Cantor's set theory paper 1: human nature based on set theory.
Abstract: As human beings, it is necessary for us to know ourselves, so that we can go further. Humanity is the humanity that fundamentally determines and explains human behavior. This paper discusses this by using the idea of set theory.
Keywords: humanity; Rationality; Sociality; Nature; Set theory thought
I. Introduction
In the long-term life, the human brain will store the information of some things under the unconscious action. Because there is no rigorous thinking in the brain, most of this information is external, just some morphological features on the surface of things. This information is not scattered and there is no connection between them. But there is a certain correlation between them. Although the structure is not rigorous, there may be mistakes. But sometimes it can work. But we can't just rely on this ideology, because we have self-awareness and need to constantly improve and progress. It is impossible to see the essence of things clearly by relying on such consciousness.
Sometimes when you ask someone why, he may answer:? Intuition? . I don't deny that intuition brings? Is it convenient? But what about this? Is it convenient? It's an excuse not to think about the nature of things. Intuition is also an ideology, but this kind of consciousness is subconscious, so it takes a long time to form consciousness. The brain can constantly adjust and improve itself, but this process is quite slow. You can't rely on this idea to make progress.
What I want to say now is that we must reduce our dependence on these consciousness. Because these consciousnesses are not the products of rigorous thinking, it is easy to make mistakes in making some reactions with such consciousnesses. This will also hinder our exploration of the real world. We should dig out this kind of consciousness, analyze its ideological structure, eliminate bad ideas, and constantly strengthen and improve defective ideas. In this way, we will be more rational. People have such a rational nature. Therefore, human beings can progress and civilization can develop.
Second, theoretical analysis.
Suppose A={a 1, a2,? ，an}，B={b 1，b2，？ , bm}. If a? B, which means that all n elements in A can be found in B, m >；; N. On the contrary, all elements in the description can be found in A and N >: M. If A=B, all elements in the description are the same as those in B, and n = m, if an element can be found in the set A, is it recorded as A? Answer.
Combining the above ideas, analyzing human and animals, animals = {frogs, fish, dogs, cats, people,}, we can see that people belong to animals, that is, people and animals. And such a set is called a common set to distinguish the set of attributes described below. Since frogs, fish, dogs, cats and people all belong to animals, that is to say, they have the same properties, such as no cell wall, and they must use ready-made organic matter to obtain energy, and they can move freely without chloroplasts. But people have other attributes besides these same attributes. That is to say, from the point of view of attribute set, the attribute set of animals is included in the attribute set of people. That is, all the attributes of animals are owned by human beings. We name the elements in the attribute set? What's the difference? And name the ordinary series? Kind? The elements in the public collection are named. Genus? .
If the property set of B is included in the property set of A, then A and B have the same genus difference, and all genus differences of B are genus differences of A. The more genus differences, the smaller the expression range of the property set, that is, the more restricted it is. Then B is obviously more extensive than A, which means that B can describe A, that is, A is B, where A is the subject and B is the object, then all differences of B are differences of A. ..
So according to the above, animals can express people, that is, people are animals. ? People? Is the difference? Animals? More, which means more restrictive conditions.
Some things exist in the subject, and their definitions cannot be used to express a subject. For example, for white people, white people? Depending on the subject of the body, it is used to express the subject of the body, that is, the body can be said to be white, but pay attention. White? The definition of cannot be used to describe the body.
The difference between genus and species can be applied to the first entity, and the difference between species can be applied to genus, so genus and species determine the nature of the entity. For example:? People? And then what? Animals? All differences apply to individuals. It can be said that people are animals, individual people are people, and individual people are animals. You can also think like this: right? Animals? The definition of must also apply, right? People? The definition of, because? People? Belong to? Animals? Yes So-called? The first entity? , for example? Personal? 、? Individual tigers? And so on, are real individuals, independent of other individuals. [ 1]
The definition of genus difference also applies to genera and individuals, and can also be used to express genera and individuals. For example:? With your feet? 、? By hand? The definition of can also be applied to? People? Being with someone. You can also say. People? With someone else? By hand? . Since the definition of species difference can be applied to individuals, species difference can also determine the nature of individuals. Moreover, these properties can be expressed as individuals through genus difference.
At this point, we should feel a little thinking. In other words, we now need to find such genus differences, and then express individuals according to the definitions of these genus differences.
But there is also a premise, that is, are individual people entities? Because we have just come to a conclusion that genus and species determine the nature of entities. In other words, these analyses are all based on entities. So we need to know whether an individual is an entity. In fact, from the most primitive and fundamental definition of entities, individuals do belong to entities, because they are real and do not depend on other subjects.
