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The Values of Mathematical Proofs

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Handbook of the History and Philosophy of Mathematical Practice

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Abstract

Proofs are central, and unique, to mathematics. They establish the truth of theorems and provide us with the most secure knowledge we can possess. It is thus perhaps unsurprising that philosophers once thought that the only value proofs have lies in establishing the truth of theorems. However, such a view is inconsistent with mathematical practice. If a proof’s only value is to show a theorem is true, then mathematicians would have no reason to reprove the same theorem in different ways, yet this is a practice they frequently engage in. This suggests that proofs can have a wide variety of values. My purpose in this paper is to provide a survey of some of them. I will discuss how proofs can have practical value, for example, by hel** mathematicians to make new discoveries or learn new mathematical tools, and how they can have abstract value, for example, by being beautiful or explanatory. I will also explore the relationships between different proof values.

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References

  • Arana A (2015) On the depth of Szemerédi’s theorem. Philos Math III 23(2):163–176

    Article  MathSciNet  MATH  Google Scholar 

  • Avigad J (2006) Mathematical method and proof. Synthese 153(1):105–159

    Article  MathSciNet  MATH  Google Scholar 

  • Avigad J (2018) Opinion: the mechanization of mathematics. Not Am Math Soc 65(06):1

    Article  MATH  Google Scholar 

  • Avigad J (2020) Modularity in mathematics. Rev Symb Logic 13(0):47–79

    Article  MathSciNet  MATH  Google Scholar 

  • Baron S, Colyvan M, Ripley D (2019) A counterfactual approach to explanation in mathematics. Philos Math III, December

    Google Scholar 

  • Breitenbach A (2015) Beauty in proofs: Kant on aesthetics in mathematics. Eur J Phil Sci 23(4):955–977

    Article  Google Scholar 

  • Breitenbach A, Rizza D (2018) Introduction to special issue: aesthetics in mathematics. Philos Math III 26(2):153–160

    Article  MathSciNet  MATH  Google Scholar 

  • Cellucci C (2008) The nature of mathematical explanation. Stud Hist Philos Sci B 39(2):202–210

    Article  Google Scholar 

  • Cellucci C (2014) Explanatory and non-explanatory demonstrations. In: Logic, methodology and philosophy of science. Proceedings of the Fourteenth International Congress. College Publications, pp 201–218

    Google Scholar 

  • Cellucci C (2015) Mathematical beauty, understanding, and discovery. Found Sci 20(4):339–355

    Article  MathSciNet  MATH  Google Scholar 

  • Corrádi K, Szadó S (1993) A generalized form of Hajós’ theorem. Commun Algebra 21(11):4119–4125

    Article  MATH  Google Scholar 

  • D’Alessandro W (forthcoming) Mathematical explanation beyond explanatory proof. Br J Philos Sci

    Google Scholar 

  • Dawson JW (2006) Why do mathematicians re-prove theorems? Philos Math III 14(3):269–286

    Article  MathSciNet  MATH  Google Scholar 

  • Dawson JW Jr (2015) Why prove it again? Alternative proofs in mathematical practice. Birkhäuser, Cham

    Book  MATH  Google Scholar 

  • de La Vallée Poussin CJ (1896) Recherches analytiques de la théorie des nombres premiers. Ann Soc Sci Bruxelles 20(0):183–256

    MATH  Google Scholar 

  • de Villiers M (2017) An explanatory, transformation geometry proof of a classic treasure-hunt problem and its generalization. Int J Math Educ Sci Technol 48(2):260–267

    Article  MathSciNet  MATH  Google Scholar 

  • Derks J (2005) A new proof for weber’s characterization of the random order values. Math Soc Sci 49(3):327–334

    Article  MathSciNet  MATH  Google Scholar 

  • Detlefsen M (2008) Purity as an ideal of proof. In: Mancosu P (ed) The philosophy of mathematical practice. Oxford University Press, Oxford, pp 179–197

    Chapter  Google Scholar 

  • Detlefsen M, Arana A (2011) Purity of methods. Philosophers’ Imprint 11

    Google Scholar 

  • Dutilh Novaes C (2019) The beauty (?) of mathematical proofs. In: Aberdein A, Inglis M (eds) Advances in experimental philosophy of logic and mathematics. Bloomsbury Academic, London, pp 63–93

    Google Scholar 

  • Easwaran K (2009) Probabilistic proofs and transferability. Philos Math 17(3):341–362

    Article  MathSciNet  MATH  Google Scholar 

  • Frans J, Weber E (2014) Mechanistic explanation and explanatory proofs in mathematics. Philos Math III 22(2):231–248

