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（“ElementaryCalculus”），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。
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 tothe traditional (or e, 3) calculus? To get the proper perspective we shallsketch the history of the calculus. Many problems involving slopes, areas, andvolumes, which we would today call calculus problems, were solved by theancient Greek mathematicians. The greatest of them was Archimedes (287-212B.C.). Archimedes anticipated both the infinitesimal and thee, 3 approach tocalculus. 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 theearly 1600's with the development of physics and astronomy. The basic rules fordifferentiation and integration were discovered in that period by informalreasoning with infinitesimals. Kepler, Galileo, Fermat, and Barrow were amongthe contributors. In the 1660's and 1670's Sir Isaac Newton and GottfriedWilhelm Leibniz independently "invented" the calculus. They took themajor step of recognizing the importance of a collection of isolated resultsand organizing them into a whole. Newton, at different times, described thederivative of y (which he called the "fluxion" of y) in threedifferent ways, roughly
(1) The ratio of an infinitesimal change in yto an infinitesimal change in x. (The infinitesimal method.) (2) The limit ofthe ratio of the change in y to the change in x, l'ly/ l'lx, as l'lx approacheszero. (The limit method.) (3) The velocity of y where x denotes time. (Thevelocity method.)
In his later writings Newton sought to avoidinfinitesimals and emphasized the methods (2) and (3). Leibniz ratherconsistently favored the infinitesimal method but believed (correctly) that thesame results could be obtained using only real numbers. He regarded theinfinitesimals as "ideal" numbers like the imaginary numbers. Tojustify them he proposed his law of continuity: "In any supposedtransition, ending in any terminus, it is permissible to institute a generalreasoning, in which the terminus may also be included."1 This"law" is far too imprecise by present standards. But it was aremarkable forerunner of the Transfer Principle on which modern infinitesimalcalculus is based. Leibniz was on the right track, but 300 years too soon! Thenotation developed by Leibniz is still in general use today, even though it wasmeant to suggest the infinitesimal method: dyjdx for the derivative (to suggestan 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 infinitesimalquantities f(x) dx). All three approaches had serious inconsistencies whichwere criticized most effectively by Bishop Berkeley in 1734. However, a precisetreatment of the calculus was beyond the state of the art at the time, and thethree intuitive descriptions (1H3) of the derivative competed with each otherfor the next two hundred years. Until sometime after 1820, the infinitesimalmethod (1) of Leibniz was dominant on the European continent, because of itsintuitive appeal and the convenience of the Leibniz notation. In England thevelocity method (3) predominated; it also has intuitive appeal but cannot bemade rigorous. In 1821 A. L. Cauchy published a forerunner of the moderntreatment of the calculus based on the limit method (2). He defined theintegral as well as the derivative in terms of limits, namely
fb f(x)dx = lim If(x) Llx. a Ax-o+ a
He still used infinitesimals, regarding themas variables which approach zero. From that time on, the limit method graduallybecame the dominant approach to calculus, while infinitesimals and appeals tovelocity survived only as a manner of speaking. There were two important pointswhich still had to be cleared up in Cauchy's work, however. First, Cauchy'sdefinition of limit was not sufficiently clear; it still relied on theintuitive use of infinitesimals. Second, a precise definition of the realnumber system was not yet available. Such a definition required a betterunderstanding of the concepts of set and function which were then evolving. Acompletely rigorous treatment of the calculus was finally formulated by KarlWeierstrass in the 1870's. He introduced the~>,[) condition as thedefinition of limit. At about the same time a number of mathematicians,including Weierstrass, succeeded in constructing the real number system fromthe positive integers. The problem of constructing the real number system alsoled to development of set theory by Georg Cantor in the 1870's. Weierstrass'approach has become the traditional or "standard" treatment ofcalculus as it is usually presented today. It begins with the e, (3 conditionas the definition of limit and goes on to develop the calculus entirely interms of the real number system (with no mention of infinitesimals). However,even when calculus is presented in the standard way, it is customary to argueinformally in terms of infinitesimals, and to use the Leibniz notation whichsuggests infinitesimals. From the time of Weierstrass until very recently, itappeared that the limit method (2) had finally won out and the history ofelementary calculus was closed. But in 1934, Thoralf Skolem constructed what wehere call the hyperintegers and proved that the analogue of the TransferPrinciple holds for them. Skolem's construction (now called the ultraproductconstruction) was later extended