- School of Arts & Communication,
Florida Institute of Technology,
150 West University Blvd,
Melbourne, Florida 32901-6975, U.S.A. - 321 674 8368
Andrew Aberdein
Florida Institute of Technology, Arts and Communication, Faculty Member
- Informal Logic, Philosophy Of Mathematics, Argumentation Theory, Logic, Virtue Epistemology, Argumentation, and 27 moreArgument Structure, Argumentation, Critical Thinking, DIscourse, Philosophy of Science, History of Mathematics, Logic And Foundations Of Mathematics, History of Logic, Communities of practice, Mathematical Practice, Imre Lakatos, Charles S. Peirce, Persuasive Definition, Pornography Studies, Bullshit Studies, Proof and Proving in Mathematics Education, Explanation, Rhetoric of Science, Mathematical Cognition, History of Science, Argumentation Theory and Critical Thinking, Philosophy of Logic, Practical Reasoning, Epistemic Justification, Fallacies, Metaphysical grounding, Mathematical reasoning and proof, Virtue Theory of Argumentation, and Visual Argumentationedit
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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.-... 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.
Research Interests: History of Mathematics, Philosophy Of Mathematics, Mathematics Education, Rhetoric of Science, Argumentation, and 8 moreMathematical Practice, Argumentation, Critical Thinking, DIscourse, Argumentation Theory, Proof and Reasoning, Argument Structure, Informal Logic, Proof and Proving in Mathematics Education, and Mathematical Cognition
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... 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...
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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... 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...
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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... 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.
Research Interests: Philosophy and Topoi
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... 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.
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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... 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’.
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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... 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.
Research Interests: History of Mathematics, Philosophy Of Mathematics, Monster Theory, Group Theory, Imre Lakatos, and 12 moreMathematical Practice, Monsters and Monster Theory, The Monstrous and Otherness, Monstrosity, Henri Poincare, Philosophy of Mathematical Practice, Monsters, Mary Douglas, Monsters and the Monstrous, Henri Poincaré, Mathematical Practices, and Jeffrey Jerome Cohen
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... 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.
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.
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Presented at 2nd European Conference on Argumentation (ECA 2017), University of Fribourg, June 2017.
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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... 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.
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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... 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.
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.
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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... 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.
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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... 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.
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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:... 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.
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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... 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.
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A list of resources for virtue theories of argumentation. Last updated October 31st, 2023. Please send suggestions and corrections to aberdein@fit.edu.
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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... 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.
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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,... 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.
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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... 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.
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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... 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.
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The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been developed. Perhaps owing to... more
The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been developed. Perhaps owing to their diverse backgrounds, there are several connections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce’s development of abductive reasoning [Peirce, 1931-58], Toulmin’s argumentation layout [Toulmin, 1958], Lakatos’s theory of reasoning in mathematics [Lakatos, 1976], Pollock’s notions of counterexample [Pollock, 1995], and argumentation schemes constructed by Walton et al. [Walton, 2008], and explore some connections between, as well as within, the theories. For instance, we investigate Peirce’s abduction to deal with surprising situations in mathematics, represent Pollock’s examples in terms of Toulmin’s layout, discuss connections between Toulmin’s layout and Walton’s argumentation schemes, and suggest new argumentation schemes to cover the sort of reasoning that Lakatos describes, in which arguments may be accepted as faulty, but revised, rather than being accepted or rejected. We also consider how such theories may apply to reasoning in mathematics: in particular, we aim to build on ideas such as Dove’s [Dove, 2007], which help to show ways in which the work of Lakatos fits into the informal reasoning community.
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This paper argues that new light may be shed on mathematical reasoning in its non-pathological forms by careful observation of its pathologies. The first section explores the application to mathematics of recent work on fallacy theory,... more
This paper argues that new light may be shed on mathematical reasoning in its non-pathological forms by careful observation of its pathologies. The first section explores the application to mathematics of recent work on fallacy theory, specifically the concept of an ‘argumentation scheme’: a characteristic pattern under which many similar inferential steps may be subsumed. Fallacies may then be understood as argumentation schemes used inappropriately. The next section demonstrates how some specific mathematical fallacies may be characterized in terms of argumentation schemes. The third section considers the phenomenon of correct answers which result from incorrect methods. This turns out to pose some deep questions concerning the nature of mathematical knowledge. In particular, it is argued that a satisfactory epistemology for mathematical practice must address the role of luck.
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A widely circulated list of spurious proof types may help our understanding of informal mathematical reasoning. An account in terms of argumentation schemes is proposed.
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Virtue theories have become influential in ethics and epistemology. This paper argues for a similar approach to argumentation. Several potential obstacles to virtue theories in general, and to this new application in particular, are... more
Virtue theories have become influential in ethics and epistemology. This paper argues for a similar approach to argumentation. Several potential obstacles to virtue theories in general, and to this new application in particular, are considered and rejected. A first attempt is made at a survey of argumentational virtues, and finally it is argued that the dialectical nature of argumentation makes it particularly suited for virtue theoretic analysis.
