Carl Theodor William Weierstrass

German mathematician

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synonym Carl Weierstrass (Carl Weierstrass) Generally refers to Carl Theodor William Weierstrass

Carl Theodor William Weierstrass (October 31, 1815 - February 19, 1997), Germany mathematician , known as the "father of modern analysis". Born Westphalia Ostenfeld, died in Berlin.

Weierstrass in mathematical analysis The biggest contribution in the field is that in the trend of rigorous mathematical analysis initiated by Cauchy, Abel, etc., with ε - δ language, the real analysis and Complex analysis The arithmetic of analysis is basically completed. He introduced uniform convergence The concept of Itemized differentiation and Item by item integration Theorem.

In the process of establishing the basis of analysis Real number axis And a series of topological concepts in n-dimensional Euclidean space, and Riemann integral Promote to a Countable set No on continuous function above. In 1872, Weierstrass gave the first Nondifferentiable function To make people realize Continuity And Differentiability This leads to a series of Peano curve The study of the function with the same abnormal behavior.

Hilbert's comment on him is: "With his passion for criticism and profound insight, Weierstrass has established a solid foundation for mathematical analysis. By clarifying the concepts of minima, maxima, functions, derivatives, etc., he excluded Calculus The various wrong expressions still appearing in Infinitesimal All kinds of confused ideas, such as, have decisively overcome the difficulties stemming from the dim ideas of infinity and infinitesimal. today, Analytics The achievement of such a harmonious, reliable and perfect degree is essentially due to Weierstrass's Scientific activities ”。

Chinese name: Carl Theodor William Weierstrass
Nationality: Germany
one's native heath: Ostenfeld of Westphalia
date of birth: October 31, 1815
Date of death: February 19, 1897
University one is graduated from: University of Bonn, University of Munster

Occupation: mathematician
Key achievements: Put forward ε - N language and ε - δ language
Put forward Weierstrass function
Cultivated a large number of famous mathematicians
Foreign name: Karl Theodor Wilhelm Weierstrass
Representative works: On Abel's Integral Theory

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Weierstrass, a German mathematician, was born in Germany on October 31, 1815 Westphalia Osdenfeldt of the region died in Berlin on February 19, 1897. As the father of modern analysis, Weierstrass's work covers: power series theory, real analysis Complex function , Abel function Infinite product 、 Variational Learning, double line and Quadratic form 、 Entire function Etc. stay Fundamentals of Mathematics On the other hand, he accepted Cantor's idea (and even broke off with his longtime friend Kronecker). His thesis and teaching influenced the style and features of analysis (even the whole mathematics) throughout the 20th century.

French mathematician Cauchy

Weierstrass analytic function Theory is the same as Cauchy and Riemann Complex function theory The founder of. Klein said when comparing Weierstrass and Riemann, "Riemann has extraordinary intuitive ability, and his genius for understanding is better than mathematicians of all ages. Weierstrass is mainly a logician, who advances slowly and systematically. In the branches of his work, he tries to reach a certain form."

Pongale wrote in his evaluation: "Riemann's method is first and foremost a method of discovery, while Weierstrass's is first and foremost a method of proof."

In addition, Weierstrass also studied elliptic function theory, variational method Algebra And many other fields. Moreover, he has trained a large number of famous mathematicians, including Engel, Bolza, Frobenius, Hensel, Holder, Hurwitz, Klein, Killing, Lie, Minkowsky, Runge, Schwarz, Stolz, etc.

Character experience

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His father, William Weierstrass, was a customs officer employed by France. William was very strict and arbitrary at home. At the age of 14, Carl entered a Catholic church in the nearby Paderborn City prep school Study, there Learn German 、 Latin 、 Greek And mathematics. When he graduated from middle school, he got excellent grades and won 7 awards, including mathematics, but Karl was not allowed to have half a sentence to argue. His father sent him to University of Bonn To study law and business, I hope he will become a civil servant in the Prussian Ministry of Civil Affairs in the future.

Weierstrass had no interest in business or law. At the University of Bonn, he spent a considerable part of his time on self-study of his favorite mathematics, including Laplace "Celestial Mechanics". Another part of his time in Bonn was spent on fencing. Weierstrass, with a strong physique, accurate shooting and whirlwind speed, soon became a fencing star in the eyes of the Bonn people. After spending four years at Bonn University in this way, Weierstrass returned home and did not get the law doctor's degree his father hoped for, nor even the master's degree. This made his father furious and scolded him as a "sick man from body to soul". Thank you other people's homes A friend of Teacher qualification examination 。

University of Bonn

In 1841, he officially passed Teacher qualification examination 。 During this period, his math teacher Goodman recognized his talent. C. Gudermann is a elliptic function As an expert, his theory of elliptic function had a great influence on Weierstrass. Weierstrass submitted a paper to pass the teacher qualification exam on the topic of finding elliptic function power series open. In his comments on the paper, Goodman wrote: "The paper shows that a rare mathematical talent will make contributions to the progress of science as long as he is not buried and abandoned.".

