Content
Abstraction in mathematics is the process of extracting the Underlying gasoline of a mathematical concept Removing Any dependence is real world objects with qui it might Originally-have-been connected, and generalizing it so That It HAS ‘wider gold applications matching Among other abstract descriptions of equivalent phenomena . [1] [2] [3] Two of the most highly abstract areas of modern mathematics are theory theory and model theory .
Description
Many areas of mathematics Began with the study of real world problems, before the Underlying rules and concepts Were APPROBATION 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 Processes of earlier axiomatization by Hippocrates of Chios . [4] 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 Generalized 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 the properties of invariantunder 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 the other.
One disadvantage of abstraction is that highly abstract concepts can be difficult to learn. [5] A degree of mathematical maturity and experience can be used for conceptual assimilation of abstractions. One of the principles of the Montessori approach to mathematics education is encouraging children to move from concrete examples to abstract thinking. [6]
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. to say. ”
See also
- Abstract detail
- generalization
- Abstract thinking
- Abstract logic
- Abstract algebraic logic
- Abstract model theory
- Abstract nonsense
References
- Jump up^ Bertrand Russell, inThe Principles of MathematicsVolume 1 (pg 219), refers to “theprinciple of abstraction”.
- Jump up^ Robert B. Ash. A Primer of Abstract Mathematics. Cambridge University Press, Jan 1, 1998
- Jump up^ The New American Encyclopedic Dictionary. Edited by Edward Thomas Roe, The Roy Hooker, Thomas W. Handford. Pg34
- Jump up^ Proclus’ Summary
- Jump up^ “… introducing pupils to abstract mathematics is not an easy task and requires a long-term effort that must take into account the variety of contexts in which mathematics is used”, PL Ferrari,Abstraction in Mathematics, Phil. Trans. R. Soc. Lond. B July 29, 2003, vol. 358 no. 1435 1225-1230
- Jump up^ Montessori Philosophy: Moving from Concrete to Abstract, North American Montessori Center