Keywords: Ad hominem - Logical universality - Virtue epistemology - Virtue ethics
Keywords: Ad hominem - Logical universality - Virtue epistemology - Virtue ethics
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This paper explores the surprising historical connections between philosophical and pornographic writing (such as pornography written by or about philosophers, and works that are both philosophical and pornographic). Examples discussed... more
This paper explores the surprising historical connections between philosophical and pornographic writing (such as pornography written by or about philosophers, and works that are both philosophical and pornographic). Examples discussed include Diderot's Les Bijoux Indiscrets, Argens's Therésè Philosophe, Aretino's Ragionamenti, Andeli's Lai d'Aristote and the Gor novels of John Norman (aka John Lange). It observes that these works frequently dramatize a tension between reason and emotion, and concludes that their existence poses a problem for philosophical arguments against pornography.
Keywords: French appetite for clandestine literature - risky endeavor, lucrative for the determined and ingenious; enlightenment classics or disreputable libertine smut - improbable marriages of philosophy and pornography; pornography and philosophy, connections exploring their shared history; historical “pornography” - similar effects on consumers; L'Ecole des filles, pretensions to philosophy - explicit in its subtitle, La Philosophie des dames; device of a young woman receiving sexual education - from more experienced women; Thérèse philosophe and La Philosophie dans le boudoir; Aristotle and Phyllis story, and its comic denouement - subjects of medieval and Renaissance art.
Keywords: French appetite for clandestine literature - risky endeavor, lucrative for the determined and ingenious; enlightenment classics or disreputable libertine smut - improbable marriages of philosophy and pornography; pornography and philosophy, connections exploring their shared history; historical “pornography” - similar effects on consumers; L'Ecole des filles, pretensions to philosophy - explicit in its subtitle, La Philosophie des dames; device of a young woman receiving sexual education - from more experienced women; Thérèse philosophe and La Philosophie dans le boudoir; Aristotle and Phyllis story, and its comic denouement - subjects of medieval and Renaissance art.
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Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and... more
Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.
Keywords: Argumentation - Mathematical practice
Keywords: Argumentation - Mathematical practice
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This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of... more
This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals.
Keywords: classical logic, logical theory, intuitionistic logic, quantum logic, relevance logic, paraconsistent logic
Keywords: classical logic, logical theory, intuitionistic logic, quantum logic, relevance logic, paraconsistent logic
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Is it possible to distinguish communities of arguers by tracking the argumentation schemes they employ? There are many ways of relating schemes to communities, but not all are productive. Attention must be paid not only to the... more
Is it possible to distinguish communities of arguers by tracking the argumentation schemes they employ? There are many ways of relating schemes to communities, but not all are productive. Attention must be paid not only to the admissibility of schemes within a community of argumentational practice, but also to their comparative frequency. Two examples are discussed: informal mathematics, a convenient source of well-documented argumentational practice, and anthropological evidence of nonstandard reasoning.
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In March 1615 His Majesty King James the VI of Scotland and I of England participated in a debate concerning the use of logic by dogs. The debate was one of several entertainments provided for the King during a visit to the University of... more
In March 1615 His Majesty King James the VI of Scotland and I of England participated in a debate concerning the use of logic by dogs. The debate was one of several entertainments provided for the King during a visit to the University of Cambridge. At first glance, this event might seem wholly frivolous. Certainly the King’s enthusiasm for hunting played a part in the choice of topic. But King James was no idle, anti-intellectual prince. He had assembled a hothouse of Protestant theologians to produce the English Bible that still bears his name, and his own collected works were to appear the following year. Moreover, as we shall see, the arguments rehearsed before King James echo down the history of logic, from antiquity to the twenty-first century.
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Virtue ethics is perhaps the fastest growing field in ethical theory. Virtue theories have also been proposed in other disciplines, such as epistemology and jurisprudence. This paper stakes a claim in another area: argumentation.
This paper considers the application to mathematical fallacies of techniques drawn from informal logic, specifically the use of ‘argument schemes’. One such scheme, for Appeal to Expert Opinion, is considered in some detail.
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Informal and formal logic are complementary methods of argument analysis. Informal logic provides a pragmatic treatment of features of argumentation which cannot be reduced to logical form. This paper shows how paying attention to aspects... more
Informal and formal logic are complementary methods of argument analysis. Informal logic provides a pragmatic treatment of features of argumentation which cannot be reduced to logical form. This paper shows how paying attention to aspects of mathematical argumentation captured by informal, but not formal, logic can offer a more nuanced understanding of mathematical proof and discovery.
Keywords: dialectic - four colour theorem - informal logic - mathematical proof - Stephen Toulmin - Douglas Walton
Keywords: dialectic - four colour theorem - informal logic - mathematical proof - Stephen Toulmin - Douglas Walton
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Much work in MKM depends on the application of formal logic to mathematics. However, much mathematical knowledge is informal. Luckily, formal logic only represents one tradition in logic, specifically the modeling of inference in terms of... more
Much work in MKM depends on the application of formal logic to mathematics. However, much mathematical knowledge is informal. Luckily, formal logic only represents one tradition in logic, specifically the modeling of inference in terms of logical form. Many inferences cannot be captured in this manner. The study of such inferences is still within the domain of logic, and is sometimes called informal logic. This paper explores some of the benefits informal logic may have for the management of informal mathematical knowledge.
