Abstract Objects

Edited by Sam Cowling (Denison University)
About this topic

Abstract entities such as numbers, propositions, and universals are usually contrasted with concrete entities such as tables, tadpoles, and quasars. While concrete entities are typically held to exist contingently, stand in causal relations, and occupy space and time, abstract entities are usually (though controversially) held to be necessary existents that are causally inert and "outside" of space and time. According to platonists, there are at least some abstract entities, but, given the peculiar metaphysical status of abstract entities, concerns arise about how we might come to know or refer to them. Further controversies surround the nature of the abstract-concrete distinction, the essential features of abstract entities, and the plausibility of nominalism, according to which there are no abstract entities.

Key works The literature on abstract entities is vast and closely connected to debates in the philosophy of mathematics, philosophy of science, philosophy of language, and epistemology. Burgess & Rosen 1997 is the leading survey of nominalist views regarding mathematical entities and includes an excellent survey of the nominalist-platonist debate. Two papers by Paul Benacerraf—Benacerraf 1973 and Benacerraf 1965—are hugely influential. Notable recent work regarding nominalist options in the philosophy of mathematics and other domains include Field 2016, Azzouni 2004, Leng 2010, and Hofweber 2016.  Guides to the specialist literature regarding specific kinds of abstract entities can be found in other entries, but some useful starting points are as follows. On possible worlds, see Divers 2002. On types, see Wetzel 2009. On propositions, see King et al 2014. On fictional entities, see Thomasson 1998.
Introductions Rosen 2008 and Szabó 2003 are useful survey articles of the issues raised by abstract entities. There are also a host of good entries in the Stanford Encyclopedia of Philosophy on topics related to abstract entities. Balaguer 2008, Linnebo 2009, and Rodriguez-Pereyra 2008 are excellent places to start. Armstrong 1989 is a leading introduction to the metaphysics of properties, which are usually held to be abstract entities. MacBride 1999 and MacBride 2003 survey fictionalist and neo-Fregean approaches to abstract mathematical entities. General introductions to metaphysics with useful sections on abstract entities include Jubien 1997 and Loux 1998. Cowling 2017 is a fairly comprehensive and accessible discussion of the metaphysics of abstract entities.
Related categories
Properties* (1,824 | 86)
Ontology of Music* (489 | 325)
Ontology of Mathematics* (2,150 | 300)
Words* (527)
The Nature of Sets* (216 | 95)
Numbers* (324)
See also:

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Material to categorize
  1. IX.—Locke's Theory of Universals.R. I. Aaron - 1933 - Proceedings of the Aristotelian Society 33 (1):173-202.
  2. Not Necessarily.Raziel Abelson - 1961 - Philosophical Review 70 (1):67-84.
  3. Universals, Explanation and Realism.Adler Jonathan - 1983 - der 16. Weltkongress Für Philosophie 2:98-105.
    If one had all the true particular statements that there are would such a collection be deficient for the purpose of science? In particular, would we still require a type of explanation that requires irreducible appeal to universals, and modalities. An argument to this conclusion is examined. In the situation envisaged, the realists needed distinctions such as between accidental and essential properties, or generalizations that are accidently true and those that are lawful and true, cannot be made. The argument then (...)
  4. Individuals and Their Masks.Belen Altuna - 2009 - Ideas Y Valores 58 (140):33-52.
  5. Frege, Boolos, and Logical Objects.David J. Anderson & Edward N. Zalta - 2004 - Journal of Philosophical Logic 33 (1):1-26.
    In this paper, the authors discuss Frege's theory of "logical objects" and the recent attempts to rehabilitate it. We show that the 'eta' relation George Boolos deployed on Frege's behalf is similar, if not identical, to the encoding mode of predication that underlies the theory of abstract objects. Whereas Boolos accepted unrestricted Comprehension for Properties and used the 'eta' relation to assert the existence of logical objects under certain highly restricted conditions, the theory of abstract objects uses unrestricted Comprehension for (...)
  6. Expressibility of Properties of Relations.Hajnal Andréka, Ivo Düntsch & István Németi - 1995 - Journal of Symbolic Logic 60 (3):970-991.
    We investigate in an algebraic setting the question of which logical languages can express the properties integral, permutational, and rigid for algebras of relations.
