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Table of Contents: Introduction.- Part I. What are Mathematical Arguments?.- Chapter 1. Non-Deduc... more Table of Contents: Introduction.- Part I. What are Mathematical Arguments?.- Chapter 1. Non-Deductive Logic in Mathematics: The Probability of Conjectures; James Franklin.- Chapter 2. Arguments, Proofs, and Dialogues; Erik C. W. Krabbe.- Chapter 3. Argumentation in Mathematics; Jesús Alcolea Banegas.- Chapter 4. Arguing Around Mathematical Proofs; Michel Dufour.- Part II. Argumentation as a Methodology for Studying Mathematical Practice.- Chapter 5. An Argumentative Approach to Ideal Elements in Mathematics; Paola Cantù.- Chapter 6. How Persuaded Are You? A Typology of Responses; Matthew Inglis and Juan Pablo Mejía-Ramos.- Chapter 7. Revealing Structures of Argumentations in Classroom Proving Processes; Christine Knipping and David Reid.- Chapter 8. Checking Proofs; Jesse Alama and Reinhard Kahle.- Part III. Mathematics as a Testbed for Argumentation Theory.- Chapter 9. Dividing by Zero—and Other Mathematical Fallacies; Lawrence H. Powers.- Chapter 10. Strategic Maneuvering in Mathematical Proofs; Erik C. W. Krabbe.- Chapter. 11 Analogical Arguments in Mathematics; Paul Bartha.- Chapter 12. What Philosophy of Mathematical Practice Can Teach Argumentation Theory about Diagrams and Pictures; Brendan Larvor.- Part IV. An Argumentational Turn in the Philosophy of Mathematics.- Chapter 13. Mathematics as the Art of Abstraction; Richard L. Epstein.- Chapter 14. Towards a Theory of Mathematical Argument; Ian J. Dove.- Chapter 15. Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics; Alison Pease, Alan Smaill, Simon Colton and John Lee.- Chapter 16. Mathematical Arguments and Distributed Knowledge; Patrick Allo, Jean Paul Van Bendegem and Bart Van Kerkhove.- Chapter 17. The Parallel Structure of Mathematical Reasoning; Andrew Aberdein.- Index.
Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and frag... more 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 math...
Records of online collaborative mathematical activity provide us with a novel, rich, searchable, ... more 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...
Deep disagreements are characteristically resistant to rational resolution. This paper explores t... more Deep disagreements are characteristically resistant to rational resolution. This paper explores the contribution a virtue theoretic approach to argumentation can make towards settling the practical matter of what to do when confronted with apparent deep disagreement, with particular attention to the virtue of courage.
The traditional view of evidence in mathematics is that evidence is just proof and proof is just ... more 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.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019
Analysis of online mathematics forums can help reveal how explanation is used by mathematicians; ... more 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. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.
In Monsters, monstrosities, and the monstrous in culture and society, Diego Compagna & Stefanie Steinhart, edd. (Wilmington, DE: Vernon Press), 2019
Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways be... more 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 (Cohen 1996). The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. I will describe these in much greater detail as they arise below, but here is a brief preview. 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 1976). He reconstructs the emergence during the nineteenth century of a proof of the Euler Conjecture, which ascribes a certain property to polyhedra. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify the conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the so-called sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed.
In The Kuhnian Image of Science: Time for a Decisive Transformation? (Moti Mizrahi, ed.), Rowman and Littlefield, London, 2018, pp. 133-154, 2018
In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglor... more 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.
This paper proposes that virtue theories of argumentation and theories of visual argumentation ca... more This paper proposes that virtue theories of argumentation and theories of visual argumentation can be of mutual assistance. An argument that adoption of a virtue approach provides a basis for rejecting the normative independence of visual argumentation is presented and its premisses analysed. This entails an independently valuable clarification of the contrasting normative presuppositions of the various virtue theories of argumentation. A range of different kinds of visual argument are examined, and it is argued that they may all be successfully evaluated within a virtue framework, without invoking any novel virtues.
