The emergence and development of limit

(1) origin

Like all scientific thinking methods, extreme thinking is also the product of abstract thinking in social practice. The idea of limit can be traced back to ancient times. For example, the secant technology in Liu Hui, the motherland, is an original and reliable application of the limit idea of "approaching constantly" based on the research of intuitive graphics; The ancient Greeks' exhaustive method also contained the idea of limit, but because of their "fear of infinity", they obviously avoided artificially "taking the limit" and completed the relevant proof by indirect proof-reduction to absurdity.

/kloc-in the 6th century, Steven, a Dutch mathematician, improved the ancient Greek exhaustive method in the process of investigating the center of gravity of a triangle. With the help of geometric intuition, he boldly used limit thought to think about problems and gave up the proof of reduction law. In this way, he inadvertently "pointed out the direction of the development of the limit method into a practical concept."

(2) development

The further development of limit thought is closely related to the establishment of calculus. /kloc-in the 0/6th century, Europe was in the embryonic stage of capitalism and its productive forces developed greatly. There are many problems in production and technology. At first, people could not solve them with elementary mathematics. It is the social background to promote the development of extreme thinking and establish calculus to require mathematics to break through the traditional scope of "only studying constants" and find new tools that can describe and study the process of motion change.

At first, Newton and Leibniz established calculus based on the concept of infinitesimal. Later, due to logical difficulties, everyone accepted the idea of limit to varying degrees in their later years. Newton used the ratio of distance change δs' to time change δt'

"Representing the average speed of a moving object makes Δ t infinitely approach zero to get the instantaneous speed of the object, from which the concept of derivative and differential theory are derived. He realized the importance of the concept of limit and tried to use it as the basis of calculus. He said: "If two quantities and the ratio of quantities remain equal for a limited time and are close to each other before the end of this time, so that the difference is less than any given difference, then it will eventually become equal." "But Newton's concept of limit is also based on geometric intuition, so he can't get the strict expression of limit. Newton's concept of limit is only close to the following intuitive language description: "If n increases indefinitely,

Infinitely close to the constant a, and then say

Take a as the limit.

It was precisely because of the lack of strict limit definition at that time that calculus theory was doubted and attacked by people on scientific theory. For example, in the concept of' instantaneous velocity' in physics, is Δ t (variation) equal to zero? If it is zero, (because if the truth is infinitely expanded, its scope of application will become wrong): How can it be used for division? (In fact, the variation cannot be zero). But people think that if it is not zero, how can computers and functions get rid of those "tiny quantities" that contain it when they are deformed? At that time, people didn't understand and hoped that there was no error in calculating variables, which led to the paradox, which was the cause of infinitesimal paradox in the history of mathematics. British philosopher and archbishop Becquerel attacked calculus the most violently. He said that the derivation of calculus was "obvious sophistry". The history and success of scientific development show that his view is wrong.

Becker's fierce attack on calculus, on the one hand, served religion, on the other hand, was due to the lack of solid theoretical foundation and flexible solution of calculus at that time, even Newton, a famous figure, could not get rid of the confusion in the concept of limit. This fact shows that understanding the concept of "limit" is a dynamic and infinitely variable process, and of course, tiny variables can be approximately equal to a constant very accurately in the trend direction. This is the ideological basis of establishing strict calculus theory, and it is of great significance as a tool of scientific research in epistemology.

(3) perfection

The perfection of limit thought is closely related to the rigor of calculus. For a long time, many people have tried to solve the theoretical problems of calculus, but they have failed. This is because the research object of mathematics has expanded from constant to variable, and people are used to thinking and analyzing problems with constant constants. The understanding of the unique concept of "variable" is not clear; There is still a lack of understanding of the differences and relations between "variable mathematics" and "constant mathematics"; The unity of opposites between "finite" and "infinite" is still unclear. In this way, people can't adapt to the new development of variable mathematics with the traditional thinking method of dealing with constant mathematics. People in ancient times used to explain the dialectical relationship of this scientific conclusion with the old concept constant [at an artificially small distance, "zero" and "infinite non-zero value near zero" can jump to equal mutual transformation].