    Article  MathSciNet  MATH  Google Scholar 

  • Giaquinto M (2016) Mathematical proofs: the beautiful and the explanatory. J Humanist Math 6(1):52–72

    Article  MathSciNet  Google Scholar 

  • Granville A, Harper AJ, Soundararajan K (2018) A more intuitive proof of a sharp version of halász’s theorem. Proc Am Math Soc 146(10):4099–4104

    Article  MATH  Google Scholar 

  • Gray J (2015) Depth – a Gaussian tradition in mathematics. Philos Math III 23(2):177–195

    Article  MathSciNet  MATH  Google Scholar 

  • Hadamard J (1896) Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bulletin Soc Math France 24(0):199–220

    Article  MathSciNet  MATH  Google Scholar 

  • Hafner J, Mancosu P (2005) The varieties of mathematical explanation. In: Mancosu P, Jørgensen KF, Pedersen SA (eds) Visualization, explanation and reasoning styles in mathematics. Springer, Dordrecht, pp 215–250

    Chapter  MATH  Google Scholar 

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

    Chapter  Google Scholar 

  • Haigh G (1980) A ‘Natural’ approach to pick’s theorem. Math Gaz 64(429):173–177

    Article  MathSciNet  Google Scholar 

  • Hanna G (1995) Challenges to the importance of proof. Learn Math 15(3):42–49

    Google Scholar 

  • Hanna G (2018) Reflections on proof as explanation. In: Stylianides AJ, Harel G (eds) Advances in mathematics education research on proof and proving: an international perspective. Springer International Publishing, Cham, pp 3–18

    Chapter  Google Scholar 

  • Hanna G, Larvor B (2020) As thurston says? On using quotations from famous mathematicians to make points about philosophy and education. ZDM, March

    Google Scholar 

  • Hardy GH (1967) A Mathematician’s apology. Cambridge University Press, reissue edition edition

    Google Scholar 

  • Hardy GH (1992) A Mathematician’s apology. Cambridge University Press, January

    Google Scholar 

  • Heawood PJ (1890) Map-Color theorem. Quar J Math 24(0):332–338

    MATH  Google Scholar 

  • Hoffman P (1998) The man who loved only numbers: the story of Paul Erdös and the search for mathematical truth. Hyperion, 1st edn, January

    Google Scholar 

  • Hua LK (2012) Introduction to number theory. Springer Science & Business Media

    Google Scholar 

  • Inglis M, Aberdein A (2015) Beauty is not simplicity: an analysis of mathematicians’ proof appraisals. Philos Math III 23(1):87–109

    Article  MathSciNet  MATH  Google Scholar 

  • Inglis M, Aberdein A (2016) Diversity in proof appraisal. In: Mathematical cultures. Springer International Publishing, pp 163–179

    Google Scholar 

  • Inglis M, Meja-Ramos JP (2019) Functional explanation in mathematics. Synthese, May

    Google Scholar 

  • Kempe AB (1879) On the geographical problem of the four colours. Am J Math 2(3):193–200

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  • Knopp M, Robins S (2001) Easy proofs of Riemann’s functional equation for ζ(s) and of lipschitz summation. Proc Am Math Soc

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Lange M (2015) Depth and explanation in mathematics. Philos Math III 23(2):196–214

    Article  MathSciNet  MATH  Google Scholar 

  • Lange M (2016a) Because without cause: non-causal explanations in science and mathematics. Oxford University Press, New York

    Book  MATH  Google Scholar 

  • Lange M (2016b) Explanatory proofs and beautiful proofs. J Humanist Math 6(1):8–51

    Article  MathSciNet  Google Scholar 

  • Lemmermeyer F (n.d.) Proofs of the quadratic reciprocity law. https://www.rzuser.uni-heidelberg.de/hb3/rchrono.html. Accessed 30 Mar 2020

  • Levinson N (1969) A motivated account of an elementary proof of the prime number theorem. Am Math Monthly 76(3):225–245

    Article  MathSciNet  MATH  Google Scholar 

  • Loomis ES (1968) The Pythagorean proposition. National Council of Teachers of Mathematics

    Google Scholar 

  • Marsden JE (1985) Review of Mill, F., “An effective potential for classical Yang-Mills fields as outline for bifurcation on gauge orbit space”. Ann Phys Rehabil Med 149:179–202. (MR# 85c:58080)

    Google Scholar 

  • Montaño U (2012) Ugly mathematics: why do mathematicians dislike computer-assisted proofs? Math Intell 34(4):21–28

    Article  MathSciNet  MATH  Google Scholar 

  • Morris RL (2015) Appropriate steps: a theory of motivated proofs. PhD thesis, Carnegie Mellon University

