# 无穷小进课堂，历史在召唤

当今，在国内微积分教科书中，对无穷小理论的误解与偏见，比比皆是，不胜枚举。

为正视听，我们特别推荐本文附件文字（无穷小理论简介），供给读者参阅袁萌 陈启清79

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Inﬁnitesimal Modeling, Part I

Chapter 1.

INTRODUCTION

1.1 A Brief History.

Scientists who use mathematical analysis as a tool have traditionallyrelied upon a vague process called “inﬁnitesimal reasoning” - a process that from the time of Archimedes until 1961 had no ﬁxed rules nor consistent language. However, application of this intuitive process is the exact cause that has led to our great analytical successes both in scientiﬁc and engineering endeavors. Unfortunately, it also led to great controversy. Beginning in about 1600 a schism developed between some mathematicians and the foremost appliers of this analytical tool. Leibniz approved entirely of the concept of the inﬁnitely small or inﬁnitely large numbers but stated that they should be treated as “ideal” elements rather than real numbers. He also believed that they should be governed by the same laws that then controlled the behavior of the ordinary numbers. He claimed, but could not justify the assertion, that all arguments involving such ideal numbers could be replaced by arguing in terms of objects that are large enough or small enough to make error as small as one wished. De l’Hospital [1715] when he wrote the ﬁrst Calculus textbook used the terminology exclusively and utilized a formal “deﬁnition - axiom” process supposedly delineating the notion of the inﬁnitesimal. Unfortunately, his ﬁrst axiom is logically contradictory. D’Alembert insisted that the Leibniz concepts were without merit and only a process using a modiﬁed “limit” idea was appropriate. Euler contended in opposition to D’Alembert that the Leibniz approach was the best that could be achieved and fought diligently for the acceptance of these ideal numbers. Due to what appeared to be logical inconsistencies within the methods, those mathematicians trained in classical logic began to demand that applied mathematicians produce “proofs” of their derivations. In answer to this criticism Kepler wrote, “We could obtain absolute and in all respects perfect demonstrations from the books of Archimedes themselves, were we not repelled by the thorny reading thereof.” The successes of these vague methods and those scientists and mathematicians such as Leibniz, Euler and Gauss who championed their continued use quieted the “unbelievers.” It should be noted that the concern of the critics was based upon the fact that they used the same vague processes and terminology in their assumed rigorous demonstrations. The major diﬃculty was the fact that mathematicians had not as yet developed a precise language for general mathematical discourse, nor had they even decided upon accepted deﬁnitions for such things as the real numbers. Within their discussions they conjoined terms such as “inﬁnitely small” with the term limit in the hopes of bringing some logical consistency to their discipline. The situation changed abruptly in 1821. Cauchy, the foremost mathematician of this period, is believed by many to be the founder of the modern limit concept that was eventually formalized by Weierstrass in the 1870’s. A reading of Cauchy’s Cours d’Analyse (Analyse Alg´ebrique)[1821] yields the fact, even to the causal observer, that he relied heavily upon this amalgamation of terms and in numerous cases utilized inﬁnitesimal reasoning entirely for his “rigorous” demonstrations. He claimed to establish an important theorem using his methods - a theorem that Abel [1826] showed by a counterexample to be in error. No matter how mathematicians of that time period described their vague inﬁnitesimal methods they failed to produce the appropriately altered theorem - a modiﬁed theorem that is essential to Fourier and Generalize Fourier Analysis. Beginning in about 1870, all of the language and methods of inﬁnitesimal reasoning were replaced in the mathematical textbooks by the somewhat nonintuitive approximation methods we

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Inﬁnitesimal Modeling, Part I

term the “δ −” approach. These previous diﬃculties are the direct causes that have led to the modern use of axiom techniques and the great linguistic precision exhibited throughout modern mathematical literature. However, scientists and engineers continued to use the old incorrect inﬁnitesimal terminology. As an example, Max Planck wrote in his books on theoretical mechanics that “a ﬁnite change in Nature always occurs in a ﬁnite time, and hence resolves into a series of inﬁnitely small changes which occur in successive inﬁnitely small intervals of time.” He then attempts to instruct the student in how to obtain mathematical models from this general description. Unfortunately, at that time, such terms as “inﬁnitely small” had no mathematical counterpart. In many textbooks that claim to bridge the gape between abstract analysis and applications, students often receive the impression that there is no consistent and ﬁxed method to obtain applied mathematical expressions and indeed it takes some very special type of “intuition” that they do not possess. In fact, Spiegel in his present day textbook “Applied Diﬀerential Equations” writes the following when he discusses how certain partial diﬀerential equations should be “derived.” He states that rigorous methods should not be attempted by the student, but “it makes much more sense, however, to use plausible reasoning, intuition, ingenuity, etc., to obtain such equations and then simply postulate the equations.” In 1961, Abraham Robinson of Yale solved the inﬁnitesimal problem of Leibniz and discovered how to correct the concept of the inﬁnitesimal. This has enabled us to return to the more intuitive analytical approach of the originators of the Calculus. Keisler writes that this achievement “will probably rank as one of the major mathematical advances of the twentieth century.” Robinson, who from 1944 - 1954 developed much of the present supersonic aerofoil theory, suggested that his discovery would be highly signiﬁcant to the applied areas. Such applied applications began in 1966, but until 1981 were conﬁned to such areas as Brownian motion, stochastic analysis, ultralogic cosmogonies, quantum ﬁeld theory and numerous other areas beyond the traditional experience of the student. In 1980, while teaching basic Diﬀerential Equations, this author was disturbed by the false impression given by Spiegel in the above quotation relative to the one dimensional wave equation. It was suggested that I apply my background in these new inﬁnitesimal methods and ﬁnd a more acceptable approach. The approach discovered not only gives the correct derivation for the ndimensional general wave equation but actually solves the d’Alembert - Euler problem and gives a ﬁxed derivation method to obtain the partial diﬀerential equations for mechanics, hydrodynamics and the like. These rigorous derivation methods will bridge the gape between a student’s laboratory, and basic textbook descriptions for natural system behavior, and the formal analytical expressions that mirror such behavior. Indeed, slightly more reﬁned procedures can even produce the relativistic alteration taught in modern physics. Moreover, pure nonstandard models are now being used to investigate the properties of a substratum world that is believed to directly or indirectly eﬀort our standard universe. These include pure nonstandard models for the fractal behavior of a natural system, nonstandard quantum ﬁelds, a necessary and purely nonstandard model for a cosmogony (or pregeometry) that generates many diﬀerent standard cosmologies as well as automatically yielding a theory of ultimate entities termed subparticles. The major goal for writing this and subsequent manuals is to present to the faculty, and through them to the student, these rigorous alterations to the old inﬁnitesimal terminologyso that the student can once again beneﬁt from the highly intuitive processes of inﬁnitesimal reasoning - so that they can better grasp and understand exactly why inﬁnitesimal models are or are not appropriate and

