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Epistemology of Mathematics

Edited by Alan Baker (Swarthmore College)
Assistant editor: Sam Roberts (University of Sheffield)
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  1. 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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  2. Joseph Auslander (2008). On the Roles of Proof in Mathematics. In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. 61--77.
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  3. Sidney Axinn (1968). Mathematics as an Experimental Science. Philosophia Mathematica (1-2):1-10.
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  4. Jody Azzouni (2009). Empty de Re Attitudes About Numbers. 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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  5. Jody Azzouni (2000). Stipulation, Logic, and Ontological Independence. 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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  6. Jody Azzouni (1994). Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences. Cambridge University Press.
    This original and exciting study offers a completely new perspective on the philosophy of mathematics. Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similiar 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 (...)
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  7. Savas Basturk (2010). First‐Year Secondary School Mathematics Students' Conceptions of Mathematical Proofs and Proving. 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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  8. Stephen Bedding, Mal Coad, Jane Forrest, Beryl Fussey & Paula Waldman de Tokman (2007). IB Course Companion: Mathematical Studies. OUP Oxford.
    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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  9. John L. Bell (2000). Hermann Weyl on Intuition and the Continuum. 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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  10. Evert Willem Beth (1966). Mathematical Epistemology and Psychology. New York, Gordon and Breach.
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  11. with Izabela Bondecka-Krzykowska (2010). Remarks on the Structuralistic Epistemology of Mathematics. In Roman Murawski (ed.), Essays in the Philosophy and History of Logic and Mathematics. Rodopi.
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  12. Ray Brassier (2005). Badiou's Materialist Epistemology of Mathematics. 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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  13. Manuel Bremer (2010). C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge Reviewed By. Philosophy in Review 30 (3):205-207.
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  14. James Robert Brown (1996). Book Reviews. [REVIEW] Philosophia Mathematica 4 (3):251-253.
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  15. Anthony Brueckner (2011). Comments on Crispin Wright on Basic Arithmetical Knowledge. Philosophical Studies 156 (1):149-154.
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  16. Bernd Buldt, Benedikt Löwe & Thomas Müller (2008). Towards a New Epistemology of Mathematics. 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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  17. John P. Burgess (1993). Book Reviews. [REVIEW] Philosophia Mathematica 1 (2):637-639.
  18. J. R. Cameron (1976). Mathematical Knowledge. Philosophical Books 17 (3):137-139.
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  19. Daniel G. Campos (2007). Peirce on the Role of Poietic Creation in Mathematical Reasoning. 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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  20. Susan Carey (2009). Where Our Number Concepts Come From. Journal of Philosophy 106 (4):220-254.
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  21. Susan Carey (2008). Math Schemata and the Origins of Number Representations. 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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  22. William R. Caspary (2004). Beyond Epistemology. Newsletter of the Society for the Advancement of American Philosophy 32 (99):57-59.
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  23. Mannis Charosh (1974). Number Ideas Through Pictures. New York,T. Y. Crowell.
  24. Charles Chihara (1989). Tharp's 'Myth and Mathematics'. Synthese 81 (2):153 - 165.
  25. L. Jonathan Cohen (1979). 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] Philosophy 54 (208):247-.
  26. Robert Alan Coleman & Herbert Korté (1995). A New Semantics for the Epistemology of Geometry I: Modeling Spacetime Structure. [REVIEW] Erkenntnis 42 (2):141 - 160.
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  27. Robert Alan Coleman & Herbert Korté (1995). A New Semantics for the Epistemology of Geometry II: Epistemological Completeness of Newton—Galilei and Einstein—Maxwell Theory. [REVIEW] Erkenntnis 42 (2):161 - 189.
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  28. J. Confrey (2011). The Transformational Epistemology of Radical Constructivism: A Tribute to Ernst von Glasersfeld. 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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  29. D. Corfield (1997). Assaying Lakatos's Philosophy of Mathematics. Studies in History and Philosophy of Science Part A 28 (1):99-121.
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  30. David Corfield (2005). 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] Philosophia Mathematica 13 (1):106-111.
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  31. David Corfield (2003). Towards a Philosophy of Real Mathematics. 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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  32. Paul Cortois (1996). The Structure of Mathematical Experience According to Jean Cavaillèst. 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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  33. Richard Cowan (2008). Differences Between the Philosophy of Mathematics and the Psychology of Number Development. 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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  34. Gabriella Crocco & Eva-Maria Engelen (forthcoming). Kurt Gödel's Philosophical Remarks (Max Phil). In Gabriella Crocco & Eva-Maria Engelen (eds.), Kurt Gödel: Philosopher-Scientist. Presses Universitaires de Provence.
    Kurt Gödel left Philosophical Remarks in his Nachlass that he himself entitled Max Phil (Maximen Philosophie). The opus originally comprised 16 notebooks but one has been lost. The content is on the whole the outline of a rational metaphysics able to relate the different domains of knowledge and of moral investigations to each other. The notebooks were at first started as an intellectual diary in which Gödel writes an account of what he does and especially about what he should do (...)
