Philosophy Abstract Objects
by
Mark Balaguer
  • LAST REVIEWED: 01 September 2022
  • LAST MODIFIED: 15 January 2019
  • DOI: 10.1093/obo/9780195396577-0384

Introduction

An abstract object is a non-physical, non-mental object that exists outside of space and time and is wholly unextended. For example, one might think that numbers are abstract objects; e.g., it is plausible to think that if the number 3 exists, then it is not a physical or mental object, and it does not exist in space and time. Likewise, one might think that properties and relations are abstract objects; e.g., it is plausible to think that if redness exists, over and above the various red balls and red houses and so on, then it is an abstract object—i.e., it is non-physical, non-mental, non-spatiotemporal, and so on. Other kinds of objects that are often taken by philosophers to be abstract objects are propositions, sentence types, possible worlds, logical objects, and fictional objects. The view the that there are abstract objects—known as platonism—is of course extremely controversial. Many philosophers think there are just no such things as abstract objects. Philosophers who endorse this antiplatonist view have to endorse some other view of objects of the above kinds—i.e., numbers, properties, propositions, etc.; in particular, in connection with each of these kinds of objects, they have to say either that these objects are physical or mental objects or that there are just no such things. There is a vast literature on the existence and nature of abstract objects. This article focuses mostly (but not entirely) on the existence question—that is, the question of whether there are any such things as abstract objects. In addition, it focuses to some extent (though, again, not entirely) on the specific version of this question that is concerned with the existence of abstract mathematical objects.

Classical Articulations of Platonism

As is clear from its name, the platonist view can be traced back to the works of Plato (e.g., Plato 1997a and Plato 1997b). In more recent times, the view has been developed in Frege 1953, Frege 1964, Frege 1968, Frege 1990, Russell 1959, and Gödel 1983. Finally, while Quine was often opposed to platonism, he was also a sometime supporter of the view, and the development of the contemporary platonist view owes much to his work, even his antiplatonist work (see, e.g., Quine 1948, cited under the One Over Many Argument, and Quine 2013, cited under Paraphrase Nominalism (Other Versions)).

  • Frege, Gottlob. The Foundations of Arithmetic. Translated by J. L. Austin. Oxford: Blackwell, 1953.

    Save Citation »Export Citation » Share Citation »

    German original, Die Grundlagen der Arithmetik (Breslau, Germany: Koebner, 1884). Classic work developing Frege’s logicist/platonist philosophy of mathematics. Also contains several classical arguments against various antiplatonistic views.

    Find this resource:

  • Frege, Gottlob. The Basic Laws of Arithmetic. Translated by Montgomery Furth. Berkeley: University of California Press, 1964.

    Save Citation »Export Citation » Share Citation »

    Another classic work developing Frege’s logicist/platonist philosophy of mathematics. The book provides a more rigorous version of the theory developed in Frege 1953 (The Foundations of Arithmetic). And once again, the book also contains some classical arguments against antiplatonistic views of mathematics. German original, Grundgesetze der Arithmetik (Hildesheim, Germany: Olms, 1893–1903).

    Find this resource:

  • Frege, Gottlob. “The Thought: A Logical Inquiry.” Reprinted in Essays on Frege. Edited by E. D. Klemke, 507–535. Urbana: University of Illinois Press, 1968.

    Save Citation »Export Citation » Share Citation »

    Classic paper developing Frege’s platonistic view of propositions and propositional attitudes. Originally published in 1919.

    Find this resource:

  • Frege, Gottlob. “On Sense and Reference.” In The Philosophy of Language. Edited by A. P. Martinich. Translated by H. Feigl. Oxford: Oxford University Press, 1990.

    Save Citation »Export Citation » Share Citation »

    Classic paper developing Frege’s view that words and sentences have two semantic values—sense and reference—as well as Frege’s view of propositions and propositional attitudes. Originally published in 1892 as “Über Sinn und Bedeutung.”

    Find this resource:

  • Gödel, Kurt. “What Is Cantor’s Continuum Problem?” In Philosophy of Mathematics. 2d ed. Edited by Paul Benacerraf and Hilary Putnam, 470–485. Cambridge, UK: Cambridge University Press, 1983.

    Save Citation »Export Citation » Share Citation »

    An earlier version of this paper was published in 1947; this later, expanded version of the paper (originally published in 1964) contains a supplement that provides a suggestion for how platonists might develop an epistemology of abstract mathematical objects.

    Find this resource:

  • Plato. The Parmenides. In Plato: Complete Works. Edited by J. M. Cooper, 359–397. Indianapolis, IN: Hackett, 1997a.

    Save Citation »Export Citation » Share Citation »

    Classic ancient dialogue that provides a critical examination of Plato’s theory of the forms; this is relevant to the development of the platonistic view of abstract objects.

    Find this resource:

  • Plato. The Phaedo. In Plato: Complete Works. Edited by J. M. Cooper, 49–100. Indianapolis, IN: Hackett, 1997b.

    Save Citation »Export Citation » Share Citation »

    Another classic dialogue relevant to the development of the platonistic view of abstract objects; includes a discussion of Plato’s recollection-based theory of knowledge, which is relevant to Plato’s response to the epistemological objection to platonism.

    Find this resource:

  • Russell, Bertrand. The Problems of Philosophy. Oxford: Oxford University Press, 1959.

    Save Citation »Export Citation » Share Citation »

    Classic book developing the platonistic view of properties and relations. Originally published in 1912 and available in many editions.

    Find this resource:

Varieties of Platonism

There are many different varieties of platonism. Two that deserve mention are plenitudinous platonism and structuralism.

