The role of testimony in mathematics

Abstract

Mathematicians appear to have quite high standards for when they will rely on testimony. Many mathematicians require that a number of experts testify that they have checked the proof of a result p before they will rely on p in their own proofs without checking the proof of p. We examine why this is. We argue that for each expert who testifies that she has checked the proof of p and found no errors, the likelihood that the proof contains no substantial errors increases because different experts will validate the proof in different ways depending on their background knowledge and individual preferences. If this is correct, there is much to be gained for a mathematician from requiring that a number of experts have checked the proof of p before she will rely on p in her own proofs without checking the proof of p. In this way a mathematician can protect her own work and the work of others from errors. Our argument thus provides an explanation for mathematicians’ attitude towards relying on testimony.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    In an important paper, Geist et al. (2010) focus on the trustworthiness of peer review, but that is a different discussion.

  2. 2.

    Empirical accounts of mathematical practice that imply that different experts validate proofs differently include Weber (2008), Mejia-Ramos and Weber (2014) and Weber et al. (2014).

  3. 3.

    The interview quotes have been cleaned for readability.

  4. 4.

    Yet, efforts to completely formalize mathematical theorems in interactive theorem provers such as Isabelle/HOL or Lean is considered by some formal verificationists to be of particular importance in rooting out errors in the mathematical literature.

  5. 5.

    In many fields, a set of paradigmatic exemplars exist, but readers of a proof are likely to go beyond these exemplars when testing the claims as they can assume that the author of the proof has also tested the claims against the exemplars.

  6. 6.

    We thank an anonymous referee for pointing out the connection between our argument and this remark by Hume.

  7. 7.

    It is beyond the scope of this paper to discuss when a mathematician can rationally rely on a result p in her own proof without checking the proof of p. But we note that it is not only relevant to consider the likelihood that the proof of p is correct. It is also relevant to consider the cost of building on p in the new proof if the proof of p is incorrect (see, for example, Kitcher 1993, pp. 334–339; Fantl and McGrath 2002; Wilholt 2013). The more essential p is to the new proof, the higher is the cost of the proof of p being incorrect, everything else being equal.

  8. 8.

    Several scholars have drawn on game theory and decision theory in investigating benefits and risks of engaging in scientific misconduct, see e.g. Sun and Tian (2016), Lacetera and Zirulia (2009) and Wible (1992). Similarly, recent empirical studies have shown that authors of retracted papers tend to suffer a decline in overall citations, see e.g. Lu et al. (2013) and Azoulay et al. (2017).

  9. 9.

    Since there is a possibility that the author has only checked the proof of p superficially or is even lying about having proved p, a mathematician who relies on the truth of p solely on the basis of the testimony of the author will often to some extent have to just trust that the author has been thorough and is not lying. Often the mathematician will not know the author well enough to know for sure that the author has been thorough and is not lying, or otherwise be in a position to know for sure that the author has been thorough and is not lying (see Andersen 2014).

  10. 10.

    A way of incentivizing referees to be more thorough is to open up the peer review process and have non-anonymous referees. Open peer review would also make it possible to give mathematicians more credit for their work as peer reviewers. The possibility of opening up the peer review process in mathematics deserves careful consideration, but open peer review comes with problems of its own, and it is beyond the scope of this paper to discuss the practice of peer review in mathematics.

  11. 11.

    We thank a referee for pointing out the relevance of Condorcet’s jury theorem to our argument.

  12. 12.

    We thank an anonymous referee for pressing us to clarify this point.

  13. 13.

    Fallis (2011, pp. 166–172) and Easwaran (2015) examine how some level of epistemic autonomy of individual mathematicians can be valuable to the individual mathematicians and the mathematical community as a whole.

  14. 14.

    The exceptions are cases where a result is used a lot but only by the same few mathematicians which may lead to ‘bubbles’ of which a few have formed and burst through the history of mathematics.

References

  1. Andersen, H. (2014). Co-author responsibility. EMBO Reports,15, 914–918.

    Article  Google Scholar 

  2. Andersen, L. E. (2017). On the nature and role of peer review in mathematics. Accountability in Research,24, 177–192.

    Article  Google Scholar 

  3. Andersen, L. E. (2020). Acceptable gaps in mathematical proofs. Synthese,197, 233–247.

    Article  Google Scholar 

  4. Andersen, L. E., Johansen, M. W., & Sørensen, H. K. (2019). Mathematicians writing for mathematicians. Synthese.

  5. Arbib, M. A. (1990). A Piagetian perspective on mathematical construction. Synthese,84, 43–58.

    Article  Google Scholar 

  6. Azoulay, P., Bonatti, A., & Krieger, J. L. (2017). The career effects of scandal: Evidence from scientific retractions. Research Policy,46, 1552–1569.

    Article  Google Scholar 

  7. Baez, J. C. (2010). Math blogs. Notices of the AMS,57, 333.

    Google Scholar 

  8. Davis, P. J. (1972). Fidelity in mathematical discourse: Is one and one really two? The American Mathematical Monthly,79, 252–263.

