In the history of mathematical development, there have been three crises, involving irrational numbers, calculus and sets, and some even overthrew famous mathematical theories and caused a sensation.

The first crisis, about Hippocrates and Pythagoras.

Pythagoras was a famous mathematician and philosopher in the 5th century BC. What is it? Pythagoras School is the cornerstone of philosophy. What are their mathematical beliefs in Pythagoras school? Everything can be expressed as an integer or a ratio of integers? But when hippasus appeared, everything changed.

Hippasus is a member of the Pythagorean School. One day, he was thinking? What is the diagonal length of a square with a side length of 1? In this problem, it is found that this length can not be expressed by integers and fractions, but only by a brand-new number, thus finding the first irrational number.

The discovery of the first irrational number caused an uproar among mathematicians at that time, which directly overthrew Pythagoras' famous theory and greatly impacted the common concept of Greeks at that time. To make matters worse, the Greeks could do nothing about it, which directly led to the cognitive crisis at that time.

For this storm, because of its great influence, it is called? The first mathematical crisis? The emergence of the second mathematical crisis stems from the use of calculus tools.

With the development of social productive forces at that time and the improvement of human understanding in scientific practice, calculus in17th century was discovered by Newton and Leibniz. The appearance of calculus solved many difficult problems. However, because the calculus theory founded by Newton and Leibniz is not perfect, calculus has been opposed and criticized by some people, among which British Archbishop Becquerel is the main representative.

They attacked Newton and Leibniz's calculus theory, although it was based on the analysis of infinitesimal, but their understanding and application of infinitesimal was very confusing. This kind of attack and questioning on the rationality of calculus almost subverts the whole calculus theory.

Fortunately, Cauchy stood up and defined infinitesimal with the method of limit, which made the calculus theory perfect and continue to develop.

Time came to the19th century, and mathematician Cantor founded the famous set theory. As soon as this new theory was born, it was fiercely attacked by many people. But soon, this theory was accepted by many mathematicians, because they found that the whole mathematical building was established from natural numbers and Cantor's set theory.

As a result, set theory has become the cornerstone of modern mathematics and has been highly praised by many mathematicians. However, the good times did not last long. 1903, the British mathematician Russell put forward his paradox, saying that set theory was flawed and shocked the whole mathematical world.

In Russell's paradox, he constructed an S, and proposed that if S is composed of all elements that do not belong to him, does S also include S? This seemingly reasonable question puts the respondent in a dilemma. If S belongs to S, then according to the definition of S, S does not belong to S .. and S does not belong to S, according to the definition, S belongs to S.

Russell's paradox, which is easy to understand, caused a shock in the fields of mathematics and logic as soon as it came out, and the huge response caused this third mathematical crisis.

After the crisis, mathematicians actively proposed solutions. Finally, in 1908, Zemelo put forward the first axiomatic set theory system, which was later perfected by other mathematicians and called ZF system, which largely made up for the defects of Cantor's set theory. When the paradox is successfully eliminated, the third mathematical crisis can be solved satisfactorily.

In fact, the three crises in mathematics seem to have almost subverted the common sense of mathematical theory and had a great influence at that time, but in fact, the three crises promoted the development and progress of mathematics in disguise and endowed the mathematical community with new vitality and vitality.