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Summary Ontology of mathematics is concerned with the existence and nature of objects that mathematics is about. An important phenomenon in the field is the need of balancing between epistemological and ontological challenges. For instance, prima facie, the ontologically simplest option is to postulate the existence of abstract mathematical objects (like numbers or sets) to which mathematical terms refer. Yet, explaining how we, mundane beings, can have knowledge of such aspatial and atemporal objects, turns out to be quite difficult. The ontologically parsimonious alternative is to deny the existence of such objects. But then, one has to explain what it is that makes mathematical theories true (or at least, correct) and how we can come to know mathematical facts. Various positions arise from various ways of addressing questions of these two sorts. 
Key works Many crucial papers are included in the following anthologies: Benacerraf & Putnam 1964, Hart 1996 and Shapiro 2005.
Introductions A good introductory survey is Horsten 2008. A readable introduction to philosophy of mathematics is Shapiro 2000. A nice, albeit somewhat biased survey of ontological options can be found in the first few chapters of Chihara 1990. A very nice introduction to the development of foundations of mathematics and the interaction between foundations, epistemology and ontology of mathematics is Giaquinto 2002.
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  1. Does Set Theory Really Ground Arithmetic Truth?Alfredo Roque Freire - manuscript
    We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth (even in a relative sense). Our method is to emphasize the incomplete picture of both theories and treat models as their syntactical counterparts. Insisting on the incomplete picture will allow us to argue in favor of the revisability of the standard model interpretation. We then show that it is hopeless to expect that the (...)
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  2. Forms of Structuralism: Bourbaki and the Philosophers.Jean-Pierre Marquis - forthcoming - In Alberto Peruzzi & Silvano Zipoli Caiani (eds.), Structures Meres, Semantics, Mathematics, and Cognitive Science. New York, NY, USA:
    In this paper, we argue that, contrary to the view held by most philosophers of mathematics, Bourbaki’s technical conception of mathematical structuralism is relevant to philosophy of mathematics. In fact, we believe that Bourbaki has captured the core of any mathematical structuralism.
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  3. The Ontological Import of Adding Proper Classes.Alfredo Roque Freire & Rodrigo de Alvarenga Freire - 2019 - Manuscrito 42 (2):85-112.
    In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to NBG. We thus consider the standard techniques of reducing one system to the other. Novak proved that from a model of ZF we can build a model of NBG (and vice versa), while Shoenfield have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF of (...)
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  4. Mathematical Models of Abstract Systems: Knowing Abstract Geometric Forms.Jean-Pierre Marquis - 2013 - Annales de la Faculté des Sciences de Toulouse 22 (5):969-1016.
    Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy theory. (...)
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  5. Constructibility and Mathematical Existence.Michael D. Resnik - 1992 - Journal of Philosophy 89 (12):648.
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  6. Language and Other Abstract Objects. J. J. Katz.Mike Dillinger - 1984 - Philosophy of Science 51 (1):175-176.
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  7. Applying Mathematics to Empirical Sciences: Flashback to a Puzzling Disciplinary Interaction.Raphaël Sandoz - 2018 - Synthese 195 (2):875-898.
    This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three (...)
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  8. Access Problems and Explanatory Overkill.Silvia Jonas - 2017 - Philosophical Studies 174 (11):2731-2742.
    I argue that recent attempts to deflect Access Problems for realism about a priori domains such as mathematics, logic, morality, and modality using arguments from evolution result in two kinds of explanatory overkill: the Access Problem is eliminated for contentious domains, and realist belief becomes viciously immune to arguments from dispensability, and to non-rebutting counter-arguments more generally.
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  9. Mathematical Spandrels.Alan Baker - 2017 - Australasian Journal of Philosophy 95 (4):779-793.
    The aim of this paper is to open a new front in the debate between platonism and nominalism by arguing that the degree of explanatory entanglement of mathematics in science is much more extensive than has been hitherto acknowledged. Even standard examples, such as the prime life cycles of periodical cicadas, involve a penumbra of mathematical features whose presence can only be explained using relatively sophisticated mathematics. I introduce the term ‘mathematical spandrel’ to describe these penumbral properties, and focus on (...)
