Generalization of mathematical objects is an opposite process to specialization. If objects A and B are in phase, B is called the generalized product of A under D, for example, from circle to ellipse, from the diameter of circle to the chord of circle, from the quartic equation in the form of x4+AX2+B = 0 to the quartic equation in the form of x4+A 1x3+A2X2+A3X+A4 = 0. From Abelian group to ring, from linear metric space to linear topological space, from group to topological group, etc. It belongs to the generalization under non-trivial standards (what is the standard, which will be involved later). For the proposition (or a general statement) about object X, we can get a generalized proposition by replacing X with a more general object and adjusting the statement appropriately.
For example, "there are infinitely many natural numbers n, so that 2n+ 1 and 3n+ 1 are complete squares" can be summarized as "there are infinitely many natural numbers n, so that for a given natural number m, Mn+ 1, (m+ 1) n+65433.
What needs to be pointed out is that when popularizing a proposition, how to treat the objects involved in the proposition directly affects the authenticity of the popularized proposition. For example, if the "three" in "the sum of the internal angles of a triangle is equal to 180" is replaced by a general natural number n(n≥3), then the generalized proposition that "the sum of the internal angles of an n-polygon is equal to 180" is obviously not. However, if 180 is written as (3-2) × 180, and then 3 is changed to n, the generalized proposition that the sum of the internal angles of n polygons is equal to (n-2) 180 holds.
Like specialization, generalization is multi-directional, measured (hierarchical) and conditional, as well as the diversity of specific paths from special object A to general object B. At the same time, for a mathematical object, it is both the starting point and the end point of generalization-the end point of generalization in different directions.
Generalization is multi-directional and comes from the multi-faceted starting point of object generalization. The starting point includes both objective and subjective factors. Objective factors refer to all aspects of the composition of objects, while subjective factors refer to the explanation of objects-how to treat a given object, including the exertion of people's subjective initiative. Starting from different starting points, we can get different generalized products.
Example 1 A=34 This object has two basic components: the base number is 3 and the exponent is 4. If 4 is extended to the variable x, a is extended to 3x; Extending 3 to variable X and A to x4.3x and x4 has no special and general relationship, and is the product of extending A to different directions.
The generalization of components leads to the generalization of the object itself. It is worth pointing out that the generalization of components cannot lead to the generalization of objects under any circumstances. An object can actually be regarded as a system composed of some parts according to certain constraints. The change of one part is restricted by other parts to some extent, and the change of parts is not absolutely free. For example, after 2- 1 is extended to x- 1, x is subject to the following constraints of index-1: X ≠ 0. Generalization is meaningful only within a certain range. This is in line with the philosophy of "everything has a degree-keep its quality". The generalization of components will not only lead to the generalization of objects, but also weaken the relationship between components, which is an important generalization method: weak abstraction. We will talk about this in detail later. Here we only give an example to illustrate that this is also one of the reasons for the generalization of multidirectional (as a system, an object has two basic components: element components and their connections. Changes in components and connections are two basic objective aspects of object changes).
2. General application
Generalization is a way of economic thinking. Generalization is conducive to improving the efficiency of thinking. When general problems are solved, special problems can often be solved. The characteristics of general objects are those of special objects. When people understand the characteristics of general objects, there is no need to prove that special objects have this property one by one. As long as it is clear that these objects are special, we can assert that they must have this property. In this way, the understanding of a general object (in terms of its characteristics) actually includes the understanding of many related special objects, that is, one is equivalent to many, thus saving people's thinking power. For example, after people know that "for real number A, a2≥0", it is not necessary to verify 22 ≥ 0, 1.52 ≥ 0, (-0.02). ... and so on. In fact, it is impossible to complete this verification procedure, because the number of real numbers is infinite, even uncountable. Moreover, if people are limited to this kind of verification work, they will only get some experience in the end. Without infinity, there would be no science of the universe (Poincare). Without generalization, people will not transition from poverty to infinity, mathematics will not be produced, and other sciences will not be produced.
It should be pointed out that the substitution of general objects for special objects is in one aspect, not in any aspect. In fact, a special object is called a special object because it has its own characteristics or "personality". For example, A in a2≥0 can only replace 2(22≥0), and other properties of 2 (such as even number) are not necessarily derived from A. 。
Generalization is a way of academic research. It leads people from the special to the general. For example, Pascal's hexagon theorem (now called) discovered by the French mathematician Pascal in the 65,438+07 century at the age of 65,438+06 went through a process of popularization (if a hexagon is inscribed with a conic curve, then every two opposite sides intersect to form a three-point * * * line). First, he studied the special forces. Then, the generalization from circle to conic curve is realized by projection and section, which proves to be effective for all conic curves. For another example, Hilbert, a master of mathematics, is often associated with formalism and axiomatic method, which is regarded as the essence of his thought. In fact, he also has a very important research path-from special to general-generalization. In an article written for the Royal Society, the famous mathematician Weil said, "Hilbert is always lucky to strike a balance between mastering a specific problem and forming a general abstract concept." "When solving special problems, Hilbert can always keenly grasp the signs that reveal the general relationship to him. During the study of number theory, Hilbert expounded the general theorem and the general law of reciprocity about class domain. This is also an excellent example to illustrate the above factors. " "Hilbert's logarithm field theory ... was studied during 1892- 1898. After the paper came out, it developed step by step from the special to the general, involving many useful concepts and methods, revealing the "intrinsic relationship of essence." Lagrange and Hamilton also found the general from the special.
