mathematical object

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A mathematical object is an abstract concept arising in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do
deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively Validity (logic), valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the concl ...
and
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every p ...
s. Typically, a mathematical object can be a value that can be assigned to a variable, and therefore can be involved in formulas. Commonly encountered mathematical objects include
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s, sets, functions, expressions, geometric objects, transformations of other mathematical objects, and spaces. Mathematical objects can be very complex; for example,
theorem In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the th ...
s, proofs, and even theories are considered as mathematical objects in proof theory. The ontological status of mathematical objects has been the subject of much investigation and debate by philosophers of mathematics. Burgess, John, and Rosen, Gideon, 1997. ''A Subject with No Object: Strategies for Nominalistic Reconstrual of Mathematics''.
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
.

# List of mathematical objects by branch

*
Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
**
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s, operations *
Combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other ar ...
**
permutation In mathematics, a permutation of a Set (mathematics), set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers ...
s, derangements,
combination In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
s * Set theory ** sets, set partitions ** functions, and relations *
Geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
** points, lines,
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
s, ** polygons (
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
s, squares, pentagons, hexagons, ...), circles, ellipses, parabolas, hyperbolas, ** polyhedra (
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex ( ...
s, cubes, octahedrons, dodecahedrons, icosahedrons),
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s, ellipsoids, paraboloids, hyperboloids,
cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a Prism (geometry), prism with a circle as its base. A cylinder ...
s, cones. *
Graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' ( ...
** graphs,
tree In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondar ...
s, nodes, edges *
Topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
** topological spaces and manifolds. *
Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
** scalars, vectors, matrices,
tensor In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vect ...
s. *
Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
** groups, ** rings, modules, ** fields,
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s, ** 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
theorem In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the th ...
s are also mathematical objects.

* Abstract object * Mathematical structure

# References

* Azzouni, J., 1994. ''Metaphysical Myths, Mathematical Practice''. Cambridge University Press. * Burgess, John, and Rosen, Gideon, 1997. ''A Subject with No Object''. Oxford Univ. Press. * Davis, Philip and Reuben Hersh, 1999 981 ''The Mathematical Experience''. Mariner Books: 156–62. * Gold, Bonnie, and Simons, Roger A., 2011.
Proof and Other Dilemmas: Mathematics and Philosophy
'. Mathematical Association of America. * Hersh, Reuben, 1997. ''What is Mathematics, Really?'' Oxford University Press. * Sfard, A., 2000, "Symbolizing mathematical reality into being, Or how mathematical discourse and mathematical objects create each other," in Cobb, P., ''et al.'', ''Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools and instructional design''. Lawrence Erlbaum. * Stewart Shapiro, 2000. ''Thinking about mathematics: The philosophy of mathematics''. Oxford University Press.