Abstraction (mathematics)

Abstraction in mathematics is the process of extracting the underlying 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.[1][2][3][4] Two of the most highly abstract areas of modern mathematics are category theory and model theory.[4]

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 structures. For example, geometry has its origins in the calculation of distances and areas in the 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 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 axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios.[5] 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 connections between geometry and abstract algebra.[6]

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 conjectures in another related area.
Techniques and methods from one area can be applied to 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.[7] A degree of mathematical maturity and experience may be needed for conceptual assimilation of abstractions. As such, one of the underlying principles of the Montessori approach to mathematics education is to encourage children to move from concrete examples to abstract thinking.[8]

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."[9]
See also

Abstract detail
Generalization
Abstract thinking
Abstract logic
Abstract algebraic logic
Abstract model theory
Abstract nonsense
Concept

References

Bertrand Russell, in The Principles of Mathematics Volume 1 (pg 219), refers to "the principle of abstraction".
Robert B. Ash. A Primer of Abstract Mathematics. Cambridge University Press, Jan 1, 1998
The New American Encyclopedic Dictionary. Edited by Edward Thomas Roe, Le Roy Hooker, Thomas W. Handford. Pg 34
"The Definitive Glossary of Higher Mathematical Jargon — Abstraction". Math Vault. 2019-08-01. Retrieved 2019-10-22.
Proclus' Summary Archived 2015-09-23 at the Wayback Machine
Torretti, Roberto (2019), Zalta, Edward N. (ed.), "Nineteenth Century Geometry", The Stanford Encyclopedia of Philosophy (Fall 2019 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-10-22
"... 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
Montessori Philosophy: Moving from Concrete to Abstract, North American Montessori Center

"Quotations by Russell". MacTutor History of Mathematics archive. Retrieved 2019-10-22.