to a wide class of structures, including theconstruction 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 paperin 1948. The hyperreal numbers were known for over a decade before they wereapplied to the calculus. Finally in 1961 Abraham Robinson discovered that thehyperreal numbers could be used to give a rigorous treatment of the calculuswith infinitesimals. The presentation of the calculus which was given in thisbook is based on Robinson's treatment (but modified to make it suitable for a firstcourse). Robinson's calculus is in the spirit of Leibniz' old method ofinfinitesimals. There are major differences in detail. For instance, Leibnizdefined the derivative as the ratio fly/ tlX where flx is infinitesimal, whileRobinson defines the derivative as the standard part of the ratio flyjflx whereflx is infinitesimal. This is how Robinson avoids the inconsistencies in theold infinitesimal approach. Also, Leibniz' vague law of continuity is replacedby the precisely formulated Transfer Principle. The reason Robinson's work wasnot done sooner is that the Transfer Principle for the hyperreal numbers is atype of axiom that was not familiar in mathematics until recently. It arose inthe subject of model theory, which studies the relationship between axioms andmathematical structures. The pioneering developments in model theory were notmade until the 1930's, by Godel, Malcev, Skolem, and Tarski; and the subjecthardly existed until the 1950's. Looking back we see that the method ofinfinitesimals was generally preferred over the method of limits for over 150years after Newton and Leibniz invented the calculus, because infinitesimalshave greater intuitive appeal. But the method of limits was finally adoptedaround 1870 because it was the first mathematically precise treatment of thecalculus. Now it is also possible to use infinitesimals in a mathematicallyprecise way. Infinitesimals in Robinson's sense have been applied not only tothe calculus but to the much broader subject of analysis. They have led to newresults and problems in mathematical research. Since Skolem's infinitehyperintegers are usually called nonstandard integers, Robinson called the newsubject "'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 pointfor this course was a pair of intuitive pictures of the real and hyperrealnumber systems. These intuitive pictures are really only rough sketches thatare not completely trustworthy. In order to be sure that the results arecorrect, the calculus must be based on mathematically precise descriptions ofthese number systems, which fill in the gaps in the intuitive pictures. Thereare two ways to do this. The quickest way is to list the mathematicalproperties of the real and hyperreal numbers. These properties are to beaccepted as basic and are called axioms. The second way of mathematicallydescribing the real and hyperreal numbers is to start with the positive integersand, step by step, construct the integers, the rational numbers, the realnumbers, and the hyperreal numbers. This second method is better because itshows that there really is a structure with the desired properties. At the endof this epilogue we shall briefly outline the construction of the real andhyperreal numbers and give some examples of infinitesimals. We now turn to thefirst way of mathematically describing the real and hyperreal numbers. We shalllist two groups of axioms in this epilogue, one for the real numbers and onefor the hyperreal numbers. The axioms for the hyperreal numbers will just bemore careful statements of the Extension Principle and Transfer Principle ofChapter 1. The axioms for the real numbers come in three sets: the AlgebraicAxioms, the Order Axioms, and the Completeness Axiom. All the familiar factsabout the real numbers can be proved using only these axioms. （以下请看原文）
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（“ElementaryCalculus”），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 tothe traditional (or e, 3) calculus? To get the proper perspective we shallsketch the history of the calculus. Many problems involving slopes, areas, andvolumes, which we would today call calculus problems, were solved by theancient Greek mathematicians. The greatest of them was Archimedes (287-212B.C.). Archimedes anticipated both the infinitesimal and thee, 3 approach tocalculus. 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 theearly 1600's with the development of physics and astronomy. The basic rules fordifferentiation and integration were discovered in that period by informalreasoning with infinitesimals. Kepler, Galileo, Fermat, and Barrow were amongthe contributors. In the 1660's and 1670's Sir Isaac Newton and GottfriedWilhelm Leibniz independently "invented" the calculus. They took themajor step of recognizing the importance of a collection of isolated resultsand organizing them into a whole. Newton, at different times, described thederivative of y (which he called the "fluxion" of y) in threedifferent ways, roughly
(1) The ratio of an infinitesimal change in yto an infinitesimal change in x. (The infinitesimal method.) (2) The limit ofthe ratio of the change in y to the change in x, l'ly/ l'lx, as l'lx approacheszero. (The limit method.) (3) The velocity of y where x denotes time. (Thevelocity method.)
