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Mathematical Wit and Mathematical Cognition
Abstract
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.
... This means that argumentation schemes can provide a unified treatment of a wide range of arguments employed in mathematics. Indeed, a number of authors have applied argumentation schemes to mathematical reasoning [1,4,5,6,7,20,50,51,54]. ...
... In addition, mathematical applications have been found for several schemes that are missing from Table 4 but which are found in the more exhaustive (but less structured) list in [75]. These include linguistic arguments, such as arguments from arbitrariness or vagueness of a verbal classification [54] and argument from definition to verbal classification [5], but also source-dependent arguments, such as ethotic argument [5], practical reasoning arguments, such as argument from positive consequences [5], discovery arguments, such as abductive argument [54] and argument from evidence to a hypothesis [5,7,54], and arguments applying rules to cases, such as argument from an exceptional case [54]. Conversely, not all of the individual schemes in Table 4 have yet been found useful in discussing mathematics. ...
... In addition, mathematical applications have been found for several schemes that are missing from Table 4 but which are found in the more exhaustive (but less structured) list in [75]. These include linguistic arguments, such as arguments from arbitrariness or vagueness of a verbal classification [54] and argument from definition to verbal classification [5], but also source-dependent arguments, such as ethotic argument [5], practical reasoning arguments, such as argument from positive consequences [5], discovery arguments, such as abductive argument [54] and argument from evidence to a hypothesis [5,7,54], and arguments applying rules to cases, such as argument from an exceptional case [54]. Conversely, not all of the individual schemes in Table 4 have yet been found useful in discussing mathematics. ...
Douglas Walton's multitudinous contributions to the study of argumentation seldom, if ever, directly engage with argumentation in mathematics. Nonetheless, several of the innovations with which he is most closely associated lend themselves to improving our understanding of mathematical arguments. I concentrate on two such innovations: dialogue types ({\S}1) and argumentation schemes ({\S}2). I argue that both devices are much more applicable to mathematical reasoning than may be commonly supposed.
... Hence, we can use mathematics as a tool to better understand humor. In this article, I follow Aberdein (2013) and suggest that the converse can be true as well. In thinking about mathematical humor, we can better understand the nature of mathematical reasoning. ...
... In mathematical parlance, there are large numbers of jokes that are frequently shared between mathematicians. Some authors have argued that these mathematical jokes are not merely amusing diversions to entertain mathematicians, but that these jokes also play important roles for the mathematical community, including building social cohesion, gently prodding members of the mathematical community to modify undesirable behavior, and identifying fallacious reasoning that mathematicians use (for instance, see Aberdein, 2013). Hence, jokes that are common in mathematical folklore can be used as a lens to show what it means to do mathematics and what it is like to be a mathematician. ...
Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings of the word ``argument''; to some of the specific argumentation-theoretic tools that have been applied to mathematics, notably Toulmin layouts and argumentation schemes; to some of the different ways that argumentation is implicated in mathematical practices; and to the social aspects of mathematical argumentation.
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.
In argumentation studies, almost all theoretical proposals are applied, in general, to the analysis and evaluation of argumentative products, but little attention has been paid to the creative process of arguing. Mathematics can be used as a clear example to illustrate some significant theoretical differences between mathematical practice and the products of it, to differentiate the distinct components of the arguments, and to emphasize the need to address the different types of argumentative discourse and argumentative situation in the practice. I consider some issues of recent papers associated with mathematical argumentation in an attempt to contribute to the discussion about the role of arguing in mathematical practice and in the evaluation of the products of this practice. I apply this discussion to learning environments to defend the thesis that argumentative practice should be encouraged when teaching technical subjects to convey a better understanding and to improve thought and creativity.
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 Jesus 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 Cantu.- Chapter 6. How Persuaded Are You? A Typology of Responses Matthew Inglis and Juan Pablo Mejia-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.
