Abstract
We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation.
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Notes
An excellent overview of the relevant literature is provided in Aberdein (2014).
A similar point was made by Kuhn (1977, pp. 320–339).
A point already made in Andrew Aberdein’s “Observations on Sick Mathematics”; Aberdein (2010).
Indeed, this is a possible explanation for numerous cases where the diagnosis rates differ between sexes for some condition. For example, cluster headaches are far more commonly diagnosed for men than women, but there are indicators that the actual incidence rates are roughly equal, with female sufferers often diagnosed with migraines instead, which are much less severe. Testimonial injustice in healthcare generally is discussed in Carel and Kidd (2014, p. 338).
Note that Fricker distinguishes between incidental injustices, which do not render the subject vulnerable to other kinds of injustices (legal, economic, political, sexual, etc.) and systematic injustices that do; cf. Fricker (2007, p. 27). According to her, “The importance of systematicity is simply that if a testimonial injustice is not systematic, then it is not central from the point of view of an interest in the broad pattern of social justice” (ibid., 29). We will argue that a certain type of folk theorem is a potential source of epistemic injustice in mathematics. Because this injustice does not necessarily render its subjects vulnerable to other forms of injustice in the relevant way, it may be incidental in Fricker’s sense. Whether or not these will then contribute to further injustices will depend on the specific cases. For example, they may contribute to systematic discriminatory practices in mathematics, or they may be incidental. Nonetheless, we submit, understanding the kind of injustices that may arise from folk theorems is relevant to an understanding and betterment of mathematical practices. Our interest in a specific nexus of epistemic practices (mathematics) is mirrored in the interest in epistemic injustices as they occur in other scientific practices as discussed in Kidd and Pohlhaus (2017). We are indebted to Alessandra Tanesini for pushing us on this point.
We thank an anonymous referee for encouraging us to be clearer on these matters.
For a discussion of the intertwined nature of case studies and our philosophical views, see also Chang (2011).
De Cezaro and Johansson (2012) includes an example.
See the seminal (Inglis and Aberdein 2014) for an argument that beauty in mathematics is not simplicity.
There are other reasons why a proof may not be published. For example, Thurston (1994) suggests that time-constraints on the mathematician may interfere with write-up and hence publication. This point will be discussed below.
This happens for example in Robertson et al. (1997).
See De Millo et al. (1979).
See Andersen (2017).
Easwaran (2009, p. 343). Indeed, Easwaran calls this a social virtue in mathematics, which we endorse as part of a virtue-based perspective on mathematics.
Such rejections occur not only in Caramello’s case. Consider Harel (1980), writing for a mathematical audience: “We would like the reader to recall the last time he received a referee’s report in which the referee dismissed his latest achievement as ‘a piece of folklore that has been around for 10 years’” (p. 379).
Caramello read a draft version of this section and offered some constructive criticism in private communication. According to her, the accessible evidence for the case under discussion is not incomplete: those voices that are not present on Caramello’s web page have chosen to remain silent.
http://www.oliviacaramello.com/Unication/InitiativeOfClaricationResults.html; accessed January 2018 (emphasis in original).
This theme will be engaged with in more detail in the following sections.
In private communication, Caramello attests that she did not feel gender was at the root of the hostility to her work.
As Caramello reports, the relationship between her and Johnstone has deteriorated: “I was a very respectful Ph.D. student who greatly admired her Ph.D. supervisor… I still cannot understand why he has chosen to betray the trust that I had in him”; http://www.oliviacaramello.com/Unification/InitiativeOfClarificationResults.html#AttitudeTwoCambridgeProfessors.
http://www.oliviacaramello.com/Unification/JohnstoneResponses.html, the conference was the International Category Theory Conference 2010.
From http://www.oliviacaramello.com/Unification/AwodeyResponse.html. Compare also Thurston (1994): “It was hard to find the time to write to keep up with what I could prove, and I built up a backlog”, p. 173.
Recall here that in this paper we do not pass judgement on the Caramello case.
See Aberdein (2016).
Addendum to the Caramello case: in private communication Caramello stressed how difficult it was for her to find an academic position. She was evaluated by the very mathematicians who had judged her work as unoriginal and who Caramello had decided to fight. Caramello furthermore attested that this episode in her life was very stressful to her and that she could have published “twice as much” otherwise. These points harken back to the negative feedback loop and epistemic silencing mentioned in Sect. 2.
Other instances of epistemic luck in mathematical practices include simultaneous discoveries with unevenly distributed credit; see Whitty (2017). A discussion of these matters is beyond the scope of this paper.
http://www.oliviacaramello.com/Unification/AwodeyResponse.html. Compare this also to Thurston’s (1994) “I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field”; p. 173.
Available online: https://www.youtube.com/watch?v=lWk5qeeb_0A.
See Wolchover (2017).
