研读俄罗斯数学分析教材有感

研读俄罗斯数学分析教材有感

附件1：十年前，短文“无穷小微积分的现实意义”发表。

附件2：二十年前，俄罗斯数学分析教材出版发行，公开倡导无穷小分析的教学。

在数学数学思想上，我们落后于俄罗斯十年.

注：我们的机器人今天说出了“无穷小微积分的现实意义”。巧合也。

袁萌 陈启清  4月11日

附件1：

无穷小微积分的现实意义

        2011年6月，J.Keisler撰写了”无穷小微积分的基础“（电子版），英文名称是”FOUNDATIONSOFINFINITESIMALCALCULUS“（本文简称为”无穷小基础“），作为教材《基础微积分》（2000年电子版）的教学参考书。

        实际上，从1969年起，J.Keisler教授就专注于将A.Robinson的现代无穷小理论用于微积分学的教学实践，四十多年从不间断，精神确实可嘉也。在漫长而多变的这四十多年里，作者在教学与科学研究的实践中究竟获得了什么”心得“，有什么认识，对此，我感到很好奇。

        在”无穷小基础“的序言中，作者说：”In1960AbrahamRobinson(1918–1974)solvedthethreehundredyearoldproblemofgivingarigorousdevelopmentofthecalculusbasedoninfinitesimals.Robinson’sachievementwasoneofthemajormathematicaladvancesofthetwentiethcentury.”意思是说：“在上世纪60年代，A.Robinson(1918-1974)解决了一个存在三百多年的老问题，给基于无穷小的微积分学一个严谨的处理。Robinson所取得的成就是二十世纪数学的重大进展之一。

        回顾过去，我们这一代数学人，是在上世纪50年代菲氏《微积分学教程》（非集合论）的培育下成长起来的。说句实在话，近五十年来，Robinson的现代无穷小思想并没有影响到我国数学发展的历史进程。推出无穷小微积分袖珍电子书就是想改变这一现实状况，把现代无穷小的理论引入国内的微积分学。

        实际上，J.Keisler教授是上世纪数理逻辑模型论的领头人之一，在模型论方面有很高的造诣与建树。说得准确一点，无穷小是在模型论的护送下重返现代微积分学的，再度赶走它几乎是不可能的事情。世界著名数理逻辑学家哥德尔称：无穷小分析“willbetheanalysisofthefuture”(无穷小分析将是未来的数学分析）。

        上世纪五十年代，在国内”全盘苏化“路线的影响下，用菲氏《微积分学教程》培育了一大批数学人才。现在，菲氏的徒子、徒孙占据着重要的社会位置与教学岗位，正在春风得意，要想撼动他们绝非易事。

        现代无穷小微积分的理论基础与历史渊源非常之深厚，决不是某个人的喜好。实际上，要讲现代无穷小的故事，就是三天三夜也讲不完。在历史上，微积分学的原始名字就叫”无穷小演算”(即“无穷小微积分”)。现在，是该为无穷小恢复名誉的时候了。我们坚信，微积分袖珍电子书的发布将为此立下汗马功义 。

