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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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    Epistemology of Mathematics
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    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
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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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    Epistemology of Mathematics
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     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
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    Ontology of Mathematics
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     The Nature of Sets
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    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
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  3. Mark Atten (2003). Critical Studies/Book Reviews. [REVIEW] Philosophia Mathematica 11 (2):241-244.
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  4. Patricia Blanchette (2003). Critical Studies / Book Reviews. [REVIEW] Philosophia Mathematica 11 (3):358-362.
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  5. Ross T. Brady (2012). The Consistency of Arithmetic, Based on a Logic of Meaning Containment. Logique Et Analyse 55 (219).
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     Visualization in Mathematics
     Phenomenology of Mathematics
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    Ontology of Mathematics
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  6. 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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    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
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  7. F. K. C. (1974). Meaning and Existence in Mathematics. Review of Metaphysics 27 (4):790-791.
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    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
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  8. J. D. C. (1971). Philosophie der Arithmetik. Review of Metaphysics 25 (1):127-128.
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    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
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  9. L. C. (1967). Mathematics and Logic in History and in Contemporary Thought. Review of Metaphysics 21 (1):154-154.
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     Numbers
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    Set Theory
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  10. Emily Carson (1998). Maoist Mathematics? [REVIEW] Philosophia Mathematica 6 (3):345-350.
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  11. Ernst Cassirer (1923/2003). Substance and Function. Dover Publications.
    In this double-volume work, a great modern philosopher propounds a system of thought in which Einstein's theory of relativity represents only the latest (albeit the most radical) fulfillment of the motives inherent to mathematics and the physical sciences. In the course of its exposition, it touches upon such topics as the concept of number, space and time, geometry, and energy; Euclidean and non-Euclidean geometry; traditional logic and scientific method; mechanism and motion; Mayer's methodology of natural science; Richter's definite proportions; relational (...)
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  12. Carlo Cellucci (2007). La Filosofia della Matematica del Novecento. Laterza.
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  13. 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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  14. Ferenc Csatari (2012). Some Remarks on the Physicalist Account of Mathematics. Open Journal of Philosophy 2 (2):165.
    The paper comments on a rather uncommon approach to mathematics called physicalist formalism. According to this view, the formal systems mathematicians concern with are nothing more and nothing less than genuine physical systems. I give a brief review on the main theses, then I provide some arguments, concerning mostly with the practice of mathematics and the uniqueness of formal systems, aiming to show the implausibility of this radical view.
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  15. A. J. Dale (1990). The Reality of Numbers: A Physicalist's Philosophy of Mathematics. Philosophical Books 31 (1):61-62.
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  16. M. Davis (1998). Review of Dawson [1997]. [REVIEW] Philosophia Mathematica 3 (6).
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  17. Walter Dean (2015). Arithmetical Reflection and the Provability of Soundness. Philosophia Mathematica 23 (1):31-64.
    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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  18. Hartry Field (1994). Are Our Logical and Mathematical Concepts Highly Indeterminate? Midwest Studies in Philosophy 19 (1):391-429.
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  19. Juliet Floyd (forthcoming). Depth and ClarityFelix Mühlhölzer. Braucht Die Mathematik Eine Grundlegung? Eine Kommentar des Teils III von Wittgensteins Bemerkungen Über Die Grundlagen der Mathematik [Does Mathematics Need a Foundation? A Commentary on Part III of Wittgenstein's Remarks on the Foundations of Mathematics]. Frankfurt: Vittorio Klostermann, 2010. ISBN: 978-3-465-03667-8. Pp. Xiv + 602. [REVIEW] Philosophia Mathematica:nku037.
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  20. 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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  21. 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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  22. Gottlob Frege (2013). Basic Laws of Arithmetic, Derived Using Concept-Script: Volumes I & Ii. Oxford University Press.
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  23. 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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  24. 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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  25. Intuitionism As Generalization (1990). Fred Richman New Mexico State University. Philosophia Mathematica 5 (124):128.
