Therefore, in the development of calculus, there is such a situation: on the one hand, calculus was applied to science and technology immediately after it was produced, which developed rapidly; On the other hand, the theory of calculus at that time was not rigorous, and there were more and more paradoxes and fallacies. The development of mathematics has encountered a profound and disturbing crisis. For example, sometimes the infinitesimal is regarded as a finite quantity that is not zero, which is eliminated from both ends of the equation, and sometimes the infinitesimal is ignored as zero. Because of these contradictions, there is a big debate in the field of mathematics For example, Becquerel, the Irish bishop and idealist philosopher at that time, laughed at "infinitesimal" as "you damn fool". Becker criticized the definition of Newton's derivative.
Newton's definition of derivative at that time was:
When the growth is, the cube (marked as) becomes the cube (marked as), which is the cubic result of. The increments of and are and respectively. Divide the increment of by the increment of, and then substitute h=0 to make the increment disappear. Their final result is. We know that this result is correct, but there are obvious mistakes in the derivation process: in the first part of the argument, it is assumed that it is not 0, and in the second part of the argument, it is taken as 0. So is it 0 after all? This is the famous Becquerel paradox. This crisis caused by calculus is called the second mathematical crisis in the history of mathematics, which is directly related to Newton. Historical requirements give calculus a strict foundation.
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D'Alembert was the first person to put forward a truly insightful view to remedy the second mathematical crisis. He pointed out in 1754 that the rough limit theory used at that time must be replaced by reliable theory. But he himself failed to provide such a theory. Lagrange was the first person to make calculus strict. In order to avoid the use of infinitesimal reasoning and the ambiguity of the concept of limit at that time, Lagrange tried to establish the whole calculus on the basis of Taylor formula. However, the range of functions considered in this way is too narrow, and there is no convergence of infinite series without the concept of limit. So Lagrange's algebraic method with power series as a tool can't solve the basic problems of calculus.
/kloc-A group of outstanding mathematicians appeared in the 20th century, who actively devoted themselves to the basic work of calculus, including the Czech philosopher Porzano, who wrote Infinite Paradox, clearly put forward the concept of series convergence, and gained a deeper understanding of limit, continuity and variables.
French mathematician Cauchy, the founder of analytical science, published The Course of Analysis and Lectures on Infinitesimal Calculation between 182 1- 1823, which are epoch-making works in the history of mathematics. There, he gave a series of basic concepts and precise definitions of mathematical analysis.
The requirement for a deeper understanding of the analytical basis occurred in 1874. At that time, the German mathematician Wilstrass constructed a continuous function without derivative, that is, a continuous curve without tangent, which contradicted the intuitive concept. It makes people realize that the dependence of limit concept, continuity, differentiability and convergence on real number system is much deeper than people think. Riemann found that Cauchy did not need to limit his definite integral to a continuous function. Riemann proved that the integrand function is discontinuous and its definite integral may exist. That is, Cauchy integral is improved to Riemann integral.
These facts make us understand that when establishing a perfect analysis foundation, we need to dig deeper: understand the deeper essence of real number system. This work was finally completed by Wilstrass, which made the mathematical analysis completely come from the real number system and divorced from perceptual knowledge and geometric intuition. In this way, all the basic concepts of mathematical analysis can be expressed by real numbers and their basic operations. Finally, the rigor of calculus has reached its peak, and only the concept of infinity has not been fully understood. In this field, German mathematician Cantor has made outstanding contributions.
In a word, the core of the second mathematical crisis is the unstable foundation of calculus. Cauchy's contribution lies in establishing calculus on the basis of limit theory. Wilstrass's contribution lies in logically constructing the theory of real numbers. To this end, the logical order of establishing the analysis basis is
Real number system-limit theory-calculus
18th century analysis
/kloc-The motive force for the continuous development of calculus in the 8th century is the need of physics, and the expression of physical problems is generally in the form of differential equations. 18th century is called the heroic century in the history of mathematics. They applied calculus to astronomy, mechanics, optics, heat and other fields, and achieved fruitful results. In mathematics itself, the theories of multivariate differential calculus, multiple integral calculus, differential equation, infinite series and variational method have been developed, which greatly expanded the scope of mathematical research.
The most famous one is the steepest descent line problem: the steepest descent curve problem. This once difficult problem can be easily solved by the theory of variational method.