In his later writings Newton sought to avoidinfinitesimals and emphasized the methods (2) and (3). Leibniz ratherconsistently favored the infinitesimal method but believed (correctly) that thesame results could be obtained using only real numbers. He regarded theinfinitesimals as "ideal" numbers like the imaginary numbers. Tojustify them he proposed his law of continuity: "In any supposedtransition, ending in any terminus, it is permissible to institute a generalreasoning, in which the terminus may also be included."1 This"law" is far too imprecise by present standards. But it was aremarkable forerunner of the Transfer Principle on which modern infinitesimalcalculus is based. Leibniz was on the right track, but 300 years too soon! Thenotation developed by Leibniz is still in general use today, even though it wasmeant to suggest the infinitesimal method: dyjdx for the derivative (to suggestan 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 infinitesimalquantities f(x) dx). All three approaches had serious inconsistencies whichwere criticized most effectively by Bishop Berkeley in 1734. However, a precisetreatment of the calculus was beyond the state of the art at the time, and thethree intuitive descriptions (1H3) of the derivative competed with each otherfor the next two hundred years. Until sometime after 1820, the infinitesimalmethod (1) of Leibniz was dominant on the European continent, because of itsintuitive appeal and the convenience of the Leibniz notation. In England thevelocity method (3) predominated; it also has intuitive appeal but cannot bemade rigorous. In 1821 A. L. Cauchy published a forerunner of the moderntreatment of the calculus based on the limit method (2). He defined theintegral as well as the derivative in terms of limits, namely
fb f(x)dx = lim If(x) Llx. a Ax-o+ a
He still used infinitesimals, regarding themas variables which approach zero. From that time on, the limit method graduallybecame the dominant approach to calculus, while infinitesimals and appeals tovelocity survived only as a manner of speaking. There were two important pointswhich still had to be cleared up in Cauchy's work, however. First, Cauchy'sdefinition of limit was not sufficiently clear; it still relied on theintuitive use of infinitesimals. Second, a precise definition of the realnumber system was not yet available. Such a definition required a betterunderstanding of the concepts of set and function which were then evolving. Acompletely rigorous treatment of the calculus was finally formulated by KarlWeierstrass in the 1870's. He introduced the~>,[) condition as thedefinition of limit. At about the same time a number of mathematicians,including Weierstrass, succeeded in constructing the real number system fromthe positive integers. The problem of constructing the real number system alsoled to development of set theory by Georg Cantor in the 1870's. Weierstrass'approach has become the traditional or "standard" treatment ofcalculus as it is usually presented today. It begins with the e, (3 conditionas the definition of limit and goes on to develop the calculus entirely interms of the real number system (with no mention of infinitesimals). However,even when calculus is presented in the standard way, it is customary to argueinformally in terms of infinitesimals, and to use the Leibniz notation whichsuggests infinitesimals. From the time of Weierstrass until very recently, itappeared that the limit method (2) had finally won out and the history ofelementary calculus was closed. But in 1934, Thoralf Skolem constructed what wehere call the hyperintegers and proved that the analogue of the TransferPrinciple holds for them. Skolem's construction (now called the ultraproductconstruction) was later extended to a wide class of structures, including theconstruction 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 paperin 1948. The hyperreal numbers were known for over a decade before they wereapplied to the calculus. Finally in 1961 Abraham Robinson discovered that thehyperreal numbers could be used to give a rigorous treatment of the calculuswith infinitesimals. The presentation of the calculus which was given in thisbook is based on Robinson's treatment (but modified to make it suitable for a firstcourse). Robinson's calculus is in the spirit of Leibniz' old method ofinfinitesimals. There are major differences in detail. For instance, Leibnizdefined the derivative as the ratio fly/ tlX where flx is infinitesimal, whileRobinson defines the derivative as the standard part of the ratio flyjflx whereflx is infinitesimal. This is how Robinson avoids the inconsistencies in theold infinitesimal approach. Also, Leibniz' vague law of continuity is replacedby the precisely formulated Transfer Principle. The reason Robinson's work wasnot done sooner is that the Transfer Principle for the hyperreal numbers is atype of axiom that was not familiar in mathematics until recently. It arose inthe subject of model theory, which studies the relationship between axioms andmathematical structures. The pioneering developments in model theory were notmade until the 1930's, by Godel, Malcev, Skolem, and Tarski; and the subjecthardly existed until the 1950's. Looking back we see that the method ofinfinitesimals was generally preferred over the method of limits for over 150years after Newton and Leibniz invented the calculus, because infinitesimalshave greater intuitive appeal. But the method of limits was finally adoptedaround 1870 because it was the first mathematically precise treatment of thecalculus. Now it is also possible to use infinitesimals in a mathematicallyprecise way. Infinitesimals in Robinson's sense have been applied not only tothe calculus but to the much broader subject of analysis. They have led to newresults and problems in mathematical research. Since Skolem's infinitehyperintegers are usually called nonstandard integers, Robinson called the newsubject "'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 pointfor this course was a pair of intuitive pictures of the real and hyperrealnumber systems. These intuitive pictures are really only rough sketches thatare not completely trustworthy. In order to be sure that the results arecorrect, the calculus must be based on mathematically precise descriptions ofthese number systems, which fill in the gaps in the intuitive pictures. Thereare two ways to do this. The quickest way is to list the mathematicalproperties of the real and hyperreal numbers. These properties are to beaccepted as basic and are called axioms. The second way of mathematicallydescribing the real and hyperreal numbers is to start with the positive integersand, step by step, construct the integers, the rational numbers, the realnumbers, and the hyperreal numbers. This second method is better because itshows that there really is a structure with the desired properties. At the endof this epilogue we shall briefly outline the construction of the real andhyperreal numbers and give some examples of infinitesimals. We now turn to thefirst way of mathematically describing the real and hyperreal numbers. We shalllist two groups of axioms in this epilogue, one for the real numbers and onefor the hyperreal numbers. The axioms for the hyperreal numbers will just bemore careful statements of the Extension Principle and Transfer Principle ofChapter 1. The axioms for the real numbers come in three sets: the AlgebraicAxioms, the Order Axioms, and the Completeness Axiom. All the familiar factsabout the real numbers can be proved using only these axioms. （以下请看原文）