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Summary Mathematical platonism is the view on which mathematical objects exist and are abstract (aspatial, atemporal and acausal) and independent of human minds and linguistic practices. According to mathematical platonism, mathematical theories are true in virtue of those objects possessing (or not) certain properties. One important challenge to platonism is explaining how biological organisms such as human beings could have knowledge of such objects. Another is to explain why mathematical theories about such objects should turn out to be applicable in sciences concerned with the physical world. 
Key works One of the most famous platonists was Frege (see e.g. Frege & Beaney 1997) and his line of thought is currently continued by neologicists (Wright 1983Hale 2001). Other famous platonists were Quine 2004 and Gödel 1947. Another group of platonists are structuralists, see the category summary for mathematical structuralism.
Introductions It's good to start with Linnebo 2009 and references therein. 
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  1. Miloš Adžić (2010). Ontological and Epistemological Dimensions of Gödel's Platonism. Theoria 53 (2):41-52.
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  2. Alan Ross Anderson (1974). What Do Symbols Symbolize?: Platonism. Philosophia Mathematica (1-2):11-29.
    The dispute between nominalists and Platonic realists has been with us for a long time — long enough to have assumed many forms. I don't want to rehearse the history of these various debates, or even to look at the matter from a historical point of view. But I would like to begin by distinguishing two quite different skirmishes in the general battle, one of which is new, and one of which is very old. We begin with the new one, (...)
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  3. Annarita Angelini (2006). « Un autre ordre du monde » : Science et mathématiques d'après les commentateurs de Proclus au Cinquecento. Revue d'Histoire des Sciences 2 (2):265-283.
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  4. Harold W. Attridge (1991). Gnostic Platonism. Proceedings of the Boston Area Colloquium of Ancient Philosophy 7 (1):1-30.
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  5. Jeremy Avigad (2010). Proof Theory. Gödel and the Metamathematical Tradition. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
  6. Jody Azzouni (2008). A Cause for Concern: Standard Abstracta and Causation. 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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  7. Alan Baker (2010). No Reservations Required? Defending Anti-Nominalism. Studia Logica 96 (2):127-139.
    In a 2005 paper, John Burgess and Gideon Rosen offer a new argument against nominalism in the philosophy of mathematics. The argument proceeds from the thesis that mathematics is part of science, and that core existence theorems in mathematics are both accepted by mathematicians and acceptable by mathematical standards. David Liggins (2007) criticizes the argument on the grounds that no adequate interpretation of “acceptable by mathematical standards” can be given which preserves the soundness of the overall argument. In this discussion (...)
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  8. M. Balaguer (forthcoming). Can We Know That Platonism is True? Philosophical Forum.
    ? Mark BALAGUER Philosophical forum 34:3-43-4, 459-475, Blackwell, 2003.
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  9. Mark Balaguer, Platonism in Metaphysics. Stanford Encyclopedia of Philosophy.
    Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and nonmental. Platonism in this sense is a contemporary view. It is obviously related to the views of Plato in important ways, but it is not entirely clear that Plato endorsed this view, as it is defined here. In order to remain neutral on this question, the (...)
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  10. Mark Balaguer (2008). Mathematical Platonism. In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. 179--204.
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  11. Mark Balaguer (2000). Reply to Dieterle. Philosophia Mathematica 8 (3):310-315.
    In this paper, I respond to an objection that Jill Dieterle has raised to two arguments in my book, Platonism and Anti-Platonism in Mathematics. Dieterle argues that because I reject the notion of metaphysical necessity, I cannot rely upon the notion of supervenience, as I in fact do in two places in the book. I argue that Dieterle is mistaken about this by showing that neither of the two supervenience theses that I endorse requires a notion of metaphysical necessity.
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  12. Mark Balaguer (1998). Non-Uniqueness as a Non-Problem. Philosophia Mathematica 6 (1):63-84.
    A response is given here to Benacerraf's (1965) non-uniqueness (or multiple-reductions) objection to mathematical platonism. It is argued that non-uniqueness is simply not a problem for platonism; more specifically, it is argued that platonists can simply embrace non-uniqueness—i.e., that one can endorse the thesis that our mathematical theories truly describe collections of abstract mathematical objects while rejecting the thesis that such theories truly describe unique collections of such objects. I also argue that part of the motivation for this stance is (...)
