Abstract Objects

by

Mark Balaguer

• LAST REVIEWED: 02 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.

  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.

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

  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).

• 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.

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

• 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.

  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.”

• 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.

  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.

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

  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.

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

  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.

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

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