Goodman's comments did not attract any attention. Weierstrass began a long life as a middle school teacher after he obtained the qualification of a middle school teacher. He spent his golden years as a mathematician from 30 to 40 in two remote middle schools. In middle school, he not only taught mathematics, but also physics German , geography and even physical education and calligraphy salary Can't even afford the postage for scientific correspondence. But Weierstrass lived a double life with amazing perseverance. He teaches during the day and studies at night Abel And wrote many papers. A few of them were published at that time German middle school It is published on an irregular journal, Introduction to Teaching, but just like the later students of Weierstrass Sweden Mathematician Mita Leffler said: "No one will go to the middle school's" Introduction to Teaching "to look for an epoch-making mathematical paper". However, Weierstrass's amateur research during this period laid the foundation for his mathematical creation in his life.

Mita Leffler

Moreover, this period of life that seemed to be unknown at that time, in fact, contained great power - which had to mention one of the greatest characteristics of Weierstrass: he was not only a great mathematician, but also an outstanding educator! He loves education so much and loves his students so much that we should not mention a large number of successful mathematical talents he has cultivated in the future (the most famous ones are Kovalevskaya (1850.1.15-1891.210, Russian female mathematician, writer, political commentator), H.A Schwartz （Schwarz， Hermann Amandus，1843.1.25-1921.11.30， French mathematician), I.L Fuchs （Fuchs， Immanuel Lazarus，1833.5.5-1902.4.26， French mathematician), M.G Mita Leffler （Mittag-Leffler， Magnus Gustaf，1846.3.16-1927.7.7， Swedish mathematician), F.H Schottky （Schottky， Friedrich Hermann，1851.7.24-1935.8.12， French mathematician), L. Konigsberger, Leo，1837.10.15-1921.12.15， French mathematician), even when he was a math teacher in the preparatory class in this remote middle school, he wanted to make his students better understand Calculus The most important Limit concept And changed Cauchy and others' definition of limit at that time, creating the famous mathematical analysis The ε - δ definition of limit, which has been used in textbooks, and a complete set of similar Representation So that the description of mathematical analysis has finally reached true accuracy.

Until 1853, Weierstrass sent a paper on Abel function to German mathematicians Crayle Sponsored《 Journal of Pure and Applied Mathematics 》(often abbreviated as《 Journal of Mathematics 》）, which made his fortune turn. Crayle's magazine is known for its openness to creative young mathematicians. Abel's paper was published by Crail magazine in 1827 when he was ignored by famous scholars such as Cauchy; Jacobian Paper on Elliptic Function Theory, Green's Potential theory Thesis, etc History of mathematics With the help of Crayle, all the important documents on the Internet were published in his magazine without being published elsewhere. This time Clare came on again. He accepted Weierstrass's paper and published it the following year, which caused a sensation immediately. Koenigsburg A mathematics professor of the university went to Brunsburg Middle School where Weierstrass was teaching at that time and awarded him the doctor's degree certificate of the University of Gothenburg. The Ministry of Education of Prussia announced the promotion of Weierstrass, and gave him a year off to take a post and engage in research. After that, he never returned to Brunsburg.

In 1856, after he became a middle school teacher for 15 years, Weierstrass was appointed professor of mathematics at the Berlin Institute of Technology, and was elected to the Berlin Academy of Sciences in the same year. He later transferred to Berlin University as a professor until his death.

Academic contribution

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1. In terms of analytic functions

German mathematician Riemann

He used power series to define analytic functions and established a set of analytic function theories, Augustin-Louis，1789.8.21-1857.5.23）、 Riemann (Georg Friedrich Bernhard, 1826.9.17-1866.7.20) is called Function theory The founder of. Starting from a known power series that defines a function in a limited region, and according to the relevant theorem of power series, he deduces other power series that define the same function in other regions, which is an important discovery of his. He defined the whole function as a function that can be expressed as the sum of convergent power series on the whole plane; It is also concluded that if the entire function is not polynomial , then Infinity point There is one Natural singularity 。 Weierstrass' research results on analytic functions constitute the main content of the complex variable function theory in today's college mathematics majors.

