First, what is the mathematical abstract method?
Mathematical abstract method is a scientific abstract method. It is based on the consideration of problems and the analysis of various empirical facts.
Observing, analyzing, synthesizing and comparing, putting aside the phenomena, externalities and contingency of things in people's minds, extracting the essence, immanence and inevitability of things, revealing the essence and law of objective things from the spatial form and quantitative relationship, or extracting one of their attributes as a new mathematical object on the basis of existing mathematical knowledge, so as to achieve the purpose of understanding the essence and law of things. For example, the concept of "point" in geometry is abstracted from concrete things such as water points, raindrops, starting points and ending points in the real world. It abandons all kinds of physical and chemical properties of things, regardless of their size, and only retains the nature of their position.
Second, the basic characteristics of mathematical abstraction
Mathematical abstraction has three basic characteristics:
1. In mathematical abstraction, all other attributes of objective objects are discarded, and only quantitative attributes are kept. It was measured here.
The concept is that with the development of human practice, its content is getting richer and richer. The so-called quantity in classical mathematics usually refers to the two basic meanings of "shape" and "number", while the quantity in modern mathematics usually refers to the relational structure system of mathematics.
2. Mathematical abstraction is a constructive activity, that is, to "construct" the corresponding mathematical objects with clear definitions.
It is called "logical construction" of mathematical objects. Only through this logical construction can mathematical objects be transformed from internal thinking activities into "external" independent existence, and the corresponding mathematical conclusions can get rid of the "individuality" that thinking activities must have and obtain the "universality" that scientific knowledge must have. For example, the concept of verticality may have different mental images for different people, but mathematics studies logical conclusions that can be derived from the definition of this concept, so it is an objective knowledge.
3. Mathematical abstraction has rich levels, which can be abstracted from objective things in the real world or from existing numbers.
On the basis of learning knowledge, its abstract height far exceeds the general abstraction of other sciences. An important feature of the development of modern mathematics is that its research object has expanded from the intuitive relationship and form of quantity to the possible relationship and form of quantity. This shows the special height of mathematical abstraction. These highly abstract concepts are far from the real world and are often called "free imagination and creative thinking".
Third, the types of mathematical abstraction.
The common methods of mathematical abstraction are idealized abstraction, equivalent abstraction, strong abstraction and weak abstraction, which are described as follows:
1. Idealized abstraction
Idealized abstraction is a special mathematical abstraction, which simplifies and perfects objective things or phenomena from the quantitative aspect.
The processing of goodness abstracts the nature and relationship of the quantity that an objective thing or phenomenon must have in its actual reality, and introduces the characteristics of the quantity that cannot belong to its actual original image in principle into the connotation of the concept. For example, the introduction of basic concepts such as points, lines and surfaces in geometry is the result of idealization and abstraction.
Mathematical concepts obtained through idealized abstraction may be inconsistent with the prototype. For example, in the real world, it is impossible to find a point without size, a line without thickness and width, and a surface without thickness. These mathematical concepts of points, lines and surfaces reflect the attributes of objective things more deeply, correctly and completely, so they are closer to things than away from them. It can be seen that idealized abstraction is the dialectical unity of subjective abstract form and objective concrete content. This method is not only very important for mathematical concepts, but also essential for establishing mathematical models. Euler used idealized and abstract methods to transform the problem of the seven bridges in Konigsberg into a mathematical model of the one-line problem.
The results of idealization and abstraction show various structural forms in mathematics, including both graphics and analytical expressions; There are both concrete mathematics and general abstract symbol systems.
2. Equivalent abstraction
Equivalence abstraction is to divide a known set by means of equivalence relation, and then "identify" the equivalent elements in it.
And a way to get a new set. Its specific meaning is that if a binary relation in a set meets the following three conditions:
(1) reflexivity is related to arbitrary sum, that is;
(2) If the symmetry is, then, where;
(3) transitivity If,, then, among them,
It is called the equivalence relation on. Thus, the obtained division makes it a union of several "equivalence classes". Equivalent elements are in the same equivalence class, and unequal elements are in different equivalence classes. Then, the elements in the same equivalence class are "identical", that is, the equivalent elements are regarded as the same thing in the abstract sense, and such an equivalence class vividly condenses a new abstract element. All these elements constitute a new set, that is, the quotient set about. The process from to is the process of equivalent abstraction. For example, in elementary number theory, if the sum of integers is divisible and has the same remainder, it is called modular congruence and recorded as. Obviously, congruence relation is an equivalent relation based on integer system. For another example, rational numbers can be regarded as equivalent classes of integer pairs.
The equivalent abstraction method is one of the commonly used methods based on the new mathematical system, which is widely used in mathematical research. Many important concepts in mathematics are caused by this, and this method can often play a role in solving problems.
3. Strong abstraction
Strong abstraction is also called enhanced structured abstraction. It refers to strengthening the original structure by introducing new features, thus completing the abstraction.
The new structure is a special case of the original structure. In other words, strong abstraction is an abstract method to establish a new concept by expanding the connotation of the original concept. For example, based on the concept of arbitrary triangle, if the attribute restriction of "edge" is strengthened and two sides are equal or three sides are equal, two new concepts of isosceles triangle or equilateral triangle will be obtained; If we strengthen the restriction on the attribute of "angle", such as requiring an angle to be a right angle, we can get the concept of right triangle through such strong abstraction. For another example, introducing the concept of continuity into the concept of function constitutes the concept of continuous function.
4. Weak abstraction
If abstract, it is also called extended abstraction of concepts. It refers to selecting a feature from the prototype and weakening the limitations of this feature.
Abstract, so as to obtain a wider structural process than the original structure. Prototype is a special case of its weak abstraction. Weak abstraction is a mathematical abstraction method to establish a new concept by narrowing the connotation of the original concept. For example, congruence has the properties of equal area and similar shape. If we start from this concept, we can weaken the restriction of "equal area", retain the nature of "similar shape" and get the concept of similar shape by weak abstraction.
Generally speaking, some concrete and intuitive things and objects that are first recognized by people, if their content structure is very rich, then we can adopt the method of weak abstraction to introduce new concepts.
Generally speaking, if people know that the content structure of things and objects is poor or not rich enough, then we can use strong abstraction to introduce new concepts. Of course, it can also be used to analyze the hierarchical structure of mathematical concepts and understand the relationship between mathematical knowledge according to the completely opposite characteristics of weak abstract thinking mode. For example, in quadrilateral, the concept of parallelogram can be obtained through strong abstraction by adding the condition that "two pairs are parallel respectively"; By removing the restriction of "two groups of opposite sides are parallel respectively" in the concept of parallelogram, the concept of quadrilateral can be obtained by weak abstraction. It can be seen that the concept of parallelogram in elementary geometry occupies a particularly important position in all kinds of quadrilateral concepts: it is not only the result of the strong abstraction of arbitrary quadrilateral and trapezoid, but also the starting point of other concepts such as rectangle, diamond and square. At the same time, it is also the starting point of weak abstraction such as trapezoid and quadrilateral.