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  1. Andrew Aberdein (2013). Mathematical Wit and Mathematical Cognition. Topics in Cognitive Science 5 (2):231-250.
    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 (...)
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  2. Andrew Aberdein (2012). The Parallel Structure of Mathematical Reasoning. In 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 7--14.
    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. (...)
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  3. Andrew Aberdein (2011). The Dialectical Tier of Mathematical Proof. In Frank Zenker (ed.), Argumentation: Cognition & Community. Proceedings of the 9th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 18--21, 2011. OSSA
    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.
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  4. Andrew Aberdein (2010). Rationale of the Mathematical Joke. In Alison Pease, Markus Guhe & Alan Smaill (eds.), Proceedings of AISB 2010 Symposium on Mathematical Practice and Cognition. AISB 1-6.
    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.
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  5. Andrew Aberdein (2010). Observations on Sick Mathematics. In Bart van Kerkhove, Jean Paul van Bendegem & Jonas de Vuyst (eds.), Philosophical Perspectives on Mathematical Practice. College Publications 269--300.
    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 (...)
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  6. Andrew Aberdein (2009). Argumentation Schemes and Communities of Argumentational Practice. In Juho Ritola (ed.), Argument Cultures: Proceedings of OSSA 2009. OSSA
    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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  7. Andrew Aberdein (2009). Mathematics and Argumentation. Foundations of Science 14 (1-2):1-8.
    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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  8. Andrew Aberdein (2007). Fallacies in Mathematics. Proceedings of the British Society for Research Into Learning Mathematics 27 (3):1-6.
    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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  9. Andrew Aberdein (2006). Introduction to the New Edition. In The Elements: Books I-XIII by Euclid. Barnes & Noble
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  10. Andrew Aberdein (2006). The Informal Logic of Mathematical Proof. In Reuben Hersh (ed.), 18 Unconventional Essays About the Nature of Mathematics. Springer-Verlag 56-70.
    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 (...)
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  11. Andrew Aberdein (2006). Managing Informal Mathematical Knowledge: Techniques From Informal Logic. Lecture Notes in Artificial Intelligence 4108:208--221.
    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 (...)
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  12. Andrew Aberdein (2006). Proofs and Rebuttals: Applying Stephen Toulmin's Layout of Arguments to Mathematical Proof. In Marta Bílková & Ondřej Tomala (eds.), The Logica Yearbook 2005. Filosofia 11-23.
    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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  13. Andrew Aberdein (2005). The Uses of Argument in Mathematics. Argumentation 19 (3):287-301.
    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 (...)
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  14. Andrew Aberdein & Ian J. Dove (eds.) (2013). The Argument of Mathematics. Springer.
    Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations. -/- The book begins by first challenging the (...)
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  15. Dennis Almeida & Paul Ernest (1996). Editorial: Teaching and the Nature of Mathematics. Philosophy of Mathematics Education Journal 9.
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  16. A. Arana (2012). Jeremy Gray. Plato's Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press, 2008. Isbn 978-0-69113610-3. Pp. VIII + 515. [REVIEW] Philosophia Mathematica 20 (2):252-255.
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  17. Andrew Arana (2007). Review of D. Corfield, Toward a Philosophy of Real Mathematics. [REVIEW] Mathematical Intelligencer 29 (2).
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  18. Tatiana Arrigoni (2003). Foundational Instances and Attention to Practices in the Philosophy of Contemporary Mathematics. Rivista di Filosofia Neo-Scolastica 95 (2):199-232.
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  19. J. Azzouni (2013). The Relationship of Derivations in Artificial Languages to Ordinary Rigorous Mathematical Proof. Philosophia Mathematica 21 (2):247-254.
    The relationship is explored between formal derivations, which occur in artificial languages, and mathematical proof, which occurs in natural languages. The suggestion that ordinary mathematical proofs are abbreviations or sketches of formal derivations is presumed false. The alternative suggestion that the existence of appropriate derivations in formal logical languages is a norm for ordinary rigorous mathematical proof is explored and rejected.
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  20. Mark Balaguer (2002). Review: Stewart Shapiro, Thinking About Mathematics. The Philosophy of Mathematics. [REVIEW] Bulletin of Symbolic Logic 8 (1):89-91.
