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  1. James Brown (2011). Does Mathematics Have a Life of its Own?: Review of P. Maddy, Second Philosophy: A Naturalistic Method. [REVIEW] Metascience 20 (3):487-493.
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  2. Eduardo Castro (2008). Review of P. Maddy, Second Philosophy: a Naturalistic Method. [REVIEW] Disputatio 2 (24):349-355.
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  3. Eduardo Castro (2008). Review of P. Maddy, Second Philosophy: A Naturalistic Method. [REVIEW] Disputatio.
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  4. Chris Daly (2006). Mathematical Fictionalism – No Comedy of Errors. Analysis 66 (291):208–216.
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  5. Chris Daly & David Liggins (2011). Deferentialism. Philosophical Studies 156 (3):321-337.
    There is a recent and growing trend in philosophy that involves deferring to the claims of certain disciplines outside of philosophy, such as mathematics, the natural sciences, and linguistics. According to this trend— deferentialism , as we will call it—certain disciplines outside of philosophy make claims that have a decisive bearing on philosophical disputes, where those claims are more epistemically justified than any philosophical considerations just because those claims are made by those disciplines. Deferentialists believe that certain longstanding philosophical problems (...)
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  6. J. M. Dieterle (1999). Mathematical, Astrological, and Theological Naturalism. Philosophia Mathematica 7 (2):129-135.
    persuasive argument for the claim that we ought to evaluate mathematics from a mathematical point of view and reject extra-mathematical standards. Maddy considers the objection that her arguments leave it open for an ‘astrological naturalist’ to make an analogous claim: that we ought to reject extra-astrological standards in the evaluation of astrology. In this paper, I attempt to show that Maddy's response to this objection is insufficient, for it ultimately either (1) undermines mathematical naturalism itself, leaving us with only scientific (...)
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  7. Paul Ernest (1997). The Legacy of Lakatos: Reconceptualising the Philosophy of Mathematics. Philosophia Mathematica 5 (2):116-134.
    Kitcher and Aspray distinguish a mainstream tradition in the philosophy of mathematics concerned with foundationalist epistemology, and a ‘maverick’ or naturalistic tradition, originating with Lakatos. My claim is that if the consequences of Lakatos's contribution are fully worked out, no less than a radical reconceptualization of the philosophy of mathematics is necessitated, including history, methodology and a fallibilist epistemology as central to the field. In the paper an interpretation of Lakatos's philosophy of mathematics is offered, followed by some critical discussion, (...)
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  8. James Franklin (2014). Aristotelian Realist Philosophy of Mathematics. Palgrave MacMillan.
    An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...)
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  9. James Franklin (2011). Aristotelianism in the Philosophy of Mathematics. Studia Neoaristotelica 8 (1):3-15.
    Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for example, is (...)
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  10. James Franklin (2009). Aristotelian Realism. In A. Irvine (ed.), The Philosophy of Mathematics (Handbook of the Philosophy of Science series). North-Holland Elsevier.
    Aristotelian, or non-Platonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity, or numerosity, or symmetry. Let us start with an example, as Aristotelians always prefer, an example that introduces the essential themes of the Aristotelian view of mathematics. A typical mathematical truth is (...)
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  11. Luca Incurvati (2008). Too Naturalist and Not Naturalist Enough: Reply to Horsten. Erkenntnis 69 (2):261 - 274.
    Leon Horsten has recently claimed that the class of mathematical truths coincides with the class of theorems of ZFC. I argue that the naturalistic character of Horsten’s proposal undermines his contention that this claim constitutes an analogue of a thesis that Daniel Isaacson has advanced for PA. I argue, moreover, that Horsten’s defence of his claim against an obvious objection makes use of a distinction which is not available to him given his naturalistic approach. I suggest a way out of (...)
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  12. Luca Incurvati & Peter Smith (2012). Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory. [REVIEW] Mind 121 (481):195-200.
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  13. A. Irvine (ed.) (2009). The Philosophy of Mathematics (Handbook of the Philosophy of Science Series). North-Holland Elsevier.
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  14. David Liggins (2007). Anti-Nominalism Reconsidered. Philosophical Quarterly 57 (226):104–111.
    Many philosophers of mathematics are attracted by nominalism – the doctrine that there are no sets, numbers, functions, or other mathematical objects. John Burgess and Gideon Rosen have put forward an intriguing argument against nominalism, based on the thought that philosophy cannot overrule internal mathematical and scientific standards of acceptability. I argue that Burgess and Rosen’s argument fails because it relies on a mistaken view of what the standards of mathematics require.
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  15. David Liggins (2006). Is There a Good Epistemological Argument Against Platonism? Analysis 66 (290):135–141.
    Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti-platonist argument proposed by Hartry Field avoids both horns of their dilemma.
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  16. Øystein Linnebo (2012). Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory. [REVIEW] Philosophy 87 (01):133-137.
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  17. Bernard Linsky & Edward N. Zalta (2006). What is Neologicism? Bulletin of Symbolic Logic 12 (1):60-99.
    In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic (...)
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  18. Penelope Maddy (2007). Second Philosophy: A Naturalistic Method. Oxford University Press.
    Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In Second Philosophy, Penelope Maddy describes and practices a particularly austere form of naturalism called "Second Philosophy". Without a definitive criterion for what counts as "science" and what doesn't, Second Philosophy can't be specified directly ("trust only the methods of science" for example), so Maddy proceeds instead by illustrating the behaviors of an idealized inquirer she calls the "Second Philosopher". (...)
