/kloc-in the 0/9th century, a group of outstanding mathematicians appeared, who worked hard for the basic work of calculus. First, Czech philosopher and mathematician Porzano began to introduce strict argument into mathematical analysis. In 18 16, he clearly put forward the concept of convergence and divergence of series in the proof of binomial expansion, and at the same time had a deeper understanding of limit, continuity and variables.
The founder of Analytics is recognized as French prolific mathematician Cauchy, who has done pioneering work in mathematical analysis and permutation group theory and is one of the greatest modern mathematicians. Cauchy's "Analysis Course" and "Notes on Infinitesimal Calculation" published in 182 1 ~ 1823 are epoch-making works in the history of mathematics, in which he gave a series of precise definitions of basic concepts of mathematical analysis. For example, he gave an accurate definition of limit, and then defined the continuity, derivative, differential, definite integral and convergence of infinite series with limit. Then, Wilstrass introduced the precise limit definition of "". In this way, calculus is based on strict limit theory. The definition used in our calculus textbook today is basically Cauchy's, but now it is written more strictly.
17 and 18 century, the fierce debate about calculus is called the second mathematical crisis. From a historical or logical point of view, its occurrence is also inevitable. The germination of this crisis appeared around 450 BC. Zhi Nuo noticed the contradiction brought by the understanding of infinity, and put forward four paradoxes about the finiteness and infinity of time and space: "Dichotomy": an object moving to its destination must first pass through the midpoint of the journey, and to pass through this point, it must first pass through the 1/4 point of the journey, and so on until infinity. The conclusion is that infinity is an endless process and movement is impossible. "Achilles (the running hero in Homer's epic) can't catch up with the tortoise": Achilles always reaches the starting point of the tortoise first, so the tortoise must always run ahead. This argument is the same as the dichotomy paradox, except that it is not necessary to divide the required distance equally again and again. "The arrow doesn't move": It means that the arrow must be in a certain position at any time during the movement, so it is stationary, so it can't be moving. "Playground or team": Two objects, A and B, move in opposite directions at the same speed. From the point of view of static C, for example, A and B both moved 2 kilometers in 1 hour, but from the point of view of A, B moved 4 kilometers in 1 hour. Exercise is contradictory, so exercise is impossible. The contradiction revealed by Zhi Nuo is profound and complicated. The first two paradoxes challenge the view that time and space are infinitely separable, so motion is continuous, while the last two paradoxes challenge the view that time and space are infinitely inseparable, so motion is discontinuous. Zeno paradox may have a deeper background, not necessarily for mathematics, but they have caused an uproar in the mathematics kingdom. They show that the Greeks saw the contradiction between infinitesimal and infinitesimal, but they could not solve these contradictions. Therefore, infinitesimal has been excluded from Greek geometric proof. After years of hard work, calculus was finally formed in the late 7th century. Newton and Leibniz are recognized founders of calculus, and their achievements mainly lie in: unifying the solutions of various related problems into differential method and integral method; There are clear calculation steps; Differential method and integral method are reciprocal operations. Because of the completeness of operation and the universality of application, calculus became an important tool to solve problems at that time. At the same time, the problems about the basis of calculus are becoming more and more serious. The key question is whether infinitesimal competition is zero. Is infinitesimal and its analysis reasonable? This caused a century-and-a-half-long mathematical and even philosophical debate, which led to the second mathematical crisis. Is infinitesimal zero or not? Both answers will lead to contradictions. Newton made three different interpretations of it: 1669 said it was a constant; 167 1 is a variable that tends to zero; 1676 It was replaced by "the final ratio of two vanishing quantities". However, he has never been able to solve the above contradictions. Leibniz tried to replace infinitesimal with finite difference proportional to infinitesimal, but he also failed to find a bridge from finite to infinitesimal. British Archbishop Becker wrote an article in 1734, attacking the number (derivative) of flow as "the ghost of disappearing number" ... People who can digest second-and third-order numbers will not vomit because they swallow theological arguments. He said that the original mistake was eliminated by ignoring the higher-order infinitesimal? What is this? What's the matter with you? Ciqin? Return to admiration? Thin 1 pure death is caused by gray, so what about the pen cover? ⑽ ⑽ 934 Slow down? Is it a teacher? What's the point? Killing people is hard. ぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃぃ 1/p & gt； At that time, some mathematicians and other scholars also criticized some problems of calculus, pointing out that it lacked the necessary logical foundation. For example, Rolle once said, "Calculus is a clever collection of fallacies." At the beginning of the creative era, science existed logical problems, not individual phenomena. /kloc-the mathematical thought of the 0/8th century is really not rigorous and intuitive, emphasizing 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 unclear; Arbitrariness of summation of divergent series, etc. The lax use of symbols; Differential does not consider continuity, the existence of derivatives and integrals, whether the function can be expanded into power series and so on. 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 beginning of the work of Porzano, Abel, Cauchy, De Ricchelli and others to the end of the work of Wilstras, Dedekind and Cantor, which basically solved the contradiction and laid a strict foundation for mathematical analysis. Porzano gave a correct definition of continuity; Abel pointed out that it is necessary to strictly limit the abuse of series expansion and summation; 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 grasped the concept of limit and pointed out that infinitesimal and infinitesimal are not fixed quantities but variables, and infinitesimal is a variable with zero as the limit; Derivative and integral are defined. De Reichley gave a modern definition of function. On the basis of these works, Wilstrass eliminated the inaccuracies, gave the current universal definition of limit and continuous definition, and strictly established derivatives and integrals on the basis of limit. From 65438 to the early 1970s, Wilstrass, Dedekind, Cantor and others independently established the real number theory, and established the basic theorem of the limit theory on the basis of the real number theory, thus making the mathematical analysis based on the strict basis of the real number theory.