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 article 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 humor 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. (shrink)
Ralph 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.
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 reasoning in mathematics [23], Pollock’s notions of counterexample [44], and argumentation (...) schemes constructed by Walton et al. [54], 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 [13], which help to show ways in which the work of Lakatos fits into the informal reasoning community. (shrink)
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.
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. (shrink)
A widely circulated list of spurious proof types may help to clarify our understanding of informal mathematical reasoning. An account in terms of argumentation schemes is proposed.
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 existence poses a problem for philosophical arguments against pornography.
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.
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.
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. (shrink)
Imagine a dog tracing a scent to a crossroads, sniffing all but one of the exits, and then proceeding down the last without further examination. According to Sextus Empiricus, Chrysippus argued that the dog effectively employs disjunctive syllogism, concluding that since the quarry left no trace on the other paths, it must have taken the last. The story has been retold many times, with at least four different morals: (1) dogs use logic, so they are as clever as humans; (2) (...) dogs use logic, so using logic is nothing special; (3) dogs reason well enough without logic; (4) dogs reason better for not having logic. This paper traces the history of Chrysippus's dog, from antiquity up to its discussion by relevance logicians in the twentieth century. (shrink)
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.
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. (shrink)
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.
Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is that a more nuanced understanding of mathematical proof and discovery may be achieved by paying attention to the aspects of mathematical argumentation which can be captured by informal, rather than formal, logic. Two accounts of argumentation are considered: the pioneering work of (...) Stephen Toulmin [The uses of argument, Cambridge University Press, 1958] and the more recent studies of Douglas Walton, [e.g. The new dialectic: Conversational contexts of argument, University of Toronto Press, 1998]. The focus of both of these approaches has largely been restricted to natural language argumentation. However, Walton’s method in particular provides a fruitful analysis of mathematical proof. He offers a contextual account of argumentational strategies, distinguishing a variety of different types of dialogue in which arguments may occur. This analysis represents many different fallacious or otherwise illicit arguments as the deployment of strategies which are sometimes admissible in contexts in which they are inadmissible. I argue that mathematical proofs are deployed in a greater variety of types of dialogue than has commonly been assumed. I proceed to show that many of the important philosophical and pedagogical problems of mathematical proof arise from a failure to make explicit the type of dialogue in which the proof is introduced. (shrink)
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. (shrink)
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. (shrink)
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.
