Abstraction inmathematics is the process of extracting the underlyingstructures, 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 equivalentphenomena.[1][2][3] In other words, to be abstract is to remove context and application.[4] Two of the most highly abstract areas of modern mathematics arecategory theory andmodel theory.
Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined asabstract structures. For example,geometry has its origins in the calculation of distances andareas in the real world, andalgebra started with methods of solving problems inarithmetic.
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, withEuclid's Elements being the earliest extant documentation of theaxioms of plane geometry—though Proclus tells of an earlieraxiomatisation byHippocrates of Chios.[5] In the 17th century,Descartes introducedCartesian co-ordinates which allowed the development ofanalytic geometry. Further steps in abstraction were taken byLobachevsky,Bolyai,Riemann andGauss, who generalised the concepts of geometry to developnon-Euclidean geometries. Later in the 19th century, mathematicians generalised geometry even further, developing such areas as geometry inn dimensions,projective geometry,affine geometry andfinite geometry. FinallyFelix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study ofproperties invariant under a givengroup ofsymmetries. This level of abstraction revealed connections between geometry andabstract algebra.[6]
In mathematics, abstraction can be advantageous in the following ways:
On the other hand, abstraction can also be disadvantageous in that highly abstract concepts can be difficult to learn.[7] A degree ofmathematical maturity and experience may be needed forconceptual assimilation of abstractions.
Bertrand Russell, inThe 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."[8]
