Papers

I’m happy to provide copies of any of my papers. Email me at F.Tanswell (at symbol) lboro.ac.uk

(2016) “Saving Proof from Paradox: Gödel’s Paradox and the Inconsistency of Informal Mathematics”
In Andreas, H. & Verdée, P. (eds.) Logical Studies of Paraconsistent Reasoning in Science and Mathematics, Trends in Logic 45, Springer International Publishing. Link

Abstract: In this paper I shall consider two related avenues of argument that have been used to make the case for the inconsistency of mathematics: firstly, Gödel’s paradox which leads to a contradiction within mathematics and, secondly, the incompatibility of completeness and consistency established by Gödel’s incompleteness theorems. By bringing in considerations from the philosophy of mathematical practice on informal proofs, I suggest that we should add to the two axes of completeness and consistency a third axis of formality and informality. I use this perspective to respond to the arguments for the inconsistency of mathematics made by Beall and Priest, presenting problems with the assumptions needed concerning formalisation, the unity of informal mathematics and the relation between the formal and informal.

(2017) “Conceptual Engineering for Mathematical Concepts”
Inquiry: An Interdisciplinary Journal of Philosophy 61, pp. 881-913.
Link

Abstract: In this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next, I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory.

(2018) “Epistemic Injustice in Mathematics”
Synthese, online access. With Colin Rittberg and Jean Paul Van Bendegem.
Link

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.

(2019) “Journeys in Mathematical Landscapes: Genius or Craft?”
In Gila Hanna, David Reid, and Michael de Villiers (eds.) Proof technology: Implications for teaching, Springer, pp. 197-212.
With Ursula Martin, Alison Pease, Lorenzo Lane, and Dave Murray-Rust.
Link

Abstract: We look at how Anglophone mathematicians have, over the last hundred years or so, presented their activities using metaphors of landscape and journey. We contrast romanticised self-presentations of the isolated genius with ethnographic studies of mathematicians at work, both alone, and in collaboration, looking particularly at on-line collaborations in the “polymath” format. The latter provide more realistic evidence of mathematicians daily practice, consistent with the “growth mindset” notion of mathematical educators, that mathematical abilities are skills to be developed, rather than fixed traits. We place our observations in a broader literature on landscape, social space, craft and wayfaring, which combine in the notion of the production of mathematics as crafting the exploration of an unknown landscape. We indicate how “polymath” has a two-fold educational role, enabling participants to develop their skills, and providing a public demonstration of the craft of mathematics in action.

(2020) “Mathematical Practice and Epistemic Virtue and Vice”
Synthese. With Ian James Kidd.
Link

Abstract: What sorts of epistemic virtues are required for effective mathematical practice? Should these be virtues of individual or collective agents? What sorts of corresponding epistemic vices might interfere with mathematical practice? How do these virtues and vices of mathematics relate to the virtue-theoretic terminology used by philosophers? We engage in these foundational questions, and explore how the richness of mathematical practices is enhanced by thinking in terms of virtues and vices, and how the philosophical picture is challenged by the complexity of the case of mathematics. For example, within different social and interpersonal conditions, a trait often classified as a vice might be epistemically productive and vice versa. We illustrate that this occurs in mathematics by discussing Gerovitch’s historical study of the aggressive adversarialism of the Gelfand seminar in post-war Moscow. From this we conclude that virtue epistemologies of mathematics should avoid pre-emptive judgments about the sorts of epistemic character traits that ought to be promoted and criticised.

(2020) “Epistemic Injustice in Mathematics Education”
ZDM. With Colin Rittberg.
Link

Abstract: Equity and ethics in the learning of mathematics is a major topic for mathematics education research. The study of ethics and injustice in relation to epistemic pursuits, such as mathematics, is receiving a great deal of interest within contemporary philosophy. We propose a bridging project between these two disciplines, importing key ideas of “epistemic injustice” and “ethical orders” from philosophy into mathematics education to address questions of ethics, equity, values and norms. We build on Dawkins and Weber’s (Educ Stud Math 95:123–142, 2017) “apprenticeship model” of learning proofs and proving, which says that mathematics education should reflect the practices of research mathematicians. Focusing on the norms and values implicit in mathematical proving, we argue that deploying this model unreflectively can lead to “epistemic injustices” in which learners are disadvantaged based on their cultural or class background. We propose thinking about the problem in terms of Max Weber’s “ethical orders”, and the clash that arises between the ethical orders of mathematics and the existing ethical orders of the learners and teachers of mathematics. Weber’s lesson is that sometimes these clashes have no overarching resolution, and so the mathematics classroom may also have to settle for tailored pragmatic measures to combat individual cases of epistemic injustice.

