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OPERADS, ALGEBRAS AND MODULES
In this paper we argue that questions about which mathematical ideas mathemati- cians are exposed to and choose to pay attention to are epistemologically relevant and entangled with power dynamics and social justice concerns. There is a consider- able body of literature that discusses the dissemination and uptake of ideas as social justice issues. We argue that these insights are also relevant for the epistemology of mathematics. We make this visible by a journalistic exploration of relevant cases and embed our insights into the larger question how mathematical ideas are taken up in mathematical practices. We argue that epistemologies of mathematics ought to account for questions of exposure to and choice of attention to mathematical ideas, and remark on the political relevance of such epistemologies. Keywords Social power · Social justice · Epistemology · Mathematics · Ethics of mathematics · Epistemic exclusion 1 Introduction In this paper we argue that questions about which mathematical ideas mathemati- cians are exposed to and choose to pay attention to are epistemologically relevant and entangled with power dynamics and social justice concerns. We have in mind here issues such as: who may make relevant mistakes in published mathemati- cal papers publicly known (turns out: some are discouraged from doing so)? Who may shape fundamental concepts of a field (turns out: differing proposals can lead * Colin Jakob Rittberg c.j.rittberg@vu.nl Eugenie Hunsicker E.Hunsicker@lboro.ac.uk Loughborough University, Loughborough, UK Vrije Universiteit Amsterdam, Amsterdam, The Netherlands Vrije Universiteit Brussel, Brussels, Belgium Vol.:(0123456789) 1 3 S1148 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 to camps and animosity)? Who determines what counts as new enough to publish (turns out: the novelty of a mathematical result is no straight forward matter)? As a contribution to a special issue on disagreement in mathematics, this paper carves out an epistemologically relevant conceptual space in which such disagree- ments may play out. Our piece makes the point that before mathematicians assess the quality of mathematical ideas (e.g., Is this proof correct? Should these axioms be accepted?) they need to have been exposed to the idea and have chosen to pay attention to it. We show how questions of exposure and attention are entangled with issues of social power and justice. The insight to learn from this is that there are epistemologically relevant disagreements amongst mathematicians that do not con- cern the quality of a mathematical idea. We claim that epistemologies of mathematics that concern themselves with “the author” who wrote “the proof” to “the theorem” which is then read by “the reader” without considering what these personae have been exposed to and chose to pay attention to, remain thin epistemologies of mathematics. In Sect. 2 we draw on an argumentation-theoretic model by Dutilh Novaes (2020a) to flesh this out. There is currently comparatively little engagement with issues of social justice and power in the philosophy of mathematics literature. There is, however, a large body of diversity, equity and inclusion (DEI) literature that discusses these issues in mathematics and related fields. The DEI literature discusses these issues primarily as social justice concerns and does not remark on their epistemological relevance. It is part of the aim of this paper to reveal the epistemological relevance of some of the issues discussed in the DEI literature. To this end, we provide an introductory over- view over the DEI literature in Sect. 3. Informed by the DEI literature we provide some explorations of how epistemo- logically relevant social justice and power issues play out in mathematics. We focus on two dimensions. First: Exposure. Who gets to participate? Whose mathematical ideas are disseminated? Second: Attention. Whose work is read and used? In Sect. 4 we present our journalistic approach to access relevant cases and report what we found. So far, we have explored our question from the perspective of individuals: what mathematical ideas am I exposed to and which of those do I choose to pay attention to? In Sect. 5 we turn to the question how these issues impact on the uptake of ideas in mathematical practices. We argue that assuming that “the best mathematical ideas will make it” is naïve by showing how the issue is entangled with the questions of Exposure and Attention we have been tracing in this paper. By way of conclusion in Sect. 6 we remark on the relevance of developing epis- temologies of mathematics that account for the Exposure and Attention dimensions of knowledge-making, what we call thick epistemologies of mathematics. Our find- ings suggest that thick epistemologies are more realistic than the thin epistemologies currently on offer. There is thus philosophical merit to thick epistemologies of math- ematics. The issue also has political relevance. We report on a current policy debate Our study received ethics approval from the Loughborough University Research Ethics Sub-commit- tee. 1 3 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 S1149 in California, USA, to show that thick epistemologies of mathematics are needed for informed policy-making. 2 Thick Epistemology In this section we introduce Dutilh Novaes’ (2020a) three-tiered model of epistemic exchange and use it to show that contemporary epistemologies of mathematics are thin in a way we specify in this section. It is commonly thought that to engage in argumentation can be a way to acquire more accurate beliefs. By critically examining reasons for and against a given position, we would be able to weed out weaker, poorly justified beliefs (more likely to be false) and end up with stronger, suitably justified beliefs (more likely to be true). (Dutilh Novaes 2020a, 210) Dutilh Novaes traces this thought back to Mill’s (1895) free exchange of ideas, i.e., the thought that argumentative engagement with one’s dissenters will yield the epis- temic benefit of overcoming wrong opinions. Mill derives from this an argument for free speech: dissenters should be allowed to voice their opinions because this will yield epistemic benefits. Dutilh Novaes adds that not only need dissenters be allowed to voice their views, the receivers of these views also need to be willing to engage with the dissenters “in good faith and with an open mind” (2020a, 212). This is where things go wrong for the common view on argumentation. There is much evidence that argument is not an effective means to change peoples’ minds (Gordon- Smith 2019; Kolbert 2017). From confirmation bias (Nickerson 1998) to echo cham- bers and epistemic bubbles (Nguyen 2020), neither are people typically inclined to change their deeply held beliefs just because somebody presents them with a (well- reasoned) argument against them, nor are people typically willing to listen to such arguments in the first place. The free exchange of ideas ideal is descriptively inac- curate, Dutilh Novaes (2020a) shows. Dutilh Novaes conceptualises argumentation as a form of social exchange: epis- temic exchange. Exchanged are epistemic resources, a broad concept which encom- passes such examples as reasons for beliefs but also the payment for a tutor. This helps her to overcome some of the descriptive inaccuracies with the free exchange of ideas ideal. Prior to the engagement with the content of any argument come the two questions (1) which kind of epistemic resources and sources an arguer is exposed to and (2) which of these resources they choose to pay attention to; Mill derived his argument for free speech from the first question but failed to consider the second. Dutilh Novaes presents this in a three-tiered model of epistemic exchange: Stage 1. Exposure: who are potential exchange partners? Stage 2. Attention: given potential candidates (stage 1) and restrictions (e.g., time), whose epistemic resources are engaged with? In a not yet published contribution to the Oxford Handbook of Argumentation, Dutilh Novaes shows how other accounts of argumentation suffer from similarly damaging idealisations. 