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Summary One way to avoid epistemic challenges that mathematical platonism runs into (how can mundane human beings have knowledge of aspatial and atemporal abstract objects?) and to develop a more naturalistically acceptable account of mathematical knowledge is to deny the existence of mathematical objects. The main challenge, if you follow this path, is to make sense of mathematics, of mathematical practice and of the applicability of mathematics without reference to abstract objects.  
Key works In the twentieth century early serious attempts at constructing nominalistic foundations of mathematics are due to S.Leśniewski (see Simons 2008 for a survey, Leśniewski et al 1991 and Urbaniak 2013 for details). The second major attempt is Goodman & Quine 1947. Nominalistic literature started flourishing in 1980s. The main proposals include: Chihara 1990 (see also a later book Chihara 2004), Field 1980, Gottlieb 1980, Hellman 1989 and  Azzouni 2004. See Burgess & Rosen 1997 for further references.
Introductions A well-written, although somewhat hostile, survey of nominalistic options is Burgess & Rosen 1997. A reasoned overview of philosophical motivations of nominalism can be found in Chihara 1990 and Chihara 2004
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  1. Ken Akiba (2000). Indefiniteness of Mathematical Objects. Philosophia Mathematica 8 (1):26--46.
    The view that mathematical objects are indefinite in nature is presented and defended, hi the first section, Field's argument for fictionalism, given in response to Benacerraf's problem of identification, is closely examined, and it is contended that platonists can solve the problem equally well if they take the view that mathematical objects are indefinite. In the second section, two general arguments against the intelligibility of objectual indefiniteness are shown erroneous, hi the final section, the view is compared to mathematical structuralism, (...)
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  2. Frank Arntenius & Cian Dorr (2012). Calculus as Geometry. In Frank Arntzenius (ed.), Space, Time and Stuff. Oxford University Press.
    We attempt to extend the nominalistic project initiated in Hartry Field's Science Without Numbers to modern physical theories based in differential geometry.
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  3. J. Azzouni (2012). Taking the Easy Road Out of Dodge. Mind 121 (484):951-965.
    I defend my nominalist account of mathematics from objections that have been raised to it by Mark Colyvan.
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  4. J. Azzouni (2005). How to Nominalize Formalism. Philosophia Mathematica 13 (2):135-159.
    Formalism shares with nominalism a distaste for abstracta. But an honest exposition of the former position risks introducing abstracta as the stuff of syntax. This article describes the dangers, and offers a new escape route from platonism for the formalist. It is explained how the needed role of derivations in mathematical practice can be explained, not by a commitment to the derivations themselves, but by the commitment of the mathematician to a practice which is in accord with a theory of (...)
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  5. Jody Azzouni (2010). Talking About Nothing: Numbers, Hallucinations, and Fictions. Oxford University Press.
    Numbers -- Hallucinations -- Fictions -- Scientific languages, ontology, and truth -- Truth conditions and semantics.
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  6. Jody Azzouni (2006). Deflating Existential Consequence: A Case for Nominalism. OUP USA.
    If we must take mathematical statements to be true, must we also believe in the existence of abstract eternal invisible mathematical objects accessible only by the power of pure thought? Jody Azzouni says no, and he claims that the way to escape such commitments is to accept (as an essential part of scientific doctrine) true statements which are about objects that don't exist in any sense at all. Azzouni illustrates what the metaphysical landscape looks like once we avoid a militant (...)
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  7. Jody Azzouni (2004). Deflating Existential Commitment: A Case for Nominalism. OUP USA.
    If we must take mathematical statements to be true, must we also believe in the existence of abstract invisible mathematical objects accessible only by the power of pure thought? Jody Azzouni says no, and he claims that the way to escape such commitments is to accept (as an essential part of scientific doctrine) true statements which are about objects that don't exist in any sense at all. Azzouni illustrates what the metaphysical landscape looks like once we avoid a militant Realism (...)
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  8. Jody Azzouni (2004). The Derivation-Indicator View of Mathematical Practice. Philosophia Mathematica 12 (2):81-106.
    The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.
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  9. Jody Azzouni (1994). Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences. Cambridge University Press.
    This original and exciting study offers a completely new perspective on the philosophy of mathematics. Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similiar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special (...)
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  10. A. Baker (2003). Does the Existence of Mathematical Objects Make a Difference? Australasian Journal of Philosophy 81 (2):246 – 264.
    In this paper I examine a strategy which aims to bypass the technicalities of the indispensability debate and to offer a direct route to nominalism. The starting-point for this alternative nominalist strategy is the claim that--according to the platonist picture--the existence of mathematical objects makes no difference to the concrete, physical world. My principal goal is to show that the 'Makes No Difference' (MND) Argument does not succeed in undermining platonism. The basic reason why not is that the makes-no-difference claim (...)
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  11. 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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  12. Alan Baker (2006). Book Review: Charles S. Chihara. A Structural Account of Mathematics. [REVIEW] Notre Dame Journal of Formal Logic 47 (3):435-442.
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  13. 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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  14. 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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  15. 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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  16. 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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  17. Tomasz Bigaj (1994). Kilka uwag w sprawie nezbędności matematyki w nauce. Filozofia Nauki 3.
