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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 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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  1. added 2020-05-18
    Spacetime, Ontology, and Structural Realism.Edward Slowik - 2005 - International Studies in the Philosophy of Science 19 (2):147 – 166.
    This essay explores the possibility of constructing a structural realist interpretation of spacetime theories that can resolve the ontological debate between substantivalists and relationists. Drawing on various structuralist approaches in the philosophy of mathematics, as well as on the theoretical complexities of general relativity, our investigation will reveal that a structuralist approach can be beneficial to the spacetime theorist as a means of deflating some of the ontological disputes regarding similarly structured spacetimes.
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  2. added 2020-02-12
    Ontology and the Vicious Circle Principle. [REVIEW]Mark Steiner - 1975 - Journal of Philosophy 72 (7):184-196.
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  3. added 2020-02-12
    Ontology and the Vicious Circle Principle.Tom Richards - 1975 - Philosophical Quarterly 25 (98):68-79.
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  4. added 2020-02-08
    Deflating Existential Consequence: A Case for Nominalism.Thomas Hofweber - 2007 - Philosophical Review 116 (3):465-467.
    As the title says, this is a book review of Azzouni’s book. I complain that Azzouni proposes an answer to a question, but it is unclear what question he is trying to answer.
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  5. added 2020-01-28
    Ptolemy’s Philosophy: Mathematics as a Way of Life. By Jacqueline Feke. Princeton: Princeton University Press, 2018. Pp. Xi + 234. [REVIEW]Nicholas Danne - 2020 - Metaphilosophy 51 (1):151-155.
  6. added 2019-12-19
    Einführung in die Philosophie der Mathematik.Jörg Neunhäuserer - 2019 - Wiesbaden, Deutschland: Springer Spektrum.
    Welche Art von Gegenständen untersucht die Mathematik und in welchem Sinne existieren diese Gegenstände? Warum dürfen wir die Aussagen der Mathematik zu unserem Wissen zählen und wie lassen sich diese Aussagen rechtfertigen? Eine Philosophie der Mathematik versucht solche Fragen zu beantworten. In dieser Einführung stellen wir maßgeblichen Positionen in der Philosophie der Mathematik vor und formulieren die Essenz dieser Positionen in möglichst einfachen Thesen. Der Leser erfährt, auf welche Philosophen eine Position zurückgeht und in welchem historischen Kontext diese entstand. Ausgehend (...)
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  7. added 2019-11-16
    The Problem of Fregean Equivalents.Joongol Kim - 2019 - Dialectica 73 (3):367-394.
    It would seem that some statements like ‘There are exactly four moons of Jupiter’ and ‘The number of moons of Jupiter is four’ have the same truth-conditions and yet differ in ontological commitment. One strategy to resolve this paradoxical phenomenon is to insist that the statements have not only the same truth-conditions but also the same ontological commitments; the other strategy is to reject the presumption that they have the same truth-conditions. This paper critically examines some popular versions of these (...)
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  8. added 2019-11-15
    Counterfactual Scheming.Sam Baron - 2020 - Mind 129 (514):535-562.
    Mathematics appears to play a genuine explanatory role in science. But how do mathematical explanations work? Recently, a counterfactual approach to mathematical explanation has been suggested. I argue that such a view fails to differentiate the explanatory uses of mathematics within science from the non-explanatory uses. I go on to offer a solution to this problem by combining elements of the counterfactual theory of explanation with elements of a unification theory of explanation. The result is a theory according to which (...)
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  9. added 2019-11-15
    Mathematical Explanation by Law.Sam Baron - 2019 - British Journal for the Philosophy of Science 70 (3):683-717.
    Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...)
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  10. added 2019-11-15
    How Mathematics Can Make a Difference.Sam Baron, Mark Colyvan & David Ripley - 2017 - Philosophers' Imprint 17.
