1. For people who don't understand philosophy, they often myth philosophy, saying that philosophy is the highest science, which is higher than all sciences; This is not objective and practical;

2. For people who are obsessed with mathematics, it is not objective to say that mathematics is "the language of God (Gauss)";

3. Mathematics and philosophy are the knowledge of understanding and transforming nature in human development. This kind of cognition can only be found that it does not change with the change of people, that is to say, mathematics and philosophy have objective characteristics and are not transferred by human will;

4. Mathematics and philosophy are both related and different: because they are both reactions to objective things, mathematics and philosophy are both discoveries of the material world, and they are bound to be related; And there are differences between them, because objective things are developing and their appearances are not only the same, so they must be different in mathematics and philosophy;

5. It is not that mathematics is a subject that studies concepts such as quantity, structure, change and spatial model, and some research methods in mathematics are also applicable to philosophy; Similarly, the methodology in philosophy also inspires and helps the study of mathematics. So mathematics and philosophy can complement and transform to some extent, because objective things can also complement and transform.

6. It is unscientific to say that philosophy is ontology and a summary of all thinking and methods; Things are constantly developing, and the methods of studying things need to be constantly developed. But focusing on the development of things will form an independent discipline, and there will be new research methods and thinking summaries, which is not the category of philosophy; Therefore, philosophy and all disciplines are equal, not antagonistic, nor superior to other disciplines; They all focus on the objective cognition of their respective fields, with the continuous development of objective things.

7. To correctly understand the relationship between mathematics and philosophy, we must oppose the view that mathematics is a tool and means of philosophy. This view obliterates the characteristics of mathematics as a methodology, virtualizes the "highest scientific theory" of philosophy, fails to understand the metaphysical theory of philosophy, challenges the cognition of different aspects of objective things, and is ignorant of the dialectical unity characteristics of objective things.

Summary:

Mathematics is a logical system, concise and clear, with few ambiguities. There are not only various numbers and functions in mathematics, but also symbols such as addition, subtraction, multiplication, division, greater than, less than, equal to, exponent, derivative and integral. It is also an established concept with almost no ambiguity. In particular, geometric methods can express complex logical relations with clear and intuitive coordinates or graphics. In school study, we often use the method of geometry to describe the application problems of various disciplines, so as to clearly see the logical relationship between various factors, and then list the appropriate mathematical formulas to solve the required problems.

Formal logic can use geometric figures to express the complex logical relations of various concepts. Philosophy is also a science, and of course it can be expressed in this scientific way. Formal logic requires all concepts to be definite, so that it can carry out normal reasoning and operation.

Dialectics holds that any concept is determined under certain conditions, and different conditions may lead to different results, so we must study different conditions and different results when determining concepts. Specific research on several different conditions and different results can only be done one by one by limited means and following metaphysical methods. Simply put, the essence of dialectics is to point out the different results of things under different conditions. The relationship between the conditions for determining a concept and the defined concept is similar to the functional relationship in mathematics.

y = f ( x)

In mathematical terms, Marx said this. "The function of one variable is another variable, and its value changes with the value of the former, which means it depends on the former." We can use formulas and concrete examples to express the above concepts. For example, in Y=X+ 1, when x is greater than 1, y is greater than 2. In Y=X+ 1, when x is less than 1, y is less than 2. In Y=X+ 1, when x equals 1, then y equals 2. Each of the above three sentences is a metaphysical expression, expressing a certain concept under certain conditions. When we analyze the above three metaphysical expressions together, there will be qualitative changes. We say that this is both a metaphysical expression and a dialectical expression. Because it points out the different results of things under different conditions. We can also say that y is greater than 2 under some conditions, less than 2 under some conditions and equal to 2 under some conditions. This is also a dialectical expression. It can be seen that some so-called dialectical expressions only omit specific conditions in several metaphysical expressions and replace them with an uncertain concept.

