Three mathematical crises, the first mathematical crisis, the Pythagorean school of ancient Greece. They believe that "everything is important" and that mathematical knowledge is reliable and accurate and can be applied to the real world. Knowledge of mathematics is acquired through pure thinking, without observation, intuition and daily experience. Pythagoras' numbers refer to integers, and one of their great discoveries in mathematics is to prove Pythagoras' theorem. They know the general formula to satisfy the three-sided length of right-angled triangles, but they also find that the three-sided ratio of some right-angled triangles cannot be expressed by integers, that is, the hook length or the node length is not commensurable with the chord length. This negates the Pythagorean creed that all phenomena in the universe can be attributed to integers or the ratio of integers. The discovery of incommensurability triggered the first mathematical crisis. The product of the first crisis-classical logic and Euclid geometry The second mathematical crisis In ancient Greek mathematics, there was no concept of irrational number and no operation of rational number except integer, but there was a proportion of quantity. Although the Greeks did not have a clear concept of limit, they had strict approximation steps when dealing with the problems of area and volume. This is the so-called "exhaustive method". It relies on indirect proof to prove many important and difficult theorems. Newton and Leibniz are recognized as the founders of calculus. Their achievements mainly lie in: 1, unifying the solutions of various problems into one method, differential method and integral method; 2, there are clear steps to calculate the differential method; 3. Differential method and integral method are reciprocal operations. Calculus has become an important tool to solve problems because of its completeness of operation and universality of application. At the same time, the problems about the basis of calculus are becoming more and more serious. Take speed as an example. The instantaneous velocity is the value of δ s/δ t when δ t tends to zero. Δ t is zero, is it a small quantity, or what? Is this infinitesimal quantity zero? This caused great controversy and led to the second mathematical crisis. Porzano denied the existence of infinite decimals and infinite numbers, and gave a correct definition of continuity. Cauchy started with the definition of variables in the algebra analysis course of 182 1, and realized that functions don't have to have analytic expressions. He mastered the concept of limit, pointed out that infinitesimal and infinitesimal are not fixed quantities but variables, and defined derivatives and integrals. Abel pointed out that it is necessary to strictly limit the abuse of series expansion and summation; Dirichlet gave a modern definition of function. On the basis of these mathematical works, Wilstrass eliminated the inaccuracy, gave the limit and continuous definition of ε-δ, and strictly established the concepts of derivative and integral on the basis of limit, thus overcoming the crisis and contradiction. The Third Mathematical Crisis After the first and second mathematical crises, people attributed the non-contradiction of the basic theory of mathematics to the non-contradiction of set theory, which has become the logical basis of the whole modern mathematics, even though the magnificent building of mathematics has been built. It seems that set theory is not contradictory, and the goal of strict mathematics has almost been achieved, and mathematicians are almost complacent about this achievement. Russell (1872-1970), a famous British mathematical logician and philosopher, announced an amazing news: set theory is self-contradictory and has no absolute rigor! History is called "Russell Paradox". The discovery of Russell's paradox is tantamount to breaking the fog in a sunny day and waking people up from their dreams. Russell paradox and other paradoxes in set theory go deep into the theoretical basis of set theory, thus fundamentally endangering the certainty and rigor of the whole mathematical system. So it caused an uproar in the fields of mathematics and logic, and formed the third crisis in the history of mathematics. The product of the third mathematical crisis-the development of mathematical logic and the emergence of a number of modern mathematics. Due to different starting points and different ways to solve problems, different schools of mathematical philosophy were formed at the beginning of this century, namely, the logicism school headed by Russell, the intuitionism school headed by Brouwer (188 1- 1966) and the formalism school headed by Hilbert. The formation and development of these three schools have pushed the research of basic mathematics theory to a new stage. The mathematical achievements of the three schools are first manifested in the formation of mathematical logic and its modern branch proof theory.