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62The 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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504Raising the tone: Definition, bullshit, and the definition of bullshitIn Hardcastle Reisch (ed.), 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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542Mathematics 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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188Beauty 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. 2010.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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74Fallacy 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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85Advances 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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39Ralph 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.
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Proofs and rebuttals: Applying Stephen Toulmin's layout of arguments to mathematical proofIn Marta Bílková & Ondřej Tomala (eds.), The Logica Yearbook 2005, Filosofia. pp. 11-23. 2006.This paper explores some of the benefits informal logic may have for the analysis of mathematical inference. It shows how Stephen Toulmin’s pioneering treatment of defeasible argumentation may be extended to cover the more complex structure of mathematical proof. Several common proof techniques are represented, including induction, proof by cases, and proof by contradiction. Affinities between the resulting system and Imre Lakatos’s discussion of mathematical proof are then explored.
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74In defence of virtue: The legitimacy of agent-based argument appraisalInformal Logic 34 (1): 77-93. 2014.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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480The Vices of ArgumentTopoi 35 (2): 413-422. 2016.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 s…Read more
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12Dale Jacquette, Meinongian Logic: The Semantics of Existence and Nonexistence (review)Philosophy in Review 17 (3): 176-178. 1997.
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193The Uses of Argument in MathematicsArgumentation 19 (3): 287-301. 2005.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 l…Read more
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492Strange bedfellows: The interpenetration of philosophy and pornographyIn Dave Monroe (ed.), Porn: How to Think with Kink, Wiley-blackwell. pp. 22-34. 2010.This paper explores some surprising historical connections between philosophy and pornography (including 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. It observes that these works frequently dramatize a tension between reason and emotion, and argues that their existen…Read more
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464Managing Informal Mathematical Knowledge: Techniques from Informal LogicLecture Notes in Artificial Intelligence 4108 208--221. 2006.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 inf…Read more
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 |