It is through research that these so-called dialectical and uncertain concepts can be turned into definite and metaphysical forms, and scientific progress can be realized. Dialectics holds that any concept is determined under certain conditions. In the eyes of dialectics, any constant is determined as a constant under certain conditions, and the concept of any mathematical symbol is also determined under the condition of unity, which is a function of determining its conditions.

All the conditions in the application problems in school are assumed to be certain, and any certain concept in real life is determined under certain conditions. Therefore, it is necessary to find out the functional relationship between these concepts and determine their conditions. The functional relationship between a concept and what conditions in a specific problem can only be determined according to the specific situation. Conditions themselves are also composed of concepts. The concept that constitutes the condition itself has a functional relationship with another set of concepts that determine it.

This cycle is endless.

Theoretically, we can infer that in practice, people's energy is limited, and we can only decide whether to further study a concept and determine the functional relationship between its conditions according to the specific situation and on the premise of meeting the actual needs.

The relative meaning of opposing concepts.

To understand the law of unity of opposites, we must understand the relative meaning of the concept of opposites. We can draw a coordinate axis. Concrete things are like a point on an axis, and each point has a concrete value. However, only concrete values cannot determine the nature of opposition. The concept of opposites has definite meaning only if it is discussed by comparing two or more figures.

Up and down, left and right, front and back, depth, height, distance, size, weight and other concepts of the opposite object's orientation, volume and weight are better understood. Sometimes we feel that there is no second point to refer to, but it is actually based on a conventional ellipsis condition. For example, people are used to distinguishing the upper, lower and left sides with the reference point in front of the observer, measuring the house price and food price with their own income, and measuring the stock price with the company's net assets or price-earnings ratio. Without a reference point, we can't draw a definite conclusion about a certain point on the coordinate axis. Coordinate axis and reference point are both indispensable conditions for determining opposing concepts. The same is true of abstract concepts such as good and evil, truth and falsehood, beauty and ugliness, and good and evil. There are standards in people's minds, and it is meaningless to discuss the concept of opposition without standards. Unfortunately, some people still don't understand this point, thinking that the relative meaning of opposing concepts is just to reverse black and white, and they are talking nonsense without facts. Black and white are concepts that can only be determined by comparing two things with different gray levels. Just like everything is in a certain gray scale, everyone is the unity of positive factors and negative factors, all in a certain definition domain on the coordinate axis. The bad guy is a concept that can only be determined by comparing with others. From the perspective of opposing Taiwan independence, Chiang Kai-shek is good and Chen Shui-bian is bad.

So is the relationship between metaphysical method and dialectics. Each concrete method is a point on the method axis. In practice, people can neither use absolute dialectical method nor absolute metaphysical method, but have both. It depends on which method you compare.

From the perspective of understanding the shape of cattle, there are two methods: partial touch and whole photo. Compared with them, taking photos is a dialectical method to understand the whole, and touching is a partial metaphysical method. Compared with the method of establishing a three-dimensional model, taking photos is a metaphysical one-sided way to look at problems, and a three-dimensional model is a dialectical and comprehensive way to look at problems. Compared with stereoscopic perspective method, stereoscopic modeling is only a metaphysical method to observe things from the surface, and perspective is a dialectical method to deeply understand the internal morphology of cattle.

Compared with the method of understanding geometric shape, it is a scientific method to understand and improve cattle in essence by deeply understanding the domestication, hybridization, breeding, variety and quality of cattle, using genetics, molecular biology, transgenic and other methods, although these scientific methods have more superficial characteristics of metaphysical methods. Any scientific progress can only be achieved by metaphysical and conceptual methods. Dialectics and metaphysical methods do not have the problem of who is good and who is bad. It's all tools. It is a user's choice to use the right tools in the right place according to different needs. Whether it is used well or not is entirely the responsibility of the user.

Quantitative and qualitative relationship

Simple quantitative changes, at a certain point, will become qualitative differences. In the process of derivation, when the arc length and chord length tend to zero, the tangent slope of the arc becomes the slope of chord length. When time and distance tend to zero, the average speed becomes instantaneous and the finite becomes infinite. In algebra, adding a negative number equals subtracting a positive number. In multiplication, the multiplication of two negative numbers equals a positive number, and the negative number is a positive number.

