Epistemology of Mathematics

Edited by Alan Baker (Swarthmore College)
Assistant editor: Sam Roberts (University of Sheffield)
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  1. Observations on Sick Mathematics.Andrew Aberdein - 2010 - In Bart van Kerkhove, Jean Paul van Bendegem & Jonas de Vuyst (eds.), Philosophical Perspectives on Mathematical Practice. College Publications. pp. 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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  2. Computers, Justification, and Mathematical Knowledge.Konstantine Arkoudas & Selmer Bringsjord - 2007 - Minds and Machines 17 (2):185-202.
    The original proof of the four-color theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depend heavily for their soundness on large amounts of computation-intensive custom-built software. Contra Tymoczko, we argue that the justification provided by certain computerized mathematical proofs is not fundamentally different from that provided by surveyable (...)
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  3. Intuitionistic Remarks on Husserl's Analysis of Finite Number in the Philosophy of Arithmetic.Mark van Atten - 2004 - Graduate Faculty Philosophy Journal 25 (2):205-225.
  4. Mathematical Knowledge. [REVIEW]David D. Auerbach - 1977 - Philosophical Review 86 (2):247.
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  5. On the Roles of Proof in Mathematics.Joseph Auslander - 2008 - In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. pp. 61--77.
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  6. Mathematics as an Experimental Science.Sidney Axinn - 1968 - Philosophia Mathematica (1-2):1-10.
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  7. Empty de Re Attitudes About Numbers.Jody Azzouni - 2009 - Philosophia Mathematica 17 (2):163-188.
    I dub a certain central tradition in philosophy of language (and mind) the de re tradition. Compelling thought experiments show that in certain common cases the truth conditions for thoughts and public-language expressions categorically turn on external objects referred to, rather than on linguistic meanings and/or belief assumptions. However, de re phenomena in language and thought occur even when the objects in question don't exist. Call these empty de re phenomena. Empty de re thought with respect to numeration is explored (...)
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  8. Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences.Jody Azzouni - 2008 - Cambridge University Press.
    Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...)
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  9. Tracking Reason: Proof, Consequence, and Truth.Jody Azzouni - 2005 - Oup Usa.
    When ordinary people - mathematicians among them - take something to follow from something else, they are exposing the backbone of our self-ascribed ability to reason. Jody Azzouni investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians. One claim of the book is that, despite our apparent intuitive grasp of consequence, we do not introspect rules by which we reason, nor do we grasp the scope and range of the domain, as it were, (...)
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  10. Stipulation, Logic, and Ontological Independence.Jody Azzouni - 2000 - Philosophia Mathematica 8 (3):225-243.
    A distinction between the epistemic practices in mathematics and in the empirical sciences is rehearsed to motivate the epistemic role puzzle. This is distinguished both from Benacerraf's 1973 epistemic puzzle and from sceptical arguments against our knowledge of an external world. The stipulationist position is described, a position which can address this puzzle. Methods of avoiding the stipulationist position by using pure logic to provide knowledge of mathematical abstracta are discussed and criticized.
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  11. First‐Year Secondary School Mathematics Students' Conceptions of Mathematical Proofs and Proving.Savas Basturk - 2010 - Educational Studies 36 (3):283-298.
    The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students . The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and mathematics education. The students’ studying methods for (...)
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  12. IB Course Companion: Mathematical Studies.Stephen Bedding, Mal Coad, Jane Forrest, Beryl Fussey & Paula Waldman de Tokman - 2007 - Oxford University Press.
    This book has been designed specifically to support the student through the IB Diploma Programme in Mathematical Studies. It includes worked examples and numerous opportunities for practice. In addition the book will provide students with features integrated with study and learning approaches, TOK and the IB learner profile. Examples and activities drawn from around the world will encourage students to develop an international perspective.
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  13. Hermann Weyl on Intuition and the Continuum.John L. Bell - 2000 - Philosophia Mathematica 8 (3):259-273.
    Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.
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  14. The Applicability of Mathematics to Physical Modality.Nora Berenstain - 2017 - Synthese 194 (9):3361-3377.
    This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...)
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  15. Proof and the Virtues of Shared Enquiry.Don Berry - forthcoming - Philosophia Mathematica:nkw022.
    This paper investigates an important aspect of mathematical practice: that proof is required for a finished piece of mathematics. If follows that non-deductive arguments — however convincing — are never sufficient. I explore four aspects of mathematical research that have facilitated the impressive success of the discipline. These I call the Practical Virtues: Permanence, Reliability, Autonomy, and Consensus. I then argue that permitting results to become established on the basis of non-deductive evidence alone would lead to their deterioration. This furnishes (...)
