學習微積分60年有感（I）

現在可以說，我這一輩子幾乎都是在與微積分打交道，分不開?？嗫嘌辛?/span>60年，人都快學傻了?，F在，人老了，還整天嘮叨不停，真煩人。

1957年秋，我進入南京大學數學天文系學習。當時，何旭初教授主講一年級微積分課程，指定教學參考書是原蘇聯菲赫金哥爾茨的《微積分學教程》（共計三卷9大本），內容幾乎涵蓋了上世紀前半期世界數學分析的全部成就。

在此期間，對基于（ε，δ）方法的極限論，頂禮膜拜，視為神圣，從不懷疑。

在20年之后，1978年，全國改革開放，轟轟烈烈。我看準了Keisler的無窮小微積分，觀點全變了。普及無窮小微積分，而且堅持不斷，矢志不移。這樣，又一路走過了人生的40年。

明天就是新年了。我把Keisler關于微積分的肺腑之言獻給廣大讀者，作為新年的一份禮物。

袁萌? 12月31日

附：基礎微積分教材的“后記”

（讀者請注意：在下面的英語句子中，有些空格被編輯器“吃”掉了。）

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How does the infinitesimalcalculus（無窮小微積分） as developed in this book relate to the traditional (or ε, δ ) calculus?To get the proper perspective we shall sketch the history of ?the calculus.

Many problems involving slopes, areas and volumes, which we would todaycall calculus problems, were solved by the ancient Greek mathematicians. Thegreatest of themwas Archimedes( 287-212 B.C.) . Archimedes anticipated both theinfinitesimaland the ε, δ approach to calculus. He sometimes discovered hisresults byreasoning with infinitesimals, but always published his proofs usingthe“method of exhaustion,” which is similar to the ε, δ approach.

Calculusproblemsbecame important in the early 1600’s with the development of physicsandastronomy. The basic rules for differentiation and integration werediscoveredin that period by informal reasoning with infinitesimals. Kepler,Galileo,Fermat, and Barrow were among the contributors.

In the 1660’sand1670’s Sir Isaac Newton and Gottfried Wilhelm Leibniz independently“invented”the calculus. They took the major step of recognizing the importanceof acollection of isolated results and organizing them into a whole.

Newton,atdifferent times, described the derivative of γ (which he called the“fluxion”of γ) in three different ways, roughly

(1) The ratio ofaninfinitesimal change in y to an infinitesimal change in x ( Theinfinitesimalmethod.)

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(2) The limitofthe ratio of the change in y to the change in x, Δy/Δx, as Δx approacheszero.(The limit method.)

(3) The velocityofy where x denotes time. (The velocity method.)

In hislaterwritings Newton sought to avoid infinitesimals and emphasized the methods(2)and (3).

Leibnizratherconsistently favored the infinitesimal method but believed (correctly)that thesame results could be obtained using only real numbers. He regardedtheinfinitesimals as “ideal” numbers like the imaginary numbers. To justifythemhe proposed his law of continuity: “In any supposed transition, ending inanyterminus, it is permissible to institute a general reasoning, in whichtheterminus may also be included.” This “law” is far too imprecise bypresentstandards. But it was a remarkable forerunner of the Transfer Principleonwhich modern infinitesimal calculus is based. Leibniz was on the righttrack,but 300 years too soon!

Thenotationdeveloped by Leibniz is still in general use today, even though it wasmeant tosuggest the infinitesimal method: dy /dx for the derivative (to suggestaninfinitesimal change in y divided by an

infinitesimal change in x ), and ∫ba?(x )dx for theintegral (to suggestthe sum of infinitely many infinitesimal quantities ?(x ) dx ).

Allthreeapproaches had serious inconsistencies which were criticized mosteffectivelyby Bishop Berkeley in 1734. However, a precise treatment of thecalculus was beyondthe state of the art at the time, and the three intuitivedescriptions(1)-(3)of the derivative competed with each other for the next twohundred years.Until sometime after 1820, the infinitesimal method (1) ofLeibniz was dominanton the European continent, because of its intuitive appealand the convenienceof the Leibniz notation. In England the velocity method (3)predominated; italso has intuitive appeal but cannot be made rigorous.

In 1821 A.L.Cauchy published a forerunner of the modern treatment of the calculus basedonthe limit method (2). He defined the integral as well as the derivativeinterms of limits, namely

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∫ba?(χ)dχ=lim Σ?(χ)Δχ

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He stillusedinfinitesimals, regarding them as variables which approach zero. From thattimeon, the limit method gradually became the dominant approach to calculus,whileinfinitesimals and appeals to velocity survived only as a manner ofspeaking.There were two important points which still had to be cleared up inCauchy’swork, however. First, Cauchy’s definition of limit was not sufficientlyclear;it still relied on the intuitive use of infinitesimals. Second, aprecisedefinition of the real number system was not yet available. Such adefinitionrequired a better understanding of the concepts of set and functionwhich werethen evolving.

Acompletelyrigorous treatment of the calculus was finally formulated by KarlWeierstrassin the 1870’s. He introduced the ε, δ condition as the definition oflimit. Atabout the same time a number of mathematicians, includingWeierstrass,succeeded in constructing the real number system from the positiveintegers.The problem of constructing the real number system also led todevelopment ofset theory by Georg Cantor in the 1870’s. Weierstrass’ approachhas become thetraditional or “standard” treatment of calculus as it is usuallypresentedtoday. It begins with the ε, δ condition as the definition of limitand goes onto develop the calculus entirely in terms of the real numbersystem(with nomention of infinitesimals). However, even when calculus ispresented in thestandard way, it is customary to argue informally in terms ofinfinitesimals,and to use the Leibniz notation which suggests infinitesimals.

