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Philosophy of Mathematics

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  1. Oscar João Abdounur, Vecchio Junior & Jacintho Del (2013). Sobre os números transfinitos. Scientiae Studia 11 (2):417-426.
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  2. Andrew Arana (forthcoming). On the Depth of Szemerédi's Theorem. Philosophia Mathematica:nku036.
    Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...)
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  3. Ross T. Brady (2012). The Consistency of Arithmetic, Based on a Logic of Meaning Containment. Logique Et Analyse 55 (219).
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  4. 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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  5. F. K. C. (1974). Meaning and Existence in Mathematics. Review of Metaphysics 27 (4):790-791.
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  6. J. D. C. (1971). Philosophie der Arithmetik. Review of Metaphysics 25 (1):127-128.
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  7. L. C. (1967). Mathematics and Logic in History and in Contemporary Thought. Review of Metaphysics 21 (1):154-154.
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  8. Carlo Cellucci (2007). La Filosofia della Matematica del Novecento. Laterza.
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  9. S. Centrone (2014). Mirja Hartimo Ed. Phenomenology and Mathematics. Phaenomenologia; 195. Dordrecht: Springer, 2010. ISBN 978-90-481-3728-2 ; 978-90-481-3728-2 ; 978-94-007-3196-7 . Pp. Xxv + 222. [REVIEW] Philosophia Mathematica 22 (1):126-129.
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  10. M. Davis (1998). Review of Dawson [1997]. [REVIEW] Philosophia Mathematica 3 (6).
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  11. Walter Dean (forthcoming). Arithmetical Reflection and the Provability of Soundness. Philosophia Mathematica:nku026.
    Proof-theoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truth-theoretic extensions thereof. This paper challenges this consensus by exploring relationships between reflection principles (...)
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  12. Hartry Field (1994). Are Our Logical and Mathematical Concepts Highly Indeterminate? Midwest Studies in Philosophy 19 (1):391-429.
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  13. Thomas Forster (forthcoming). Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF. Philosophia Mathematica:nku005.
    Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no (...)
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  14. Joachim Frans & Erik Weber (2014). Mechanistic Explanation and Explanatory Proofs in Mathematics. Philosophia Mathematica 22 (2):231-248.
    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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  15. Gottlob Frege (2013). Basic Laws of Arithmetic, Derived Using Concept-Script: Volumes I & Ii. Oxford University Press.
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  16. Greg Frost-Arnold (2013). Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science. Open Court Press.
    During the academic year 1940-1941, several giants of analytic philosophy congregated at Harvard, holding regular private meetings, with Carnap, Tarski, and Quine. Carnap, Tarski, and Quine at Harvard allows the reader to act as a fly on the wall for their conversations. Carnap took detailed notes during his year at Harvard. This book includes both a German transcription of these shorthand notes and an English translation in the appendix section. Carnap’s notes cover a wide range of topics, but surprisingly, the (...)
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  17. Mihai Ganea (2014). Finitistic Arithmetic and Classical Logic. Philosophia Mathematica 22 (2):167-197.
    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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  18. Intuitionism As Generalization (1990). Fred Richman New Mexico State University. Philosophia Mathematica 5 (124):128.
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  19. Eduard Glas (2014). A Role for Quasi-Empiricism in Mathematics Education. In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. 731-753.
    Although there are quite a few directions in modern philosophy of mathematics that invoke some essential role for (quasi-)empirical material, this chapter will be devoted exclusively to what may be considered the seminal tradition. This enabled me to present the subject as one coherent whole and to forestall the discussion getting scattered in a diversity of directions without doing justice to any one of them. -/- Quasi-empiricism in this tradition is the view that the logic of mathematical inquiry is based, (...)
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  20. Jeremy Gray (forthcoming). Depth — A Gaussian Tradition in Mathematics. Philosophia Mathematica:nku035.
    Mathematicians use the word ‘deep’ to convey a high appreciation of a concept, theorem, or proof. This paper investigates the extent to which the term can be said to have an objective character by examining its first use in mathematics. It was a consequence of Gauss's work on number theory and the agreement among his successors that specific parts of Gauss's work were deep, on grounds that indicate that depth was a structural feature of mathematics for them. In contrast, French (...)
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  21. 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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  22. E. R. Grosholz (2004). Lorenzo Magnani. Philosophy and Geometry: Theoretical and Historical Issues. Philosophia Mathematica 12 (1):79-80.
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  23. Marie Grossi, Montgomery Link, Katalin Makkai & Charles Parsons (1998). A Bibliography of Hao Wang. Philosophia Mathematica 6 (1):25-38.
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  24. G. E. R. Haddock (2003). Anastasio aleman. Logica, matematicas Y realidad. Philosophia Mathematica 11 (1):108-119.
