Eds.), Explaining understanding: New perspectives from epistemology and philosophy of science (pp. 251–271). London: Routledge.
Khalifa, K. (2013). The role of explanation in understanding. British Journal for the Philosophy of Science, 64, 161–187.
Kitcher, P. (1989). Explanatory unification and the causal structure of the world. In P. Kitcher & W. Salmon (Eds.), Scientific explanation (Minnesota studies in the philosophy of science) (Vol. XIII, pp. 410–505). Minneapolis: University of Minnesota Press.
Knuuttila, T., & Merz, M. (2009). Understanding by modeling: An objectual approach. In H. W. De Regt, S. Leonelli, & K. Eigner (Eds.), Scientific understanding: Philosophical perspectives (pp. 146–168). Pittsburgh, PA: University of Pittsburgh Press.
Koch, H. (1991). Introduction to classical mathematics I: From the quadratic reciprocity law to the uniformization theorem (J. Stillwell, Trans.). Dordrecht: Springer.
Lange, M. (2009). Why proofs by mathematical induction are generally not explanatory. Analysis, 69, 203–211.
Lange, M. (2014). Aspects of mathematical explanation: Symmetry, unity, and salience. Philosophical Review, 123, 485–531.
Lange, M. (2015). Depth and explanation in mathematics. Philosophia Mathematica, 23, 196–214.
Lange, M. (2016). Because without cause: Non-causal explanations in science and mathematics. New York: Oxford University Press.
Lauenbacher, R. C., & Pengelley, D. J. (1994). Eisenstein’s misunderstood geometric proof of the quadratic reciprocity theorem. College Mathematics Journal, 25, 29–34.
Legendre, A.-M. (1830). Théorie des nombres (3rd ed., Vol. 1). Paris: Duprat.
Lemmermeyer, F. (2000). Reciprocity laws: From Euler to Eisenstein. Berlin: Springer.
Lemmermeyer, F. (2019). “Proofs of the quadratic reciprocity law.” Online at http://www.rzuser.uni-heidelberg.de/~hb3/rchrono.html. Accessed 18 June 2019.
Mancosu, P. (2008). Mathematical explanation: Why it matters. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 134–150). New York: Oxford University Press.
Manin, Y., & Panchishkin, A. (2005). Introduction to modern number theory: Fundamental problems, ideas and theories (2nd ed.). Springer: Berlin.
McLarty, C. (2007). The rising sea: Grothendieck on simplicity and generality. In J. J. Gray & K. H. Parshall (Eds.), Episodes in the history of modern algebra (1800–1950). Providence, RI: American Mathematical Society.
Nadler, S. B. (2010). A proof of Darboux’s theorem. American Mathematical Monthly, 117, 174–175.
Narins, B. (Ed.). (2001). World of mathematics (Vol. 1). New York: Gale Group.
Neukirch, J. (1999). Algebraic number theory (N. Schappacher, Trans.). Berlin: Springer.
Pincock, C. (2015). The unsolvability of the quintic: A case study in abstract mathematical explanation. Philosophers’ Imprint, 15(3), 1–19.
Raatikainen, P. (2018). “Gödel’s incompleteness theorems.” In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy (Fall 2018 Edition). https://plato.stanford.edu/archives/fall2018/entries/goedel-incompleteness/.
Raman-Sundström, M. (2016). The notion of fit as a mathematical value. In B. Larvor (Ed.), Mathematical cultures: The London meetings 2012–2014 (pp. 271–286). Cham: Springer.
Raman-Sundström, M., & Öhman, L.-D. (2016). Mathematical fit: A case study. Philosophia Mathematica, 26, 184–210.
Resnik, M. D., & Kushner, D. (1987). Explanation, independence and realism in mathematics. British Journal for the Philosophy of Science, 38, 141–158.
Rogawski, J. (2000). The nonabelian reciprocity law for local fields. Notices of the AMS, 47, 35–41.
Rousseau, G. (1991). On the quadratic reciprocity law. Journal of the Australian Mathematical Society (Series A), 51, 423–425.
Rowe, D. E. (1988). Gauss, Dirichlet, and the law of biquadratic reciprocity. Mathematical Intelligencer, 10, 13–26.
