Around the 5th century BC, the discovery of incommensurable measures led to the Pythagorean paradox. At that time, the Pythagorean school attached importance to the study of the invariable factors in nature and society, and called geometry, arithmetic, astronomy and music "four arts", in which it pursued the harmony and law of the universe. They think that everything in the universe can be summed up as an integer or a ratio of integers. A great contribution of the Pythagorean school is to prove the Pythagorean theorem, but it is also found that the hypotenuse of some right-angled triangles cannot be expressed as integers or the ratio of integers (incommensurability), such as right-angled triangles with one side long. This paradox directly violated the fundamental creed of Pythagoras school, which led to the "crisis" of cognition at that time, thus resulting in the first mathematical crisis.

In 370 BC, eudoxus of Pythagoras School solved this contradiction by giving a new definition of proportion. His method of dealing with incommensurable measures appeared in the fifth volume of Euclid's Elements of Geometry. Eudoxus and Dai Dejin's interpretation of irrational numbers in 1872 is basically consistent with the modern interpretation. The treatment of similar triangles in today's middle school geometry textbooks still reflects some difficulties and subtleties brought by incommensurable measurement. The first mathematical crisis had a great influence on the concept of mathematics in ancient Greece. This shows that some truths of geometry have nothing to do with arithmetic. Geometric quantities can not be expressed by integers and their ratios, but by geometric quantities. The authoritative position of integers began to shake, while the identity of geometry rose. The crisis also shows that intuition and experience are not necessarily reliable, but reasoning is reliable. Since then, the Greeks began to attach importance to deductive reasoning, and thus established a geometric axiom system, which is a great revolution in mathematical thought! Is infinity zero? -The second mathematical crisis

18th century, differential method and integral method have been widely and successfully applied in production and practice, and most mathematicians have no doubt about the reliability of this theory.

1734, the British philosopher and archbishop Becker published An Analyst or Mathematician who doesn't believe in orthodoxy, pointing directly at the infinitesimal problem, the foundation of calculus, and putting forward the so-called Becker paradox. He pointed out: "Newton first gives X an increment of 0, applies binomial (x+0)n, subtracts xn from it to get the increment, then divides it by 0 to get the ratio of xn's increment to X's increment, and then makes 0 disappear, thus getting the final increment ratio. Here Newton went through the formalities of violating the law of contradiction-let X have an increment first, and then let the increment be zero, that is, let X have no increment. " He thinks that the infinitesimal dx is equal to zero but not equal to zero, and it is absurd to come and go at once. "dx is a lost soul". Is infinitesimal zero or not? Is infinitesimal and its analysis reasonable? This has caused a debate in mathematics and even philosophy for a century and a half. It led to the second mathematical crisis in the history of mathematics.

/kloc-the mathematical thought of the 0/8th century is really not rigorous, and it intuitively emphasizes formal calculation without considering the reliability of the foundation. In particular, there is no clear concept of infinitesimal, so the concepts of derivative, differential and integral are not clear, the concept of infinity is not clear, and the sum of divergent series is arbitrary, the symbols are not used strictly, the differential does not consider continuity, the existence of derivative and integral and whether the function can be expanded into power series.

It was not until the 1920s of 19 that some mathematicians paid more attention to the strict foundation of calculus. It took more than half a century from the work of Porzano, Abel, Cauchy, De Lecelli and others to the work of Wilstras, Dydykin and Cantor, which basically solved the contradiction and laid a strict foundation for mathematical analysis. The Birth of Paradox —— The Third Mathematical Crisis

The third crisis in the history of mathematics was caused by the sudden impact of 1897. So far, on the whole, it has not been solved to a satisfactory degree. This crisis is caused by the paradox found on the edge of Cantor's general set theory. Because the concept of set has penetrated into many branches of mathematics, in fact, set theory has become the basis of mathematics, and the discovery of paradox in set theory naturally raises doubts about the validity of the whole basic structure of mathematics.

1897, Forcy revealed the first paradox in set theory. Two years later, Cantor discovered a very similar paradox. In 1902, Russell found another paradox, which involves no other concepts except the concept of set itself. Russell's paradox has been popularized in many forms. The most famous one was given by Russell in 19 19, which involved the plight of a country barber. The barber announced a principle that he would shave all the people who don't shave themselves, only the people in the village. When people try to answer the following question, they realize the contradiction of this situation: "Does the barber shave himself?" If he doesn't shave himself, then shave himself according to the principle; If he shaves himself, then he doesn't conform to his principles.

Russell's paradox shook the whole math building. No wonder Frege wrote at the end of the second volume of his forthcoming Basic Law of Arithmetic after receiving Russell's letter: "A scientist will not encounter anything more embarrassing than this, that is, when his work is completed, his foundation collapses. When this book was waiting for printing, a letter from Mr. Russell put me in this position. " Thus ended nearly 12 years of efforts.

Admitting infinite sets and infinite cardinality is like all disasters, which is the essence of the third mathematical crisis. Although paradoxes can be eliminated and contradictions can be solved, mathematical certainty is gradually lost. There are a lot of axioms in modern axiom set theory. It is hard to say whether these axioms are true or false, but they cannot be eliminated. They are closely related to the whole mathematics. So the third crisis was solved on the surface, but in essence it continued in other forms.