Plenitudinous Platonism

Plenitudinous platonism is (a bit roughly) the view that there exist abstract objects of all possible kinds, or the view that all the abstract objects that possibly could exist actually do exist. Articulations of this view can be found in Balaguer 1995, Balaguer 1998a, and Linsky and Zalta 1995. Beall 1999 argues for a version of platonism with an even more robust ontology than plenitudinous platonism. Discussions of Balaguer’s version of plenitudinous platonism can be found in Colyvan and Zalta 1999 and in Liston 2003–2004 (cited under Psychologism). An objection to Balaguer’s articulation of plenitudinous platonism can be found in Restall 2003.

  • Balaguer, Mark. “A Platonist Epistemology.” Synthese 103.3 (1995): 303–325.

    DOI: 10.1007/BF01089731Save Citation »Export Citation » Share Citation »

    Develops the plenitudinous platonist view. Also uses the plenitudinous view to provide an account of how human beings could acquire knowledge of abstract objects, thus providing a response to Benacerraf’s epistemological argument against platonism.

    Find this resource:

  • Balaguer, Mark. Platonism and Anti-platonism in Mathematics. New York: Oxford University Press, 1998a.

    Save Citation »Export Citation » Share Citation »

    Develops and defends plenitudinous platonism; also develops a nominalist (fictionalist) view of mathematics and argues that there’s no fact of the matter between the fictionalist and plenitudinous platonist views of mathematics.

    Find this resource:

  • Beall, J. C. “From Full-Blooded Platonism to Really Full-Blooded Platonism.” Philosophia Mathematica 7.3 (1999): 322–325.

    DOI: 10.1093/philmat/7.3.322Save Citation »Export Citation » Share Citation »

    Argues that if Balaguer’s version of plenitudinous platonism (namely, full-blooded platonism) solves the Benacerrafian epistemoogical problem, then another view does too, namely, really full-blooded platonism, which entails the existence of the objects of inconsistent mathematical theories as well as those of consistent mathematical theories.

    Find this resource:

  • Colyvan, M., and E. Zalta. “Mathematics: Truth and Fiction?” Philosophia Mathematica 7.3 (1999): 336–349.

    DOI: 10.1093/philmat/7.3.336Save Citation »Export Citation » Share Citation »

    Book review of Balaguer 1998a (Platonism and Anti-platonism in Mathematics). Discusses and critiques various aspects of Balaguer’s version of plenitudinous platonism.

    Find this resource:

  • Linsky, Bernard, and Edward Zalta. “Naturalized Platonism and Platonized Naturalism.” Journal of Philosophy 92.10 (1995): 525–555.

    DOI: 10.2307/2940786Save Citation »Export Citation » Share Citation »

    Develops the plenitudinous platonist view. Also uses the plenitudinous platonist view to provide an epistemology of abstract objects.

    Find this resource:

  • Restall, G. “Just What Is Full-blooded Platonism?” Philosophia Mathematica 11.1 (2003): 82–91.

    DOI: 10.1093/philmat/11.1.82Save Citation »Export Citation » Share Citation »

    Argues that Balaguer’s attempts to characterize plenitudinous platonism fail; in particular, according to Restall, the various characterizations are all either too strong or too weak.

    Find this resource:

Structuralism

Structuralism is usually articulated as a view of abstract mathematical objects in particular. A bit roughly, it is the view that abstract mathematical objects are positions in structures—where a structure is something like a pattern, or an “objectless template” (i.e., a system of positions that can be “filled” by any system of objects that exhibits the given structure). On some versions of this view, mathematical objects do not have any properties aside from those that they have in virtue of occupying the places that they occupy in the given structure. Articulations of the structuralist view of mathematics can be found in Resnik 1997 and Shapiro 1997.

  • Resnik, Michael. Mathematics as a Science of Patterns. Oxford: Oxford University Press, 1997.

    Save Citation »Export Citation » Share Citation »

    Book-length defense of a platonistic version of mathematical structuralism, a view that contends that mathematics is a science of patterns, and patterns are abstract objects.

    Find this resource:

  • Shapiro, Stewart. Philosophy of Mathematics: Structure and Ontology. New York: Oxford University Press, 1997.

    Save Citation »Export Citation » Share Citation »

    Book-length treatment and defense of the structuralistic/platonistic philosophy of mathematics.

    Find this resource:

The Aristotelian View of Abstract Objects

In addition to the platonistic view of abstract objects, there is also an Aristotelian view that holds that abstract objects exist but that they are not wholly non-spatiotemporal. The most prominent version of this view is the Aristotelian view of properties, which holds that, e.g., redness exists in red things (and so in space and time). This view goes back to Aristotle (see, e.g., Aristotle 1984), although scholars might dispute the claim that Aristotle endorsed the view defined in the preceding sentence. A recent articulation of the Aristotelian view of properties and relations can be found in Armstrong 1978. Aristotle also (arguably) held a view like this of mathematical objects, i.e., a view that takes mathematical objects to be something like abstract objects that exist in space and time (see, again, Aristotle 1984). A recent view of mathematics that is at least roughly Aristotelian can be found in Maddy 1990.

  • Aristotle. Metaphysics. In The Complete Works of Aristotle. Vol. 2. Edited by Jonathan Barnes. Princeton, NJ: Princeton University Press, 1984.

    Save Citation »Export Citation » Share Citation »

    Classic ancient work that investigates the question: “What is fundamentally real?”

    Find this resource:

  • Armstrong, David. A Theory of Universals. Cambridge, UK: Cambridge University Press, 1978.

    Save Citation »Export Citation » Share Citation »

    Book-length argument for the existence of Aristotelian properties and relations.

    Find this resource:

  • Maddy, Penelope. Realism in Mathematics. Oxford: Oxford University Press, 1990.

    Save Citation »Export Citation » Share Citation »

    Develops and defends a philosophy of mathematics that takes sets to be something like abstract objects that exist in space and time. The view has physicalistic leanings, but in the end, it is best interpreted as endorsing the existence of abstract objects, since it takes sets to be non-physical and non-mental in obvious ways.