    Article  Google Scholar 

  9. Easwaran, K. (2015). Rebutting and undercutting in mathematics. Philosophical Perspectives,29, 146–162.

    Article  Google Scholar 

  10. Fallis, D. (2011). Probabilistic proofs and the collective epistemic goals of mathematicians. In H. B. Schmid, M. Weber, & D. Sirtes (Eds.), Collective epistemology (pp. 157–175). Frankfurt am Main: Ontos Verlag.

    Google Scholar 

  11. Fantl, J., & McGrath, M. (2002). Evidence, pragmatics, and justification. The Philosophical Review,111, 67–94.

    Article  Google Scholar 

  12. 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 

  13. Hardwig, J. (1985). Epistemic dependence. Journal of Philosophy,82, 335–349.

    Article  Google Scholar 

  14. Hardwig, J. (1991). The role of trust in knowledge. Journal of Philosophy,88, 693–708.

    Article  Google Scholar 

  15. Hume, D. (1740/2009). A treatise of human nature: Being an attempt to introduce the experimental method of reasoning into moral subjects. The Floating Press.

  16. Kitcher, P. (1993). The advancement of science: Science without legend, objectivity without illusions. New York: Oxford University Press.

    Google Scholar 

  17. Kowalski, E. (2009). Quoting the great unknown. Blog post. http://blogs.ethz.ch/kowalski/2009/02/14/quoting-the-great-unknown/. Retrieved September 20, 2019.

  18. Lacetera, N., & Zirulia, L. (2009). The economics of scientific misconduct. Journal of Law Economics and Organization,27, 568–603.

    Article  Google Scholar 

  19. List, C. (2013). Social choice theory. In E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. https://plato.stanford.edu/entries/social-choice/. Retrieved January 26, 2020.

  20. Lu, S. F., Jin, G. Z., Uzzi, B., & Jones, B. (2013). The retraction penalty: Evidence from the Web of Science. Scientific Reports,3, 1–5.

    Google Scholar 

  21. Mejia-Ramos, J. P., & Weber, K. (2014). Why and how mathematicians read proofs: Further evidence from a survey study. Educational Studies in Mathematics,85, 161–173.

    Article  Google Scholar 

  22. Müller-Hill, E. (2011). Die epistemische Rolle formalisierbarer mathematischer Beweise. Inaugural-Dissertation. Bonn: Rheinischen Friedrich-Wilhelms-Universität. http://hss.ulb.uni-bonn.de/2011/2526/2526.htm. Retrieved September 20, 2019.

  23. Nias, V. (2012). How often do people read the work that they cite? MathOverflow. http://mathoverflow.net/questions/98821/. Retrieved September 20, 2019.

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

    Article  Google Scholar 

  25. Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica,15, 291–320.

    Article  Google Scholar 

  26. Sauvaget, T. (2010). Published results: when to take them for granted? MathOverflow. http://mathoverflow.net/questions/23758/. Retrieved September 20, 2019.

  27. Stillwell, J. (2016). Is it possible to have a research career while checking the proof of every theorem that you cite? https://mathoverflow.net/questions/237987/. MathOverflow. Retrieved October 12, 2019.

  28. Sun, Y., & Tian, R. (2016). Dishonest academic conduct: From the perspective of the utility function. Accountability in Research,23, 139–162.

    Article  Google Scholar 

  29. Thom, R. (1994). Response to ‘Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics’ by A. Jaffe and F. Quinn. Bulletin of the American Mathematical Society,30, 203–204.

    Google Scholar 

  30. Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education,39, 431–459.

    Google Scholar 

  31. Weber, K., Inglis, M., & Mejia-Ramos, J. P. (2014). How mathematicians obtain conviction: Implications for mathematics instruction and research on epistemic cognition. Educational Psychologist,49, 36–58.

    Article  Google Scholar 

  32. Wible, J. R. (1992). Fraud in science: An economic approach. Philosophy of the Social Sciences,22, 5–27.

    Article  Google Scholar 

  33. Wilholt, T. (2013). Epistemic trust in science. British Journal for the Philosophy of Science,64, 233–253.

    Article  Google Scholar 

Download references

Acknowledgements

We are very grateful to the interviewees for their time and support. The paper has benefited greatly from the feedback we received from three anonymous referees. We also thank Mikkel Willum Johansen for valuable feedback. Earlier versions of the paper were presented at: the OZSW Graduate Conference in Theoretical Philosophy at the University of Twente in 2016; the Centre for Logic and Philosophy of Science Colloquium at Vrije Universiteit Brussel in 2016; the Virtue Epistemology of Mathematical Practices workshop at Vrije Universiteit Brussel in 2018; and the Mathematical Collaboration III workshop at the University of Bristol in 2019. We thank the audiences for valuable feedback. Part of the research for this paper was conducted while LEA was a postdoctoral researcher at the Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel, Brussels, Belgium. At Aarhus University, she is supported by K. Brad Wray’s Grant, AUFF-E-2017-FLS-7-3.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Line Edslev Andersen.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Andersen, L.E., Andersen, H. & Sørensen, H.K. The role of testimony in mathematics. Synthese (2020). https://doi.org/10.1007/s11229-020-02734-9

Download citation

Keywords

  • Mathematics
  • Mathematical practice
  • Epistemic dependence
  • Testimony