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Formal logic has been profoundly valuable in capturing the content of mathematics. However, mathematicians seldom write proofs in formal logic. Although most mathematical proofs may in principle be formalized, the process is often arduous... more
Formal logic has been profoundly valuable in capturing the content of mathematics. However, mathematicians seldom write proofs in formal logic. Although most mathematical proofs may in principle be formalized, the process is often arduous and can dramatically reduce intelligibility. For this reason such formalization is rarely attempted, and most mathematicians regard formal logic as of little relevance for their work. But formal logic only represents one tradition in logic, specifically the modeling of inference in terms of logical form. Many inferences, especially in natural language, cannot be captured in this manner. The study of such inferences is still within the domain of logic, and is sometimes called informal logic. In this paper I explore some of the benefits this tradition may have for the analysis of informal mathematical inference. Specifically, I show how Stephen Toulmin’s treatment of (primarily) non-mathematical arguments may be extended to cover mathematical proofs. I also exhibit affinities between Toulmin’s pioneering treatment of defeasible argumentation and Imre Lakatos’s discussion of the role played by such arguments in the development of mathematical proofs.
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Stephen Toulmin once observed that ‘it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate’ [Toulmin et al., 1979, An Introduction to Reasoning, Macmillan, London, p. 89]. Might the... more
Stephen Toulmin once observed that ‘it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate’ [Toulmin et al., 1979, An Introduction to Reasoning, Macmillan, London, p. 89]. Might the application of Toulmin’s layout of arguments to mathematics remedy this oversight? Toulmin’s critics fault the layout as requiring so much abstraction as to permit incompatible reconstructions. Mathematical proofs may indeed be represented by fundamentally distinct layouts. However, cases of genuine conflict characteristically reflect an underlying disagreement about the nature of the proof in question.
Keywords Euclid - mathematical argumentation - proof - rebuttal - Stephen Toulmin - undercutter
Keywords Euclid - mathematical argumentation - proof - rebuttal - Stephen Toulmin - undercutter
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The status and limits of science are the focus of urgent public debate. This paper contributes a philosophical analysis of representations of science and the supernatural in popular culture. It explores and critiques a threefold taxonomy... more
The status and limits of science are the focus of urgent public debate. This paper contributes a philosophical analysis of representations of science and the supernatural in popular culture. It explores and critiques a threefold taxonomy of supernatural narratives: (1) reduction of the supernatural to contemporary science; (2) reduction to a `future science' methodologically continuous with contemporary science; (3) the supernatural as irreducible. The means by which the TV series Buffy the Vampire Slayer adroitly negotiates the borderlines between these narratives is related to the `science wars', the two cultures debate, and the ancients vs. moderns dispute.
The recapture relationship is an important element to any understanding of the connexion between different systems of logic. Loosely speaking, one system of logic recaptures another if it is possible to specify a subsystem of the former... more
The recapture relationship is an important element to any understanding of the connexion between different systems of logic. Loosely speaking, one system of logic recaptures another if it is possible to specify a subsystem of the former system which exhibits the same patterns of inference as the latter system. In particular if a relationship of this kind can be shown to exist between a non-classical logic and classical logic, the non-classical system is said to exhibit classical recapture. This has been invoked by several proponents of non-classical logics to argue that their system retains classical logic as a limit case, and is therefore a methodologically progressive successor to classical logic. In this paper I advance and defend a new and more precise account of recapture and the character of its reception by the proponents of the recapturing system. I then indicate some of the applications of classical recapture which this account makes possible.
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Charles Stevenson introduced the term 'persuasive definition' to describe a suspect form of moral argument 'which gives a new conceptual meaning to a familiar word without substantially changing its emotive meaning'. However, as Stevenson... more
Charles Stevenson introduced the term 'persuasive definition' to describe a suspect form of moral argument 'which gives a new conceptual meaning to a familiar word without substantially changing its emotive meaning'. However, as Stevenson acknowledges, such a move can be employed legitimately. If persuasive definition is to be a useful notion, we shall need a criterion for identifying specifically illegitimate usage. I criticize a recent proposed criterion from Keith Burgess-Jackson and offer an alternative.
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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.
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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.
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Douglas Walton , One-Sided Arguments: A Dialectical Analysis of Bias Reviewed by.
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Dale Jacquette , Meinongian Logic: The Semantics of Existence and Nonexistence
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Edouard Morot-Sir , The Imagination of Reference II: Perceiving, Indicating, Naming Reviewed by.
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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... 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.
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Presidential Address to the 61st Annual Conference of the Florida Philosophical Association, Flagler College, St Augustine, FL, 6th November 2015.
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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... 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 try to find a countermodel, that is an assignment of truth values to the individual letters that makes the premisses true but the conclusion... 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.
Research Interests: Logic and Logic Teaching
In researching your essay, you find a passage that says exactly what you want to say. What should you do?
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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... 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.
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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... 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.