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Mathematics is the reflection of the human brain on the real world, and the essence of the real world lies in its material unity. As a reflection of the material world, mathematics must be unified. With the development of mathematics and the deepening of understanding, the understanding of the nature of the real world will show this unity more and more.
The great development of science and technology and the great differentiation of scientific theories have promoted the development of mathematics and the differentiation of mathematics. Geometry, algebra, analysis, probability theory, trigonometry and so on began to become independent disciplines, and they all developed in depth. But with differentiation, the tendency of synthesis will inevitably play a role. For example, generating analytic geometry on the basis of full differentiation between geometry and algebra can be regarded as an attempt to unify mathematics. This unity of algebra and geometry promoted the development of mathematics and laid the foundation for the generation of calculus.
The method of calculus has made great achievements in mathematics and its application. Mathematicians use analytical methods in various mathematical fields, which makes differentiation bring new possibilities for unity. The rigor of analytical theory links it with real number theory and natural number operation. Axiom method has gradually become a widely used expression method in mathematics, and the consistency of expression methods in various branches of mathematics also reflects their essential unity. The research on the basic problems of mathematics shows the consistency of mathematics on this basis.
Whether it is the basic concept, basic method or problem-solving method, there are great differences between algebra and geometry in the early days. Modern algebra and geometry theories are rapidly "merged", just like the homology group theory of reconciling algebra and topology has some same concepts and methods. In addition, the new algebraic geometry theory has been unable to determine whether it belongs to algebra or a branch of geometry. Other branches, such as statistical mathematics, which studies random phenomena, are increasingly combined with algebra, geometry and analysis, resulting in a series of comprehensive disciplines and marginal disciplines. In modern mathematics, the comprehensive application of various branching methods is more common.
In fact, there is no obvious boundary between pure mathematics and applied mathematics, especially in modern times. It can develop from concrete application to "pure" theory, such as some theoretical problems in graphic theory; Many achievements of pure mathematics have been applied again, such as the uniqueness theorem of factorization in number theory and the pure theoretical problem of factorization of large numbers in cryptography.