1 3 S1150 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 Stage 3. Content: only after the dust of stages 1 and 2 has settled does engage- ment with the argument proper occur. Note that we slightly deviate from Dutilh Novaes’ (2020a, 225) original labelling of the stages (stage 1: exposure/attention; stage 2: choosing whom to engage with; stage 3: engagement with content). We prefer to call stage 2 (rather than stage 1) Attention because it is a natural term to capture what is happening in the cases we will present in this paper. Furthermore, it avoids the double use of engagement (with a person in stage 2 and the content of an argument in stage 3). These are purely stylistic choices rather than philosophical disagreement with Dutilh Novaes’ model. Dutilh Novaes (2020a, 226) explicitly remarks that her three-tiered model of epis- temic exchange “offers resources to account for various forms of epistemic injus- tice”. Our paper employs these resources to study the impact of social power and justice issues on the epistemology of mathematics. Epistemologies of mathematics currently on offer in the philosophy of mathemat- ics literature largely focus on questions of validity, rigour, and correctness of proofs, that is on stage 3 (e.g., Manders 2008). Even explicitly agent-focussed accounts, such as Hamami and Morris (2020), concentrate on these stage 3 questions. The exploration of the relevance of stages 1 and 2 to the epistemology of mathematics remains an under-researched topic in the field. Interestingly, Dutilh Novaes’ earlier work on the epistemology of mathematics led her to propose the Prover-Skeptic model of mathematical proof, which accounts for deductive reasoning in terms on the adversarial collaboration between these two personae (Dutilh Novaes 2016). That Skeptic gets to speak (Exposure) and Prover is willing to listen (Attention) is assumed by the model. In recent work, Dutilh Novaes (2020b, chp 11) discusses the abc-conjecture case as an instance where this well- functioning of her model breaks down. Her discussion reveals that her Prover-Skep- tic model presupposes that stages 1 and 2 have already occurred. Dutilh Novaes’ insights about the three stages of epistemic exchange align remarkably well with a recent anthropological exploration of class privilege in business: for “merit” to land it has to be given the opportunity to be demonstrated, it has to be performed in a way that aligns with dominant ideas about the “right” way to work, and it has to be recognised as valuable by those holding the keys to progression. (Friedman and Laurison 2019, 210) Prior to the assessment of “merit”, much like the assessment of the Content of an argument (stage 3), stand questions of Exposure (stage 1) and Attention (stage 2). It is time for the philosophy of mathematics to also recognise the relevance of stages Noteworthy contributions include Barany (2021) and Rittberg et al. (2020). Dutilh Novaes employs the abc case to go beyond what her Prover-Skeptic model covers: she discusses who counts as relevant expert, which is a stage 2 question. This is an enrichment of the Prover-Skeptic model and a step towards the kind of thick epistemology we are arguing for in this paper. 1 3 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 S1151 1 and 2. We need thick epistemologies of mathematics that account for the fact that the merit of a mathematical idea or the correctness of a proof is only assessed after the community has been Exposed to and has chosen to pay Attention to it. Our points align with the stress on the “person- and cultural/social relatedness” of mathematical knowing in Burton’s (1995, 287) proposal of a feminist epistemology of mathematics. Her piece remains theoretical; Burton did not provide a detailed case analysis of how this relatedness plays out in mathematics. It is part of the aim of this paper to supply this lack. We claim that issues of social power and justice impact the Exposure and Atten- tion stages (1, 2) in Dutilh Novaes’ model in the case of mathematics. We under- stand social power as a relational feature between two (groups of) agent(s) that arises when one (group of) agent(s) is in possession of resources relevant to the other (group of) agents(s) (Emerson 1962). Power itself is thus neither just nor unjust; relations of power often help social structures to function properly. Issues of justice can arise when power is abused or when power is unevenly distributed. Our par- ticular interest in social justice is motivated by our intention to reveal the relevance of the DEI literature to the philosophy of mathematics. In Sect. 6 we use this con- nection to argue for the political relevance of a thick epistemology of mathematics. 3 Background In the last section we saw that who our potential epistemic exchange partners are (stage 1) and who we choose to pay attention to (stage 2) are epistemologically rel- evant questions. In this section we introduce some DEI literature on mathematics and related fields which discusses these issues as social justice issues. The DEI lit- erature gives evidence of exclusionary mechanisms that affect whose scientific work the community is Exposed to: e.g., through harassment that causes them to leave research altogether; through not publishing their work; or through not funding work. The DEI literature furthermore gives evidence that there are differential outcomes for members of underrepresented groups as compared to majority groups in science that come from a combination of the second two tiers: what others choose to pay Attention to and the result of their engagement with the Content. In this section, we present some of the literature on DEI considerations in sci- ence. These go back to at least the late 1940s, with the publication of a series of pamphlets by the United States Department of Labor, Women’s Bureau with titles, Anthropologist Geertz (1973) argued that factual accounts of human action without interpretation are “thin descriptions” (e.g. “she winked”). To understand the meaning of social events, human behaviour, and processes (“why did she wink?”) we need “thick descriptions” of cultural contexts (nervous tick, conspiratorial signal, mockery,…). Our proposal of a thick epistemology of mathematics makes the related claim that to understand how mathematical knowledge is formed, transmitted, and passed on we need thick descriptions of mathematical knowledge-making that cover all three stages of Dutilh Novaes’ model of epistemic exchange. Fricker (2007, 3) helpfully remarks that not all justice issues are reducible to power concerns. Here and throughout we capitalise the words Exposure, Attention, and Content to connect to the three stages of Dutilh Novaes’ model of argumentation presented in Sect. 2. 