  7. Is Michelis a ‘Platonist’?Dimitri Z. Andriopoulos - 1972 - British Journal of Aesthetics 12 (4):395-402.
  8. The Solo Numero Paradox.István Aranyosi - 2011 - American Philosophical Quarterly 48 (4):347.
    Leibniz notoriously insisted that no two individuals differ solo numero, that is, by being primitively distinct, without differing in some property. The details of Leibniz’s own way of understanding and defending the principle –known as the principle of identity of indiscernibles (henceforth ‘the Principle’)—is a matter of much debate. However, in contemporary metaphysics an equally notorious and discussed issue relates to a case put forward by Max Black (1952) as a counter-example to any necessary and non-trivial version of the principle. (...)
  9. Revisions, and Quiddities.David Armstrong - unknown
    I used to think of the connection between a particular and a universal that it instantiates as a contingent one. Now I think that this is not quite right. This revision, as I now see it, is not a very large one. I still think that the states of affairs that unite particulars and universals are contingent beings. But the connection within states of affairs is, in a certain way, necessary.
  10. Resemblance Nominalism: A Solution to the Problem of Universals.David M. Armstrong - 2003 - Australasian Journal of Philosophy 81 (2):285-286.
  11. A Defense of Naturalism.Keith Augustine - 2001 - Dissertation, University of Maryland, College Park
    The first part of this essay discusses what naturalism in the philosophy of religion should entail for one's ontology, considers various proposed criteria for categorizing something as natural, uses an analysis of these proposed criteria to develop theoretical criteria for both the natural and nonnatural, and develops a set of criteria for identifying a potentially supernatural event in practice. The second part of the essay presents a persuasive empirical case for naturalism based on the lack of uncontroversial evidence for any (...)
  12. On Postulating Universals.Bruce Aune - 1973 - Canadian Journal of Philosophy 3 (2):285 - 294.
  13. Fisk on Capacities and Natures.Bruce Aune - 1970 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1970:83 - 87.
  14. The Nature of Universals.D. M. Azraf - 1958 - Pakistan Philosophical Journal 2 (1):60.
  15. Why Deflationary Nominalists Shouldn’T Be Agnostics.Jody Azzouni - 2015 - Philosophical Studies 172 (5):1143-1161.
    A feature of agnostic views—views that officially express ignorance about the existence of something —is that they are widely perceived to be epistemically more cautious than views that are committed to the entities in question. This is often seen as giving agnostics a debating advantage: all things being equal, fence-sitters have smaller argumentative burdens. Otávio Bueno argues in this way for what he calls “agnostic nominalism,” the view that we don’t know whether ontologically-independent Platonic objects exist. I show that agnostic (...)
  16. II—Jody Azzouni: Singular Thoughts.Jody Azzouni - 2011 - Aristotelian Society Supplementary Volume 85 (1):45-61.
  17. Abstract Objects: An Introduction to Axiomatic Metaphysics by Edward N. Zalta.John Bacon - 1986 - Critical Philosophy 3 (3):218.
  18. Masks.Krystyna Baker - 1981 - Texas Tech University Press.
  19. Between the Concretist and Abstract Concepts of Being.Bogdan Bakies - 1982 - Roczniki Filozoficzne 30 (1):91.
  20. Fictionalism in the Philosophy of Mathematics.Mark Balaguer - 2008 - Stanford Encyclopedia of Philosophy.
    Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...)
  21. Platonism in Metaphysics.Mark Balaguer - 2008 - Stanford Encyclopedia of Philosophy.
    Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and nonmental. Platonism in this sense is a contemporary view. It is obviously related to the views of Plato in important ways, but it is not entirely clear that Plato endorsed this view, as it is defined here. In order to remain neutral on this question, the (...)
  22. A Theory of Mathematical Correctness and Mathematical Truth.Mark Balaguer - 2001 - Pacific Philosophical Quarterly 82 (2):87–114.
  23. On Bernshtein's "Solution of a Mathematical Problem in the Theory of Heredity".P. Ballonoff - 1976 - Social Science Information 15 (4-5):793-795.
  24. Collapse, Plurals and Sets.Eduardo Alejandro Barrio - 2014 - Principia: An International Journal of Epistemology 18 (3):419.
  25. Realism and Nominalism Revisited.J. D. Bastable - 1955 - Philosophical Studies 5:164-165.
  26. Exemplification Postulates.Robert W. Beard - 1968 - Philosophical Studies 19 (3):33 - 37.
  27. 7. Universals, Tropes, Substance, and Events.Beards Andrew - 2008 - In Andrew Beards (ed.), Method in Metaphysics. University of Toronto Press. pp. 193-242.
