Limit thoughtThe bud of "limit" can be traced back to the ancient Greek period and the Warring States Period of China, but the first appearance of the concept of "limit" in the true sense was in theWallisIn Infinite Arithmetic, Newton《Mathematical Principles of Natural Philosophy》The word limit is clearly used and explained in the book.But it was not until the second half of the 18th century that D'Alembert and other people realized thatCalculusBased on the concept of limit, calculus is perfect,CauchyThe descriptive definition of limit was first given, and then Weierstrass gave the strict definition of limit (ε - δ and ε - N definitions).
Since then, all kinds of limit problems have practical criteria, which makes limit theory become the tool and foundation of calculus.
CalculusSince its birth, it has played a great role in mechanics and astronomy, and can easily solve many problems that were thought to be helpless.Later, calculus has achieved fruitful results in more fields.It is generally recognized that calculus is the highest achievement of mathematics in the 17th and 18th centuries, but the arguments made by its founders Newton and Leibniz are not clear and rigorous.Whether it is Newton's instantaneous sumNumber of streamsOr Leibniz's dx and“Infinitesimal quantity", but they did not give a definite and consistent definition in their respective discussions. In the derivation and operation of calculus, the infinitesimal quantity is often used firstdenominatorDivide, and then treat the infinitesimal quantity as zero to eliminate the items that contain it.So is "infinitesimal" zero or non-zero?If it is zero, how can it be usedDivisorWhat about?If it is not zero, how can we eliminate the items that contain it?Newton and Leibniz were aware of this logical contradiction.Newton used the initial ratio and the final ratio of finite difference to explain the meaning of stream number. But when the difference value has not reached zero, its ratio is not final. When the difference value reaches zero, their ratio becomes. How to understand such final ratio?It's really confusing.Newton admitted that he only made a "brief explanation, not a correct argument" for his method. Leibniz once putInfinitesimal quantityIt is described as an "ideal quantity", but as some mathematicians said: "It is more a mystery than an explanation."
Berkeley
Strangely, calculus itself has obvious logic confusion, but it is an effective tool in practical application.Thus, calculus has“Mystery"At first," mystery "focused on"Infinitesimal quantity"The understanding of this concept has been attacked by all kinds of people. Mathematicians cannot tolerate that the theory of this new method itself is so vague and absurd. French mathematicianMichel Rolle Calling calculus "a collection of ingenious fallacies";Famous thinkerVoltaireCalculus is said to be "the art of accurately calculating and measuring something that cannot be imagined to exist".In the midst of doubts and questions, the British bishop and philosopherBerkeleyHe ridiculed the infinitesimal number as "ghosts of a lost quantity" and said that calculus contains "a lot of emptiness, darkness and chaos", which is "clear sophistry".
Darumbel
Marx once made a historical investigation on calculus. He called this period the "mysterious calculus" period, and made such a comment: "Therefore, people themselves believed in the mystery of the newly discovered algorithm. This algorithm must have obtained the correct (and surprising in geometric application) results through incorrect mathematical methods.In this way, people mystified themselves and gave a higher evaluation to the new discovery, which made a group of old orthodox mathematicians more angry and aroused hostile clamors, which even reverberated outside the mathematical worldnew thingsIt is inevitable to open up new roads."
The logic defects of calculus and people's fierce attacks encourage mathematicians to work hard to eliminate the mystery of calculus, that is, to establish a reasonable theoretical basis for calculus.In the 18th century, the main representatives who made contributions in this regard were D'AlembertEulerAnd Lagrange.But“Infinitesimal quantity"The nature of,Infinite seriesThe problem of "harmony" is becoming increasingly prominent.In calculus, a typical basic algorithm is to add infinite multiple items, which is called summation of infinite series.stayElementary mathematicsIn, the sum of a finite number of items always has a definite sum.While the addition of infinite multiple items cannot be completed, what is the "sum" of infinite series is unclear.For a long time, people used to apply the operation rules of finite multinomial addition to infinite series. Although many problems have been solved, sometimes absurd results like 1/2=0 have appeared.
Cauchy
Since the 19th century, with the wider and deeper application of calculus, the quantitative relationship encountered has become more complex, and many problems, such asheat conductionThe study of phenomena has gone beyond that of early mechanicsIntuitiveness。In this case, clear concepts, logical reasoning andAlgorithmIt becomes more important and urgent.In fact, calculus, as variable mathematics, uses "infinity" to describe and study the process of movement and change, and has achieved success, but has not given a correct explanation of the concept of "infinity" for a long time, even leading to logical confusion. The mystery of calculus comes from this, and this is exactly the problem to be solved by the theoretical basis of calculus.
After more than 100 years of hard exploration, mathematicians have finally established the theoretical basis of calculus on the basis of a large number of achievements (including many failed attempts) accumulated by predecessors.Cauchy(1789-1857) published in 1821《Analysis Tutorial》We began to have a basic and clear description of the concept of limit, and based on the concept of limitInfinite seriesThe concepts of "sum" and "sum" are clearly defined.For example, from the perspective of limit“Infinitesimal quantity"Is a variable with zero limit. In the process of change, it can be" non-zero ", but its change tends to be" zero ", infinitely close to" zero ".Limit theoryIt is from the trend of change that the internal relationship between "infinitesimal quantity" and "zero" is explained, thus clarifying the confusion in logic and tearing off the mysterious veil of early calculus.Later, afterPolchano, Weierstrass, Dedekin, Cantor and others further established limit theory on the basis of strictset theoryandReal number theoryOn the basis of the above, the ε - δ language describing the limit process is formed.The rigorous foundation of calculus theory has made calculus leap forward and expand tomodern mathematics Important areas.
Weierstrass
Development of calculusHistory tells usA discipline can not only stay in the perceptual stage, if it does not rise to the rational stage and does not have a solid theoretical foundation, its application is limited, and the discipline itself is difficult to continue to develop.However, in many movements in our country in the last century, "mathematics isidealismUnder the influence of the wrong idea of "hereditary territory", limit theory and ε - δ language have been criticized and expelled from the classroom for many times. "After the Cultural Revolution", a teacher said with emotion: "When I was a student, I also took part in criticism with great enthusiasm, but after graduation, I did teaching work for several years, and I realized that what I criticized in the past was actually correct and significant.However, when I told the students these truths, I myself became the object of criticism of the students."
Engels has long pointed out that "if a nation wants to stand at the peak of science, it can't stop for a momentTheoretical thinking。"