1 3 S1152 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 “The Outlook for Women in…” Mathematics and Statistics, Chemistry, Biologi- cal Sciences, Physics and Astronomy (Zapoleon et al. 1948a, 1948b, 1948c, 1948d, 1948e, 1948f, 1948g, 1948h). The subsequent literature has considered both statisti- cal studies of underrepresentation of various groups, and a wide range of hypotheses explaining the observed inequalities in representation. On one end of the spectrum, some authors have attributed this entirely to barriers to participation by members of these groups (García-Holgado et al. 2019, 2020; Blackburn 2017; Metcalf 2010). On the other end, some authors argue for an explanation rooted in biological basis for mathematical ability (Herrnstein and Murray 1996). There are many explanations offered between these extremes, such as work from the neuroscience and psychol- ogy literature related to stress and anxiety responses. The motivation for all of this research comes from the observation that among the most recognised and celebrated members of these professions, women and other groups are underrepresented com- pared to their representation in the general population. For example, in Mathemat- ics, among winners of the awards commonly considered to be the top prizes in math- ematics, the Fields Medal and the Abel Prize, there have been only 1 out of 60 and 1 out of 22 women awardees, respectively, and no awardees have been of sub-Saharan 8, 9 African ethnicities. There is some ambiguity in determining which part of the thick epistemology model each observed effect relates to. For example, there are gatekeeper practices, in which the results of the Attention of a gatekeeper with work determines what work the broader community is Exposed to. This ambiguity also arises when it is not pos- sible from the design of a study to determine if a lack of uptake of work relates to a decision not to pay Attention or to a decision after engagement that the Content of the work is not worth building upon. 3.1 W ho Gets to Participate? We first consider three examples from the DEI literature of ways in which indi- viduals’ engagement with the research community is limited. These can be seen as related to stage 1 of the model (cf. Section 2). Early focus on barriers to participation was on policies that directly excluded or disadvantaged certain groups. An example of this is presented in Murray’s (2000): the case of Lida Barrett who, due to anti-nepotism rules, was unable to obtain a permanent position at the University of Tennessee in the 1960s until after her math- ematician husband’s death, despite holding a series of temporary positions in the department in which she taught, undertook research and supervised PhD students. As described in Walker (2014) African-American mathematician Raymond John- son was not able to formally enrol at Rice University in his first year in 1963 until a court case required them to admit black students, and many other southern US uni- versities only began to admit Black students for PhDs in the 1960s and 1970s. https:// www. mathu nion. org/ imu- awards/ fields- medal. https:// maths histo ry. st- andre ws. ac. uk/ Honou rs/ AbelP rize/. 1 3 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 S1153 More recently, however, as explicitly discriminatory policies have been changed and nevertheless representation of women and blacks remains low, focus has shifted to evidence of barriers in the form of cultural bias in the way that the careers and work of members of different social groups are evaluated. There is evidence of ineq- uity in outcomes across a wide range of practices that influence career progression. For example, Moss-Racusin et al. (2012) conducted a cv study that asked science faculty from research intensive universities to rate the application materials of a stu- dent for a lab manager position, where the gender of the applicant was manipulated and randomized. Faculty participants rated the male applicant as significantly more compe- tent and hireable than the (identical) female applicant. These participants also selected a higher starting salary and offered more career mentoring to the male applicant. (Moss-Racusin et al. 2012, 16474) Finally, bullying and harassment affects underrepresented groups throughout their careers. A recent report by the National Academies in the US has concluded that “the cumulative result of sexual harassment in academic sciences, engineering, and medicine is significant damage to research integrity and a costly loss of talent in these fields” (Ngun 2019; Johnson et al. 2018). 3.2 Gatekeeper Cases Editorial practices determine whose work is published and the prestige of the jour- nal in which work appears, as well as how much time it takes for a given individual to get work published, which impacts on the time available for further work. Hen- gel (2017) studies evidence from the discipline of economics that women authors are held to different standards by journal editors than male authors. She finds that papers by women spend on average six months longer in peer review, and also evi- dence that women’s papers are held to higher standards of “readability”. Grant funding is another critical component of career success. Bornmann et al.’s (2007) meta-analysis of 21 studies on gender and grant funding indicated that men have statistically significant greater odds of receiving a grant in science than women by about 7%. Van der Lee and Ellemers’ (2015) study of research funding for early career researchers in the Netherlands finds that women are disadvantaged in com- parison to men in the funding process particularly through differences in evaluation of “quality of researcher” as opposed to in evaluation of “quality of proposal”. This effect was seen also in a study of the Canadian Institutes of Health Research (Witte- man et al. 2019). Finally, we have evidence of a lack of uptake of the work of minority groups in Mathematics which may arise either from stage 2 or stage 3, and it is not possible from the evidence to determine which. Budrikis (2020) examines gendered citation https:// www. natio nalac ademi es. org/ our- work/ sexual- haras sment- in- acade mia. 1 3 S1154 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 patterns in neuroscience and concludes that scientific work led by women tends to be cited less than comparable work by men. Hofstra et al. (2020) find in a natural language processing study of about 1.2 million US doctoral theses from 1977 to 2015 that women and underrepresented ethnic groups produce higher rates of nov- elty in their PhD theses, but that their novel contributions were less likely to be cited in subsequent literature, and equally impactful work by these groups was less likely to result in successful scientific careers than for majority groups, which also feeds back into who is able in the long term to participate in the research community. 