附件2：

Foreword

Nonstandard methods of analysis consist generally in comparative study of two interpretations of a mathematical claim or construction given as a formal symbolic expression by means of two di?erent set-theoretic models: one, a “standard” model and the other, a “nonstandard” model. The second half of the 20th century is a period of signi?cant progress in these methods and their rapid development in a few directions. The ?rst of the latter appears often under the name minted by its inventor, A.Robinson. Thismemorableterm nonstandard analysis oftenswapsplaceswithits synonymous versions like robinsonian or classical nonstandard analysis, remaining slightly presumptuous and de?ant. The characteristic feature of robinsonian analysis is a frequent usage of many controversial concepts appealing to the actual in?nitely small and in?nitely large quantities that have resided happily in natural sciences from ancient times but were strictly forbidden in mathematics for many decades of the 20th century. The present-day achievements revive the forgotten term in?nitesimal analysis which reminds us expressively of the heroic bygones of the Calculus. In?nitesimal analysis expands rapidly, bringing about radical reconsideration of the general conceptual system of mathematics. The principal reasons for this progress are twofold. Firstly, in?nitesimal analysis provides us with a novel understanding for the method of indivisibles rooted deeply in the mathematical classics. Secondly, it synthesizes both classical approaches to di?erential and integral calculus which belong to the noble inventors of the latter. In?nitesimal analysis ?nds newer and newest applications and merges into every section of contemporary mathematics. Sweeping changes are on the march in nonsmooth analysis, measure theory, probability, the qualitative theory of di?erential equations, and mathematical economics. The second direction, Boolean valued analysis, distinguishes itself by ample usage of such terms as the technique of ascending and descending, cyclic envelopes and mixings, B-sets and representation of objects in V(B). Boolean valued analysis originated with the famous works by P. J. Cohen on the continuum hypothesis.

viii Foreword

Progressinthisdirectionhasevokedradicallynewideasandresultsinmanysections of functional analysis. Among them we list Kantorovich space theory, the theory of von Neumann algebras, convex analysis, and the theory of vector measures. The book [1], printed by the Siberian Division of the Nauka Publishers in 1990 and translated into English by Kluwer Academic Publishers in 1994 (see [2]), gave a ?rst uni?ed treatment of the two disciplines forming the core of the present-day nonstandard methods of analysis. The reader’s interest as well as successful research into the ?eld assigns us the task of updating the book and surveying the state of the art. Implementation of the task has shown soon that it is impossible to compile new topics and results in a single book. Therefore, the Sobolev Institute Press decided to launch the series Nonstandard Methods of Analysis which willconsist ofmonographs onvarious aspects of this direction in mathematical research. The series started with the book [3] whose English edition [4] appeared quite simultaneously. The second in the series was the collection [5] and its English counterpart [6]. This book continues the series and addresses in?nitesimal analysis. The antique treasure-trove contains the idea of an in?nitesimal or an in?nitely small quantity. In?nitesimals have proliferated for two millennia, enchanting scientists and philosophers but always raising controversy and sometimes despise. After about half a century of willful neglect, contemporary mathematics starts paying rapt attention to in?nitesimals and related topics. In?nitely large and in?nitely small numbers, alongside the atoms of mathematics, “indivisibles”or“monads,” resurrectinvariouspublications, becomingpartand parcel of everyday mathematical practice. A turning point in the evolution of in?nitesimal concepts is associated with an outstanding achievement of A. Robinson, the creation of nonstandard analysis now called robinsonian and in?nitesimal. Robinsonian analysis was ranked long enough as a rather sophisticated, if not exotic, logical technique for corroborating the possibility of use of actual in?nites and in?nitesimals. This technique has also been evaluated as hardly applicable and never involving any signi?cant reconsideration of the state-of-the-art. By the end of the 1970s, the views of the place and role of in?nitesimal analysis had been drastically changed and enriched after publication of the so-called internal set theory IST by E. Nelson and the external set theories propounded soon after IST by K. Hrb′ aˇcek and T. Kawai. In the light of the new discoveries it became possible to consider nonstandard elements as indispensable members of all routine mathematical objects rather than some “imaginary, ideal, or surd entities” we attach to conventional sets by ad hoc reasons of formal convenience. This has given rise to a new doctrine claiming that every set is either standard or nonstandard. Moreover, the standard sets constitute some frame of reference

