1、極限思想外文翻譯pdfBSHMBulletin,2014DidWeierstrasssdifferentialcalculushavealimit-avoidingcharacter?His,definitionofalimitinstyleMICHIYONAKANENihonUniversityResearchInstituteofScience&Technology,JapanInthe1820s,Cauchyfoundedhiscalculusonhisoriginallimitconceptand,developedhisthe-orybyusinginequalities,b

2、uthedidnotapplytheseinequalitiesconsistentlytoallpartsofhistheory.Incontrast,Weierstrassconsistentlydevelopedhis1861lecturesondifferentialcalculusintermsofepsilonics.HislectureswerenotbasedonCauchyslimitandaredistin-guishedbytheirlimit-avoidingcharacter.Dugacspartialpublicationofthe1861lecturesmakes

3、thesedifferencesclear.Butintheunpublishedportionsofthelectures,Weierstrassactu-allydefinedhislimitintermsofinequalities.Weierstrassslimitwasaprototypeofthemodernlimitbutdidnotserveasafoundationofhiscalculustheory.Forthisreason,hedidnotprovidethebasicstructureforthemodernedstyleanalysis.ThusitwasDini

4、s1878text-bookthatintroducedthe,definitionofalimitintermsofinequalities.IntroductionAugustinLouisCauchyandKarlWeierstrassweretwoofthemostimportantmathematiciansassociatedwiththeformalizationofanalysisonthebasisoftheeddoctrine.Inthe1820s,Cauchywasthefirsttogivecomprehensivestatementsofmathematicalana

5、lysisthatwerebasedfromtheoutsetonareasonablycleardefinitionofthelimitconcept(Edwards1979,310).Heintroducedvariousdefinitionsandtheoriesthatinvolvedhislimitconcept.Hisexpressionsweremainlyverbal,buttheycouldbeunderstoodintermsofinequalities:givenane,findnord(Grabiner1981,7).Asweshowlater,Cauchyactual

6、lyparaphrasedhislimitconceptintermsofe,d,andn0inequalities,inhismorecomplicatedproofs.ButitwasWeierstrasss1861lectureswhichusedthetechniqueinallproofsandalsoinhisdefi-nition(Lutzen?2003,185-186).Weierstrasssadoptionoffullepsilonicarguments,however,didnotmeanthatheattainedaprototypeofthemoderntheory.

7、Modernanalysistheoryisfoundedonlimitsdefinedintermsofedinequalities.HislectureswerenotfoundedonCauchyslimitorhisownoriginaldefinitionoflimit(Dugac1973).Therefore,inordertoclarifytheformationofthemoderntheory,itwillbenecessarytoidentifywheretheeddefinitionoflimitwasintroducedandusedasafoundation.Wedo

8、notfindthewordlimitinthepublishedpartofthe1861lectures.Accord-ingly,Grattan-Guinness(1986,228)characterizesWeierstrasssanalysisaslimit-avoid-ing.However,Weierstrassactuallydefinedhislimitintermsofepsilonicsintheunpublishedportionofhislectures.Histheoryinvolvedhislimitconcept,althoughtheconceptdidnot

9、functionasthefoundationofhistheory.Basedonthisdiscovery,thispaperreexaminestheformationofedcalculustheory,notingmathematicianstreat-mentsoftheirlimits.Werestrictourattentiontotheprocessofdefiningcontinuityandderivatives.Nonetheless,thisfocusprovidessufficientinformationforourpurposes.First,weconfirm

10、thatepsilonicsargumentscannotrepresentCauchyslimit,thoughtheycandescriberelationshipsthatinvolvedhislimitconcept.Next,weexaminehowWeierstrassconstructedanovelanalysistheorywhichwasnotbased2013BritishSocietyfortheHistoryofMathematicss resu lts. Then52BSHMBulletinonCauchyslimitsbutcouldhaveinvolvedCau

11、chyWeierstrasssdefinitionoflimit.Finally,wenotethatDiniorganizedhisanalysistextbookin1878basedonanalysisperformedintheedstyle.CauchyslimitandepsilonicargumentsCauchysseriesoftextbooksoncalculus,Coursdanalyse(1821),Resumedeslecons?donneesalEcoleroyalepolytechniquesurlecalculinfinitesimaltomepremier(1

