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Is God a mathematician? This question has made the greatest mathematicians (philosophers) ponder for centuries. As the British physicist James Jones once pointed out, ∶ "the universe seems to have been designed by a theoretical mathematician." Mathematics seems to be not only the most effective tool to describe and explain the whole universe, but also can be used to explain the most complex human activities.

Today, mathematics is used in almost every field. What gives mathematics such incredible power?

How can mathematics, a product of human thinking independent of human experience, be so perfectly consistent with matter in physical reality? -- Einstein

Thousands of years of impressive mathematical research and extensive philosophical thinking have not really explained the mystery of mathematical power, and it can even be said that, in a sense, the mystery of mathematics has intensified. For example, Roger Penrose, a famous Oxford mathematical physicist, realized that there is not only one world around human beings, but three mysterious worlds. According to Penrose's division, these three worlds are the world perceived by ∶ consciousness, the world of physical reality and the Platonic world of mathematical form.

For Penrose, like the spiritual world and the physical world, the world is real. The operation of the physical world seems to follow certain laws, which really exist in the mathematical world. Second, human insightful thinking itself seems to come from the physical world. How on earth does thinking come from matter? can we raise the working mechanism of thinking to a theory as clear and convincing as today's electromagnetic field theory? Finally, the three worlds are mysteriously linked together to form a closed circle. By discovering or creating abstract mathematical formulas and concepts and expressing them clearly, insightful thinking can miraculously enter the realm of mathematics.

Using mathematical success to explain the world around us can actually be understood in two ways, both of which are equally amazing. The first is its "active" side. When physicists lose their way in the labyrinth of nature, mathematics illuminates the way ahead for them, the tools they use and create, the models they build, and the explanations they expect, all of which are inseparable from mathematics. Maxwell, a Scottish physicist, expanded the scope of classical physics in the 1860s. He explained all the known electromagnetic phenomena by using only four mathematical formulas. Einstein's general theory of relativity is even more amazing. it is a perfect example of an extremely accurate and self-consistent mathematical theory, which reveals basic things such as the structure of time and space.

The second is its "passive" side. When mathematicians study and explore mathematical concepts and the relationship between various concepts, sometimes only for the purpose of theoretical research, never consider the practicality of the theory. But decades later, and sometimes hundreds of years later, people suddenly find that their theories provide unexpected solutions to physical problems. Hardy is a mathematician who studies pure theory. One of his research results is called Hardy-Weinberg Law, which is the basis for geneticists to study population evolution. Hardy-Weinberg law states that if a large population population randomly mating (without migration, genetic mutation and selective mating), the genetic composition will remain constant and will not change from generation to generation. On the surface, Hardy studied abstract number theory, but unexpectedly he was found to be able to solve practical problems.

In 1973, Clifford Cox, a British mathematician, made a breakthrough in the field of cryptography by using number theory. But Hardy once said that ∶ "no one can apply number theory to war." It is obvious that cryptography is absolutely indispensable in modern military information transmission.

This is just the tip of the iceberg. In a classic lecture in 1854, George Friedrich Bernhardt Riemann summed up the main contents of several emerging geometry, which happened to be the necessary tools for Einstein to explain the structure of the universe. There is also a mathematical "language" called group theory, which was founded by Galois, a young mathematical genius. At first, it was only used to determine the solvability of algebraic equations, but today it has been widely used by physicists, engineers, linguists and even human ecologists to study almost all symmetry problems.

In addition, the concept of mathematical symmetry subverts the whole scientific research process to some extent. For centuries, the first step for scientists to understand the universe is to collect data and results after repeated experiments and observations, and then sum up the general laws of nature from them. This combing process begins with local observation, and then is put together piece by piece like a jigsaw puzzle. After entering the 20th century, people realized that well-organized mathematical design described the basic structure of the subatomic world, and contemporary physicists began to do the opposite. They put mathematical symmetry in the first place and insist that the laws of nature and the basic elements of things should follow a specific pattern, so according to this requirement, they deduce the general law. How does nature know that it should follow the principle of mathematical symmetry?

One day in 1975, young mathematical physicist Mitch Feigenbaum used his portable calculator to work out a simple equation at Los Alamos National Laboratory. He gradually noticed that the number 11 on the calculator was getting closer and closer to a specific number ∶ 4.669. He was surprised to find that this magical number appeared again as he worked out other equations. Although Feigenbaum could not explain why, he quickly concluded that the number he had discovered seemed to mark some universal rule in the transition from order to chaos.

What is the reason why there are the same mathematical characteristics behind systems that seem to be very different? After half a year of expert review, Feigenbaum's first paper on the subject was rejected. Soon after, experiments showed that when liquid helium was heated from below, the process of change was exactly the same as that predicted by the Fagenbaum universal solution. It has been found that this is not the only system that behaves this way. The surprising number found by Feigenbaum occurs not only in the transition of fluid from orderly to disorderly flow, but also in the dripping process of faucets.

There are still many examples of the necessity of first "predicting" the existence of the law in mathematics, and then confirmed by later generations, and it is still going on. The mysterious and unexpected interaction between the mathematical world and the real (physical) world is vividly reflected in knot theory, a subject that studies knots. The mathematical "knot" is very similar to the knot on the rope in reality, except that the head and tail of the rope must be spliced together. In other words, the mathematical knot is on a closed curve with no freely moving rope end.

Oddly enough, the main reason for the creation of knot theory is a wrong model of atomic structure developed in the 19th century. This model has been proved wrong 20 years after it was proposed, but knot theory, as a relatively difficult branch of theoretical mathematics, is constantly developing and evolving. Surprisingly, the abstract explorations made by mathematicians in the field of knot theory have suddenly been widely used in modern scientific research. Its application range covers DNA molecular structure, string theory and so on.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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