2. In terms of elliptic functions

Elliptic function is biperiodic Meromorphic function , is to find the ellipse from arc length Caused by. Relevant research is a hot topic in the 19th century. After Abel and Jacobi, Weierstrass made great contributions in this regard. In 1882, he transformed elliptic functions into square root The three different forms of, the elliptic function obtained by the first integral of "inversion" is taken as the basic elliptic function, and it is also proved that this is the simplest Biperiodic function 。 He proved that every elliptic function can use this basic elliptic function and its Derivative function Simply express it. In a word, Weierstrass pushed the research of elliptic function theory to a new level, further completing, rewriting and beautifying it Theoretical system 。

3. In the field of algebra

In 1858, he gave a general method for simultaneously generating the square sum of two quadratic forms, and proved that if one of the quadratic forms is positive definite, even if some characteristic value Equally, this simplification is also possible. In 1868, he had completed the theoretical system of quadratic form and extended these results to bilinear form.

4. In variational theory

In 1879, he proved three conditions of weak variation, that is, the function obtained Minimum Of sufficient condition 。 After that, he turned to Qiang Variational problem And obtained strongly variational maximum value The sufficient conditions of. Many other achievements have been made in variational calculus.

5. In differential geometry

Weierstrass studied the geodesic and minimal surface.

6. In mathematical analysis

Calculus

In the history of mathematics, Weierstrass' contribution to the rigorization of analysis earned him the title of "father of modern analysis". He is a master of introducing strict argumentation into analytics, has made indelible contributions to the rigorousness of analysis, and is one of the pioneers of the movement of arithmetical analysis. The outstanding manifestation of this strictness is the creation of a set of languages to rebuild the analysis system. He criticized Cauchy and other predecessors for the obvious kinematics In this way, we redefined the basic concepts of analysis such as limit, continuity and derivative, especially by introducing the previously neglected Uniform convergence It eliminates various objections and confusions that constantly appear in calculus. It can be said that the rigorous form of mathematical analysis is essentially attributed to the work of Weierstrass.

He proved that (1860): any bounded infinity point set , there must be one Limit point 。 In a speech as early as 1860, he derived rational numbers from natural numbers, and then used increasing numbers bounded sequence To define irrational numbers, we get the whole real number system. This is a theory that has successfully laid a theoretical foundation for calculus.

In order to illustrate the unreliability of intuition, on July 18, 1872, in a speech at the Berlin Academy of Sciences, Weierstrass constructed an example of continuous functions that are nondifferentiable everywhere, thus changing the existing "continuous functions must be derivable" Major misunderstanding , shocked the whole mathematical world! This example has pushed people to construct more functions, such functions are continuous on an interval or everywhere, but in a Dense and dense Or nondifferentiable at any point, thus promoting the development of function theory.

As early as 1842, Weierstrass had uniform convergence And use this concept to give the series Item by item integration And the conditions for differentiation under the integral sign.

In 1885, Weierstrass proved the theorem of arbitrarily approximating continuous functions with polynomials, which is a broad research field in the 20th century Function construction theory , which is one of the starting points of function approximation and interpolation theory.

In addition, Weierstrass also studied the N-body problem And the theory of light.

Contribution to education

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Weierstrass loved mathematics all his life education , guiding students enthusiastically and tirelessly throughout their lives. Regardless of personal fame and wealth, he allowed students or others to spread his research results in various ways, regardless of the merits. This noble character is also very valuable. He has cultivated a large number of successful mathematical talents, especially World history Top 1 Math Female doctor ： Sofia Kovalevskaya (Софжя ВасилжевнаКовалевская, January 15, 1850 - February 10, 1891, Russia Female mathematician, Germany University of Gettingen Doctor of Philosophy, former Sweden Stockholms universitet Professor. stay partial differential equation And rigid body rotation theory. Obtained in 1888 for solving the problem of rigid body rotating around a fixed point Academy of France Botting Award, and became St.Petersburg Academician of the Academy of Sciences is the first woman to win this title in Russian history.

Kovalevskaya, a famous Russian female mathematician

You know, in Europe at that time Social ethos Most people are against women's acceptance regular education Women can't enter the university! Kovalevskaya, in order to Petersburg Going to college to listen to lectures is the price of "fake marriage", and it is only at the expense of parents' guardianship and control. Even so, she can only be a sneak in Petersburg Auditorian Under the cover of her husband, relatives and well intentioned classmates, she dodged the eyes of the school supervisors again and again. When she and her husband came to Germany, although they basically had the freedom to attend classes, as a woman, they faced extremely severe discrimination whether they were formally enrolled or taking exams. Although Kovalevskaya showed excellent personality and mathematical talent, there was no mathematics professor in Heidelberg who could accept her as a student because of her fearsome words.

So Kovalevskaya had to come directly Berlin He turned to Weierstrass, whose character was well known. I personally investigated and wrote a letter to ask Heidelberg, a special girl student Mathematics After his ability and character, Weierstrass was deeply moved by Sophia Kovalevskaya's ambition. So he decided to teach her at home alone (because many of his own students are firmly opposed to women's admission) - this teaching is four years!

