In This Article Expand or collapse the "in this article" section Ontology of Mathematics

  • Introduction
  • Textbooks and Surveys
  • Anthologies
  • Platonism
  • The Indispensability Argument
  • Nominalism and Fictionalism
  • Structuralism
  • Logicism and Neo-Logicism
  • Social Constructivism
  • Higher-Order Logic
  • Potential Infinity, Indefinite Extensibility, and Absolute Generality

Philosophy Ontology of Mathematics
Roy T. Cook
  • LAST REVIEWED: 23 March 2022
  • LAST MODIFIED: 23 March 2022
  • DOI: 10.1093/obo/9780195396577-0427


Ontology is the study of being. Thus, the study of the ontology of mathematics involves asking questions about whether special objects exist to serve as the subject matter of mathematical theories, what the nature of such objects is (if they exist), what role such objects play in making mathematical claims true (or whether and how mathematical claims are true if there are no such objects), and how we obtain knowledge regarding such objects (again, if such objects exist). As a result, it is often difficult to sharply separate questions regarding the ontology of mathematics (and its metaphysics more generally) from philosophy of mathematics more broadly—especially with regard to questions about epistemology and semantics. Nevertheless, some approaches to the philosophy of mathematics place ontological concerns more squarely at the center than others, and this article will focus on such accounts. Attitudes toward the ontology of mathematics divide into roughly two camps: nominalists and fictionalists deny that there is a special realm of objects (usually understood to be abstract objects) that serve as the subject matter of mathematics, while ontological or object realists (which includes platonists) affirm the existence of such objects. Importantly, some strands of research in the philosophy of mathematics—including both social constructivist and structuralist accounts—come in both realist and nominalist varieties. Finally, there are a number of topics that are not explicitly subdisciplines of the philosophy of mathematics—for example, the status of higher-order logic and various questions about the nature of, and our ability to grasp, infinite collections—that are nevertheless directly relevant to the ontology of mathematics.

Textbooks and Surveys

While there are no textbooks or general surveys devoted specifically to the ontology of mathematics or its metaphysics more generally, there are a number of textbooks on the philosophy of mathematics that cover much that is of relevance. Shapiro 2000 and Linnebo 2017 are extremely accessible introductions to the subject, with the former taking a somewhat historical approach to the subject and the latter focusing more on Gottlob Frege’s influence on 20th- and 21st-century work on the topic. Bostock 2009 provides a nice overview of the history of the philosophy of mathematics, while Brown 2008 and Colyvan 2012 focus on recent developments in the field. Giaquinto 2002 is somewhat more mathematically advanced and focuses on the role that the set-theoretic paradoxes and the foundational crises in real analysis played in the development of the philosophy of mathematics throughout the 20th century. Finally, Horsten 2017 is an excellent survey article covering both historical and contemporary work in the field.

  • Bostock, David. Philosophy of Mathematics: An Introduction. Oxford: Wiley-Blackwell, 2009.

    A survey of the philosophy of mathematics that focuses on its history, beginning with Plato and Aristotle, continuing with the work of John Stuart Mill and Immanuel Kant, and ending with contemporary approaches.

  • Brown, James Robert. Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. 2d ed. London: Routledge, 2008.

    An engaging and accessible textbook that focuses on topics that gained attention in the latter half of the 20th century, including picture proofs, the role of empirical evidence in mathematics, computer proofs, and work on large infinities.

  • Colyvan, Mark. An Introduction to the Philosophy of Mathematics. Cambridge Introductions to Philosophy. Cambridge, UK: Cambridge University Press, 2012.

    DOI: 10.1017/CBO9781139033107

    An excellent introduction to the philosophy of mathematics focusing on 20th-century developments, including an extensive discussion of philosophically relevant mathematical results.

  • Giaquinto, Marcus. The Search for Certainty: A Philosophical Account of Foundations of Mathematics. Oxford: Oxford University Press, 2002.

    A somewhat technically focused survey of 20th-century philosophy of mathematics that focuses on the role played by the set-theoretic paradoxes and the foundational crises that plagued mathematics in the 19th century.

  • Horsten, Leon. “Philosophy of Mathematics.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. Stanford, CA: Stanford University, 2017.

    An excellent survey article on the philosophy of mathematics, with a strong focus on the metaphysics and ontology of mathematics.

  • Linnebo, Øystein. Philosophy of Mathematics. Princeton Foundations of Contemporary Philosophy. Princeton, NJ: Princeton University Press, 2017.

    DOI: 10.2307/j.ctt216687n

    An accessible introduction to contemporary work in the philosophy of mathematics, with a special focus on ontology, realism, and platonism, logicism, and the role that Gottlob Frege’s work played in the development of the philosophy of mathematics.

  • Shapiro, Stewart. Thinking about Mathematics: The Philosophy of Mathematics. Oxford: Oxford University Press, 2000.

    An accessible introduction to the philosophy of mathematics which includes more pre-20th-century philosophy of mathematics than many other textbooks, including chapters devoted to Plato, Aristotle, Immanuel Kant, and John Stuart Mill.

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