However, this is also one of the greatness and charm of mathematics. When we solve problems, we will form new knowledge theories, and at the same time, new problems will arise in the process of solving problems, which will push mathematics forward in a cycle. To some extent, the solution of the problem has promoted the formation and development of mathematics.
The emergence of problems means that there are contradictions within or between things, and the struggle or solution of these contradictions requires the essence of mathematics.
Therefore, in a sense, learning mathematics means learning how to solve problems and finally solve contradictions.
For example, the famous Fermat's last theorem: when the integer n > 2, the indefinite equation xn+yn= zn about x, y, z y, z has no positive integer solution.
In the hands of early mathematicians, they can prove that Fermat's Last Theorem holds in special cases such as n=3, 4, 5 and 6. But the number of integers is infinite, and the proof one by one will never be finished, which is also very unrealistic. Even if you can prove Fermat's Last Theorem from n=3 to a very large integer, you may encounter a larger integer that makes the theorem untenable, and even such an integer may exist in many cases.
At this time, the most important task of all mathematicians is how to solve infinite problems with finite steps, that is, to prove the establishment of Fermat's last theorem with a complete finite step.
After entering the 20th century, with the continuous development of computer technology, mathematicians can prove a large number of Fermat's Last Theorem with the help of computers, but in the end, they also need to simplify an infinite number of integers into the case of finite step proof, and there is no finite step proof process. The so-called computer proof is only a special case.
Therefore, all mathematicians and scientists realize that solving mathematical problems always needs to solve the contradiction between "finiteness and infinity". As long as there is infinity in a mathematical problem, then we need to take the initiative to solve it, which can be said to be one of the roots to promote the development of mathematics.
It took nearly three centuries to put forward and solve Fermat's last theorem, and countless mathematicians participated in it. For example, after many mathematicians, including Riemann and Mo Deer, did the follow-up work, Fermat's Last Theorem was connected with rational points (points whose coordinates are rational numbers) on algebraic curves. These changes have promoted the development of related fields of mathematics and the process of proving Fermat's Last Theorem.
Wiles, a young British mathematician, finally proved Fermat's last theorem by using the elliptic function theory developed by his predecessors and its research results.
The proof of Fermat's Last Theorem not only provides inspiration for us to solve the contradiction between finite and infinite, but also reminds the world that if we want to solve problems, we sometimes need to make some changes, such as turning unsolved problems into new problems in known or easy-to-solve fields.
Therefore, mathematicians will process and create new knowledge theories when dealing with problems. For example, early human beings solved the existing problems to a certain extent after inventing natural numbers, but with the continuous development of society and trade, debts appeared. At this time, in order to solve new problems better, people must create knowledge concepts like 0 and negative numbers.
A series of knowledge such as rational number, irrational number, real number and complex number appeared because of new contradictions and problems in the process of social development at that time, and people created new knowledge theories in the process of solving these problems.
The most famous contradiction in the history of mathematics should belong to "three mathematical crises". The first two mathematical crises have been successfully solved, but the third mathematical crisis has not been completely solved.
The third mathematical crisis is mainly due to the discovery of paradox on the edge of set theory, and the whole mathematical kingdom is essentially based on set theory, which has penetrated into many branches of mathematics, so the discovery of paradox of set theory naturally raises doubts about the validity of the whole mathematical basic structure.
Frankly speaking, when we realize infinite set and infinite cardinal number, we need to solve the contradiction between "finite and infinite", otherwise many mathematical problems will follow, which is the essence of the third mathematical crisis.
The pursuit of mathematics is to solve contradictions and problems. To put it bluntly, it is not contradictory. However, what do you mean by non-contradiction? From a logical point of view, existence is reasonable and there is no contradiction, but this is only the law of formal logic. However, what mathematics wants to solve is not as simple as formal logic, because it wants to prove that infinity is not contradictory, and formal logic only comes from limited human experience.
Although the third mathematical crisis has been apparently solved, it exists in mathematics in other forms. We can't throw away all the set theories that are considered contradictory, because they play a very important role in some fields.
Mathematics has never been afraid of contradictions and problems, because with the solution of contradictions and problems, it can bring a lot of new knowledge and cognition to mathematics and other fields, and even bring revolutionary changes to human society.
For example, in the past two centuries, human beings have gained more knowledge and achievements in mathematics and understanding of things than in the previous centuries combined. Especially after the Second World War, many disciplines, including mathematics, ushered in a big explosion and rapid development, and many new achievements emerged one after another.
Since the birth of set theory, modern mathematics has created important branches of mathematics, such as abstract algebra, topology, functional analysis and measure theory, especially traditional algebraic geometry, differential geometry and complex analysis. , has been extended to higher dimensions. Algebraic number theory, for example, has been perfected by many mathematicians.
Many times, the solution of a problem will certainly enrich the relevant knowledge theory and even create new problems, which is one of the essence of learning and studying mathematics.