For four years, Weierstrass has been lecturing in class all the time, and then lecturing for Kovalevskaya alone at home. In four years, Sophia Kovalevskaya not only completed all university courses, but also completed three important mathematical papers, when she was only 23 years old. Each of these three papers is enough to earn her the title of "mathematician". Therefore, it was Weierstrass who made arrangements in person to let Kovalevskaya, who was both a discriminated woman and a "foreigner" who was not proficient in German, successfully obtain a doctorate in mathematics. The achievements of this female student in the future are enough to prove that her mentor has a clear vision and broad mind to cherish and cultivate talents.

Weierstrass seldom officially published his research results, and many of his ideas and methods were mainly through his tu berlin and University of Berlin Some of them were later arranged and published by his students. Started in 1857 Analytic function theory In the course, Weierstrass gave the first strict definition of real number, which is to the effect that Natural number Starting definition positive Rational number , and then define real numbers through the set of infinite rational numbers. As in most cases, Weierstrass only gave lectures in class. In 1872, someone suggested that he publish this definition, but Weierstrass refused.

tu berlin

His noble demeanor and exquisite Teaching Art It is a shining example worthy of learning by mathematics teachers all over the world. In 1873, Weierstrass became president of the University of Berlin and has been a busy man ever since. Apart from teaching, his official duties took up almost all his time, making him exhausted. The intense work has affected his health, but his intelligence has not declined. The scale of his 70th birthday celebration was quite large, and students from all over Europe came to pay homage to him. Ten years later, the celebration of his 80th birthday is more grand. To some extent, he is simply regarded as Germany Our national heroes. In early 1897, he contracted Influenza , later converted to pneumonia He died on February 19 at the age of 82.

except Berlin Academy Besides, Weierstrass is Gettingen Member of the Royal Society of Sciences (1856), Academician of the Paris Academy of Sciences (1868) Royal Society Member (1881).

Arithmetization of analysis

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Conceived in ancient Greek The thought and method of calculus of the era, after a long period of gestation, arrived in the seventeenth century industrial revolution Under the stimulation of Newton and Leibniz, they finally stood out. The birth of calculus creatively pushed mathematics to a new height. It declared the basic end of classical mathematics, and marked that Modern mathematics Start of.

Newton, British mathematician and physicist

Although the concept of early calculus is still relatively rough and its reliability is still questionable, its outstanding strength in computing technology makes all traditional mathematics pale in the face. Through the invention of calculus, people saw a new blessed land of mathematics. Throughout the seventeenth and eighteenth centuries, almost all European mathematicians showed great interest and positive dedication to calculus. The criticism of tradition, the pursuit of new methods, and the expansion of new fields made them compose a "heroic symphony" in the history of mathematics!

Just like R Courant pointed out: "Calculus... this discipline is the crystallization of a mind shaking intellectual struggle; this struggle has gone through more than 2500 years, and it is deeply rooted in human activity In many areas. Moreover, as long as people make efforts to know themselves and nature for more than one day, this struggle will continue. "

"Arithmetization of analysis" is a vivid manifestation of this struggle.

The lingering "ghost"

At the beginning of the foundation of calculus, Newton and Leibniz did not clearly understand or strictly define the basic concepts of calculus. The first criticism comes from Netherlands Physicist and geometer B Nieuwentijdt， He criticized the vagueness of the new method and complained that it was incomprehensible Infinitesimal quantity How to distinguish from 0, and ask why the sum energy of infinitesimal quantity is finite. He also questioned Higher order differential The meaning and existence of "," and the reason why infinitesimal quantities are abandoned in the process of reasoning.

German mathematician Leibniz

Leibniz made various answers to this in an article in the Journal of Teachers in 1695, and he admitted that Infinitesimal It is not a simple and absolute zero, but a relative zero. In other words, it is a disappearing quantity, but it still retains its disappearing characteristics. Leibniz emphasized the value of the things he created in practice or algorithm. He was convinced that as long as he clearly stated and properly used his algorithm, he would get reasonable and correct results, regardless of how suspicious the meaning of the symbols used was.

With the expansion of the concept and skills of calculus, people try to supplement the missing foundation. When Newton's English followers tried to connect calculus with geometric or physical concepts, they confused Newton's "moments" (indivisible increments) with "fluxions" (continuous variables); Mainland scholars who follow Leibniz are committed to formal calculus, and they can not strictly define the concept. French mathematician M Rolle cautioned that calculus is a collection of ingenious fallacies.