  21. Aristides Baltas (1995). Do Mathematics Constitute a Scientific Continent? Neusis 3:97-108.
  22. A. Barabashev (1988). Empiricism as a Historical Phenomenon of Philosophy of Mathematics. Revue Internationale de Philosophie 42 (167):509-517.
  23. A. G. Barabashev (1988). On the Impact of the World Outlook on Mathematical Creativity. Philosophia Mathematica (1):1-20.
  24. A. G. Barabashev, S. S. Demidov & M. I. Panov (1987). Regularities and Modern Tendencies of the Development of Mathematics. Philosophia Mathematica (1):32-47.
  25. Alexei G. Barabashev (1986). The Philosophy of Mathematics in U.S.S.R. Philosophia Mathematica (1-2):15-25.
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  26. Timothy Bays (2004). Review of David Corfield, Towards a Philosophy of Real Mathematics. [REVIEW] Notre Dame Philosophical Reviews 2004 (1).
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  27. JC Beall (2010). The Philosophy of Mathematical Practice. Australasian Journal of Philosophy 88 (2):376-376.
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  28. Otávio Bueno & Jody Azzouni (2005). Review of D. Mac Kenzie, Mechanizing Proof: Computing, Risk, and Trust. [REVIEW] Philosophia Mathematica 13 (3):319-325.
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  29. Otávio Bueno & Jody Azzouni (2005). Review of D. Mac Kenzie, Mechanizing Proof: Computing, Risk, and Trust. Philosophia Mathematica 13 (3):319-325.
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  30. Stefan Buijsman (forthcoming). Accessibility of Reformulated Mathematical Content. Synthese:1-18.
    I challenge a claim that seems to be made when nominalists offer reformulations of the content of mathematical beliefs, namely that these reformulations are accessible to everyone. By doing so, I argue that these theories cannot account for the mathematical knowledge that ordinary people have. In the first part of the paper I look at reformulations that employ the concept of proof, such as those of Mary Leng and Ottavio Bueno. I argue that ordinary people don’t have many beliefs about (...)
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  31. Begoña Carrascal (2015). Proofs, Mathematical Practice and Argumentation. Argumentation 29 (3):305-324.
    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 (...)
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  32. Begoῆa Carrascal, The Practice of Arguing and the Arguments: Examples From Mathematics.
    In argumentation studies, almost all theoretical proposals are applied, in general, to the analysis and evaluation of written argumentative texts. I will consider mathematics to illustrate some differences between argumentative practice and the products of it, to emphasize the need to address the different types of argumentative discourse and argumentative situation. Argumentative practice should be encouraged when teaching technical subjects to convey a better understanding and to improve thought and creativity.
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  33. Jessica Carter (2014). Mathematics Dealing with 'Hypothetical States of Things'. Philosophia Mathematica 22 (2):209-230.
    This paper takes as a starting point certain notions from Peirce's writings and uses them to propose a picture of the part of mathematical practice that consists of hypothesis formation. In particular, three processes of hypothesis formation are considered: abstraction, generalisation, and an abductive-like inference. In addition Peirce's pragmatic conception of truth and existence in terms of higher-order concepts are used in order to obtain a kind of pragmatic realist picture of mathematics.
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  34. Carlo Cellucci (forthcoming). Is Mathematics Problem Solving or Theorem Proving? Foundations of Science:1-17.
    The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...)
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  35. Karine Chemla (2006). Artificial Languages in the Mathematics of Ancient China. Journal of Indian Philosophy 34 (1-2):31-56.
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  36. Roy T. Cook (2013). Patricia A. Blanchette. Frege's Conception of Logic. Oxford University Press, 2012. ISBN 978-0-19-926925-9 (Hbk). Pp. Xv + 256. [REVIEW] Philosophia Mathematica (1):nkt029.
  37. John Corcoran (2014). Corcoran Reviews Boute’s 2013 Paper “How to Calculate Proofs”. MATHEMATICAL REVIEWS 14:444-555.
    Corcoran reviews Boute’s 2013 paper “How to calculate proofs”. -/- There are tricky aspects to classifying occurrences of variables: is an occurrence of ‘x’ free as in ‘x + 1’, is it bound as in ‘{x: x = 1}’, or is it orthographic as in ‘extra’? The trickiness is compounded failure to employ conventions to separate use of expressions from their mention. The variable occurrence is free in the term ‘x + 1’ but it is orthographic in that term’s quotes (...)