In differentiation, we take the difference first, then discard the difference, so that dx/dy becomes 0/0, and then we can deduce the dialectical result by metaphysical rules. Engels said in Dialectics of Nature, "Our subjective thinking and the objective world follow the same laws, so we can't contradict each other in the end, but must be consistent with each other. This fact absolutely supports our whole theoretical thinking. This fact is the instinctive and unconditional premise of our theoretical thinking. "

"Dialectics is considered to be the science about the most universal laws of all movements. In other words, the law of dialectics must be equally applicable to the movement in nature and human history, or to the movement of thinking. " Only differential calculus can make natural science not only express the state with mathematics, but also express the process and movement. I agree with Engels' above views. The laws of philosophy are consistent with all natural laws, including the laws of human society and thinking. The laws of philosophy can only be called real science if they are consistent with other scientific laws. Unifying philosophical concepts with other scientific concepts is the premise of maintaining the consistency of scientific laws. Engels also said, "Calculus is essentially nothing more than the application of dialectics in mathematics". I not only agree with Engels' assertion, but also think that the opposite is true. I think the methods and laws of function and calculus are, in a sense, the methods and laws of dialectics. Mathematics includes arithmetic, algebra and advanced mathematics. The unity of arithmetic law, function law and calculus law in mathematics proves the unity of dialectics and metaphysics law. The rules of mathematics are consistent with those of philosophy.

First, I only give a solution here.

In my opinion, the so-called Russell paradox cannot exist forever, or there is no solution, or it is not a problem at all. When rules are made or practiced, there will be these results.

1, thinking that the rules are unreasonable, modify the rules and change them to "Barbers can shave their own hair";

2. The rules do not limit the generation of hairdressers. Then, the hairdresser can take an apprentice and shave his head.

3, the barber can't bear it, but he sticks to his principles, but there is no new barber. At this time, the barber will simply find a person who is not a barber to shave at will;

Stick to the principle but don't want new methods, so the barber keeps his hair and never shaves it.

Therefore, there is nothing wrong with Russell paradox. If there is a problem, that is, the barber didn't come to shave his head, that can only be the reason: you are stuck in a rut and don't want to change it or change it in other ways. Then Russell's paradox is indeed unsolvable. But there is no solution to this problem because the premise restricts the result, and the premise stipulated by Russell paradox contains contradictions. The result can only be to either modify the premise, adapt it, or insist that the premise will never be solved. No solution is not a problem, not all problems have solutions. X+y=6, is there a solution? There are countless solutions, limited to x= 1 Is there a solution to this equation? Yes Is there a solution to the restriction that x= 1 and y cannot be equal to 5? No. There is no solution. In fact, the Russell paradox, which insists on rules and is inflexible, is indeed unsolvable.

Second, one of the crises of mathematics: no time.

There is no time factor in set theory, only basic concepts such as element and inclusion. There can be a mapping relationship between sets, but there is no concept of time. In the view of set theory, the element itself is unchangeable. However, the objective world mapped by set theory changes with time. At this time, when describing the objective world, set theory will face the situation that cannot be described because of the variability of the world or its own limitations. Expressed vividly by Russell's paradox, the rules in Russell's paradox are immutable, the elements of the village, namely the attributes of hairdressers and other villagers, are immutable, and villagers cannot become hairdressers. But the objective world is actually developing and changing, and it is not against the rules for villagers to be barbers. However, Russell's paradox does not envisage this kind of accommodation without contradiction. In fact, any rules must interact with the objective. The rules of WTO, the laws of all countries in the world, and even the reimbursement system are constantly being modified by feedback. In addition to the loopholes in the rules themselves, there is another reason that the objective situation has changed.