The most powerful criticism of calculus in the 18th century came from Bishop George Berkeley. In 1734, he published The Analyst, or to an unbelieving mathematician, in which he examined whether the objects, principles and reasoning of modern analysis were more clearly conceived or more clearly reasoned than the mysteries and creeds of religion. Berkeley correctly criticized many of Newton's arguments. He said that Newton first gave an increment of x and then made it zero, which violated the "law of contradiction", and the "stream number" obtained was actually 00. As for the ratio of dy to dx, Berkeley said that they are "neither limited nor infinitesimal, but neither nonexistent", and these change rates are just "the ghosts of disappeared quantities". Berkeley also attacked the proposal made by l'Hospital and other European scholars Differential method 。 Berkeley said that the differential ratio should be determined secant Rather than decide tangent The way to rely on "ignoring advanced infinitesimal to eliminate errors" is to "compensate each other for errors". Berkeley's criticism hit the nail on the head.

British Bishop George Berkeley

According to the modern view of the essence of mathematics, the imagination of philosophy in Berkeley's criticism is more strict than mathematics, but many terms used by Newton really need logical clarification. The significance of Berkeley's criticism is that this fact has attracted attention. As a result, over the next seven years, more than 30 pamphlets and papers appeared in an attempt to correct this situation. For example, James Jurin published Geometry, a Non religious Friend in 1734, and Benjamin Roberts wrote On Sir Newton's Stream Number Method and the Essence and Reliability of Initial and Final Comparison Methods in 1735. In response to Berkeley's objection to Newton's method of finding stream numbers in the Quadrature, Jurin said that in this case, instead of making the increment zero, the increment "becomes vanishing" or "in Vanishing point And claimed that "there is a final ratio for the disappearing increment". Jurin's answer indicates that He did not have. Sufficiently understand the essence of Berkeley's argument or limit concept. Berkeley criticized Jurin for "defending what he did not know" in "defending free thought in mathematics" (1735). In this work, Berkely once again grasped the contradiction in Newton's view to explain the ambiguity of concepts such as instantaneous, stream number and limit. Jurin's answer in The Little Mathematician the same year was still a evasive repetition of his words. He said that "a nascent increment is an increment that has just begun to exist in nothingness, or an increment that has just begun to grow, but has not reached any small amount that can be specified." His final ratio to Newton is literally understood as "their ratio at the moment of disappearance." Jurin does not use limits to explain Newton's "instant" of product lemma Instead, he entangled himself in an infinitesimal amount of entanglement, which shows that this "lost ghost" is very difficult to wave away.

English mathematician McLaughlin

In order to hit back at Berkeley and Colin Maclaurin, he tried to establish the rigor of calculus in his Stream Number Theory (1742). Maclaurin loved geometry, so he tried to Exhaustion method He hopes to avoid the concept of limit by establishing the theory of flow mathematics. This is a commendable but incorrect effort.

New Pathfinder

When British mathematicians were busy demonstrating various viewpoints involved in stream number method Effectiveness Calculus Continental Europe Shanghai is rapidly gaining popularity.

Swiss mathematician Euler

Mathematicians in continental Europe rely more on algebra expression Formal calculus, not geometry. The representative of this method is Euler. Euler refused to take geometry as the basis of calculus, but studied functions purely in form.

The real contribution of Euler's formal method is to liberate calculus from geometry and make it based on arithmetic and algebra. This step at least opens the way for the fundamental demonstration of calculus based on the real number system. However, some people are still worried about this formalistic approach. In 1743, d'Alembert said, "Up to now, what shows more concern is to expand the building rather than to put up lights at the entrance; it is to build the house higher rather than to add appropriate strength to the foundation." However, he encouraged students studying calculus: "Persists and trust will come to you."

Lagrange is also determined to provide full rigor for calculus, which is based on the theory of analytic function (1797) Subtitle "Contains Calculus The main theorem of, instead of concepts such as infinitesimal, or vanishing quantity, or limit and stream number, comes down to the limited art of algebraic analysis. "We can see his ambition. Indeed, the stream number method did not interest Lagrange because it quoted the irrelevant idea of "motion". Euler's statement that dx and dy are zero does not satisfy him, because he lacks a clear and definite understanding of the ratio of two terms that become zero. Lagrange is committed to finding a simple algebraic method 。 In 1759, he seemed satisfied that he had found this method, because in that year, he wrote to Euler that he believed that mechanics and Differential calculus The real theoretical basis of the principle as deep as possible. However, it should be noted that Lagrange's work is purely formal, and he uses Symbolic expression It does not involve such fundamental concepts as limit and continuity.