The last century has seen many disciplines place a greater prior-ity on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been devel-oped. Perhaps owing to their diverse backgrounds, there are several con-nections 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 argumen-tation schemes constructed by Walton et al. [54], and explore some connec-tions between, as well as within, the theories. For instance, we investigate Peirce's abduction to deal with surprising situations in mathematics, rep-resent Pollock's examples in terms of Toulmin's layout, discuss connections between Toulmin's layout and Walton's argumentation schemes, and sug-gest 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.
This paper builds a method to help a student of informal logic to go through a text of discourse in a natural language and identify common types of arguments that occur in the text. It is shown how this procedure is very helpful for students learning informal logic skills, as they sometimes misidentify arguments. The paper presents the state-of-the-art on what resources are available to build a useful argument identification procedure, and includes a survey of work done on automated argument mining tools used in artificial intelligence.
The peer review process has been the topic of many studies in the medical sciences, but not so in mathematics. Given that mathematicians refer to results from the literature without checking the proofs in detail, it is interesting to see how the mathematical refereeing process affects the epistemic certainty of this type of mathematical knowledge by testimony. We give a description of the mathematical refereeing process and some results of empirical studies.
This study asked 101 preservice elementary teachers enrolled in a sophomore-level mathematics course to judge the mathematical correctness of inductive and deductive verifications of either a familiar or an unfamiliar statement. For each statement, more than half the students accepted an inductive argument as a valid mathematical proof. More than 60% accepted a correct deductive argument as a valid mathematical proof; 38% and 52% accepted an incorrect deductive argument as being mathematically correct for the familiar and unfamiliar statements, respectively. Over a third of the students simultaneously accepted an inductive and a correct deductive argument as being mathematically valid.
Three experiments are reported that investigate the extent to which an authority figure influences the level of persuasion undergraduate students and research-active mathematicians invest in mathematical arguments. We demonstrate that, in some situations, both students and researchers rate arguments as being more persuasive when they are associated with an expert mathematician than when the author is anonymous. We develop a model that accounts for these data by suggesting that, for both students and researchers, an authority figure only plays a role when there is already some uncertainty about the argument's mathematical status. Implications for pedagogy, and for future research, are discussed.
In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which
mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing
mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t
matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification:
in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts.
I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided
proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places
derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when
the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against,
the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical
argument.
Research on humor is carried out in a number of areas in psychology, including the cognitive (What makes something funny?), developmental (when do we develop a sense of humor?), and social (how is humor used in social interactions?) Although there is enough interest in the area to have spawned several societies, the literature is dispersed in a number of primary journals, with little in the way of integration of the material into a book. Dr. Martin is one of the best known researchers in the area, and his research goes across subdisciplines in psychology to be of wide appeal. This is a singly authored monograph that provides in one source, a summary of information researchers might wish to know about research into the psychology of humor. The material is scholarly, but the presentation of the material is suitable for people unfamiliar with the subject-making the book suitable for use for advanced undergraduate and graduate level courses on the psychology of humor-which have not had a textbook source. 2007 AATH Book Award for Humor/Laughter Research category! *Up-to-date coverage of research on humor and laughter in every area of psychology *Research findings are integrated into a coherent conceptual framework *Includes recent brain imaging studies, evolutionary models, and animal research *Draws on contributions from sociology, linguistics, neuroscience, and anthropology *Provides an overview of theories of humor and early research *Explores applications of humor in psychotherapy, education, and the workplace *Points out interesting topics for further research and promising research methodologies *Written in a scholarly yet easily accessible style * 2007 AATH Book Award for Humor/Laughter Research category.