12、823),andLecons?surlecalculdifferentiel(1829),areoftenconsideredasthemainreferen-cesformodernanalysistheory,therigourofwhichisrootedmoreinthenineteenththanthetwentiethcentury.AtthebeginningofhisCoursdanalyse,Cauchydefinedthelimitconceptasfollows:Whenthesuccessivelyattributedvaluesofthesamevariableind

13、efinitelyapproachafixedvalue,sothatfinallytheydifferfromitbyaslittleasdesired,thelastiscalledthelimitofalltheothers(1821,19;EnglishtranslationfromGrabiner1981,80).Startingfromthisconcept,Cauchydevelopedatheoryofcontinuousfunc-tions,infiniteseries,derivatives,andintegrals,constructingananalysisbasedo

14、nlim-its(Grabiner1981,77).Whendiscussingtheevolutionofthelimitconcept,Grabinerwrites:This con -cept,translatedintothealgebraofinequalities,wasexactlywhatCauchyneededforhiscalculus(1981,80).Fromthepresent-daypointofview,Cauchydescribedratherthandefinedhiskineticconceptoflimits.Accordingtohisdefinitio

15、nwhichhasthequalityofatranslationordescriptionhecoulddevelopanyaspectofthetheorybyreducingittothealgebraofinequalities.Next,Cauchyintroducedinfinitelysmallquantitiesintohistheory.Whenthesuc-cessiveabsolutevaluesofavariabledecreaseindefinitely,insuchawayastobecomelessthananygivenquantity,thatvariable

16、becomeswhatiscalledaninfinitesimal.Suchavariablehaszeroforitslimit(1821,19;EnglishtranslationfromBirkhoffandMerzbach1973,2).Thatistosay,inCauchysframeworkthelimitofvariablexiscisintuitivelyunderstoodasxindefinitelyapproachesc,andisrepresentedasjxcjisaslittleasdesiredorjxcjisinfinitesimal.Cauchysidea

17、ofdefininginfinitesimalsasvariablesofaspecialkindwasoriginal,becauseLeibnizandEuler,forexample,hadtreatedthemasconstants(Boyer1989,575;Lutzen?2003,164).InCoursdanalyseCauchyatfirstgaveaverbaldefinitionofacontinuousfunc-tion.Then,herewroteitintermsofinfinitesimals:Inotherwords,thefunctionfexTwillrema

18、incontinuousrelativetoxinagivenintervalif(inthisinterval)aninfinitesimalincrementinthevariablealwayspro-ducesaninfinitesimalincrementinthefunctionitself.(1821,43;Englishtransla-tionfromBirkhoffandMerzbach1973,2).Heintroducedtheinfinitesimal-involvingdefinitionandadoptedamodifiedversionofitinResume(1

19、823,1920)andLecons?(1829,278).FollowingCauchysdefinitionofinfinitesimals,acontinuousfunctioncanbedefinedasafunctionfexTinwhichthevariablefextaTfexTisaninfinitelysmallquantity(aspreviouslydefined)wheneverthevariableais,thatis,thatfextaTfexTapproachestozeroasadoes,asnotedbyEdwards(1979,311).Thus,thede

20、finitioncanbetranslatedintothelanguageofedinequalitiesfromamodernviewpoint.Cauchysinfinitesimalsarevariables,andwecanalsotakesuchaninterpretation.Volume29(2014)53Cauchyhimselftranslatedhislimitconceptintermsofedinequalities.HechangedIfthedifferencefext1TfexTconvergestowardsacertainlimitk,forincreasi

21、ngvaluesofx,(.)toFirstsupposethatthequantitykhasafinitevalue,anddenotebyeanumberassmallaswewishwecangivethenumberhavaluelargeenoughthat,whenxisequaltoorgreaterthanh,thedifferenceinquestionisalwayscontainedbetweenthelimitske;kte(1821,54;EnglishtranslationfromBradleyandSandifer2009,35).InResume,Cauchy