/kloc-in the 8th century, Robbins, D'Alembert, Lorrell and others clearly put forward that limit must be the basic concept of calculus, and they all made their own definitions of limit. Among them, D'Alembert's definition is: "One quantity is the limit of another quantity, and if the second quantity is closer to the first quantity than any given value", its description connotation is close to the correct definition of "limit"; However, none of these people's definitions can get rid of their dependence on geometric intuition. This is the only view, because most of the concepts of arithmetic and geometry before19th century are based on the concept of geometric quantities. In fact, "concretization" is not synonymous with reverse thinking, nor is the study of geometric intuition synonymous with reverse thinking, because today, functions can still be "mapped" into graphs to study more complex trend problems. If there is a trend, the concept of limit can be established. For example, replacing functions with "figurative" graphics can kidnap and intuitively prove that a proposition that has no rules to describe and cannot be attacked by users for a long time is untenable; (or another function can be established), and then make concrete mathematical proofs of "symbol pattern" respectively.

Czech mathematician Porzano first gave the correct definition of derivative with the concept of limit. He defined the derivative of the function f(x) as the difference quotient.

Limit f'(x), he stressed that f'(x) is not the quotient of two zeros. Porzano's thought is valuable, but he still hasn't described the nature of limit clearly.

/kloc-in the 9th century, French mathematician Cauchy expounded the concept of limit and its theory completely on the basis of predecessors' work. He pointed out in Analysis Tutorial: "When the value of a variable infinitely approaches a fixed value, the difference between the value of the variable and the fixed value will be as small as possible, and this fixed value is called the limit value of all other values. In particular, when the variable value (?

Cauchy regards infinitesimal as "a variable with a limit of 0" and correctly establishes the concept of infinitesimal as "seemingly zero is not zero, but it can be artificially treated as equal to 0". That is to say, in the process of variable change, its value is actually not equal to zero, but its changing trend is "zero" and it can be infinitely close to zero. Then people can use "equal to 0" to deal with it, and there will be no wrong result.

Cauchy tried to eliminate the geometric intuition in the concept of limit (but "geometric intuition" is not a negative thing, and we can also use our imagination when studying functions-"If the variable image of dynamic trend is enlarged to a huge astronomical multiple, we will never see the variable value' overlapping with 0', so it will be more" clear "to express it with inequality), and then make a clear definition of limit, thus fulfilling Newton's wish. However, there are still descriptive words such as "infinite approximation" and "the smaller the better" in Cauchy's narrative, which are relatively easy to understand, so it is easier to understand the concept, so its definition still retains the intuitive traces of geometry and physics. If it is split in two, it will help to have a more intuitive trace, but it is easier to understand the concept of' limit' by combining the following abstract definitions.

In order to eliminate the intuitive trace in the concept of limit, Wilstrass put forward the static abstract definition of limit, which provided a strict theoretical basis for calculus. so-called

If you are interested in anything

There is always a natural number n, so when

Time and inequality

Keep building. "

With the help of inequality, this definition quantitatively and concretely describes the relationship between two "infinite processes" through the relationship between ε and n. Therefore, this definition should be a strict definition at present, which can be used as the basis for scientific argumentation and is still used in mathematical analysis books. In this definition, only' number and its size relationship' are involved, and only words such as given, existence and any are used, and the word' close' is got rid of, and the intuition of movement is no longer resorted to. (However, understanding the concept of' limit' cannot abandon the concept of' motion trend', otherwise it will easily lead to the unscientific introduction of' constant concept into the field of calculus')

Constant can be understood as' constant'. Before the appearance of calculus, people used to study mathematical objects with static images. Since the appearance of analytic geometry and calculus, the thinking mode of motion considering' variation' has entered the field of mathematics, and people have mathematical tools to dynamically study the changing process of physical quantities and other things. Later, the ε-N language established by Wilstras described the changing trend of variables with static definitions. This spiral evolution of "static-dynamic-static" reflects the dialectical law of mathematical development.