Foreword ix

“dense” everywhere in the universe of all objects of set-theoretic mathematics, which guarantees healthy translation of mathematical facts from the collection of standard sets to the whole universe. At the same time many familiar objects of in?nitesimal analysis turn out to be “cantorian” sets falling beyond any of the canonical universes in ample supply by formal set theories. Among these “external” sets we list the monads of ?lters, the standard part operations on numbers and vectors, the limited parts of spaces, etc. The von Neumann universe fails to exhaust the world of classical mathematics: this motto is one of the most obvious consequences of the new stances of mathematics. Therefore, the traditional views of robinsonian analysis begin to undergo revision, requiring reconsideration of its backgrounds. Thecrucialadvantageofnewwaystoin?nitesimalsistheopportunitytopursue an axiomatic approach which makes it possible to master the apparatus of the modern in?nitesimal analysis without learning prerequisites such as the technique and toolbox of ultraproducts, Boolean valued models, etc. The suggestedaxiomsareverysimpletoapply,whileadmittingcomprehensible motivation at the semantic level within the framework of the “naive” set-theoretic stance current in analysis. At the same time, they essentially broaden the range of mathematical objects, open up possibilities of developing a new formal apparatus, and enable us to diminish signi?cantly the existent dangerous gaps between the ideas, methodological credenda, and levels of rigor that are in common parlance in mathematics and its applications to the natural and social sciences. In other words, the axiomatic set-theoretic foundation of in?nitesimal analysis has a tremendous signi?cance for science in general. In 1947 K. G¨odel wrote: “There might exist axioms so abundant in their veri?able consequences, shedding so much light upon the whole discipline and furnishing such powerful methods for solving given problems (and even solving them, as far as that is possible, in a constructivistic way), that quite irrespective of their intrinsic necessity they would have to be assumed at least in the same sense as any well established physical theory” [129, p. 521]. This prediction of K. G¨odel turns out to be a prophecy. The purpose of this book is to make new roads to in?nitesimal analysis more accessible. To this end, we start with presenting the semantic qualitative views of standard and nonstandard objects as well as the relevant apparatus at the “naive” level of rigor which is absolutely su?cient for e?ective applications without appealing to any logical formalism. We then give a concise reference material pertaining to the modern axiomatic expositionsof in?nitesimal analysiswithinthe classical cantorian doctrine. We have found it appropriate to allot plenty of room to the ideological and historical facets of our topic, which has determined the plan and style of exposition.

x Foreword

Chapters 1 and 2 contain the historical signposts alongside the qualitative motivation of the principles of in?nitesimal analysis and discussion of their simplest implications for di?erential and integral calculus. This lays the “naive” foundation of in?nitesimal analysis. Formal details of the corresponding apparatus of nonstandard set theory are given in Chapter 3. The following remarkable words by N. N. Luzin contains a weighty argument in favor of some concentricity of exposition: “Mathematical analysis is a science far from the state of ultimate completion with unbending and immutable principles we are only left to apply, despite common inclination to view it so. Mathematical analysis di?ers in no way from any other science, having its own ?ux of ideas which is not only translational but also rotational, returning every now and then to various groups of former ideas and shedding new light on them” [335, p. 389]. Chapters 4 and 5 set forth the in?nitesimal methods of general topology and subdi?erential calculus. Chapter 6 addresses the problem of approximatingin?nite-dimensional Banach spaces and operators between them by ?nite-dimensional spaces and ?nite-rank operators. Naturally, some in?nitely large number plays the role of the dimension or such an approximate space. The next of kin is Chapter 7 which provides the details of the nonstandard technique for “hyperapproximation” of locally compact abelian groups and Fourier transforms over them. The choice ofthesetopicsfromthe varietyofrecent applicationsofin?nitesimal analysis is basically due to the personal preferences of the authors. Chapter 8 closes exposition, collecting some exercises for drill and better understanding as well as a few open questions whose complexity varies from nil to in?nity. We cannot bear residing in the two-element Boolean algebra and indulge occasionally in playing with general Boolean valued models of set theory. For the reader’s convenience we give preliminaries to these models in the Appendix. This book is in part intended to submit the authors’ report about the problems we were deeply engrossed in during the last quarter of the 20th century. We happily recall the ups and downs of our joint venture full of inspiration and friendliness. It seems appropriate to list the latter among the pleasant manifestations and consequences of the nonstandard methods of analysis.