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  25. D. M. Hausman (2003). E. Roy Weintraub. How Economics Became a Mathematical Science. Philosophia Mathematica 11 (3):354-357.
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  26. James Higginbotham (1993). McGinn's Logicisms. Philosophical Issues 4:119-127.
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  27. David Hilbert (1996). On the Concept of Number. In William Bragg Ewald (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford University Press. 2--1089.
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  28. Carlo Ierna (2014). Burt C. Hopkins. The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein. Studies in Continental Thought. Bloomington: University of Indiana Press, 2011. ISBN 978-0-253-35671-0 (Hbk). Pp. Xxxi + 559. [REVIEW] Philosophia Mathematica 22 (2):249-262.
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  29. A. D. Irvine (1995). Bertrand Russell, Philosophical Papers 1896-99. Philosophia Mathematica 3:301-301.
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  30. A. D. Irvine (1995). John Allen Paulos, Beyond Numeracy. Philosophia Mathematica 3:307-307.
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  31. A. D. Irvine (1995). Kurt Goedel, Collected Works. Volumes I and II. Philosophia Mathematica 3:299-299.
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  32. Andrew David Irvine (2014). Mark Colyvan. An Introduction to the Philosophy of Mathematics. Cambridge: Cambridge University Press, 2012. ISBN 978-0-521-82602-0 (Hbk); 978-0-521-53341-6 (Pbk). Pp. Ix + 188. [REVIEW] Philosophia Mathematica 22 (1):124-125.
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  33. J. Kim (2010). Yi on 2. Philosophia Mathematica 18 (3):329-336.
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  34. Ernst Kleinert (2012). Studien Zur Mathematik Und Philosophie. Leipziger Universitätsverlag.
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  35. Israel Krakowski (1980). The Four Color Problem Reconsidered. Philosophical Studies 38 (1):91 - 96.
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  36. G. Landini (2014). Gregory Landini. Zermelo and Russell's Paradox: Is There a Universal Set? Philosophia Mathematica 22 (1):142-142.
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  37. André Lebel (2014). Jean-Michel Salanskis. Philosophie des Mathématiques. Problèmes & Controverses. Paris: Vrin, 2008. ISBN 978-2-7116-1988-7. Pp. 304. [REVIEW] Philosophia Mathematica 22 (2):262-270.
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  38. Renren Liu (2010). Duo Zhi Luo Ji Han Shu Jie Gou Li Lun Yan Jiu. Ke Xue Chu Ban She.
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  39. Giuseppe Longo (1999). Mathematical Intelligence, Infinity and Machines: Beyond Godelitis. Journal of Consciousness Studies 6 (11-12):11-12.
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  40. Minxia Luo (2010). Fan Luo Ji Xue Yu Gou Li Lun. Ke Xue Chu Ban She.
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  41. William Marias Malisoff (1935). An Examination of the Quantum Theories. IV. Philosophy of Science 2 (3):334-343.
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  42. William Marias Malisoff (1934). An Examination of the Quantum Theories. I. Philosophy of Science 1 (1):71-77.
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  43. William Marias Malisoff (1934). An Examination of the Quantum Theories. II. Philosophy of Science 1 (2):170-175.
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  44. William Marias Malisoff (1934). An Examination of the Quantum Theories III. Philosophy of Science 1 (4):398-408.
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  45. Paolo Mancosu (forthcoming). William Ewald and Wilfried Sieg, Eds, David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917–1933. Heidelberg: Springer, 2013. ISBN: 978-3-540-69444-1 ; 978-3-540-20578-4 . Pp. Xxv + 1062. [REVIEW] Philosophia Mathematica:nku031.
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  46. Alex Miller (ed.) (2013). Logic, Language and Mathematics: Essays for Crispin Wright. Oxford University Press.
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  47. M. M. Muntersbjorn (2003). Meir Buzaglo. The Logic of Concept Expansion. Philosophia Mathematica 11 (3):341-348.
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  48. Gianluigi Oliveri (forthcoming). Book Review.'I Fondamenti della Matematica nel Logicismo di Bertrand Russell'. Stefano Donati. Firenze (Firenze Atheneum). 2003. ISBN: 88-7255-204-4. 988 pages.€ 39.00. [REVIEW] Philosophia Mathematica.
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  49. Roland Omnès (2011). Wigner's “Unreasonable Effectiveness of Mathematics”, Revisited. Foundations of Physics 41 (11):1729-1739.
    A famous essay by Wigner is reexamined in view of more recent developments around its topic, together with some remarks on the metaphysical character of its main question about mathematics and natural sciences.
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  50. C. Parsons (1996). Jean Dieudonne. Mathematics-The Music of Reason. Philosophia Mathematica 4:190-195.
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