About this topic
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 S. Chihara 2003), 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 S. Chihara 2003
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199 found
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  1. added 2018-11-10
    Quine’s Intuition: Why Quine’s Early Nominalism is Naturalistic.James Andrew Smith - forthcoming - Erkenntnis:1-20.
    According to a growing consensus in the secondary literature on Quine, the judgment Quine makes in favor of the nominalism outlined in “Steps Toward a Constructive Nominalism” ) is in tension with the naturalism he later adopts. In this paper, I show the consensus view is mistaken by showing that Quine’s judgment is rooted in a naturalistic standard of clarity. Moreover, I argue that Quine late in his career is committed to accepting one plausible reading of his judgment in 1947. (...)
  2. 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 (...)
  3. added 2018-09-06
    Nominalism and Causal Theories of Reference.Jeffrey W. Roland - 2009 - SATS 10 (2):51-67.
  4. added 2018-04-03
    Existence, Mathematical Nominalism, and Meta-Ontology: An Objection to Azzouni on Criteria for Existence.Farbod Akhlaghi-Ghaffarokh - 2018 - Philosophia Mathematica 26 (2):251-265.
    Jody Azzouni argues that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use ‘exist’ that we can employ to argue for positions in ontology. I raise and defend a novel objection to Azzouni: his view has the counterintuitive consequence that the facts regarding what exists can and will change when users of the word ‘exist’ change what criteria they associate with its usage. Considering three responses, I argue Azzouni (...)
  5. added 2018-03-22
    A Nominalist's Dilemma and its Solution.Otávio Bueno & Edward N. Zalta - 2005 - Philosophia Mathematica 13 (3):294-307.
    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 (...)
  6. added 2018-02-17
    Platonism, Naturalism, and Mathematical Knowledge.James Robert Brown - 2011 - Routledge.
    This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does (...)
  7. added 2018-02-17
    Revolutionary Fictionalism: A Call to Arms.Mary Leng - 2005 - Philosophia Mathematica 13 (3):277-293.
    This paper responds to John Burgess's ‘Mathematics and _Bleak House_’. While Burgess's rejection of hermeneutic fictionalism is accepted, it is argued that his two main attacks on revolutionary fictionalism fail to meet their target. Firstly, ‘philosophical modesty’ should not prevent philosophers from questioning the truth of claims made within successful practices, provided that the utility of those practices as they stand can be explained. Secondly, Carnapian scepticism concerning the meaningfulness of _metaphysical_ existence claims has no force against a _naturalized_ version (...)
  8. added 2017-12-12
    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 (...)
  9. 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 (...)
  10. 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.
  11. 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.
  12. added 2017-08-13
    Hilbert's Program Then and Now.Richard Zach - 2007 - In Dale Jacquette (ed.), Philosophy of Logic. Amsterdam: North Holland. pp. 411–447.
    Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial (...)
  13. added 2017-08-13
    Numbers and Functions in Hilbert's Finitism.Richard Zach - 1998 - Taiwanese Journal for History and Philosophy of Science 10:33-60.
    David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so far received (...)
  14. added 2017-07-10
    Boarding Neurath's Boat: The Early Development of Quine's Naturalism.Sander Verhaegh - 2017 - Journal of the History of Philosophy 55 (2):317-342.
    W. V. Quine is arguably the intellectual father of contemporary naturalism, the idea that there is no distinctively philosophical perspective on reality. Yet, even though Quine has always been a science-minded philosopher, he did not adopt a fully naturalistic perspective until the early 1950s. In this paper, I reconstruct the genesis of Quine’s ideas on the relation between science and philosophy. Scrutinizing his unpublished papers and notebooks, I examine Quine’s development in the first decades of his career. After identifying three (...)
  15. added 2017-06-17
    Optimal Representations and the Enhanced Indispensability Argument.Manuel Barrantes - 2017 - Synthese:1-17.
    The Enhanced Indispensability Argument (EIA) appeals to the existence of Mathematical Explanations of Physical Phenomena (MEPPs) to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP —the explanation of the 13-year and 17-year life cycle of magicicadas— and argue that this case cannot be used to justify mathematical Platonism. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on (...)
  16. added 2017-06-01
    Number Words as Number Names.Friederike Moltmann - 2017 - Linguistics and Philosophy 40 (4):331-345.
    This paper critically evaluates the view according to which number words in argument position retain the meaning they have when occurring as determiners or adjectives. In particular, it argues against syntactic evidence from German given by myself in support of that view.
  17. added 2017-03-13
    A Generic Russellian Elimination of Abstract Objects.Kevin C. Klement - 2015 - Philosophia Mathematica:nkv031.
    In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in a system (...)
  18. added 2017-02-09
    Scientific Realism: Between Platonism and Nominalism.Stathis Psillos - 2010 - Philosophy of Science 77 (5):947-958.
  19. added 2017-02-07
    The Medium of Signs: Nominalism, Language and the Philosophy of Mind in the Early Thought of Dugald Stewart.M. D. Eddy - 2006 - Studies in History and Philosophy of Science Part C 37 (3):373-393.