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  10. Deflating Cold War Rationality.Michael Pettit - 2016 - Studies in History and Philosophy of Science Part A 58:46-49.
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  11. ¿ES LA MATEMÁTICA LA NOMOGONÍA DE LA CONCIENCIA? REFLEXIONES ACERCA DEL ORIGEN DE LA CONCIENCIA Y EL PLATONISMO MATEMÁTICO DE ROGER PENROSE / Is Mathematics the “nomogony” of Consciousness? Reflections on the origin of consciousness and mathematical Platonism of Roger Penrose.Miguel Acosta - 2016 - Naturaleza y Libertad. Revista de Estudios Interdisciplinares 7:15-39.
    Al final de su libro “La conciencia inexplicada”, Juan Arana señala que la nomología, explicación según las leyes de la naturaleza, requiere de una nomogonía, una consideración del origen de las leyes. Es decir, que el orden que observamos en el mundo natural requiere una instancia previa que ponga ese orden específico. Sabemos que desde la revolución científica la mejor manera de explicar dicha nomología ha sido mediante las matemáticas. Sin embargo, en las últimas décadas se han presentado algunas propuestas (...)
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  12. What is the Nature of Mathematical–Logical Objects?Stathis Livadas - 2017 - Axiomathes 27 (1):79-112.
    This article deals with a question of a most general, comprehensive and profound content as it is the nature of mathematical–logical objects insofar as these are considered objects of knowledge and more specifically objects of formal mathematical theories. As objects of formal theories they are dealt with in the sense they have acquired primarily from the beginnings of the systematic study of mathematical foundations in connection with logic dating from the works of G. Cantor and G. Frege in the last (...)
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  13. Beyond Quantities and Qualities: Frege and Jevons on Measurement.Raphaël Sandoz - 2016 - Hopos: The Journal of the International Society for the History of Philosophy of Science 6 (2):212-238.
    On which philosophical foundations is the attribution of numerical magnitudes to qualitative phenomena based? That is, what is the philosophical basis for attributing, through measurement operations, numbers to empirical qualities that our senses perceive in the outside world? This question, nowadays rarely addressed in such a way, actually refers to an old debate about the quantification of qualities. A historical analysis reveals that it was a major issue in the “context of discovery” of the first attempts to mathematize new fields (...)
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  14. Science Without Numbers: A Defense of Nominalism. Hartry H. Field.Michael Friedman - 1981 - Philosophy of Science 48 (3):505-506.
  15. La naturaleza del conocimiento matemático. Crítica a un libro de Philip Kitcher.Francisco Miró Quesada - 1987 - Critica 19 (57):109-136.
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  16. Early Modern Mathematical Practice in the Round. [REVIEW]Richard J. Oosterhoff - 2012 - Studies in History and Philosophy of Science Part A 43 (1):224-227.
  17. Foundation of Mathematics Between Theory and Practice.Giorgio Venturi - 2014 - Philosophia Scientiae 18 (1):45-80.
    Je me propose dans cet article de traiter de la théorie des ensembles, non seulement comme fondement des mathématiques au sens traditionnel, mais aussi comme fondement de la pratique mathématique. De ce point de vue, je marque une distinction entre un fondement ensembliste standard, d'une nature ontologique, grâce auquel tout objet mathématique peut trouver un succédané ensembliste, et un fondement pratique, qui vise à expliquer les phénomènes mathématiques, en donnant des conditions nécessaires et suffisantes pour prouver les propositions mathématiques. Je (...)
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  18. Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
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  19. An Anti-Realist Account of the Application of Mathematics.Otávio Bueno - 2016 - Philosophical Studies 173 (10):2591-2604.
    Mathematical concepts play at least three roles in the application of mathematics: an inferential role, a representational role, and an expressive role. In this paper, I argue that, despite what has often been alleged, platonists do not fully accommodate these features of the application of mathematics. At best, platonism provides partial ways of handling the issues. I then sketch an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features of the application (...)
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  20. Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences. [REVIEW]Alexander George - 1996 - Philosophical Review 105 (1):89.
