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Mathematical object

From left to right, top to bottom: A tesseract or four-dimensional hypercube, the graph of a binary function, a trefoil knot (a kind of mathematical knot), and a common hierarchy of sets of numbers.[a]

A mathematical object is an abstract concept arising in mathematics.[1] Typically, a mathematical object can be a value that can be assigned to a symbol, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs, and even theories are considered as mathematical objects in proof theory.

In Philosophy of mathematics, the concept of "mathematical objects" touches on topics of existence, identity, and the nature of reality.[2] In metaphysics, objects are often considered entities that possess properties and can stand in various relations to one another.[3] Philosophers debate whether mathematical objects have an independent existence outside of human thought (realism), or if their existence is dependent on mental constructs or language (idealism and nominalism). Objects can range from the concrete: such as physical objects usually studied in applied mathematics, to the abstract, studied in pure mathematics. What constitutes an "object" is foundational to many areas of philosophy, from ontology (the study of being) to epistemology (the study of knowledge). In mathematics, objects are often seen as entities that exist independently of the physical world, raising questions about their ontological status.[4][5] There are varying schools of thought which offer different perspectives on the matter, and many famous mathematicians and philosophers each have differing opinions on which is more correct.[6]


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  1. ^ Oxford English Dictionary, s.v. “Mathematical (adj.), sense 2,” September 2024. "Designating or relating to objects apprehended not by sense perception but by thought or abstraction."
  2. ^ Rettler, Bradley; Bailey, Andrew M. (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Object", The Stanford Encyclopedia of Philosophy (Summer 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
  3. ^ Carroll, John W.; Markosian, Ned (2010). An introduction to metaphysics. Cambridge introductions to philosophy (1. publ ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-82629-7.
  4. ^ Burgess, John, and Rosen, Gideon, 1997. A Subject with No Object: Strategies for Nominalistic Reconstrual of Mathematics. Oxford University Press. ISBN 0198236158
  5. ^ Falguera, José L.; Martínez-Vidal, Concha; Rosen, Gideon (2022), Zalta, Edward N. (ed.), "Abstract Objects", The Stanford Encyclopedia of Philosophy (Summer 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
  6. ^ Horsten, Leon (2023), Zalta, Edward N.; Nodelman, Uri (eds.), "Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-29

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