# Mathematical Explanations that are Not Proofs

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## Abstract

Explanation in mathematics has recently attracted increased attention from philosophers. The central issue is taken to be how to distinguish between two types of mathematical proofs: those that explain why what they prove is true and those that merely prove theorems without explaining why they are true. This way of framing the issue neglects the possibility of mathematical explanations that are not proofs at all. This paper addresses what it would take for a non-proof to explain. The paper focuses on a particular example of an explanatory non-proof: an argument that mathematicians regard as explaining why a given theorem holds regarding the derivative of an infinite sum of differentiable functions. The paper contrasts this explanatory non-proof with various non-explanatory proofs (and non-explanatory nonproofs) of the same theorem. The paper offers an account of what makes the given non-proof explanatory. This account is motivated by investigating the difficulties that arise when we try to extend Mark Steiner’s influential account of explanatory proofs to cover this explanatory non-proof.

## References

- Baker, A. (2009). Mathematical accidents and the ends of explanation. In O. Bueno & Ø. Linnebo (Eds.),
*New waves in the philosophy of mathematics*(pp. 137–159). Basingstoke: Palgrave Macmillan.CrossRefGoogle Scholar - Berndt, B. C., & Rankin, R. A. (1995).
*Ramanujan: Letters and commentary*. Providence: American Mathematical Society.CrossRefGoogle Scholar - Bressoud, D. M. (1994).
*A radical approach to real analysis*. Washington, DC: Mathematical Association of America.Google Scholar - D’Alessandro, W. (forthcoming). Mathematical explanation beyond explanatory proof.
*British Journal for the Philosophy of Science.*Google Scholar - D’Angelo, J. P. (2002).
*Inequalities from complex analysis*. Washington, DC: Mathematical Association of America.CrossRefGoogle Scholar - Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra.
*Journal for Research in Mathematics Education,**31,*396–428.CrossRefGoogle Scholar - Hempel, C. G. (2001). Explanation and prediction by covering laws (1963). In J. H. Fetzer (Ed.),
*The philosophy of Carl G. Hempel*(pp. 69–86). Oxford: Oxford University Press.Google Scholar - Lange, M. (2010). What are mathematical coincidences and why does it matter?
*Mind, 119*, 301–340.CrossRefGoogle Scholar - Lange, M. (2014). Aspects of mathematical explanation: Symmetry, unity, and salience.
*Philosophical Review, 123*, 485–531.CrossRefGoogle Scholar - Lange, M. (2016).
*Because without cause*. New York: Oxford University Press.CrossRefGoogle Scholar - Leng, M. (2005). Mathematical explanation. In C. Cellucci & D. Gillies (Eds.),
*Mathematical reasoning and heuristics*(pp. 167–189). London: King’s College Publications.Google Scholar - Mancosu, P. (2008). Mathematical explanation: Why it matters. In P. Mancosu (Ed.),
*The philosophy of mathematical practice*(pp. 134–150). Oxford: Oxford University Press.CrossRefGoogle Scholar - Mudaly, V., & de Villiers, M. (2000). Learners’ needs for conviction and explanation within the context of dynamic geometry.
*Pythagoras,**52*(8), 20–23.Google Scholar - Nahin, P. (2009).
*Mrs. Perkins’s electric quilt*. Princeton: Princeton University Press.CrossRefGoogle Scholar - Peled, I., & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-examples that (also) explain.
*Focus on Learning Problems in Mathematics,**19,*49–61.Google Scholar - Resnik, M., & Kushner, D. (1987). Explanation, independence and realism in mathematics.
*The British Journal for the Philosophy of Science,**38,*141–158.CrossRefGoogle Scholar - Smith, G. H., & McLelland, G. J. (2003).
*On the shoulders of giants: A course in single variable calculus*. Sydney, NSW: University of New South Wales Press.Google Scholar - Spivak, M. (1980).
*Calculus*(2nd ed.). Berkeley: Publish or Perish.Google Scholar - Steiner, M. (1978). Mathematical explanation.
*Philosophical Studies,**34,*135–151.CrossRefGoogle Scholar