The first is assembly. The rally seems to be a clearing, a blank sheet of paper, and a group of actors who have not been assigned roles.
Once some relationships are introduced between the elements of a set, the elements of the set have their own personalities, and according to the nature of the relationships, structures begin to appear on the set.
Structure is not arbitrarily designated by people subjectively, nor does it exist in the world of ideas forever. It is the result of a large number of perceptual experience summaries rising to concepts.
According to the Bourbaki school, there are three basic structures in mathematical research, namely, the matrix structure:
A structure called algebra. By operating on a set, a third element can be generated from two elements, which is called algebraic structure. The group we just talked about is a basic algebraic structure.
A structure called order. There is an ordered relationship between some elements in a set, which is called having an ordered structure. Ordered structure is also a widely used structure. The relationship between the size of numbers, the parent-child relationship of creatures and the inclusion relationship of classes are all ordinal relationships.
There is also a topology. It is used to describe the continuity, separability, proximity and boundary of space.
We see that these structures are only the reflection of the relationships and forms in the real world in our minds:
Algebraic structure-operation-comes from quantitative relationship;
Sequence structure-continuously-comes from the concept of time;
Topological structure-continuity-comes from space experience.
But once these things are abstracted into mathematical concepts and become * structures * divorced from specific contents, they can be used in any system with similar properties, not necessarily related to time, space and numbers.
A system can have several structures. As a real number system, it has two interrelated algebraic structures: addition, subtraction, multiplication and division. It is an ordered structure with a difference in size, and its continuity reflects the topological structure.
Some axiomatic derived substructures can be added to the basic structure, and connection conditions can be added to more than two structures to produce composite structures. For real numbers, if a>b, then A+C > B+c, which shows that algebraic structure is associated with ordered structure. Through the change, combination and intersection of structures, various branches of mathematics are formed, which show a colorful mathematical world.
When mathematicians meet a new research object, they naturally think, can what they meet be put into a known structure? If possible, immediately use all the known properties of this structure as a weapon to defeat the enemy.
There has been such an example in history: mathematicians can't understand complex numbers for a long time and call them imaginary numbers. Later, it was found that complex numbers can be represented by points on a plane, which is equivalent to linking the algebraic structure of complex numbers with the topological structure of a plane. The study of complex numbers immediately had practical significance, found applications and made rapid development. This shows how important it is to integrate new and unfamiliar objects into known structures.
The Bourbaki school also admitted that it is still a rough approximation of the present situation of mathematics to regard mathematics as a science to study various structures, which are constantly growing and developing with several parent structures as the skeleton.
Mathematics can be regarded as a big developing city. The buildings in the city are separated by streets and connected by streets. Streets form structures, and buildings grow in the norms of structures. But there are many distinctive buildings whose characteristics cannot be explained by the structure of streets. This is the universality of the structural viewpoint. Some local situations that it can't care about have little to do with structure, and sometimes have great significance. For example, a large number of isolated problems in number theory (such as Goldbach problem) are difficult to connect with known structures well.
The Bourbaki school also believes that the structure should not be static, and the development of mathematics may find new and important basic structures. Because mathematics is a booming science, it cannot be "closed" and there will be no ultimate truth.
Generally speaking, the Bourbaki school regards mathematics as a science with structure as its object, which is consistent with dialectical materialism. Because: it denies the transcendental view of mathematical knowledge, advocates that structure comes from people's practical experience, and correctly describes the abstract formation process of structural concepts in mathematics; It looks at mathematics from the viewpoint of holism, focuses on the internal relations of various departments of mathematics, and explains what makes mathematics both unified and diverse; It looks at mathematics from the viewpoint of change and development, and advocates that the structure is not static; It advocates that the truth of mathematics should be tested by scientific practice in the end, and the structural viewpoint should be supported by scientific successful experience.
The emergence of structuralism is not accidental. The Bourbaki school itself pointed out that this is the result of mathematical progress in more than half a century (that is, from the end of 19 to the middle of the 20th century). In fact, it can also be said that it is the result of more than two thousand years of mathematical progress. Axiomatic method began with Euclid. After the appearance of non-Euclid geometry, mathematicians began to have modern axiomatic views. This method has passed the test of the third mathematical crisis, especially because of the vigorous advocacy of the formalism school Albert, which took root and blossomed in mathematical practice, and finally formed the concept of "structure" through the next level.
At first, people pursued the completeness or completeness of axioms. That is to say, in the axiomatic system, whether any proposition is true or not can only have a unique answer. In this way, an axiomatic system with completeness can only describe one kind of object in essence. For example, Euclid's geometric axiom only describes an object in essence, although it can be varied in form, which weakens the universality of axiomatic system application. Without parallel axioms, the geometric axiom system loses its completeness, but its application scope is wider. In the geometric system without parallel axioms, the proved theorems are valid in Euclidean geometry and Roche geometry. If some axioms are removed, the theorems derived from the remaining axioms are all established in Euclid, Roche and Riemannian geometry, which are called "absolute geometry" theorems.
Mathematicians have found that the imperfection of axiomatic system is not a bad thing, but a good thing. Incomplete, can accommodate more abundant objects. Axiom is a restriction on the object of study. The more restrictions, the narrower the research field; If the restrictions are appropriately reduced, the scope of application of the research results will be more abundant.
Inspired by this knowledge, mathematicians have studied many incomplete axiomatic systems, such as groups, rings, fields, linear spaces, probability theory, measure theory and so on. Mathematical practice has proved that the study of incomplete axiomatic systems has great vitality, which urges people to decompose axiomatic systems into some more basic and incomplete axiomatic systems, and finally promotes the emergence of structural viewpoints.