Three crises in the history of mathematics

Author: Yang Yinzhou

 

There are economic crises, and there are also three mathematical crises in history. The first crisis occurred in 580-BC ~ In the ancient Greek period between 568, the mathematician bidaras established the bidaras School. This school integrates religion, science, and philosophy. The school has a fixed number of students, and its knowledge is kept confidential. All inventions and creations are attributed to school leaders. At that time, people were still very limited in understanding rational numbers, and they knew nothing about the concept of irrational numbers. The number mentioned by the bildagos school was originally an integer and they did not regard scores as a kind of numbers, instead, it is regarded as the ratio of two integers. They mistakenly believe that all phenomena in the universe are attributed to the ratio of integers or integers. The member of this school, Hebrew, found through logical reasoning that the diagonal length of a square with a side length of L is neither an integer nor an integer ratio. The discovery of Hebrew is regarded as "Absurd" and a violation of common sense. It not only seriously violates the creed of the bildagos school, but also impacts the traditional views of the Greek people at that time. Greek mathematicians were deeply disturbed at the time. It is said that the discovery of the Hebrew family was drowned in the sea, the first Mathematical Crisis. This crisis has been solved by introducing the concept of non-accessibility in geometry. Two geometric line segments. If a third line segment can be used to complete them at the same time, the two line segments are allowed to pass through. Otherwise, the two are called inaccessible. The side and diagonal lines of a square do not have the third line that can be used to accumulate them at the same time. Therefore, they cannot be interconnected. Obviously, the so-called Mathematical Crisis no longer exists as long as you acknowledge that there is no allow for the existence of the approximate number so that the geometric number is no longer limited by integers. The amount of research that could not be passed began in ocus, 4 century BC. The results were absorbed by Euclidean, and some were included in his "geometric original. The Second Mathematical crisis occurred in the 17th century. After the birth of calculus in the 17th century, due to the theoretical basis of calculus, the mathematics field encountered chaos, that is, the Second Mathematical Crisis. The formation of calculus has brought about revolutionary changes in the field of mathematics and has been widely used in various scientific fields. However, calculus has contradictions in theory. An infinite number is one of the basic concepts of calculus. In some typical derivation processes, Newton, the principal founder of calculus, used an infinitely small number as the denominator for Division. Of course, an infinitely small number cannot be zero. In the second step, Newton considered an infinitely small number as zero, remove the items that contain it to obtain the expected formula. The application of mechanics and geometry proves that these formulas are correct, but their mathematical derivation process is logically self-contradictory. The focus is: whether the infinitely small quantity is zero or non-zero? If it is zero, how can we use it for division? If it is not zero, how can we remove the items that contain an infinitely small number? Until the 19th century, kherk had developed the Limit Theory in detail and systematically. The number of Infinitely small numbers is regarded as a definite number. Even if it is zero, it cannot be said, and it will conflict with the definition of the limit. An infinitely small volume should be a small amount. Therefore, it is essentially a variable and the limit is zero. At this point, we have clarified the concept of an infinitely small number, in addition, the infinite number is freed from the constraints of metaphysics, and the Second Mathematical Crisis is basically solved.

 

 

The solution to the Second Mathematical Crisis improved calculus.

 

 

The third mathematical crisis occurred at the end of the 19th century. At that time, the British mathematician Russell divided the collection into two types.

 

 

The first type of set: the set itself is not its element, that is, a A; the second type of set: the set itself is one of its elements, a, a, for example, a set composed of all sets. For any Set B, either the first set or the second set.

 

 

Suppose that the whole of the first set constitutes a set m, then M belongs to the first set or belongs to the second set.

 

 

If M belongs to the first set, m should be an element of M, that is, m in m, but the set that satisfies the m in M relationship should belong to the second set, and there is a conflict.

 

 

If M belongs to the second set, m should satisfy the relationship between M and M, so m is the first set conflict.

 

 

The so-called Russell paradox formed by the above reasoning process. Due to the establishment of strict limit theory, the first second crisis in mathematics has been solved, but the limit theory is based on the real number theory, while the real number theory is based on the set theory, now the set theory has appeared the Russell paradox, thus forming a greater crisis in the history of mathematics. Since then, mathematicians have begun to find a solution to this crisis. One of them is to establish the set theory on a set of principles to avoid paradox. First of all, this work was done by the German mathematician timerro. He proposed seven theorems and established a set theory that would not generate a paradox, which was improved by another German mathematician, fazikel, A non-contradictory set theory system is formed. The so-called ZF justice system. This mathematical crisis has now eased. The mathematical crisis has brought new impetus to the development of mathematics. In this crisis, the theory of set theory has developed rapidly, the foundation of mathematics has improved faster, and the logic of mathematics has become more mature. However, conflicts and unexpected things continue to emerge, and will continue in the future.