It is a pleasure to return as guest editor for another issue of The Reasoner. I am particularly g... more It is a pleasure to return as guest editor for another issue of The Reasoner. I am particularly grateful to Prof. Daniel Cohen of Colby College for agreeing to be this month’s interviewee. Dan is a well-known figure in the informal logic community, and beyond: his TEDx talk, “For argument’s sake,” has received more than one million views. In that talk he addresses the challenge of how to make arguments fully satisfying for all the parties involved—something confrontational styles of argumentation all too often fail to achieve. He concludes that better arguments will require better arguers. This focus on the arguer has also characterised much of Dan’s recent scholarly work: he may be best known for his work on the application of virtue theory to argumentation. The relationship between virtues and arguments was the theme of the most recent Ontario Society for the Study of Argumentation conference, at which Dan was a keynote speaker (2013, “Virtue, in context,” Informal Logic, 33(4), pp. 471– 85). In that paper, which sums up his work of the previous decade, Dan defends a virtue theory of argumentation as the best theoretical basis for the pursuit of fully satisfying arguments. Virtue argumentation theory has enjoyed a recent surge of attention (this bibliography identifies more than 150 relevant works). In particular, it is the theme of a forthcoming special issue of Topoi which Dan and I have just finished editing. It contains some excellent papers, and we hope that it will broaden and deepen what is already a rich debate. My thanks again to Dan for an engaging discussion and to the editors of The Reasoner for the invitation to edit this issue.
What should a virtue theory of argumentation say about fallacious reasoning? If good arguments ar... more What should a virtue theory of argumentation say about fallacious reasoning? If good arguments are virtuous, then fallacies are vicious. Yet fallacies cannot just be identified with vices, since vices are dispositional properties of agents whereas fallacies are types of argument. Rather, if the normativity of good argumentation is explicable in terms of virtues, we should expect the wrongness of bad argumentation to be explicable in terms of vices. This approach is defended through analysis of several fallacies, with particular emphasis on the ad misericordiam.
We investigated whether mathematicians typically agree about the qualities of mathematical proofs... more 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 (2014). 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.
What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By ap... more What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians’ appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.
Several authors have recently begun to apply virtue theory to argumentation. Critics of this prog... more Several authors have recently begun to apply virtue theory to argumentation. Critics of this programme have suggested that no such theory can avoid committing an ad hominem fallacy. This criticism is shown to trade unsuccessfully on an ambiguity in the definition of ad hominem. The ambiguity is resolved and a virtue-theoretic account of ad hominem reasoning is defended.
A list of resources for virtue theories of argumentation. Last updated July 9th, 2024. Please sen... more A list of resources for virtue theories of argumentation. Last updated July 9th, 2024. Please send suggestions and corrections to aberdein@fit.edu.
Virtues of Argumentation: Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 22–25, 2013, Dima Mohammed & Marcin Lewinski, edd., 2013
If good argument is virtuous, then fallacies are vicious. Yet fallacies cannot just be identified... more If good argument is virtuous, then fallacies are vicious. Yet fallacies cannot just be identified with vices, since vices are dispositional properties of agents whereas fallacies are types of argument. Rather, if the normativity of good argumentation is explicable in terms of virtues, we should expect the wrongness of fallacies to be explicable in terms of vices. This approach is defended through case studies of several fallacies, with particular emphasis on the ad hominem.
The published works of scientists often conceal the cognitive processes that led to their results... more The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This paper analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humour is associated with risky forms of inference which are essential to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic neutral reasoning. These in turn are classified under seven headings: retroduction; citation; intuition; meta-argument; closure; generalization; and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed.
Table of Contents: Introduction.- Part I. What are Mathematical Arguments?.- Chapter 1. Non-Deduc... more Table of Contents: Introduction.- Part I. What are Mathematical Arguments?.- Chapter 1. Non-Deductive Logic in Mathematics: The Probability of Conjectures; James Franklin.- Chapter 2. Arguments, Proofs, and Dialogues; Erik C. W. Krabbe.- Chapter 3. Argumentation in Mathematics; Jesús Alcolea Banegas.- Chapter 4. Arguing Around Mathematical Proofs; Michel Dufour.- Part II. Argumentation as a Methodology for Studying Mathematical Practice.- Chapter 5. An Argumentative Approach to Ideal Elements in Mathematics; Paola Cantù.- Chapter 6. How Persuaded Are You? A Typology of Responses; Matthew Inglis and Juan Pablo Mejía-Ramos.- Chapter 7. Revealing Structures of Argumentations in Classroom Proving Processes; Christine Knipping and David Reid.- Chapter 8. Checking Proofs; Jesse Alama and Reinhard Kahle.- Part III. Mathematics as a Testbed for Argumentation Theory.- Chapter 9. Dividing by Zero—and Other Mathematical Fallacies; Lawrence H. Powers.- Chapter 10. Strategic Maneuvering in Mathematical Proofs; Erik C. W. Krabbe.- Chapter. 11 Analogical Arguments in Mathematics; Paul Bartha.- Chapter 12. What Philosophy of Mathematical Practice Can Teach Argumentation Theory about Diagrams and Pictures; Brendan Larvor.- Part IV. An Argumentational Turn in the Philosophy of Mathematics.- Chapter 13. Mathematics as the Art of Abstraction; Richard L. Epstein.- Chapter 14. Towards a Theory of Mathematical Argument; Ian J. Dove.- Chapter 15. Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics; Alison Pease, Alan Smaill, Simon Colton and John Lee.- Chapter 16. Mathematical Arguments and Distributed Knowledge; Patrick Allo, Jean Paul Van Bendegem and Bart Van Kerkhove.- Chapter 17. The Parallel Structure of Mathematical Reasoning; Andrew Aberdein.- Index.
Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and frag... more 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 math...
Records of online collaborative mathematical activity provide us with a novel, rich, searchable, ... more 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...
Deep disagreements are characteristically resistant to rational resolution. This paper explores t... more Deep disagreements are characteristically resistant to rational resolution. This paper explores the contribution a virtue theoretic approach to argumentation can make towards settling the practical matter of what to do when confronted with apparent deep disagreement, with particular attention to the virtue of courage.
The traditional view of evidence in mathematics is that evidence is just proof and proof is just ... more 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.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019
Analysis of online mathematics forums can help reveal how explanation is used by mathematicians; ... more 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. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.
In Monsters, monstrosities, and the monstrous in culture and society, Diego Compagna & Stefanie Steinhart, edd. (Wilmington, DE: Vernon Press), 2019
Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways be... more 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 (Cohen 1996). The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. I will describe these in much greater detail as they arise below, but here is a brief preview. 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 1976). He reconstructs the emergence during the nineteenth century of a proof of the Euler Conjecture, which ascribes a certain property to polyhedra. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify the conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the so-called sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed.
In The Kuhnian Image of Science: Time for a Decisive Transformation? (Moti Mizrahi, ed.), Rowman and Littlefield, London, 2018, pp. 133-154, 2018
In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglor... more 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.
This paper proposes that virtue theories of argumentation and theories of visual argumentation ca... more This paper proposes that virtue theories of argumentation and theories of visual argumentation can be of mutual assistance. An argument that adoption of a virtue approach provides a basis for rejecting the normative independence of visual argumentation is presented and its premisses analysed. This entails an independently valuable clarification of the contrasting normative presuppositions of the various virtue theories of argumentation. A range of different kinds of visual argument are examined, and it is argued that they may all be successfully evaluated within a virtue framework, without invoking any novel virtues.
It is a pleasure to return as guest editor for another issue of The Reasoner. I am particularly g... more It is a pleasure to return as guest editor for another issue of The Reasoner. I am particularly grateful to Prof. Daniel Cohen of Colby College for agreeing to be this month’s interviewee. Dan is a well-known figure in the informal logic community, and beyond: his TEDx talk, “For argument’s sake,” has received more than one million views. In that talk he addresses the challenge of how to make arguments fully satisfying for all the parties involved—something confrontational styles of argumentation all too often fail to achieve. He concludes that better arguments will require better arguers. This focus on the arguer has also characterised much of Dan’s recent scholarly work: he may be best known for his work on the application of virtue theory to argumentation. The relationship between virtues and arguments was the theme of the most recent Ontario Society for the Study of Argumentation conference, at which Dan was a keynote speaker (2013, “Virtue, in context,” Informal Logic, 33(4), pp. 471– 85). In that paper, which sums up his work of the previous decade, Dan defends a virtue theory of argumentation as the best theoretical basis for the pursuit of fully satisfying arguments. Virtue argumentation theory has enjoyed a recent surge of attention (this bibliography identifies more than 150 relevant works). In particular, it is the theme of a forthcoming special issue of Topoi which Dan and I have just finished editing. It contains some excellent papers, and we hope that it will broaden and deepen what is already a rich debate. My thanks again to Dan for an engaging discussion and to the editors of The Reasoner for the invitation to edit this issue.
