About this topic
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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372 found
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  1. added 2018-12-22
    Plato’s Philosophy of Cognition by Mathematical Modelling.Roman S. Kljujkov & Sergey Kljujkov - 2014 - Dialogue and Universalism 24 (3):110-115.
    By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a long time, (...)
  2. added 2018-11-14
    Why Do Certain States of Affairs Call Out for Explanation? A Critique of Two Horwichian Accounts.Dan Baras - 2018 - Philosophia:1-15.
    Motivated by examples, many philosophers believe that there is a significant distinction between states of affairs that are striking and therefore call for explanation and states of affairs that are not striking. This idea underlies several influential debates in metaphysics, philosophy of mathematics, normative theory, philosophy of modality, and philosophy of science but is not fully elaborated or explored. This paper aims to address this lack of clear explanation first by clarifying the epistemological issue at hand. Then it introduces an (...)
  3. added 2018-11-04
    Set-Theoretic Pluralism and the Benacerraf Problem.Justin Clarke-Doane - forthcoming - Philosophical Studies.
    Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...)
  4. added 2018-10-01
    Deflationary Nominalism and Puzzle Avoidance.David Mark Kovacs - forthcoming - Philosophia Mathematica:nky019.
    In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...)
  5. added 2018-09-21
    The Adverbial Theory of Numbers: Some Clarifications.Joongol Kim - forthcoming - Synthese:1-20.
    In a forthcoming paper in this journal, entitled “Bad company objection to Joongol Kim’s adverbial theory of numbers”, Namjoong Kim presents an ingenious Russell-style paradox based on an analogue of Kim’s definition of the number 1, and argues that Kim’s theory needs to provide a criterion of demarcation between acceptable and unacceptable definitions of adverbial entities. This paper addresses this ‘bad company’ objection and some other related issues concerning Kim’s adverbial theory by clarifying the purposes and uses of the formal (...)
  6. added 2018-09-21
    Easy Ontology Without Deflationary Metaontology.Daniel Z. Korman - forthcoming - Philosophy and Phenomenological Research.
    This is a contribution to a symposium on Amie Thomasson’s Ontology Made Easy (2015). Thomasson defends two deflationary theses: that philosophical questions about the existence of numbers, tables, properties, and other disputed entities can all easily be answered, and that there is something wrong with prolonged debates about whether such objects exist. I argue that the first thesis (properly understood) does not by itself entail the second. Rather, the case for deflationary metaontology rests largely on a controversial doctrine about the (...)
  7. added 2018-07-24
    Towards a Theory of Singular Thought About Abstract Mathematical Objects.James E. Davies - forthcoming - Synthese.
    This essay uses a mental files theory of singular thought—a theory saying that singular thought about and reference to a particular object requires possession of a mental store of information taken to be about that object—to explain how we could have such thoughts about abstract mathematical objects. After showing why we should want an explanation of this I argue that none of three main contemporary mental files theories of singular thought—acquaintance theory, semantic instrumentalism, and semantic cognitivism—can give it. I argue (...)
  8. added 2018-06-12
    Execution of the Universal Dream.Sergey Kljujkov - manuscript
    Even the ancient Greeks defined the Dream as a happy πόλις, Heraclitus - κόσμοπόλις, Socrates - ethical anthropology, Plato - Good, Hegel - absolute idea, Marx - communism... All of Humanity has made a lot of its survival experience for the realization of Dreams. Without any plan, to the touch to, only by the method of "trial and error" it aspired the Dream on unknown roads, which often stymied deadlocks. Among the many achieved results of Humanity by Plato's prompts, the (...)
  9. added 2018-06-11
    Mathematical Structural Realism.Christopher Pincock - 2011 - In Alisa Bokulich & Peter Bokulich (eds.), Scientific Structuralism. Springer Science+Business Media. pp. 67--79.
    Epistemic structural realists have argued that we are in a better epistemic position with respect to the structural claims made by our theories than the non-structural claims. Critics have objected that we cannot make the structure/non-structure distinction precise. I respond that a focus on mathematical structure leads to a clearer understanding of this debate. Unfortunately for the structural realist, however, the contribution that mathematics makes to scientific representation undermines any general confidence we might have in the structural claims made by (...)
  10. added 2018-03-22
    Mathematical Descriptions.Bernard Linsky & Edward N. Zalta - 2019 - Philosophical Studies 176 (2):473-481.
