Andrew Aberdein

Virtue-based approaches have attracted significant recent interest in argumentation, including a recent anthology of Chinese translations of important articles in the field. In this article, adapted from the introduction to that anthology, we discuss the origins of virtue argumentation and some of the challenges it has faced, as well as attempt to provide an overview of recent work on the virtues and vices relevant to argumentation. In the final section we discuss the articles that were selected and motivate their selection.

A queue‐jumping argument concludes that some course of action is impermissible by likening it to the presumptively impermissible act of jumping a queue. Arguments of this sort may be found in a disparate range of contexts and in support of policies favoured by both left and right. Examples include arguments against private education and private health care but also arguments against accommodations for learning disabilities, refugee resettlement, and birthright citizenship. We infer that, although queue‐jumping arguments are strictly analogies, they constitute a sufficiently distinct class of arguments to justify their separate treatment. The paper proposes an argumentation scheme for queue‐jumping arguments and demonstrates its applicability to some existing arguments of this type.

The virtues and vices of argument are now an established part of argumentation theory. They have helped direct attention to hitherto neglected aspects of how we argue. However, it remains controversial whether a virtue theory can contribute to some of the central questions of argumentation theory. Notably, Harvey Siegel disputes whether what he calls ‘arguments in the abstract propositional sense’ can be evaluated meaningfully within a virtue theory. This paper explores the prospects for grounding an account of argument evaluation in arguers’ virtues and vices by examination of a corresponding debate in virtue ethics: Can an ethics of virtue guide our actions? It is thereby argued that an affirmative answer is possible: virtues suffice for argument evaluation.

Anonymous argumentation has recently been the focus of public controversy: flash points include the outing of pseudonymous bloggers by newspapers and the launch of an academic journal that expressly permits pseudonymous authorship. However, the controversy is not just a recent one—similar debates took place in the nineteenth century over the then common practice of anonymous journalism. Amongst the arguments advanced by advocates of anonymous argumentation in either era is the contention that it is essential if the widest range of voices are to be heard; amongst the counterarguments of its critics, that it weakens the credibility of individual arguments and irresponsibly cheapens the standard of public discourse. This paper explores some implications of the controversy for the ethics of argumentation in general and virtue theories of argumentation in particular.

“Is every definition persuasive?” If essentialist views on definition are rejected and a pragmatic account adopted, where defining is a speech act which fixes the meaning of a term, then a problem arises: if meanings are not fixed by the essence of being itself, is not every definition persuasive? To address the problem, we refer to Douglas Walton’s impressive intellectual heritage—specifically on the argumentative potential of definition. In finding some non-persuasive definitions, we show not every definition is persuasive. The persuasiveness lies not in syntactic or semantic properties, but the context. We present this pragmatic account and provide rules for analysing and evaluating persuasive definition—a promising direction for further research.

Disagreements that resist rational resolution, often termed “deep disagreements”, have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki’s work on the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice.

The fallacy fallacy is either the misdiagnosis of fallacy or the supposition that the conclusion of a fallacy must be a falsehood. This paper explores the relevance of these and related errors of reasoning for the appraisal of arguments, especially within virtue theories of argumentation. In particular, the fallacy fallacy exemplifies the Owl of Minerva problem, whereby tools devised to understand a norm make possible new ways of violating the norm. Fallacies are such tools and so are vices. Hence a similar problem arises with argumentative vices. Fortunately, both instances of the problem have a common remedy.

This chapter argues that a virtue-theoretic account of argumentation can enhance our understanding of the phenomenon of populism and offer some lines of response. Virtue theories of argumentation emphasize the role of arguers in the conduct and evaluation of arguments and lay particular stress on arguers’ acquired dispositions of character, otherwise known as intellectual virtues and vices. One variety of argumentation of particular relevance to democratic decision-making is group deliberation. There are both theoretical and empirical reasons for maintaining that intellectual humility is pivotal to the practice of group deliberation. Several factors to which the rise of populism has been attributed may be understood as failures of the virtues of argumentation in general, and intellectual humility in particular.

“Is every definition persuasive?” If essentialist views on definition are rejected and a pragmatic account adopted, where defining is a speech act which fixes the meaning of a term, then a problem arises: if meanings are not fixed by the essence of being itself, is not every definition persuasive? To address the problem, we refer to Douglas Walton’s impressive intellectual heritage—specifically on the argumentative potential of definition. In finding some non-persuasive definitions, we show not every definition is persuasive. The persuasiveness lies not in syntactic or semantic properties, but the context. We present this pragmatic account and provide rules for analysing and evaluating persuasive definition—a promising direction for further research.

Virtue theories of argumentation (VTA) emphasize the roles arguers play in the conduct and evaluation of arguments, and lay particular stress on arguers’ acquired dispositions of character, that is, virtues and vices. The inspiration for VTA lies in virtue epistemology and virtue ethics, the latter being a modern revival of Aristotle’s ethics. Aristotle is also, of course, the father of Western logic and argumentation. This paper asks to what degree Aristotle may thereby be claimed as a forefather by VTA.

Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend.

We investigated whether mathematicians typically agree about the qualities of mathematical proofs. Between-mathematician consensus in proof appraisals is an implicit assumption of many arguments made by philosophers of mathematics, but to our knowledge the issue has not previously been empirically investigated. We asked a group of mathematicians to assess a specific proof on four dimensions, using the framework identified by Inglis and Aberdein (2015). We found widespread disagreement between our participants about the aesthetics, intricacy, precision and utility of the proof, suggesting that a priori assumptions about the consistency of mathematical proof appraisals are unreasonable.

Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed.

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.

Many of the methods commonly used to research mathematical practice, such as analyses of historical episodes or individual cases, are particularly well-suited to generating causal hypotheses, but less well-suited to testing causal hypotheses. In this paper we reflect on the contribution that the so-called hypothetico-deductive method, with a particular focus on experimental studies, can make to our understanding of mathematical practice. By way of illustration, we report an experiment that investigated how mathematicians attribute aesthetic properties to mathematical proofs. We demonstrate that perceptions of the aesthetic properties of mathematical proofs are, in some cases at least, subject to social influence. Specifically, we show that mathematicians’ aesthetic judgements tend to conform to the judgements made by others. Pedagogical implications are discussed.

I intend to bring recent work applying virtue theory to the study of argument to bear on a much older problem, that of disagreements that resist rational resolution, sometimes termed "deep disagreements". Just as some virtue epistemologists have lately shifted focus onto epistemic vices, I shall argue that a renewed focus on the vices of argument can help to illuminate deep disagreements. In particular, I address the role of arrogance, both as a factor in the diagnosis of deep disagreements and as an obstacle to their mutually acceptable resolution. Arrogant arguers are likely to make any disagreements to which they are party seem deeper than they really are and arrogance impedes the strategies that we might adopt to resolve deep disagreements. As a case in point, since arrogant or otherwise vicious arguers cannot be trusted not to exploit such strategies for untoward ends, any policy for deep disagreement amelioration must require particularly close attention to the vices of argument, lest they be exploited by the unscrupulous.

Virtue theories have lately enjoyed a modest vogue in the study of argumentation, echoing the success of more far-reaching programmes in ethics and epistemology. Virtue theories of argumentation (VTA) comprise several conceptually distinct projects, including the provision of normative foundations for argument evaluation and a renewed focus on the character of good arguers. Perhaps the boldest of these is the pursuit of the fully satisfying argument, the argument that contributes to human flourishing. This project has an independently developed epistemic analogue: eudaimonistic virtue epistemology. Both projects stress the importance of widening the range of cognitive goals beyond, respectively, cogency and knowledge; both projects emphasize social factors, the right sort of community being indispensable for the cultivation of the intellectual virtues necessary to each project. This paper proposes a unification of the two projects by arguing that the intellectual good life sought by eudaimonistic virtue epistemologists is best realized through the articulation of an account of argumentation that contributes to human flourishing.

In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglorious' revolutions--only the former preserves all 'the key components of a theory' [1]. A widespread view, expressed in these terms, is that empirical science characteristically exhibits inglorious revolutions but that revolutions in mathematics are at most glorious [2]. Here are three possible responses: 0. Accept that empirical science and mathematics are methodologically discontinuous; 1. Argue that mathematics can exhibit inglorious revolutions; 2. Deny that inglorious revolutions are characteristic of science. Where Aberdein and Read take option 1, option 2 is preferred by Mizrahi [3]. This paper seeks to resolve this disagreement through consideration of some putative mathematical revolutions. [1] Andrew Aberdein and Stephen Read, The philosophy of alternative logics, The Development of Modern Logic (Leila Haaparanta, ed.), Oxford University Press, Oxford, 2009, pp. 613-723. [2] Donald Gillies (ed.), Revolutions in Mathematics, Oxford University Press, Oxford, 1992. [3] Moti Mizrahi, Kuhn's incommensurability thesis: What's the argument?, Social Epistemology 29 (2015), no. 4, 361-378.

The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning.

Analysis of online mathematics forums can help reveal how explanation is used by mathematicians; we contend that this use of explanation may help to provide an informal conceptualization of simplicity. We extracted six conjectures from recent philosophical work on the occurrence and characteristics of explanation in mathematics. We then tested these conjectures against a corpus derived from online mathematical discussions. To this end, we employed two techniques, one based on indicator terms, the other on a random sample of comments lacking such indicators. Our findings suggest that explanation is widespread in mathematical practice and that it occurs not only in proofs but also in other mathematical contexts. Our work also provides further evidence for the utility of empirical methods in addressing philosophical problems.

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book.