Abstract
In this paper we consider the role of trust in relation to mathematical proof. After some basic remarks on the main philosophical issues involved, we give material for five case studies, dealing in particular with a few recent disputes on controversial proofs (or purported proofs) of important conjectures. We try to assess some criteria of evaluation in order to obtain a possible philosophical lesson.
A good proof is one that makes us wiser
Yu. I. Manin
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Notes
- 1.
For a thorough discussion of these points one can see Bundy et al. (2005).
- 2.
See also Alama’s Formal proofs and refutations, unpublished Ph.D. Thesis, Stanford University, 2009.
- 3.
- 4.
As Ruelle (2000, 254) comments in an incisive way, ‘human mathematics is a sort of dance around an unwritten formal text, which if written would be unreadable’.
- 5.
It is very usual to find in published papers a footnote in which anonymous referees are acknowledged for their important contribute to get an improved version of the work.
- 6.
Hardwig (1991) is now a “classic” paper on the subject; to a related topic he had already devoted the paper (1985).
- 7.
We just recall Mercier and Sperber (2019).
- 8.
- 9.
And also that its refinement could yield the right consistency strength for NF, which is apparently much less than the strength of ZF, being that of Zermelo set theory with comprehension restricted to bounded formulas.
- 10.
The main references are given there. But see also, for a dissenting voice on the topic, Ivan Fesenko’s papers: https://www.maths.nottingham.ac.uk/plp/pmzibf/nov.html.
- 11.
As Pierre Deligne once famously quipped, ‘I believe in a proof if I understand it’.
References
Alama J, Kahle R (2013) Checking proofs. In: Aberdein A, Dove IJ (eds) The argument of mathematics. Springer, Dordrecht
Bundy A et al (2005) The nature of mathematical proof. Philos Trans R Soc A 363(1835)
Chow T (2018) FOM mailing list, Jan 2018. https://cs.nyu.edu/pipermail/fom/2018-January/
Coleman E (2009) The surveyability of long proofs. Found Sci 14:27–43
Detlefsen M, Luker M (1980) The four color theorem and mathematical proof. J Philos 77:803–820
Frans J, Kosolosky L (2014) Mathematical proofs in practice: revisiting the reliability of published mathematical proofs. Theoria (Revista) 81:345–360
Friedman H (2018) FOM mailing list, Jan 2018. https://cs.nyu.edu/pipermail/fom/2018-January/
Hales T et al (2008) A special issue on formal proof. Not Am Math Soc 55(11)
Hales T et al (2017) A formal proof of the Kepler conjecture. Forum Math \(\Pi \) 5:1-29
Hardwig J (1985) Epistemic dependence. J Philos 82:335–349
Hardwig J (1991) The role of trust in knowledge. J Philos 88:693–708
Klarreich E (2018) Titans of mathematics clash over epic proof of ABC conjecture. Quanta Magazine, 20 Sept 2018. https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/
Klimpel T (2015) Mathoverflow, 29 July 2015. https://mathoverflow.net/questions/132103/
Lakatos I (1976) Proofs and refutations. CUP, Cambridge
MacKenzie D (2001) Mechanizing proof. Computing, risk, and trust. The MIT Press, Cambridge, Mass
Mercier H, Sperber D (2019) The enigma of reason. A new theory of human understanding. Harvard UP, Cambridge, Mass
Primiero G, Kosolosky L (2013) The semantics of untrustworthiness. Topoi 35:253–266
Primiero G, Taddeo M (2012) A modal type theory for formalizing trusted communications. J Appl Logic 10:92–114
Ruelle D (2000) Conversations on mathematics with a visitor from outer space. In: Arnold VI et al (eds) Mathematics: frontiers and perspectives. American Mathematical Society, Providence, pp 251–260
Scanlon T (2010) Mathoverflow, 22 April 2010. https://mathoverflow.net/questions/22142/
Smith J (2018) FOM mailing list, Jan 2018. https://cs.nyu.edu/pipermail/fom/2018-January/
Voevodsky V (2014) The origins and motivations of Univalent foundations, 26 March 2014. https://www.ias.edu/ideas/2014/voevodsky-origins
Acknowledgements
We wish to thank Roberto Gronda, Paolo Mancosu and Giuseppe Primiero for helpful suggestions.
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Bellotti, L., Moriconi, E. (2020). On Trust in Mathematics: Some Case Studies. In: Fabris, A. (eds) Trust. Trust 2020. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-030-44018-3_7
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