    In this paper, the authors briefly summarize how object theory uses definite descriptions to identify the denotations of the individual terms of theoretical mathematics and then further develop their object-theoretic philosophy of mathematics by showing how it has the resources to address some objections recently raised against the theory. Certain ‘canonical’ descriptions of object theory, which are guaranteed to denote, correctly identify mathematical objects for each mathematical theory T, independently of how well someone understands the descriptive condition. And to have (...)
  11. added 2018-03-19
    R. Schmit, Husserls Philosophie der Mathematik. Platonistische Und Konstruktivistische Momente in Husserls Mathematikbegriff. [REVIEW]B. Smith - 1983 - History and Philosophy of Logic 4 (2):230.
  12. added 2018-03-06
    Gödel's Argument for Cantorian Cardinality.Matthew W. Parker - 2017 - Noûs.
    On the first page of “What is Cantor's Continuum Problem?”, Gödel argues that Cantor's theory of cardinality, where a bijection implies equal number, is in some sense uniquely determined. The argument, involving a thought experiment with sets of physical objects, is initially persuasive, but recent authors have developed alternative theories of cardinality that are consistent with the standard set theory ZFC and have appealing algebraic features that Cantor's powers lack, as well as some promise for applications. Here we diagnose Gödel's (...)
  13. added 2018-02-17
    On Field’s Epistemological Argument Against Platonism.Ivan Kasa - 2010 - Studia Logica 96 (2):141-147.
    Hartry Field's formulation of an epistemological argument against platonism is only valid if knowledge is constrained by a causal clause. Contrary to recent claims (e.g. in Liggins (2006), Liggins (2010)), Field's argument therefore fails the very same criterion usually taken to discredit Benacerraf's earlier version.
  14. added 2018-02-17
    De regreso a la fuente del platonismo en la filosofía de las matemáticas: la crítica de Aristóteles a los números eidéticos.Burt C. Hopkins - 2010 - Areté. Revista de Filosofía 22 (1):27-50.
    De acuerdo con la así llamada concepción platonista de la naturalezade las entidades matemáticas, las afirmaciones matemáticas son análogas alas afirmaciones acerca de objetos físicos reales y sus relaciones, con la diferencia decisiva de que las entidades matemáticas no son ni físicas ni espaciotemporalmente individuales, y, por tanto, no son percibidas sensorialmente. El platonismo matemático es, por lo tanto, de la misma índole que el platonismo en general, el cual postula la tesis de un mundo ideal de entidades –eídē– que (...)
  15. added 2018-02-17
    Skolem's Paradox and Platonism.Carlo Cellucci - 1970 - Critica 4 (11/12):43-54.
  16. added 2017-12-13
    An Intrinsic Theory of Quantum Mechanics: Progress in Field's Nominalistic Program, Part I.Eddy Keming Chen - manuscript
    In this paper, I introduce an intrinsic account of the quantum state. This account contains three desirable features that the standard platonistic account lacks: (1) it does not refer to any abstract mathematical objects such as complex numbers, (2) it is independent of the usual arbitrary conventions in the wave function representation, and (3) it explains why the quantum state has its amplitude and phase degrees of freedom. -/- Consequently, this account extends Hartry Field’s program outlined in Science Without Numbers (...)
  17. added 2017-11-28
    The Semantic Plights of the Ante-Rem Structuralist.Bahram Assadian - 2018 - Philosophical Studies 175 (12):1-20.
    A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante- rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist (...)
  18. added 2017-11-20
    Autonomy Platonism and the Indispensability Argument. By Russell Marcus. Lanham, Md.: Lexington Books, 2015. Pp. Xii + 247. [REVIEW]Nicholas Danne - 2017 - Metaphilosophy 48 (4):591-594.
  19. added 2017-11-01
    Can Mathematical Objects Be Causally Efficacious?Seungbae Park - 2018 - Inquiry: An Interdisciplinary Journal of Philosophy:00-00.
    Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and how (...)
  20. added 2017-10-21
    Mathematics and Its Applications, A Transcendental-Idealist Perspective.Jairo Da Silva - 2017 - Springer.
    This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...)
  21. added 2017-10-05
    Rejecting Mathematical Realism While Accepting Interactive Realism.Seungbae Park - 2018 - Analysis and Metaphysics 17:7-21.