3.3 Where Does this Come From? Literature from psychology on the topic of persuasion is relevant to why often the decision about merit of an idea is made at stage 2 (Attention) rather than stage 3 (Content). This work, referred to as the Elaboration Likelihood Model (Petty and Cacioppo 1986), proposes there are two routes to attitude change: the “central” and “periph- eral” routes. Central persuasion occurs when there is careful and engaged consid- eration of the quality of information presented. Peripheral persuasion occurs when cues other than actual information, such as the sorts of social cues associated with implicit bias, are most influential. In terms of a thick model of epistemology, the central route involves engagement at the third stage (Content), whereas the periph- eral route relates to decisions made at the second stage (Attention). We might expect that experts that are evaluating the merit of some intellectual material will be engaged with the central route: the assessment of Content. This expectation may be too idealistic. In West et al. (2012), the authors found that although people of high “cognitive sophistication” were less likely than others to make decisions based on the peripheral rather than central route, when they did they were also less likely to realise they had. Although they note that this runs coun- ter to other literature, which found most biases were negatively or uncorrelated to cognitive abilities, the effect is also reflected in observations about the impact of blind auditions on gender diversity in major orchestras. In an interview with Gar- diner Morse for the Harvard Business Review, Iris Bohnet from the Harvard Ken- nedy School has pointed out that moves towards blind auditions met with resistance from orchestra directors: ‘Note that this [change in proportion of women] didn’t result from changing mindsets. In fact, some of the most famous orchestra directors at the time were convinced that they didn’t need curtains because they, of all people, certainly focused on the quality of the music and not whether somebody looked the part. The evidence told a different story’. It has been suggested that the same effect may occur in mathematics: Hofstra et al. (2020, 9285) explain: “To measure scientific innovation, we first identify the set of sci- entific concepts being employed in theses. For this, we use natural language processing techniques”. They then filter for meaningful concept pairs which are related for the first time. Novelty is then opera- tionalised as “the number of conceptual co-occurrences within each [PhD] thesis” (p. 9285). https:// hbr. org/ 2016/ 07/ desig ning-a- bias- free- organ izati on; accessed 16.6.2022. 1 3 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 S1155 In academia, and especially math, objectivity is an ideal quality: scholars must separate themselves from their work. But humans, by nature, make subjective and biased decisions even if they are striving for objectivity. Those committed to scholarly objectivity may pass off their personal beliefs as ultimate truths without recognizing their own biases may have crept in. (Hu 2016) Belief in academia as a meritocratic system also may contribute to biases. Castilla and Benard (2010) report the results of a randomised CV study in which the genders given on the CVs were manipulated and also the instructions given to evaluators. One condition emphasised meritocracy in the decision process and the other did not. The authors found that when primed with the instructions emphasising meritoc- racy, the participants were more biased in favour of CVs labelled with male names than when they were primed with the instructions that did not. Note that this work underlines the unreliability of the testimony of experts about their own judgement processes. We now consider why social power is relevant to the peripheral route to persua- sion, and which social dimensions may be relevant to an expert rejecting work at stage 2 (Attention) rather than engaging in stage 3 (Content). We start with gender. One explanation for the bias in evaluation of women’s track records offered by the literature is that there is an expectation of career trajectory, referred to by Murray (2000) as “myth of the Mathematical Life Course”. This tra- jectory involves early and steady success in the career with no disruptions—a tra- jectory that is not consistent with the lives of women with families. Valian (1999) discusses the role that gender schemas can play in both the way people are perceived by others and by how they perceive themselves. They refer to the 1995 study of the awarding of medical fellowships in Sweden which were shown to have a large gen- der bias: The judges did not intend to discriminate. But their schemas represent women in general as less scientifically qualified than men. Those schemas affected how the judges viewed information that they probably considered to be objec- tive but was in fact ambiguous and open to interpretation. (Valian 1999, 1047) The paper goes on to discuss how gender schemas can also influence perception of the self: “One way that gender schemas affect women, then, is in their perception of themselves as worth less and entitled to less” which can result in their advocating less strongly for their work and careers than men. We now move to affiliation. Recent work studying patterns in awarding of the Fields Medal has shown that it tends to be awarded within particular “mathemati- cal families”, with frequent connections through advisory relationships between awardees (Chang and Fu 2021; Barany 2018). This also suggests that the evalua- tion of the track record of mathematicians is influenced by social factors associated to affiliation. A 2010 paper analysing the outcomes of EU grant applications found that “the probability to get funded increases significantly for [applicants] that have a nearby panelist from the host institution. At the same time, the effect differs between 1 3 S1156 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 disciplines and countries, and men profit more of it than women do” (Mom and Van den Besselaar 2020). Finally, we consider social class. The foundational work (Friedman and Lauri- son 2019) is an investigation of the mechanisms through which social class influ- ence mobility within four different careers. The authors conclude that people do not have an equal capacity to turn their merit in a role into career progression. Further, Cooper and Berry (2020) found that in Australia, socioeconomic class is the largest barrier to STEM participation in secondary school students, and link this to issues related to access to cultural and social capital. In the next two sections we seek to bridge the gap between DEI and philosophy of mathematics. Informed by the DEI literature we will present some evidence that helps to make visible how the first two stages of Dutilh Novaes’ model impact the epistemology of mathematics. 4 Uptake of Ideas of Individuals At this point we have seen the considerable body of DEI literature on numerous issues that impact whose work the scientific community is Exposed to, whose work is paid Attention to, and its entanglement with the assessment of merit of the Con- tent of work (cf. the three stages of Dutilh Novaes’ model introduced in Sect. 2). We now move to an exploration of how these issues play out in mathematics. To do this, we were looking for suitable case studies. The most suitable cases for us to present in this paper would show how issues of social justice and power influence the epistemology of mathematics. They would be akin to the Caramello case presented by the second author and his co-authors in (Rittberg et al. 2020). Caramello’s submitted mathematical paper was rejected on the grounds that the results of the piece were already known, they were mathematical folklore. Closer inspection revealed that this bit of folklore knowledge was neither published anywhere, nor was it known to all relevant actors in the field. The case shows that what counts as a new mathematical result is no straightforward matter. Rittberg et al. argue that the practice of using folk theorems puts those who know them in positions of power which can be abused to the detriment of the field: they are a source of epistemic injustices. Another example, also discussed in Rittberg et al. (2020), is the case of Royen who proved the Gauss correlation inequality but failed to use means of communica- tion considered standard in the mathematical community. He risked being regarded as a crank and his result discarded on that assumption. This case raises questions about just exclusions from mathematical practices. Rittberg et al. (2020) argue that Royen did indeed not suffer unjust exclusion because he failed to meet standards that