Cauchy's mother heard the rumor and wrote to ask him the truth. Cauchy wrote back: "If Christians become mental patients, the madhouse will be full of philosophers. Dear mother, your child is like a windmill on a leaf, and mathematics and faith are his wings. As soon as the wind blows, the windmill will rotate in a balanced way, generating the power to help others. 』
18 16, Cauchy returned to Paris and became a professor of mathematics at her alma mater. Cauchy wrote: "I am as excited as a salmon that has found its own river. Soon he got married, and a happy married life helped him communicate with others. Bernoulli, a master of mathematics, once said, "Only mathematics can explore infinity, and infinity is one of God's attributes". Physics, chemistry and biology are all limited disciplines, and "infinity" can represent the limit that can never be measured. The concept of infinity makes philosophers crazy, makes theologians sigh and makes many people deeply afraid. On the other hand, Cauchy uses infinity to define a more precise mathematical meaning. He regarded the differential of mathematics as "the change of infinite hours" and expressed the integral as "the sum of infinite infinitesimals". Cauchy redefined calculus with infinity, which is still the beginning of every calculus textbook.
182 1 year, Cauchy's reputation spread far and wide. Students from as far away as Berlin, Madrid and St. Petersburg came to his classroom to attend classes. He also published a very famous "eigenvalue" theory and wrote: "In the field of pure mathematics, it seems that there is no actual physical phenomenon to prove it, and there is nothing in nature to explain it, but it is a paradise that mathematicians can see from a distance. Theoretical mathematicians are not discoverers, but reporters of promised land. After forty, Cauchy was unwilling to be loyal to the new government. He believes that academics should not be influenced by politics. He gave up his job and his motherland and took his wife to teach in Switzerland and Italy. He is welcomed by universities all over the world. But he wrote: "the stimulus of mathematics is that the body can't bear the load for a long time and is very tired!" After Cauchy was forty, she stopped doing research after class.
His health is gradually weakening. 1838, he returned to teach at the University of Paris, but left again because of political loyalty. Because of his insistence, 1848 the academic freedom of French university professors is based on personal conscience and is not subject to political restrictions. Since then, universities all over the world have followed this system, and universities have become places of academic freedom. Let the function f(x) be defined in the centripetal neighborhood of point X. If there is a constant A, there is always a positive number δ for any given positive number ε (no matter how small it is), so that when x satisfies inequality 0.
| f(x)-A | & lt； ε
Then the constant a is called the time limit of the function f(x) when x → x
"Strictly speaking, there is no such thing as mathematical proof. In the end, we will do nothing but do something; ..... It proves that this is what Litowood and I said God blew. It is a touching rhetoric, enough pictures on the blackboard in class, and a way to stimulate students' imagination. "-Hardy.
Mathematics is so important that it has the same status as China literature in China. The reason is that mathematics itself is a language and a universal world language. Therefore, it is very necessary to strictly distinguish the parts of speech of mathematical concepts, which is not only the requirement of mathematics itself, but also the requirement of language science.
Speaking of language and part of speech, it is necessary to know some basic knowledge of Chinese.
1. Noun: a word indicating the name of a person or thing, place, position, etc.
2. Verbs: words expressing actions, development and changes, psychological activities, etc.
Calculus has never left contradiction and refutation since the first day of its birth. For example, Becker refutation (infinitesimal refutation), Zeno paradox and so on. If, through these arguments, we can find that they are actually just discussing the final form in disguise! Just as Leibniz cares about the ultimate fate of particles. Some people say that Cauchy-Wilstrass's definition of limit has the phenomenon of "limit avoidance". This statement is one-sided and not objective, but it still points out some problems (it should be said that it is the ultimate form of avoidance). Cauchy-Wilstrass's definition of limit was very classic when it was translated into China. Cauchy-Weierstrass's definition of limit not only defines the limit, but also describes a movement phenomenon-the movement approaching the limit (the final form). Finally, make the finishing point and call the final form a (if it exists, it is not clear how it came from) limit.
Grammatically speaking, this statement essentially gives the "final form" a title (name)-restriction. Therefore, in Cauchy-Wilstrass's definition of limit, limit is a noun, not a verb.
Therefore, the movement close to the limit is called the limit phenomenon. Many people understand Cauchy-Wilstrass's definition of limit, confuse limit phenomenon and limit, and generally call "limit phenomenon" and "limit" limit.
About the final form of learning, I once briefly talked about it in the secret report of Calculus 4. Because the modern definition of function limit does not explain the final form (avoidance)! So, what is the story of the limit definition of function? What is the relevant mathematical proof?
In fact, it is saying one thing: if there is a limit (final form), there must be a limit phenomenon; On the contrary, if there is a limit phenomenon, there must be a limit! Simply put, limit phenomenon is a necessary and sufficient condition for limit (final form). Therefore, to prove the existence of limit (without studying how it came from) is enough to prove the existence of limit phenomenon, which is indeed suspected of opportunism!
Because of this, the modern definition of limit can not tell you where the limit comes from, but can only tell you that the limit exists (and can be proved). Limit phenomenon is essentially a movement phenomenon. What is the ideal tool to describe the motion phenomenon-function? Therefore, in the modern definition of function (professional) limit, it is not surprising that some functions have flavor (one-to-one correspondence, there is always ε and δ correspondence).
Some people are also quite outrageous, saying that limit is a verb. The reason is that the essence of limit is: "a variable quantity is infinitely close to a fixed quantity." This is the essence of extreme phenomena, not extremes.
However, to describe the limit phenomenon. Must there be Cauchy-Wilstrass model? Of course not, the model can be changed, and elementary calculus has changed this model. It simplifies some complicated mathematical proofs, such as uniqueness of limit and monotonicity of function.
In Cauchy's works, there is no common language, and his statements seem inaccurate, which sometimes leads to errors, such as those caused by the failure to establish the concepts of uniform continuity and uniform convergence. But regarding the principle of calculus, his concept is mainly correct, and its clarity is unprecedented. For example, his definition of continuous function and its integral is accurate. He first proved Taylor formula accurately, and he gave the definition of convergence and divergence of series and some discrimination methods. Although Cauchy mainly studies analysis, he has made contributions in all fields of mathematics. As for other disciplines applying mathematics, his achievements in astronomy and optics are secondary, but he is one of the founders of mathematical elasticity theory. In addition to the above, his other contributions to mathematics are as follows:
1. Analysis: the basic concept of traveling characteristic line in the theory of first-order partial differential equation; Realize the function of Fourier transform in solving differential equations.
2. Geometry: The integral geometry is established, and the formula for expressing the length of plane convex curve by some orthogonal projections on a plane straight line is obtained.
3. Algebra: First, it is proved that the matrix with order exceeding has eigenvalue; Firstly, the concept of permutation group is clearly put forward, and some unconventional results in group theory are obtained. Independent discovery of the so-called "algebraic essence", that is, grassmann's external algebraic principle.