From the timeofWeierstrass until very recently, it appeared that the limit method(2)hadfinally won out and the history of elementary calculus was closed. But in1934,Thoralf Skolem constructed what we here call the hyperintegers and provedthatthe analogue of the Transfer Principle holds for them. Skolem’sconstruction(now called the ultraproduct construction) was later extended to awide classof structures, including the construction of the hyperreal numbersfrom thereal numbers.

'See Kline,p.385.Boyer, p. 217.

The name“hyperreal”was first used by E. Hewitt in a paper in 1948. The hyperrealnumbers were knownfor over a decade before they were applied to the calculus.

Finally in1961Abraham Robinson discovered that the hyperreal numbers could be used togive arigorous treatment of the calculus with infinitesimals. The presentationof thecalculus which was given in this book is based on Robinson’s treatment(butmodified to make it suitable for a first course).

Robinson’scalculusis in the spirit of Leibniz’ old method of infinitesimals. There aremajordifferences in detail. For instance, Leibniz defined the derivative asthe ratioΔy/Δx where Δx is infinitesimal, while Robinson defines thederivative as thestandard part of the ratio Δy/Δx where Δx is infinitesimal.This is how Robinsonavoids the inconsistencies in the old infinitesimalapproach. Also, Leibniz’vague law of continuity is replaced by the preciselyformulated TransferPrinciple.

ThereasonRobinson’s work was not done sooner is that the Transfer Principle forthe hyperrealnumbers is a type of axiom that was not familiar in mathematicsuntil recently.It arose in the subject of model theory, which studies therelationship betweenaxioms and mathematical structures. The pioneeringdevelopments in model theorywere not made until the 1930’s, by G?del, Malcev,Skolem, and Tarski; and thesubject hardly existed until the 1950’s.

Looking back weseethat the method of infinitesimals was generally preferred over the methodoflimits for over 150 years after Newton and Leibniz invented thecalculus,because infinitesimals have greater intuitive appeal. But the methodof limitswas finally adopted around 1870 because it was the firstmathematically precisetreatment of the calculus. Now it is also possible to useinfinitesimals in amathematically precise way. Infinitesimals in Robinson’ssense have beenapplied not only to the calculus but to the much broader subjectof analysis.They have led to new results and problems in mathematical research.SinceSkolem’s infinite hyperintegers are usually called nonstandardintegers.Robinson called the new subject “nonstandard analysis.” (he called therealnumbers “standard” and the other hyperreal numbers “nonstandard.” This istheorigin of the name “standard part.”)

The startingpointfor this course was a pair of intuitive pictures of the real andhyperrealnumber systems. These intuitive pictures are really only roughsketches that arenot completely trustworthy. In order to be sure that theresults are correct,the calculus must be based on mathematically precisedescriptions of thesenumber systems, which fill in the gaps in the intuitivepictures. There are twoways to do this. The quickest way is to list themathematical properties of thereal and hyperreal numbers. These properties areto be accepted as basic and arecalled axioms. The second way of mathematicallydescribing the real andhyperreal numbers is to start with the positiveintegers and, step by step,construct the integers, the rational numbers, the realnumbers, and thehyperreal numbers. This second method is better because itshows that therereally is a structure with the desired properties. At the endof this epiloguewe shall briefly outline the construction of the real andhyperreal numbers andgive some examples of infinitesimals.

We now turn tothefirst way of mathematically describing the real and hyperreal numbers. Weshalllist two groups of axioms in the epilogue, one for the real numbers andone forthe hyperreal numbers. The axioms for the hyperreal numbers will justbe morecareful statements of the Extension Principle and Transfer Principle ofChapter1. The axioms for the real numbers come in three sets: the AlgebraicAxioms, theOrder Axioms, and the Completeness Axiom. All the familiar factsabout the realnumbers can be proved using only these axioms.

1.ALGEBRAICAXIOMSFOR THE REAL NUMBERS

?? A??? Closure?laws 0 and 1 arerealnumbers. If a and b are real numbers, then so are a + b, ab and -a. If a isareal number and a ≠ 0, then 1/a is a realnumber.

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B??? Commutative laws???? a+b=b+a? ab = ba

C??? Associative laws?????a+(b+c)= (a+b) +c???? a(bc) = (ab) c.

D ?? Identity laws ?????? 0+a=a???????????????1·a = a .

E??? Inverselaws???????? a+(-a)=0???? if a ≠0,???? a.1?????????????????????????????????????????????????????

F??? Distributivelaw???????a·(b + c) = ab + ac

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DEFINITION

?????????The?positiveintegers are the real numbers 1,2 = 1+1, 3 = 1+1+1 ,

????????? 4=?1+1+1+1,and so on.

║.? ORDER AXIOMS FORREAL NUMBERS

A??? 0<1.

B??? Transitive law? if a< b and b< c.

C??? Trichotomy law Exactly one of the relations a

D??? Sum law? If a< b , then a+c < b+c.

E??? product law? If a < bc.

F??? Root axiom?? For every real number a>0and every positive integer n, there is a real number b>0 such that _____=a

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Ⅲ.??COMPLETENESS? AXIOM

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??? Let?A be a set of real numbers such that wheneverxand y are in A, any real number between x and y is in A. Then A is aninterval.（全文待續）

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總結

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