无穷小微积分教材的结束语

五年前的今天，无穷小放飞互联网行动正式开启。时至今日，已经整整五年了。

现在，与以往不同的是，无穷小微积分就在你的指尖之上，任何人只要进入“无穷小微积分”网站，用指尖点击按钮（“Elementary Calculus”），相关信息就会飘然而至（只需等待数秒钟）。

在该教材的第902页，有一个“Epsilogue”（结束语），坦诚地回顾了微积分的历史发展，并且构建了无穷小微积分的理论基础。

当今，任何人可以不接受无穷小微积分，但是，，没有人可以反对它，除非理论“混混儿”、数学文盲。

袁萌  6月17日

附：基础微积分的结束语。

EPILOGUE （结束语）

How does the infinitesimal calculus as developed in this book relate to the traditional (or e, 3) calculus? To get the proper perspective we shall sketch the history of the calculus. Many problems involving slopes, areas, and volumes, which we would today call calculus problems, were solved by the ancient Greek mathematicians. The greatest of them was Archimedes (287-212 B.C.). Archimedes anticipated both the infinitesimal and thee, 3 approach to calculus. He sometimes discovered his results by reasoning with infinitesimals, but always published his proofs using the "method of exhaustion," which is similar to thee, 3 approach. Calculus problems became important in the early 1600's with the development of physics and astronomy. The basic rules for differentiation and integration were discovered in that period by informal reasoning with infinitesimals. Kepler, Galileo, Fermat, and Barrow were among the contributors. In the 1660's and 1670's Sir Isaac Newton and Gottfried Wilhelm Leibniz independently "invented" the calculus. They took the major step of recognizing the importance of a collection of isolated results and organizing them into a whole. Newton, at different times, described the derivative of y (which he called the "fluxion" of y) in three different ways, roughly

(1) The ratio of an infinitesimal change in y to an infinitesimal change in x. (The infinitesimal method.) (2) The limit of the ratio of the change in y to the change in x, l'ly/ l'lx, as l'lx approaches zero. (The limit method.) (3) The velocity of y where x denotes time. (The velocity method.)

In his later writings Newton sought to avoid infinitesimals and emphasized the methods (2) and (3). Leibniz rather consistently favored the infinitesimal method but believed (correctly) that the same results could be obtained using only real numbers. He regarded the infinitesimals as "ideal" numbers like the imaginary numbers. To justify them he proposed his law of continuity: "In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning, in which the terminus may also be included."1 This "law" is far too imprecise by present standards. But it was a remarkable forerunner of the Transfer Principle on which modern infinitesimal calculus is based. Leibniz was on the right track, but 300 years too soon! The notation developed by Leibniz is still in general use today, even though it was meant to suggest the infinitesimal method: dyjdx for the derivative (to suggest an infinitesimal change in y divided by an infinitesimal change in x), and s~ f(x) dx for the integral (to suggest the sum of infinitely many infinitesimal quantities f(x) dx). All three approaches had serious inconsistencies which were criticized most effectively by Bishop Berkeley in 1734. However, a precise treatment of the calculus was beyond the state of the art at the time, and the three intuitive descriptions (1H3) of the derivative competed with each other for the next two hundred years. Until sometime after 1820, the infinitesimal method (1) of Leibniz was dominant on the European continent, because of its intuitive appeal and the convenience of the Leibniz notation. In England the velocity method (3) predominated; it also has intuitive appeal but cannot be made rigorous. In 1821 A. L. Cauchy published a forerunner of the modern treatment of the calculus based on the limit method (2). He defined the integral as well as the derivative in terms of limits, namely