It is obvious that generalization is helpful to enhance the universality of cognition and expand the scope of cognition. Because the direct embodiment of generalization is the extension of object extension, which is also one of the purposes of generalization. Due to the increase in the adaptability of facts (or concepts), it has laid a foundation for the application of this concept in a wide range. For example, the boundedness theorem and intermediate value theorem of continuous functions on closed intervals can be used in many fields after being extended to a certain extent. Formula means solving equation x2+5x-7 = 0.
Then we can solve any quadratic equation with real coefficient with quadratic coefficient 1 (for example, x2-3x-5 = 0). In more specific generalization methods, symbol and abstraction are two important ways to enhance cognitive universality.
Mathematics (mainly) is a (symbolic) language, which is characterized by the extensive use of various symbols, and with the development of history, this feature is increasingly strongly manifested (for example, this trend is further aggravated after Hilbert's formal view is put forward). Perhaps it can be said that mathematical logic is particularly important. Generalization of mathematical content (object, proposition, etc.). ) with the change of mathematical language-or words (for example, real number → complex number; Continuous function → Lebesgue integrable function; Wait a minute. ), or semantic changes (such as continuous function in ordinary calculus → continuous function in topology, also called continuous function, but the former is more special than the latter). The concepts of function and series convergence have also experienced a process from narrow sense to broad sense, that is, from special to general. To some extent, the introduction of symbols has laid a linguistic foundation for generalization. For example, before F. Vieta consciously and systematically used letters, algebra (theory of equations) was basically the algebra of language expression. At that time, equations were described in language, not written in such a concise form as AX2+BX+C = 0, and the equations people dealt with were only various very specific equations expressed in language. After David introduced symbols, the situation changed substantially. He used letters to represent unknown quantities and their powers. Today, letters are also used to represent so-called general coefficients (constant variables). Usually he uses consonants to represent known quantities and vowels to represent unknown quantities. With the help of symbols, the general formula AX2+BX+C = 0 of a quadratic equation can be given, which is a homonym of a class of equations and a general element, not a concrete equation. When the equation is generalized, people can consider its general solution and seek the solution of quadratic equation. This leads to the sublimation of people's understanding of solving equations. Here, it is obvious that the transformation from literal algebra to symbolic algebra, from the study of individual equations to the study of general equations, is based on the introduction of symbols. On the other hand, symbols are sometimes introduced to expand the existing cognitive scope, and the introduced symbols are new elements added in form. This often appears in the application of the principle of adding elements to be complete. For example, the natural number {65438+, in the range of 2, …, n, …}, addition and multiplication are closed and unimpeded, but their inverse operation, subtraction and division are not. In order to eliminate or break through this limitation, people introduce symbols 0,-1, -2, …, -n, …, so that A+x = b is always solvable, that is, subtraction is closed (eliminating no), so that the original laws of addition and multiplication are formal solutions of corresponding equations. Of course, symbols cannot be introduced casually, and the general elements of the corresponding interval are generalized. Here, the introduced symbols are the direct implementers of popularization, and popularization is an important form of mathematics popularization.
The main means to expand the scope of knowledge in abstract form is axiomatization (axiom can be regarded as the product of separating and summarizing the characteristics of specific things), including the modern axiomatization of form. After people have studied the axiomatic system, the corresponding properties of various concrete systems (satisfying axioms) will be clear. Algebraic structure is a typical axiomatization. Axiom A given object, no matter what its specific constituent elements are, is abstract as long as the relationship between the elements meets the axiom, because it is defined by nature (it is not the object that restricts nature, but the opposite). The conclusion of the axiomatic system is applicable to any concrete system that satisfies these axioms, and the conclusion drawn by the concrete system is only applicable to itself (whether other systems are established or not needs to be verified), so the axiomatic conclusion is more universal.
Generalization helps to enhance the profundity of understanding (universality and profundity are two basic characteristics of science). People generalize not only for generalization, but also for better and deeper understanding of particularity.
Accuracy and clarity are important signs of deepening cognition. Generalization is beneficial to the accuracy of cognition. For example, regarding the rank rk of a matrix, there are the following theorems in higher algebra:
For matrices An×m 1, Bn×m2, there are
max{rk(A)，rk(B)}≤rk(A，B)
≤min{n，rk(A)+rk(B)}。
It is impossible to give the expression of rk(A, b) by the common methods of higher algebra, but with the help of the generalized inverse matrix, we can do this and realize the accuracy of rk(A, b) formula:
rk(A，B)=rk(A)+rk[(I-AA+)B]
=rk(B)+rk[(I-BB+)A]。
Where I is identity matrix, A+ and B+ are Moore-Penrose inverses of A and B, respectively. Here, the generalization of the concept leads to the accuracy and quantification of the proposition.
Generalization is a way of systematic learning. If people list the concepts and propositions of a certain subject or teaching material in order from special to general, it will help people to systematically remember and learn. Theoretically, this table also has a certain guiding role in scientific research. We will explain these in detail in the next section.