E. Gordon A. Kusraev S. Kutateladze

Foreword xi

References 1. Kusraev A. G. and Kutateladze S. S., Nonstandard Methods of Analysis [in Russian], Nauka, Novosibirsk (1990). 2. Kusraev A.G.andKutateladzeS.S., Nonstandard MethodsofAnalysis, Kluwer Academic Publishers, Dordrecht etc. (1994). 3. Kusraev A. G. and Kutateladze S. S., Boolean Valued Analysis [in Russian], Sobolev Institute Press, Novosibirsk (1999). 4. Kusraev A. G. and Kutateladze S. S., Boolean Valued Analysis, Kluwer Academic Publishers, Dordrecht etc. (1999). 5. Gutman A. E. et al., Nonstandard Analysis and Vector Lattices [in Russian], Sobolev Institute Press, Novosibirsk (1999). 6. Kutateladze S. S. (ed.), Nonstandard Analysis and Vector Lattices, Kluwer Academic Publishers, Dordrecht etc. (2000).

Chapter 1 Excursus into the History of Calculus

The ideas of di?erential and integral calculus are traceable from the remote ages, intertwining tightly with the most fundamental mathematical concepts. Weadmit readilythattopresent theevolutionofviewsofmathematicalobjects and the history of the processes of calculation and measurement which gave an impetus to the modern theory of in?nitesimals requires the Herculean e?orts far beyond the authors’ abilities and intentions. The matter is signi?cantly aggravated by the fact that the history of mathematics has always fallen victim to the notorious incessant attempts at providing an apologia for all stylish brand-new conceptions and misconceptions. In particular, many available expositions of the evolution of calculus could hardly be praised as complete, fair, and unbiased. One-sided views of the nature of the di?erential and the integral, hypertrophy of the roleof the limitand neglect of the in?nitesimal have been spread so widely in the recent decades that we cannot ignore their existence. It has become a truism to say that “the genuine foundations of analysis have for a long time been surrounded with mystery as a result of unwillingness to admit that the notion of limit enjoys an exclusive right to be the source of new methods”(cf. [65]). However, Pontryagin was right to remark: “In a historical sense, integral and di?erential calculus had already been among the established areas of mathematics long before the theory of limits. The latter originated as superstructure over an existent theory. Many physicists opine that the so-called rigorous de?nitionsofderivativeandintegralareinnowaynecessary forsatisfactorycomprehension ofdi?erential and integral calculus. I share thisviewpoint”[401,pp. 64–65]. Considering the above, we ?nd it worthwhile to brief the reader about some turningpointsandcrucialideasintheevolutionofanalysisasexpressed inthewords of classics. The choice of the corresponding fragments is doomed to be subjective. We nevertheless hope that our selection will be su?cient for the reader to acquire a critical attitude to incomplete and misleading delineations of the evolution of in?nitesimal methods.

2 Chapter 1

1.1. G. W. Leibniz and I. Newton The ancient name for di?erential and integral calculus is “in?nitesimal analysis.” ? The ?rst textbook on this subject was published as far back as 1696 under the title Analyse des in?niment petits pour l’intelligence des lignes courbe. The textbook was compiled by de l’H?opital as a result of his contacts with J. Bernoulli (senior), one of the most famous disciples of Leibniz. The history of creation of mathematical analysis, the scienti?c legacy of its founders and their personal relations have been studied in full detail and even scrutinized. Each fair attempt is welcome at reconstructing the train of thought of the men of genius and elucidating the ways to new knowledge and keen vision. We must however bearin mind theprincipal di?erences between draft papers and notes, personalletterstocolleagues, andthearticleswrittenespeciallyforpublication. Itis therefore reasonable to look at the “o?cial” presentation of Leibniz’s and Newton’s views of in?nitesimals. The ?rst publication on di?erential calculus was Leibniz’s article “Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractals nec irrationales quantitates moratur, et singulare pro illis calculi genus” (see [311]). This article was published in the Leipzig journal “Acta Eruditorum” more than three centuries ago in 1684. Leibniz gave the following de?nition of di?erential. Considering a curve YY and a tangent at a ?xed point Y on the curve which corresponds to a coordinate X on the axis AX and denoting by D the intersection point of the tangent and axis, Leibniz wrote: “Now some straight line selected arbitrarily is called dx and another line whose ratio to dx is the same as of ... y ... to XD is called ... dyor di?erence (di?erentia) ... of y....” The essential details of the picture accompanying this text are reproduced in Fig. 1. By Leibniz, given an arbitrary dx and considering the function x →y(x) ata point x, we obtain dy := YX XD dx. In other words, the di?erential of a function is de?ned as the appropriate linear mapping in the manner fully acceptable to the majority of the today’s teachers of analysis. Leibniz was a deep thinker and polymath who believed that “the invention of the syllogistic form ranks among the most beautiful and even the most important