Third, the result analysis
1. People are rational: Are there any articles about fish? Suicide? This report. I was thinking, what about fish? Suicide? What about yours? Suicide means that fish have self-awareness and can choose their own death. But science shows that all animals in nature except human beings (not the whole universe here) have direct consciousness, but no self-consciousness. Is science not objective? Actually, it's not like this. It's just a deliberate exaggeration by the media. Fish just react instinctively because of the change of environment. This instinct is direct consciousness. Fish didn't think about whether it would lead to death, just instinctively. Then, compared with other animals, the difference is that people are rational.
For example, when a tiger is hungry, it will pounce on food. But when people are hungry, they don't jump on food when they see it, but think about whether they can eat it. This is the difference from other animals. That is to say? Rational? what's up People? One of them is poor.
2. People are social: people communicate and exchange information with other individuals in society. Share, divide and exchange materials. Society is interactive and cannot be supported by individuals. In other words, we are in a society, and only when we get together can we share, share and communicate together. Some people say that they are lonely, but in fact this is not true loneliness, and there can be no true loneliness. Because people cannot exist without sociality. Maybe someone will say to me just now? There will be no real loneliness? If they have an opinion, they will say: since there is no loneliness, isn't it meaningless to create the word? Loneliness is only a human feeling, and feeling can't reflect the real law of things. So as I said before, we must give up some wrong ideas. So as not to be blinded by feelings and superficial phenomena.
In the huge group activities of human society, no matter what simple activities, it is inevitable to exchange information with other individuals. Only in this way can human beings develop and multiply. In this way, animals should also be social. This is obviously for sure. Some animals also have this property, such as ants, bees and so on. Visible? Sociality? What else? People? One of them is poor.
3. Man is natural: as a member of nature, man cannot be unnatural. Some basic characteristics produced by human organizational structure, physiological structure and natural communication process all show human naturalness. It is impossible for human beings to exist independently from nature. And other creatures also have this property. So what? Natural? What else? People? One of them is poor.
Four. Concluding remarks
As human beings, it is necessary for us to know ourselves so that we can make greater progress. It is the highlight of this paper to analyze human nature with the thought of set theory. In addition to these three attributes, there are other attributes. Because of my limited wisdom, I won't give more properties here, but the focus of this paper is to provide a feasible analysis method. Through mathematical logic, analysis will become more rigorous and systematic. This is a bold attempt of this paper.
References:
[1] Aristotle. The Complete Works of Aristotle (Volume I) [M]. Miao, translate. Beijing: Renmin University of China Press, 1990.
Cantor's Set Theory Paper 2: Set Theory and the Third Mathematical Crisis.
The emergence and development of mathematics has always been closely related to the production and life of human society. In the new textbook, the introduction of any new concept emphasizes its realistic background, the development background of mathematical theory or the historical background of mathematical development. Only in this way can students feel that the development of knowledge is natural. Therefore, it is especially hoped that the relevant knowledge of the history of mathematics can be infiltrated from time to time in teaching, and the educational value of the history of mathematics can be fully exerted and utilized, so that students can understand mathematics more comprehensively and deeply through understanding the history of mathematics.
First, the birth of set theory
It is generally believed that set theory was born at the end of 1873. 1873165438+1October 29th, Cantor (1845- 19 18) gave it to Dai Dejin (18365438). 19 16)? Whether the positive integer set can correspond to the real number set one by one is a big problem that leads to the emergence of set theory. A few days later, Cantor proved the negative result of this question by reducing to absurdity. Are real numbers uncountable sets? And this result was published in the German "Clare Journal of Mathematics" with the title "A property of all real algebraic number sets". The first revolutionary paper on infinite set theory? In his series of papers, he first defined set, infinite set, derived set, ordinal number and set operation. This article by Cantor marks the birth of set theory.
Secondly, set theory has become the foundation of modern mathematics building.
Cantor's set theory is the most revolutionary and creative theory in the history of mathematics. He dealt with the infinite set of the most difficult objects in mathematics and made countless reasons? Infinite? Mathematicians who have been troubled for a long time have found their spiritual home in this magical mathematical world. Its concepts and methods have penetrated into many branches of mathematics such as algebra, topology and analysis, and even into other natural disciplines such as physics, providing basic methods for these disciplines. It can almost be said that it is difficult to have a profound understanding of modern mathematics without the viewpoint of set theory.
