Ontology of Mathematics

Abstract

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.