can reasonably be expected. What makes cases such as these so suitable for our purposes is that they show directly how issues of social power and justice impact what kind of mathematical knowledge agents are Exposed to and which pieces of knowledge they choose to pay Attention to; the Caramello case shows that rejecting someone’s work because they have not paid Attention to results they could not reasonably have been Exposed to is 1 3 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 S1157 a justice issue with an epistemic component; the Royen case shows that even for val- uable mathematical Content questions of Exposure and Attention are entangled with questions of exclusion and social justice (it is only after working through the specif- ics of the Royen case that it becomes clear that no injustice was suffered). These cases show how social justice and power impacts on mathematical epistemology. This is unlike arguments such as “if the pipeline were not so leaky, we would have more non-male mathematicians and that would be beneficial to mathematics”. Such arguments take a relevant social justice issue and connect it to epistemology by means of counter-factual reasoning. Such connections are somewhat tenuous, because they tend to have little on offer to convince those who say “I don’t think so”. That is: there are genuine social justice issues that ought to be taken seriously, but to serve our purposes in this paper we are looking for concrete cases where issues of social power and justice impact on the epistemology of mathematics in direct ways. To find such cases, a small qualitative study was carried out under the approval of the Loughborough University Ethics Committee. A text explaining the concepts of thick and thin epistemology was sent, together with a participant information and consent form, to a collection of academic mathematicians with a request for rele- vant anecdotes. The choice was made to restrict to those in permanent positions so as to minimize any risks associated with potential identification of the individuals through their stories. We also performed a literature review spanning over academic publications in philosophy and history, policy documents, newspaper articles, blog posts, and other internet resources. In this section we present the insights we gained from these anecdotes. We approached these discussions as investigative journalists with no social science methodology to follow. Our hope was that our respondents would tell us about a case suitable for the purposes of this paper, which we would then study based on further material. Only one case permitted further investigation: the operad case. Peter May introduced the term operad in the 1970s as a portmanteau of the words operator and monad; (May 1997). The concept is still studied today; e.g., Mandell (2019). On 4 May 2021 Martin Markl gave an online talk for the Intensive Research Programme on Higher Homotopical Structures (HHS) at the Centre de Recerca Matematica in Barcelona, entitled “Operadic categories and their operads: May vs Markl”. In minute 8 of the recording Markl remarks: When I published the paper [in which] I introduced this notation or this set- up [which amounts to a new definition of operads] I got a kind of agitated, I would even say angry, email from Peter May who said that I am not sup- posed to use this because it completely obscures the nature of operads and he is going to ignore everything which is based on this notation. I was 36 by that time [1996], so you can imagine how I felt. In similarity to arguments in Alexander (2021). Available here: https:// sites. google. com/ view/ hhs20 21- semin ar/ past- talks- videos. 1 3 S1158 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 May seems to have made good on his threat of not engaging with Markl’s work: we are not aware of any work by May which refers to Markl. Notice that this is not for lack of mathematical relevance. Markl’s 2021 HHS talk discusses connections between his and May’s definition of the concept. Today there are largely two camps, those who employ the definition that goes back to May and those who employ the definition proposed by Markl. These camps focus on different areas of study: May’s definition is primarily used to study applications of operads to homotopy theory, spectra, and localizations, whereas Markl’s definition is applied in studies of geom- etry, category theory, algebra and combinatorics. There is only little interaction between the two camps, Markl told us in private email communication. One exam- ple of this interaction is Markl’s 2021 HHS talk. The operad case thus reveals that stage two (Attention) of Dutilh Novaes’ model also plays out in mathematics: mathematicians sometimes wilfully ignore relevant mathematical ideas. In the operad case the notions of social power and social justice come apart. In 1996 May was an established figure in the study of operads with a relevant network amongst which he could distribute new ideas. This network is an epistemic resource which Markl lacked at the time, which put May in a position of social power over Markl. May chose not to distribute the ideas of then-young Markl, which carries the potential for injustice. However, Markl’s concept enjoyed uptake regardless of May’s attitude. Markl told us in email communication that May’s dismissive attitude towards his idea did not negatively affect his career. To call this episode an issue of social justice may hence stretch the concept too far. The operad case was the only case we learned about for which we could har- vest publicly available resources. All other cases our respondents reported on were personal experiences covered by the anonymity agreement with our respondents. This affects the quality of the evidence we can provide. We will report on what we learned here, but the readers will not be able to scrutinise our case studies for them- selves. This is perhaps most clearly visible in the flawed fundamental papers case. One of our respondents reported how during their PhD studies they found a flaw in some fundamental papers which they were discouraged to make publicly known. They had been working on certain types of action on some algebraic structures (we avoid detail here to ensure anonymity). There are two conceptualisations of these actions—two ways of looking at these actions—that coincide for finite versions of these algebraic structures. These actions also coincide for some infinite versions of these structures, but not for all. This is where the mistakes in the fundamental papers that study these actions slip in: at times these actions are regarded as interchangea- ble where, in fact, they are not. More precisely, features of one action were exploited in these papers whilst the other action was under consideration. This was noticed by the PhD student, who had to work hard to show that the desired feature actually also holds for the action under consideration in their specific case (an infinite case in which the two actions do not coincide). The way to do this was to reprove many of the results of the fundamental papers but paying close attention to which action is used and which features can be exploited. This meant coming up with numerous new proofs which took many months. In the end they succeeded. That is, the results 1 3 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 S1159 of the fundamental papers are correct, but the originally published proofs to obtain them are not. Here we have a clear epistemic achievement (spotting flaws and managing to cor - rect them) of relevance (the flaws occurred in the fundamental papers of a field). Nonetheless the PhD student was discouraged from making their findings publicly known. Their supervisors argued that the PhD student did not yet have the academic standing needed for such a publication. To this day, the flaws in the fundamental papers are only engaged with in the PhD thesis; no other academic publication on these matters ever followed. It is unclear who would have been harmed if the flaws would have been made publicly known. The fundamental papers were all written by one author. At the time of the events, the author was an established mathematician and had moved on to another field of study. When the PhD student contacted the author with their find- ings, the author acknowledged the correctness of the PhD student’s findings and gave no indication that the author wishes that they are not made publicly known. Perhaps the PhD supervisors were worried that publishing these flaws would reveal that they, notable experts in the field, had not noticed these flaws. This is speculation for which neither we nor the PhD student has convincing evidence. This episode reveals how those in position of power (in this case: PhD supervi- sors) can impact the take-up of relevant epistemic achievements in mathematics (in this case by dissuading their PhD student from publication). The episode suggests that to publish in mathematics it is not enough for your piece of mathematics to sat- isfy Littlewood’s three precepts (Krantz 1997, 125): (1) the PhD student’s work was new in that it provided a missing proof; (2) the work is correct as judged by a PhD committee; (3) the work is surprising in that it reveals mistakes in the fundamental papers of a field. Academic standing is needed for the publication of some math- ematical results; cf. also the Royen case mentioned above. Another case of relevance to the claims we make in this paper has to do with funding. In this case our respondent served on a funding panel discussing the joint work of a male and a female tenured researcher. This joint work was part of their separate funding applications. The initial evaluation of the proposals marked them as speculative but potentially ground-breaking. This was then incorporated into the individual evaluations of the two proposals by different reviewers. In the case of the male researcher, the work was considered innovative, ground-breaking and with far- reaching implications. For the female researcher, the work was deemed speculative, ungrounded and lacking in motivation. When the drafts of the evaluations were read to the entire panel, nobody seemed to notice this difference in judgement, much to the surprise of our anonymous respondent. When the difference was pointed out lan- guage ultimately changed, but our respondent is certain that without this interven- tion the female researcher’s proposal would have had no chance of funding. The relevance of this episode for our argument lies in the fact that the worth of the same piece of mathematical work was judged differently depending on the author. The effects of authority on the persuasiveness of mathematical arguments Geist et al. add a fourth precept, “Is it well-written?”, which cannot be assessed here because the proofs were never written up for stand-alone publication. 1 3 S1160 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 have been empirically investigated by Inglis and Mejia-Ramos (2009). Their study shows that when a reader is certain that a mathematical argument is persuasive, then the author of the argument plays no role; this is primarily the case for deductive mathematical arguments. When the reader is not certain of the persuasiveness of the argument, then authority considerations can come in. Inglis and Meija-Ramos focus on proofs. The funding case suggests that these insights also apply to the evaluation of grant proposals. Inglis’ and Mejia-Ramos’ (2009) echoes the notions of peripheral and central routes to persuasion we introduced in Sect. 3.3. It is noted by the authors of that work that peripheral persuasion is more likely in settings in which there is more uncertainty. The relevance of peripheral routes to persuasion was also highlighted by another respondent of ours. They reported that much of the spoken mathematical communication between mathematicians employs larger concepts and references to results about these concepts, rather than detailed arguments why these results hold. Mathematicians gauge the mathematical ability of their peers by how proficient they are at this jargon-throwing. This affects mathematical careers, our respondent reported. If you do not talk the talk, then you are less likely to have a successful aca- demic career, even if your written work is solid and has obvious merit. Our respond- ent reported how this conflicted with their initial view that mathematical ability is measured in the quality of one’s work. That is, our respondent initially thought that what matters in mathematical activity are central routes to persuasion. This is not so: peripheral routes to persuasion play important roles, so their report suggests. This may be exacerbated by further social factors. For example, in their research com- munity persons who do not align with the dominant social behaviours of their col- leagues are regarded with suspicion, and members of the community are less willing to pay Attention to such a person’s mathematical work or engage in collaboration. In this section we reported on our journalistic findings about how social power impacts the take-up and passing on of mathematical ideas: 1. Relevant actors can wilfully ignore relevant mathematical ideas (operad case) 2. Publication of relevant results can require academic standing (a fl wed papers case) 3. The perceived worth of a piece of mathematical knowledge can depend on its author (funding case and peripheral persuasion) These insights flesh out aspects of how stage 1 (Exposure) and stage 2 (Attention) of Dutilh Novaes’ model (cf. Section 2) play out for the individual in mathematics. In the next section we explore what these insights mean for the uptake of mathemat- ical ideas by a mathematical community. 5 Community Level Uptake of Ideas In the last section we presented particular episodes of mathematical knowledge- making as experienced by individual mathematicians to indicate how issues of social power and justice may impact the epistemology of mathematics. We focussed on 1 3 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 S1161 individuals and small groups. In this section we explore how these insights impact on the uptake of ideas in mathematical practices. Tao (2007) asks what good mathematics could be. He considers the example of Szemerédi’s Theorem. After a discussion of the history leading to and from this result, he concludes, the very best examples of good mathematics…are more importantly part of a greater mathematical story, which then unfurls to generate many further pieces of good mathematics of many different types. He situates this piece of good mathematics within a programme starting with a result of Van der Waerden (1927). The authors investigated the uptake of this work and related papers using mathscinet. The next result Tao mentions, which builds on Van der Waerden’s work, is from 1936, then the next one from 1946. The theo- rem of Szemerédi itself appears only in 1975, with Szemerédi’s first paper on the topic appearing in 1969. Altogether by decade, the references in (Tao 2007) include: 1930s: 3; 1940s: 1; 1950s: 1; 1960s: 2; 1970s: 4; 1980s: 4; 1990s: 4; 2000s: 17, including 6 preprints. Of course, these are the main results on the topic, not the entire literature. The 1936 paper of Erdős and Turán, which first proposed the con- jecture