In the 20 years before and after the birth of set theory, it experienced many hardships, but it was finally recognized by the world. By the beginning of the 20th century, set theory had been generally accepted by mathematicians. Everyone agrees that all mathematical achievements can be based on set theory. In short, with the help of the concept of set theory, the whole mathematical building can be established. Even Jules Henri Poincare, a famous mathematician who strongly opposed the birth of set theory, (1854- 19 12) happily announced at the second international congress of mathematicians in 1900. With the help of the concept of set theory, we can build the whole mathematics building. Today, it can be said that it has been absolutely strict. ? However, the good times did not last long, and a news that shocked the mathematics community came out that set theory was flawed! If so, it means that there is a hole in the foundation of the math building. How terrible it will be for the mathematics community!
Thirdly, the paradox of Bertrand Russell (1872- 1970) led to the third mathematical crisis.
1903, the British mathematician Russell gave a paradox in his book Principles of Mathematics, which clearly showed the contradiction of set theory, thus shaking the foundation of the whole mathematics and leading to the emergence of mathematical crisis. The third mathematical crisis? .
Russell constructs a set r that does not belong to himself (that is, it does not contain itself as an element). Now, does R belong to R? If R belongs to R and R meets the definition of R, then R does not belong to itself, that is, R does not belong to R. On the other hand, if R does not belong to R, then R does not meet the definition of R, then R should belong to itself, that is, R belongs to R, so it is contradictory in any case, which is the famous Russell paradox (also known as Barber paradox).
Russell's paradox not only shook the foundation of the whole mathematical building, but also spread to the field of logic. Frith, a famous German logician, received a letter from Russell about this paradox when his Foundation of Set Theory was about to be printed. He immediately found that a series of achievements that he had been busy for a long time were ruined by this paradox. He could only write at the end of the book: The worst thing for a scientist is to find that the foundation of his work has collapsed when his work is about to be completed. ? In this way, Russell's paradox has influenced mathematics and logic, which have always been considered extremely rigorous.
Fourth, eliminate the paradox and resolve the crisis.
The existence of Russell's paradox clearly shows that there is something wrong with set theory. Since mathematics in the 20th century was based on set theory, many mathematicians began to devote themselves to eliminating contradictions and solving crises. Mathematicians put forward their own solutions, hoping to reform Cantor's set theory and eliminate the paradox by limiting the definition of set, which requires the establishment of new principles.
At the beginning of the 20th century, there were probably two methods. One is the axiomatic set theory put forward by mathematician zermelo (zermelo, Ernst Friedrich, 187 1 ~ 1953) in 1908. The original intuitive concept of set is based on strict axioms, which fully restricts the set to eliminate known contradictions, thus avoiding the emergence of paradox.
Prior to this, Russell, the maker of the crisis, put forward a hierarchy theory to solve this contradiction in his works, also known as branching. But this hierarchy theory is very complicated, and Zemelo simplified this method and proposed? Zermelo-fraenkel, axiom of elementary set, axiom of separation, axiom of power set, axiom of union, axiom of choice, axiom of infinity? By introducing these seven axioms, some inappropriate sets are eliminated, thus eliminating the conditions for Russell paradox. Later, Zermelo's axiom system was revised and supplemented by others, especially frankl and Scoland, and became a modern standard? Zermelo and frankl axiomatic system (ZF system for short)? In this way, mathematics has returned to a rigorous but not contradictory field, and it has also promoted the rapid development of a new branch of mathematics-basic mathematics.
Enlightenment of verb (abbreviation of verb) crisis
Cantor's set theory has been put forward for more than one hundred years, and great changes have taken place in mathematics, all of which are inseparable from Cantor's pioneering work and the hard work of mathematicians. From the emergence to the solution of the crisis, we can see that the development of mathematics can not be separated from asking questions and facing difficulties. During this period, we will experience numerous setbacks and failures, but as long as we persist, we will eventually succeed.
The elimination of contradictions and the resolution of crises often bring new contents, new changes and even revolutionary changes to mathematics, which also embodies the basic principle that contradictions and struggles are the historical driving force for the development of things. As the mathematician Felix Christian Klein (1849-1925) said in The Loss of Mathematical Determinism:? Uncertainty and doubt related to future mathematics will replace past certainty and complacency. Although this paradox has been explained and the crisis has been resolved, it is more unknown, because as long as it is carefully analyzed, the contradiction will be discovered by researchers with deeper understanding. This discovery should not be considered as? Crisis? We should feel that the opportunity for the next breakthrough has arrived. ?
References:
1. "Compulsory Mathematics Course 1", standard experimental textbook for senior high schools, used by teachers, People's Education Press.
