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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 (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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  3. 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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  4. Andrew Aberdein (2010). 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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  5. 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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  6. 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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  7. 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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  8. 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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  9. Andrew Aberdein (2006). Introduction to the New Edition. In The Elements: Books I-XIII by Euclid. Barnes & Noble.
  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 (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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  13. 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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  14. Andrew Arana (2007). Review of D. Corfield, Toward a Philosophy of Real Mathematics. [REVIEW] Mathematical Intelligencer 29 (2).
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  15. 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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  16. 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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  17. 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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  18. Karine Chemla (2006). Artificial Languages in the Mathematics of Ancient China. Journal of Indian Philosophy 34 (1-2):31-56.
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  19. 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.
  20. 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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  21. Donald Gillies (2014). 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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  22. 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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  23. L. Horsten (2011). Review of M. Leng, Mathematics and Reality. [REVIEW] Analysis 71 (4):798-799.
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  24. 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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  25. 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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  26. Doug Jesseph (2008). Review of Emily R. Grosholz, Representation and Productive Ambiguity in Mathematics and the Sciences. [REVIEW] Notre Dame Philosophical Reviews 2008 (5).
  27. Juliette Kennedy (2011). Review of The Autonomy of Mathematical Knowledge. Bulletin of Symbolic Logic 17 (1):119-122.
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  28. Penelope Maddy (1992). Indispensability and Practice. Journal of Philosophy 89 (6):275-289.
  29. Paolo Mancosu (ed.) (2008). The Philosophy of Mathematical Practice. OUP Oxford.
    Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice. This book is the first attempt to give a coherent and unified presentation (...)
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  30. Paolo Mancosu (1996). Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford University Press.
    The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...)
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  31. Paolo Mancosu (1991). On the Status of Proofs by Contradiction in the Seventeenth Century. Synthese 88 (1):15 - 41.
    In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations (...)
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  32. Tyler Marghetis & Rafael Núñez (2013). The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics. Topics in Cognitive Science 5 (2):299-316.
    The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, (...)
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  33. Jean-pierre Marquis (1997). Abstract Mathematical Tools and Machines for Mathematics. Philosophia Mathematica 5 (3):250-272.
    In this paper, we try to establish that some mathematical theories, like K-theory, homology, cohomology, homotopy theories, spectral sequences, modern Galois theory (in its various applications), representation theory and character theory, etc., should be thought of as (abstract) machines in the same way that there are (concrete) machines in the natural sciences. If this is correct, then many epistemological and ontological issues in the philosophy of mathematics are seen in a different light. We concentrate on one problem which immediately follows (...)
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  34. András Máté (2006). Árpád Szabó and Imre Lakatos, or the Relation Between History and Philosophy of Mathematics. Perspectives on Science 14 (3):282-301.
    The thirty year long friendship between Imre Lakatos and the classic scholar and historian of mathematics Árpád Szabó had a considerable influence on the ideas, scholarly career and personal life of both scholars. After recalling some relevant facts from their lives, this paper will investigate Szabó's works about the history of pre-Euclidean mathematics and its philosophy. We can find many similarities with Lakatos' philosophy of mathematics and science, both in the self-interpretation of early axiomatic Greek mathematics as Szabó reconstructs it, (...)
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  35. Edward A. Maziarz (1986). Roles Mathematicians Play. Philosophia Mathematica (1-2):37-62.
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  36. Edward A. Maziarz (1969). Book Review:Problems in the Philosophy of Mathematics Imre Latakos. [REVIEW] Philosophy of Science 36 (3):324-.
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  37. Tuyoshi Mori (1978). The Social History of Mathematics in Modern Japan. Philosophia Mathematica (1):88-105.
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  38. John Mumma & Marco Panza (2012). Diagrams in Mathematics: History and Philosophy. Synthese 186 (1):1-5.
    Diagrams are ubiquitous in mathematics. From the most elementary class to the most advanced seminar, in both introductory textbooks and professional journals, diagrams are present, to introduce concepts, increase understanding, and prove results. They thus fulfill a variety of important roles in mathematical practice. Long overlooked by philosophers focused on foundational and ontological issues, these roles have come to receive attention in the past two decades, a trend in line with the growing philosophical interest in actual mathematical practice.
