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Structural method
Mathematical method
The construction method means that when solving some mathematical problems, it is difficult to solve the problem by using the usual method according to the directional thinking, the condition sum should be set according to the problem
conclusion
Observe, analyze and understand the object from a new perspective, firmly grasp the internal connection between the conditions reflecting the problem and the conclusion, use the data, shape, coordinates and other features of the problem, use the known conditions in the problem as raw materials, and use the known mathematical relations and theories as tools to construct mathematical objects satisfying the conditions or conclusions in thinking. Thus, the relationship and nature implied in the original problem can be clearly displayed in the newly constructed mathematical object, and the method of solving the mathematical problem can be easily and quickly with the help of the mathematical object.
- Chinese name
- Structural method
- Foreign name
- structured approach
- tool
- The efficiency of simplifying the problem solving process way
Intuitive mathematical stage
Intuitionism was pioneered in the late 19th century
Germany
Klonnick, he clearly put forward and emphasized the possibility, arguing that without the possibility, it can not be recognized as the existence of.
He argued that "the definition should include the method of calculating the object defined by a finite number of steps, and the proof of existence should permit the calculation to an arbitrary degree of precision for the quantity whose existence is to be established." He had planned to arithmeticize mathematics and rid the field of mathematics of all non-constructive elements and their roots. The second strong advocate was Poincare, who argued that the natural numbers were the most basic intuitions that could be believed without further analysis, and "like Kronnick, he insisted that all definitions and proofs must be constructive." The systematic founder of modern constructivism is Brouwer, who fully and thoroughly carried out and developed the view that "existence must be constructed" from both philosophical and mathematical aspects. Other major figures in this school are Hayding and Weyl.
Their basic positions in mathematical work are as follows: first, they believe that the starting point of mathematics is not set theory, but natural number theory. This is what Hayding said: "Mathematics began after the formation of the concept of natural numbers and their equality." So they did not allow the general concept of set theory into mathematics, but reduced all mathematics to natural number arithmetic and a constructiveness built with "spread shapes"
continuum
Conceptual assumptions. Second, it denies the universal validity of traditional logic and reconstructs intuitionistic logic. Thirdly, he criticizes the lack of structure of traditional mathematics and establishes the intuitionistic mathematics with structure. This began the first phase of the construction method, the period of intuitive mathematics.
Algorithmic mathematical stage
discover
Paradox of set theory
Later, some mathematicians decided that the only radical way to solve the problems raised by these paradoxes was to eliminate all general set theory concepts from mathematics and to limit the study to those objects that could be defined or constructed in a workable way." To this end, he abandoned many of the usual mathematical terms and introduced various supermathematical principles, thus making intuitive mathematics difficult to read. At the same time, intuitive mathematics absolutely excludes non
Constructive mathematics
And the wrong approach of traditional logic can not explain the validity of the latter in a certain range of applications. On this point, it is opposed by the vast majority of mathematicians. For mathematicians, Brouwer's theory has always been a curiosities, of interest mainly to logicians. Several other constructional tendencies have thus emerged, which, less extreme than intuitive mathematics, aim at limiting the range of allowable mathematical objects to a more or less arbitrarily selected class, rather than challenging the traditional rules of proof, as intuitive mathematics does. Among them, the "algorithmic mathematics" created by Markov and his collaborators is particularly remarkable. Algorithmic mathematics is a constructive method that reduces all the concepts of mathematics to one basic concept-algorithm. It begins with
Recursive function
Based on theory, therefore, its concept has a very strict definition: every function is treated by its Godel number method, every real number is a specific recursive function, and so on. The method used is standard constructive, and the logic adopted is intuitive logic. Thus, Markov's theory is one that limits not only the class of objects, but also what is permissible
Proof method
In the class of "strictly terminalist" theories, Shanin continued Markov's work by studying various classical theories in mathematical simulations of Markov's algorithm. He was even able to describe the analytic image
Hilbert space
and
Lebesgue integral
The constructional theory. Because of Markov's work, the constructional method entered the stage of "algorithmic mathematics". However, because this construction method depends on
Recursive function
The terminology of the theory, which makes it difficult for the mathematical layman to read, and the fact that Markov's successors seem to be more interested in complex theories and their applications to computer science than in the practice of algorithmic mathematics itself, has left algorithmic mathematics in a state of hibernation due to the lack of a suitable framework for mathematical practice.