    Find this resource:

The One Over Many Argument

One of the oldest and most famous arguments for the existence of properties and relations (i.e., for the existence of abstract objects) is the One Over Many argument. While this argument is often thought of as an argument for platonism, it is really just an argument for the existence of properties and relations (it does not apply to other kinds of objects that one might think are abstract, e.g., numbers, and it can be combined with antiplatonistic views of the nature of properties and relations). There are many ways to articulate the One Over Many argument, but one fairly standard way is as follows: “Red roses and red houses and red balls all resemble one another; therefore, they have something in common; but what they have in common is clearly a property, namely, redness; therefore, redness exists.” This argument goes all the way back to Plato (see, e.g., Plato 1997a, cited under Classical Articulations of Platonism). It has been discussed by numerous philosophers since then; perhaps the most notable recent articulation of the argument can be found in Armstrong 1978 (cited under the Aristotelian View of Abstract Objects). The famous “third-man response” to the One Over Many can be found in Aristotle 1984 (cited under the Aristotelian View of Abstract Objects). Another famous response to the One Over Many can be found in Quine 1948. And the Quinean response is spelled out in more detail in Devitt 1980.

The Singular-Term Argument for Platonism

Perhaps the most important argument for platonism—i.e., for the existence of abstract objects—is the singular term argument. The argument can be put like this: “We should endorse platonism because there are sentences that are obviously true that make reference to abstract objects—and, hence, that could not be true unless abstract objects existed.” The first clear articulation of arguments of this general kind are due to Frege (see, e.g., Frege 1953, Frege 1968, and Frege 1990 [all cited under Classical Articulations of Platonism]). Versions of the singular term argument have been put forward in support of platonistic views of many different kinds of objects—e.g., mathematical objects, propositions, sentence types, and possible worlds.

The Mathematical Object Version of the Singular Term Argument

This version of the singular term argument is based on the claim that our mathematical theories are best interpreted as being about abstract objects. Put differently, the argument proceeds from the claim that sentences like “3 is prime” are obviously true to the conclusion that 3 exists (and that it could only be an abstract object). An argument of this kind can be found in Frege 1953 (cited under Classical Articulations of Platonism). Other advocates of the argument include Putnam 2011 (cited under Formulations of the Indispensability Argument), Steiner 1975, Wright 1983, and Hale 1987.

  • Hale, Robert. Abstract Objects. Oxford: Basil Blackwell, 1987.

    Save Citation »Export Citation » Share Citation »

    Develops and defends the platonistic view that there are abstract objects, especially abstract mathematical objects. The version of platonism developed is neo-Fregean and neologicist.

    Find this resource:

  • Steiner, Mark. Mathematical Knowledge. Ithaca, NY: Cornell University Press, 1975.

    Save Citation »Export Citation » Share Citation »

    Book-length defense of the platonistic philosophy of mathematics.

    Find this resource:

  • Wright, Crispin. Frege’s Conception of Numbers as Objects. Aberdeen, Scotland: Aberdeen University Press, 1983.

    Save Citation »Export Citation » Share Citation »

    Develops and defends a neo-Fregean, neologicist, platonistic philosophy of mathematics.

    Find this resource:

The Proposition Version of the Singular Term Argument

This version of the singular term argument is based on the claim that certain kinds of sentences containing that-clauses—e.g., belief ascriptions like “Jane believes that snow is white”—are best interpreted as making claims about propositions (e.g., the proposition that-snow-is-white). The central claim is that since some of these sentences are true, it follows that the propositions that they talk about must exist (and that they could only be abstract objects). The first articulation of this argument is due to Frege (see Frege 1968 and Frege 1990 [cited under Classical Articulations of Platonism]). Other advocates of arguments of this general kind include Church 1950, Bealer 1993, Salmon 1986, and Schiffer 1994. One response to the argument can be found in Balaguer 1998b.

  • Balaguer, Mark. “Attitudes without Propositions.” Philosophy and Phenomenological Research 58.4 (1998b): 805–826.

    DOI: 10.2307/2653723Save Citation »Export Citation » Share Citation »

    Develops and defends an antiplatonistic (in particular, fictionalistic) view of belief ascriptions like “Jane believes that snow is white.”

    Find this resource:

  • Bealer, George. “Universals.” Journal of Philosophy 60.1 (1993): 5–32.

    DOI: 10.2307/2940824Save Citation »Export Citation » Share Citation »

    Gives an argument from intentional logic for the existence of propositions, where propositions are taken to be abstract objects.

    Find this resource:

  • Church, Alonzo. “On Carnap’s Analysis of Statements of Assertion and Belief.” Analysis 10.5 (1950): 97–99.

    DOI: 10.1093/analys/10.5.97Save Citation »Export Citation » Share Citation »

    Develops Church’s famous translation-test argument against certain kinds of antiplatonist views of belief ascriptions; the paper can be seen as motivating a platonistic view of these sentences.

    Find this resource:

  • Salmon, Nathan. Frege’s Puzzle. Cambridge, MA: MIT Press, 1986.

    Save Citation »Export Citation » Share Citation »

    Develops and defends the direct reference theory of sentences containing proper names and argues against Fregean views of these sentences; in the process, the book argues that “that” clauses refer to singular (or Russellian) propositions.

    Find this resource:

  • Schiffer, Stephen. “A Paradox of Meaning.” Nous 28.3 (1994): 279–324.

    DOI: 10.2307/2216061Save Citation »Export Citation » Share Citation »

    Develops an argument for the claim that “that” clauses (in sentences like belief ascriptions) refer to propositions; also develops a deflationary view of the nature of propositions and argues against the hypothesis that every natural language has a compositional semantics.

    Find this resource:

The Sentence-Type Version of the Singular Term Argument

This version of the singular term argument is based on the claim that our best linguistic theories are best interpreted as being about sentence types (and that sentence types could only be abstract objects). An argument of this kind can be found in Katz 1981. See also Soames 1985.

