In a sense, mathematics in the modern sense (that is, pure mathematics as a deductive system) comes from Pythagoras school in ancient Greece. This school flourished around 500 BC and is an idealistic school. They attach importance to the study of the invariable factors in nature and society, and call geometry, arithmetic, astronomy and music "four arts", in which they pursue the harmony and law of the universe. 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. Some people say that this property was discovered by hippasus around 400 BC, so his companions threw him into the sea. But it is more likely that Pythagoras knew the fact that hippasus was executed for leaking information. In any case, this discovery 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 cannot be completely expressed by integers and their ratios, but numbers can be expressed by geometric quantities. The lofty position of integers was challenged, so geometry began to occupy a special position in Greek mathematics.
At the same time, it also reflects that intuition and experience are not necessarily reliable, but reasoning proves to be reliable. From then on, the Greeks set out from the axiom of "self-evident" and established the geometric system through deductive reasoning, which was a great revolution in mathematical thought and a natural product of the first mathematical crisis.
Looking back at the previous mathematics, it is nothing more than "calculation", that is, providing algorithms. Even in ancient Greece, mathematics was applied to practical problems from reality. For example, Thales predicted the solar eclipse, calculated the pyramid height by using the shadow distance, and measured the offshore distance of ships. All belong to the category of computing technology. As for Egypt, Babylon, China, Indian and other countries, mathematics has never experienced such a crisis and revolution, so it still stays in the stage of "mathematization". Greek mathematics, on the other hand, took a completely different road, forming the axiomatic system of Euclid's Elements of Geometry and Aristotle's logical system.
The second mathematical crisis
As early as ancient times, people were interested in the measurement of length, area and volume. Eudoxus, an ancient Greek, introduced the concept of quantity to consider things that change continuously, and treated continuous quantity strictly according to geometry. This leads to the long-term separation of number and quantity. In ancient Greek mathematics, except for integers, there was no concept of irrational numbers, or even the operation of rational numbers, but there was a proportion of numbers. They are interested in the relationship between continuity and discreteness, especially the four famous paradoxes put forward by Zhi Nuo:
The first paradox is that motion does not exist, because a moving object must reach half before reaching its destination, and must reach half before reaching half ... If this goes on, it must pass through an infinite number of points, which is impossible in a limited time.
The second paradox is that Ashley runs fast, but she can't catch up with the tortoise in front. Because when the tortoise is in front of him, he must first reach the starting point of the tortoise, and then use the logic of the first paradox, the tortoise is in front of him. These two paradoxes oppose the view that space and time are infinitely separable.
The third and fourth paradoxes oppose the fact that space and time are composed of inseparable intervals. The third paradox is "the arrow does not move", because in a certain time interval, the arrow is always in a certain position in a certain space interval, so it is static. The fourth paradox is the military parade, which is similar in content. This shows that the Greeks have seen the contradiction between infinitesimal and tiny. Of course, they can't solve these contradictions.
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.
In the 16th and 17th centuries, in addition to finding the length of the curve and the area surrounded by the curve, many new problems appeared, such as finding the speed, tangent, maximum value and minimum value. After years of hard work, 17 century finally formed calculus, which is the beginning of mathematical analysis.
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.
/kloc-mathematicians in the 0 th and 8 th centuries successfully solved many practical problems with calculus, so some people are not interested in discussing these basic problems. D'Alembert, for example, said that it is now "to build the house higher, not to lay a more solid foundation". Many people think that the so-called strictness is cumbersome.
But because of this, the basic problems of calculus have been criticized and attacked by some people, the most famous of which is the attack of Bishop Becquerel in 1734.
/kloc-the mathematical thought of the 0/8th century is really not rigorous and intuitive, and pays attention to formal calculation, regardless of whether the foundation is reliable or not. 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 also unclear; Arbitrariness of summation of divergent series; Inaccurate use of symbols; Differential does not consider continuity, the existence of derivatives and integrals, and whether it can be expanded into power series.
It was not until11920s that some mathematicians began to pay more attention to the strict foundation of calculus. They started with the work of Porzano, Abel, Cauchy, Dirichlet and others, and were finally completed by Wilstes, Dedeking and Cantor. After more than half a century, they basically solved the contradiction and laid a strict foundation for mathematical analysis.
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.
In the early 1970s, Wilstrass, Dai Dejin, Cantor and others independently established the real number theory, and established the basic theorem of limit theory on the basis of the real number theory, so that the mathematical analysis was finally based on the strict basis of the real number theory.
At the same time, Wilstrass gives an example of a continuous function that can be differentiated everywhere. This discovery and many examples of morbid functions later fully show that intuition and geometric thinking are unreliable and must resort to strict concepts and reasoning. Results The second mathematical crisis made mathematics explore the problem of real number theory, which was the basis of mathematical analysis. This not only led to the birth of set theory, but also reduced the non-contradictory problem of mathematical analysis to the non-contradictory problem of real number theory, which was the primary problem in the mathematical foundation of the twentieth century.
1-6 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.