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  38. J. Ellenberg & E. Sober (2011). Objective Probabilities in Number Theory. Philosophia Mathematica 19 (3):308-322.
    Philosophers have explored objective interpretations of probability mainly by considering empirical probability statements. Because of this focus, it is widely believed that the logical interpretation and the actual-frequency interpretation are unsatisfactory and the hypothetical-frequency interpretation is not much better. Probabilistic assertions in pure mathematics present a new challenge. Mathematicians prove theorems in number theory that assign probabilities. The most natural interpretation of these probabilities is that they describe actual frequencies in finite sets and limits of actual frequencies in infinite sets. (...)
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  39. Donald Gillies (2013). Should Philosophers of Mathematics Make Use of Sociology? Philosophia Mathematica 22 (1):12-34.
    This paper considers whether philosophy of mathematics could benefit by the introduction of some sociology. It begins by considering Lakatos's arguments that philosophy of science should be kept free of any sociology. An attempt is made to criticize these arguments, and then a positive argument is given for introducing a sociological dimension into the philosophy of mathematics. This argument is illustrated by considering Brouwer's account of numbers as mental constructions. The paper concludes with a critical discussion of Azzouni's view that (...)
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  40. Eduard Glas (1995). Kuhn, Lakatos, and the Image of Mathematics. Philosophia Mathematica 3 (3):225-247.
    In this paper I explore possibilities of bringing post-positivist philosophies of empirical science to bear on the dynamics of mathematical development. This is done by way of a convergent accommodation of a mathematical version of Lakatos's methodology of research programmes, and a version of Kuhn's account of scientific change that is made applicable to mathematics by cleansing it of all references to the psychology of perception. The resulting view is argued in the light of two case histories of radical conceptual (...)
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  41. Jeremy Gray & Jose Ferreiros (eds.) (2006). The Architecture of Modern Mathematics. Oxford University Press.
    This edited volume, aimed at both students and researchers in philosophy, mathematics and history of science, highlights leading developments in the overlapping areas of philosophy and the history of modern mathematics. It is a coherent, wide ranging account of how a number of topics in the philosophy of mathematics must be reconsidered in the light of the latest historical research, and how a number of historical accounts can be deepened by embracing philosophical questions.
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  42. L. Horsten (2011). Review of M. Leng, Mathematics and Reality. [REVIEW] Analysis 71 (4):798-799.
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  43. Matthew Inglis & Andrew Aberdein (2014). Beauty Is Not Simplicity: An Analysis of Mathematicians' Proof Appraisals. Philosophia Mathematica 23 (1):87-109.
    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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  44. Matthew Inglis & Juan Pablo Mejía-Ramos (2009). On the Persuasiveness of Visual Arguments in Mathematics. Foundations of Science 14 (1-2):97-110.
    Two experiments are reported which investigate the factors that influence how persuaded mathematicians are by visual arguments. We demonstrate that if a visual argument is accompanied by a passage of text which describes the image, both research-active mathematicians and successful undergraduate mathematics students perceive it to be significantly more persuasive than if no text is given. We suggest that mathematicians’ epistemological concerns about supporting a claim using visual images are less prominent when the image is described in words. Finally we (...)
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  45. Matthew Inglis, Juan Pablo Mejia-Ramos, Keith Weber & Lara Alcock (2013). On Mathematicians' Different Standards When Evaluating Elementary Proofs. Topics in Cognitive Science 5 (2):270-282.
    In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, (...)
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  46. Doug Jesseph (2008). Review of Emily R. Grosholz, Representation and Productive Ambiguity in Mathematics and the Sciences. [REVIEW] Notre Dame Philosophical Reviews 2008 (5).
  47. Juliette Kennedy (2011). Review of The Autonomy of Mathematical Knowledge. Bulletin of Symbolic Logic 17 (1):119-122.
  48. Gianluca Longa (2014). Pappus of Alexandria in the 20th Century. Analytical Method and Mathematical Practice. Dissertation, University of Milan
  49. Penelope Maddy (1992). Indispensability and Practice. Journal of Philosophy 89 (6):275-289.
  50. Paolo Mancosu (ed.) (2008). The Philosophy of Mathematical Practice. OUP Oxford.
    There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.
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