1734, Archbishop George Berkeley published a long book entitled "Analyst: or a Paper for an Atheist Mathematician" in the name of "A Little Philosopher", in which he examined whether the objects, principles and conclusions of modern analytical science were more clear or obvious than the mysteries of religion and the main points of belief.
In this book, Becker attacked Newton's theory. Because in Newton's theory, infinitesimal is said to be zero for a while and not zero for a while. Therefore, Becquerel ridiculed infinitesimal as "the ghost of death". Although Becker's attack came from the purpose of maintaining theology, it did seize the defects in Newton's theory and hit the nail on the head.
George berkeley was born in kilkenny in March 1685, and died in Oxford in October 1753+ 14. Teenagers are precocious They were admitted to Trinity College in Dublin at the age of 15, and got their bachelor's degree in 1704 and master's degree in 1707. They stay in school as lecturers and junior researchers. The publication of New Vision in 1709, Principles of Human Knowledge in17/0, and the three dialogues between Heiles and Philo North in17/3 all became heated discussions in British universities at that time. 1734 was appointed bishop of kilkenny, Ireland, and served for 18 years, still devoted to philosophical speculation. 1752 Transfer to a new college near Oxford.
Brief introduction of Becker paradox
In the history of mathematics, the Becquerel problem is called "Becquerel Paradox". Generally speaking, Becker's paradox can be expressed as "whether infinitesimal is zero": as far as the practical application of infinitesimal at that time is concerned, it must be both zero and non-zero. But as far as formal logic is concerned, this is undoubtedly a contradiction.
Mathematicians who have been engaged in calculus research for a long time have long thought about the illogical and imprecise problems of mathematics itself brought about by infinitesimal quantity, and there have been heated discussions and debates among them. From a mathematical point of view, how to better understand this problem may be considered as a purely technical problem; However, from a cultural point of view, only by examining infinitesimal operation and its connotation from a broader perspective, especially in close contact with Christian culture, which played an important role in European life at that time, can we better understand this fierce debate about infinitesimal operation.
The influence of Becker's paradox
With the improvement of people's understanding of scientific theory and practice, calculus, a sharp mathematical tool, was discovered independently by Newton and Leibniz almost simultaneously in the seventeenth century. As soon as this tool came out, it showed its extraordinary power. After using this tool, many difficult problems have become easy. But Newton and Leibniz's calculus theory is not strict. Their theories are all based on infinitesimal analysis, but their understanding and application of the basic concept of infinitesimal is confusing. Therefore, calculus has been opposed and attacked by some people since its birth. Among them, the British Archbishop Becquerel is the most fierce attacker, and it is the question of whether infinitesimal is zero in Becquerel's paradox that triggered the second mathematical crisis.
"Is the infinitesimal zero?" The answer to the question
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 not only acknowledged the existence of infinite decimals and infinite numbers, but also 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.