Cryptologist Wang Xiaoyun recently won the 2019 Future Science Award for Mathematics and Computer Science. She puts forward the collision attack theory of cryptographic hash function, which promotes the design of a new generation of cryptographic hash function standards, has been widely used in important fields such as finance and transportation, and has made important contributions to supporting network information security.
Mathematics, as the foundation of natural science, its strength often affects the national strength. Almost all major discoveries are related to the development of mathematics, which is an indispensable basic tool for studying and studying modern science and technology.
A few days ago, the Ministry of Science and Technology, the Ministry of Education, the Chinese Academy of Sciences and the Natural Science Foundation of China jointly formulated the Work Plan on Strengthening Mathematical Science Research, which emphasized the importance of mathematical research.
Mathematics is the foundation of natural science and major technological innovation and development, and has become an indispensable and important support in the fields of aerospace, national defense security, biomedicine, information, energy, ocean, artificial intelligence, advanced manufacturing and so on. So, what is the essence of mathematics? The Economic Daily reporter interviewed professional math researchers and math enthusiasts, trying to solve it.
How to understand mathematical thinking
Zhou Chuan, an associate researcher at the Institute of Mathematics and Systems Science of China Academy of Sciences, believes that the essence of mathematics lies in mathematical thinking. "Mathematical thinking refers to the ability to use mathematical ideas and methods reasonably in the process of thinking and solving problems. Mathematical thinking is not a kind of knowledge, but a kind of ability. Mathematical thinking is the most important foundation for building the mathematical world. It plays an important role whether it is pure mathematical learning and mathematical research or applying mathematical tools to other fields. " Zhou Chuan said.
Specifically, mathematical thinking includes logical thinking, image thinking and spatial abstract thinking. It is like the huge root system of the towering tree of mathematics. Although it is invisible from the outside, it provides an important source of nutrition for mathematics. "We often say that mathematics is beautiful, and this beauty is mainly reflected in the beauty of its thinking." Zhou Chuan said.
The beauty of mathematical thinking lies in the balance between practicality and rationality. Zhang Jian, a math enthusiast, told reporters that taking statistics, one of the branches of mathematics, as an example, in the process of solving practical problems, statistics gives people a concise and lively aesthetic feeling. Basic statistical laws, such as the law of large numbers, the central limit law and Bayesian probability, show the harmony after the conceptual world and the perceptual world are consistent. A series of theorems behind it play an important role in the balance and unity of reason and experience, theory and practice, deduction and induction, axiomatic system and algorithm program. "The data show that some precise sciences can be developed by clear definition and logic, and some problems need to be solved by approximate measurement, which requires statistical wisdom such as error theory, probability theory and mathematical statistics." Zhang Jian said.
The beauty of mathematical thinking lies in the beauty of "overlord" above other disciplines. According to the data, the foundation of modern science is the Elements of Geometry written by Greece 2,500 years ago. Any discipline that can be called science must have two characteristics: first, it has an axiomatic system like the Elements of Geometry; Second, the hypothesis can be verified by experiments. Physics is based on mathematics, chemistry is based on physics and biology is based on physics and chemistry. In the final analysis, all modern science is based on mathematics. "For example, in the field of artificial intelligence, all the mathematical foundations of machine learning based on neural networks are partial differential equations and linear algebra, while other schools of artificial intelligence involve probability theory and stochastic processes. Today’s artificial intelligence is difficult to leave the blessing of big data and its related statistical variables. In order to make achievements in this field, solid and advanced mathematics literacy is indispensable. " Zhang Jian said.
Zhou Chuan believes that the beauty of mathematics lies in its profound definition, clear logic and concise result. For example, the famous Goldbach conjecture that "any even number greater than 4 can be written as the sum of two prime numbers" is concise but profound and wonderful. Hardy, a great British mathematician, once said, "Beauty is the touchstone of good mathematics, and ugly mathematics cannot last forever". Hardy’s idea is related to his advocating the artistry of mathematics, but it also reflects the mathematicians’ aesthetic pursuit of mathematical theory to some extent.
How to have mathematical thinking
Mathematical thinking is so important, how can we get it? Doing the problem is the first way.