What should a virtue theory of argumentation say about fallacious reasoning? If good arguments ar... more What should a virtue theory of argumentation say about fallacious reasoning? If good arguments are virtuous, then fallacies are vicious. Yet fallacies cannot just be identified with vices, since vices are dispositional properties of agents whereas fallacies are types of argument. Rather, if the normativity of good argumentation is explicable in terms of virtues, we should expect the wrongness of bad argumentation to be explicable in terms of vices. This approach is defended through analysis of several fallacies, with particular emphasis on the ad misericordiam.
We investigated whether mathematicians typically agree about the qualities of mathematical proofs... more 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 (2014). 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.
What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By ap... more What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians’ appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.
Several authors have recently begun to apply virtue theory to argumentation. Critics of this prog... more Several authors have recently begun to apply virtue theory to argumentation. Critics of this programme have suggested that no such theory can avoid committing an ad hominem fallacy. This criticism is shown to trade unsuccessfully on an ambiguity in the definition of ad hominem. The ambiguity is resolved and a virtue-theoretic account of ad hominem reasoning is defended.
A list of resources for virtue theories of argumentation. Last updated July 9th, 2024. Please sen... more A list of resources for virtue theories of argumentation. Last updated July 9th, 2024. Please send suggestions and corrections to aberdein@fit.edu.
Virtues of Argumentation: Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 22–25, 2013, Dima Mohammed & Marcin Lewinski, edd., 2013
If good argument is virtuous, then fallacies are vicious. Yet fallacies cannot just be identified... more If good argument is virtuous, then fallacies are vicious. Yet fallacies cannot just be identified with vices, since vices are dispositional properties of agents whereas fallacies are types of argument. Rather, if the normativity of good argumentation is explicable in terms of virtues, we should expect the wrongness of fallacies to be explicable in terms of vices. This approach is defended through case studies of several fallacies, with particular emphasis on the ad hominem.
The published works of scientists often conceal the cognitive processes that led to their results... more The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This paper analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humour is associated with risky forms of inference which are essential to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic neutral reasoning. These in turn are classified under seven headings: retroduction; citation; intuition; meta-argument; closure; generalization; and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed.
This paper proposes an account of mathematical reasoning as parallel in structure: the arguments ... more This paper proposes an account of mathematical reasoning as parallel in structure: the arguments which mathematicians use to persuade each other of their results comprise the argumentational structure; the inferential structure is composed of derivations which offer a formal counterpart to these arguments. Some conflicts about the foundations of mathematics correspond to disagreements over which steps should be admissible in the inferential structure. Similarly, disagreements over the admissibility of steps in the argumentational structure correspond to different views about mathematical practice. The latter steps may be analysed in terms of argumentation schemes. Three broad types of scheme are distinguished, a distinction which is then used to characterize and evaluate four contrasting approaches to mathematical practice.
In notes accompanying the published radio scripts for Hitchhiker's, Douglas Adams explains the pr... more In notes accompanying the published radio scripts for Hitchhiker's, Douglas Adams explains the predicament which led to his invention of the infinite improbability drive. Having ended the pilot with his protagonists being thrown into space, he struggled to extricate them in a manner which did not appear utterly improbable. But, watching a documentary on judo, he had a breakthrough: use the problem against itself, by making the improbability of their rescue the means of their rescue. This principle, of using problems against themselves, has an important place in philosophical methodology. Perhaps the most conspicuous example is Descartes's ‘Cogito’: an audacious attempt to use scepticism as a foundation for certainty. The Cogito has a place in Hitchhiker's too, satirized in Deep Thought's deduction of the existence of rice pudding and income tax before its data banks had been connected (a priori, as philosophers say). Adams once sought to discourage a thesis on his work, telling the prospective author that his ideas ‘come from the logic of jokes’. But, as the judo principle demonstrates, the logic of jokes can be a valuable source of philosophical inspiration. This chapter will explore the insights into philosophical method which Adams's style reveals.
Review of Mohan Ganesalingam, The language of mathematics: A linguistic and philosophical investi... more Review of Mohan Ganesalingam, The language of mathematics: A linguistic and philosophical investigation, FoLLI Publications on Logic, Language and Information, Springer, 2013. Philosophia Mathematica, 25(1), 2017, pp. 143–7.
Review of Leonard Nelson: A theory of philosophical fallacies. Translated by Fernando Leal and Da... more Review of Leonard Nelson: A theory of philosophical fallacies. Translated by Fernando Leal and David Carus (Argumentation Library, Vol. 26) Springer, Cham, Switzerland, 2016, vi + 211 pp. Argumentation 31(2), 2017, pp. 455–61.