Examples include the German Frankfurter Allgemeine Zeitung (FAZ), the Spiegel and the English Independent. We will refer to the FAZ article found here: http://plus.faz.net/feuilleton/2017-04-07/der-beweis/338154.html/ accessed 26/01/2018.
“Polytechnic Bingen” (our translation).
Ibid.
Roughly translated to “the weakness and perils of the academic system”.
“Anfänglich besuchte Royen noch ab und zu Konferenzen, doch wirklich engen Kontakt hatte Royen nicht zu Kollegen. In Deutschland hatte er mit dem niedrigen Ansehen der Fachhochschulen zu kämpfen. „Wenn man in Deutschland nicht an einer Universität ist, dann gilt man ja nichts.” Er lacht.” (Anderl 2017). Our translation: “At first Royen still attended conferences from time to time, but he did not have close contact with colleagues. In Germany he struggled with the low reputation of polytechnics. ‘In Germany, when one is not at a university, then one is considered a nobody. He laughs.”
It is interesting to make the comparison with the accepted practice in mathematics of attributing important theorems to specific mathematicians (often, but certainly not always the actual author). We all know the examples: the Cauchy–Schwarz inequality, Brouwer’s fixed-point theorem, the Radon–Nikodym theorem, Fermat’s Last Theorem, … This is not to be seen as a case of authority acknowledgement but rather a recognition of the fact that that mathematician was the first one to prove the theorem. Apart from this “property claim”, nothing needs to be known about his or her life and times.
A nice example are formulas that express the number of ways, p(n), into which a natural number n can be partitioned. If anybody would come up with a simple formula for p(n), that would be too ‘nice’ to be true. An alternative example would be any statement that talks about objects in a particular domain without any exceptions. So instead of “For all functions f, satisfying conditions C1, C2, …, Cn, X holds”, one boldly states: “For all functions f, X holds”. This, by the way, corresponds nicely to one of the ten signs a claimed mathematical breakthrough can be wrong, summed up by Scott Aaronson on his blog https://www.scottaaronson.com/blog/?p=304 (consulted on May 15, 2018). The third sign is that “the approach seems to yield something much stronger and maybe even false (but the authors never discuss that).”
At least that surely was Hilbert’s intention, witness the opening statement of the 1900 lecture: “Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries?” (Hilbert 1902, p. 407).
It is important to remark here that oracles are not intended to replace mathematical proofs and proof search. This has been made quite clear by Yehuda Rav (see Rav 1999, pp. 5–6): the existence of a perfectly reliable yes–no oracle—called PYTHIAGORA, by the way—would eliminate the need for proofs, thus marking the end of mathematics as such. The main argument is that lack of insight would prevent us from thinking up new conjectures and ideas worth pursuing.
A famous example is no doubt Pierre de Fermat. In Mahoney (1994) we read that: “His notion of how to share mathematical results had not, however, changed much since the early 1640’s. He still preferred to let his correspondents wrestle with the problems before revealing his solutions and methods of solution. If he did not like controversy, he did enjoy intellectual combat.” (pp. 62–63).
Menary tells us that he “seeks to outline the phylogenetic and ontogenetic conditions for the process of enculturation” (Menary 2015, p. 20).
See e.g. Kroeber and Kluckhohn (1952).
Cf. Aberdein (2016).
Hardy and Wright (1975) call (a + a′)/(b + b′) the “mediant” (p. 23); we are indebted to Peter Cameron for drawing our attention to this. Notice that there is a slight abuse of notation in our presentation: on the left-hand side the + sign denotes the newly defined function, on the right-hand side the + sign has its usual meaning. This need not bother us as it is clear from the context what + is intended to denote.
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Acknowledgements
The authors are indebted to Ladislav Kvasz for a very fruitful discussion about folk theorems. We thank Olivia Caramello for her comments and discussion on the folk theorem section and Katherine Hawley for her helpful comments on Sect. 2. We would also like to thank Andrew Aberdein, Neil Barton, Catarina Dutilh-Novaes, Alessandra Tanesini, Benedikt Löwe, Josh Habgood-Coote, and the audiences at St Andrews, Munich, Hamburg and the CLWF in Brussels for their helpful criticisms on presentations of drafts of this paper. Furthermore, we thank two anonymous referees for their very helpful remarks. Research for this paper by the first author has been funded by the Research Foundation—Flanders (FWO), Project G056716N. Research for this paper by the second author was supported by the EPSRC grant for the project ‘The Social Machine of Mathematics’ led by Prof. Ursula Martin [EP/K040251/2].
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Rittberg, C.J., Tanswell, F.S. & Van Bendegem, J.P. Epistemic injustice in mathematics. Synthese 197, 3875–3904 (2020). https://doi.org/10.1007/s11229-018-01981-1
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DOI: https://doi.org/10.1007/s11229-018-01981-1