(1) There have been some major breakthroughs in pure mathematics. For example, continuum hypothesis, large cardinality problem and so on. "Mandatory Law", "Model Theory" and "Generalized Function Theory" in mathematical logic; "Singular Sphere Theorem" in Topology, axiom of choice, Discussion on Deterministic Axiom. There are various new trends of thought in mathematics. Such as nonstandard analysis, fuzzy mathematics, catastrophe theory, structural mathematics, structural mathematics and so on.
(2) Mathematics permeates almost all academic fields (not only natural science) and plays an increasingly important role.
In fact, with the continuous development and progress of science, it is required to quantify or mathematicize the research object. The maturity of a subject can even be determined by quantitative description. For example, mathematics was rarely used in biology before, but now it is different. There are biomathematics, biostatistics, mathematical biology and other disciplines. Economics, psychology and history also use mathematical methods. Even the research and analysis of literary works "Dream of Red Mansions" and "Shakespeare's Plays" created by vivid thinking depends on mathematics.
On the other hand, new disciplines of applied mathematics have sprung up like mushrooms after rain, such as game theory, planning theory, queuing theory and optimization methods (such as optimization methods and master planning methods). ), management science and operational research. Comprehensive disciplines such as cybernetics, information theory and system theory have emerged and developed one after another.
(3) The position of the viewpoint of set theory is gradually improved, and the axiomatic method is becoming more and more perfect.
Set is the basic concept of modern mathematics. Based on this concept, mathematics can have new development. Through the perfection of axiomatic method, people have made in-depth research on basic problems of mathematics.
(4) The entry of electronic computers into the field of mathematics has had an immeasurable impact.
Wu Wenjun, a famous mathematician in China, has made gratifying achievements in studying machine proof. He pointed out that we should pay attention to an immeasurable aspect that has a decisive influence on the future development of mathematics, that is, the influence of computers on mathematics. With the development and application of microcomputers, especially, mathematicians must be fully prepared for this prospect.
Finally, we are sure that the prospect of mathematics is bright. It advances in contradictions and even sweeps in many aspects. As JeanDieudonnè of Bourbaki School reiterated Hilbert's maxim in a speech: "We must know, and we will know, that a mathematics will not leave a place for agnosticism".
The development of mathematics in the twentieth century witnessed the transformation, and the dimensions became infinite. Physicists took it to the next level. In quantum field theory, they really want to study infinite dimensional space, in which infinite dimensional space is standard function space. Therefore, just as most mathematics in the 20th century paid attention to the development of geometry, topology, algebra and analysis in finite dimensional Lie groups and manifolds, this part of physics was treated similarly in infinite dimensions.
Historical summary:
18 and 19 centuries are the same, which can be called the era of classical mathematics and is related to Euler and Gauss. Classical mathematics has made good achievements and development, which is almost the conclusion of mathematics, but in the 20th century, on the contrary, it was really productive.
The first half of the 20th century was an era dominated by "special times", in which Hilbert had to formulate everything and define it carefully, with far-reaching influence. The second half of the year definitely surpassed the "integration era", and technology entered other fields from this field, and it was mixed to an alarming extent.
Perhaps it is the era of quantum mechanics, or it can be said to be infinite-dimensional mathematics. This means an understanding of the analysis, geometry, topology and algebra of various nonlinear function spaces.