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. 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 geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios. 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.
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.
One disadvantage of abstraction is that highly abstract concepts can be difficult to learn. A degree of mathematical maturity and experience may be needed for conceptual assimilation of abstractions. One of the underlying principles of the Montessori approach to mathematics education is encouraging children to move from concrete examples to abstract thinking.
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."