    In 1792 Dugald Stewart published Elements of the philosophy of the human mind. In its section on abstraction he declared himself to be a nominalist. Although a few scholars have made brief reference to this position, no sustained attention has been given to the central role that it played within Stewart’s early philosophy of mind. It is therefore the purpose of this essay to unpack Stewart’s nominalism and the intellectual context that fostered it. In the first three sections I aver (...)
  20. added 2017-02-01
    Survey Article. Listening to Fictions: A Study of Fieldian Nominalism.F. MacBride - 1999 - British Journal for the Philosophy of Science 50 (3):431-455.
    One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers.
  21. added 2017-01-31
    Review. A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. JP Burgees & G Rosen.B. Hale - 1998 - British Journal for the Philosophy of Science 49 (1):161-167.
  22. added 2017-01-28
    Walter H. Burgess, John Robinson, Pastor of the Pilgrim Fathers. [REVIEW]Rendel Harris - 1920 - Hibbert Journal 19:588.
  23. added 2017-01-20
    Book Review:Nominalistic Systems Rolf A. Eberle. [REVIEW]Fred Wilson - 1972 - Philosophy of Science 39 (4):556-.
  24. added 2017-01-19
    From Weird Wonders to Stem Lineages: The Second Reclassification of the Burgess Shale Fauna.Keynyn Brysse - 2008 - Studies in History and Philosophy of Science Part C 39 (3):298-313.
    The Burgess Shale, a set of fossil beds containing the exquisitely preserved remains of marine invertebrate organisms from shortly after the Cambrian explosion, was discovered in 1909, and first brought to widespread popular attention by Stephen Jay Gould in his 1989 bestseller Wonderful life: The Burgess Shale and the nature of history. Gould contrasted the initial interpretation of these fossils, in which they were ‘shoehorned’ into modern groups, with the first major reexamination begun in the 1960s, when the creatures were (...)
  25. added 2017-01-18
    Nominalismus Und Gesellschaft.Friedel Weinert - 1986 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 17 (2):322-345.
    Ever since the so-called linguistic revolution in philosophy, the problem of universals has become the question of whether or not abstract/general terms refer. Nominalism gives a negative answer to that question. But there is, let us say, a Continental side to nominalism which this paper sets out to explore. It examines the social consequences of a nominalist approach to questions of knowledge. In particular it looks in detail at 17th century science and Merton's scientific ethos and describes the effects of (...)
  26. added 2017-01-18
    Nominalistic Set Theory.David Lewis - 1970 - Noûs 4 (3):225-240.
  27. added 2017-01-16
    Nominalistic Systems.Michael Jubien & Rolf A. Eberle - 1973 - Philosophical Review 82 (4):540.
  28. added 2017-01-15
    Rigor and Structure, by John P. Burgess. [REVIEW]Toby Meadows - 2017 - Australasian Journal of Philosophy 95 (2):397-400.
  29. added 2017-01-15
    Reply to Dr. Ellen Burgess.Kathleen Cranley Glass - 1995 - Journal of Law, Medicine and Ethics 23 (2):212-212.
  30. added 2016-12-19
    John P. Burgess , Philosophical Logic . Reviewed By. [REVIEW]Stephen McLeod - 2011 - Philosophy in Review 31 (1):4-7.
  31. added 2016-12-12
    Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences.Jody Azzouni - 2008 - Cambridge University Press.
    Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar 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 kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...)
  32. added 2016-12-08
    Neologicist Nominalism.Rafal Urbaniak - 2010 - Studia Logica 96 (2):149-173.
    The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those (...)
  33. added 2016-12-08
    No Reservations Required? Defending Anti-Nominalism.Alan Baker - 2010 - 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 (...)
  34. added 2016-10-15
    Science Nominalized?Susan C. Hale & Michael D. Resnik - 1987 - Philosophy of Science 54 (2):277-280.
    We argue that Horgan's program for nominalizing science fails, because its translation of quantitative statements destroys the inferential structures of explanations, predictions and retrodictions of nonquantitative scientific facts.
  35. added 2016-09-15
    Numerical Cognition and Mathematical Realism.Helen De Cruz - 2016 - Philosophers' Imprint 16.
    Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...)
  36. added 2016-09-09
    Carnap on Abstract and Theoretical Entities.Gregory Lavers - 2016 - In Ontology After Carnap.
    Carnap’s ‘Empiricism, Semantics, and Ontology’ (Carnap (1950a), ESO hereafter) is certainly a classic of twentieth century analytic philosophy. For decades now, most undergraduates are expected to read it at some point in their studies. Lately, it is being seen as the inspiration for a host of positions in the field of metaontology. Despite the widespread agreement on the importance of the paper, there is a lack of agreement on what Carnap attempts to do in the paper. My main aim in (...)
  37. added 2016-09-09
    Carnap, Quine, Quantification and Ontology.Gregory Lavers - 2015 - In Alessandro Torza (ed.), Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics, and Language. Springer.