    Standard approaches to counterfactuals in the philosophy of explanation are geared toward causal explanation. We show how to extend the counterfactual theory of explanation to non-causal cases, involving extra-mathematical explanation: the explanation of physical facts by mathematical facts. Using a structural equation framework, we model impossible perturbations to mathematics and the resulting differences made to physical explananda in two important cases of extra-mathematical explanation. We address some objections to our approach.
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  11. added 2019-11-11
    Classless.Sam Roberts - 2020 - Analysis 80 (1):76-83.
    Classes are a kind of collection. Typically, they are too large to be sets. For example, there are classes containing absolutely all sets even though there is no set of all sets. But what are classes, if not sets? When our theory of classes is relatively weak, this question can be avoided. In particular, it is well known that von Neuman–Bernays–Godel class theory is conservative over the standard axioms of set theory ): anything NGB can prove about the sets is (...)
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  12. added 2019-09-20
    Platonistic Physicalism Without Tears.D. G. Witmer - 2017 - Journal of Consciousness Studies 24 (9-10):72-90.
    Susan Schneider argues that the entities to be identified as part of the 'physical base' for physicalism must be in part abstract and that this fact either falsifies physicalism or renders it so problematic as to be 'no physicalism worth having'. I accept the abstractness of the entities but argue both that physicalism is consistent with such and that none of the alleged problems for Platonistic physicalism are serious.
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  13. added 2019-07-05
    Optimal Representations and the Enhanced Indispensability Argument.Manuel Barrantes - 2019 - Synthese 196 (1):247-263.
    The Enhanced Indispensability Argument appeals to the existence of Mathematical Explanations of Physical Phenomena 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 defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what I will call ‘optimal (...)
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  14. added 2019-06-07
    Nominalistic Systems. Rolf A. Eberle.Fred Wilson - 1972 - Philosophy of Science 39 (4):556-558.
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  15. added 2019-06-06
    Think About the Consequences! Nominalism and the Argument From the Philosophy of Logic.Torsten Wilholt - 2006 - Dialectica 60 (2):115-133.
    Nominalism faces the task of explaining away the ontological commitments of applied mathematical statements. This paper reviews an argument from the philosophy of logic that focuses on this task and which has been used as an objection to certain specific formulations of nominalism. The argument as it is developed in this paper aims to show that nominalism in general does not have the epistemological advantages its defendants claim it has. I distinguish between two strategies that are available to the nominalist: (...)
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  16. added 2019-06-06
    Ontological Independence as the Mark of the Real. Jody Azzouni. Deflating Existential Consequence: A Case for Nominalism. New York: Oxford University Press, 2004. Pp. Viii + 241. ISBN 0-19-515988-8. [REVIEW]Mark Colyvan - 2005 - Philosophia Mathematica 13 (2):216-225.
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  17. added 2019-06-06
    How to Nominalize Formalism &Dagger.Jody Azzouni - 2005 - 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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  18. added 2019-06-06
    Book Review: John P. Burgess and Gideon Rose. A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. [REVIEW]Stewart Shapiro - 1998 - Notre Dame Journal of Formal Logic 39 (4):600-612.
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  19. added 2019-06-06
    Constructibility and Mathematical Existence.John P. Burgess - 1992 - Philosophical Review 101 (4):916.
  20. added 2019-06-06
    Chihara, Charles S. [1990]: Constructibility and Mathematical Existence. [REVIEW]Donald Gilles - 1992 - British Journal for the Philosophy of Science 43 (2):263.
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  21. added 2019-06-05
    Number Words as Number Names.Friederike Moltmann - 2017 - Linguistics and Philosophy 40 (4):331-345.
    This paper criticizes the view that number words in argument position retain the meaning they have on an adjectival or determiner use, as argued by Hofweber :179–225, 2005) and Moltmann :499–534, 2013a, 2013b). In particular the paper re-evaluates syntactic evidence from German given in Moltmann to that effect.
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  22. added 2019-06-05
    Logical Consequence for Nominalists.Marcus Rossberg & Daniel Cohnitz - 2009 - Theoria : An International Journal for Theory, History and Fundations of Science 24 (2):147-168.