Zhou Chuan told reporters that the purpose of doing the questions is to test the mastery of knowledge, strengthen the cognitive understanding of knowledge and develop thinking. Any scientific research is a process of trial and error, and mathematical research is no exception. "When faced with a mathematical problem, we usually make bold assumptions, divergent thinking, try a variety of different ideas, and carefully deduce and prove to see which idea is feasible and beautiful." Zhou Chuan said, "Doing problems can help mathematicians deepen their knowledge understanding, develop innovative thinking and stimulate innovative ideas, which are of great benefit to scientific research."
Zhou Chuan believes that mathematical research will not be smooth sailing. For real problems, the ideas and methods that are usually easy to think of are often not effective. This requires maintaining enough courage and perseverance in mathematical research. When encountering difficulties, we must have the courage to "open the road and meet the water" and strive to move forward until the destination, although this process may be very long.
In addition, it is also important to read more famous mathematical books. Sunny, a math researcher, told reporters that his desk has kept classic works such as Introduction to Advanced Mathematics by Hua Luogeng, Ancient and Modern Mathematics Thoughts by Maurice Klein and Princeton Mathematics Guide all the year round. "The whimsy in these books often gives people inspiration, which makes us professional mathematics researchers shine and gain valuable innovative ideas."
In Sunny’s eyes, most famous mathematical works are concise and to the point, which will inspire both professional researchers and staff in other fields. For example, in the third volume of the Princeton Mathematical Guide, there is a saying: "Curiosity is the driving force for doing mathematical work. When will a special result be true? Is this the best proof, or is there a more natural or beautiful proof? … … If you keep asking yourself such questions, sooner or later there will be a flash of answers — — Discover the possible path of research. " "It is tempting to think of mathematics as a collection of separate branches, such as geometry, algebra, analysis, number theory and so on. Geometry is mainly an attempt to understand ‘ Space ’ The concept of algebra is to understand the art of manipulating symbols, and analysis is to contact ‘ Infinite ’ And ‘ Continuum ’ , and so on. " Sunny said.
How to apply mathematical thinking
The application of mathematical thinking can be described as flexible, magical and extraordinary. Its application is more reflected in how to think and solve problems in a mathematical way.
For example, Zhou Chuan’s current research on graph data modeling and algorithm is one of the most competitive research directions. A basic idea of this research direction is to organize, model and analyze data in the form of graphs. The reporter learned that this way of thinking can be traced back to the famous problem of the Seven Bridges in Konigsberg: In the 18th century, in the suburb of Konigsberg, East Prussia, Europe, the Pregiel River crossed the city. There are two islands in the river, and there are seven bridges between the banks and the two islands. At that time, the residents in the city discussed such a question enthusiastically: how can a walker walk from one place to all seven bridges at once without repeating, and finally return to the original starting point?
"At first glance, this question doesn’t seem too difficult. Many people want to give it a try, but no one can find the answer. At that time, Euler, a great mathematician, thought from all the failures that such a move might not exist at all. How did Euler model and analyze this problem? " Zhou Chuan explained that, in fact, the modeling tool used by Euler is "graph". In this picture, there are 4 nodes and 7 edges. Nodes represent two banks and two islands, while edges represent bridges. Euler transformed the problem of seven bridges into the problem of finding special paths on the graph. Later, through mathematical methods, he strictly proved that such a special path did not exist, which brought a satisfactory end to the Seven Bridges problem.
Euler’s way of thinking is a very representative thinking paradigm, which still has a strong guiding significance today. Because there are often complex relationships between data entities, modeling data organization from the perspective of graph usually leads to higher analysis and mining accuracy. "For example, in the problem of ranking the importance of web pages, web pages are usually regarded as nodes in the graph and hyperlinks as edges in the graph, and Markov process in mathematics is used to describe users’ web browsing behavior on the Internet. Under the framework of this mathematical model, the problem of ranking the importance of web pages is transformed into the problem of solving the stationary distribution of the corresponding Markov process. This transformation provides a new way of thinking and a more effective solution for the problem of ranking the importance of web pages. " Zhou Chuan said.
"When you really experience the beauty of applying mathematical thinking to solve practical problems, you will feel ‘ Mathematics is the source of natural science ’ The discussion is really incisive. " Zhang Jian said. (Economic Daily China Economic Net reporter Liang Jianwei)