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  1. From a Logical Point of View.W. V. O. Quine - 1953 - Harvard University Press.
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  • Philosophy of Logic.John Corcoran - 1973 - Philosophy of Science 40 (1):131-133.
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  • Realism, Mathematics and Modality.Hartry Field - 1988 - Philosophical Topics 16 (1):57-107.
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  • Mathematics and Reality.Mary Leng (ed.) - 2010 - Oxford University Press.
    Mary Leng defends a philosophical account of the nature of mathematics which views it as a kind of fiction. On this view, the claims of our ordinary mathematical theories are more closely analogous to utterances made in the context of storytelling than to utterances whose aim is to assert literal truths.
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  • Grundlagen der Arithmetik.Gottlob Frege - 1884 - Breslau: Wilhelm Koebner Verlag.
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  • Platonism and Anti-Platonism in Mathematics.Matthew McGrath - 2001 - Philosophy and Phenomenological Research 63 (1):239-242.
    Mark Balaguer has written a provocative and original book. The book is as ambitious as a work of philosophy of mathematics could be. It defends both of the dominant views concerning the ontology of mathematics, Platonism and Anti-Platonism, and then closes with an argument that there is no fact of the matter which is right.
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  • Fictionalism, Theft, and the Story of Mathematics.Mark Balaguer - 2009 - Philosophia Mathematica 17 (2):131-162.
    This paper develops a novel version of mathematical fictionalism and defends it against three objections or worries, viz., (i) an objection based on the fact that there are obvious disanalogies between mathematics and fiction; (ii) a worry about whether fictionalism is consistent with the fact that certain mathematical sentences are objectively correct whereas others are incorrect; and (iii) a recent objection due to John Burgess concerning “hermeneuticism” and “revolutionism”.
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  • Should Scientific Realists Be Platonists?Jacob Busch & Joe Morrison - 2016 - Synthese 193 (2):435-449.
    Enhanced indispensability arguments claim that Scientific Realists are committed to the existence of mathematical entities due to their reliance on Inference to the best explanation. Our central question concerns this purported parity of reasoning: do people who defend the EIA make an appropriate use of the resources of Scientific Realism to achieve platonism? We argue that just because a variety of different inferential strategies can be employed by Scientific Realists does not mean that ontological conclusions concerning which things we should (...)
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  • In Defense of Mathematical Inferentialism.Seungbae Park - 2017 - Analysis and Metaphysics 16:70-83.
    I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism.
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  • Nominalism, Naturalism, Epistemic Relativism.Gideon Rosen - 2001 - Noûs 35 (s15):69 - 91.
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  • Revolutionary Fictionalism: A Call to Arms.Mary Leng - 2005 - Philosophia Mathematica 13 (3):277-293.
    This paper responds to John Burgess's ‘Mathematics and _Bleak House_’. While Burgess's rejection of hermeneutic fictionalism is accepted, it is argued that his two main attacks on revolutionary fictionalism fail to meet their target. Firstly, ‘philosophical modesty’ should not prevent philosophers from questioning the truth of claims made within successful practices, provided that the utility of those practices as they stand can be explained. Secondly, Carnapian scepticism concerning the meaningfulness of _metaphysical_ existence claims has no force against a _naturalized_ version (...)
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  • The Conceivability of Platonism.Benjamin Callard - 2007 - Philosophia Mathematica 15 (3):347-356.
    It is widely believed that platonists face a formidable problem: that of providing an intelligible account of mathematical knowledge. The problem is that we seem unable, if the platonist is right, to have the causal relationships with the objects of mathematics without which knowledge of these objects seems unintelligible. The standard platonist response to this challenge is either to deny that knowledge without causation is unintelligible, or to make room for causal interactions by softening the platonism at issue. In this (...)
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  • Philosophy of Logic.Leslie Stevenson - 1973 - Philosophical Quarterly 23 (93):366-367.
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  • Realism, Mathematics & Modality.Hartry Field - 1989 - Blackwell.
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  • Platonism in the Philosophy of Mathematics.Øystein Linnebo - 2009 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy.
    Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. In this survey article, the view is clarified and distinguished from some related views, and arguments for and against the view are discussed.
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  • Science Without Numbers.Hartry Field - 1980 - Synthese 51 (2):283-291.
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  • Science Without Numbers.Hartry Field - 1980 - Princeton University Press.
    Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.
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  • The Indispensability of Mathematics.Mark Colyvan - 2001 - Oxford University Press.
    This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.
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  • Mathematics as a Science of Patterns.Michael D. Resnik - 1997 - New York ;Oxford University Press.
    This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...)
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  • Platonism and Anti-Platonism in Mathematics.Mark Balaguer - 1998 - Oxford University Press.
    In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument (...)
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  • Pursuit of Truth.Gary Ebbs - 1994 - Philosophical Review 103 (3):535.
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  • Science Without Numbers.Michael D. Resnik - 1983 - Noûs 17 (3):514-519.
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  • Philosophy of Logic.Hilary Putnam - 1971 - London: Allen & Unwin.
    First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.
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  • Mathematical Explanation in Science.Alan Baker - 2009 - British Journal for the Philosophy of Science 60 (3):611-633.
    Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss (...)