?This term was used in 1748 by Leonhard Euler in Introductio in Analysin In?nitorum [109] (cf. [239, p. 324]).

Excursus into the History of Calculus 3

discoveries of the human mind. This is a sort of universal mathematics whose signi?cance has not yet been completely comprehended. It can be said to incarnate the art of faultlessness ....” [313, pp. 492–493]. Leibniz understood de?nitely that the description and substantiation of the algorithm of di?erential calculus (in that way he referred to the rules of di?erentiation) required clarifying the concept of tangent. He proceeded with explaining that “we have only to keep in mind that to ?nd a tangent means to draw the line that connects two points of the curve at an in?nitely small distance, or the continued side of a polygon with an in?nite number of angles which for us takes the place of the curve.” We may conclude that Leibniz rested his calculus on appealing to the structure of a curve “in the small.”

DA X x

y

Y

Y

dx

Fig. 1

At that time, there were practically two standpoints as regards the status of in?nitesimals. According to one of them, which seemed to be shared by Leibniz, an in?nitely small quantity was thought of as an entity “smaller than any given or assignable magnitude.” Actual “indivisible” elements comprising numerical quantities and geometrical ?gures are the perceptions corresponding to this concept of the in?nitely small. Leibniz did not doubt the existence of “simple substances incorporated into the structure of complex substances,” i.e., monads. “It is these monads that are the genuine atoms of nature or, to put it short, elements of things” [312, p. 413]. For the other founder of analysis, Newton, the concept of in?nite smallness is primarily related to the idea of vanishing quantities [384, 408]. He viewed the indeterminate quantities “not as made up of indivisible particles but as described by a continuous motion” but rather “as increasing or decreasing by a perpetual motion, in their nascent or evanescent state.” The celebrated “method of prime and ultimate ratios” reads in his classical treatise Mathematical Principles of Natural Philosophy (1687) as follows: “Quantities, and the ratios of quantities, which in any ?nite time converge continuously

