Mathematics takes the spatial form and quantitative relationship of the real world as the research object. In order to study these forms and relationships in pure form, it must be separated from the content of the real world. However, there is no form and relationship without content. Therefore, mathematics tries to achieve this separation according to its essence, which is an attempt to achieve an impossible thing. This is the fundamental contradiction in the essence of mathematics and the special expression of the general contradiction of cognition in mathematics. By constantly solving and repeating the above contradictions in various stages of understanding that are getting closer to reality, mathematics will continue to advance and develop, from simple to complex, from low to high.
Humans first recognized natural numbers and experienced a struggle to introduce zero and negative numbers: either introducing these numbers or subtracting a large number will not work. Similarly, the introduction of fraction makes multiplication have the inverse operation-division, otherwise many practical problems can not be solved.
But then the question comes: can all quantities be expressed by rational numbers? The discovery of irrational numbers finally solved the first mathematical crisis and promoted the development of logic and the systematization of geometry. The problem of equation solution leads to the appearance of imaginary number, which is considered as "unreal" from the beginning, but this unreal number solves the problem that real number can't solve, thus winning the right to exist for itself. This is how mathematics develops in the struggle of contradictions. Geometry has developed from Euclidean geometry to many kinds of geometry, and so on.
19th century, many problems that can't be solved by traditional methods were found, such as algebraic equations of quintic or above, which can't be solved by adding, subtracting, multiplying and dividing; The three major problems of ancient Greek geometry can not be solved by drawing with compasses and rulers, and so on. These negative results show the limitations of traditional methods and also reflect the deepening of human understanding.
These discoveries have brought great impact to related disciplines and almost completely changed their direction. For example, algebra has developed from that time to abstract algebra, and solving the roots of equations has become a subject of analytical mathematics and computational mathematics. In the third mathematical crisis, this situation also appeared many times, especially the incompleteness of formal system including integer operation and the undecidability of many problems, which greatly improved people's understanding and promoted the great development of mathematical logic.
The second mathematical crisis caused by infinitesimal contradiction reflects the contradiction between finite and infinite in mathematics. The third mathematical crisis involves set theory and mathematical logic, but it involves infinite sets from the beginning, and modern mathematics cannot be separated from infinite sets. One extreme view is to consider only finite sets or at most countable sets, but then most mathematics will cease to exist.
Even the contents of these finite mathematics involve many infinite methods, and many mathematical proofs need to solve infinite problems with limited steps. With the help of computer to prove the four-color theorem, we should first simplify the infinite possible map into a limited situation. For infinity, computers are powerless. It can be seen that mathematics can never avoid the contradiction between finite and infinite, which can be said to be one of the root causes of mathematical contradictions.
There is always a contradiction between clear application and strict logic in mathematics. In this respect, practical mathematicians pay more attention to blind application, while more careful mathematicians criticize it. Only when these two aspects are in harmony can the contradiction be solved. For example, operator calculus and δ function, which are formal calculus at first, are applied arbitrarily. It was not until Schwal that the rigorous system of generalized function theory was established. The application of calculus and the establishment of limit theory are well known.
In the history of mathematics, two important trends have always played a role: one is the trend of continuous differentiation of disciplines, and the other is the trend of continuous integration of disciplines. The dialectical movement of these two contradictory trends is a process of negation of negation.
As a unified whole with infinite diversity, nature enters people's consciousness through feeling and perception. In ancient times, mathematics grasped nature on the basis of the relationship between total number and shape. Arithmetic, algebra and geometry are not separated from each other. Any famous math book contains these three aspects and integrates them into one. Therefore, ancient mathematics is essentially a perceptual and intuitive comprehensive science about number and reason.
Since the emergence of analytic geometry and calculus in17th century, the trend of discipline differentiation has been dominant. A single undifferentiated discipline has developed into many specialized branches, and each discipline studies a certain aspect of numbers and shapes in concrete and complete mathematics. By the second half of the19th century, this constant differentiation has reached a very fine level, and algebra, geometry, analysis and other disciplines have formed their own different research fields, especially the development of analysis field is booming. All disciplines can develop independently and are not related to each other, and all disciplines can be unrelated to each other in theory, language and methods, which is far from a unified mathematical picture.
From 1872, when Klein unified all kinds of geometry with the viewpoint of "group", to Cantor's establishment of set theory and axiomatic movement, the trend of increasingly divided mathematics towards synthesis gradually became obvious. By the beginning of the 20th century, the differentiation and synthesis of mathematics had obviously accelerated. Since1920s, especially after World War II, the comprehensive trend has been dominant. The continuous differentiation of disciplines is actually a manifestation of the trend of integration, because the continuous emergence of new disciplines is increasingly eliminating the traditional boundaries between disciplines. For the in-depth understanding of numbers and shapes, multidisciplinary methods adopt a more comprehensive understanding form. Therefore, the links between various disciplines are closer. This is a dialectical method for the development of modern mathematics. The more we understand all aspects of the whole, the closer we are to grasping the whole comprehensively.
Maybe there will be a recognized new viewpoint in the future to unify the present mathematics. However, this unification is only temporary and relative. With the development of production and science and technology, new problems will appear, new branches will be formed and new differentiation will be promoted. Mathematics will continue to advance in this constant differentiation and synthesis.