In this great era, Newton and Leibniz independently invented calculus, but infinitesimal calculus caused the second mathematical crisis. Infinitely small is like a wild horse galloping on the mathematical wasteland, and no one can make it obedient. However, a great mathematician finally tamed "infinitesimal quantity" with the definition of "limit". This is Cauchy.
Cauchy 1789 was born in Paris. His father was a lawyer, and he had a close personal relationship with the great mathematicians Lagrange and Laplace at that time. When he was a teenager, Cauchy showed outstanding mathematical talent, which was greatly appreciated by two mathematicians. He believed that he would make great achievements in mathematics in the future.
However, every genius is lonely, and so is Cauchy. Cauchy didn't like to talk when she was a student, so she didn't have any friends around. In that proud era of "reading philosophy books", Cauchy often read Lagrange's math books after class, and people laughed at him as a "bitter gourd" and a mental derangement.
From 1807 to 18 10, Cauchy gave up his identity as a traffic road engineer and devoted himself to the study of pure mathematics. A few years later, Cauchy returned to Paris and became a professor of mathematics at her alma mater. Cauchy wrote in her diary: "I am as excited as a salmon that has found its own river." It is this excellent position of "professor of mathematics" that makes Cauchy's mathematics research work like a duck to water, and also ushers in a brand-new spring for Cauchy's mathematics career.
17th century is a special era. Newton and Leibniz's "calculus" based on the concept of "infinitesimal quantity" has precipitated thousands of years of human civilization, and is ready to bloom in the sky of human civilization.
However, because people are satisfied with the results of calculus and eager to open up new fields with great strides, they ignore the logic structure at the bottom of calculus, which leads people to question calculus: Is infinitesimal equal to zero? If it is "zero", how to divide it? If it is not "zero", then why do you remove those "tiny quantities" when "the function is deformed"?
The chaotic logic at the bottom of Calculus caused authorities from all walks of life, led by British philosopher and Archbishop Becquerel, to launch a fierce attack on Calculus, and the "second mathematical crisis" broke out.
/kloc-at the beginning of the birth of calculus in the 0/7th century, for more than 100 years, the free wild horse "Infinitely Small" ran freely in the wasteland of mathematics, full of great energy and infinite vitality, but people could not grasp it.
To solve the problem of "infinitesimal quantity", we have to start with the concept of "infinity". People's deep thinking on the concept of infinity can be traced back to ancient Greece. In that remote and barren era, Zhi Nuo put forward the famous Zeno paradox, such as the paradox of "the arrow does not move", one of the four paradoxes.
The paradox of "the arrow does not move" is this: an arrow flying in the air must be at a specific infinitesimal point in space at any infinitesimal point in time. However, at every specific infinitesimal time point, the arrow can only be "static". Because the flight path of an arrow consists of "infinite" such points, the flying arrow is always stationary, so the movement of the arrow is impossible.
The paradox of "the arrow does not move" put forward by Zhi Nuo is also the first time that human beings sharply put the problem of "infinity" on the table for discussion.
The paradox of "fixed arrow" is actually a question about "infinite set". There are two kinds of early infinite sets: one is infinite process, which people call potential infinity, and the other is infinite whole, which people call real infinity. Aristotle, a great mathematician at that time, thought that there was only "potential infinity" in the world, but "real infinity" did not exist. For example, the universe is "infinite" in human eyes, but it is "limited" from God's perspective, so there is no "real infinity" in the world, so there is no "infinite set". Because of Aristotle's authoritative position, people's research on "infinite set" has been stagnant for more than two thousand years.
In this long time, "infinity" only exists as an idea, and no one regards it as a number, let alone participates in "mathematical operation".
But when Newton and Leibniz founded calculus, they took infinitesimal as the most basic quantity and established a new discipline-infinitesimal calculus. Adding "infinite" infinitesimals together is the "integral method", and dividing "two" and "infinitesimal" is the "differential method".
At this time, the "infinitesimal quantity", which has been feared by traditional ideas, entered the field of mathematics on a large scale. This makes the authoritative figures in mathematics full of doubts, even Gauss, who is known as the "king of mathematicians", is no exception.
Gauss is a "potential infinitist" who does not recognize "infinite set". In his letter to his friends, he strongly opposed people taking the concept of infinity and the symbol of infinity as "ordinary numbers" to participate in "mathematical operations".
Early Cauchy, like most great mathematicians, denied the existence of "infinite set", but with the continuous emergence of mathematical achievements about "infinite set", he finally accepted "real infinity" and thus "infinitesimal quantity". He said: "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 promised land that mathematicians can see from a distance. Theoretical mathematicians are not discoverers, but reporters of this promised land. "
Cauchy began to try to redefine calculus with the concepts of "strictness" and "infinity", and started the work of "strictness" in mathematical analysis with mathematicians all over the world at that time. Cauchy regards infinitesimal as a variable with a limit of 0, which fully explains why infinitesimal can sometimes be regarded as zero and sometimes as zero. That's because infinitesimal is a variable. In the process of its change, although its value is not directly equal to zero, its changing trend is infinitely close to zero, so people sometimes directly regard "infinitesimal" as "equal to zero" and will not produce wrong results.
In 182 1, Cauchy put forward the method of defining "limit" and described "limit process" as "inequality". Cauchy successfully established the "limit theory" for the first time, and "infinitesimal" was completely tamed.
Under the concept of limit, curve and straight line can be transformed into each other, that is, straight line and curve are equivalent in differentiation. With the help of "extreme" thinking method, people have solved many problems that "elementary mathematics" cannot solve. A series of important concepts in mathematical analysis such as continuity, derivative and definite integral of function are defined by limit method. If you want to ask: What is the subject of "Mathematical Analysis"? Then it can generally be said that "mathematical analysis" is a subject that studies "function" with the idea of "limit".
The definitions of limit, continuity, derivative and convergence in calculus textbooks today are all defined by Cauchy and others. He first strictly proved the "Basic Theorem of Calculus" with the "Mean Value Theorem". Through the efforts of Cauchy and others, the basic concept of "mathematical analysis" has been "strict". It makes "calculus" no longer depend on "geometric concept", "movement" and "intuitive understanding", and develops into the most basic and huge mathematics discipline in "modern mathematics".
Cauchy has made brilliant achievements in mathematics all his life. His Complete Works of Cauchy has 27 volumes and more than 800 works. In the history of mathematics, his "number of academic achievements" is second only to Euler. His name is remembered in many textbooks along with many "theorems" and "guidelines". However, the most remarkable mathematical achievement in his life is undoubtedly the definition of "infinitesimal quantity" with the concept of "limit", which makes "calculus" completely get rid of the confusion of the underlying logic, leads "modern mathematics" out of the "second mathematical crisis" successfully, and puts modern mathematics on the road of healthy development.
1857 On May 23rd, the great mathematician Cauchy died suddenly at the age of 68. His last words before he died were:
"People will die, but achievements will last forever."