Editing Abstraction (mathematics)

{{Short description|Process of extracting the underlying essence of a mathematical concept}}

'''Abstraction''' in [[mathematics]] is the process of extracting the underlying [[Mathematical structure|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]].<ref>[[Bertrand Russell]], in ''The Principles of Mathematics'' Volume 1 (pg 219), refers to "the principle of abstraction".</ref><ref>Robert B. Ash. A Primer of Abstract Mathematics. Cambridge University Press, Jan 1, 1998</ref><ref>The New American Encyclopedic Dictionary. Edited by Edward Thomas Roe, Le Roy Hooker, Thomas W. Handford. Pg [https://books.google.com/books?id=KhQLAQAAMAAJ&pg=PA34 34]</ref> In other words, to be abstract is to remove context and application.<ref>{{Cite book |last=Donaldson |first=Neil |title=Introduction to Group Theory |page=1 |language=en}}</ref> Two of the most highly abstract areas of modern mathematics are [[category theory]] and [[model theory]].

==Description==

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 [[area]]s in the real world, 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. 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 [[axiom]]s of plane geometry—though Proclus tells of an earlier [[Axiomatic system|axiomatisation]] by [[Hippocrates of Chios]].<ref>[http://www-gap.dcs.st-and.ac.uk/~history/Extras/Proclus_history_geometry.html Proclus' Summary] {{webarchive|url=https://web.archive.org/web/20150923114020/http://www-gap.dcs.st-and.ac.uk/~history/Extras/Proclus_history_geometry.html |date=2015-09-23 }}</ref> In the 17th century, [[Descartes]] introduced [[Cartesian co-ordinates]] which allowed the development of [[analytic geometry]]. Further steps in abstraction were taken by [[Nikolai Lobachevsky|Lobachevsky]], [[Bolyai]], [[Riemann]] and [[Carl Friedrich Gauss|Gauss]], who generalised the concepts of geometry to develop [[non-Euclidean geometry|non-Euclidean geometries]]. Later in the 19th century, mathematicians generalised geometry even further, developing such areas as geometry in [[N-dimensional space|''n'' dimension]]s, [[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 [[Invariant (mathematics)|properties invariant]] under a given [[group (mathematics)|group]] of [[Symmetry|symmetries]]. This level of abstraction revealed connections between geometry and [[abstract algebra]].<ref>{{Citation|last=Torretti|first=Roberto|title=Nineteenth Century Geometry|date=2019|url=https://plato.stanford.edu/archives/fall2019/entries/geometry-19th/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2019|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-10-22}}</ref>

In mathematics, abstraction can be advantageous in the following ways:

* It reveals deep connections between different areas of mathematics.
* Known results in one area can suggest [[conjecture]]s in another related area.
* Techniques and methods from one area can be applied to [[mathematical proof|prove]] results in other related areas.
*Patterns from one mathematical object can be generalized to other similar objects in the same class.

On the other hand, abstraction can also be disadvantageous in that highly abstract concepts can be difficult to learn.<ref>"... introducing pupils to abstract mathematics is not an easy task and requires a long-term effort that must take into account 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</ref> A degree of [[mathematical maturity]] and experience may be needed for [[Assimilation (psychology)|conceptual assimilation]] of abstractions. 

[[Bertrand Russell]], in ''The Scientific Outlook'' (1931), writes that "Ordinary language is totally unsuited for expressing 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."<ref>{{Cite web|url=https://www-history.mcs.st-andrews.ac.uk/Quotations/Russell.html|title=Quotations by Russell|last=|first=|date=|website=[[MacTutor History of Mathematics archive]]|url-status=live|archive-url=https://web.archive.org/web/20020117002740/http://www-history.mcs.st-andrews.ac.uk:80/Quotations/Russell.html |archive-date=2002-01-17 |access-date=2019-10-22}}</ref>

==See also==
* [[Abstract detail]]
* [[Generalization]]
* [[Abstract thinking]]
* [[Abstract logic]]
* [[Abstract algebraic logic]]
* [[Abstract model theory]]
* [[Abstract nonsense]]
* [[Concept]]
* [[Mathematical maturity]]

==References==
{{Reflist}}

==Further reading==
{{wikiquote}}
*{{cite book
 | last = Bajnok
 | first = Béla
 | title = An Invitation to Abstract Mathematics
 | publisher = Springer
 | year = 2013
 | isbn = 978-1-4614-6635-2}}

[[Category:Mathematical terminology]]
[[Category:Abstraction]]