Stage of modern structural mathematics
In 1967, after the publication of Bishop's book, it was declared that the construction method had entered the stage of "modern structural mathematics". Bishop reinvigorated the tectonic method by reconstructing an important part of modern analysis. His research topics ranged from measure theory,
Duality theory
,
functional
Micro product. In particular, his and Chin's new constructional measure theory, based on Daniel integrals, easily eliminated the question of where
Real line
The possibility of constructively countably additive measures is a matter of concern, and also proves that constructively
continuum
Uncountable in a strong sense. Although Bishop's work was rooted in the work of Brouwer, he was able to free himself from the self-imbibe concept of intuitive mathematics, avoiding the hypermathematical principles of intuitionism and getting rid of algorithmic mathematical pairs
Recursive function
The unnecessary reliance of the theoretical method has removed any constraints on the formal system and thus left room for further innovation. At the same time, Chope uses familiar customary terminology and symbols in mathematics, so it is easy for ordinary mathematicians to understand.
General steps of the construction method:
(1) Analyze the meaning of the question and find out what the problem involved in the question is.
(2) According to the questions involved in the question to find the key knowledge points involved in mathematics.
(3) Integrate relevant knowledge points into the topic meaning to find the basic form necessary to construct this knowledge point.
(4) Using the constructed form, combined with this knowledge point to explore the problem and find out the idea of solving the problem.
(5) Give detailed and correct answers.
Broadly speaking, there are two types of uses for mathematical construction:
1. It is used to find constructive explanations for the concepts and theorems of classical mathematics. In most cases, it is not easy to guess the constructive content of the classical theorems, if they exist at all. Let's give you an example.
Example 1 How to define the concept of real numbers in the sense of constructibility?
The specific approach of intuitive number scholars is to introduce the so-called "species" concept to replace the Cantor sense
Set concept
. Then Brouwer introduced the concept of "selection sequence" and replaced the rational numbers in classical analysis with "selection sequence of rational numbers"
Cauchy sequence
Concept, called "real number generator". Corresponding to classical analysis, the real numbers are defined as Cauchy sequences of rational numbers
Equivalence class
A single real number in the constructible sense is defined as an equivalent species of the real number generator. As seen above, there is no substantial difficulty in establishing the notion of constructability real numbers, because Cauchy-Weierstrass's entire limit theory is based on it
Potential infinity
The idea. Thus, in essence, the intuitionist is merely restating the Cauchy sequence under the requirement of feasibility.
The practice of modern constructivist numbers is that in order to construct a real number, we must give a finite method by which each of them can be combined
Positive integer
n is converted to a rational number xn ', and such that x1 ', x2 ',... Be a
Cauchy sequence
, which converges to the real number to be constructed. We must also give a clear estimate of the convergence rate of this sequence. Thus, modern constructional mathematics has detached itself from the concepts (such as selection sequences, species concepts) that seem to kill intuitive numerologists.
The classical statement of the fundamental theorem of algebra is: any non-normal number of complex coefficients
polynomial
f has at least one complex root. (1)
The most famous traditional proof for (1) is that, assuming f does not take a zero value, take
Liouville theorem
Applied to the reciprocal of f, it is concluded that 1/f is a constant and therefore f is a constant, completing the proof of the contradiction.
Constructivists would argue, however, that this proves not the fundamental theorem, but the weaker claim:
A polynomial on a complex number that does not take zero is a constant. (2)
At the same time, the above proof does not suggest how to find roots for polynomials.
Fundamental theorem of algebra
The constructional statement is given by Brouwer:
There is a finite method for any nonconstant polynomial f with complex coefficients that we can use to compute the roots of f. (3)
Now give Brouwer's proof of the fundamental theorem of algebra for polynomials with a leading coefficient of 1: he first proves that f can be assumed to be a polynomial of positive order in the Gaussian number field Q [i], then chooses the radius R to be large enough so that f(x) is dominated by its leading term, and then uses the number of turns of f around a circle centered on O and with a radius of R equal to f
Rank
For this fact, he constructed a Gaussian number z so that f(z) is minimal and f '(z) is relatively large. Finally, the complex root of f is constructed by Newton-Rafson iteration.
Comparing the constructive proof with the traditional proof, it can be seen that although Brouwer's proof is indeed more than used
Liouville theorem
The proof is longer, but the constructive proof is better than the traditional proof.