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  26. 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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  27. 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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  28. 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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  29. E. R. Grosholz (2004). Lorenzo Magnani. Philosophy and Geometry: Theoretical and Historical Issues. Philosophia Mathematica 12 (1):79-80.
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  30. Marie Grossi, Montgomery Link, Katalin Makkai & Charles Parsons (1998). A Bibliography of Hao Wang. Philosophia Mathematica 6 (1):25-38.
    A listing is given of the published writings of the logician and philosopher Hao Wang , which includes all items known to the authors, including writings in Chinese and translations into other languages.
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  31. G. E. R. Haddock (2003). Anastasio aleman. Logica, matematicas Y realidad. Philosophia Mathematica 11 (1):108-119.
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  32. B. Hale (1993). Physicalism and Mathematics. In Howard M. Robinson (ed.), Objections to Physicalism. Oxford University Press. 39--59.
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  33. D. M. Hausman (2003). E. Roy Weintraub. How Economics Became a Mathematical Science. Philosophia Mathematica 11 (3):354-357.
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  34. James Higginbotham (1993). McGinn's Logicisms. Philosophical Issues 4:119-127.
    Russian translation of Higginbotham J. McGinn's Logicisms // Philosophical Issues, 4, 1993. Translated by Kristina Goncharenko with kind permission of the author.
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  35. 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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  36. 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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  37. A. D. Irvine (1995). Bertrand Russell, Philosophical Papers 1896-99. Philosophia Mathematica 3:301-301.
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  38. A. D. Irvine (1995). John Allen Paulos, Beyond Numeracy. Philosophia Mathematica 3:307-307.
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  39. A. D. Irvine (1995). Kurt Goedel, Collected Works. Volumes I and II. Philosophia Mathematica 3:299-299.
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  40. 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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  41. J. Kim (2010). Yi on 2. Philosophia Mathematica 18 (3):329-336.
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  42. Ernst Kleinert (2012). Studien Zur Mathematik Und Philosophie. Leipziger Universitätsverlag.
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  43. Israel Krakowski (1980). The Four Color Problem Reconsidered. Philosophical Studies 38 (1):91 - 96.
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  44. 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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  45. 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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  46. David Lewis (1989). John Bigelow: "The Reality of Numbers: A Physicalist's Philosophy of Mathematics". [REVIEW] Australasian Journal of Philosophy 67:487.
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  47. Renren Liu (2010). Duo Zhi Luo Ji Han Shu Jie Gou Li Lun Yan Jiu. Ke Xue Chu Ban She.
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  48. Giuseppe Longo (1999). Mathematical Intelligence, Infinity and Machines: Beyond Godelitis. Journal of Consciousness Studies 6 (11-12):11-12.
    We informally discuss some recent results on the incompleteness of formal systems. These theorems, which are of great importance to contemporary mathematical epistemology, are proved using a variety of conceptual tools provably stronger than those of finitary axiomatisations. Those tools require no mathematical ontology, but rather constitute particularly concrete human constructions and acts of comprehending infinity and space rooted in different forms of knowledge. We shall also discuss, albeit very briefly, the mathematical intelligence both of God and of computers. We (...)
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  49. Minxia Luo (2010). Fan Luo Ji Xue Yu Gou Li Lun. Ke Xue Chu Ban She.
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  50. William Marias Malisoff (1935). An Examination of the Quantum Theories. IV. Philosophy of Science 2 (3):334-343.
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     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
    Areas of Mathematics
     Algebra
     Analysis
     Category Theory
     Geometry
     Logic and Phil of Logic
     Number Theory
     Set Theory
     Topology
     Areas of Mathematics, Misc
    Theories of Mathematics
     Logicism in Mathematics
     Formalism in Mathematics
     Intuitionism and Constructivism
     Predicativism in Mathematics
     Mathematical Naturalism
     Mathematical Finitism
     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
     The Application of Mathematics
     History of Mathematics
     Mathematical Practice
     Phil of Mathematics, General Works
     Mathematical Explanation
     Phil of Mathematics, Misc
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