Samuel, P. (1970). Algebraic theory of numbers (A. J. Silberger, Trans.). Paris: Hermann.
Severini, T. A. (2005). Elements of distribution theory. New York: Cambridge University Press.
Shanks, D. (1978). Solved and unsolved problems in number theory (2nd ed.). New York: Chelsea Publishing Company.
Smith, D. E. (1959). A source book in mathematics (Vol. 1). New York: Dover.
Smith, H. J. S. (1894). The collected mathematical papers of Henry John Stephen Smith, (Vol. I), ed. J.W.L. Glaisher. Oxford: Clarendon Press.
Stanley, R. P. (2012). Enumerative Combinatorics (2nd ed., Vol. I). New York: Cambridge University Press.
Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135–151.
Stillwell, J. (2015). What does ‘depth’ mean in mathematics? Philosophia Mathematica, 23, 215–232.
Swan, R. G. (1963). An application of graph theory to algebra. Proceedings of the American Mathematical Society, 14, 367–373.
Tangedal, B. A. (2000). Eisenstein’s lemma and quadratic reciprocity for Jacobi symbols. Mathematics Magazine, 73, 130–134.
Tappenden, J. (2005). Proof style and understanding in mathematics I: Visualization, unification and axiom choice. In P. Mancosu, K. Jørgensen, & S. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics. Berlin: Springer.
Tappenden, J. (2008). Mathematical concepts and definitions. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 256–275). New York: Oxford University Press.
Trout, J. D. (2002). Scientific explanation and the sense of understanding. Philosophy of Science, 69, 212–233.
Turri, J. (2015). Understanding and the norm of explanation. Philosophia, 43, 1171–1175.
Urquhart, A. (2015). Mathematical depth. Philosophia Mathematica, 23, 233–241.
Waskan, J., Harmon, I., Horne, Z., Spino, J., & Clevenger, J. (2014). Explanatory anti-psychologism overturned by lay and scientific case classifications. Synthese, 191, 1013–1035.
Weber, E., & Verhoeven, L. (2002). Explanatory proofs in mathematics. Logique at Analyse, 179–180, 299–307.
Weyl, H. (1968). Algebraic theory of numbers. Princeton, NJ: Princeton University Press.
Wilkenfeld, D. A. (2014). Functional explaining: A new approach to the philosophy of explanation. Synthese, 191, 3367–3391.
Yap, A. (2011). Gauss’ quadratic reciprocity theorem and mathematical fruitfulness. Studies in History and Philosophy of Science, 42, 410–415.
Zagier, D. (1990). A one-sentence proof that every prime \(p\equiv 1\) (mod \(4\)) is a sum of two squares. American Mathematical Monthly, 97, 144.
Acknowledgements
An earlier version of this paper was my MS thesis in mathematics at the University of Illinois at Chicago. I’m grateful to my advisor, Ramin Takloo-Bighash, and my other committee members, Nathan Jones and Jamie Tappenden, for their help and support. Thanks especially to Ramin, who introduced me to quadratic reciprocity and whose faith, advice and good cheer were invaluable over the many months I worked on this project. Thanks to Dick Gross, Jordan Ellenberg and Keith Conrad for correspondence with Ramin and I about proofs of QR. Many thanks to Marc Lange for his characteristically perceptive and generous comments and for his ongoing support. Many thanks as well to Kenny Easwaran and the Texas A&M philosophy and math departments for their hospitality. As usual, Daniel Sutherland and Lauren Woomer gave me helpful feedback. Finally, thanks to the 2019 Midwest Philosophy of Mathematics Workshop participants—including John Baldwin, Paddy Blanchette, Bernd Buldt, Curtis Franks, Doug Marshall, Rebecca Morris, Daniel Nolan, Patrick Ryan, and Susan Vineberg—for their comments, discussion and encouragement.
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D’Alessandro, W. Proving quadratic reciprocity: explanation, disagreement, transparency and depth. Synthese 198, 8621–8664 (2021). https://doi.org/10.1007/s11229-020-02591-6
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DOI: https://doi.org/10.1007/s11229-020-02591-6