French mathematician d'Alembert

At the end of the 18th century, Lazare Carnot's Philosophical Thoughts on Infinite Calculus appeared in 1797, which may be the most famous attempt to solve difficulties. In view of the lack of clarity and unity of the popular papers on calculus at that time, Carnot wanted to come up with a set of strict and accurate theories. Considering many contradictory understandings of this discipline, Carnot's purpose is to clarify "what is the real spirit of infinitesimal analysis." When choosing the principle of unity, he made a regrettable choice. He concluded that "the real Philosophical Principles ... is still... error Compensation principle 。” In elaborating this view, he essentially returned to the ideas expressed by Leibniz. He advocates that two specified quantities must be strictly equal, as long as it is proved that their difference cannot be a "specified quantity". Carnot further commented Leibniz's point of view that: we can replace any quantity with another quantity that is infinitely different from it; The infinitesimal method simply simplifies the exhaustion method into a calculation method; The "imperceptible quantity" only plays an auxiliary role. It is introduced only for the convenience of calculation, and can be eliminated after the final result is obtained.

French mathematician Lagrange

Carnot even repeated Leibniz's preferred explanation with the principle de continuity. He said that infinitesimal analysis can be understood from two perspectives, depending on whether you regard infinitesimal as "effective quantity" or "absolute zero". In the first case, Carnot believes that differential calculus can be used error compensation As a basis for explanation, the "imperfect equation" becomes "perfectly accurate" by eliminating some quantities called errors; In the second case, Carnot believes that differential calculus is an "art" of comparing vanishing quantities with each other, and finds out the relationship between those given quantities from these comparisons. For the vanishing quantity is zero and not zero Objections Carnot replied, "The so-called infinitesimal quantity is not an arbitrary zero, but a zero given by the continuity law that determines the relationship." Although Carnot's work was widely welcomed, it was published in France until 1921 and translated into several languages, but it is difficult to evaluate whether it correctly guided people to Analytics The difficulties involved are clearly understood.

French mathematician and engineer Lazar Cano

Carnot is a famous soldier Administrative staff , and received French Parliament Awarded the title of "Organizer of Victory". As a mathematician who emphasizes that the value of mathematics is related to the application of science, analytics is an equation rather than a function concept in his thought, although his book title shows that it is biased towards theory, proviso Middle pair Algorithm It is more convenient in application than that involved logical reasoning More attention.

Almost every mathematician in the 18th century made some efforts on the logical basis of calculus. Although one or two approaches were right, all efforts were fruitless. In the absence of foundation, how to analyze and calculate various functions? That is: relying on physics and intuition in their hearts, they have simple Algebraic function ——Find properties from simple and specific functions, and then extend them to all functions. They have displayed superb skills, explored and enhanced the power of calculus, and aggressively expanded new territories: Infinite series 、 differential equation 、 differential geometry and Variational method So as to establish the broadest field in mathematics - mathematical analysis.

German mathematician Gauss

Mathematicians in the 18th century were completely intoxicated with their great achievements and were mostly indifferent to the loss of rigor. It is precisely because mathematicians in the 18th century still rushed forward so bravely without logical support, this period is called the "heroic age" of mathematics.

Injection tightness in analysis

On February 12, 1826, Lobatchevshy Kazan University He read out his paper "On Principles of Geometry", which was considered to be Non Euclidean geometry The day of birth. The concept of mathematics is destined to change fundamentally in the nineteenth century. Perhaps by historical coincidence, Abel expressed his anxiety about analysis in his letter to a friend in 1826:

Russian mathematician Lobachevsky

"People have indeed found amazing ambiguities in the analysis. It is a miracle that so many people can study such an analysis without plan and system. The worst is that analysis has never been treated strictly. Only a few principles in advanced analysis are proved in a logically sound way. People find this kind of unreliable Reasoning method It is very strange that this method only leads to a few so-called paradoxes. "

The work to really inject rigor into analysis started with the work of Bolzano, Cauchy, Abel and Dirichlet, and was further developed by Weierstrass.

Bolzano is a priest and philosopher of Bohemia. In 1799, Gauss considered geometry and gave a proof of the basic theorem of algebra that every rational integral equation must have one root. Bolzano wanted to have a proof derived from arithmetic, algebra and analysis. Just as Largerange believes that it is unnecessary to introduce time and motion into mathematics, Bolzano tries to avoid involving spatial intuition in his proof. In this way, it is necessary to have an appropriate continuity definition first.