  • Katz, Jerrold. Language and Other Abstract Objects. Totowa, NJ: Rowman & Littlefield, 1981.

    Save Citation »Export Citation » Share Citation »

    Develops the platonistic view that linguistics is a science that gives us theories of sentence types, where sentence types are abstract objects.

    Find this resource:

  • Soames, Scott. “Semantics and Psychology.” In Philosophy of Linguistics. Edited by Jerry Katz, 204–226. Oxford: Oxford University Press, 1985.

    Save Citation »Export Citation » Share Citation »

    Argues against psychologistic views of semantic theories and in favor of a platonistic view of these theories.

    Find this resource:

The Possible-Worlds Version of the Singular Term Argument

This version of the singular term argument is based on the claim that ordinary modal claims (e.g., “There are two ways the election could turn out”) are best interpreted as being about possible worlds (and that possible worlds are abstract objects). Plantinga 1974, Adams 1974, Stalnaker 1976, and van Inwagen 1986 can be seen as defending this argument in one way or another.

The Fictional-Object Version of the Singular Term Argument

This version of the singular term argument is based on the claim that ordinary claims about fictional characters and objects (e.g., “Sherlock Holmes is a detective”) are best interpreted as true claims about abstract objects of a certain kind (namely, fictional objects, like Sherlock Holmes). Arguments in this general vicinity can be found in van Inwagen 1977, Wolterstorff 1980, and Zalta 1988. Salmon 1998 and Thomasson 1999 also take fictional objects to be abstract, but their views are a bit different; they maintain that abstract fictional objects are created by humans.

  • Salmon, Nathan. “Nonexistence.” Noûs 32.3 (1998): 277–319.

    DOI: 10.1111/0029-4624.00101Save Citation »Export Citation » Share Citation »

    Develops a sort of neo-Meinongian view of sentences containing vacuous singular terms; also develops a nonstandard platonist view of fictional characters.

    Find this resource:

  • Thomasson, Amie. Fiction and Metaphysics. Cambridge, UK: Cambridge University Press, 1999.

    Save Citation »Export Citation » Share Citation »

    Develops an artifactual theory of fictional objects and characters according to which fictional objects and characters are abstract objects but also artifacts, created by human beings.

    Find this resource:

  • van Inwagen, Peter. “Creatures of Fiction.” American Philosophical Quarterly 14.4 (1977): 299–308.

    Save Citation »Export Citation » Share Citation »

    Argues for the view that there are fictional characters (and that they are abstract objects) by arguing that there are true sentences about these objects.

    Find this resource:

  • Wolterstorff, Nicholas. Works and Worlds of Art. Oxford: Clarendon, 1980.

    Save Citation »Export Citation » Share Citation »

    Book-length development of the view that art is an action performed by the artist, who is an agent. Also develops the view that fictional objects are best thought of as abstract objects.

    Find this resource:

  • Zalta, Edward N. Intensional Logic and the Metaphysics of Intentionality. Cambridge, MA: MIT Press, 1988.

    Save Citation »Export Citation » Share Citation »

    Book-length treatment of the problems that arise in connection with intentional logic; develops an axiomatized theory of abstract objects.

    Find this resource:

The Indispensability Argument for Mathematical Platonism

One of the most important arguments for mathematical platonism—i.e., for the existence of abstract mathematical objects—is the indispensability argument (sometimes called “the Quine-Putnam Argument”). A very simple version of this argument proceeds as follows: (1) our empirical scientific theories are true; and (2) these theories make indispensable reference to mathematical objects like numbers and sets; therefore, (3) mathematical objects like numbers and sets exist. The Quine-Putnam argument can be thought of as a version of the Singular Term argument. But it has received so much attention in recent years that it deserves special attention.

Formulations of the Indispensability Argument

The indispensability argument is usually attributed to Quine and Putnam. One formulation of the argument can be found in Putnam 2011 (originally published in 1971). People often cite Quine 1948 (cited under the One Over Many Argument) and Quine 1951 in this regard, but neither contains a clear formulation of the argument. The argument receives more thorough treatments in Resnik 1997 (cited under Structuralism) and Colyvan 2001. A somewhat nonstandard version of the argument can be found in Azzouni 2009. Finally, while Maddy has often argued against the indispensability argument, her work contains interesting formulations of the argument (see, e.g., Maddy 1992, Maddy 2007).

The Hard-Road Response to the Indispensability Argument

Broadly speaking, antiplatonists have offered two different responses to the indispensability argument. The first one, the hard-road response, is based on the claim that mathematics is, in fact, not indispensable to empirical science; more specifically, the idea here is that our empirical theories can be nominalized—that is, reformulated so that they do not contain any reference to, or quantification over, mathematical objects. The hard-road response was originally formulated in Field 1980. Field 1989 develops and defends the view further. Objections to Field’s view are given in Malament 1982, Shapiro 1983, and Resnik 1985. A response to one of Malament’s objections is given in Balaguer 1996a, and a response to Balaguer’s argument is given in Bueno 2003. Some considerations in favor of a Field-style hard-road response are given in Arntzenius and Dorr 2012.

The Easy-Road Response to the Indispensability Argument

The second main response to the indispensability argument is the easy-road response. This response is based on the idea that even if mathematics is indispensable to empirical science, it does not matter because we can account for these applications of our mathematical theories without supposing that mathematical objects exist. This view is developed in Sober 1993, Balaguer 1996b, Balaguer 1998a (cited under Plenitudinous Platonism), Maddy 1997, Mortensen 1998, Melia 2000, Rosen 2001, Yablo 2002, Azzouni 2004 (cited under Deflationary-Truth Nominalism), Bueno 2009 (cited under Fictionalism), and Leng 2010.

