Information about Abstraction (mathematics)
Abstraction in mathematics 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 structures. For example, geometry has its origins in the calculation of distances and areas in the real world; statistics has its origins in the calculation of probabilities in gambling; and algebra started with methods of solving problems in arithmetic.
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 being the first person (as far as we know) to document the axioms of plane geometry. In the 17th century Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry. Further steps in abstraction were taken by Lobachevsky, Bolyai and Gauss who generalised the concepts of geometry to develop non-Euclidean 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. 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. This level of abstraction revealed deep connections between geometry and abstract algebra.
Two of the most highly abstract areas of modern mathematics are category theory and model theory.
The advantages of abstraction are :
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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. For example, geometry has its origins in the calculation of distances and areas in the real world; statistics has its origins in the calculation of probabilities in gambling; and algebra started with methods of solving problems in arithmetic.
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 being the first person (as far as we know) to document the axioms of plane geometry. In the 17th century Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry. Further steps in abstraction were taken by Lobachevsky, Bolyai and Gauss who generalised the concepts of geometry to develop non-Euclidean 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. 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. This level of abstraction revealed deep connections between geometry and abstract algebra.
Two of the most highly abstract areas of modern mathematics are category theory and model theory.
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
See also
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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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.
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Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.
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gambling has had many different meanings depending on the cultural and historical context in which it is used. Currently, in Western societies, it has an economic definition, referring to "wagering money or something of material value on an event with an uncertain outcome with the
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Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Arabic[1] mathematician, astronomer, astrologer and geographer,
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Arithmetic or arithmetics (from the Greek word αριθμός = number) 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
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Euclid
Born fl. 300 BC
Residence Alexandria, Egypt
Nationality Greek
Field Mathematics
Known for Euclid's Elements Euclid (Greek:
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Born fl. 300 BC
Residence Alexandria, Egypt
Nationality Greek
Field Mathematics
Known for Euclid's Elements Euclid (Greek:
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René Descartes (French IPA: [ʁə'ne de'kaʁt]) (March 31, 1596 – February 11, 1650), also known as Renatus Cartesius
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Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra.
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Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский
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János Bolyai (December 15, 1802 – January 27, 1860) was a Hungarian mathematician, known for his work in non-Euclidean geometry.
Bolyai was born in Kolozsvár, Transylvania (today Cluj-Napoca, Romania), the son of a well-known mathematician, Farkas Bolyai.
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Bolyai was born in Kolozsvár, Transylvania (today Cluj-Napoca, Romania), the son of a well-known mathematician, Farkas Bolyai.
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Johann Carl Friedrich Gauss
Carl Friedrich Gauss, painted by Christian Albrecht Jensen
Born 30 March 1777
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Carl Friedrich Gauss, painted by Christian Albrecht Jensen
Born 30 March 1777
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non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines.
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Projective geometry is a non-metrical form of geometry. First developed by Desargues in the 17th century, it did not achieve prominence as a field of mathematics until the early 19th century through the work of Poncelet and others.
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In geometry, affine geometry is geometry not involving any notions of origin, length or angle, but with the notion of subtraction of points giving a vector.
It occupies a place intermediate between Euclidean geometry and projective geometry.
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It occupies a place intermediate between Euclidean geometry and projective geometry.
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A finite geometry is any geometric system that has only a finite number of points. 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.
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Felix Klein
Felix Christian Klein
Born March 25 1849
Düsseldorf, Prussia
Died May 22 1925 (aged 76)
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Felix Christian Klein
Born March 25 1849
Düsseldorf, Prussia
Died May 22 1925 (aged 76)
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Erlangen Program (Erlanger Programm) — Klein was then at Erlangen — proposed a new kind of solution to the problems of geometry of the time.
At the time, geometry contained a very large number of theorems.
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At the time, geometry contained a very large number of theorems.
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Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most authors nowadays simply write algebra instead of abstract algebra.
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In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
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- This article discusses model theory as a mathematical discipline and not the informally used term mathematical model as used in other parts of mathematics and science.
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Mathematical maturity is a loose term used by mathematicians that refers to a mixture of mathematical experience and insight that cannot be directly taught, but instead comes from repeated exposure to complex mathematical concepts.
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Hypostatic abstraction, also known as hypostasis or subjectal abstraction, is a formal operation that takes an element of information, such as might be expressed in a proposition of the form X is Y
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Prescisive abstraction or prescision, variously spelled as precisive abstraction or prescission, is a formal operation that marks, selects, or singles out one feature of a concrete experience to the disregard of others.
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