(2020) “Using Crowd-Sourced Maths to Understand Mathematical Practice”
ZDM. With Alison Pease, Ursula Martin, and Andrew Aberdein.
Link

Abstract: Records of online collaborative mathematical activity provide us with a novel, rich, searchable, accessible and sizeable source of data for empirical investigations into mathematical practice. In this paper we discuss how the resources of crowdsourced mathematics can be used to help formulate and answer questions about mathematical practice, and what their limitations might be. We describe quantitative approaches to studying crowdsourced mathematics, reviewing work from cognitive history (comparing individual and collaborative proofs); social psychology (on the prospects for a measure of collective intelligence); human–computer interaction (on the factors that led to the success of one such project); network analysis (on the differences between collaborations on open research problems and known-but-hard problems); and argumentation theory (on modelling the argument structures of online collaborations). We also give an overview of qualitative approaches, reviewing work from empirical philosophy (on explanation in crowdsourced mathematics); sociology of scientific knowledge (on conventions and conversations in online mathematics); and ethnography (on contrasting conceptions of collaboration). We suggest how these diverse methods can be applied to crowdsourced mathematics and when each might be appropriate.

(2021) “Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups“.
Episteme. Joint work with Joshua Habgood-Coote.
Link

Abstract: In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects.

(2022) “Instructions and Recipes in Mathematical Proofs”.
Educational Studies in Mathematics. Joint work with Keith Weber.
Link

Abstract: In mathematics education research, proofs are often conceptualized as sequences of mathematical assertions. We argue that this ignores proofs that contain instructions to perform mathematical actions, often in the form of imperatives, which are common both in mathematical practice and in undergraduate mathematics textbooks. We consider in detail a specific type of proof which we call a recipe proof, which is comprised of sequence of instructions that direct the reader to produce mathematical objects with desirable properties. We present a model of what it means to understand a recipe proof and use this model in conjunction with process-object theories from mathematics education research, to explain why recipe proofs are inherently difficult for students to understand.

(2022) “The Concept of Extinction: Epistemology, Responsibility, and Precaution”. Ethics, Policy & Environment. OPEN ACCESS.
Link

Extinction is a concept of rapidly growing importance, with the world currently in the sixth mass extinction event and a biodiversity crisis. However, the concept of extinction has itself received surprisingly little attention from philosophers. I will first argue that in practice there is no single unified concept of extinction, but instead that its usage divides between descriptive, epistemic, and declarative concepts. I will then consider the epistemic challenges that arise in ascertaining whether a species has gone extinct, and how these lead to serious limitations on responsibility and accountability for preventing extinctions. I will propose two conceptual engineering changes to our understanding of extinction. Firstly, to use twin epistemic concepts of extinction, corresponding to high and low epistemic demands. Secondly, to explicitly recognize that in many contexts extinction is a thick concept that contains an evaluative component, which therefore motivates explicit ethical considerations such as the use of precautionary principles.

PhD Thesis

(2016) PhD Thesis: Proofs, Rigour and Informality: A Virtue Account of Mathematical Knowledge.
Link

Book Reviews

(2020) Book Review of 99 Variations on a proof by Philip Ording, British Journal for the History of Mathematics. Link

(2020) Book Review of A Concise History of Mathematics by John Stillwell, Metascience. Link

Drafts

In the following link you can find the draft papers listed below. Link

“Go Forth and Multiply: On Actions, Instructions and Imperatives in Mathematical Proofs”, to appear in Joshua Brown and Otávio Bueno (eds.) Essays on the Philosophy of Jody Azzouni, Cham: Springer.

“The Language of Proofs: A Philosophical Corpus Linguistics Study of Instructions and Imperatives in Mathematical Texts”. Joint work with Matthew Inglis. To appear in the Handbook of the History and Philosophy of Mathematical Practice.