    One effect of W. V. Quine’s assault on the analytic-synthetic distinction is pressure on the boundaries between mathematics and empirical science. Assumptions about reference and knowledge that are natural in the context of the empirical sciences have been exported to the case of mathematics. Problems then arise when we ask how, given the abstractness of mathematical entities, we can refer to them or know anything about them. For if abstractness entails causal impotence, and if reference and knowledge require causal intercourse, (...)
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  21. Realism in Mathematics.Joan Weiner - 1993 - Philosophical Review 102 (2):281.
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  22. Constructibility and Mathematical Existence.John P. Burgess - 1992 - Philosophical Review 101 (4):916.
  23. Mathematics Without Numbers: Towards a Modal-Structural Interpretation.Bob Hale & Geoffrey Hellman - 1992 - Philosophical Review 101 (4):919.
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  24. Maudlin's Mathematical Maneuver: A Case Study in the Metaphysical Implications of Mathematical Representations.Robbie Hirsch - 2017 - Philosophy and Phenomenological Research 94 (1):170-210.
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  25. The Foundations of Arithmetic. A Logico-Mathematical Enquiry Into the Concept of Number.E. N. - 1951 - Journal of Philosophy 48 (10):342.
  26. IV—Mathematical Tennis.J. R. Lucas - 1985 - Proceedings of the Aristotelian Society 85 (1):63-72.
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  27. Zvi Artstein. Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics. Translated by Alan Hercberg. 426 Pp., Illus., Figs., Tables, Bibl., Index. Amherst, N.Y.: Prometheus Books, 2014. $26. [REVIEW]Kevin Kuhl - 2015 - Isis 106 (4):889-890.
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  28. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. Paul Hoffman.Judith V. Grabiner - 2000 - Isis 91 (4):804-805.
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  29. A Challenge to Divine Psychologism.Greg Fried - 2016 - Theology and Science 14 (2):175-189.
    Alvin Plantinga proposes that mathematical objects and propositions are divine thoughts. This position, which I call divine psychologism, resonates with some remarks by contemporary thinkers. Plantinga claims several advantages for his position, and I add another: it helps to explain the glory of mathematics. But my main purpose is to issue a challenge to divine psychologism. I argue that it has an implausible consequence: it identifies an entity with God’s relation to that entity. I consider and rebut several ways in (...)
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  30. Ortega y Gasset on Georg Cantor’s Theory of Transfinite Numbers.Lior Rabi - 2016 - Kairos (15):46-70.
    Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a (...)
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  31. Towards Mathematical Philosophy.Jacek Malinowski David Makinson & Wansing Heinrich (eds.) - 2009 - Springer.
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  32. The Varieties of Indispensability Arguments.Marco Panza & Andrea Sereni - 2016 - Synthese 193 (2):469-516.
    The indispensability argument comes in many different versions that all reduce to a general valid schema. Providing a sound IA amounts to providing a full interpretation of the schema according to which all its premises are true. Hence, arguing whether IA is sound results in wondering whether the schema admits such an interpretation. We discuss in full details all the parameters on which the specification of the general schema may depend. In doing this, we consider how different versions of IA (...)
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  33. Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08.John Corcoran - 1972 - Philosophy of Science 39 (1):106-108.
    Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08. -/- Constance Reid was an insider of the Berkeley-Stanford logic circle. Her San Francisco home was in Ashbury Heights near the homes of logicians such as Dana Scott and John Corcoran. Her sister Julia Robinson was one of the top mathematical logicians of her generation, as was Julia’s husband Raphael Robinson for whom Robinson Arithmetic was named. Julia was a Tarski PhD and, in recognition of (...)
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  34. Edward Arthur Milne—The Relations of Mathematics to Science.S. Rebsdorf & H. Kragh - 2002 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 33 (1):51-64.
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  35. A Survey of Leśniewski's Logic.V. Frederick Rickey - 1977 - Studia Logica 36 (4):407-426.
  36. The Systems of Leśniewski in Relation to Contemporary Logical Research.Andrzej Grzegorczyk - 1955 - Studia Logica 3 (1):77-95.
  37. The Sensible Foundation for Mathematics: A Defense of Kant's View.Mark Risjord - 1990 - Studies in History and Philosophy of Science Part A 21 (1):123-143.