Czech mathematician Boercano

In fact, when Pythagoras School replaced by number Geometric quantity What we encounter is the difficulty of continuity; Newton tried to avoid this difficulty by virtue of the intuition of continuous motion, and Leibniz used his continuity postulate to bypass this problem. Now, analytics has brought mathematicians back to the beginning of history. What puzzles mathematical historians is why this historic breakthrough took place far away from the European mathematical center Bragg ！ Bolzano explicitly pointed out for the first time that the basis of the concept of continuity lies in the concept of limit: if the function f (x), for any value x in an interval, and the sufficiently small Δ x, whether positive or negative, the difference f (x+Δ x) - f (x) is always less than any given quantity, Bolzano defined this function as continuous in this interval. There is no major difference between this definition and Cauchy's later definition. In 1843, Bolzano gave an undifferentiable continuous function This example works in mathematics, like Judgmental experiment As in science, it clarifies the impression created by geometric or physical intuition for centuries, indicating that continuous functions may not have derivatives! However, because most of Bolzano's work was forgotten, his views did not have a decisive impact on calculus at that time. The problem of nondifferentiability of continuous functions will not attract people's attention again until a third century later, when Weierstrass's famous example.

Norwegian mathematician Abel

Perhaps Weierstrass's example did not appear early, but it was a blessing in the history of calculus. As Emile Picard said in 1905, "If Newton and Leibniz knew that continuous functions are not necessarily derivable, differential calculus would not come into being." Indeed, rigorous thinking can sometimes hinder creativity.

In the controversy over the chaos of the foundation of calculus, Cauchy saw that the core problem was the limit. Cauchy's concept of limit is based on the consideration of arithmetic. However, his statement that "one variable is infinitely close to one limit" in his definition was criticized by Weierstrass that "this statement unfortunately reminds people of time and motion". In order to eliminate the descriptive language used by Bolzano and Cauchy in defining the continuity and limit of functions, which "becomes and remains less than any given quantity" Uncertainty Weierstrass gave the famous definition of "ε - N (ε - δ)". For the first time, the definition of "ε - N (ε - δ)" frees limits and continuity from any involvement with geometry and motion, and gives a clear definition based only on the concept of number and function, thus turning an ambiguous dynamic description into a strictly stated static concept, which cannot but be regarded as a major innovation in the history of variable mathematics. Today, the essence of "ε - N (ε - δ)" language has penetrated into every blood vessel of modern mathematics, affecting every nerve. For this reason, Hilbert believes that "Weierstrass has established a solid foundation for mathematical analysis with his passion for criticism and profound insight. By clarifying the concepts of minimum, maximum, function, derivative, etc., he eliminated all kinds of wrong expressions still appearing in calculus, cleared up all kinds of confused ideas about infinity, infinitesimal, etc., and decisively overcame the difficulties stemming from the obscure ideas of infinity and infinitesimal······ Today, analytics can achieve such a harmonious, reliable and perfect degree... Essentially, it should be attributed to Weierstrass's scientific activities ".

French mathematician Cauchy

After the strict definition of limit, infinitesimal, as a variable with limit of 0, is classified into the category of function, and is no longer mixed in Archimedes Number field A rebellious ghost.

German mathematician Hilbert

At limit, infinitesimal and Continuity of function After the concepts were clarified, some important properties in the analysis came on stage. Weierstrass applied Bolzano's“ Minimum upper bound Principle "proves that“ Accumulation point Principles ", Weierstrass proved that Closed interval Of upper continuous function Maximum theorem 。 In 1870, Heine defined Uniform continuity Sex, then prove Bounded Continuous functions on closed intervals are uniformly continuous. In Heine's proof, he used“ Limited coverage ”This property was later described as an independent theorem by Emile Borel (1895). The nature of "interval set" was not recognized by Bachmann until 1892. Continuity and Differentiability , continuity and Integrability , infinite series astringency They have also been studied in depth.

arithmetization of analysis

The work of Bolzano, Cauchy, Weiestrass and others provided rigor for analysis. These works liberated calculus and its generalization from the complete dependence on geometric concepts, motion and intuitive understanding. These studies caused a great sensation at the beginning. It is said that Cauchy put forward the theory of series convergence at a scientific conference of the Paris Academy of Sciences. After the conference, Laplace He hurried home to avoid people and checked the series he used in Celestial Mechanics. Fortunately, every series used in the book was convergent.

Pierre Simon de Laplace

The rigorous analysis promotes the understanding that Number system The lack of a clear understanding of the matter itself must be remedied. For example, Bolzano's“ Zero theorem ”One of the key mistakes is that Real number system Lack of adequate understanding; For further study of limits, it is also necessary to understand the real number system. Cauchy could not prove his own adequacy , which is also due to his lack of in-depth understanding of the structure of the real number system. Weierstrass pointed out that in order to establish the properties of continuous functions in detail, arithmetic is needed Continuum This is the fundamental foundation of the arithmetization of analysis.

1872 is the most memorable year in the history of modern mathematics. This year, F. Kline put forward the famous "Erlanger Program", and Weierstrass gave a famous example of continuous but nondifferentiable functions everywhere. It was also in this year that the three major schools of real number theory: Dedekind's "segmentation" theory; The "basic sequence" theory of Cantor, Henie, Meray, and the "bounded monotone sequence" theory of Weierstrass also appeared in Germany.