2. Hu Zuoxuan, "The Third Mathematical Crisis"
Cantor set theory paper 3: metaphor from the perspective of fuzzy set theory.
From the perspective of fuzzy set theory, this paper studies the logical truth in the process of metaphor understanding, and reveals that the fuzziness of metaphor is inherent and objective, which plays an important role in human understanding of the world and literary creation.
Fuzzy set theory; Metaphor; Literary works
Fuzziness is one of the essential features of natural language. The fuzziness of objective things, the limitations of human cognition and different discourse contexts will all lead to the formation of fuzzy language. Since the birth of fuzzy set theory, it has been crossed with many disciplines, and the combination with linguistics has given us a new perspective of semantic research. Metaphor, as a special semantic phenomenon, shows the characteristics of vague language in its interpretation. The fuzziness of metaphor reflects the potential logical laws of human beings, which is objective and implicit. It is not only the result of human psychological categorization, but also the product of human fuzzy thinking, so fuzzy set theory opens a new window for us to study and analyze metaphor [1].
1965 Zadeh, an American cybernetic expert, was inspired by linguistic fuzziness and published a paper "Fuzzy Sets" in Information and Control, which was first proposed? Fuzzy set theory? The concept of. Traditional set theory emphasizes that any member of a set either belongs to it (membership degree is 1) or does not belong to it (membership degree is 0), and there are only two truth cases [2]. However, if we classify many objects in nature, we often can't find the basis for accurately determining their identities. Therefore, Zadeh's definition of fuzzy sets in his paper "Fuzzy Sets" is: Let X be an interval composed of points, and the class attribute elements in the interval are represented by X, that is, X ={x}. In the interval X, the fuzzy set A is represented by the membership function fA(x), which has the attributes of the elements that make up the set. This function is associated with any real number in the interval [0, 1], and the corresponding value represents the qualification degree of X forming a ... If two critical points are set in the interval, namely 0.
Fuzzy set theory provides the basis for the legitimacy of metaphorical truth. The understanding of metaphor depends on the understanding of two different categories. What if we want to put? A is b? As metaphor, not literal meaning, then we need to determine the meanings of A and B. Syntax, semantics and context can help us to determine their meanings, but the interpretation of the final meaning determines the screening results of similar attributes and different attributes [3]. In order to understand the comparison process of metaphor semantic attributes, we can turn to the concept of fuzzy set theory. By blurring the boundaries of different sets, the attributes of one set referred to by metaphor can be combined with the attributes of other sets, thus overcoming the obstacles caused by accurate definition. From the surface structure of language, the ontology set of metaphor is incompatible with the vehicle set. If we apply the open principle of fuzzy logic, we can compare and distinguish the attributes in these two different sets, and find out the similar attributes and incomparable attributes.
In the name of Shakespeare? Juliet is the sun. ? Juliet is the sun. For example, the sun? Is a subset of inanimate semantic markers. Juliet. It is a subset of life semantic markers. Because this metaphor points out the similarity between the importance of the sun to human beings and the importance of Juliet to Romeo, the membership function of the related element attribute is a value less than 1, which makes this metaphor more enlightening and suggestive. Generally speaking, according to logical truth, metaphor can be divided into denotative metaphor (figurative metaphor) and suggestive metaphor (suggestive metaphor). In Metaphor and Reality published by 1962, P. Wheelwright pointed out that the basic function of epiphor is expression, while diaphor's main function is suggestion [4]. The juxtaposition of metaphors will cause contradictions in semantic sets, so some scholars regard metaphor as an entity that does not conform to grammar and logic. But if we explain metaphor through the ternary logic in fuzzy set theory, we can prove that its use is legitimate. According to Zadeh's standards, 0
The essence of metaphor is to blur the boundary between ontology set and vehicle set, so as to find the coincidence point between them. Because fuzzy set theory sets three interval boundaries? 、? And then what? , and 0.
refer to
[1]Earl R. MacCORMAC, Metaphor and Fuzzy Sets [J]. Fuzzy Sets and Systems. 1982(7).
[2] L.A. Zadeh. Fuzzy sets. Information and control. 1965(8).
[3] An Jun. The logical characteristics of metaphor [J]. Philosophical Research, 2007(2).
[4] Su Lianbo. A Study on the Cognitive Mechanism of Metaphor Fuzziness [J]. Journal of Chengdu University (Social Science Edition) 20 1 1(5).
[5] Shu. On the basic types and syntactic and semantic features of metaphor [J]. Foreign Languages, 2000( 1).
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