Research on humor is carried out in a number of areas in psychology, including the cognitive (What makes something funny?), developmental (when do we develop a sense of humor?), and social (how is humor used in social interactions?) Although there is enough interest in the area to have spawned several societies, the literature is dispersed in a number of primary journals, with little in the way of integration of the material into a book. Dr. Martin is one of the best known researchers in the area, and his research goes across subdisciplines in psychology to be of wide appeal. This is a singly authored monograph that provides in one source, a summary of information researchers might wish to know about research into the psychology of humor. The material is scholarly, but the presentation of the material is suitable for people unfamiliar with the subject-making the book suitable for use for advanced undergraduate and graduate level courses on the psychology of humor-which have not had a textbook source. 2007 AATH Book Award for Humor/Laughter Research category! *Up-to-date coverage of research on humor and laughter in every area of psychology *Research findings are integrated into a coherent conceptual framework *Includes recent brain imaging studies, evolutionary models, and animal research *Draws on contributions from sociology, linguistics, neuroscience, and anthropology *Provides an overview of theories of humor and early research *Explores applications of humor in psychotherapy, education, and the workplace *Points out interesting topics for further research and promising research methodologies *Written in a scholarly yet easily accessible style * 2007 AATH Book Award for Humor/Laughter Research category.
From a psychological perspective, humor is essentially a positive emotion called mirth, which is typically elicited in social contexts by a cognitive appraisal process involving the perception of playful, nonserious incongruity, and which is expressed by the facial and vocal behavior of laughter. In social interactions, humor takes on many different forms, including canned jokes, spontaneous witticisms, and unintentionally funny utterances and actions. Psychological functions of humor include the cognitive and social benefits of the positive emotion of mirth, and its uses as a mode of social communication and influence, and as a way of relieving tension, regulating emotions, and coping with stress. Popular conceptions of laughter have changed dramatically over the past two or three centuries, from being viewed as essentially aggressive and somewhat socially inappropriate to being seen as positive, psychologically, and physically healthy, and socially desirable. The meaning of the word humor has also evolved from a narrow focus on benign and sympathetic sources of mirth distinguished from more aggressive types of wit, to its use as a broad umbrella term to refer to all sources of laughter. Although humor has important psychological functions and touches on all branches of psychology, and there is a sizable and growing research literature on the topic, mainstream psychology has paid relatively little attention to it until now.
According to recent theory, many of the interpersonal functions of humor derive from its inherently ambiguous nature due to the multiple concurrent meanings that it conveys. Because of this ambiguity, humor is a useful vehicle for communicating certain messages and dealing with situations that would be more difficult to handle using a more serious, unambiguous mode of communication. Importantly, a message communicated in a humorous manner can be retracted more easily than if it were expressed in the serious mode, allowing both the speaker and the listener to save face if the message is not well received. These insights concerning the ambiguity and face-saving potential of humor have been applied by theorists and researchers to account for a wide variety of social uses of humor, including self-disclosure and social probing, decommitment and conflict de-escalation, enforcing social norms and exerting social control, establishing and maintaining status, enhancing group cohesion and identity, discourse management, and social play. The multiple interpersonal functions of humor suggest that it may be viewed as a type of social skill or interpersonal competence. Employed in an adept manner, humor can be a very useful tool for achieving one's interpersonal goals.
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. Otherwise, we risk being unable to distinguish successful informal mathematical reasoning from chance. We shall see that argumentation schemes may play a role in resolving this problem, too.
As Dr Maxwell writes in his preface to this book, his aim has been to instruct through entertainment. 'The general theory is that a wrong idea may often be exposed more convincingly by following it to its absurd conclusion than by merely announcing the error and starting again. Thus a number of by-ways appear which, it is hoped, may amuse the professional, and help to tempt back to the subject those who thought they were losing interest.' The standard of knowledge expected is fairly elementary. In most cases a straightforward statement of the fallacious argument is followed by an exposure in which the error is traced to the most elementary source, and this process often leads to an analysis which is often of unexpected depth. Many students will discover just how mathematically minded they are when they read this book; nor is that the only discovery they will make. Teachers of mathematics in schools and technical schools, colleges and universities will also be sure to find something here to please them.
