General mathematical analysis, also called standard analysis, mainly calculus, is a branch of mathematics that studies continuous variables and their relationships in the real world. Its basic concept is the concept of variables and functions with values in the range of real number system, and its research method is limit theory. Therefore, standard analysis refers to the calculus theory established by Cauchy, Wilstras and others in the19th century. They used limit method instead of infinitesimal method in mathematical proof, and made a more rigorous logical demonstration of calculus theory. Their theory is a big step forward compared with 17 and 18 century calculus theory. This is manifested in that it creates a series of discriminant rules and finds some important results about the continuity and differentiability of functions. ["Standard Analysis" is based on strict limit theory. ]
/kloc-At the beginning of the 0/8th century, a debate about calculus was fierce. Words can start from Newton's time. Try to find the derivative of y=x2. Take the infinitesimal ⊿x first, then take ⊿y=(x+⊿x)2-x2=2x⊿x+⊿x2, that is ⊿ y/⊿x = 2x+⊿ X. Because ⊿ The infinitesimal ⊿x here is not 0 (divisible by ⊿x), but equal to 0 (finally ignored, ⊿x disappears). This method looks a bit like magic. Marx called the neglect of ⊿x "violent repression", while Archbishop Becquerel called it "the ghost of departure" or "the ghost of departure". This kind of mystification of infinitesimal is really not good. "Come quickly and go quickly" is a complete set of ghosts and gods. However, no matter how to attack, its operation result is always right. The great mathematician Euler once made brilliant achievements with this lax calculus. Gradually people no longer have objections.
/kloc-in the 9th century, French mathematician Cauchy realized that the correct conclusion does not mean the integrity of the system, so he set out to make "infinitesimal analysis" strict. These are the famous expressions ε-N and ε-δ, which were not completed until the 65438+70s in Weierstrass. This description of the limit process expressed by static motion removes the mysterious coat: the so-called infinitesimal is just a variable with a limit of 0. It is not "a number", but a changing process, that is, a variable that keeps approaching the constant 0 with any small error. Its representation is completely arithmetic, ε, δ and so on. Is unmistakable. However, "infinitesimal" is not a number, which cannot be directly divided or ignored. Fresh operations are submerged in the ocean of tables, and people complain that calculus is becoming more and more difficult to learn. Ignoring the criticism of infinitesimal, engineers still follow the convenient practice of Newton-Euler era, hold infinitesimal in their hands and refuse to throw it away. However, "infinitesimal" cannot remain on the altar after all. Since the 20th century, it has almost disappeared, and the occasional mention of it is only a habitual noun introduction.
Things turned around in the autumn of 1960. Abraham Robinson (19 18 ~ 1974, German Jew, 1962 went to the United States) pointed out in a report of Princeton University that the concepts and methods of modern mathematical logic can be "infinitesimal" and "infinite". 196 1 year, Robinson published an article entitled "nonstandard analysis" in the Journal of the Royal Academy of Sciences in Amsterdam, the Netherlands, which heralded the birth of this new branch of mathematics.
In standard analysis, the set of rational numbers and irrational numbers studied is called real number set. Real number sets correspond to points on a straight line one by one, and real number sets are continuous. In nonstandard analysis, Robinson's basic idea is: Since infinitesimal is not a "number", that is, it has no place in the real number set, can the real number set be expanded to make it a new super real number set, and the intuition and simplicity of Newton-Euler era can be maintained when calculus is realized in the super real number set? Robinson did this with the method of model theory in mathematical logic. In the hyperreal number set, every common real number is a standard number, and there are many "infinitesimals" (nonstandard real numbers) around it, just like electrons around the nucleus. There is no Archimedes property in the set of hyperreal numbers, that is, if you choose integers α and β, you may not always find the natural number n, so that nα > β, because infinitesimal is a nonstandard real number greater than 0, and its arbitrary integer multiple is still infinitesimal and cannot be greater than the positive standard number β. [The elements in the hyperreal set must be hyperreal! ]
From a macro point of view, the number axis of super real number set is the same as that of real number set. But from a microscopic point of view, it is not the same. There are many nonstandard real numbers at every point on the hyperreal axis. These nonstandard real numbers differ from each other by an infinitesimal amount to form a point with internal structure, which is called a "monad", and each monad has only one standard real number. From the standard real number, the point is continuous, and from the super real number axis, the point is the unity of opposites of continuity and discontinuity.
In its physical sense, it is like a light, which is continuous from a macro point of view, and not only discontinuous from a micro point of view, but also uneven. Quantum theory proves that light has fluctuation and particle duality, which shows that light is the unity of opposites of continuity and discontinuity.
Non-standard analysis has opened a new world for us-the world of "points". Any point is a "world"; Any world is a "point", just like there is a day outside, and there are some points inside. In the solar system, the earth is a "point", which is structured and separable. The same molecule can also be used as a "point", which is structured and separable. Mathematically speaking, from a smaller level, any "point" can establish a coordinate system because it is a "world", and from a larger level, any "world" can be just a point in the coordinate system. Non-standard analysis accepted the dialectic of separability of "point".
This method of mathematical logic is quite complicated, and it is much more difficult to understand than to understand the concept of calculus. However, infinitesimal has returned to the digital altar and become a member of logically tenable mathematics. This is the "revolutionary" information brought by non-standard analysis, which is a happy thing. Philosophically, it also has its own significance. The negation of negation, the foundation of calculus has made new development, and it is really "another village with a bright future!" "Non-standard analysis" was established by Robinson's redefinition of calculus with infinitesimal. ]
1In April, 965, Robinson wrote a book, Nonstandard Analysis, which was widely circulated. Many mathematicians support this point, and many people doubt it. 1973, Robinson met the most famous mathematician of this century-the famous Godel at the Princeton Institute for Advanced Studies. Godel made this evaluation:
"Non-standard analysis can not only simplify the proof of elementary theorems, but also simplify the proof of difficult conclusions. For example, the theorem that compact operators have "invariant subspace" can be greatly simplified. ..... We have reason to believe that nonstandard analysis will become the mathematical analysis in the future, no matter from any aspect. ...... In the next century, we will consider an important event in the history of mathematics, which is why the first strict infinitesimal theory was developed 300 years after the invention of calculus. "
Godel's evaluation makes nonstandard analysis more important. Nonstandard group theory, nonstandard functional analysis and nonstandard topology have come out one after another. Kessler wrote a calculus textbook of nonstandard analysis. After the trial teaching, it is said that it has been accepted well and is ready to expand the experiment. Recently, however, more and more people are skeptical about this. The reason is that "all the results that can be obtained by non-standard analysis can be obtained by the original standard method. Since there is nothing new and it is so difficult to understand, why do you have to learn? " Some people even think that nonstandard analysis is just a "dream" and "fantasy" of mathematical logicians, which is really unnecessary. As for whether nonstandard analysis can become "mathematical analysis in the future century", I am afraid it will be tested by practice and history. It takes a process for people to accept a new thing, especially a new sentence and a new decoration, and it takes more time. It is almost as difficult for people to use nonstandard analysis as it is for people to speak another foreign language. Whether Godel's prediction is correct or not depends on the future! However, Robinson's contribution to infinitesimal regeneration will not be erased, and he will definitely occupy a place in the history of mathematics.