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.
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In an important paper, Geist et al. (2010) focus on the trustworthiness of peer review, but that is a different discussion.
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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.
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.
We thank an anonymous referee for pointing out the connection between our argument and this remark by Hume.
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.
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).
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).
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.
We thank a referee for pointing out the relevance of Condorcet’s jury theorem to our argument.
We thank an anonymous referee for pressing us to clarify this point.
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.
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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.
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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
- Mathematical practice
- Epistemic dependence