22、gaveadefinitionofaderivative:iffexTiscontinuous,thenitsderivativeisthelimitofthedifferencequotient,yf(x，i),f(x),xiasitendsto0'(1823,22-23).Healsotranslatedtheconceptofderivativeasfollows:Designatebydandetwoverysmallnumbers;thefirstbeingchoseninsuchawaythat,fornumericalvaluesofilessthand,.,therat

23、iofextiTfexT=ialwaysremainsgreaterthanfexTeandlessthanfexTte(1823,44-45;Englishtransla-tionfromGrabiner1981,115).TheseexamplesshowthatCauchynotedthatrelationshipsinvolvinglimitsorinfinitesimalscouldberewrittenintermofinequalities.CauchysargumentsaboutinfiniteseriesinCoursdanalyse,whichdealtwiththere

24、lationshipbetweenincreasingnumbersandinfinitesimals,hadsuchacharacter.Laugwitz(1987,264;1999,58)andLutzen?(2003,167)havenotedCauchysstrictuseoftheeNcharacterizationofconvergenceinseveralofhisproofs.BorovickandKatz(2012)indicatethatthereisroomtoquestionwhetherornotourrepresentationusingedinequalities

25、conveysmessagesdifferentfromCauchysoriginalintention.Butthispaperacceptstheinter-pretationsofEdwards,Laugwitz,andLutzen?.Cauchyslecturesmainlydiscussedpropertiesofseriesandfunctionsinthelimitprocess,whichwererepresentedasrelationshipsbetweenhislimitsorhisinfinitesi-mals,orbetweenincreasingnumbersand

26、infinitesimals.Hiscontemporariespresum-ablyrecognizedthepossibilityofdevelopinganalysistheoryintermsofonlye,d,andn0inequalities.Withafewnotableexceptions,allofCauchyslecturescouldberewrit-tenintermsofedinequalities.Cauchylimitsandhisinfinitesimalswerenotfunc-tionalrelationships,1sotheywerenotreprese

27、ntableintermsofedinequalities.Cauchyslimitconceptwasthefoundationofhistheory.Thus,WeierstrasssfullepsilonicanalysistheoryhasadifferentfoundationfromthatofCauchy.Weierstrasss1861lecturesWeierstrasssconsistentuseofedargumentsWeierstrassdeliveredhislecturesOnthedifferentialcalculusattheGewerbeInsti-tut

28、Berlin2inthesummersemesterof1861.NotesoftheselecturesweretakenbylEdwards(1979,310),Laugwitz(1987,260-261,271-272),andFisher(1978,16-318)pointoutthatCauchy'sinfinitesimalsequatetoadependentvariablefunctionoraehTthatapproacheszeroash!0.Cauchyadoptedthelatterinfinitesimals,whichcanbewrittenintermso

29、fedarguments,whenheintro-ducedaconceptofdegreeofinfinitesimals(1823,250;1829,325).EveryinfinitesimalofCauchysisavari-ableinthepartsthatthepresentpaperdiscusses.2AforerunneroftheTechnischeUniversit?atBerlin.54BSHMBulletinHermanAmandusSchwarz,andsomeofthemhavebeenpublishedintheoriginalGermanbyDugac(19

30、73).Notingthenewaspectsrelatedtofoundationalconceptsinanalysis,fulleddefinitionsoflimitandcontinuousfunction,anewdefinitionofderivative,andamoderndefinitionofinfinitesimals,Dugacconsideredthatthenov-eltyofWeierstrass'slectureswasincontestable(1978,372,1976,6-7).3Afterbeginninghislecturesbydefini

31、ngavariablemagnitude,Weierstrassgavethedefinitionofafunctionusingthenotionofcorrespondence.Thisbroughthimtothefollowingimportantdefinition,whichdidnotdirectlyappearinCauchystheory:(D1)Ifitisnowpossibletodetermineforhabounddsuchthatforallvaluesofhwhichintheirabsolutevaluearesmallerthand,fexthTfexTbec

32、omessmallerthananymagnitudee,howeversmall,thenonesaysthatinfinitelysmallchangesoftheargumentcorrespondtoinfinitelysmallchangesofthefunction.(Dugac1973,119;EnglishtranslationfromCalinger1995,607)Thatis,Weierstrassdefinednotinfinitelysmallchangesofvariablesbutinfinitelysmallchangesoftheargumentscorres

