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694Five theories of reasoning: Interconnections and applications to mathematicsLogic and Logical Philosophy 20 (1-2): 7-57. 2011.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 [39], Toulmin’s argumentation layout [52], Lakatos’s theory of reasoni…Read more
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1The companions and Socrates: Is Inara a hetaera?In Rhonda V. Wilcox & Tanya Cochran (eds.), Investigating Firefly and Serenity: Science Fiction on the Frontier, I. B. Tauris. pp. 63-75. 2008.
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265Observations on Sick MathematicsIn Bart van Kerkhove, Jean Paul van Bendegem & Jonas de Vuyst (eds.), Philosophical Perspectives on Mathematical Practice, College Publications. pp. 269--300. 2010.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 speci…Read more
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63Fallacies in MathematicsProceedings of the British Society for Research Into Learning Mathematics 27 (3): 1-6. 2007.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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60The parallel structure of mathematical reasoningIn Alison Pease & Brendan Larvor (eds.), Proceedings of the Symposium on Mathematical Practice and Cognition Ii: A Symposium at the Aisb/Iacap World Congress 2012, Society For the Study of Artificial Intelligence and the Simulation of Behaviour. pp. 7--14. 2012.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 admissibil…Read more
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464Raising the tone: Definition, bullshit, and the definition of bullshitIn G. Reisch & G. Hardcastle (eds.), Bullshit and Philosophy, Open Court. pp. 151-169. 2006.Bullshit is not the only sort of deceptive talk. Spurious definitions are another important variety of bad reasoning. This paper will describe some of these problematic tactics, and show how Harry Frankfurt’s treatment of bullshit may be extended to analyze their underlying causes. Finally, I will deploy this new account of definition to assess whether Frankfurt’s definition of bullshit is itself legitimate.
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445Mathematics and argumentationFoundations of Science 14 (1-2): 1-8. 2009.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.
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179Beauty Is Not Simplicity: An Analysis of Mathematicians' Proof AppraisalsPhilosophia Mathematica 23 (1): 87-109. 2015.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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60Argumentation schemes and communities of argumentational practiceIn Juho Ritola (ed.), Argument Cultures: Proceedings of OSSA 2009, Ossa. 2009.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 …Read more
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71Fallacy and argumentational viceIn Dima Mohammed & Marcin Lewinski (eds.), Virtues of argumentation: Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 22–25, 2013, Ossa. 2014.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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81Advances in Experimental Philosophy of Logic and Mathematics (edited book)Bloomsbury Academic. 2019.This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews…Read more
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36Ralph Johnson argues that mathematical proofs lack a dialectical tier, and thereby do not qualify as arguments. This paper argues that, despite this disavowal, Johnson’s account provides a compelling model of mathematical proof. The illative core of mathematical arguments is held to strict standards of rigour. However, compliance with these standards is itself a matter of argument, and susceptible to challenge. Hence much actual mathematical practice takes place in the dialectical tier.
University of St. Andrews
PhD, 2001
Melbourne, Florida, United States of America
Areas of Specialization
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Disagreement |
Epistemic Virtues |