4 Chapter 1

to equality, and before the end of that time approach nearer to each other than by any given di?erence, become ultimately equal” [408, p. 101]. Propounding the ideas which are nowadays attributed to the theory of limits, Newton exhibited the insight, prudence, caution, and wisdom characteristic of any great scientist pondering over the concurrent views and opinions. He wrote: “To institute an analysis after this manner in ?nite quantities and investigate the prime or ultimate ratios of these ?nite quantities when in their nascent state is consonant to the geometry of the ancients, and I was willing to show that in the method of ?uxions there is no necessity of introducing in?nitely small ?gures into geometry. Yet the analysis may be performed in any kind of ?gure, whether ?nite or in?nitely small, which are imagined similar to the evanescent ?gures, as likewise in the ?gures, which, by the method of indivisibles, used to be reckoned as in?nitely small provided you proceed with due caution” [384, p. 169]. Leibniz’s viewswereasmuch pliableandin-depth dialectic. Inhisfamousletter to Varignion of February 2, 1702 [408], stressing the idea that “it is unnecessary to make mathematical analysis depend on metaphysical controversies,” he pointed out the unity of the concurrent views of the objects of the new calculus: “Ifanyopponenttriestocontradictthisproposition,itfollowsfromourcalculus that the error will be less than any possible assignable error, since it is in our power tomakethisincomparablysmallmagnitudesmallenoughforthispurpose, inasmuch as we can always take a magnitude as small as we wish. Perhaps this is what you mean, Sir, when you speak on the inexhaustible, and the rigorous demonstration of the in?nitesimal calculus which we use undoubtedly is to be found here.... So it can also be said that in?nites and in?nitesimals are grounded in such a way that everything in geometry, and even in nature, takes place as if they were perfect realities. Witness not only our geometrical analysis of transcendental curves but also my law of continuity, in virtue of which it is permitted to consider rest as in?nitely small motion (that is, as equivalent to a species of its own contradictory), and coincidence as in?nitely small distance, equality as the last inequality, etc.” Similar views were expressed by Leibniz in the following quotation whose end in italics is often cited in works on in?nitesimal analysis in the wake of Robinson [421, pp. 260–261]: “There is no need to take the in?nite here rigorously, but only as when we say in optics that the rays of the sun come from a point in?nitely distant, and thus are regarded as parallel. And when there are more degrees of in?nity, or in?nitely small, it is as the sphere of the earth is regarded as a point in respect to the distance of the sphere of the ?xed stars, and a ball which we hold in the hand is also a point in comparison with the semidiameter of the sphere of the earth. And then the distance to the ?xed stars is in?nitely in?nite or an in?nity of in?nities in relation

Excursus into the History of Calculus 5

to the diameter of the ball. For in place of the in?nite or the in?nitely small we can take quantities as great or as small as is necessary in order that the error will be less than any given error. In this way we only di?er from the style of Archimedes in the expressions, which are more direct in our method and better adapted to the art of discovery.” [311, p. 190].

1.2. L. Euler The 18th century is rightfully called the age of Euler in the history of mathematical analysis (cf. [45]). Everyone looking through his textbooks [112] will be staggered by subtle technique and in-depth penetration into the essence of the subject. It is worth recalling that an outstanding Russian engineer and scientist Krylov went into raptures at the famous Euler formula eiπ =?1 viewing it as the quintessential symbol of integrity of all branches of mathematics. He noted in particular that “here 1 presents arithmetic; i, algebra;π, geometry; ande, analysis.” Euler demonstrated an open-minded approach, which might deserve the epithet “systemic” today, to studying mathematical problems: he applied the most sophisticated tools of his time. We must observe that part and parcel of his research was the e?ective and productive use of various in?nitesimal concepts, ?rst of all, in?nitely large and in?nitely small numbers. Euler thoroughly explained the methodological background of his technique in the form of the “calculus of zeros.” It is a popular ?xation to claim that nothing is perfect and to enjoy the imaginary failures and follies of the men of genius (“to look for sun-spots” in the words of a Russian saying). For many years Euler had been incriminated in the “incorrect” treatment of divergent series until his ideas were fully accepted at the turn of the 20th century. You may encounter such a phrase in the literature: “As to the problem of divergent series, Euler was sharing quite an up-to-date point of view.” It would be more fair to topsy-turvy this phrase and say that the mathematicians of today have ?nally caught up with some of Euler’s ideas. As will be shown in the sections to follow (see 2.2 and 2.3), the opinion that “we cannot admire the way Euler corroborates his analysis by introducing zeros of various orders” is as selfassured as the statement that “the giants of science, mainly Euler and Lagrange, have laid false foundations of analysis.” It stands to reason to admit once and for ever that Euler was in full possession of analysis and completely aware what he had created.

1.3. G. Berkeley The general ideas of analysis greatly a?ected the lineaments of the ideological outlook in the 18th century. The most vivid examples of the depth of penetration of the notions of in?nitely large and in?nitely small quantities into the cultural media of that time are in particular Gulliver’s Travels by Jonathan Swift published