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Philosophy of Mathematics

Edited by Øystein Linnebo (Birkbeck College)
Assistant editor: Sam Roberts (Birkbeck College, University of Oslo)
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  1. added 2014-04-09
    T. Parent, Paradox with Just Self-Reference.
    If a semantically open language allows self-reference, one can show there is a predicate which is both satisfied and unsatisfied by a self-referring term. The argument requires something akin to diagonalization on substitution-instances of a definition-scheme (*): ‘x is Lagadonian iff the variable in the nth substitution-instance of (*) is replaced with x’. Given an appropriate enumeration of the instances, a self-referring term is counted as “Lagadonian” if the variable in (*) is replaced with the term itself. But the same (...)
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  2. added 2014-04-06
    G. Aldo Antonelli (2012). A Note on Induction, Abstraction, and Dedekind-Finiteness. Notre Dame Journal of Formal Logic 53 (2):187-192.
    The purpose of this note is to present a simplification of the system of arithmetical axioms given in previous work; specifically, it is shown how the induction principle can in fact be obtained from the remaining axioms, without the need of explicit postulation. The argument might be of more general interest, beyond the specifics of the proposed axiomatization, as it highlights the interaction of the notion of Dedekind-finiteness and the induction principle.
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  3. added 2014-04-06
    Irving H. Anellis (1987). The Conference on the History of Mathematics University of Wisconsin-Lacrosse, April 1, 1986. Philosophia Mathematica 2 (1):123-125.
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  4. added 2014-04-05
    Editor The (1988). The Poverty of Philosophy, I: Ethics and Mathematics. Philosophia Mathematica 2 (1):55-74.
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  5. added 2014-04-04
    John L. Bell (2014). Review of G. H. Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence. [REVIEW] Philosophia Mathematica 22 (1):131-134.
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  6. added 2014-04-03
    Intuitionism As Generalization (1990). Fred Richman New Mexico State University. Philosophia Mathematica 5 (124):128.
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  7. added 2014-04-02
    Jessica Carter (forthcoming). Mathematics Dealing with 'Hypothetical States of Things'. Philosophia Mathematica:nkt040.
    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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  8. added 2014-04-02
    L. Luna & W. Taylor (forthcoming). Taming the Indefinitely Extensible Definable Universe. Philosophia Mathematica:nkt044.
    In previous work in 2010 we have dealt with the problems arising from Cantor's theorem and the Richard paradox in a definable universe. We proposed indefinite extensibility as a solution. Now we address another definability paradox, the Berry paradox, and explore how Hartogs's cardinality theorem would behave in an indefinitely extensible definable universe where all sets are countable.
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  9. added 2014-04-02
    Andrea Sereni (forthcoming). Frege, Indispensability, and the Compatibilist Heresy. Philosophia Mathematica:nkt046.
    In Grundgesetze, Vol. II, §91, Frege argues that ‘it is applicability alone which elevates arithmetic from a game to the rank of a science’. Many view this as an in nuce statement of the indispensability argument (IA) later championed by Quine. Garavaso has questioned this attribution. I argue that even though Frege's applicability argument is not a version of IA, it facilitates acceptance of suitable formulations of IA. The prospects for making the empiricist IA compatible with a rationalist Fregean framework (...)
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  10. added 2014-04-02
    Mihai Ganea (forthcoming). Finitistic Arithmetic and Classical Logic. Philosophia Mathematica:nkt042.
    It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition. The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's ‘predicative arithmetic’ program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly (...)
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  11. added 2014-04-02
    André Lebel (forthcoming). Jean-Michel Salanskis. Philosophie des Mathématiques. Problèmes & Controverses. Paris: Vrin, 2008. ISBN 978-2-7116-1988-7. Pp. 304. [REVIEW] Philosophia Mathematica:nku001.
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  12. added 2014-04-02
    Russell Marcus (forthcoming). How Not to Enhance the Indispensability Argument. Philosophia Mathematica:nku004.
