# Acceptable gaps in mathematical proofs

## Abstract

Mathematicians often intentionally leave gaps in their proofs. Based on interviews with mathematicians about their refereeing practices, this paper examines the character of intentional gaps in published proofs. We observe that mathematicians’ refereeing practices limit the number of certain intentional gaps in published proofs. The results provide some new perspectives on the traditional philosophical questions of the nature of proof and of what grounds mathematical knowledge.

## Keywords

Mathematical practice Gaps in proofs Peer review in mathematics The nature of proofs## Notes

### Acknowledgements

Part of the research for this paper was conducted while I was a postdoc at the Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel, Brussels, Belgium. Two years ago, I had the pleasure of interviewing eight mathematicians about their refereeing practices. I thank them for their time and openness. The paper has benefited greatly from comments from Henrik Kragh Sørensen, Karen Francois, Mikkel Willum Johansen, and two anonymous referees. I also thank Henrik and Mikkel for their help in developing the interview questions. Earlier versions of the paper were presented at the 2016 Society for Philosophy of Science in Practice conference (Glassboro, New Jersey), the 2016 Novembertagung on the history and philosophy of mathematics (Sønderborg, Denmark), the workshop ‘Mathematical evidence and argument: Historical, philosophical, and educational perspectives’ (Bremen, Germany, 2017), the workshop ‘Group knowledge and mathematical collaboration’ workshop (Oxford, UK, 2017), and the 2017 Nordic Network for Philosophy of Science meeting (Copenhagen, Denmark). I would like to thank the audiences for useful comments.

### Compliance with ethical standards

### Conflict of interest

The author declares that she has no conflict of interest.

## References

- Andersen, L. E. (2017). On the nature and role of peer review in mathematics.
*Accountability in Research,**24,*177–192.CrossRefGoogle Scholar - Azzouni, J. (2004). The derivation-indicator view of mathematical practice.
*Philosophia Mathematica,**12,*81–105.CrossRefGoogle Scholar - Azzouni, J. (2009). Why do informal proofs conform to formal norms?
*Foundations of Science,**14,*9–26.CrossRefGoogle Scholar - Baker, A. (2015). Non-deductive methods in mathematics. In E. N. Zalta (Ed.),
*The Stanford encyclopedia of philosophy*. https://plato.stanford.edu/archives/fall2015/entries/mathematics-nondeductive/. Accessed 14 December 2016. - Davis, P. J. (1972). Fidelity in mathematical discourse: Is one and one really two?
*The American Mathematical Monthly,**79,*252–263.CrossRefGoogle Scholar - Devlin, K. (2003). When is a proof? In
*Devlin’s angle*. A column written for the Mathematical Association of America. https://www.maa.org/external_archive/devlin/devlin_06_03.html. Accessed 14 December 2016. - Dutilh Novaes, C. (2016). Reductio ad absurdum from a dialogical perspective.
*Philosophical Studies,**173,*2605–2628.CrossRefGoogle Scholar - Dutilh Novaes, C. (2017). A dialogical conception of explanation in mathematical proofs
**(forthcoming)**.Google Scholar - Easwaran, K. (2009). Probabilistic proofs and transferability.
*Philosophia Mathematica,**17,*341–362.CrossRefGoogle Scholar - Ernest, P. (1994). The dialogical nature of mathematics. In P. Ernest (Ed.),
*Mathematics, education and philosophy: An international perspective*(pp. 33–48). London: Falmer Press.Google Scholar - Fallis, D. (2003). Intentional gaps in mathematical proofs.
*Synthese,**134,*45–69.CrossRefGoogle Scholar - Geist, C., Löwe, B., & Van Kerkhove, B. (2010). Peer review and knowledge by testimony in mathematics. In B. Löwe & T. Müller (Eds.),
*PhiMSAMP. Philosophy of mathematics: Sociological aspects and mathematical practice*(pp. 155–178). London: College Publications.Google Scholar - Grcar, J. (2013). Errors and corrections in mathematics literature.
*Notices of the American Mathematical Society,**60,*418–425.CrossRefGoogle Scholar - Grice, H. P. (2001).
*Aspects of reason*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Hamami, Y. (2014). Mathematical rigor, proof gap and the validity of mathematical inference.
*Philosophia Scientiæ,**18,*7–26.CrossRefGoogle Scholar - Hardwig, J. (1991). The role of trust in knowledge.
*Journal of Philosophy,**88,*693–708.CrossRefGoogle Scholar - Johansen, M. W., & Misfeldt, M. (2016). An empirical approach to the mathematical values of problem choice and argumentation. In B. Larvor (Ed.),
*Mathematical cultures: The London meetings 2012–2014*(pp. 259–269). Basel: Birkhäuser.Google Scholar - Müller-Hill, E. (2009). Formalizability and knowledge ascriptions in mathematical practice.
*Philosophia Scientiæ,**13,*21–43.CrossRefGoogle Scholar - Müller-Hill, E. (2011). Die epistemische Rolle formalisierbarer mathematischer Beweise. Formalisierbarkeitsorientierte Konzeptionen mathematischen Wissens und mathematischer Rechtfertigung innerhalb einer sozio-empirisch informierten Erkenntnistheorie der Mathematik. Inaugural-Dissertation. Bonn: Rheinische Friedrich-Wilhelms-Universität. http://hss.ulb.uni-bonn.de/2011/2526/2526.htm. Accessed 14 December 2016.
- Paseau, A. C. (2011). Mathematical instrumentalism, Gödel’s theorem, and inductive evidence.
*Studies in History and Philosophy of Science,**42,*140–149.CrossRefGoogle Scholar - Paseau, A. C. (2016). What’s the point of complete rigour?
*Mind,**125,*177–207.CrossRefGoogle Scholar - Pelc, A. (2009). Why do we believe theorems?
*Philosophia Mathematica,**17,*84–94.CrossRefGoogle Scholar - Prawitz, D. (2012). The epistemic significance of valid inference.
*Synthese,**187,*887–898.CrossRefGoogle Scholar - Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices.
*Philosophia Mathematica,**15,*291–320.CrossRefGoogle Scholar - Tanswell, F. (2015). A problem with the dependence of informal proofs on formal proofs.
*Philosophia Mathematica,**23,*295–310.CrossRefGoogle Scholar - Thurston, W. P. (1994). On proof and progress in mathematics.
*Bulletin of the American Mathematical Society,**30,*161–177.CrossRefGoogle Scholar - Weber, K. (2008). How mathematicians determine if an argument is a valid proof.
*Journal for Research in Mathematics Education,**39,*431–459.Google Scholar