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  13. Mark Balaguer (1998). Platonism and Anti-Platonism in Mathematics. 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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  14. Mark Balaguer (1995). A Platonist Epistemology. Synthese 103 (3):303 - 325.
    A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about mathematical (...)
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  15. Mark Balaguer (1994). Against (Maddian) Naturalized Platonism. Philosophia Mathematica 2 (2):97-108.
    It is argued here that mathematical objects cannot be simultaneously abstract and perceptible. Thus, naturalized versions of mathematical platonism, such as the one advocated by Penelope Maddy, are unintelligble. Thus, platonists cannot respond to Benacerrafian epistemological arguments against their view vias Maddy-style naturalization. Finally, it is also argued that naturalized platonists cannot respond to this situation by abandoning abstractness (that is, platonism); they must abandon perceptibility (that is, naturalism).
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  16. Mark Augustan Balaguer (1992). Knowledge of Mathematical Objects. Dissertation, City University of New York
    This dissertation provides a refutation of the epistemological argument against mathematical platonism; that is, it provides an epistemology of abstract objects, in particular, of mathematical objects. ;After an introductory first chapter, I formulate what I argue is the strongest version of the epistemological argument against platonism. It is stronger than Paul Benacerraf's version because the only plausible way for a platonist to respond to it is to actually provide an epistemology of mathematical objects. ;In chapters three and four, I argue (...)
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  17. Mark Balaguer & J. M. Dieterle (1999). Reviews-Platonism and Anti-Platonism in Mathematics. British Journal for the Philosophy of Science 50 (4):775-780.
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  18. Platonism Balaguer (forthcoming). Balaguer, Mark,„Platonism in Metaphysics “. Stanford Encyclopedia of Philosophy.
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  19. Sorin Ioan Bangu (2008). Inference to the Best Explanation and Mathematical Realism. Synthese 160 (1):13-20.
    Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.
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  20. Sam Baron (2013). A Truthmaker Indispensability Argument. Synthese 190 (12):2413-2427.
    Recently, nominalists have made a case against the Quine–Putnam indispensability argument for mathematical Platonism by taking issue with Quine’s criterion of ontological commitment. In this paper I propose and defend an indispensability argument founded on an alternative criterion of ontological commitment: that advocated by David Armstrong. By defending such an argument I place the burden back onto the nominalist to defend her favourite criterion of ontological commitment and, furthermore, show that criterion cannot be used to formulate a plausible form of (...)
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  21. Sam Baron (2013). Can Indispensability‐Driven Platonists Be (Serious) Presentists? Theoria 79 (3):153-173.
    In this article I consider what it would take to combine a certain kind of mathematical Platonism with serious presentism. I argue that a Platonist moved to accept the existence of mathematical objects on the basis of an indispensability argument faces a significant challenge if she wishes to accept presentism. This is because, on the one hand, the indispensability argument can be reformulated as a new argument for the existence of past entities and, on the other hand, if one accepts (...)
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  22. Sam Baron (2013). Optimisation and Mathematical Explanation: Doing the Lévy Walk. Synthese (3):1-21.
    The indispensability argument seeks to establish the existence of mathematical objects. The success of the indispensability argument turns on finding cases of genuine extra-mathematical explanation (the explanation of physical facts by mathematical facts). In this paper, I identify a new case of extra-mathematical explanation, involving the search patterns of fully-aquatic marine predators. I go on to use this case to predict the prevalence of extra-mathematical explanation in science.
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  23. Jc Beall (2001). Existential Claims and Platonism. Philosophia Mathematica 9 (1):80-86.
    This paper responds to Colin Cheyne's new anti-platonist argument according to which knowledge of existential claims—claims of the form such-tmd-so exist—requires a caused connection with the given such-and-so. If his arguments succeed then nobody can know, or even justifiably believe, that acausal entities exist, in which case (standard) platonism is untenable. I argue that Cheyne's anti-platonist argument fails.
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  24. Jc Beall (1999). Prom Full Blooded Platonism to Really Full Blooded Platonism. Philosophia Mathematica 7 (3):322-325.