Is bias an obstacle to a virtue theory of argumentation? Virtue theories seem vulnerable to a sit... more Is bias an obstacle to a virtue theory of argumentation? Virtue theories seem vulnerable to a situationist challenge, analogous to similar challenges in virtue ethics and epistemology, that behavioural dispositions are too situation-specific for virtues to be psychologically plausible. This paper argues that virtue argumentation may respond to this challenge by combining a defence of the virtue of humility with a demonstration of the role of attitude strength, as exhibited by deep-seated virtues.
Presidential Address to the 61st Annual Conference of the Florida Philosophical Association, Flag... more Presidential Address to the 61st Annual Conference of the Florida Philosophical Association, Flagler College, St Augustine, FL, 6th November 2015.
Virtues of Argumentation: Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 22–25, 2013, Dima Mohammed & Marcin Lewinski, edd., 2013
Virtues of Argumentation: Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 22–25, 2013, Dima Mohammed & Marcin Lewinski, edd., 2013
Intuitively, a pluralist solution is one in which a single question receives multiple answers. Su... more Intuitively, a pluralist solution is one in which a single question receives multiple answers. Such pluralist solutions have been proposed in many widely disparate contexts. This paper restates the concept of pluralism with greater precision; distinguishes it from, and establishes its independence of, some other notions with which it is frequently confused; and briefly lays out some of the benefits that this more nuanced approach to pluralism may yield for the debates in which it may be invoked.
How do we use the one-line, or reverse, truth table method to test an argument for validity? We t... more How do we use the one-line, or reverse, truth table method to test an argument for validity? We try to find a countermodel, that is an assignment of truth values to the individual letters that makes the premisses true but the conclusion false. If we find a countermodel it shows that the argument is invalid. (Remember: the argument is not your friend! Your success in finding a countermodel means that the argument fails.) On the other hand, if we can demonstrate that the argument cannot have a countermodel, then we have shown that it is valid.
In researching your essay, you find a passage that says exactly what you want to say. What should... more In researching your essay, you find a passage that says exactly what you want to say. What should you do?
This chapter argues that a virtue-theoretic account of argumenta-tion can enhance our understandi... more This chapter argues that a virtue-theoretic account of argumenta-tion 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. Several factors to which the rise of populism has been attributed may be understood as arising from vices of argumentation, including arrogance, emulousness, and insouciance. Conversely, virtues of argument such as humility and good listening offer some prospect of a constructive response to populism.
What are the prospects (if any) for a virtue-theoretic account of inference? This paper compares ... more What are the prospects (if any) for a virtue-theoretic account of inference? This paper compares three options. Firstly, assess each argument individually in terms of the virtues of the participants. Secondly, make the capacity for cogent inference itself a virtue. Thirdly, recapture a standard treatment of cogency by accounting for each of its components in terms of more familiar virtues. The three approaches are contrasted and their strengths and weaknesses assessed.
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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 relationship between virtues and arguments was the theme of the most recent Ontario Society for the Study of Argumentation conference, at which Dan was a keynote speaker (2013, “Virtue, in context,” Informal Logic, 33(4), pp. 471– 85). In that paper, which sums up his work of the previous decade, Dan defends a virtue theory of argumentation as the best theoretical basis for the pursuit of fully satisfying arguments. Virtue argumentation theory has enjoyed a recent surge of attention (this bibliography identifies more than 150 relevant works). In particular, it is the theme of a forthcoming special issue of Topoi which Dan and I have just finished editing. It contains some excellent papers, and we hope that it will broaden and deepen what is already a rich debate. My thanks again to Dan for an engaging discussion and to the editors of The Reasoner for the invitation to edit this issue.
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 relationship between virtues and arguments was the theme of the most recent Ontario Society for the Study of Argumentation conference, at which Dan was a keynote speaker (2013, “Virtue, in context,” Informal Logic, 33(4), pp. 471– 85). In that paper, which sums up his work of the previous decade, Dan defends a virtue theory of argumentation as the best theoretical basis for the pursuit of fully satisfying arguments. Virtue argumentation theory has enjoyed a recent surge of attention (this bibliography identifies more than 150 relevant works). In particular, it is the theme of a forthcoming special issue of Topoi which Dan and I have just finished editing. It contains some excellent papers, and we hope that it will broaden and deepen what is already a rich debate. My thanks again to Dan for an engaging discussion and to the editors of The Reasoner for the invitation to edit this issue.