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Do mathematical explanations have instrumental value?

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Abstract

Scientific explanations are widely recognized to have instrumental value by helping scientists make predictions and control their environment. In this paper I raise, and provide a first analysis of, the question whether explanatory proofs in mathematics have analogous instrumental value. I first identify an important goal in mathematical practice: reusing resources from existing proofs to solve new problems. I then consider the more specific question: do explanatory proofs have instrumental value by promoting reuse of the resources they contain? In general, I argue that the answer to this question is “no” and demonstrate this in detail for the theory of mathematical explanation developed by Marc Lange.

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Notes

  1. Zelcer (2013) is an exception.

  2. Weber and Verhoeven (2002) briefly discuss the value of mathematical explanations, focusing on unification. Mancosu (2008) has addressed the philosophical importance of mathematical explanations, highlighting the significance of this topic not only to philosophy of mathematics and science, but also to metaphysics and epistemology more broadly. My focus here, however, will be on the significance of mathematical explanations to mathematics and mathematicians. In other words, I will be concerned with the extent to which such explanations may be of benefit to mathematics and the mathematical community.

  3. I will not consider mathematical explanations of scientific facts. See Mancosu (2008) for a discussion of this terminology.

  4. For a summary of the basic concepts, see e.g. Sutherland (1975, Ch 1).

  5. I am grateful to an anonymous referee for pointing out this difference between reuse and prediction and control.

  6. I am grateful to an anonymous referee for pointing out this difference between reuse and explanatory goals in science.

  7. On Steiner’s account, “an explanatory proof depends on a characterizing property of something mentioned in the theorem: if we ‘deform’ the proof, substituting the characterizing property of a related entity, we get a related theorem” (Steiner 1978, p. 147). This seems to suggest that explanatory proofs, on Steiner’s account, will promote reuse, since the proof resources are reused when the proof is deformed to obtain new theorems. However, the situation is not clear cut (two anonymous referees had conflicting views, for example). It seems plausible, for instance, for a proof to be explanatory on Steiner’s account and yet badly presented or designed, making it difficult for mathematical agents to obtain the deformations, i.e. to reuse the resources in practice. In other words, it seems plausible for there to be explanatory proofs, on Steiner’s account, which are deformable in principle but not in practice. If that is correct, then explanatory proofs, on Steiner’s account, can’t be said to promote reuse after all.

  8. Kitcher’s account of explanation is a unificationist one. As we have seen in Sect. 2.3, unification has a quantitative nature, while reuse is more qualitative and agent-focused. Thus while Kitcher’s account of explanation may at first seem to have a strong connection to reuse, under closer inspection any such connection appears much weaker. In fact, Hafner and Mancosu (2008) argue that, according to his account, proofs that instantiate the decision procedure argument-scheme discussed in Sect. 2.3 are explanatory. However, we have already seen that such proofs manage information inefficiently and thus fail to promote reuse.

  9. While this proof is in the style of Lagrange, I have modified it to make it manage information even less efficiently than the original.

  10. The recurrence relation is given by the equations: \(A' = \frac{p(p-1)}{2}, 2A'' = \frac{p(p-1)(p-2)}{2\cdot 3} + \frac{(p-1)(p-2)}{2}A', \ldots \).

  11. Lagrange (1773) does break out lemmas in his original proof. This is one way in which the information management in the Lagrange style proof I gave above is worse than the information management in the original.

  12. This is not a criticism of Lagrange. He could not have chosen to use the resources of congruence notation because they were not invented when he wrote his proof.

References

  • Anguswamy, R. (2013). Factors affecting the design and use of reusable resources. Ph.D. dissertation, Virginia Polytechnic Institute and State University.

  • Avigad, J. (2018). Modularity in mathematics. The Review of Symbolic Logic. https://doi.org/10.1017/S1755020317000387.

  • Bressoud, D. (1999). Proofs and confirmations: The story of the alternating-sign matrix conjecture. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Douglas, H. (2009). Reintroducing prediction to explanation. Philosophy of Science, 76(4), 444–463.