    Mark Balaguer argues for full blooded platonism (FBP), and argues that FBP alone can solve Benacerraf's familiar epistemic challenge. I note that if FBP really can solve Benacerraf's epistemic challenge, then FBP is not alone in its capacity so to solve; RFBP—really full blooded platonism—can do the trick just as well, where RFBP differs from FBP by allowing entities from inconsistent mathematics. I also argue briefly that there is positive reason for endorsing RFBP.
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  25. Edward G. Belaga, Halfway Up To the Mathematical Infinity I: On the Ontological & Epistemic Sustainability of Georg Cantor’s Transfinite Design.
    Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability (...)
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  26. Boran Berčić (2005). Zašto 2+2=4? Filozofska Istrazivanja 25 (4):945-961.
    The starting point of this article is the ontological question: What makes it true that2+2=4?, that is, what are the truth makers of mathematical propositions? Of course,the satisfactory theory in the philosophy of mathematics has to answer semantical question: What are mathematical propositions about? Also, epistemological question:How do we know them?, as well. Author compares five theories in the philosophy of mathematics, that is, five accounts of the nature of truth makers in mathematical discourse: fictionalism ; nominalism ; physicalism ; (...)
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  27. George Berry (1968). Logic with Platonism. Synthese 19 (1-2):215 - 249.
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  28. John Bigelow (1988). The Reality of Numbers: A Physicalist's Philosophy of Mathematics. Oxford University Press.
    Challenging the myth that mathematical objects can be defined into existence, Bigelow here employs Armstrong's metaphysical materialism to cast new light on mathematics. He identifies natural, real, and imaginary numbers and sets with specified physical properties and relations and, by so doing, draws mathematics back from its sterile, abstract exile into the midst of the physical world.
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  29. Max Black (1951). Review: Gottlob Frege, J. L. Austin, The Foundations of Arithmetic. A Logico-Mathematical Enquiry Into the Concept of Number. [REVIEW] Journal of Symbolic Logic 16 (1):67-67.
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  30. Ulrich Blau (2009). The Self in Logical-Mathematical Platonism. Mind and Matter 7 (1):37-57.
    A non-classical logic is proposed that extends classical logic and set theory as conservatively as possible with respect to three domains: the logic of natural language, the logcal foundations of mathematics, and the logical-philosophical paradoxes. A universal mechanics of consciousness connects these domains, and its best witness is the liar paradox. Its solution rests formally on a subject-object partition, mentally arising and disappearing perpetually. All deep paradoxes are paradoxes of consciousness. There are two kinds, solvable ones and unsolvable ones. The (...)
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  31. H. J. Blumenthal (1991). Platonism and Mathematics. The Classical Review 41 (01):101-.
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  32. Dougal Blyth (2000). Platonic Number in the Parmenides and Metaphysics XIII. International Journal of Philosophical Studies 8 (1):23 – 45.
    I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic (...)
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  33. Crispin Wright Bob Hale (2002). Benacerraf's Dilemma Revisited. European Journal of Philosophy 10 (1):101-129.
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  34. Izabela Bondecka-Krzykowska (2004). Strukturalizm jako alternatywa dla platonizmu w filozofii matematyki. Filozofia Nauki 1.
    The aim of this paper is to analyze structuralism as an alternative view to platonism in the philosophy of mathematics. We also try to find out if ontological and epistemological problems of platonism can be avoided by admitting the principles of structuralism. Structuralism claims that mathematical objects are merely positions in structures and have no identity or in general any important features outside these structures. Such view allows to avoid problems of the nature of numbers and other mathematical objects. But (...)
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  35. B. Borstner (2002). Mathematical Structuralism is a Kind of Platonism. Filozofski Vestnik 23 (1):7-24.
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  36. Jacques Bouveresse (2005). III *-on the Meaning of the Word 'Platonism' in the Expression 'Mathematical Platonism'. Proceedings of the Aristotelian Society 105 (1):55-79.
    The expression 'platonism in mathematics' or 'mathematical platonism' is familiar in the philosophy of mathematics at least since the use Paul Bernays made of it in his paper of 1934, 'Sur le Platonisme dans les Math?matiques'. But he was not the first to point out the similarities between the conception of the defenders of mathematical realism and the ideas of Plato. Poincar? had already stressed the 'platonistic' orientation of the mathematicians he called 'Cantorian', as opposed to those who (like himself) (...)
