In mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.
Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations. Geometry as a branch of mathematics has such objects as hexagons, points, lines, triangles, circles, spheres, polyhedra, topological spaces and manifolds. Algebra, another branch, has groups, rings, fields, group-theoretic lattices and order-theoretic lattices. Categories are simultaneously homes to mathematical objects and mathematical objects in their own right.
The ontological status of mathematical objects has been the subject of much investigation and debate by philosophers of mathematics. On this debate, see the monograph by Burgess and Rosen (1997).
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One view that emerged around the turn of the 20th century with the work of Cantor is that all mathematical objects can be defined as sets. The set {0,1} is a relatively clear-cut example. On the face of it the group Z2 of integers mod 2 is also a set with two elements. However, it cannot simply be the set {0,1}, because this does not mention the additional structure imputed to Z2 by the operations of addition and negation mod 2: how are we to tell which of 0 or 1 is the additive identity, for example? To organize this group as a set it can first be coded as the quadruple ({0,1},+,−,0), which in turn can be coded using one of several conventions as a set representing that quadruple, which in turn entails encoding the operations + and − and the constant 0 as sets.
This approach raises the fundamental philosophical question of whether the ontology of mathematics should be beholden to practice or pedagogy. Mathematicians do not work with such codings, which are neither canonical nor practical. They do not appear in any algebra texts, and neither students nor instructors in algebra courses have any familiarity with such codings. Hence, if ontology is to reflect practice, mathematical objects cannot be reduced to sets in this way.
If, however, the goal of mathematical ontology is taken to be the internal consistency of mathematics, it is more important that mathematical objects be definable in some uniform way (for example, as sets) regardless of actual practice, in order to lay bare the essence of its paradoxes. This has been the viewpoint taken by foundations of mathematics, which has traditionally accorded the management of paradox higher priority than the faithful reflection of the details of mathematical practice as a justification for defining mathematical objects to be sets.
Much of the tension created by this foundational identification of mathematical objects with sets can be relieved without unduly compromising the goals of foundations by allowing two kinds of objects into the mathematical universe, sets and relations, without requiring that either be considered merely an instance of the other. These form the basis of model theory as the domain of discourse of predicate logic. From this viewpoint, mathematical objects are entities satisfying the axioms of a formal theory expressed in the language of predicate logic.
A variant of this approach replaces relations with operations, the basis of universal algebra. In this variant the axioms often take the form of equations, or implications between equations.
A more abstract variant is category theory, which abstracts sets as objects and the operations thereon as morphisms between those objects. At this level of abstraction mathematical objects reduce to mere vertices of a graph whose edges as the morphisms abstract the ways in which those objects can transform and whose structure is encoded in the composition law for morphisms. Categories may arise as the models of some axiomatic theory and the homomorphisms between them (in which case they are usually concrete, meaning equipped with a faithful forgetful functor to the category Set or more generally to a suitable topos), or they may be constructed from other more primitive categories, or they may be studied as abstract objects in their own right without regard for their provenance.