Abstraction (mathematics)
Encyclopedia
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 or matching among other abstract descriptions of equivalent phenomena.
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
s. 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's Elements
being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios
. 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, Riemann, 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 :
The main disadvantage of abstraction is that highly abstract concepts are more difficult to learn, and require a degree of mathematical maturity
and experience before they can be assimilated.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
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 or matching among other abstract descriptions of equivalent phenomena.
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 in mathematics 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. For example, geometry
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 the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
has its origins in the calculation of probabilities in gambling
Gambling
Gambling is the wagering 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 the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
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. It involves the study of quantity, especially as the result of combining numbers...
.
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 mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
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 Greek mathematician, , and astronomer, who lived c. 470 – c. 410 BCE.He was born on the isle of Chios, where he originally was a merchant. After some misadventures he went to Athens, possibly for litigation...
. In the 17th century Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
. Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann, and Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
who generalised the concepts of geometry to develop non-Euclidean geometries
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
. 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 that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, 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 translations...
and finite geometry
Finite geometry
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 as many points as there are real numbers...
. Finally Felix Klein
Felix Klein
Christian Felix Klein was a German 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 über 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 groups, rings, fields, modules, vector spaces, and algebras...
.
Two of the most highly abstract areas of modern mathematics are category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
and model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
.
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 an informal term used by mathematicians to refer to a mixture of mathematical experience and insight that cannot be directly taught...
and experience before they can be assimilated.