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  19. Penelope Maddy (1992). Indispensability and Practice. Journal of Philosophy 89 (6):275-289.
  20. Penelope Maddy, Second Philosophy.
    Perhaps some of the movie-goers among you have had the experience of sitting through a film with no discernable plot and no significant action, only to be accused by your companions of having missed the point. 'It's not supposed to be dramatic', they tell you, 'it's a Character Study! ' The conventions of this genre seem to require that it centre on an otherwise inconspicuous person who undergoes some familiar life passage or other with terribly subtle, if any, reactions or (...)
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  21. Penelope J. Maddy (2001). Some Naturalistic Reflections on Set Theoretic Method. Topoi 20 (1):17-27.
    My ultimate goal in this paper is to illuminate, from a naturalistic point of view, the significance of the application of mathematics in the natural sciences for the practice of contemporary set theory.
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  22. C. McLarty (2013). Penelope Maddy. Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford: Oxford University Press, 2011. ISBN 978-0-19-959618-8 (Hbk); 978-0-19-967148-9 (Pbk). Pp. X + 150. [REVIEW] Philosophia Mathematica 21 (3):385-392.
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  23. Matthew E. Moore (2007). Naturalism, Truth and Beauty in Mathematics. Philosophia Mathematica 15 (2):141-165.
    Can a scientific naturalist be a mathematical realist? I review some arguments, derived largely from the writings of Penelope Maddy, for a negative answer. The rejoinder from the realist side is that the irrealist cannot explain, as well as the realist can, why a naturalist should grant the mathematician the degree of methodological autonomy that the irrealist's own arguments require. Thus a naturalist, as such, has at least as much reason to embrace mathematical realism as to embrace irrealism.
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  24. Alexander Paseau (2008). Naturalism in the Philosophy of Mathematics. In Stanford Encyclopedia of Philosophy.
    Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed more briefly in section (...)
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  25. Alexander Paseau (2005). Naturalism in Mathematics and the Authority of Philosophy. British Journal for the Philosophy of Science 56 (2):377-396.
    Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two forms, and examine the (...)
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  26. Augustín Rayo (2003). Success by Default? Philosophia Mathematica 11 (3):305-322.
    I argue that Neo-Fregean accounts of arithmetical language and arithmetical knowledge tacitly rely on a thesis I call [Success by Default]—the thesis that, in the absence of reasons to the contrary, we are justified in thinking that certain stipulations are successful. Since Neo-Fregeans have yet to supply an adequate defense of [Success by Default], I conclude that there is an important gap in Neo-Fregean accounts of arithmetical language and knowledge. I end the paper by offering a naturalistic remedy.
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  27. Adam Rieger (2002). Patterns in the Philosophy of Mathematics. [REVIEW] Philosophical Quarterly 52 (207):247–255.
  28. Jeffrey W. Roland (2009). On Naturalizing the Epistemology of Mathematics. Pacific Philosophical Quarterly 90 (1):63-97.
    In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.
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  29. Jeffrey W. Roland (2008). Kitcher, Mathematics, and Naturalism. Australasian Journal of Philosophy 86 (3):481 – 497.
    This paper argues that Philip Kitcher's epistemology of mathematics, codified in his Naturalistic Constructivism, is not naturalistic on Kitcher's own conception of naturalism. Kitcher's conception of naturalism is committed to (i) explaining the correctness of belief-regulating norms and (ii) a realist notion of truth. Naturalistic Constructivism is unable to simultaneously meet both of these commitments.
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  30. Jeffrey W. Roland (2007). Maddy and Mathematics: Naturalism or Not. British Journal for the Philosophy of Science 58 (3):423 - 450.
    Penelope Maddy advances a purportedly naturalistic account of mathematical methodology which might be taken to answer the question 'What justifies axioms of set theory?' I argue that her account fails both to adequately answer this question and to be naturalistic. Further, the way in which it fails to answer the question deprives it of an analog to one of the chief attractions of naturalism. Naturalism is attractive to naturalists and nonnaturalists alike because it explains the reliability of scientific practice. Maddy's (...)
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  31. G. Rosen (1999). Review of P. Maddy, Naturalism in Mathematics. [REVIEW] British Journal for the Philosophy of Science 50 (3):467-474.
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  32. Marcus Rossberg (2006). Die Vertreibung aus dem Platonischen Paradies. Erwägen – Wissen – Ethik 17 (Naturalism in Mathematics):387–389.
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  33. Stewart Shapiro & Patrick Reeder (2009). A Scientific Enterprise?: A Critical Study of P. Maddy, Second Philosophy: A Naturalistic Method. [REVIEW] Philosophia Mathematica 17 (2):247-271.
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  34. Harvey Siegel (2010). Review of P. Maddy, Second Philosophy: A Naturalistic Method. [REVIEW] British Journal for the Philosophy of Science 61 (4):897-903.
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  35. B. Stroud (2009). Review of P, Maddy, Second Philosophy. [REVIEW] Mind 118 (470):500-503.
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  36. Neil Tennant (2000). What is Naturalism in Mathematics, Really?: A Critical Study of P. Maddy, Naturalism in Mathematics. [REVIEW] Philosophia Mathematica 8 (3):316-338.
    Review of PENELOPE MADDY. Naturalism in Mathematics. Oxford: Clarendon Press, 1997.
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  37. Richard Tieszen (1994). Review of P. Maddy, Realism in Mathematics. [REVIEW] Philosophia Mathematica 2 (1).
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  38. S. Vineberg (2012). Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory. [REVIEW] Analysis 72 (3):635-637.
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