33、pond(ing)toinfinitelysmallchangesoffunctionthatwerepresentedintermsofedinequalities.Hefoundedhistheoryonthiscorrespondence.Usingthisconcept,hedefinedacontinuousfunctionasfollows:(D2)Ifnowafunctionissuchthattoinfinitelysmallchangesoftheargumenttherecorrespondinfinitelysmallchangesofthefunction,onethe

34、nsaysthatitisacontinuousfunctionoftheargument,orthatitchangescontinuouslywiththisargument.(Dugac1973,119-120;EnglishtranslationfromCalinger1995,607)Soweseethatinaccordancewithhisdefinitionofcorrespondence,Weierstrassactuallydefinedacontinuousfunctiononanintervalintermsofepsilonics.Since(D2)isderived

35、bymerelychangingCauchystermproduceto,itseemsthatWeierstrasstooktheideaofthisdefinitionfromcorrespondCauchy.However,Weierstrasssdefinitionwasgivenintermsofepsilonics,whileCauchysdefinitioncanonlybeinterpretedintheseterms.Furthermore,WeierstrassachieveditwithoutCauchyslimit.Luzten?(2003,186)indicatest

36、hatWeierstrassstillusedtheconceptofinfinitelysmallinhislectures.Untilgivinghisdefinitionofderivative,Weierstrassactuallyafunctioncontinuedtousetheterminfinitesimallysmallandoftenwroteofwhichbecomesinfinitelysmallwithh.Butseveralinstancesofinfinitesimallysmallappearedinformsoftherelationshipsinvolvin

37、gthem.Definition(D1)givestherela-tionshipintermsofedinequalities.WemaythereforeassumethatWeierstrassslecturesconsistentlyusededinequalities,eventhoughhisdefinitionswerenotdirectlywrittenintermsoftheseinequalities.Weierstrassinsertedsentencesconfirmingthattherelationshipsinvolvingtheterminfinitelysma

38、llweredefinedintermsofedinequalitiesasfollows:ehTisan(D3)Ifhdenotesamagnitudewhichcanassumeinfinitelysmallvalues,arbitraryfunctionofhwiththepropertythatforaninfinitelysmallvalueofhit3ThepresentpaperalsoquotesKurtBingstranslationincludedinCalingersClassicsofmathematics.Volume29(2014)55alsobecomesinfi

39、nitelysmall(thatis,thatalways,assoonasadefinitearbitrarysmallmagnitudeeischosen,amagnitudedcanbedeterminedsuchthatforallvaluesofhwhoseabsolutevalueissmallerthand,ehTbecomessmallerthane).(Dugac1973,120;EnglishtranslationfromCalinger,1995,607)AsDugac(1973,65)indicates,somemoderntextbooksdescribeehTasi

40、nfinitelysmallorinfinitesimal.WeierstrassarguedthatthewholechangeoffunctioncaningeneralbedecomposedasDfexT?fexthTfexT?p:hthehT;e1TwherethefactorpisindependentofhandehTisamagnitudethatbecomesinfinitelysmallwithh.4However,heoverlookedthatsuchdecompositionisnotpossibleforallfunctionsandinsertedthetermi

41、ngeneral.Herewrotehasdx.OnecanmakethedifferencebetweenDfexTandp:dxsmallerthananymagnitudewithdecreasingdx.HenceWeierstrassdefineddifferentialas the changewhichafunctionundergoeswhenitsargumentchangesbyaninfinitesimallysmallmagnitudeanddenoteditasdfexT.Then,dfexT?p:dx.Weierstrasspointedoutthatthediff

42、erentialcoefficientpisafunctionofxderivedfromfexTandcalleditaderivative(Dugac1973,120-121;EnglishtranslationfromCalinger1995,607-608).InaccordancewithWeierstrasssdefinitions(D1)and(D3),helargelydefinedaderivativeintermsofepsilonics.Weierstrassdidnotadopttheterminfinitelysmallbutdirectlyusededinequalitieswhenhediscussedpropertiesofinfiniteseriesinvolvinguniformconver-gence(Dugac1973,122124).Itmaybeinfe