    Indispensablists contend that accepting scientific realism while rejecting mathematical realism involves a double standard. I refute this contention by developing an enhanced version of scientific realism, which I call interactive realism. It holds that interactively successful theories are typically approximately true, and that the interactive unobservable entities posited by them are likely to exist. It is immune to the pessimistic induction while mathematical realism is susceptible to it.
  22. added 2017-09-21
    Review of C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge[REVIEW]Neil Tennant - 2010 - Philosophia Mathematica 18 (3):360-367.
    This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’, one which ‘respects (...)
  23. added 2017-09-18
    The Reality of Field’s Epistemological Challenge to Platonism.David Liggins - 2018 - Erkenntnis 83 (5):1027-1031.
    In the introduction to his Realism, mathematics and modality, and in earlier papers included in that collection, Hartry Field offered an epistemological challenge to platonism in the philosophy of mathematics. Justin Clarke-Doane Truth, objects, infinity: New perspectives on the philosophy of Paul Benacerraf, 2016) argues that Field’s challenge is an illusion: it does not pose a genuine problem for platonism. My aim is to show that Clarke-Doane’s argument relies on a misunderstanding of Field’s challenge.
  24. added 2017-08-17
    Beyond Platonism and Nominalism? [REVIEW]Vassilis Livanios - 2016 - Axiomathes 26 (1):63-69.
    Review of James Franklin: An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure, Palgrave Macmillan, 2014, x + 308 pp.
  25. added 2017-07-12
    A Reliability Challenge to Theistic Platonism.Dan Baras - 2017 - Analysis 77 (3):479-487.
    Many philosophers believe that when a theory is committed to an apparently unexplainable massive correlation, that fact counts significantly against the theory. Philosophical theories that imply that we have knowledge of non-causal mind-independent facts are especially prone to this objection. Prominent examples of such theories are mathematical Platonism, robust normative realism and modal realism. It is sometimes thought that theists can easily respond to this sort of challenge and that theism therefore has an epistemic advantage over atheism. In this paper, (...)
  26. added 2017-07-07
    Two Criticisms Against Mathematical Realism.Seungbae Park - 2017 - Diametros 52:96-106.
    Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an (...)
  27. added 2017-07-04
    Two Criticisms Against Mathematical Realism.Seungbae Park - 2017 - Diametros 52:96-106.
    Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an (...)
  28. added 2017-03-20
    Introduction to Abstractionism.Philip A. Ebert & Marcus Rossberg - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford: Oxford University Press. pp. 3-33.
  29. added 2017-03-20
    Dummett’s Criticism of the Context Principle.A. Ebert Philip - 2015 - Grazer Philosophische Studien 92 (1):23-51.
  30. added 2017-03-11
    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.
  31. added 2017-03-06
    Referring to Mathematical Objects Via Definite Descriptions.Stefan Buijsman - 2017 - Philosophia Mathematica 25 (1):128-138.
    Linsky and Zalta try to explain how we can refer to mathematical objects by saying that this happens through definite descriptions which may appeal to mathematical theories. I present two issues for their account. First, there is a problem of finding appropriate pre-conditions to reference, which are currently difficult to satisfy. Second, there is a problem of ensuring the stability of the resulting reference. Slight changes in the properties ascribed to a mathematical object can result in a shift of reference (...)
  32. added 2017-03-02
    Science and Necessity.John Bigelow & Robert Pargetter - 1991 - Cambridge University Press.
    This book espouses a theory of scientific realism in which due weight is given to mathematics and logic. The authors argue that mathematics can be understood realistically if it is seen to be the study of universals, of properties and relations, of patterns and structures, the kinds of things which can be in several places at once. Taking this kind of scientific platonism as their point of departure, they show how the theory of universals can account for probability, laws of (...)
  33. added 2017-02-25
    Knowledge of Abstract Objects in Physics and Mathematics.Michael J. Shaffer - 2017 - Acta Analytica 32 (4):397-409.
    In this paper a parallel is drawn between the problem of epistemic access to abstract objects in mathematics and the problem of epistemic access to idealized systems in the physical sciences. On this basis it is argued that some recent and more traditional approaches to solving these problems are problematic.
  34. added 2017-02-15
    Ferran Sunyer I Balaguer. [REVIEW]Massimo Mazzotti - 1998 - British Journal for the History of Science 31 (1):63-102.
