# Abstraction (mathematics)

abstractabstractionAbstraction in mathematicsmathematical abstraction
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.wikipedia
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### Mathematics

mathematicalmathmathematician
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.
Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects.

### Abstract structure

abstract abstractabstract form
Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures.

### Phenomenon

phenomenaphenomenalphysical phenomena
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

### Category theory

categorycategoricalcategories
Two of the most highly abstract areas of modern mathematics are category theory and model theory.

### Model theory

modelmodelsmodel-theoretic
Two of the most highly abstract areas of modern mathematics are category theory and model theory.

### Arithmetic

arithmetic operationsarithmeticsarithmetic operation
For example, geometry has its origins in the calculation of distances and areas in the real world; algebra started with methods of solving problems in arithmetic.

### Euclid's Elements

ElementsEuclid's ''ElementsEuclid
For example, the first steps in the abstraction of geometry were historically made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios.

### Axiomatic system

axiomatizationaxiomatic methodaxiom system
For example, the first steps in the abstraction of geometry were historically made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios.

### Hippocrates of Chios

Hippocrates
For example, the first steps in the abstraction of geometry were historically made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios.

### René Descartes

DescartesCartesianRene Descartes
In the 17th century, Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry.

### Cartesian coordinate system

Cartesian coordinatesCartesian coordinateCartesian
In the 17th century, Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry.

### Analytic geometry

analytical geometryCartesian geometrycoordinate geometry
In the 17th century, Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry.

### Nikolai Lobachevsky

LobachevskyNikolai Ivanovich LobachevskyLobachevski
Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann and Gauss, who generalised the concepts of geometry to develop non-Euclidean geometries.

### János Bolyai

BolyaiBolyai JánosBolyai, János
Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann and Gauss, who generalised the concepts of geometry to develop non-Euclidean geometries.

### Bernhard Riemann

RiemannGeorg Friedrich Bernhard RiemannRiemann, Bernhard
Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann and Gauss, who generalised the concepts of geometry to develop non-Euclidean geometries.

### Carl Friedrich Gauss

GaussCarl GaussCarl Friedrich Gauß
Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann and Gauss, who generalised the concepts of geometry to develop non-Euclidean geometries.

### Non-Euclidean geometry

non-Euclideannon-Euclidean geometriesalternative geometries
Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann and Gauss, who generalised the concepts of geometry to develop non-Euclidean geometries.

### Dimension

dimensionsdimensionalone-dimensional
Later in the 19th century, mathematicians generalised geometry even further, developing such areas as geometry in n dimensions, projective geometry, affine geometry and finite geometry.

### Projective geometry

projectiveprojective geometriesProjection
Later in the 19th century, mathematicians generalised geometry even further, developing such areas as geometry in n dimensions, projective geometry, affine geometry and finite geometry.

### Affine geometry

affineaffine ''d''-spaceaffine geometries
Later in the 19th century, mathematicians generalised geometry even further, developing such areas as geometry in n dimensions, projective geometry, affine geometry and finite geometry.

### Finite geometry

finite geometriesfinitefinite field geometry
Later in the 19th century, mathematicians generalised geometry even further, developing such areas as geometry in n dimensions, projective geometry, affine geometry and finite geometry.

### Felix Klein

KleinFelix Christian KleinC. Felix Klein
Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries.

### Erlangen program

Erlangen programmeErlanger programnotion of geometry
Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries.

### Invariant (mathematics)

invariantinvariantsinvariance
Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries.

### Symmetry

symmetricalsymmetricsymmetries
Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries.