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.
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, and algebra started with methods of solving problems in arithmetic.
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.
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.