Abstraction is to extract the essence of mathematical conceptsprocessIn this way, we will remove the dependencies of objects in reality that are associated with the original, and generalize them to make them more widely used, so as to match the abstract description of other equivalent phenomena.The two most abstract fields in modern mathematics areCategorical theoryandModel theory。
In mathematics, abstraction is the process of extracting the essence of mathematical concepts. In this way, we can remove the dependency relationship with the original associated objects in reality, and generalize it to make it more widely used, thus matching with the abstract description of other equivalent phenomena.The two most abstract fields in modern mathematics areCategorical theoryandModel theory。[1]
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Many mathematical fields begin with the study of real world problems, and then determine the basic rules and concepts as abstract structures.For example, geometry originated from the calculation of distance and area in the real world, and algebra began from the method of solving arithmetic problems.
Abstraction is a continuous mathematical process, and the historical development of many mathematical subjects shows the development from concrete to abstract.Take the historical development of geometry as an example;ancientGreeceThe first step of human abstraction is the ancient Greek language. Although Proclus introduced the early axioms of the city of Hiokrates,EuclidThe proof of the axiom of plane geometry is the earliest existing document.In the 17th century,Descartes Cartesian coordinates are introduced to promote the development of analytical geometry.The further abstraction was carried out by Lobachevsky, Portsmouth, Riemann and Gauss, who summarized the concept of geometry as non Euclidean geometry.Later in the 19th century, mathematicians further promoted geometry and developed n-dimensional geometry, projective geometry, affine geometry and finite geometry.Finally, Felix Klein's "Erlangen program" determined the basic themes of all these geometries and defined them as the study of the invariant attributes under a given group of objects.This level of abstraction reveals the relationship between geometry and abstract algebra.[2]
The advantages of abstraction are:
(1) It reveals the deep relationship between different mathematical fields.
(2) The known results in one field can be conjectured in related fields.
(3) The technologies and methods in one field can be applied to prove the achievements in related fields.
One disadvantage of abstraction is that highly abstract concepts may be difficult to learn.Assimilation of abstract concepts may require a certain degree of mathematical maturity and experience.One of the basic principles of Montessori's mathematical education method is to encourage children to shift from concrete examples to abstract thinking.
Bertrand Russell in "Science Outlook" (1931) wrote: "Ordinary language is totally unsuitable for expressing any real assertion of physics, because the discourse of daily life is not abstract enough, and only mathematics and mathematical logic can express the meaning of physicists".
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Abstract dictionary explanation:
(1) The action or process of taking out one or several characteristics of a complex object and paying attention only to other characteristics (for example, the mind only considers the shape of the tree itself or the color of the leaves, regardless of their size and shape)
(2) The act or process of abstracting the common properties or characteristics of several different objects or considering them in isolation.Abstraction is necessary to divide things into genera and species
(3) Not specific;General.
(4) Invisible because it is invisible.The above explanation is very interesting.The first explanation shows that people need to choose and give up when considering a problem or looking at a thing.Do a good job in "giving up" and "getting".The second explanation shows that in the process of understanding things and abstracting, the comparative method is important and the classification method is important.Without comparison and classification, we cannot know things.The first two describe the abstract process and method.The following explanation shows the result after abstraction.A thing that is not concrete or general is abstract.What is invisible because it is invisible is also abstract.Ideas and concepts are invisible and abstract.When we introduce a certain thing to others, the thing itself may be concrete, but when we describe it to others in language and words, what you describe has actually become "another thing" abstracted by you, which requires the audience to connect with the specific things in his experience, reassemble and assemble the information you convey to him in his mind into specific things,Can give the impression of the audience image, otherwise others do not understand.This requires that when we introduce abstract things, we should visualize them, draw pictures, metaphors and analogies, and use the existing knowledge and experience of the audience as much as possible.[3]