    Abstract At the time of The Logical Syntax of Language (Syntax), Quine was, in his own words, a disciple of Carnap’s who read this work page by page as it issued from Ina Carnap’s typewriter. The present paper will show that there were serious problems with how Syntax dealt with ontological claims. These problems were especially pronounced when Carnap attempted to deal with higher order quantification. Carnap, at the time, viewed all talk of reference as being part of the misleading (...)
  38. added 2016-09-05
    Philosophy of Mathematics for the Masses : Extending the Scope of the Philosophy of Mathematics.Stefan Buijsman - 2016 - Dissertation, Stockholm University
    One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...)
  39. added 2016-03-03
    Accessibility of Reformulated Mathematical Content.Stefan Buijsman - 2017 - Synthese 194 (6).
    I challenge a claim that seems to be made when nominalists offer reformulations of the content of mathematical beliefs, namely that these reformulations are accessible to everyone. By doing so, I argue that these theories cannot account for the mathematical knowledge that ordinary people have. In the first part of the paper I look at reformulations that employ the concept of proof, such as those of Mary Leng and Ottavio Bueno. I argue that ordinary people don’t have many beliefs about (...)
  40. added 2015-11-30
    Number Sentences and Specificational Sentences. Reply to Moltmann.Robert Schwartzkopff - 2015 - Philosophical Studies:1-20.
    Frege proposed that sentences like ‘The number of planets is eight’ be analysed as identity statements in which the number words refer to numbers. Recently, Friederike Moltmann argued that, pace Frege, such sentences be analysed as so-called specificational sentences in which the number words have the same non-referring semantic function as the number word ‘eight’ in ‘There are eight planets’. The aim of this paper is two-fold. First, I argue that Moltmann fails to show that such sentences should be analysed (...)
  41. added 2015-10-30
    A Few Historical-Critical Glances on Mathematical Ontology Through the Hermann Weyl and Edmund Husserl Works.Giuseppe Iurato - manuscript
    From the general history of culture, with a particular attention turned towards the personal and intellectual relationships between Hermann Weyl and Edmund Husserl, it will be possible to identify certain historical-critical moments from which a philosophical reflection concerning aspects of the ontology of mathematics may be carried out. In particular, a notable epistemological relevance of group theory methods will stand out.
  42. added 2015-09-06
    The ‘Space’ at the Intersection of Platonism and Nominalism.Edward Slowik - 2015 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 46 (2):393-408.
    This essay explores the use of platonist and nominalist concepts, derived from the philosophy of mathematics and metaphysics, as a means of elucidating the debate on spacetime ontology and the spatial structures endorsed by scientific realists. Although the disputes associated with platonism and nominalism often mirror the complexities involved with substantivalism and relationism, it will be argued that a more refined three-part distinction among platonist/nominalist categories can nonetheless provide unique insights into the core assumptions that underlie spatial ontologies, but it (...)
  43. added 2015-07-21
    Dale Gottlieb, Ontological Economy: Substitutional Quantification and Mathematics. [REVIEW]Hartry Field - 1984 - Noûs 18 (1):160-165.
  44. added 2015-04-22
    Mathematics as Make-Believe: A Constructive Empiricist Account.Sarah Elizabeth Hoffman - 1999 - Dissertation, University of Alberta (Canada)
    Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics are considered and rejected. Constructive empiricism (...)
  45. added 2015-04-14
    True Nominalism: Referring Versus Coding.Jody Azzouni & Otávio Bueno - 2016 - British Journal for the Philosophy of Science 67 (3):781-816.
    One major motivation for nominalism, at least according to Hartry Field, is the desirability of intrinsic explanations: explanations that don’t invoke objects that are causally irrelevant to the phenomena being explained. There is something right about the search for such explanations. But that search must be carefully implemented. Nothing is gained if, to avoid a certain class of objects, one only introduces other objects and relations that are just as nominalistically questionable. We will argue that this is the case for (...)
  46. added 2015-04-05
    Numbers and Things: Nominalism and Constructivism in Seventeenth-Century Mathematical Philosophy.David Christopher Sepkoski - 2002 - Dissertation, University of Minnesota
    My dissertation is a reexamination of a crucial question in the history of early modern mathematical science: What was the basis for the adoption of mathematics as the primary mode of discourse for describing natural events by a large segment of the natural philosophical community during the 17 th century? In answering this question, I argue that in order to properly understand the adoption of mathematics as the 'language' of nature by early modern practitioners, it is important to examine contemporary (...)
  47. added 2015-04-05
    Jody Azzouni, Metaphysical Myths, Mathematical Practice. [REVIEW]Louis Marinoff - 1995 - Philosophy in Review 15:156-158.
  48. added 2015-04-05
    Mathematics From Several Nominalistic Points of View.Clive Ronald Taylor - 1977 - Dissertation, University of California, Berkeley
  49. added 2015-04-04
    Nominalistic Systems: The Logic and Semantics of Some Nominalistic Positions.Rolf A. Eberle - 1965 - Dissertation, University of California, Los Angeles
  50. added 2015-04-03
    Deflating Existential Consequence: A Case for Nominalism.Thomas Hofweber - 2007 - Philosophical Review 116 (3):465-467.
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