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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 valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...
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 pr ...
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 number ...
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 that has been 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 theorem is a logical consequence of t ...
s,
proofs Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...
, and even theories are considered as mathematical objects in
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
. 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 book ...
.


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 integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
**
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 number ...
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 structures. It is closely related to many other areas of mathematics and has many a ...
**
permutation In mathematics, a permutation of a 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 to the act or pro ...
s, derangements, combinations *
Set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
**
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s,
set partition In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every par ...
s ** 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 c ...
**
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points ...
, lines,
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
s, **
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
s (
triangle A triangle is a polygon with three edges and three 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, any three points, when non- colline ...
s, squares,
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be sim ...
s,
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
s, ...),
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
s,
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
s,
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
s,
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s, ** polyhedra (
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
s,
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
s,
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
s, dodecahedrons,
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
s),
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
s,
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
s, paraboloids,
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
s,
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an ...
s, cones. *
Graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
** graphs,
tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
s,
node In general, a node is a localized swelling (a " knot") or a point of intersection (a vertex). Node may refer to: In mathematics * Vertex (graph theory), a vertex in a mathematical graph * Vertex (geometry), a point where two or more curves, line ...
s,
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...
s *
Topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
**
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s and
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s. *
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 matrice ...
** scalars, vectors,
matrice Matrice is a '' comune'' (municipality) in the Province of Campobasso in the Italian region Molise, located about northeast of Campobasso. As of 31 December 2004, it had a population of 1,081 and an area of .All demographics and other statistics ...
s,
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s. *
Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
** groups, ** rings,
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
s, ** fields,
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
, proofs and
theorem In mathematics, a theorem is a statement that has been 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 theorem is a logical consequence of t ...
s are also mathematical objects.


See also

*
Abstract object In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, h ...
*
Mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additiona ...


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.


External links

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Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...
:
Abstract Objects
—by Gideon Rosen. *Wells, Charles,


AMOF: The Amazing Mathematical Object FactoryMathematical Object Exhibit
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