Every interacting social group develops, over time, a joking culture: a set of humorous references that are known to members of the group to which members can refer and that serve as the basis of further interaction. Joking, thus, has a historical, retrospective, and reflexive character. We argue that group joking is embedded, interactive, and referential, and these features give it power within the group context. Elements of the joking culture serve to smooth group interaction, share affiliation, separate the group from out-siders, and secure the compliance of group members through social control. To demonstrate these processes we rely upon two detailed ethnographic examples of continuing joking: one from mushroom collectors and the second from professional meteorologists.
study of folklore), a folk is defined as any group whatsoever that shares at least one common linking factor. The factor could be na-tionality, ethnicity, religion, or occupation. Members of a profession would also qualify as a folk group. Hence, mathematicians constitute a folk. And, like all folk groups, mathematicians have their own folk speech (slang), proverbs, limericks, and jokes, among other forms of folklore. It is pre-cisely the folklore of a group that defines that group. So, mathematicians as a group share a com-mon core of mathematical folklore. Some of this folklore tends to be quite esoteric and intelligible only to members of the group. Outsiders not pos-sessing the requisite mathematical vocabulary and knowledge rarely know such esoteric material, and even if they did they would probably not under-stand it. But there is also exoteric mathematical folklore that is known to a limited number of out-siders, for example, physicists, chemists, and en-gineers. Much of this exoteric folklore consists of classic jokes contrasting members of different but related academic disciplines. We intend to offer a brief sampling of both esoteric and exoteric math-ematical folklore, concentrating on humorous gen-res such as jokes. We are persuaded that these data not only serve as a basis for identity among mathematicians but also provide a unique window
In this paper, 28 mathematics majors who completed a transition-to-proof course were given 10 mathematical arguments. For each argument, they were asked to judge how convincing they found the argument and whether they thought the argument constituted a mathematical proof. The key findings from this data were (a) most participants did not find the empirical argument in the study to be convincing or to meet the standards of proof, (b) the majority of participants found a diagrammatic argument to be both convincing and a proof, (c) participants evaluated deductive arguments not by their form but by their content, but (d) participants often judged invalid deductive arguments to be convincing proofs because they did not recognize their logical flaws. These findings suggest improving undergraduates' comprehension of mathematical arguments does not depend on making undergraduates aware of the limitations of empirical arguments but instead on improving the ways in which they process the arguments that they read.
This book provides a systematic analysis of many common argumentation schemes and a compendium of 96 schemes. The study of these schemes, or forms of argument that capture stereotypical patterns of human reasoning, is at the core of argumentation research. Surveying all aspects of argumentation schemes from the ground up, the book takes the reader from the elementary exposition in the first chapter to the latest state of the art in the research efforts to formalize and classify the schemes, outlined in the last chapter. It provides a systematic and comprehensive account, with notation suitable for computational applications that increasingly make use of argumentation schemes.
This paper explores applications of concepts from argumentation theory to mathematical proofs. Note is taken of the various
contexts in which proofs occur and of the various objectives they may serve. Examples of strategic maneuvering are discussed
when surveying, in proofs, the four stages of argumentation distinguished by pragma-dialectics. Derailments of strategies
(fallacies) are seen to encompass more than logical fallacies and to occur both in alleged proofs that are completely out
of bounds and in alleged proofs that are at least mathematical arguments. These considerations lead to a dialectical and rhetorical
view of proofs.
The paper attempts to analyze in some detail the main problems encountered in reasoning using diagrams, which may cause errors
in reasoning, produce doubts concerning the reliability of diagrams, and impressions that diagrammatic reasoning lacks the
rigour necessary for mathematical reasoning. The paper first argues that such impressions come from long neglect which led
to a lack of well-developed, properly tested and reliable reasoning methods, as contrasted with the amount of work generations
of mathematicians expended on refining the methods of reasoning with formulae and predicate calculus. Next, two main groups
of problems occurring in diagrammatic reasoning are introduced. The second group, called diagram imprecision, is then briefly
summarized, its detailed analysis being postponed to another paper. The first group, called collectively the generalization
problem, is analyzed in detail in the rest of the paper. The nature and causes of the problems from this group are explained,
methods of detecting the potentially harmful occurrences of these problems are discussed, and remedies for possible errors
they may cause are proposed. Some of the methods are adapted from similar methods used in reasoning with formulae, several
other problems constitute new, specifically diagrammatic ways of reliable reasoning.
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