  35. added 2017-02-15
    Causal Contraints on Mathematical Knowledge.Michael John Pool - unknown
    Submitted to the Faculty of Graduate Studies and Research in partial fulfilment of the requirements for the degree of Master of Arts, Department of Philosophy.
  36. added 2017-02-14
    Many of the Basic Problems in the Philosophy of Mathematics Center Around the Positions Just Mentioned. It Will Not Be Possible to Dis-Cuss These Problems in Any Detail Here, but at Least Some General Indications Can Be Given. A Major Difficulty for Platonism has Been to Explain How It Is. [REVIEW]Richard Tieszen - 1995 - In Barry Smith & David Woodruff Smith (eds.), The Cambridge Companion to Husserl. Cambridge University Press. pp. 438.
  37. added 2017-02-13
    Mathematical Reality‖.J. Polkinghorne - 2011 - In J. C. Polkinghorne (ed.), Meaning in Mathematics. Oxford University Press. pp. 27--34.
  38. added 2017-02-09
    On the Metaphysical Status of Mathematical Entities.R. M. Martin - 1985 - Review of Metaphysics 39 (1):3 - 21.
  39. added 2017-02-08
    Aristotle on Mathematical Objects.Edward Hussey - 1991 - Apeiron 24 (4):105 - 133.
  40. added 2017-02-08
    Benacerraf's Dilemma.W. D. Hart - 1991 - Critica 23 (68):87 - 103.
  41. added 2017-02-08
    Mathematical Naturalism.Robert Bates Graber - 1989 - Southern Journal of Philosophy 27 (3):427-441.
  42. added 2017-02-08
    Mathematical Reality.James Byrnie Shaw - 1927 - The Monist 37 (1):113-119.
  43. added 2017-02-03
    Mathematical Entities.Peter Clark - 2009 - In Robin Le Poidevin (ed.), The Routledge Companion to Metaphysics. Routledge.
  44. added 2017-02-03
    Platonic Explanation: Or, What Abstract Entities Can Do for You.James Robert Brown - 1988 - International Studies in the Philosophy of Science 3 (1):51 – 67.
    (1988). Platonic explanation: Or, what abstract entities can do for you. International Studies in the Philosophy of Science: Vol. 3, No. 1, pp. 51-67. doi: 10.1080/02698598808573324.
  45. added 2017-02-02
    Practical Reason and Mathematical Argument.J. O'Neill - 1998 - Studies in History and Philosophy of Science Part A 29 (2):195-205.
  46. added 2017-02-01
    Mathematical Realism and Gödel's Incompleteness Theorems.Richard Tieszen - 1994 - Philosophia Mathematica 2 (3):177-201.
    In this paper I argue that it is more difficult to see how Godel's incompleteness theorems and related consistency proofs for formal systems are consistent with the views of formalists, mechanists and traditional intuitionists than it is to see how they are consistent with a particular form of mathematical realism. If the incompleteness theorems and consistency proofs are better explained by this form of realism then we can also see how there is room for skepticism about Church's Thesis and the (...)
  47. added 2017-02-01
    In Defense of a Modest Platonism.Richard E. Grandy - 1977 - Philosophical Studies 32 (4):359 - 369.
  48. added 2017-01-29
    Mathematical Practice as a Guide to Ontology: Evaluating Quinean Platonism by its Consequences for Theory Choice.Mary Leng - 2002 - Logique Et Analyse 45.
  49. added 2017-01-28
    Why and How Platonism?Guillermo Rosado Haddock - 2007 - Logic Journal of the IGPL 15 (5-6):621-636.
    Probably the best arguments for Platonism are those directed against its rival philosophies of mathematics. Frege's arguments against formalism, Gödel's arguments against constructivism and those against the so-called syntactic view of mathematics, and an argument of Hodges against Putnam are expounded, as well as some arguments of the author. A more general criticism of Quine's views follows. The paper ends with some thoughts on mathematics as a sort of Platonism of structures, as conceived by Husserl and essentially endorsed by the (...)
  50. added 2017-01-27
    Astride the Divided Line: Platonism, Empiricism, and Einstein's Epistemological Opportunism.Don Howard - 1998 - Poznan Studies in the Philosophy of the Sciences and the Humanities 63:143-164.
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