There are two ways to solve the problem of set theory. One is to limit the application scope of set theory to a relatively static and fixed form. Another method is to add the set theory to the time factor, thinking that the set itself and the elements themselves are also developing and changing. The latter view can be regarded as the set theory of system theory. However, it should be noted that although there are problems, it does not mean that set theory is bankrupt, but lacks wider practicability. This relationship is similar to that between Newtonian mechanics and relativity. If we solve the problem in another direction, the method is to add time factor to set theory, and classical set theory adds time factor, which is similar to Newtonian mechanics adding light speed invariance theory.

So the conclusion is: if we look at the problem from the perspective of set theory, static and unchangeable, Russell paradox is indeed unsolvable under the premise. But after adding the time factor, the rules and elements are changeable, and the problem is solved. If we look at the theoretical system of mathematics from a philosophical point of view, it must interact with the objective, but not abstract to the extent of being divorced from the objective. Otherwise, there will be paradox in objective application. In fact, Russell paradox comes from objective falsification.

From another perspective, axioms do not need to be proved, or they are proved outside the system. It goes without saying that there is no objective proof, but no theoretical proof. No theory can be used to explain its own basis. The so-called self-consistency is only internal wit, and internal wit may not prove the premise is correct. Dialectical logic is different from formal logic, which cannot be changed, so it is inevitable to fall into self-contradiction, while dialectical logic regards concepts and premises as changes and solves problems in motion. This problem, especially set theory, is also the main reason why mathematics is not competent in the social field. As far as society is concerned, its elements are not collective, but systematic, and its elements, rules and structures are changing.

Third, the abstraction of mathematics.

The current mathematical theory is abstract to the point of being divorced from objectivity, and mathematicians also regard it as beauty. For example, the elements of a set have no structure and no hierarchy, which is similar to the point in geometry that has no size, but can it have value without size? Is there a difference between the size of the objective world? Is it 2.0 or 2.00 or 2.000 ... In fact, the difference in other situations lies in subtlety. When n tends to infinity, humans can think that 1/n can be ignored, but it is infinite in 1/(nxn).

The infinite set in mathematics can be extrapolated infinitely, but it cannot be extrapolated infinitely when applied to a certain level. For example, although the number of atoms on the earth is very large, it is still limited. The reason is the same as Newton's mechanics can't be extrapolated indefinitely. In fact, gravity cannot be extrapolated. Mathematically speaking, in the calculation of gravity, the distance can be zero, so any mass of matter can shrink into a black hole. But it doesn't actually exist. It is not true that the world is inexhaustible. The objective world is not always continuous, and it is no longer a ruler after tens of thousands of times. Can't "take". The same is true for eggs first. It is generally believed that only the answer may be wrong, but in fact the premise may also be wrong, and the question itself may also be wrong. The question of who comes first lies in metaphysics and in the problem itself. The premise of the problem itself is that chickens and eggs were like this several years ago. But what about the facts? Are all evolved, and there are no such chickens and eggs.

Set theory does not pay attention to distinguishing structures, for example, elements and sets are at different levels, so it cannot be discussed side by side, and it can be discussed in layers but not side by side. For example, "whether the set containing all sets contains itself" is a problem lacking hierarchy.

How did set theory come into being? It is an objective mapping and abstraction. Objects with certain attributes form a set, and the set is constructed according to the attributes, and these attributes have the same measure. However, the term "all sets" itself contains different levels of sets, and their measurement attributes are inconsistent. If the measurement attributes are inconsistent, it cannot form a set by itself. So the problem itself is problematic. For example, people and dogs can be unified in the concept of animals, but people and animals cannot be unified side by side in a certain concept. Man belongs to a collection of animals and is at different levels. If human beings are regarded as a set and animals as a set, does the set containing human beings and animals coexist? Does not exist. So the solution is to point out the mistake of the problem, that is, the problem juxtaposes the sets of different levels. But there is another solution, that is, to transform set theory according to the viewpoint of system theory. Even if we think that there is such a combination with different levels, for example, we propose a biological collection, which includes animals and people, but the language expression is inaccurate. The accurate expression is that organisms include animals and plants, and animals include people.