Amount of information
"Much more. For example, Brouwer's method can find the root of a polynomial with a leading coefficient of 1 for any given positive degree over a complex number. In particular, with his proof, you can find roots for polynomials of order 100, whereas traditional proofs do not involve finding roots at all.
Bishop writes that every classical theorem presents a challenge: to find a constructive statement and to give it a constructive proof. But in fact, many classical theorems do not seem to have any constructive claims and proofs, such as Bolzano's -
Weierstrass theorem
zorn lemma and so on.
2. For development
Constructive mathematics
A new frontier,
Combinatorial mathematics
The mathematics involved in computer science is a new field of constructive mathematics, especially graph theory is one of the typical fields of the development of constructive mathematics. Because the definition of graph is constructive, at the same time, many application problems of graph, such as computer networks, block diagrams of programs, fraction expressions, etc., are also very constructive problems.
Example 3 gives the constructional definitions of tree, minimum tree and tree graph.
A tree is an undirected graph in which any two vertices can be joined by a unique (i.e. no loops) method by a sequence (i.e. a chain) of pinions with common vertices. Each edge of the graph has a length, so that the total length of the smallest tree, called the minimum tree. The tree is a structure on the directional graph: if you do not consider
Line segment
The direction, then it is a tree; If you consider the direction of the line segment, there is one vertex v0 without any
Directed line segment
Point to it, and every other point vi has a unique directed line segment pointing to it. We call the directed line segment an arc and the vertex v0 the root of the tree. It can be seen that their definitions are intuitive and actionable, so they are constructive definitions.
Example 4 In 1965, Zhu Yongjin and Liu Zhenhong published a study on the orientation diagram
Minimum tree diagram
"The article. Their proposed algorithm for minimum tree graphs is summarized as follows:
(1) For each point vi, except v0, a shortest ai is selected from the arcs pointing to vi, and if the selected arcs happen to form a tree, it is the shortest tree. Otherwise, the selected arcs form some disjoint loop ci.
(2) Let c be a loop, v is a point on c, then there is only one arc on c pointing to v, denoted as a(v), its length denoted as l [a(v)], let w be the vertex at c, and l(w, v) is an arc of the graph, its length is l(w, v). The length of l(w, v) is now modified and defined as l(w, v)=l(w, v)-l [a(v)]. After you do this for every point v on c, you shrink c, and you get a new graph.
Repeat (1) and (2) two steps, and finally get the tree diagram on the contraction diagram.
(3) The contraction points in this tree diagram are re-stretched into a loop in turn, and at this time, some points on the stretched diagram have two arcs pointing to it, so after removing the arc on the return road, it becomes the tree diagram. Repeat this step until there are no shrinking points. And this is the tree, which is
Minimum tree diagram
. It can be seen that this algorithm is a method that can be realized by a finite number of steps in a fixed way, so it is a constructive method.
Another branch of mathematics that benefits greatly from the application of constructive methods is numerical analysis.
Example 5 gives the theorem at the heart of numerical approximation theory:
Let X be real
Normed space
There is a finite dimension in E
Linear space
So, for every point a in E, corresponds to a point b in X, such that the distance from a to b is equal to the distance from A to X. (4)
Find an appropriately constructive alternative proposition. To this end, the following concepts are introduced:
Define 1 to be E
Distance space
X is the nonempty subset of E, and a is an element of E. If for each pair of distinct elements x, y, there is z in x such that max{d(a, x), d(a, y)} > d(a, z), then a is said to have at most one optimal approximation in x.
Definition 2 If d(a, x)≥d(a, b) is true for all x in x, then an element b in X is said to be the optimal approximation of a in X.
We then find an appropriate constructive alternative:
Let X be real
Normed space
One of E has finite dimensions
linearity
a subspace, where A is an element of E, has at most one optimal approximation in X. So, we can calculate the optimal approximation of a to X. (5)
According to classical mathematics, (4) is equivalent to (5). However, the constructional proof of (5) includes an algorithm with extremely broad applications. 1.
Also, topology, especially
dimension
Theory is also a branch of mathematics that can provide examples for the insight of the constructional method, so it is also a new field of constructional mathematics to be developed.
In order to fully appreciate the difference between constructive and non-constructive mathematics, two principles of operation prevail. The first principle is the law of exclusion of middle, which is valid in non-constructive mathematics but unacceptable in constructive mathematics. The second is what Bishop called the omnipotence limit Principle (LPO); If (an) is a sequence on {0,1}, then either an=0 for all N, or there exists N such that aN=1.