    Google Scholar 

  • Morris RL (2020) Motivated proofs: what they are, why they matter and how to write them. Rev Symb Logic 13(0):23–46

    Article  MathSciNet  MATH  Google Scholar 

  • Morris RL (forthcoming) Do mathematical explanations have instrumental value? Synthese, February

    Google Scholar 

  • Pincock C (2015) The unsolvability of the quintic: a case study in abstract mathematical explanation. Philos’ Imp 15(3):1–19

    Google Scholar 

  • Pólya G (1949) With, or without, motivation? Am Math Monthly 56(10):684–691

    Article  MathSciNet  Google Scholar 

  • Raman-Sundström M (2016) The notion of fit as a mathematical value. In: Mathematical cultures. Springer International Publishing, Berlin, pp 271–285

    Chapter  Google Scholar 

  • Raman-Sundström M, Öhman L-D (2018) Mathematical fit: a case study. Philos Math 26(2):184–210

    MathSciNet  MATH  Google Scholar 

  • Rav Y (1999) Why do we prove theorems? Philos Math 7(1):5–41

    Article  MathSciNet  MATH  Google Scholar 

  • Resnik MD, Kushner D (1987) Explanation, independence and realism in mathematics. Br J Philos Sci 38(2):141–158

    Article  MathSciNet  MATH  Google Scholar 

  • Rolen L (2015) A new construction of eisenstein’s completion of the weierstrass zeta function. Proc Am Math Soc 144(4):1453–1456

    Article  MathSciNet  MATH  Google Scholar 

  • Rota G-C (1997) The phenomenology of mathematical beauty. Synthese 111(2):171–182

    Article  MathSciNet  MATH  Google Scholar 

  • Russell B 1907 The study of mathematics. The New Quarterly 1

    Google Scholar 

  • Russell B (1919) The study of mathematics. In: Mysticism and logic. Longman, London

    Google Scholar 

  • Sandborg D (1997) Explanation in mathematical practice. PhD thesis, University of Pittsburgh

    Google Scholar 

  • Scovel JC (1985) A simple intuitive proof of a theorem in degree theory for gradient map**s. Proc Am Math Soc 93(4)

    Google Scholar 

  • Selberg A (1949) An elementary proof of the prime-number theorem. Ann Math 50(2):305–313

    Article  MathSciNet  MATH  Google Scholar 

  • Sieg W (2010) Searching for proofs (and uncovering capacities of the mathematical mind). In: Feferman S, Sieg W (eds) Proofs, categories and computations: essays in honor of Grigori Mints. College Publications, London, pp 189–215

    Google Scholar 

  • Smith SP (1991) Review of Artin, M., and Van den Bergh, M., “Twisted homogeneous coordinate rings”. J Algebra 133 (1990), 249–271. (MR# 91k:14003)

    Google Scholar 

  • Steiner M (1978) Mathematical explanation. Philos Stud 34(2):135–151

    Article  MathSciNet  Google Scholar 

  • Stillwell J (2015) What does ‘depth’ mean in mathematics? Philos Math III 23(2):215–232

    Article  MathSciNet  MATH  Google Scholar 

  • Strevens M (2011) Depth: an account of scientific explanation. Harvard University Press, Cambridge, MA

    Book  Google Scholar 

  • Urquhart A (2015) Mathematical depth. Philos Math III 23(2):233–241

    Article  MathSciNet  MATH  Google Scholar 

  • Michael De Villiers. The role and function of proof in mathematics. Pythagoras, 24(24):17–24, 1990

    Google Scholar 

  • Weber K (2010) Proofs that develop insight. Learn Math 30(1):32–36

    Google Scholar 

  • Weber E, Verhoeven L (2002) Explanatory proofs in mathematics. Log Anal 45(179–180):9–180

    MathSciNet  MATH  Google Scholar 

  • Wechsler J (1988) On aesthetics in science (Design science collection). Birkhäuser, 1988 edition, February 1988

    Google Scholar 

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Authors and Affiliations

  1. Independent Scholar, Minneapolis, MN, USA

    Rebecca Lea Morris

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  1. Rebecca Lea Morris
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Correspondence to Rebecca Lea Morris .

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Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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  1. Vrije Universiteit Brussel, Brussels, Belgium

    Bart Van Kerkhove

  2. Vrije Universiteit Brussel, Brussels, Belgium

    Joachim Frans

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Morris, R.L. (2021). The Values of Mathematical Proofs. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_34-1

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  • DOI: https://doi.org/10.1007/978-3-030-19071-2_34-1

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