# 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**

### Non-Euclidean geometry

**non-Euclideannon-Euclidean geometriesalternative 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**

### 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**