The constructive interpretation of LPO is that we have a finite method that applies to any sequence (an) on {0,1}, either to prove that for every n an is zero, or to construct aN such that aN=1. If such a method is true, then we have uniform solutions (such as Fermat's last theorem, Riemann's conjecture, and Goldbach's conjecture) to a large number of outstanding problems, and we can assert that such a broad unified solution will never be found, so LPO is not the working principle of constructive mathematics. Therefore, classical theorems that require the use of LPO to be proved constructively, or that do not require the use of LPO but require some other principle similar to the way LPO is constructed, are essentially non-constructive.
The relation between constructive mathematics and non-constructive mathematics is manifested in symbiosis and bifurcation. So far, advances in the constructive approach to mathematics have followed directly from standard non-constructive mathematical ideas. Therefore, people often have an illusion that the development of constructive mathematics is parasitic on non-constructive mathematics. In fact, constructional mathematics can often provide more natural and simpler proofs of certain theorems than non-constructional mathematics, and may even lead to some new non-constructional theorems. Therefore, the relationship between these two types of mathematics is a symbiotic relationship that complements each other.
On the other hand, a classical theorem has several classical equivalents, some of which can be proved constructively, on the premise that the law of exclusion is recognized, so that it often happens that a classical theorem has several very different constructive interpretations. Picard's last theorem in complex analysis, for example. This is the "bifurcation" of interpretation. This bifurcation is one of the most interesting and productive aspects of constructive mathematics.
Wang Hao, a Chinese-American mathematician, believes that"
Constructive mathematics
Is the mathematics of doing, non-constructive mathematics is the mathematics of being." Mathematics is about information pattern and structure, and mathematics is about information processing. Chinese mathematician Hu Shihua thinks that the tendency of constructive mathematics is to use mathematics to obtain results and construct results, to focus on the construction practice of thinking, and to use the law of exclusion in a limited way. The tendency of non-constructive mathematics is to understand problems and laws mathematically, build mathematical models and form mathematical theoretical systems. In pursuit of scientific ideals, we can freely use the law of exclusion from the middle.
Constructivism and non-constructivism mathematics not only have the above differences, but also have certain connections, they complement each other. Advances in the constructive method of mathematics are derived, consciously or unconsciously, directly from non-constructive mathematical ideas; Non-constructive mathematics always contains elements of constructive mathematics, and pure non-constructive mathematics does not exist.
High school mathematics mainly studied arithmetic series and arithmetic series, but in the usual exercises, often encountered not these two kinds of series, so sometimes need to use the construction method to convert it into arithmetic series or arithmetic series.
The construction of geometric series
Encounter a
(n)
=M×a
(n-1)
When +C (C is a constant), the geometric series can be constructed.
a
1
=1, a
(n+1)
=2a
(n)
+ 1
Get an a
(n)
+ 1 = 2
n
Thus a
(n)
= 2
n
- 1
For example, find the greatest common divisor of 525,231.
525=231 times 2+63,
231=63 times 3+42,
63=42 x 1+21,
42=21 times 2.
Finding the greatest common divisor of the above two numbers is obtained after a finite number of steps, so it is a constructive method.
Another example is the quadratic equation ax with one variable
2
The root of +bx+c=0 can be found in a finite number of steps using the root finding formula. This is also a constructive approach.
Now consider
Continuous function
the
Maximum value theorem
: A continuous function on a closed interval has a maximum (small) value. in
Mathematical analysis
When we prove this theorem, we only talk about the existence of this maximum value, and we do not give a feasible procedure to compute this maximum value in a finite number of steps, which is a non-constructive method.
A graph is some vertices and some
Line segment
In graph theory, the exact definition is given, and this definition belongs to constructivism.
Through the above examples, it can be clearly seen that the construction method has the following two basic characteristics:
1. The object under discussion can be described intuitively;
2. The concreteness of realization is not only to determine the existence of a certain solution, but also to realize the concrete solution.
The construction of arithmetic series
For a
(n+1)
=M×a
(n)
The form +f(n) (f(n) is not a constant) can be used to construct an arithmetic sequence.
Get an a
(n+1)
=2×a
n
+ 3 x 2
(n - 1)
Divide both sides by 2
(n - 1)
Have to
therefore
Is an arithmetic sequence
therefore
So we can represent a
n