f

 b f(x) dx = lim If(x) Llx. a Ax-o+ a

He still used infinitesimals, regarding them as variables which approach zero. From that time on, the limit method gradually became the dominant approach to calculus, while infinitesimals and appeals to velocity survived only as a manner of speaking. There were two important points which still had to be cleared up in Cauchy's work, however. First, Cauchy's definition of limit was not sufficiently clear; it still relied on the intuitive use of infinitesimals. Second, a precise definition of the real number system was not yet available. Such a definition required a better understanding of the concepts of set and function which were then evolving. A completely rigorous treatment of the calculus was finally formulated by Karl Weierstrass in the 1870's. He introduced the~>,[) condition as the definition of limit. At about the same time a number of mathematicians, including Weierstrass, succeeded in constructing the real number system from the positive integers. The problem of constructing the real number system also led to development of set theory by Georg Cantor in the 1870's. Weierstrass' approach has become the traditional or "standard" treatment of calculus as it is usually presented today. It begins with the e, (3 condition as the definition of limit and goes on to develop the calculus entirely in terms of the real number system (with no mention of infinitesimals). However, even when calculus is presented in the standard way, it is customary to argue informally in terms of infinitesimals, and to use the Leibniz notation which suggests infinitesimals. From the time of Weierstrass until very recently, it appeared that the limit method (2) had finally won out and the history of elementary calculus was closed. But in 1934, Thoralf Skolem constructed what we here call the hyperintegers and proved that the analogue of the Transfer Principle holds for them. Skolem's construction (now called the ultraproduct construction) was later extended to a wide class of structures, including the construction of the hyperreal numbers from the real numbers.

1 See Kline, p. 385.

904 EPILOGUE

The name "hyperreal" was first used by E. Hewitt in a paper in 1948. The hyperreal numbers were known for over a decade before they were applied to the calculus. Finally in 1961 Abraham Robinson discovered that the hyperreal numbers could be used to give a rigorous treatment of the calculus with infinitesimals. The presentation of the calculus which was given in this book is based on Robinson's treatment (but modified to make it suitable for a first course). Robinson's calculus is in the spirit of Leibniz' old method of infinitesimals. There are major differences in detail. For instance, Leibniz defined the derivative as the ratio fly/ tlX where flx is infinitesimal, while Robinson defines the derivative as the standard part of the ratio flyjflx where flx is infinitesimal. This is how Robinson avoids the inconsistencies in the old infinitesimal approach. Also, Leibniz' vague law of continuity is replaced by the precisely formulated Transfer Principle. The reason Robinson's work was not done sooner is that the Transfer Principle for the hyperreal numbers is a type of axiom that was not familiar in mathematics until recently. It arose in the subject of model theory, which studies the relationship between axioms and mathematical structures. The pioneering developments in model theory were not made until the 1930's, by Godel, Malcev, Skolem, and Tarski; and the subject hardly existed until the 1950's. Looking back we see that the method of infinitesimals was generally preferred over the method of limits for over 150 years after Newton and Leibniz invented the calculus, because infinitesimals have greater intuitive appeal. But the method of limits was finally adopted around 1870 because it was the first mathematically precise treatment of the calculus. Now it is also possible to use infinitesimals in a mathematically precise way. Infinitesimals in Robinson's sense have been applied not only to the calculus but to the much broader subject of analysis. They have led to new results and problems in mathematical research. Since Skolem's infinite hyperintegers are usually called nonstandard integers, Robinson called the new subject "'nonstandard analysis." (He called the real numbers "standard" and the other hyperreal numbers "nonstandard." This is the origin of the name "standard part.") The starting point for this course was a pair of intuitive pictures of the real and hyperreal number systems. These intuitive pictures are really only rough sketches that are not completely trustworthy. In order to be sure that the results are correct, the calculus must be based on mathematically precise descriptions of these number systems, which fill in the gaps in the intuitive pictures. There are two ways to do this. The quickest way is to list the mathematical properties of the real and hyperreal numbers. These properties are to be accepted as basic and are called axioms. The second way of mathematically describing the real and hyperreal numbers is to start with the positive integers and, step by step, construct the integers, the rational numbers, the real numbers, and the hyperreal numbers. This second method is better because it shows that there really is a structure with the desired properties. At the end of this epilogue we shall briefly outline the construction of the real and hyperreal numbers and give some examples of infinitesimals. We now turn to the first way of mathematically describing the real and hyperreal numbers. We shall list two groups of axioms in this epilogue, one for the real numbers and one for the hyperreal numbers. The axioms for the hyperreal numbers will just be more careful statements of the Extension Principle and Transfer Principle of Chapter 1. The axioms for the real numbers come in three sets: the Algebraic Axioms, the Order Axioms, and the Completeness Axiom. All the familiar facts about the real numbers can be proved using only these axioms. （以下请看原文）