    The new explanatory or enhanced indispensability argument alleges that our mathematical beliefs are justified by their indispensable appearances in scientific explanations. This argument differs from the standard indispensability argument which focuses on the uses of mathematics in scientific theories. I argue that the new argument depends for its plausibility on an equivocation between two senses of explanation. On one sense the new argument is an oblique restatement of the standard argument. On the other sense, it is vulnerable to an instrumentalist (...)
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  13. added 2014-04-02
    Joachim Frans & Erik Weber (forthcoming). Mechanistic Explanation and Explanatory Proofs in Mathematics. Philosophia Mathematica:nku003.
    Although there is a consensus among philosophers of mathematics and mathematicians that mathematical explanations exist, only a few authors have proposed accounts of explanation in mathematics. These accounts fit into the unificationist or top-down approach to explanation. We argue that these models can be complemented by a bottom-up approach to explanation in mathematics. We introduce the mechanistic model of explanation in science and discuss the possibility of using this model in mathematics, arguing that using it does not presuppose a Platonist (...)
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  14. added 2014-04-02
    Chris Daly & David Liggins (2014). Nominalism, Trivialist Platonism and Benacerraf's Dilemma. Analysis 74 (2):224-231.
    In his stimulating new book The Construction of Logical Space, Agustín Rayo offers a new account of mathematics, which he calls ‘Trivialist Platonism’. In this article, we take issue with Rayo’s case for Trivialist Platonism and his claim that the view overcomes Benacerraf’s dilemma. Our conclusion is that Rayo has not shown that Trivialist Platonism has any advantage over nominalism.
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  15. added 2014-04-02
    Gregory Landini (2014). Gregory Landini. Zermelo and Russell's Paradox: Is There a Universal Set? Philosophia Mathematica 22 (1):142-142.
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  16. added 2014-04-01
    Daniël F. M. Strauss (forthcoming). The On to Log I Cal Sta Tus of the Prin Ci Ple of the Ex Cluded Mid Dle. Philosophia Mathematica.
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  17. added 2014-03-29
    Minxia Luo (2010). Fan Luo Ji Xue Yu Gou Li Lun. Ke Xue Chu Ban She.
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  18. added 2014-03-29
    J. R. Brown (1994). John D. Barrow, Pi in the Sky: Counting, Thinking, and Being. Philosophia Mathematica 2:251-251.
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  19. added 2014-03-29
    Marie La Palme Reyes, John Macnamara, Gonzalo E. Reyes & Houman Zolfaghari (1994). The Non-Boolean Logic of Natural Language Negation. Philosophia Mathematica 2 (1):45-68.
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  20. added 2014-03-27
    A. W. Moore (1995). Shaughan Lavine, Understanding the Infinite. Philosophia Mathematica 3:294-294.
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  21. added 2014-03-27
    M. Detlefsen (1995). Janet Folina, Poincare and the Philosophy of Mathematics. Philosophia Mathematica 3:208-208.
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  22. added 2014-03-27
    M. Steiner (1995). Shlomo Sternberg, Group Theory and Physics. Philosophia Mathematica 3:313-313.
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  23. added 2014-03-27
    A. D. Irvine (1995). John Allen Paulos, Beyond Numeracy. Philosophia Mathematica 3:307-307.
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  24. added 2014-03-27
    T. Tymoczko (1995). Jerry P. King, The Art of Mathematics. Philosophia Mathematica 3:120-120.
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  25. added 2014-03-27
    A. D. Irvine (1995). Kurt Goedel, Collected Works. Volumes I and II. Philosophia Mathematica 3:299-299.
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  26. added 2014-03-27
    N. Griffin (1995). Alejandro R. Garciadiego, Bertrand Russell and the Origins of the Set-theoretic'Paradoxes'. Philosophia Mathematica 3:304-304.