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Inﬁnitesimal Modeling, Part I

when appropriate why they predict natural system behavior. Except for the basic calculus and the more advanced areas, there are no textbooks nor any properly structured documentation available which presents this material at the undergraduate level. In my opinion it will be 15 to 20 years, if not much longer, before such material is available in the commercial market and instructors properly trained. An immediate solution to this problem will give your students a substantial advantage over their contemporaries at other institutions and place your institution in the forefront of what will become a major worldwide trend in mathematical modeling.

1.2 Manual Construction.

The basic construction of these manuals will be considerably diﬀerent from the usual mathematical textbook. No proofs of any of the fundamental propositions will appear within the main body of these manuals. However, all propositions that do not require certain special models to establish are proved within the various appendices. A large amount of attention is paid to the original intuitive approaches as envisioned by the creators of the Calculus and how these are modiﬁed in order to achieve a rigorous mathematical theory. Another diﬀerence lies in the statements of the basic analytical deﬁnitions. Many deﬁnitions are formulated in terms of the original inﬁnitesimal concepts and not in terms of those classical approximations developed after 1870. Each of these deﬁnitions is shown, again in an appropriate appendix, to be equivalent to some well-known “δ−” expression. Moreover, since these manuals are intended for individuals who have a good grasp of either undergraduate analysis or its application to models of natural system behavior then, when appropriate, each concept is extended immediately to Euclidean n-spaces. Nonstandard analysis is NOT a substitute for standard analysis, it is a necessary rigorous extension. Correct and eﬃcient inﬁnitesimal modeling requires knowledge of both standard and nonstandard concepts and procedures. Indeed, the nonstandard methods that are the most proﬁcient utilize all of known theories within standard mathematics in order to obtain the basic properties of these nonstandard extensions. It is the inner play between such notions as the standard, internal and external objects that leads to a truly signiﬁcant comprehension of how mathematical structures correlate to patterns of natural system behavior. Our basic approach employs simple techniques relative to abstract model theory in order to take full advantage of all aspects of standard mathematics. The introduction of these techniques is in accordance with this author’s intent to present the simplest and direct approach to this subject. Since it is assumed that all readers of these manuals are well-versed in undergraduate Calculus, then your author believes that is it unnecessary to follow the accepted ordering of a basic Calculus course; but, rather, he will, now and then, rearrange and add to the standard content. This will tend to bring the most noteworthy aspects of inﬁnitesimal modeling to your attention at the earliest possible moment. I have this special remark for the mathematician. These manuals are mostly intended for those who apply mathematics to other disciplines. For this reason, many deﬁnitions, proofs and discussions are presented in extended form. Many would not normally appear in a mathematicians book since they are common knowledge to his discipline. Some would even be considered as “trivial.” Please be patient with my exposition. It has taken 300 years to solve what has been termed “The problem of Leibniz” and it should not be assumed that the solution is easily grasped or readily obtained. You will experience some startling new ideas and encounter procedures that may be foreign to you. Hopefully, experience, intuition and knowledge are not immutable. It is my ﬁrm belief that, though proper training, these

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Inﬁnitesimal Modeling, Part I

three all important aspects of scientiﬁc progress can be expanded in order to reveal the true, albeit considerably diﬀerent, mathematical world that underlies all aspects of rigorous scientiﬁc modeling. It has been hoped for many years that individuals who have a vast and intuitive understanding of their respective disciplines would learn these concepts and correctly apply them to enhance their mathematical models. It is through your willingness to discard the older less productive, and even incorrect, modeling language that this goal will eventually be met.

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Inﬁnitesimal Modeling, Part I

Chapter 2.

INFINITESIMALS, LIMITED AND INFINITE NUMBERS

2.1 … … … …