My view is to introduce objectivity and concreteness and transform set theory with system theory. The difference between system theory set theory and classical set theory lies in that the system has the concepts of structure and time. Fourth, mathematics is metaphysics, and metaphysics is insufficient.

In fact, what I wrote was the absolute abstraction in the philosophy of a powerful country. While writing this passage, I also thought of philosophy and system theory. Even metaphysics cannot be divorced from positivism, because the abstract basis of mathematics is objective existence. If the abstraction is incomplete, then this incompleteness will cause problems due to its own limitations when applied to specific problems. The problem with mathematics now is that it is too metaphysical and pure for beauty. But the barber case of Russell's paradox is only a falsification from an empirical point of view.

Metaphysics cannot be separated from metaphysics. The so-called ultimate reason is wishful thinking. The correct way is theory, argument, theory, argument ... metaphysics without demonstration is metaphysics. Is metaphysics meaningful? Yes, the meaning lies in the self-consistency within the category, but whether it can be consistent with the objective requires objective participation. In fact, this involves the formation of axiomatic system and basic definition. Axiomatic system and basic definition are inseparable from the objective world. 5. Abstract concept

Concrete science, mathematics and philosophy abstract objective things. The more detailed, the more detailed and vague the abstract is, the more connotation and the smaller the extension of the concept is. The abstract concept of matter in philosophy only has objective attributes. The set theory of mathematics is more abstract than the system theory, thus abstracting the interaction among time, level and elements. However, the abstraction of system theory is richer and has the above contents. Other concrete scientific abstractions are more detailed and have more attributes. What is the basis of abstraction? It is precisely the attribute.

Set theory has lost many concrete attributes because of its high abstraction, but it is still meaningful, and the concrete significance of system theory is more abundant. What does Russell Paradox mean? Explain such a problem: set theory loses the attributes of hierarchy, development and timeliness when it is abstract objectivity, so there is a problem when it needs to describe the attributes of hierarchy, development and timeliness.

Similarly, there is an ambition in philosophy. Philosophy always tries to cover everything. In fact, philosophy can cover everything, but this coverage is extension, not connotation. In other words, even if you master philosophy, you can't understand the specific attributes. As far as concreteness is concerned, concrete science covers more.

Judging from deductive reasoning, the bigger the premise, the more empty it is. In order to draw a concrete conclusion, there must be a minor premise, that is, the content of specific scientific. In fact, there is a minor premise under the minor premise. If the minor premise disappears somewhere, the more specific conclusion disappears. Therefore, we can see that the view that philosophy is higher than specific scientific is not valid, nor is the view that philosophy has a higher level. Or we can look at it this way, that is, we think that philosophy is indeed higher than concrete science, but it is higher than concrete in abstraction. So is the theoretical system, and so is the relationship between theory and objectivity. It cannot be said that theory must be higher than objectivity. On the one hand, the conclusion of the theory must be objectively verified, on the other hand, the basis of the theory is objective.

So I think it is important to establish a theoretical system based on abstraction. At the same time, whether abstract concepts can be integrated into the basic concepts of mathematics is also very important.

Fourth, Tao, another philosophy.

The first five points are all related to mathematics and western philosophy, but there is also a non-analytical philosophical concept-Tao, which is richer than the material concept and has not only material significance, but also regular significance. According to China people's understanding, Tao is all-encompassing, seamless, not only extensive, but also rich in connotation. Tao cannot be Tao, and Tao cannot be Tao. This is different from western philosophy. So how do you know Tao?

It can be seen that Tao is not defined, but matter and law are defined.

Therefore, it can be considered that matter and law are theoretical expressions of objective results, while Tao itself is objective, and objective changes will lead to changes. Perhaps it is this reason that leads to the differences between East and West. The epistemology emphasized by the west emphasizes what it looks like objectively to people. China people are directly concerned with themselves and the ever-changing Tao.

The last paragraph has nothing to do with modern science based on analysis, but it still embodies a world view, that is, China's own world view.