that became Szemerédi’s Theorem, accrued 7 citations prior to 1998, when Timothy Gowers, a 1996 Fields Medallist, wrote a paper on the topic (Gowers 1998). Our search on mathscinet.ams.org revealed that there were then 14 more cita- tions before 2006, when Tao, in the same year as his Fields Medal, first published on the topic. Subsequent to that, Erdös’ and Turán’s (1936) accrued an additional 70 citations. Finally, the paper by Szemerédi containing the proof had no citations prior to 2005, and only 10 citations prior to Tao’s paper on the subject, since when it has accrued an additional 372 citations. Tao recognises that “without the benefit of hindsight it is difficult to predict with certainty what types of mathematics will have [the property to involve and inspire further good mathematics]” (Tao 2007, 633). The uptake of ideas by the commu- nity generally takes time. What Tao fails to remark is that citations ramped up once champion mathematicians like Erdős, Turán, Gowers, or Tao himself started engag- ing with the work. Yet this is relevant for Tao’s view on the uptake of mathematical ideas: one can view the history of entire fields of mathematics as being primarily generated by a handful of…great stories, their evolution through time, and their interaction with each other. (Tao 2007, 632) This suggests that good mathematics can either be seen as fitting into an existing “great story” (or possibly more than one), or creating a new “great story”. Tao’s (2007) presentation of the life of a mathematical idea abstracts away from the mathematicians who come up with, are Exposed to, and choose to pay Attention to the idea. He overlooks that the reception of mathematics that proves established conjectures, or proposes tools for attacking them, or generalises existing work or 1 3 S1162 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 connects existing “great stories” in new ways will have different obstructions to and processes of uptake from mathematics that proposes new stories, in which case it may be particularly difficult to make a judgement of the likely impact of the work. According to the psychology research on the elaboration likelihood model, absent reliable signposts for making such a judgement, the decision to take it up or not is more likely to be influenced by peripheral routes than in cases where there are reli- able signposts, as with work that fits into an established research programme (Hen- derson 2021). This harkens back to our remark about the champion mathematicians who started work on issues surrounding Szemerédi’s. They are influential mathe- maticians, and their influence adds peripheral adornment to the “great story” Tao (2007) recounts. What makes a mathematician influential? One possible source of evidence is answers to the question “Which mathematicians have influenced you the most?” on MathOverflow. Answers cite a wide range of characteristics beyond the impact of the mathematicians’ work itself, which in fact occurs in only a handful of answers (“Saharon Shelah—His work on forcing, and singular cardinals keep me asking questions, and open up the possibility for questions I didn’t even know could be asked.”). Other characteristics mentioned include: clarity of exposition, (John Willard Milnor…for writing papers in a way that they are quite self-contained and readable.”), particular talent (such as “Hitchin—for his ability to find a new mathematical structure out of every physical context”), view of mathematics (“Dedekind, whose championing of concepts (vs. calcula- tion) left a longstanding impression on the way that I conceive mathematics”), personality (“Joseph-Louis Lagrange—for his modesty as a human being”), mathematical style (“Thurston’s work really inspired me to appreciate the role of imagination and visualization in geometry/topology.”), writings about doing mathematics (Atiyah’s (2010) “Advice to a Young Math- ematician” is cited), and personal relationship (“[My] graduate advisor at Queens College of the City Uni- versity Of New York, Nick Metas, was and continues to be my greatest influ- ence.”). The answers to this question also answer another relevant question, about when individuals are most open to influence. Almost all of the answers indicate that the respondent was in school, university or graduate school when they first came across the mathematician who was most influential to them. Of course, it is worthwhile to note that the MathOverflow population tends to be young, and this may reflect that, but there are also comments suggesting that at some point one is too old to be influenced: “From his writings I found analysis of PDE as a fascinating subject and I am really happy that I found this topic not too late,” and “Unfortunately, I’m too It is important to note that this site is already known to enforce social biases (see blog “Why I am not on MathOverflow” by woman mathematician and Accidental Mathematician blogger Izabella Laba): https:// ilaba. wordp ress. com/ 2011/ 03/ 28/ why- im- not- on- matho verfl ow/. 1 3 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 S1163 old and too off topic to be influenced by his research.” A comment to the question itself, rather than a response states: “It’s always your advisor(s) that influence you the most, aren’t they?” This point is also reinforced by responses to a Facebook post about social influences in mathematics, which suggested that it is “incredibly hard to advance a programme in mathematics for those mathematicians who do not have doctoral students and postdocs.” We note that there is evidence that mathematicians are aware of the difference between how mathematics might develop in an ideal state of unlimited resource vs how mathematics develops within the real circumstances of limited resource, and that various aspects of social power influence decisions about what work gets space in prestigious journal, gets grant funding, gets spoken about in high profile venues. For example, Volker Mehrmann, President of the European Mathematical Society said in a recent panel discussion on gender in mathematics, “Being in committees all the time, I see that people advocate their own students and field”. In terms of Dutilh Noveas’ model, as applied to the “gatekeepers” who make these decisions, after their stage 3 engagement with the work comes a step where they decide if to let the work through the particular gate or not, and this decision may be influenced by factors other than the quality they attribute to the work. In terms of broader uptake of work, this then influences stage 1 (Exposure) for a broader audience. 6 Conclusion It is time to take stock. We have seen that there is a large body of literature that explores stage 1 (Exposure) and stage 2 (Attention) of Dutilh Novaes’ model as social justice issues in scientific knowledge-making (Sect. 3). This led us to conjec- ture that knowledge-making in mathematics is just as prone to social justice issues as knowledge-making in other practices. The question became how this plays out in mathematics. We reported on our journalistic findings of how questions of Exposure and Attention of an individual mathematician are impacted by issues of social power and justice (Sect. 4) and considered the implications of this for the uptake of ideas in mathematical communities by means of a literature study (Sect. 5). Our findings support the push for a thick epistemology of mathematics. By this we mean an epistemology of mathematics that engages with all three stages of Dutilh Novaes’ three-tiered model of epistemic exchange: (1) What epistemic resources are we Exposed to? (2) Which arguments do we choose to pay Attention to? (3) Engage- ment with the Content of arguments. The epistemologies of mathematics currently on offer are largely thin in the sense that they focus nearly exclusively on stage 3 (cf. Section 3). Our findings indicate how social justice and power issues in mathemat- ics impact on stages 1 and 2. That is, they indicate how issues of social justice and power impact on the epistemology of mathematics. And that means that these social This panel discussion was held at the 8th European Congress of Mathematics 2021 and chaired by the first author; https:// 8ecm. si/ progr amme/ open- panels- and- socie ty- meeti ngs# title2. The quotation is taken from the minutes of that session. A summary of the discussion will be submitted to the European Math- ematics Society newsletter for publication. 