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  • Pursuit of Truth.W. V. O. Quine - 1990 - Philosophy 65 (253):384-385.
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  • Philosophy of Logic.Hilary Putnam - 1971 - Routledge.
    First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.
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  • Platonism and Anti-Platonism in Mathematics.Mark Balaguer - 1998 - Bulletin of Symbolic Logic 8 (4):516-518.
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  • Science-Driven Mathematical Explanation.Alan Baker - 2012 - Mind 121 (482):243-267.
    Philosophers of mathematics have become increasingly interested in the explanatory role of mathematics in empirical science, in the context of new versions of the Quinean ‘Indispensability Argument’ which employ inference to the best explanation for the existence of abstract mathematical objects. However, little attention has been paid to analysing the nature of the explanatory relation involved in these mathematical explanations in science (MES). In this paper, I attack the only articulated account of MES in the literature (an account sketched by (...)
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  • Are There Genuine Mathematical Explanations of Physical Phenomena?Alan Baker - 2005 - Mind 114 (454):223-238.
    Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...)
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  • Parsimony and Inference to the Best Mathematical Explanation.Alan Baker - 2016 - Synthese 193 (2).
    Indispensability-based arguments for mathematical platonism are typically motivated by drawing an analogy between abstract mathematical objects and concrete scientific posits. In this paper, I argue that mathematics can sometimes help to reduce our concrete ontological, ideological, and structural commitments. My focus is on optimization explanations, and in particular the case study involving periodical cicadas. I argue that in this case, stronger mathematical apparatus yields explanations that have fewer concrete commitments. The nominalist cannot accept these more parsimonious explanations without embracing the (...)
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  • Towards a Nominalization of Quantum Mechanics.Mark Balaguer - 1996 - Mind 105 (418):209-226.
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  • Comments on “Parsimony and Inference to the Best Mathematical Explanation”.Fabrice Pataut - 2016 - Synthese 193 (2):351-363.
    The author of “Parsimony and inference to the best mathematical explanation” argues for platonism by way of an enhanced indispensability argument based on an inference to yet better mathematical optimization explanations in the natural sciences. Since such explanations yield beneficial trade-offs between stronger mathematical existential claims and fewer concrete ontological commitments than those involved in merely good mathematical explanations, one must countenance the mathematical objects that play a theoretical role in them via an application of the relevant mathematical results. The (...)
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  • A Theory of Mathematical Correctness and Mathematical Truth.Mark Balaguer - 2001 - Pacific Philosophical Quarterly 82 (2):87–114.
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  • A Cause for Concern: Standard Abstracta and Causation.Jody Azzouni - 2008 - Philosophia Mathematica 16 (3):397-401.
    Benjamin Callard has recently suggested that causation between Platonic objects—standardly understood as atemporal and non-spatial—and spatio-temporal objects is not ‘a priori’ unintelligible. He considers the reasons some have given for its purported unintelligibility: apparent impossibility of energy transference, absence of physical contact, etc. He suggests that these considerations fail to rule out a priori Platonic-object causation. However, he has overlooked one important issue. Platonic objects must causally affect different objects differently, and different Platonic objects must causally affect the same objects (...)
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  • Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl.Gottlob Frege - 1996 - Wittgenstein-Studien 3 (2):993-999.
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  • Abstract Objects: A Case Study.Stephen Yablo - 2002 - Philosophical Issues 12 (1):220-240.
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  • Abstract Objects: A Case Study.Stephen Yablo - 2002 - Noûs 36 (s1):220 - 240.
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  • A Dilemma for Benacerraf’s Dilemma?Andrea Sereni - 2016 - In Fabrice Pataut (ed.), Truth, Objects, Infinity. Springer Verlag.
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  • Mathematics and Reality.Mary Leng - 2011 - Bulletin of Symbolic Logic 17 (2):267-268.
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  • Fictionalism in the Philosophy of Mathematics.Mark Balaguer - 2008 - Stanford Encyclopedia of Philosophy.
    Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...)
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  • Platonism in Metaphysics.Mark Balaguer - 2016 - Stanford Encyclopedia of Philosophy 1 (1):1.
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  • Indispensability and Explanation: An Overview and Introduction.Daniele Molinini, Fabrice Pataut & Andrea Sereni - 2016 - Synthese 193 (2):317-332.
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  • On What There Is.P. T. Geach, A. J. Ayer & W. V. Quine - 1951 - Aristotelian Society Supplementary Volume 25:125-160.
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  • What is Cantor's Continuum Problem?Kurt Gödel - 1947 - In Solomon Feferman, John Dawson & Stephen Kleene (eds.), Journal of Symbolic Logic. Oxford University Press. pp. 176--187.
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  • Die Grundlagen der Arithmetik.Gottlob Frege - 1988 - Felix Meiner Verlag.
    Die "Grundlagen" gehören zu den klassischen Texten der Sprachphilosophie, Logik und Mathematik. Frege stützt sein Programm einer Begründung von Arithmetik und Analysis auf reine Logik, indem er die natürlichen Zahlen als bestimmte Begriffsumfänge definiert. Die philosophische Fundierung des Fregeschen Ansatzes bilden erkenntnistheoretische und sprachphilosophische Analysen und Begriffserklärungen.
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