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  16. Mathematical Epistemology and Psychology.Evert Willem Beth - 1966 - New York: Gordon & Breach.
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  17. Remarks on the Structuralistic Epistemology of Mathematics* Izabela Bondecka-Krzykowska and Roman Murawski.Izabela Bondecka-Krzykowska - 2006 - Logique Et Analyse 49:31-41.
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  18. 2. Remarks On The Structuralistic Epistemology Of Mathematics.Izabella Bondecka-Krzykowska & Roman Murawski - 2006 - Logique Et Analyse 49:85-93.
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  19. Remarks on the Structuralistic Epistemology of Mathematics.with Izabela Bondecka-Krzykowska - 2010 - In Roman Murawski (ed.), Essays in the Philosophy and History of Logic and Mathematics. Rodopi.
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  20. The Experimental Mathematician : The Pleasure of Discovery and the Role of Proof.Jonathon M. Borwein - unknown
    The emergence of powerful mathematical computing environments, the growing availability of correspondingly powerful (multi-processor) computers and the pervasive presence of the internet allow for mathematicians, students and teachers, to proceed heuristically and ‘quasi-inductively’. We may increasingly use symbolic and numeric computation, visualization tools, simulation and data mining. The unique features of our discipline make this both more problematic and more challenging. For example, there is still no truly satisfactory way of displaying mathematical notation on the web; and we care more (...)
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  21. Badiou's Materialist Epistemology of Mathematics.Ray Brassier - 2005 - Angelaki 10 (2):135 – 150.
    One establishes oneself within science from the start. One does not reconstitute it from scratch. One does not found it. Alain Badiou, Le Concept de modèle1 [T]here are no crises within science, nor can there be, for science is the pure affirmation of difference. Alain Badiou, "Marque et manque" 2.
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  22. C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge Reviewed By.Manuel Bremer - 2010 - Philosophy in Review 30 (3):205-207.
  23. Book Reviews. [REVIEW]James Robert Brown - 1996 - Philosophia Mathematica 4 (3):251-253.
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  24. Comments on Crispin Wright on Basic Arithmetical Knowledge.Anthony Brueckner - 2011 - Philosophical Studies 156 (1):149-154.
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  25. Empiricism, Mathematical Truth and Mathematical Knowledge.Otavio Bueno - 2000 - Poznan Studies in the Philosophy of the Sciences and the Humanities 71:219-242.
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  26. Learning the Natural Numbers as a Child.Stefan Buijsman - forthcoming - Noûs.
    How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...)
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  27. Towards a New Epistemology of Mathematics.Bernd Buldt, Benedikt Löwe & Thomas Müller - 2008 - Erkenntnis 68 (3):309 - 329.
    In this introduction we discuss the motivation behind the workshop “Towards a New Epistemology of Mathematics” of which this special issue constitutes the proceedings. We elaborate on historical and empirical aspects of the desired new epistemology, connect it to the public image of mathematics, and give a summary and an introduction to the contributions to this issue.
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  28. Book Reviews. [REVIEW]John P. Burgess - 1993 - Philosophia Mathematica 1 (2):637-639.
  29. Mathematical Knowledge.J. R. Cameron - 1976 - Philosophical Books 17 (3):137-139.
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  30. Peirce on the Role of Poietic Creation in Mathematical Reasoning.Daniel G. Campos - 2007 - Transactions of the Charles S. Peirce Society 43 (3):470 - 489.
    : C.S. Peirce defines mathematics in two ways: first as "the science which draws necessary conclusions," and second as "the study of what is true of hypothetical states of things" (CP 4.227–244). Given the dual definition, Peirce notes, a question arises: Should we exclude the work of poietic hypothesis-making from the domain of pure mathematical reasoning? (CP 4.238). This paper examines Peirce's answer to the question. Some commentators hold that for Peirce the framing of mathematical hypotheses requires poietic genius but (...)
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  31. Where Our Number Concepts Come From.Susan Carey - 2009 - Journal of Philosophy 106 (4):220-254.
  32. Math Schemata and the Origins of Number Representations.Susan Carey - 2008 - Behavioral and Brain Sciences 31 (6):645-646.
    The contrast Rips et al. draw between and approaches to understanding the origin of the capacity for representing natural number is a false dichotomy. Its plausibility depends upon the sketchiness of the authors' own proposal. At least some of the proposals they characterize as bottom-up are worked-out versions of the very top-down position they advocate. Finally, they deny that the structures that these putative bottom-up proposals consider to be sources of natural number are even precursors of concepts of natural number. (...)
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  33. Beyond Epistemology.William R. Caspary - 2004 - Newsletter of the Society for the Advancement of American Philosophy 32 (99):57-59.
