1. Pythagoras was a famous mathematician and philosopher in ancient Greece in the fifth century BC. He once founded a school of mysticism: Pythagoras School, which integrates politics, scholarship and religion. Pythagoras' famous proposition "Everything is a number" is the philosophical cornerstone of this school. "All numbers can be expressed as integers or the ratio of integers" is the mathematical belief of this school. After the Pythagorean theorem was put forward, hippasus, a member of his school, considered a question: What is the diagonal length of a square with a side length of 1? He found that this length can not be expressed by integer or fraction, but only by a new number. Hippasus's discovery led to the birth of the first irrational number √2 in the history of mathematics. The paradox of this conclusion lies in its conflict with common sense: any quantity can be expressed as a rational number within any precision range. However, the conclusion that is convinced by our experience and completely in line with common sense is overturned by the existence of a small √2! This directly led to the crisis of people's understanding at that time, which led to a big storm in the history of western mathematics, known as the "first mathematical crisis." More than two thousand years later, the real number theory established by mathematicians destroyed it.
2. The second mathematical crisis stems from the use of calculus tools. Becker hit the nail on the head and pointed out that Newton took △x as 0 and △x as 0 when he took the derivative of x n (n is a positive integer), which was a serious contradiction and almost made calculus stagnate. Later, Cauchy and Wilstrass put forward that infinitesimal is a variable that is infinitely close to 0, but it will never be equal to 0, and firmly established calculus on the basis of strict limit theory, thus eliminating this mathematical crisis!
3./kloc-In the second half of the 9th century, Cantor founded the famous set theory. 1900, at the international congress of mathematicians, poincare, a famous French mathematician, declared cheerfully: "… with the help of the concept of set theory, we can build the whole mathematical building … today, we can say that we have reached absolute strictness …" However, the good times did not last long. 1903, a shocking news came out: set theory is flawed! This is the famous Russell paradox put forward by British mathematician Russell.
Russell built a set S: S is made up of all elements that don't belong to him. Then Russell asked: Does S belong to S? According to law of excluded middle, an element belongs to a set or not. Therefore, for a given set, it is meaningful to ask whether it belongs to itself. But this seemingly reasonable question, the answer will be in a dilemma. If s belongs to s, according to the definition of s, s does not belong to s; On the other hand, if S does not belong to S, then S also belongs to S by definition. It is contradictory in any case. It can be said that this paradox is like throwing a boulder on the calm water of mathematics, which caused great repercussions and led to the third mathematical crisis.
After the crisis, mathematicians put forward their own solutions. For example, the ZF axiomatic system. The solution to this problem is only now in progress. The root of Russell's paradox is that there is no restriction on sets in set theory, so Russell can construct such a "super-large" set of all sets. The restriction of set construction is still a huge problem in mathematics!