The purpose of trying to establish real numbers is to give a formal logical definition, which does not rely on the meaning of geometry and avoids using limits to define irrational numbers Logic error 。 With these definitions as the basis, the deduction of the basic theorem of limit in calculus will not have a theoretical cycle. Derivative and integral can be directly established on these definitions, eliminating any Perceptual knowledge The nature of the connection. Geometric concepts cannot be fully understood and accurate, which has been proved in the long years of development of calculus. Therefore, the necessary strictness can only be fully achieved through the concept of number and after cutting off the relationship between the concept of number and the concept of geometric quantity. here, Dedekind This is because the real number defined by "Dedekind segmentation" is a creation of human wisdom that does not rely on the intuition of space and time.

German mathematician Cantor

In 1858, when Dedekind taught calculus, he said To make a request The desire to seek ways to make analysis more rigorous, he said: "... We must not think that the introduction of differential calculus in this way is scientific. This has been recognized. As for me, I can not restrain this sense of dissatisfaction and resolve to study this problem until it is established as Infinitesimal analysis The principle is based on pure arithmetic and complete strictness. " Dedekind doesn't consider how to define irrational numbers to avoid Cauchy's vicious circle, but considers what makes it solve the problem in continuous geometric quantities if the arithmetic method fails obviously: that is, what is the essence of continuity? Thinking along this direction, Dedekind learned that the continuity of a line cannot be explained by fuzzy aggregation, but can only be used as the property of dividing a line by points. He saw that if the points on the line are divided into two categories, so that each point in one category is on the left of each point in the other category, there will be one point but only one point, and this cut will be generated. This pair of ordered rational number systems is not tenable. That's why Dot composition A continuum, and rational numbers are impossible. As Dedekind said, "This ordinary view exposes the secret of continuity."

DeDekin, German mathematician

The three theories of real numbers are essentially correct Irrational number A strict definition is given, thus establishing a complete Real number field 。 The successful construction of the real number field has completely filled the gap between arithmetic and geometry for more than 2000 years. Irrational numbers are no longer "irrational numbers". The ancient Greeks' idea of arithmetic continuity has finally been realized in a strict scientific sense.

Italian mathematician Piano

The next goal is to give the definition and properties of rational numbers. Ohm, Weierstrass, Kronecker and Peano have done outstanding work in this regard. Around 1859, Weierstrass and others realized that as long as natural numbers were recognized, no further axioms were needed to establish real numbers. So establish Real number theory The key of rational number system is rational number system, and the core of establishing rational number system is to construct the basis of ordinary integers and establish the properties of integers. In 1872-78, Dedekind gave a theory of integers.

In 1889, Peano was the first to use axiomatic methods to introduce integers with a set of axioms, thus establishing a complete theory of natural numbers. The symbols created by Peano, such as "∈", "", "N0", "N0", "N0", "N0", "N0", "N0", "N0", "N0", "N0", "N0", "N0", "N0", "N0", "N0", "N0", "N0". But who can believe that because he also used these symbols in class, the students rebelled. He tried to satisfy them with all the passing methods, but it didn't work. He was forced to resign his professorship at Turin University.

German mathematician Kronecker

Kronecker said, "God made the integers, and everything else is artificial." all the rest is the work of man）。 However, in the process of analysis and arithmetization, integers are not favored by God immunity 。

Seeking unity is an important driving force for the development of mathematics. Recalling the whole process of "analysis arithmetization", we found that at the starting place, people did not know where the destination was, let alone where to go How can we go there? 。 From the discovery of incommensurable measurement by the Pythagrass School to the concern about the concept of infinity caused by Zeno paradox, various studies leading to calculus have been bred. When Dedekind, Cantor, Weierstrass and others established irrational numbers on the basis of rational numbers, and finally Peano gave the logical axioms of natural numbers, the rational number theory was finally completed, so the basic problem of the real number system was finally declared complete. The basic concept of calculus -- the limit of continuous variables: derivatives and integrals, rigorous in logic, rigorous in form, like Euclid geometry General amazing! Chinese sages have a saying old saying : Nine to one! If we understand the "one" here as the first "1" of natural numbers, then the famous saying of Pythagoras about the historical development of calculus is surprisingly appropriate: all things are counted! （All is number.）

French mathematician Poincare

In 1900, at the Second International Mathematical Conference held in Paris, Poincare was not without pride in praising: "If we take pains to be strict in our analysis today, we will find that there is only syllogism or it can be attributed to Pure number It is impossible for our intuition to deceive us. Today we can declare that absolute strictness has been achieved. "^[1]

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