    This is an attempt to defend Field's nominalistic program from the criticism raised by K. Wójtowicz in his article. The author argues for the following theses: (a) that Wójtowicz uses the notion of „mathematical theory” broader than Field does it; (b) that he misinterprets the conception of the „abstract counterparts” of nominalistic statements; (c) and that his general evaluation of Field's program is based on too high methodological standards which he applies to the possible nominalistic versions of empirical theories. The (...)
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  18. George Boolos (1985). Nominalist Platonism. Philosophical Review 94 (3):327-344.
  19. Jacques Bouveresse (1992). Wittgenstein, Anti-Realism and Mathematical Propositions. Grazer Philosophische Studien 42:133-160.
    Wittgenstein is generally supposed to have abandoned in the 1930's a realistic conception of the meaning of mathematical propositions, founded on the idea of tmth-conditions which could in certain cases transcend any possibility of verification, for a realistic one, where the idea of truth-conditions is replaced by that of conditions of justification of assertability. It is argued that for Wittgenstein mathematical propositions, which are, as he says, "grammatical" propositions, have a meaning and a role which differ to a much greater (...)
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  20. 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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  21. O. Bueno (2012). An Easy Road to Nominalism. Mind 121 (484):967-982.
    In this paper, I provide an easy road to nominalism which does not rely on a Field-type nominalization strategy for mathematics (Field 1980). According to this proposal, applications of mathematics to science, and alleged mathematical explanations of physical phenomena, only emerge when suitable physical interpretations of the mathematical formalism are advanced. And since these interpretations are rarely distinguished from the mathematical formalism, the impression arises that mathematical explanations derive from the mathematical formalism alone. I correct this misimpression by pointing out, (...)
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  22. Otávio Bueno (2012). Nominalism and the Application of Mathematics. Metascience 21 (2):301-304.
    Nominalism and the application of mathematics Content Type Journal Article Category Book Review Pages 1-4 DOI 10.1007/s11016-012-9653-6 Authors Otávio Bueno, Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
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  23. Otávio Bueno (2008). Truth and Proof. Manuscrito 31 (1).
    Current versions of nominalism in the philosophy of mathematics face a significant problem to understand mathematical knowledge. They are unable to characterize mathematical knowledge as knowledge of the objects mathematical theories are taken to be about. Oswaldo Chateaubriand’s insightful reformulation of Platonism (Chateaubriand 2005) avoids this problem by advancing a broader conception of knowledge as justified truth beyond a reasonable doubt, and by introducing a suitable characterization of logical form in which the relevant mathematical facts play an important role in (...)
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  24. Otávio Bueno & Edward N. Zalta (2005). A Nominalist's Dilemma and its Solution. Philosophia Mathematica 13 (3):297-307.
    Current versions of nominalism in the philosophy of mathematics have the benefit of avoiding commitment to the existence of mathematical objects. But this comes with the cost of not taking mathematical theories literally. Jody Azzouni's Deflating Existential Consequence has recently challenged this conclusion by formulating a nominalist view that lacks this cost. In this paper, we argue that, as it stands, Azzouni's proposal does not yet succeed. It faces a dilemma to the effect that either the view is not nominalist (...)
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  25. John P. Burgess (1993). Book Reviews. [REVIEW] Philosophia Mathematica 1 (2):637-639.
  26. John P. Burgess (1992). Review: Constructibility and Mathematical Existence by Charles S. Chihara. [REVIEW] Philosophical Review 101:916-918.
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  27. John P. Burgess & Gideon A. Rosen (1997). A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford University Press.
    Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured previous (...)
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  28. Ross P. Cameron (2013). How to Be a Nominalist and a Fictional Realist. In Christy Mag Uidhir (ed.), Art and Abstract Objects. Oxford University Press. 179.
  29. Chihara Charles (2006). Burgess's ‘Scientific’ Arguments for the Existence of Mathematical Objects. Philosophia Mathematica 14 (3):318-337.
    This paper addresses John Burgess's answer to the ‘Benacerraf Problem’: How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess responds that we should look at how mathematicians come to accept: There are prime numbers greater than 1010 That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers. This paper investigates what lies behind (...)
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  30. Charles Chihara (2010). New Directions for Nominalist Philosophers of Mathematics. Synthese 176 (2):153 - 175.
    The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine's "Indispensability Argument", which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge (...)
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  31. Charles Chihara (2007). The Burgess-Rosen Critique of Nominalistic Reconstructions. Philosophia Mathematica 15 (1):54--78.
    In the final chapter of their book A Subject With No Object, John Burgess and Gideon Rosen raise the question of the value of the nominalistic reconstructions of mathematics that have been put forward in recent years, asking specifically what this body of work is good for. The authors conclude that these reconstructions are all inferior to current versions of mathematics (or science) and make no advances in science. This paper investigates the reasoning that led to such a negative appraisal, (...)
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  32. Charles Chihara (1989). Tharp's 'Myth and Mathematics'. Synthese 81 (2):153 - 165.