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  39. Marco Panza (2011). Breathing Fresh Air Into the Philosophy of Mathematics. Metascience 20 (3):495-500.
    Breathing fresh air into the philosophy of mathematics Content Type Journal Article DOI 10.1007/s11016-010-9470-8 Authors Marco Panza, IHPST, 13, rue du Four, 75006 Paris, France Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
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  40. Alison Pease & Andrew Aberdein (2011). Five Theories of Reasoning: Interconnections and Applications to Mathematics. Logic and Logical Philosophy 20 (1-2):7-57.
    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 (...)
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  41. Alison Pease, Markus Guhe & Alan Smaill (2013). Developments in Research on Mathematical Practice and Cognition. Topics in Cognitive Science 5 (2):224-230.
    We describe recent developments in research on mathematical practice and cognition and outline the nine contributions in this special issue of topiCS. We divide these contributions into those that address (a) mathematical reasoning: patterns, levels, and evaluation; (b) mathematical concepts: evolution and meaning; and (c) the number concept: representation and processing.
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  42. Alison Pease, Alan Smaill, Simon Colton & John Lee (2009). Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics. Foundations of Science 14 (1-2):111-135.
    We argue that there are mutually beneficial connections to be made between ideas in argumentation theory and the philosophy of mathematics, and that these connections can be suggested via the process of producing computational models of theories in these domains. We discuss Lakatos’s work (Proofs and Refutations, 1976) in which he championed the informal nature of mathematics, and our computational representation of his theory. In particular, we outline our representation of Cauchy’s proof of Euler’s conjecture, in which we use work (...)
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  43. S. Pollard (2013). Dianoia Left and Right. Philosophia Mathematica 21 (3):309-322.
    In Plato's Phaedrus, Socrates offers two speeches, the first portraying madness as mere disease, the second celebrating madness as divine inspiration. Each speech is correct, says Socrates, though neither is complete. The two kinds of madness are like the left and right sides of a living body: no account that focuses on just one half can be adequate. In a recent paper, Hugh Benson gives a left-handed speech about a psychic condition endemic among mathematicians: dianoia. Benson acknowledges that his account (...)
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  44. Yehuda Rav (2007). A Critique of a Formalist-Mechanist Version of the Justification of Arguments in Mathematicians' Proof Practices. Philosophia Mathematica 15 (3):291-320.
    In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine (...)
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  45. Michael D. Resnik (1995). Review of J. Azzouni, Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences. [REVIEW] Philosophia Mathematica 3 (3).
  46. Adrian Riskin (1997). Review of J. Høyrup, In Measure, Number, and Weight. [REVIEW] Philosophia Mathematica 5 (3):276-277.
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  47. Adrian Riskin (1994). On the Most Open Question in the History of Mathematics: A Discussion of Maddy. Philosophia Mathematica 2 (2):109-121.
    In this paper, I argue against Penelope Maddy's set-theoretic realism by arguing (1) that it is perfectly consistent with mathematical Platonism to deny that there is a fact of the matter concerning statements which are independent of the axioms of set theory, and that (2) denying this accords further that many contemporary Platonists assert that there is a fact of the matter because they are closet foundationalists, and that their brand of foundationalism is in radical conflict with actual mathematical practice.
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  48. Uwe Riss (2011). Objects and Processes in Mathematical Practice. Foundations of Science 16 (4):337-351.
    In this paper it is argued that the fundamental difference of the formal and the informal position in the philosophy of mathematics results from the collision of an object and a process centric perspective towards mathematics. This collision can be overcome by means of dialectical analysis, which shows that both perspectives essentially depend on each other. This is illustrated by the example of mathematical proof and its formal and informal nature. A short overview of the employed materialist dialectical approach is (...)
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  49. Davide Rizza (2011). Review of M. Leng, Mathematics and Reality. [REVIEW] Philosophical Quarterly 61 (244):655-657.
  50. D. Schlimm (2013). Axioms in Mathematical Practice. Philosophia Mathematica 21 (1):37-92.
    On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...)
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