  28. The Epistemology of Abstract Objects.D. A. Bell & W. D. Hart - 1979 - Aristotelian Society Supplementary Volume 53 (1):135-166.
  29. Comments on Maddy and Tymoczko.Paul Benacerraf - 1984 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1984:476 - 485.
  30. Mathematics and Poetry.Ermanno Bencivenga - 2006 - Inquiry : An Interdisciplinary Journal of Philosophy 49 (2):158 – 169.
    Since Descartes, mathematics has been dominated by a reductionist tendency, whose success would seem to promise greater certainty: the fewer basic objects mathematics can be understood as dealing with, and the fewer principles one is forced to assume about these objects, the easier it will be to establish a secure foundation for it. But this tendency has had the effect of sharply limiting the expressive power of mathematics, in a way that is made especially apparent by its disappointing applications to (...)
  31. The Natural Numbers From Frege to Hilbert.David Wells Bennett - 1961 - Dissertation, Columbia University
  32. Universals as Sortals in the Categories.Hugh H. Benson - 1988 - Pacific Philosophical Quarterly 69 (4):282-306.
  33. Frege's Hidden Nominalism.Gustav Bergmann - 1958 - Philosophical Review 67 (4):437-459.
  34. The Notion of Structure.S. Bernard Lonergan - 1996 - Method 14 (2):117-132.
  35. Solution of a Mathematical Problem in the Theory of Heredity.S. N. Bernshtein - 1976 - Social Science Information 15 (4-5):797-821.
  36. Readymades, Monochromes, Etc.: Nominalism and the Paradox of Modernism.J. M. Bernstein - 2004 - Diacritics 32 (1):83-100.
  37. Abstract.Alexander U. Bertland - 1996 - New Vico Studies 14:96-99.
  38. Nominalism and Inner Experience.Peter Bieri - 1982 - The Monist 65 (January):68-87.
  39. The Reality of Numbers.John Bigelow - 1991 - Mind 100 (2):283-287.
  40. Semantic Nominalism.John Bigelow - 1981 - Australasian Journal of Philosophy 59 (4):403 – 421.
  41. Resemblance Nominalism and Counterparts.A. Bird - 2003 - Analysis 63 (3):221-228.
  42. Otto Heyn Als Nominalist Dissertation.Katharina Biske - 1930 - Druckerei-Genossenschaft.
  43. The Theory of Semantic Categories and the Problem of the Typology of Universals.Bogdan Bjankov - 1983 - der 16. Weltkongress Für Philosophie 2:398-405.
    According to the basic idea of the theory of semantic categories the huge variety of expressions could be reduced to three basic classes, called basic semantic categories: names, statements, and functors. On this basis abstract objects or universale can be reduced also to three basic typest abstract objects-terms, abstract objects-statements, and abstract objects-operators. The so-called auxilliary signs, in particular brackets in formalized languages, fulfil a certain, structural function and, on this ground, can be numbered to the type of abstract objects-operators (...)
  44. 3 Mathematical Objects and Identity.Patricia Blanchette - 2007 - In Michael O'Rourke Corey Washington (ed.), Situating Semantics: Essays on the Philosophy of John Perry. pp. 73.
  45. Medieval Currencies : Nominalism and Art.C. D. Blanton - 2010 - In Andrew Cole & D. Vance Smith (eds.), The Legitimacy of the Middle Ages: On the Unwritten History of Theory. Duke University Press.
  46. The Nominalist Argument of the New Essays.Martha Brandt Bolton - 1996 - Leibniz Society Review 6:1-24.
  47. Mathematics and Metalogic.Daniel Bonevac - 1984 - The Monist 67 (1):56-71.
    In this paper I shall attempt to outline a nominalistic theory of mathematical truth. I call my theory nominalistic because it avoids a real (see [4]) ontological commitment to abstract entities. Traditionally, nominalists have found it difficult to justify any reference to infinite collections in mathematics. Even those who have tried to do so have typically restricted themselves to predicative and, thus, denumerable realms. I Indeed, many have linked impredicative definitions to platonism; nominalists have tended to agree with Weyl that (...)
  48. Frege's Theorem and the Peano Postulates.George Boolos - 1995 - Bulletin of Symbolic Logic 1 (3):317-326.
  49. Whence the Contradiction?George Boolos - 1993 - Aristotelian Society Supplementary Volume 67:211--233.
  50. P. 54-72.(1985)'Nominalist Platonism'.Reprinted Boolos - 1998 - Philosophical Review 94:327-44.
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