Mathematical object
A mathematical object is an abstract concept arising in mathematics.
In usual language of mathematics, an object is anything that has been formally defined, and with which one may do deductive reasoning and mathematical proofs. Typically, a mathematical object can be the value of a variable, and therefore can be involved in formulas. Commonly encountered mathematical objects include: numbers, integers, integer partition, or expressions. Each branch of mathematics has its own objects. Some examples are:
- Combinatorics
- *permutations, derangements, combinations
- Set theory
- *sets, set partitions
- *functions, and relations
- Geometry
- *points, lines, line segments,
- *polygons, circles, ellipses, parabolas, hyperbolas,
- *polyhedra, spheres, ellipsoids, paraboloids, hyperboloids, cylinders, cones.
- Graph theory
- *graphs, trees, nodes, edges
- Topology
- *topological spaces and manifolds.
- Linear algebra
- *scalars, vectors, matrices, tensors.
- Abstract algebra
- *groups,
- *rings, modules,
- *fields, vector spaces,
- *group-theoretic lattices, and order-theoretic lattices.
Categories are simultaneously homes to mathematical objects and mathematical objects in their own right. In proof theory, proofs and theorems are also mathematical objects.
The ontological status of mathematical objects has been the subject of much investigation and debate by philosophers of mathematics.
