Abstraction (mathematics)

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Abstraction (mathematics)

Abstraction in mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications.

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

Abstract structure

An abstract structure is a formal object that is defined by a set of laws, properties, and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects....
s.

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Abstraction in mathematics

Mathematics

Abstract structure

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
has its origins in the calculation of distances and areas in the real world; statistics

Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
has its origins in the calculation of probabilities in gambling

Gambling

Gambling is the wikt:wager#Verb of money or something of material Value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods....
; and algebra

Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
started with methods of solving problems in arithmetic

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations....
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Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. Take the historical development of geometry as an example; the first steps in the abstraction of geometry were made by the ancient Greeks, with Euclid's Elements

Euclid's Elements

Euclid's Elements is a mathematics and geometry treatise consisting of 13 books written by the Greek mathematics Euclid in Alexandria circa 300 BC....
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 of Chios was an ancient Greece mathematician, , and astronomer, who lived c. 470 – c. 410 Common Era.He was born on the isle of Chios, where he originally was a merchant....
. In the 17th century Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry

Analytic geometry

Analytic geometry, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra; the modern development of analytic geometry is thus suggestively called algebraic geometry....
. Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann, and Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss. was a Germans mathematician and scientist who contributed significantly to many fields, including number theory, statistics, mathematical analysis, Differential geometry and topology, geodesy, electrostatics, astronomy and optics....
who generalised the concepts of geometry to develop non-Euclidean geometries

Non-Euclidean geometry

In mathematics, non-Euclidean geometry describes hyperbolic geometry and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of Parallel lines....
. Later in the 19th century mathematicians generalised geometry even further, developing such areas as geometry in n dimensions, projective geometry

Projective geometry

In mathematics projective geometry is the study of geometric properties which are invariant under projective transformations. The field of projective geometry is itself divided into many subfields, two examples of which are projective algebraic geometry and projective differential geometry ....
, affine geometry

Affine geometry

In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and Translation s....
and finite geometry

Finite geometry

A finite geometry is any geometry system that has only a finite set number of point .Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact precisely the same number of points as there are real numbers....
. Finally Felix Klein

Felix Klein

Felix Christian Klein was a Germany mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory....
's "Erlangen program

Erlangen program

An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen ?ber neuere geometrische Forschungen....
" 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. This level of abstraction revealed deep connections between geometry and abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
.

Two of the most highly abstract areas of modern mathematics are category theory

Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
and model theory

Model theory

In mathematics, model theory is the study of mathematical Structure such as Group , fields, graph , or even models of set theory, using tools from mathematical logic....
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The advantages of abstraction are :

It reveals deep connections between different areas of mathematics
Known results in one area can suggest conjectures in a related area
Techniques and methods from one area can be applied to prove results in a related area

The main disadvantage of abstraction is that highly abstract concepts are more difficult to learn, and require a degree of mathematical maturity

Mathematical maturity

Mathematical maturity is a loose term used by mathematicians that refers to a mixture of mathematical experience and insight that cannot be directly taught....
and experience before they can be assimilated.

Abstraction (mathematics)

See also