    It is often claimed that nominalistic programmes to reconstruct mathematics fail, since they will at some point involve the notion of logical consequence which is unavailable to the nominalist. In this paper we use an idea of Goodman and Quine to develop a nominalistically acceptable explication of logical consequence.
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  23. added 2019-05-28
    The Enhanced Indispensability Argument, the Circularity Problem, and the Interpretability Strategy.Jan Heylen & Lars Arthur Tump - forthcoming - Synthese:1-13.
    Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker (...)
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  24. added 2019-05-13
    John P. Burgess and Gideon Rosen, A Subject with No Object. Strategies for Nominalistic Interpretation of Mathematics, Clarendon Press, Oxford University Press, Oxford and New York1997, Xi + 259 Pp. [REVIEW]Charles Parsons - 1999 - Journal of Symbolic Logic 64 (1):391-394.
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  25. added 2019-03-04
    Quine on Naturalism, Nominalism, and Philosophy’s Place Within Science.James Andrew Smith - forthcoming - Synthese:1-19.
    W.V. Quine is a well-known proponent of naturalism, the view on which reality is described only in science. He is also well-known for arguing that our current scientific theories commit us to the existence of abstract objects. It is tempting to believe that the naturalistic philosopher should think scientists outside of philosophy are in the best position to assess the merits of revising our current commitment to abstract objects. But Quine rejects this deferential view. On the reading of Quine’s philosophical (...)
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  26. added 2019-01-28
    Ontological Realism and Sentential Form.Eileen Nutting - 2018 - Synthese 195 (11):5021-5036.
    The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: some mathematical claims are true, and the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny. Those who deny typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for. Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have (...)
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  27. 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. (...)
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  28. added 2018-10-01
    Deflationary Nominalism and Puzzle Avoidance.David Mark Kovacs - 2019 - Philosophia Mathematica 27 (1):88-104.
    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 (...)
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  29. added 2018-09-06
    Nominalism and Causal Theories of Reference.Jeffrey W. Roland - 2009 - SATS 10 (2):51-67.
  30. 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 (...)
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  31. 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 (...)
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  32. 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 (...)
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  33. 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 (...)
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  34. 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 (...)
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  35. added 2017-11-01
    Can Mathematical Objects Be Causally Efficacious?Seungbae Park - 2019 - Inquiry: An Interdisciplinary Journal of Philosophy 62 (3):247–255.
    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 (...)
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  36. 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.
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  37. added 2017-08-17
    Beyond Platonism and Nominalism?: James Franklin: An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure, Palgrave Macmillan, 2014, X + 308 Pp. [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.
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  38. 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 (...)
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  39. 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 (...)
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  40. 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 (...)
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  41. added 2017-03-14
    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 (...)
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  42. added 2017-02-10
    Scientific Realism: Between Platonism and Nominalism.Stathis Psillos - 2010 - Philosophy of Science 77 (5):947-958.
  43. 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: Studies in History and Philosophy of Biological and Biomedical Sciences 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 (...)
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  44. 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.
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  45. added 2017-01-31
    Review of J. P. Burgess and G. Rosen, A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics.Bob Hale - 1998 - British Journal for the Philosophy of Science 49 (1):161-167.
  46. added 2017-01-29
    Walter H. Burgess, John Robinson, Pastor of the Pilgrim Fathers. [REVIEW]Rendel Harris - 1920 - Hibbert Journal 19:588.
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  47. 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: Studies in History and Philosophy of Biological and Biomedical Sciences 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 (...)
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  48. added 2017-01-19
    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 (...)
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  49. added 2017-01-18
    Nominalistic Set Theory.David K. Lewis - 1970 - Noûs 4 (3):225-240.
  50. added 2017-01-17
    Nominalistic Systems.Michael Jubien - 1973 - Philosophical Review 82 (4):540.
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1 — 50 / 208