  • Balaguer, Mark. “A Fictionalist Account of the Indispensable Applications of Mathematics.” Philosophical Studies 83.3 (1996b): 291–314.

    DOI: 10.1007/BF00364610Save Citation »Export Citation » Share Citation »

    Early statement of the easy-road response to the indispensability argument; the view is based on the idea that mathematical objects would be causally inert if they existed, and that mathematics plays a merely descriptive or representational role in empirical science.

    Find this resource:

  • Leng, Mary. Mathematics and Reality. Oxford: Oxford University Press, 2010.

    DOI: 10.1093/acprof:oso/9780199280797.001.0001Save Citation »Export Citation » Share Citation »

    Sustained book-length defense of mathematical fictionalism against a number of different objections. Among many other things, it develops a version of the easy-road response to the indispensability argument.

    Find this resource:

  • Maddy, Penelope. Naturalism in Mathematics. Oxford: Oxford University Press, 1997.

    Save Citation »Export Citation » Share Citation »

    This is a book-length articulation and defense of Maddy’s naturalistic view of mathematics.

    Find this resource:

  • Melia, Joseph. “Weaseling Away the Indispensability Argument.” Mind 109.435 (2000): 455–479.

    DOI: 10.1093/mind/109.435.455Save Citation »Export Citation » Share Citation »

    Develops a version of the easy-road response to the indispensability argument; argues that we can assert our platonistic empirical theories and then simply take back the platonistic/mathematical consequences of our scientific assertions.

    Find this resource:

  • Mortensen, Chris. “On the Possibility of Science without Numbers.” Australasian Journal of Philosophy 76.2 (1998): 182–197.

    DOI: 10.1080/00048409812348341Save Citation »Export Citation » Share Citation »

    Provides a critique of Field’s nominalization program and argues that it is unnecessary.

    Find this resource:

  • Rosen, Gideon. “Nominalism, Naturalism, Epistemic Relativism.” Noûs 35.s15 (2001): 60–91.

    DOI: 10.1111/0029-4624.35.s15.4Save Citation »Export Citation » Share Citation »

    Develops a version of the easy-road response to the indispensability argument; argues that fictionalism is epistemically permissible because another community of scientists could accept the very same theories that we do while (rationally) endorsing a fictionalistic attitude toward the mathematical components of their theories.

    Find this resource:

  • Sober, Elliott. “Mathematics and Indispensability.” The Philosophical Review 102.1 (1993): 35–57.

    DOI: 10.2307/2185652Save Citation »Export Citation » Share Citation »

    Develops a sort of easy-road response to the indispensability argument.

    Find this resource:

  • Yablo, Stephen. “Go Figure: A Path through Fictionalism.” Midwest Studies in Philosophy 25.1 (2002): 72–102.

    DOI: 10.1111/1475-4975.00040Save Citation »Export Citation » Share Citation »

    Develops a roughly fictionalist view according to which mathematical claims are, at least very often, analogous to claims that contain figurative speech; responds to several objections to fictionalism, including the indispensability argument.

    Find this resource:

The Explanatory Version of the Indispensability Argument

Colyvan 2002 and Baker 2005 respond to the easy-road response to the indispensability argument by arguing that mathematics does not just play a descriptive role in science. It also plays an explanatory role. And noticing this, they think, leads us to a better and more powerful version of the indispensability argument. Responses to this explanatory version of the indispensability argument are given in Melia 2002, Leng 2005, Leng 2012, and Bangu 2008. Responses to some of these objections are given in Baker 2009 and Colyvan 2010. And a response to Colyvan 2010 is given in Yablo 2012.

The Epistemological Argument against Platonism

Probably the most important argument against platonism is the epistemological argument. This argument is usually thought of as an argument against the platonist view of mathematical objects, but exactly analogous arguments can be constructed against other kinds of abstract objects. The argument is based on the claim that since abstract objects would exist outside of space and time, if they existed, we would have no way of knowing anything about them. Philosophers have known since Antiquity that there is an epistemological worry about platonism. The contemporary version of the argument was originally formulated in Benacerraf 1973. An alternative version is articulated in Field 1989 (cited under the Hard-Road Response to the Indispensability Argument). Platonists can respond to the epistemological challenge to their view by either (a) arguing that human beings can have epistemic access to (or information-transferring contact with) abstract objects, or (b) arguing that human beings can acquire knowledge of abstract objects despite the fact that they have no substantive access to, or contact with, these objects. Gödel seems to have endorsed a contact-based view, and it is arguable that Plato and Frege did as well (see Frege 1953, Frege 1964, Frege 1968, Frege 1990, Plato 1997a, and Plato 1997b, all cited under Classical Articulations of Platonism). A more recent (and more naturalistic) contact-based epistemology can be found in Maddy 1990 (cited under the Aristotelian View of Abstract Objects. No-contact responses to the epistemological argument have been developed in Steiner 1975, Wright 1983, Hale 1987 (all cited under the Mathematical Object Version of the Singular Term Argument), Balaguer 1995, Balaguer 1998a, Linsky and Zalta 1995 (all cited under Plenitudinous Platonism), Resnik 1997, Shapiro 1997 (both cited under Structuralism), Katz 1998, and Linnebo 2006.

The Multiple-Reductions Argument against Platonism

Another prominent argument against platonism is the multiple-reductions argument. The argument is formulated in Benacerraf 1965. The most prominent response to this argument is based on the acceptance of structuralism. The structuralist view and the corresponding response to the multiple-reductions argument are developed in Resnik 1981, Resnik 1997 (cited under Structuralism), Shapiro 1989, Shapiro 1997 (cited under Structuralism), and Parsons 1990. For other responses to the multiple-reductions argument, see Linsky and Zalta 1995 (cited under Plenitudinous Platonism), Maddy 1990 (cited under the Aristotelian View of Abstract Objects), and Balaguer 1998a (cited under Plenitudinous Platonism). As was the case with the epistemological argument, the multiple-reductions argument is usually discussed in connection with abstract mathematical objects. But, again, analogous arguments can be constructed against platonistic views of other kinds of objects. Balaguer 2011 can be seen as developing a version of the argument directed against propositions.

