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Abstraction (mathematics)From Wikipedia, the free encyclopedia

Abstractionin 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

generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent

phenomena.[1][2][3]Two of the most highly abstract areas of modern mathematics are category theory and model

theory.

Contents

1 Description

2 See also

3 References

4 Further reading

escription

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 inthe real world; algebra started with methods of solving problems in arithmetic.

Abstraction is an ongoing process in mathematics and the historical development of many mathematical topicsexhibits 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 geometrythough Proclus tells of an earlier

axiomatisation by Hippocrates of Chios.[4]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 dimension

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 connections between geometry and abstract algebra.

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.

http://en.wikipedia.org/wiki/Riemannhttp://en.wikipedia.org/wiki/Model_theoryhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Essencehttp://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Symmetryhttp://en.wikipedia.org/wiki/Invariant_(mathematics)http://en.wikipedia.org/wiki/Erlangen_programhttp://en.wikipedia.org/wiki/Felix_Kleinhttp://en.wikipedia.org/wiki/Finite_geometryhttp://en.wikipedia.org/wiki/Affine_geometryhttp://en.wikipedia.org/wiki/Projective_geometryhttp://en.wikipedia.org/wiki/N-dimensional_spacehttp://en.wikipedia.org/wiki/Non-Euclidean_geometryhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Riemannhttp://en.wikipedia.org/wiki/Bolyaihttp://en.wikipedia.org/wiki/Lobachevskyhttp://en.wikipedia.org/wiki/Analytic_geometryhttp://en.wikipedia.org/wiki/Cartesian_co-ordinateshttp://en.wikipedia.org/wiki/Descarteshttp://en.wikipedia.org/wiki/Hippocrates_of_Chioshttp://en.wikipedia.org/wiki/Axiomatic_systemhttp://en.wikipedia.org/wiki/Euclid%27s_Elementshttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Abstract_structurehttp://en.wikipedia.org/wiki/Model_theoryhttp://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Phenomenahttp://en.wikipedia.org/wiki/Essencehttp://en.wikipedia.org/wiki/Mathematics

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One disadvantage of abstraction is that highly abstract concepts can be difficult to learn. [5]A degree of

mathematical maturity and experience may be needed for conceptual assimilation of abstractions. One of the

underlyingprinciples of the Montessori approach to mathematics education is encouraging children to move from

concrete examples to abstract thinking.[6]

Bertrand Russell, in The Scientific Outlook(1931), writes that "Ordinary language is totally unsuited for expressin

what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and

mathematical logic can say as little as the physicist means to say."

See also

Abstract detail

Generalization

Abstract thinking

Abstract logic

Abstract algebraic logicAbstract model theory

Abstract nonsense

References

1. ^Bertrand Russell, in The Principles of MathematicsVolume 1 (pg 219), refers to "the principle of abstraction".

2. ^Robert B. Ash. A Primer of Abstract Mathematics. Cambridge University Press, Jan 1, 1998

3. ^The New American Encyclopedic Dictionary. Edited by Edward Thomas Roe, Le Roy Hooker, Thomas W.

Handford. Pg 34 (http://books.google.com/books?id=KhQLAQAAMAAJ&pg=PA34)

4. ^Proclus' Summary (http://www-gap.dcs.st-and.ac.uk/~history/Extras/Proclus_history_geometry.html)

5. ^"...introducing pupils to abstract mathematics is not an easy task and requires a long-term effort that must take

intoaccount the variety of the contexts in which mathematics is used", P.L. Ferrari,Abstraction in Mathematics,

Phil. Trans. R. Soc. Lond. B 29 July 2003 vol. 358 no. 1435 1225-1230

6. ^Montessori Philosophy: Moving from Concrete to Abstract

(http://montessoritraining.blogspot.co.uk/2008/07/montessori-philosophy-moving-from.html), North American

Montessori Center

Further reading

Bajnok, Bla (2013).An Invitation to Abstract Mathematics. Springer. ISBN 978-1-4614-6635-2.

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Categories: Mathematical terminology Abstraction

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