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  34. Mathematical Reasoning and Heuristics.Carlo Cellucci & Donald Gillies (eds.) - 2005 - College Publications.
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  35. Number Ideas Through Pictures.Mannis Charosh - 1974 - New York: T. Y. Crowell.
  36. Proof in Mathematics: Response to Jairo José da Silva.O. Chateaubriand - 2008 - Manuscrito 31 (1):197-202.
    The paper by Jairo José da Silva is mainly concerned with the character of mathematical proof and with the nature of mathematics and its ontology. Although there is a fair amount of agreement in our views, I focus my response on three issues on which we disagree. The first is his view of mathematical proof as generally unconstrained by language and by a previous proof apparatus. The second is his discussion of Brouwer’s views on proof and formalization. The third is (...)
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  37. Tharp's 'Myth and Mathematics'.Charles Chihara - 1989 - Synthese 81 (2):153 - 165.
  38. Set-Theoretic Pluralism and the Benacerraf Problem.Justin Clarke-Doane - forthcoming - Philosophical Studies.
    Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...)
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  39. Mathematical Knowledge.Nino Cocchiarella - 1978 - Philosophia 8 (2-3):471-484.
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  40. Philosophical Papers By Imre Lakatos Edited by John Worrall and Gregory Currie Vol. I, The Methodology of Scientific Research Programmes, Viii + 250 Pp., £9.00 Vol. II, Mathematics, Science and Epistemology, X + 286 Pp., £10.50 Cambridge: Cambridge University Press, 1978. [REVIEW]L. Jonathan Cohen - 1979 - Philosophy 54 (208):247-.
  41. A New Semantics for the Epistemology of Geometry I: Modeling Spacetime Structure. [REVIEW]Robert Alan Coleman & Herbert Korté - 1995 - Erkenntnis 42 (2):141 - 160.
  42. A New Semantics for the Epistemology of Geometry II: Epistemological Completeness of Newton—Galilei and Einstein—Maxwell Theory. [REVIEW]Robert Alan Coleman & Herbert Korté - 1995 - Erkenntnis 42 (2):161 - 189.
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  43. The Transformational Epistemology of Radical Constructivism: A Tribute to Ernst von Glasersfeld.J. Confrey - 2011 - Constructivist Foundations 6 (2):177-182.
    Problem: What is it that Ernst von Glasersfeld brought to mathematics education with radical constructivism? Method: Key ideas in the author’s early thinking are related to ideas that are central in constructivism, with the aim of showing their importance in math education. Results: The author’s initial thinking about constructivism began with Toulmin’s view of thinking as evolving. Ernst showed how Piaget’s genetic epistemology implied an epistemology that was not about ontology. Continuing with an analysis of the way radical and trivial (...)
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  44. Assaying Lakatos's Philosophy of Mathematics.D. Corfield - 1997 - Studies in History and Philosophy of Science Part A 28 (1):99-121.
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  45. Martin H. Krieger. Doing Mathematics: Convention, Subject, Calculation, Analogy. Singapore: World Scientific Publishing, 2003. Pp. XVIII + 454. ISBN 981-238-2003 (Cloth); 981-238-2062 (Paperback). [REVIEW]David Corfield - 2005 - Philosophia Mathematica 13 (1):106-111.
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  46. Towards a Philosophy of Real Mathematics.David Corfield - 2005 - Cambridge University Press.
    In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing (...)
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  47. The Structure of Mathematical Experience According to Jean Cavaillèst.Paul Cortois - 1996 - Philosophia Mathematica 4 (1):18-41.
    In this expository article one of the contributions of Jean Cavailles to the philosophy of mathematics is presented: the analysis of ‘mathematical experience’. The place of Cavailles on the logico-philosophical scene of the 30s and 40s is sketched. I propose a partial interpretation of Cavailles's epistemological program of so-called ‘conceptual dialectics’: mathematical holism, duality principles, the notion of formal contents, and the specific temporal structure of conceptual dynamics. The structure of mathematical abstraction is analysed in terms of its complementary dimensions: (...)
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  48. What Mathematical Cognition Could Tell Us About the Actual World.Sorin Costreie - 2012 - In Elsevier (ed.), Procedia Social and Behavioral Science 33. pp. 138-142.
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  49. Differences Between the Philosophy of Mathematics and the Psychology of Number Development.Richard Cowan - 2008 - Behavioral and Brain Sciences 31 (6):648-648.
    The philosophy of mathematics may not be helpful to the psychology of number development because they differ in their purposes.
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  50. Lakatos's Philosophy of Mathematics.Gregory Currie - 1979 - Synthese 42 (2):335 - 351.
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