  33. Charles S. Chihara (2006). Burgess's `Scientific' Arguments for the Existence of Mathematical Objects. Philosophia Mathematica 14 (3):318-337.
    This paper addresses John Burgess's answer to the ‘Benacerraf Problem’: How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess responds that we should look at how mathematicians come to accept: There are prime numbers greater than 1010That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers. This paper investigates what lies behind Burgess's (...)
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  34. Charles S. Chihara (1990). Constructibility and Mathematical Existence. Oxford University Press.
    Chihara here develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. He utilizes this system in the analysis of the nature of mathematics, and discusses many recent works in the philosophy of mathematics from the viewpoint of the constructibility theory developed. This innovative analysis will appeal to mathematicians and philosophers of logic, mathematics, and science.
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  35. Charles S. Chihara (1982). A Gödelian Thesis Regarding Mathematical Objects: Do They Exist? And Can We Perceive Them? Philosophical Review 91 (2):211-227.
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  36. Charles S. Chihara (1973). Ontology and the Vicious-Circle Principle. Ithaca [N.Y.]Cornell University Press.
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  37. Charles S. Chihara (1965). On the Possibility of Completing an Infinite Process. Philosophical Review 74 (1):74-87.
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  38. Justin Clarke-Doane, Platonic Semantics.
    If anything is taken for granted in contemporary metaphysics, it is that platonism with respect to a discourse of metaphysical interest, such as fictional or mathematical discourse, affords a better account of the semantic appearances than nominalism, other things being equal. This belief is often motivated by the intuitively stronger one that the platonist can take the semantic appearances “at face-value” while the nominalist must resort to apparently ad hoc and technically problematic machinery in order to explain those appearances away. (...)
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  39. Justin Clarke-Doane, Platonic Semantics.
    If anything is taken for granted in contemporary metaphysics, it is that platonism with respect to a discourse of metaphysical interest, such as fictional or mathematical discourse, affords a better account of the semantic appearances than nominalism, other things being equal. This belief is often motivated by the intuitively stronger one that the platonist can take the semantic appearances “at face-value” while the nominalist must resort to apparently ad hoc and technically problematic machinery in order to explain those appearances away. (...)
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  40. James Henry Collin, Nominalist's Credo.
    Introduction: I lay out the broad contours of my thesis: a defence of mathematical nominalism, and nominalism more generally. I discuss the possibility of metaphysics, and the relationship of nominalism to naturalism and pragmatism. Chapter 2: I delineate an account of abstractness. I then provide counter-arguments to claims that mathematical objects make a di erence to the concrete world, and claim that mathematical objects are abstract in the sense delineated. Chapter 3: I argue that the epistemological problem with abstract objects (...)
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  41. M. Colyvan (2005). Ontological Independence as the Mark of the Real: Review of J. Azzouni, Deflating Existential Consequence: A Case for Nominalism. [REVIEW] Philosophia Mathematica 13 (2):216-225.
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  42. M. Colyvan (1999). Review of P Maddy Naturalism in Mathematics. [REVIEW] Mind 108 (No 431 (July 1999)):586-590.
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  43. Mark Colyvan, Scientific Realism and Mathematical Nominalism: A Marriage Made in Hell.
    The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in light (...)
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  44. Mark Colyvan (2005). (Book Review) Ontological Independence as the Mark of the Real. [REVIEW] Philosophia Mathematica 13 (2):216-225.
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  45. John Corcoran (1991). REVIEW OF Alfred Tarski, Collected Papers, Vols. 1-4 (1986) Edited by Steven Givant and Ralph McKenzie. [REVIEW] MATHEMATICAL REVIEWS 91 (h):01101-4.
  46. Lieven Decock (2010). Mathematical Entities. In Robrecht Vanderbeeken & Bart D'Hooghe (eds.), Worldviews, Science and Us. World Scientific. 224-241.
  47. Cian Dorr (2010). Of Numbers and Electrons. Proceedings of the Aristotelian Society 110 (2pt2):133-181.
    According to a tradition stemming from Quine and Putnam, we have the same broadly inductive reason for believing in numbers as we have for believing in electrons: certain theories that entail that there are numbers are better, qua explanations of our evidence, than any theories that do not. This paper investigates how modal theories of the form ‘Possibly, the concrete world is just as it in fact is and T’ and ‘Necessarily, if standard mathematics is true and the concrete world (...)
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  48. Cian Dorr (2008). There Are No Abstract Objects. In Theodore Sider, John Hawthorne & Dean W. Zimmerman (eds.), Contemporary Debates in Metaphysics. Blackwell.
    I explicate and defend the claim that, fundamentally speaking, there are no numbers, sets, properties or relations. The clarification consists in some remarks on the relevant sense of ‘fundamentally speaking’ and the contrasting sense of ‘superficially speaking’. The defence consists in an attempt to rebut two arguments for the existence of such entities. The first is a version of the indispensability argument, which purports to show that certain mathematical entities are required for good scientific explanations. The second is a speculative (...)
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  49. Hartry Field (1993). The Conceptual Contingency of Mathematical Objects. Mind 102 (406):285-299.
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  50. Hartry Field (1989). Realism, Mathematics & Modality. Basil Blackwell.
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