Possible Alternatives to Platonism

There are many alternatives to platonism. In the philosophy of mathematics, there are six different views that are anti-platonistic—i.e., that avoid committing to the existence of abstract mathematical objects—namely, Fictionalism, Paraphrase Nominalism (If-Thenist Version), and Paraphrase Nominalism (Other Versions), Deflationary-Truth Nominalism, Physicalism, Psychologism, and Meinongianism. Similar views can be developed in connection with other kinds of abstract objects; but the following discussion focuses on antiplatonist works in the philosophy of mathematics.

Fictionalism

Mathematical Fictionalism is the view that (a) our mathematical sentences and theories do purport to be about abstract mathematical objects, as platonists suggest, but (b) there are no such things as abstract objects, and so (c) our mathematical theories are not true. Thus, the idea here is that sentences like “4 is even” are false, or untrue, for the same reason that, say, “Santa Claus lives at the North Pole” is false or untrue—because just as there is no such person as Santa Claus, so too there is no such thing as the number 4. The fictionalist view was originally formulated in Field 1980 (cited under the Hard-Road Response to the Indispensability Argument), and it was further developed (in various different ways) in Field 1989 (cited under the Hard-Road Response to the Indispensability Argument), Balaguer 1996b (cited under the Easy-Road Response to the Indispensability Argument), Balaguer 1998a (cited under Plenitudinous Platonism), and Rosen 2001, Yablo 2002, and Leng 2010 (all cited under the Easy-Road Response to the Indispensability Argument) as well as Bueno 2009.

  • Bueno, Otávio. “Mathematical Fictionalism.” In New Waves in Philosophy of Mathematics. Edited by Otávio Bueno and Øystein Linnebo, 59–79. Basingstoke, UK: Palgrave Macmillan, 2009.

    DOI: 10.1057/9780230245198_4Save Citation »Export Citation » Share Citation »

    Develops a version of mathematical fictionalism that contains some deflationary-truth nominalist threads.

    Find this resource:

Paraphrase Nominalism (If-Thenist Version)

Paraphrase nominalism is the view that ordinary mathematical sentences such as “3 is prime” should not be read at face value—or more specifically, that they should not be read as being of the form “Fa” and making claims about mathematical objects. There are several different versions of paraphrase nominalism. Each different version provides a different view of the logical forms of mathematical sentences. The most widely discussed version of paraphrase nominalism is if-thenism. If-thenism is the view that sentences like “3 is prime” should be interpreted as expressing conditional claims of some kind, e.g., “If there had been numbers, then 3 would have been prime.” If-thenist views are developed in Putnam 1967a, Putnam 1967b, Horgan 1984, Hellman 1989, Dorr 2008, and Yablo 2017.

  • Dorr, Cian. “There Are No Abstract Objects.” In Contemporary Debates in Metaphysics. Edited by Theodore Sider, John Hawthorne, and Dean Zimmerman, 12–64. Oxford: Blackwell, 2008.

    Save Citation »Export Citation » Share Citation »

    Develops an if-thenist view and provides a response to the indispensability argument.

    Find this resource:

  • Hellman, Geoffrey. Mathematics without Numbers. Oxford: Clarendon, 1989.

    Save Citation »Export Citation » Share Citation »

    Book-length development and defense of an if-thenist view of mathematics. The view developed is also important because it is an antiplatonist version of structuralism.

    Find this resource:

  • Horgan, Terence. “Science Nominalized.” Philosophy of Science 51.4 (1984): 529–549.

    DOI: 10.1086/289204Save Citation »Export Citation » Share Citation »

    Provides an if-thenist reading of mathematics and science that delivers an easy nominalization of science—one that does not require the vindication of Field’s program.

    Find this resource:

  • Putnam, Hilary. “Mathematics without Foundations.” Journal of Philosophy 64.1 (1967a): 5–22.

    DOI: 10.2307/2024603Save Citation »Export Citation » Share Citation »

    Important early work developing the if-thenist view of mathematics.

    Find this resource:

  • Putnam, Hilary. “The Thesis That Mathematics Is Logic.” In Bertrand Russell, Philosopher of the Century: Essays in His Honour. Edited by Ralph Schoenman, 273–303. London: Allen and Unwin, 1967b.

    Save Citation »Export Citation » Share Citation »

    Another early work developing the if-thenist view.

    Find this resource:

  • Yablo, Stephen. “If-Thenism.” Australasian Philosophical Review 1.2 (2017): 115–132.

    DOI: 10.1080/24740500.2017.1346423Save Citation »Export Citation » Share Citation »

    Defends an if-thenist view of certain kinds of claims that seem to have ontological commitments to controversial objects.

    Find this resource:

Paraphrase Nominalism (Other Versions)

Other versions of paraphrase nominalism, aside from if-thenism, are developed in Chihara 1990, Chihara 2004, Yi 2002, Hofweber 2005, Rayo 2008, Rayo 2013, and Moltmann 2013. It is also arguable that the view developed in Wittgenstein 1956 is of this kind, although this is controversial. Also, early discussion of the paraphrase strategy can be found in Quine 2013 (originally published in 1960), although it should be noted that Quine never endorsed the view. Finally, critiques of many paraphrase nominalist views are provided in Burgess and Rosen 1997.

  • Burgess, John, and Gideon Rosen. A Subject with No Object. New York: Oxford University Press, 1997.

    Save Citation »Export Citation » Share Citation »

    Surveys a number of different paraphrase-nominalist programs and provides arguments against these views.

