Since the birth of mathematics since ancient times, mathematicians have been pursuing truth and have made brilliant achievements. Besides mathematics, mathematical concepts and their inferences provide the essence for important scientific theories.
Everything contains contradictions, and the contradictions within things promote the development of things, and mathematics is no exception. Mathematics is developed by discovering and solving contradictions. There is a euphemism for contradiction in mathematics, which is called paradox. The more serious consequence of mathematical paradox is mathematical crisis. The so-called mathematical crisis is just a severe warning to the mathematical community by mathematicians in the spirit of strictly exploring mathematical truth. It is precisely because of this severe warning that many paradoxes have been solved and the continuous development of mathematics has been promoted.
There have been three paradoxes and mathematical crises in the history of mathematics.
The first mathematical crisis occurred in ancient Greece. But at that time, the Pythagorean school advocated a philosophical view called "numerology", which believed that all things and phenomena could be attributed to integers or the ratio of integers to integers, which was called "harmony of numbers". But later, when hippasus proved Pythagorean theorem, he found that the height x on the hypotenuse of right triangle existed.
1:x=x:2，x=√2，
√2 is not integer ratio. So the "harmony of numbers" was broken, which led to the emergence of irrational numbers (the word irrational numbers has been used until today, although indecent, and has become a custom).
The second crisis occurred in the17th century, involving the theoretical basis of calculus, which was caused by Becker's paradox. Becquerel pointed out that in the calculation formula
δy/δx
The infinitesimal δx is neither zero nor abandoned, which violates the law of contradiction. Infinitesimal can't be divided if it is zero, and can't be abandoned if it is not zero. This is the famous "Becquerel Paradox". This led to the confusion of mathematics and the second mathematical crisis broke out. People realize that although calculus is a good method to solve many practical problems, it lacks a rigorous theoretical basis. Therefore, many mathematicians have made unremitting efforts to establish the theoretical basis of calculus. French mathematician Cauchy first gave the definition of limit, and then established the theories of continuity, derivative, differential and integral.
Many mathematicians, such as Descartes, Newton, Leibniz, Cauchy, Lagrange, Abel, Cantor, Fermat, Bernoulli family, Euler, Laplace and Hilbert, have laid a mathematical foundation for the development of mathematics and made brilliant achievements.
The set theory established by Cantor has become the basis of mathematics. However, the British mathematician Russell put forward a famous "barber paradox"-a barber in a small town released a rhetoric: "I help and only help all people in the city who don't shave." But the question is: Should the barber shave himself? If he shaves himself, don't shave only those who don't shave themselves, as he said. But if he doesn't shave himself, he should shave himself, as he said, "shave everyone who doesn't shave himself in the city." Barber's paradox is a popular saying used by Russell to compare Russell's paradox, which was put forward by Bertrand Russell in 190 1. The appearance of Russell paradox is due to the unlimited definition of elements in naive set theory. Because set theory has become the basis of mathematical theory at that time, the appearance of this paradox directly led to the third mathematical crisis, and also triggered many mathematicians to remedy this problem, and finally formed the current axiomatic set theory. At the same time, the appearance of Russell paradox urges mathematicians to realize the necessity of axiomatization of mathematical foundation.
So mathematicians did a lot of work and devoted themselves to public physics and chemistry, and by 1930, three mathematical schools were established; Intuitionism, logicism and formalism, although still flawed, have made great contributions to the development of mathematics.
Among the three schools, the formalism school seems to be mainly established by Hilbert, which is slightly better. It once claimed to have solved the problem of "compatibility and integrity". However, in 193 1, the Austrian mathematician Godel (who later moved to the United States) proved two incompleteness theorems, "revealing the inevitable limitations of formalism." Godel's incompleteness theorem is a negation of law of excluded middle, but "if,
In a word, although mathematics has made brilliant achievements. However, there are still many problems, and the fundamental problems have not been solved, such as the definition of mathematics and the definition of numbers, which require people to devote themselves to mathematical research, especially the research of mathematical foundation, and hope that people with will will will devote themselves to this.
For this reason, Guo Dunqing studied mathematics for 40 years, and wrote papers such as Goldbach conjecture proof, mathematical program-micro-mathematics and macro-mathematics, Euclid geometry parallel problem solution, and on reduction to absurdity-living universal axiom, which were published in China Guo Dunqing's column in 2008-2009.
The first chapter of the mathematical plan-micro-mathematics and macro-mathematics was also published in Baidu. Goldbach conjecture proof included in Baidu snapshot.