    Article  Google Scholar 

  • Hafner, J., & Mancosu, P. (2008). Beyond unification. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 151–178). Oxford: Oxford University Press.

    Chapter  Google Scholar 

  • Hempel, C., & Oppenheim, P. (1948). Studies in the logic of explanation. Philosophy of Science, 15(2), 135–175.

    Article  Google Scholar 

  • Ivory, J. (1806). Demonstration of a theorem respecting prime numbers. In L. Leybourn (Ed.), The mathematical repository (pp. 6–8). Hatton Garden: Gledinning.

    Google Scholar 

  • Kitcher, P. (1998). Explanatory unification and the causal structure of the world. In P. Kitcher, & W. Salmon (Eds.), Scientific explanation (Vol. 8, pp. 410–505). Minneapolis: University of Minnesota Press, 1989.

  • Koshy, T. (2007). Elementary number theory with applications. Amsterdam: Elsevier.

    Google Scholar 

  • Lady, L. (n.d). How to do mathematical research. URL http://www.math.hawaii.edu/~lee/how-to.html. Archived at https://perma.cc/LZE6-ETTD

  • Lagrange, J.-L. (1773). Demonstration d’un théorème nouveau concernant les nombres premiers. Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres, 2, 125–137.

    Google Scholar 

  • Lange, M. (2014). Aspects of mathematical explanation: symmetry, unity, and salience. Philosophical Review, 123(4), 485–531.

    Article  Google Scholar 

  • Lange, M. (2009). Why proofs by mathematical induction are generally not explanatory. Analysis, 69(2), 203–211.

    Article  Google Scholar 

  • Lombrozo, T. (2011). The instrumental value of explanations. Philosophy Compass, 6(8), 539–551.

    Article  Google Scholar 

  • Mancosu, P. (2008). Mathematical explanation: Why it matters. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 134–149). Oxford: Oxford University Press.

    Chapter  Google Scholar 

  • Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(1), 5–41.

    Article  Google Scholar 

  • Rota, G.-C. (1997). The phenomenology of mathematical proof. Synthese, 111(2), 183–196.

    Article  Google Scholar 

  • Salmon, W. (1989). 4 decades of scientific explanation. Minnesota Studies in the Philosophy of Science, 13, 3–219.

    Google Scholar 

  • Smith, H. J. S. (1894). The collected mathematical papers of Henry John Stephen Smith. Oxford: The Clarendon Press.

    Google Scholar 

  • Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34(2), 135–151.

    Article  Google Scholar 

  • Strevens, M. (2006). Scientific explanation. In D. M. Borchert (Ed.), Encyclopedia of philosophy (2nd ed.) Reference.

  • Sutherland, W. (1975). Introduction to metric and topological spaces. Oxford: Oxford Science Publications.

    Google Scholar 

  • Weber, K. (2010). Proofs that develop insight. For the Learning of Mathematics, 30(1), 32–36.

    Google Scholar 

  • Weber, E., & Verhoeven, L. (2002). Explanatory proofs in mathematics. Logique & Analyse, 45(179/180), 299–307.

    Google Scholar 

  • Zelcer, M. (2013). Against mathematical explanation. Journal for General Philosophy of Science, 44(1), 173–192.

    Article  Google Scholar 

Download references

Acknowledgements

I am very grateful to Jeremy Avigad, Michael Friedman, Erich Kummerfeld and Wilfried Sieg for helpful feedback on drafts of this paper. I am also grateful to participants at the 2018 Stanford Workshop on Mathematical Reasoning for their helpful questions and discussions on reuse in mathematics. Finally I am grateful to the anonymous reviewers who provided helpful feedback and suggestions.

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  1. Suppes Center for History and Philosophy of Science, Stanford University, 450 Serra Mall, Main Quad, Building 90, Stanford, CA, 94305, USA

    Rebecca Lea Morris

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Morris, R.L. Do mathematical explanations have instrumental value?. Synthese 198, 1309–1328 (2021). https://doi.org/10.1007/s11229-019-02114-y

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