    Find this resource:

  • Chihara, Charles. Constructibility and Mathematical Existence. Oxford: Oxford University Press, 1990.

    Save Citation »Export Citation » Share Citation »

    Book-length development and defense of a paraphrase nominalist view. On Chihara’s view, mathematical sentences that seem to make claims about what mathematical objects exist—for example, “There is a prime number between 2 and 4”—can be paraphrased into sentences about what it is possible for us to write down.

    Find this resource:

  • Chihara, Charles. A Structural Account of Mathematics. Oxford: Oxford University Press, 2004.

    Save Citation »Export Citation » Share Citation »

    Develops a nominalistic, structuralist philosophy of mathematics.

    Find this resource:

  • Hofweber, Thomas. “Number Determiners, Numbers, and Arithmetic,” Philosophical Review 114.2 (2005): 179–225.

    DOI: 10.1215/00318108-114-2-179Save Citation »Export Citation » Share Citation »

    Develops an adjectival view of numerals and the view that arithmetic is not about objects.

    Find this resource:

  • Moltmann, Friederike. “Reference to Numbers in Natural Language.” Philosophical Studies 162.3 (2013): 499–536.

    DOI: 10.1007/s11098-011-9779-1Save Citation »Export Citation » Share Citation »

    Develops the adjectival view of numerals as well as the view that numbers are plural properties.

    Find this resource:

  • Quine, W. V. O. Word and Object. Cambridge, MA: MIT Press, 2013.

    Save Citation »Export Citation » Share Citation »

    Originally published in 1960. This is Quine’s magnum opus; it is mostly concerned with developing his philosophy of language.

    Find this resource:

  • Rayo, Agustín. “On Specifying Truth-Conditions.” Philosophical Review 117.3 (2008): 385–443.

    DOI: 10.1215/00318108-2008-003Save Citation »Export Citation » Share Citation »

    Develops and argues in favor of the view that nothing is required of the world in order for arithmetical sentences to be true.

    Find this resource:

  • Rayo, Agustín. The Construction of Logical Space. Oxford: Oxford University Press, 2013.

    DOI: 10.1093/acprof:oso/9780199662623.001.0001Save Citation »Export Citation » Share Citation »

    Develops a novel view of metaphysical possibility and a trivialist-platonist view of mathematics based on the just-is statements that we accept—e.g., statements like “For the number of Martian moons to be 2 just is for there to be two moons.”

    Find this resource:

  • Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics. Oxford: Blackwell, 1956.

    Save Citation »Export Citation » Share Citation »

    Posthumous collection of writings on the philosophy of mathematics; develops an antiplatonist view; the view is notoriously difficult to interpret, but one way to interpret it is as a version of paraphrase nominalism.

    Find this resource:

  • Yi, Byeong-uk. Understanding the Many. New York: Routledge, 2002.

    Save Citation »Export Citation » Share Citation »

    Develops a view of pluralities, and our talk of pluralities, as well as a view of natural numbers that takes them to be properties of pluralities (e.g., being two things, being three things, and so on).

    Find this resource:

Deflationary-Truth Nominalism

Deflationary-Truth nominalism is the view that (a) as platonists and fictionalists maintain, ordinary mathematical sentences such as “3 is prime” should be read at face value—that is, as being of the form “Fa” and hence as making claims about mathematical objects, and that (b) there are no such things as mathematical objects, but (c) our mathematical sentences are still true. This view was introduced by Azzouni; see Azzouni 1994, Azzouni 2004, and Azzouni 2010. It also seems to be endorsed by Bueno; see Bueno 2005 and Bueno 2009 (cited under Fictionalism).

Physicalism

Physicalism (about mathematics) is the view that our mathematical sentences and theories provide descriptions of ordinary physical objects. An early view of this general kind can be found in Mill 2011 (originally published in 1843). Kitcher 1983 develops a view with physicalistic leanings, although, in the end, one might hesitate to interpret Kitcher as a full-blown physicalist about mathematics. Hoffman 2004 develops a nonstandard version of fictionalism that is based on the physicalist-leaning view developed in Kitcher 1983. For criticisms of Mill’s view, see Frege 1953 (cited under Classical Articulations of Platonism). For another recent view with physicalistic leanings, see Maddy 1990 (cited under the Aristotelian View of Abstract Objects).

  • Hoffman, Sarah. “Kitcher, Ideal Agents, and Fictionalism.” Philosophia Mathematica 12.1 (2004): 3–17.

    DOI: 10.1093/philmat/12.1.3Save Citation »Export Citation » Share Citation »

    Develops a nonstandard fictionalist view of mathematics; Hoffman endorses Kitcher’s physicalistic reconstruction of mathematical discourse and then endorses a fictionalistic view of this reconstruction.

    Find this resource:

  • Kitcher, Philip. The Nature of Mathematical Knowledge. Oxford: Oxford University Press, 1983.

    Save Citation »Export Citation » Share Citation »

    This book argues against the view that mathematical knowledge is a priori and in favor of the view that it is empirical. The book has strong physicalistic leanings, but one might argue that the view is really a nonstandard version of paraphrase nominalism or fictionalism.

    Find this resource:

  • Mill, John Stuart. A System of Logic, Ratiocinative and Inductive. 2 vols. Cambridge, UK: Cambridge University Press, 2011.

    Save Citation »Export Citation » Share Citation »

    Originally published in 1843 (London: Longmans, Green). Classic book developing Mill’s views on a variety of topics, including language, logic, and science. The book develops an empiricist/physicalist view of mathematics.