  38. What Constitutes the Numerical Diversity of Mathematical Objects?F. MacBride - 2006 - Analysis 66 (1):63-69.
  39. The Role of Representations in Mathematical Reasoning1.Jessica Carter - 2012 - Philosophia Scientae 16:55-70.
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  40. Mathematical Practice and Naturalist Epistemology: Structures with Potential for Interaction.Bart Van Kerkhove & Jean Paul Van Bendegem - 2005 - Philosophia Scientae 9:61-78.
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  41. Maddy and Mathematics: Naturalism or Not.Jeffrey W. Roland - 2007 - 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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  42. Conjoining Mathematical Empiricism with Mathematical Realism: Maddy’s Account of Set Perception Revisited.Alex Levine - 2005 - Synthese 145 (3):425-448.
    Penelope Maddy's original solution to the dilemma posed by Benacerraf in his 'Mathematical Truth' was to reconcile mathematical empiricism with mathematical realism by arguing that we can perceive realistically construed sets. Though her hypothesis has attracted considerable critical attention, much of it, in my view, misses the point. In this paper I vigorously defend Maddy's account against published criticisms, not because I think it is true, but because these criticisms have functioned to obscure a more fundamental issue that is well (...)
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  43. A Critique of Resnik’s Mathematical Realism.Timothy John Nulty - 2005 - Erkenntnis 62 (3):379-393.
    This paper attempts to motivate skepticism about the reality of mathematical objects. The aim of the paper is not to provide a general critique of mathematical realism, but to demonstrate the insufficiency of the arguments advanced by Michael Resnik. I argue that Resnik's use of the concept of immanent truth is inconsistent with the treatment of mathematical objects as ontologically and epistemically continuous with the objects posited by the natural sciences. In addition, Resnik's structuralist program, and his denial of relational (...)
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  44. Godel's Disjunction: The Scope and Limits of Mathematical Knowledge.Leon Horsten & Philip Welch (eds.) - 2016 - Oxford University Press UK.
    To what extent can we hope to find answers to all mathematical questions? A famous theorem from Gödel entails that if our thinking capacities do not go beyond what an electronic computer is capable of, then there are indeed absolutely unsolvable mathematical problems. Thus it is of capital importance to find out whether human mathematicians can outstrip computers. Within this context, the contributions to this book critically examine positions about the scope and limits of human mathematical knowledge.
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  45. Corcoran Recommends Hambourger on the Frege-Russell Number Definition.John Corcoran - 1978 - MATHEMATICAL REVIEWS 56.
    It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in each possible world, (3) (...)
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  46. Representation of Ordinal Numbers and Derived Sets in Certain Continuous Sets.Frederick Bagemihl - 1981 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 27 (19-21):333-336.
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  47. Non-Representational Mathematical Realism.Maria Jose Frapolli - 2015 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 30 (3):331-348.
    This paper is an attempt to convince anti-realists that their correct intuitions against the metaphysical inflationism derived from some versions of mathematical realism do not force them to embrace non-standard, epistemic approaches to truth and existence. It is also an attempt to convince mathematical realists that they do not need to implement their perfectly sound and judicious intuitions with the anti-intuitive developments that render full-blown mathematical realism into a view which even Gödel considered objectionable. I will argue for the following (...)
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  48. The Construction of Logical Space, by Augustin Rayo. [REVIEW]S. Berry - 2015 - Mind 124 (496):1375-1379.
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  49. Partnership and Partition: A Case Study of Mathematical Exchange.Adrian Rice - 2015 - Philosophia Scientiæ 19:115-134.
    It is now just over one hundred years since the beginning of the mathematical partnership between the Cambridge analyst G. H. Hardy and the Indian mathematical genius Srinivasa Ramanujan, one of the most celebrated collaborations in the history of mathematics. Indeed, the story of how Ramanujan was brought from India to Cambridge and feted by the British mathematical establishment now borders on legendary. But, in the context of this collection of articles, it provides an interesting case study of mathematical exchange. (...)
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  50. Mathematics Via Symmetry.Noson Yanofsky & Mark Zelcer - unknown
    We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as satisfying such symmetry places mathematics as comparable with modern views of physics and science where, over the past century, symmetry also plays a defining role. We explore the very nature of mathematics and its relationship with natural science from this perspective. This (...)
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