1 3 S1164 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 justice issues are epistemologically relevant. Similar to Geertz (1973), who inspired our terminology (see footnote 5), we do not claim that all epistemological accounts of mathematics ought to consider social justice issues; it is not clear to us that, e.g., Manders’ (2008) would be improved by such considerations. Thinness is not a sin; it is a focus. In this paper we aimed to show some of what this focus overlooks, and how these issues impact on how mathematical knowledge is formed, transmitted, and passed on. So much for the argument that thick epistemologies of mathematics are philo- sophically relevant. We now move to our argument that there is also political rel- evance to developing thick epistemologies of mathematics. We argue that the over- focus of the philosophy of mathematics literature on questions of thin epistemology help to perpetuate social injustices. On 13 July 2021 an open letter was published that argues to replace the proposed new California mathematics curriculum framework. As of 16 August 2021, 726 people have signed the letter, the majority of which self-identified as academic mathematicians, physicists, or computer scientists. According to the letter, the pro- posed framework: Urges teachers to take a “justice-oriented perspective at any grade level, K–12” and explicitly rejects the idea that mathematics itself is a “neutral discipline.” The rhetoric that mathematics is a neutral, value-free discipline is echoed by the mathematicians Deift et al. (2021). They claim that curricular increasingly shift from actual mathematical content to courses about social justice and identity politics. We take this as evidence that amongst many mathematicians, mathematics is still seen as a value neutral discipline; see also (Ernest 2018). Contemporary trends in the philosophy of mathematics have revealed that math- ematics is not value neutral. Works on the explanatoriness of proofs (D’Alessandro 2019), the purity of methods (Detlefsen and Arana 2011), on mathematical beauty (Thomas 2016), publishability of results (Geist et al. 2010) all show this. Pérez- Escobar and Sarikaya (2021) argue that mathematics cannot be divorced from its sociological aspects; they argue that agent and community values impact on the epistemology of mathematics. Arguably, most mathematicians, even those who signed the open letter mentioned above, know this. They know that a mathematical result needs to be relevant enough to be published (Geist et al. 2010), that proofs that employ elementary methods are valued in the mathematical community (Rittberg 2021, Sect. 3), that there is value in having multiple proofs of the same theorem. https:// www. indep endent. org/ news/ artic le. asp? id= 13658. Deift et al. (2021) provide a link to https:// www. scu. edu/ ecp/ progr ams/ cours esche dules/ educa tion- course- descr iptio ns/# accor dion- panel- 402237, which lists courses such as “Effective teaching for stu- dents with disabilities”, “Social foundations of education”, and “Bilingual methods”. 1 3 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 S1165 We submit that when mathematicians claim that mathematics is a “neutral dis- cipline” they do not have the above-mentioned value-judgements in mind. Rather, they claim that the epistemology of mathematics is not affected by issues of social power and justice. It is the claim that there is an objective standard on which math- ematical knowledge-making should be judged. The open letter speaks of “judging ideas […] according to their real merit” (rather than based on their cultural origins). Deift et al. (2021) speak of “objectively exhibited levels of performance”. Such neutrality claims find (perhaps unwitting) support from the thin episte- mologies provided by philosophers of mathematics. Even the agent-focussed epis- temologies of some philosophers of mathematical practices (cf. Sect. 2) assume that “the reader” has access to the proof in question (stage 1: Exposure) and is willing to engage with the work (stage 2: Attention). Focus is solely on stage 3 (Content). Paired with the widely held belief that the quality of mathematical arguments is independent of social concerns, this can give the impression that the success of a mathematical idea is independent from social justice concerns. By failing to high- light how issues of social power and justice affect mathematical knowledge-making philosophers indirectly support the claim that mathematics is a “neutral discipline” in the above sense. There is thus political relevance to the development of thick epistemologies; they are needed to inform debates such as about the new framework for Califor- nia’s mathematics curriculum the open letter mentioned above criticises. We need accounts of how social power and justice impact the epistemology of mathematics and investigations of the mechanisms at play. Understanding these mechanisms will allow for an informed debate about how mathematical practices ought to be shaped by policies. Part of understanding how social justice impacts the epistemology of mathemat- ics is the recognition that thick epistemologies of mathematics need not undermine those standards of mathematical quality that the authors of the open letter see threat- ened by the proposed new policy. It is entirely consistent to claim that issues of mathematical Content (3rd stage) are divorced from social justice concerns, whilst acknowledging that which mathematical content we are Exposed to (1st stage) and what we choose to pay Attention to (2nd stage) is epistemologically relevant and entangled in issues of social power and justice. We need thick epistemologies of mathematics. We need case studies that reveal how social power and justice issues impact mathematical knowledge-making, and we need a philosophical analysis of the mechanisms at play. This will produce more realistic epistemologies of mathematics with the means to enrich debates about poli- cies for mathematical practices. Acknowledgements We wish to thank our anonymous respondents for their time and insights. We thank Brendan Larvor, Catarina Dutilh Novaes, Martin Markl, and two anonymous referees for helpful discus- sions and suggestions. This paper is part of a project by the second author that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska- Curie Grant Agreement No. 883724. E.g. Tao (2007) discussed in the last section. See also Ernest (2018). 1 3 S1166 Axiomathes (2022) 32 (Suppl 3):S1147–S1168 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com- mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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Axiomathes – Springer Journals
Published: Dec 1, 2022
Keywords: Social power; Social justice; Epistemology; Mathematics; Ethics of mathematics; Epistemic exclusion
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