    Find this resource:

Psychologism

Psychologism is the view that our mathematical sentences and theories provide descriptions of mental objects—that is, things existing in our heads, presumably ideas. The most famous versions of this view—those developed in Brouwer 1913, Brouwer 1983, and Heyting 1980—were developed in conjunction with a view known as intuitionism, and, indeed, these works are more widely thought of as developing the intuitionist view than as developing the psychologistic view. A psychologistic view was also defended in Husserl 1891. For criticisms of psychologism, see Frege 1894 (and see also Frege 1953 and Frege 1964, both cited under Classical Articulations of Platonism). Recently, a few philosophers have suggested that mathematical objects are mind-dependent abstract objects; in other words, on this view, numbers are non-spatiotemporal objects that came into being because of the activities of human beings. Views of this kind are developed in Liston 2003–2004, Cole 2009, and Bueno 2009 (cited under Fictionalism).

  • Brouwer, L. E. J. “Intuitionism and Formalism.” Bulletin of the American Mathematical Society 20 (1913): 90–96.

    DOI: 10.1090/S0002-9904-1913-02440-6Save Citation »Export Citation » Share Citation »

    Classic paper developing the intuitionist view of mathematics. The view developed is also psychologistic.

    Find this resource:

  • Brouwer, L. E. J. “Consciousness, Philosophy, and Mathematics.” Reprinted in Philosophy of Mathematics. 2d ed. Edited by Paul Benacerraf and Hilary Putnam, 90–96. Cambridge, UK: Cambridge University Press, 1983.

    Save Citation »Export Citation » Share Citation »

    Another classic paper developing Brouwer’s intuitionistic/psychologistic philosophy of mathematics.

    Find this resource:

  • Cole, Julian. “Creativity, Freedom, and Authority: A New Perspective on the Metaphysics of Mathematics.” Australasian Journal of Philosophy 87.4 (2009): 589–608.

    DOI: 10.1080/00048400802598629Save Citation »Export Citation » Share Citation »

    Develops and argues for the view that mathematical objects are constructed by human activity.

    Find this resource:

  • Frege, Gottlob. “Review of E. Husserl’s Philosophie der Arithmetik.” Zeitschrift für Philosophie und philosophische Kritik 103 (1894): 313–332.

    Save Citation »Export Citation » Share Citation »

    Contains some classical objections to psychologistic views of mathematics.

    Find this resource:

  • Heyting, Arend. Intuitionism: An Introduction. 3d ed. Amsterdam: North-Holland, 1980.

    Save Citation »Export Citation » Share Citation »

    Provides a book-length development of the intuitionistic/psychologistic philosophy of mathematics. The view is extremely similar to that of Brouwer, but more of the technical details are provided here. Originally published in 1956.

    Find this resource:

  • Husserl, Edmund. Philosophie der Arithmetik. Leipzig: C. E. M. Pfeffer, 1891.

    Save Citation »Export Citation » Share Citation »

    Develops a psychologistic philosophy of mathematics. English edition, Philosophy of Arithmetic, translated by Dallas Willard (Dordrecht, The Netherlands: Kluwer, 2003).

    Find this resource:

  • Liston, Michael. “Thin- and Full-Blooded Platonism.” The Review of Modern Logic 9.3–4 (2003–2004): 129–161.

    Save Citation »Export Citation » Share Citation »

    This is a long book review of Balaguer 1998a (cited under Plenitudinous Platonism). Among other things, this essay develops the view that mathematics is about mind-dependent abstract objects.

    Find this resource:

Meinongianism

Meinongianism is the view that (a) as platonists and fictionalists maintain, ordinary mathematical sentences like “3 is prime” should be read at face value—that is, as being of the form “Fa” and hence as making claims about mathematical objects—and (b) there do not exist any mathematical objects, but (c) despite the fact that mathematical objects do not exist, they still have properties, and the properties of these nonexistent objects make our mathematical sentences true. For instance, according to Meinongianism, the number 4 does not exist, but, despite this, it is still even, and so the sentence “4 is even” is true. For a statement of Alexius Meinong’s view, see Meinong 1904. The Meinongian view of mathematics is also defended by Richard Routley and Graham Priest; see, for example, Routley 1980, Priest 2003, and Priest 2005. It is often objected that despite what they say, Meinongians are committed to the existence of mathematical objects after all, so that their view collapses into a kind of platonism; versions of this argument are developed in Quine 1948 (cited under the One Over Many Argument) and Lewis 1990.

  • Lewis, David. “Noneism or Allism?” Mind 99.393 (1990): 23–31.

    Save Citation »Export Citation » Share Citation »

    Develops a classic argument against Routley’s Meinongian view.

    Find this resource:

  • Meinong, Alexius. “Über Gegenstandstheorie.” In Untersuchungen zur Gegenstandstheorie und Psychologie. Edited by Alexius Meinong, 1–51. Leipzig: J. A. Barth, 1904.

    Save Citation »Export Citation » Share Citation »

    Develops Meinong’s famous theory of objects.

    Find this resource:

  • Priest, Graham. “Meinongianism and the Philosophy of Mathematics.” Philosophia Mathematica 11.1 (2003): 3–15.

    DOI: 10.1093/philmat/11.1.3Save Citation »Export Citation » Share Citation »

    Develops a Meinongian view of mathematics and responds to several objections to that view, including the objection that it is just platonism in disguise.

    Find this resource:

  • Priest, Graham. Towards Non-being. Oxford: Oxford University Press, 2005.

    DOI: 10.1093/0199262543.001.0001Save Citation »Export Citation » Share Citation »

    Develops an account of the semantics of intentional verbs like “believes.” The view developed in the book is Meinongian.

    Find this resource:

  • Routley, Richard. Exploring Meinong’s Jungle and Beyond. Canberra: Australian National University Research School of Social Sciences, 1980.

    Save Citation »Export Citation » Share Citation »

    Book-length treatment of Meinong’s theory of objects; contains a clear statement and defense of the Meinongian view of mathematics. Note that in later years Routley changed his name to “Richard Sylvan,” and his works are sometimes listed under that name.

    Find this resource:

back to top

Article

Up

Down