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  35. Gregory Currie (1979). Lakatos's Philosophy of Mathematics. Synthese 42 (2):335 - 351.
  36. Corfield David (1998). Beyond the Methodology of Mathematics Research Programmes. Philosophia Mathematica 6 (3):272-301.
    In this paper I assess the obstacles to a transfer of Lakatos's methodology of scientific research programmes to mathematics. I argue that, if we are to use something akin to this methodology to discuss modern mathematics with its interweaving theoretical development, we shall require a more intricate construction and we shall have to move still further away from seeing mathematical knowledge as a collection of statements. I also examine the notion of rivalry within mathematics and claim that this appears to (...)
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  37. E. B. Davies (2003). Empiricism in Arithmetic and Analysis. Philosophia Mathematica 11 (1):53-66.
    We discuss the philosophical status of the statement that (9n – 1) is divisible by 8 for various sizes of the number n. We argue that even this simple problem reveals deep tensions between truth and verification. Using Gillies's empiricist classification of theories into levels, we propose that statements in arithmetic should be classified into three different levels depending on the sizes of the numbers involved. We conclude by discussing the relationship between the real number system and the physical continuum.
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  38. Philip J. Davis (1995). The Companion Guide to the Mathematical Experience, Study Edition. Birkhäuser.
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  39. William Demopoulos (2013). Logicism and its Philosophical Legacy. Cambridge University Press.
    Frege's analysis of arithmetical knowledge -- Carnap's thesis -- On extending 'empiricism, semantics and ontology' to the realism-instrumentalism controversy -- Carnap's analysis of realism -- Bertrand Russell's The analysis of matter: its historical context and contemporary interest (with Michael Friedman) -- On the rational reconstruction of our theoretical knowledge -- Three views of theoretical knowledge -- Frege and the rigorization of analysis -- The philosophical basis of our knowledge of number -- The 1910 Principia's theory of functions and classes -- (...)
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  40. William Demopoulos (1998). The Philosophical Basis of Our Knowledge of Number. Noûs 32 (4):481-503.
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  41. Michael Detlefsen (1995). Wright on the Non-Mechanizability of Intuitionist Reasoning. Philosophia Mathematica 3 (1):103-119.
    Crispin Wright joins the ranks of those who have sought to refute mechanist theories of mind by invoking Gödel's Incompleteness Theorems. His predecessors include Gödel himself, J. R. Lucas and, most recently, Roger Penrose. The aim of this essay is to show that, like his predecessors, Wright, too, fails to make his case, and that, indeed, he fails to do so even when judged by standards of success which he himself lays down.
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  42. Vladimir Drekalović (2010). Benacerraf on Mathematical Knowledge. Prolegomena 9 (1):97-121.
    Causal theory of knowledge has been used by some theoreticians who, dealing with the philosophy of mathematics, touched the subject of mathematical knowledge. Some of them discuss the necessity of the causal condition for justification, which creates the grounds for renewing the old conflict between empiricists and rationalists. Emphasizing the condition of causality as necessary for justifiability, causal theory has provided stimulus for the contemporary empiricists to venture on the so far unquestioned cognitive foundations of mathematics. However, in what sense (...)
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  43. Tommy Dreyfus & Theodore Eisenberg (1978). On Acceptance of Mathematical Theories. Philosophia Mathematica (1):56-87.
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  44. Kenny Easwaran (2008). The Role of Axioms in Mathematics. Erkenntnis 68 (3):381 - 391.
    To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide (...)
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  45. Philip A. Ebert & Stewart Shapiro (2009). The Good, the Bad and the Ugly. Synthese 170 (3):415 - 441.
    This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form (...)
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  46. Lyn D. English (ed.) (1997). Mathematical Reasoning: Analogies, Metaphors, and Images. L. Erlbaum Associates.
    Presents the latest research on how reasoning with analogies, metaphors, metonymies, and images can facilitate mathematical understanding. For math education, educational psychology, and cognitive science scholars.
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  47. J. Fang (1989). A REJOINDER: “Unum Post Aliud”. Philosophia Mathematica (2):236-245.
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  48. J. Fang (1989). “DEUS EX MACHINA” REDIVIVUS: The “Synthetic A Priori” in the Computer Age. Philosophia Mathematica (2):217-232.
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  49. Joong Fang (1997). Kant and Mathematics Today: Between Epistemology and Exact Sciences. Edwin Mellen Press.
  50. S. Feferman (2012). Curtis Franks. The Autonomy of Mathematical Knowledge: Hilbert's Program Revisted. Cambridge: Cambridge University Press, 2009. Isbn 978-0-521-51437-8. Pp. XIII+213. [REVIEW] Philosophia Mathematica 20 (3):387-400.
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