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  27. added 2014-03-27
    A. D. Irvine (1995). Bertrand Russell, Philosophical Papers 1896-99. Philosophia Mathematica 3:301-301.
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  28. added 2014-03-27
    N. Tennant (1995). Keith Devlin, Logic and Information. Philosophia Mathematica 3:179-179.
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  29. added 2014-03-27
    T. Tymoczko (1995). Review of The Art of Mathematics, by JP King. [REVIEW] Philosophia Mathematica 3:120-126.
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  30. added 2014-03-26
    Alan Mathison Turing (2012). Alan Turing's Systems of Logic: The Princeton Thesis. Princeton University Press.
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  31. added 2014-03-26
    J. -P. Marquis (1996). D. Vernant. La Philosophie Mathematique de Russell. Philosophia Mathematica 4:202-205.
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  32. added 2014-03-26
    C. Parsons (1996). Jean Dieudonne. Mathematics-The Music of Reason. Philosophia Mathematica 4:190-195.
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  33. added 2014-03-26
    R. Teiszen (1996). Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness. Philosophia Mathematica 4:281-289.
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  34. added 2014-03-26
    S. Shapiro & G. Svensson (1996). Special Issue Of. Philosophia Mathematica 3 (4).
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  35. added 2014-03-26
    E. D. Mares (1996). I. Grattan-Guinness, Editor. Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Philosophia Mathematica 4:198-201.
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  36. added 2014-03-25
    Georg Schiemer (2010). Fraenkel's Axiom of Restriction: Axiom Choice, Intended Models and Categoricity. In Benedikt L.öwe & Thomas Müller (eds.), PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. College Publications. 307{340.
  37. added 2014-03-25
    Neil Tennant (1998). Games Some People Would Have All of Us Play. Philosophia Mathematica 6 (3):90-115.
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  38. added 2014-03-25
    Marie Grossi, Montgomery Link, Katalin Makkai & Charles Parsons (1998). A Bibliography of Hao Wang. Philosophia Mathematica 6 (1):25-38.
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  39. added 2014-03-25
    Neil Tennant (1998). Jaakko Hintikka, The Principles of Mathematics Revisited. Philosophia Mathematica 6:90-115.
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  40. added 2014-03-25
    M. Davis (1998). Review of Dawson [1997]. [REVIEW] Philosophia Mathematica 3 (6).
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  41. added 2014-03-25
    Göran Sundholm (1998). Inference, Consequence, Implication: A Constructivist's Perspective. Philosophia Mathematica 6 (2):178-194.
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  42. added 2014-03-25
    J. P. Burgess & P. Ernest (1997). Philip J. Davis, Reuben Hersh, and Elena Anne Marchisotto. The Mathematical Experience Study Guide and The Companion Guide to the Mathematical Experience Study Edition. [REVIEW] Philosophia Mathematica 5:175-188.
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  43. added 2014-03-25
    Darcy Cutler (1997). Stewart Shapiro. Foundations Without Foundationalism: A Case for Second-Order Logic. Philosophia Mathematica 5:71-91.
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  44. added 2014-03-25
    A. Riskin (1997). Jen Hoeyrup, In Measure, Number, and Weight. Philosophia Mathematica 5:276-277.
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  45. added 2014-03-25
    F. Richman (1997). Christopher Ormell. Some Criteria for Sets in Mathematics. Philosophia Mathematica 5:92-96.
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  46. added 2014-03-23
    E. Carson (1998). Maoist Mathematics? Philosophia Mathematica 6 (3):345-350.
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  47. added 2014-03-14
    D. Gillies (2003). Carlo Cellucci.[Philosophy and Mathematics]. Philosophia Mathematica 11 (2):246-252.
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  48. added 2014-03-14
    G. E. R. Haddock (2003). Anastasio aleman. Logica, matematicas Y realidad. Philosophia Mathematica 11 (1):108-119.
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  49. added 2014-03-14
    M. Motterlini (2003). John Kadvany. Imre Lakatos and the Guises of Reason. Philosophia Mathematica 11 (1):120-127.
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  50. added 2014-03-14
    J. Folina (2003). Mark Greaves. The Philosophical Status of Diagrams. Philosophia Mathematica 11 (3):349-353.
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