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  37. Jacques Bouveresse (2004). On the Meaning of the Word 'Platonism' in the Expression 'Mathematical Platonism'. Proceedings of the Aristotelian Society 105 (1):55–79.
    The expression 'platonism in mathematics' or 'mathematical platonism' is familiar in the philosophy of mathematics at least since the use Paul Bernays made of it in his paper of 1934, 'Sur le Platonisme dans les Mathématiques'. But he was not the first to point out the similarities between the conception of the defenders of mathematical realism and the ideas of Plato. Poincaré had already stressed the 'platonistic' orientation of the mathematicians he called'Cantorian', as opposed to those who (like himself) were (...)
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  38. Glenn G. Parsons And James Robert Brown (2004). Platonism, Metaphor, and Mathematics. Dialogue 43 (1):47-66.
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  39. James Robert Brown (2013). Marco Panza and Andrea Sereni. Plato's Problem: An Introduction to Mathematical Platonism. London and New York: Palgrave Macmillan, 2013. ISBN 978-0-230-36548-3 (Hbk); 978-0-230-36549-0 (Pbk); 978-1-13726147-2 (E-Book); 978-1-13729813-3 (Pdf). Pp. Xi + 306. [REVIEW] Philosophia Mathematica (1):nkt031.
  40. James Robert Brown (2012/2011). Platonism, Naturalism, and Mathematical Knowledge. Routledge.
    Mathematical explanation -- What is naturalism? -- Perception, practice, and ideal agents: Kitcher's naturalism -- Just metaphor?: Lakoff's language -- Seeing with the mind's eye: the Platonist alternative -- Semi-naturalists and reluctant realists -- A life of its own?: Maddy and mathematical autonomy.
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  41. James Robert Brown (2003). Kitcher's Mathematical Naturalism. Croatian Journal of Philosophy 3 (1):1-20.
    Recent years have seen a number of naturalist accounts of mathematics. Philip Kitcher’s version is one of the most important and influential. This paper includes a critical exposition of Kitcher’s views and a discussion of several issues including: mathematical epistemology, practice, history, the nature of applied mathematics. It argues that naturalism is an inadequate account and compares it with mathematical Platonism, to the advantage of the latter.
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  42. Bernd Buldt, Mathematical Practice and Platonism: A Phenomenological Perspective.
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  43. J. P. Burgess (2001). Platonism and Anti-Platonism in Mathematics. Philosophical Review 110 (1):79-82.
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  44. John Burnet (1919). MORE, P. E. -Platonism. [REVIEW] Mind 28:96.
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  45. C. W. V. C. W. V. (1926). TAYLOR, A. E. -Platonism and its Influence. [REVIEW] Mind 35:114.
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  46. Eva H. Cadwallader & Paul D. Eisenberg (1975). Platonism-Proper Vs. Property-Platonism. Idealistic Studies 5 (1):90-95.
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  47. Benjamin Callard (2007). The Conceivability of Platonism. 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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  48. Meyrick H. Carré (1955). Platonism and the Rise of Science. Philosophy 30 (115):333 - 343.
    The scientific developments of the sixteenth and seventeenth centuries have traditionally been associated with the revival of Platonism. The natural philosophers who invented the methods of classical physics have usually been depicted as men who repudiated the principles of Aristotle and embraced conceptions provided by the writings of Plato and his school. The characteristic feature derived from Platonism was the emphasis on mathematics and it is with the application of mathematics to experience, under specially devised conditions, that modern science arose. (...)
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  49. Jessica Carter (2004). Ontology and Mathematical Practice. Philosophia Mathematica 12 (3):244-267.
    In this paper I propose a position in the ontology of mathematics which is inspired mainly by a case study in the mathematical discipline if-theory. The main theses of this position are that mathematical objects are introduced by mathematicians and that after mathematical objects have been introduced, they exist as objectively accessible abstract objects.
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  50. Pierre Cassou-Noguès (2011). On Gödel's “Platonism”. Philosophia Scientiae 15:137-172.
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