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 7月9日

附件：

… … … … …

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

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

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

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.

Inﬁnitesimal Modeling, Part I

Chapter 2.

INFINITESIMALS, LIMITED AND INFINITE NUMBERS

2.1 … … … …

无穷小进课堂，历史在召唤相关推荐

人脸识别进课堂？印度学生：不新鲜了！
By 超神经场景描述:「人工智能进课堂」的话题近日登上微博热搜,学生上课无论睡觉.玩手机还是其他小动作,都会被监控系统实时捕捉到.学生党纷纷感叹「做学生太难了!」其实,课堂监控系统在亚洲早已不是新鲜 ...
中医药文化进校园进课堂之河南省实验小学
中医是我国的国医国粹,为了弘扬中医文化,2016年9月23日金水区"中医药文化科普进校园活动"走进了河南省实验小学,100余名学生体验了一节别开生面的中医文化课堂. 2016年起, ...
网络语言进课堂:上海禁止北京面对
网络语言进课堂:上海禁止北京面对 2010年12月18日 [b][/b] [b]小葵按:[/b]上一篇转录文章――<成都最潮语文老师:用"网络语言"玩转文言文>,网络语 ...
核心素养进课堂计划单小学计算机,核心素养进课堂的计划单.doc
核心素养进课堂计划单姓名白雪学段小学教材(版本) 人教版学科语文章/单元第一单元第3课课题名称鸟的天堂学科核心素养未来的小学语文教育改革要适应社会发展变革对人才观.语文教育质 ...
音乐计算机融合课,信息技术与高效课堂融合的反思——《自制音乐带进课堂》...
构建小学体育高效课堂我认为,首先要充分建构完善的体育课堂常规,严格遵照课堂常规上好每一节课,让学生在高效的模式下形成固有的动作思维模式.用原有的基础知识加以固定,形成动作记忆模块:再融入新的知识,再固 ...
中医药文化进校园进课堂之郑东新区春华学校
中医药文化进校园进课堂之郑东新区春华学校 "花草食物也是药!"夏枯草是什么草?秋桑叶是什么叶?为什么有的人喝凉茶会难受?有的人吃羊肉嗓子会干? 我们日常吃的山药.胡椒.山楂.鱼腥草 ...
中医药文化进课堂河南省濮阳市第二实验小学
2016年9月28日,承易启慧文化传播公司(今果盛教育科技公司)讲师赴河南省濮阳市第二实验小学讲授中医药文化知识课程,开展"中医药文化进校园"活动. 2016年起,果盛教育以< ...
阿里云推出教育一体机：把云计算技术带进课堂
3月13日阿里云召开了教育一体机的发布会.教育一体机基于阿里云的技术积累,选取并融合了当今企业广泛使用的六种新技术,为高校高职学子量身打造了一系列课程及开发实验.此款产品也是为了促进教育链.人才链与产 ...
计算机720p进制,历史频道《人类大历史 Big History》第1季全17集英语中字 720P高清纪录片...
由历史频道<人类大历史 Big History> 第1集我们都知道人类无法离开空气. 第2集淘金热 Gold Fever 这一集提出了人类为什么为金痴狂的问题,并发现我们迷恋金子有着 ...

无穷小进课堂，历史在召唤

无穷小进课堂，历史在召唤相关推荐

最新文章

热门文章