极限思想外文翻译pdf.docx

极限思想外文翻译pdf.docx

《极限思想外文翻译pdf.docx》由会员分享，可在线阅读，更多相关《极限思想外文翻译pdf.docx（10页珍藏版）》请在冰点文库上搜索。

极限思想外文翻译pdf

极限思想外文翻译pdf

BSHMBulletin,2014

DidWeierstrass’sdifferentialcalculushavealimit-avoidingcharacter?

His

,,definitionofalimitinstyle

MICHIYONAKANE

NihonUniversityResearchInstituteofScience&Technology,Japan

Inthe1820s,Cauchyfoundedhiscalculusonhisoriginallimitconceptand

,,developedhisthe-orybyusinginequalities,buthedidnotapplythese

inequalitiesconsistentlytoallpartsofhistheory.Incontrast,Weierstrassconsistentlydevelopedhis1861lecturesondifferentialcalculusintermsofepsilonics.HislectureswerenotbasedonCauchy’slimitandaredistin-guishedby

theirlimit-avoidingcharacter.Dugac’spartialpublicationofthe1861lectures

makesthesedifferencesclear.Butintheunpublishedportionsofthelectures,

,,Weierstrassactu-allydefinedhislimitintermsofinequalities.Weierstrass’s

limitwasaprototypeofthemodernlimitbutdidnotserveasafoundationofhiscalculustheory.Forthisreason,hedidnotprovidethebasicstructureforthemodernedstyleanalysis.ThusitwasDini’s1878text-bookthatintroducedthe

,,definitionofalimitintermsofinequalities.

Introduction

AugustinLouisCauchyandKarlWeierstrassweretwoofthemostimportantmathematiciansassociatedwiththeformalizationofanalysisonthebasisoftheeddoctrine.Inthe1820s,Cauchywasthefirsttogivecomprehensivestatementsofmathematicalanalysisthatwerebasedfromtheoutsetonareasonablycleardefinitionofthelimitconcept（Edwards1979,310）.Heintroducedvariousdefinitionsandtheoriesthatinvolvedhislimitconcept.Hisexpressionsweremainlyverbal,buttheycouldbeunderstoodintermsofinequalities:

givenane,findnord（Grabiner1981,7）.Asweshowlater,Cauchyactuallyparaphrasedhislimitconceptintermsofe,d,andn0inequalities,inhismorecomplicatedproofs.ButitwasWeierstrass’s1861lectureswhichusedthetechniqueinallproofsandalsoinhisdefi-nition（Lutzen?

2003,185–186）.

Weierstrass’sadoptionoffullepsilonicarguments,however,didnotmeanthatheattainedaprototypeofthemoderntheory.Modernanalysistheoryisfoundedonlimitsdefinedintermsofedinequalities.HislectureswerenotfoundedonCauchy’slimitorhisownoriginaldefinitionoflimit（Dugac1973）.Therefore,inordertoclarifytheformationofthemoderntheory,itwillbenecessarytoidentifywheretheeddefinitionoflimitwasintroducedandusedasafoundation.

Wedonotfindtheword‘limit’inthepublishedpartofthe1861lectures.

Accord-ingly,Grattan-Guinness（1986,228）characterizesWeierstrass’sanalysisas

limit-avoid-ing.However,Weierstrassactuallydefinedhislimitintermsofepsilonicsintheunpublishedportionofhislectures.Histheoryinvolvedhislimitconcept,althoughtheconceptdidnotfunctionasthefoundationofhistheory.Basedonthisdiscovery,thispaperre-examinestheformationofedcalculustheory,notingmathematicians’treat-mentsoftheirlimits.Werestrictour

attentiontotheprocessofdefiningcontinuityandderivatives.Nonetheless,thisfocusprovidessufficientinformationforourpurposes.

First,weconfirmthatepsilonicsargumentscannotrepresentCauchy’slimit,

thoughtheycandescriberelationshipsthatinvolvedhislimitconcept.Next,weexaminehowWeierstrassconstructedanovelanalysistheorywhichwasnotbased

2013BritishSocietyfortheHistoryofMathematics

52BSHMBulletin

onCauchy’slimitsbutcouldhaveinvolvedCauchy’sresults.Thenweconfirm

Weierstrass’sdefinitionoflimit.Finally,wenotethatDiniorganizedhisanalysistextbookin1878basedonanalysisperformedintheedstyle.

Cauchy’slimitandepsilonicarguments

Cauchy’sseriesoftextbooksoncalculus,Coursd’analyse（1821）,Resumedes

lecons?

donneesal’Ecoleroyalepolytechniquesurlecalculinfinitesimaltome

premier（1823）,andLecons?

surlecalculdifferentiel（1829）,areoftenconsideredasthemainreferen-cesformodernanalysistheory,therigourofwhichisrootedmoreinthenineteenththanthetwentiethcentury.

AtthebeginningofhisCoursd’analyse,Cauchydefinedthelimitconceptasfol-

lows:

‘Whenthesuccessivelyattributedvaluesofthesamevariableindefinitelyapproachafixedvalue,sothatfinallytheydifferfromitbyaslittleasdesired,thelastiscalledthelimitofalltheothers’（1821,19;Englishtranslationfrom

Grabiner1981,80）.Startingfromthisconcept,Cauchydevelopedatheoryofcontinuousfunc-tions,infiniteseries,derivatives,andintegrals,constructingananalysisbasedonlim-its（Grabiner1981,77）.

Whendiscussingtheevolutionofthelimitconcept,Grabinerwrites:

‘Thiscon-

cept,translatedintothealgebraofinequalities,wasexactlywhatCauchyneededforhiscalculus’（1981,80）.Fromthepresent-daypointofview,Cauchydescribedratherthandefinedhiskineticconceptoflimits.Accordingtohis‘definition’—

whichhasthequalityofatranslationordescription—hecoulddevelopanyaspect

ofthetheorybyreducingittothealgebraofinequalities.

Next,Cauchyintroducedinfinitelysmallquantitiesintohistheory.‘Whenthesuc-

cessiveabsolutevaluesofavariabledecreaseindefinitely,insuchawayastobecomelessthananygivenquantity,thatvariablebecomeswhatiscalledaninfinitesimal.Suchavariablehaszeroforitslimit’（1821,19;Englishtranslation

fromBirkhoffandMerzbach1973,2）.Thatistosay,inCauchy’sframework‘the

limitofvariablexisc’isintuitivelyunderstoodas‘xindefinitelyapproachesc’,

andisrepresentedas‘jxcjisaslittleasdesired’or‘jxcjisinfinitesimal’.

Cauchy’sideaofdefininginfinitesimalsasvariablesofaspecialkindwasoriginal,becauseLeibnizandEuler,forexample,hadtreatedthemasconstants（Boyer1989,575;Lutzen?

2003,164）.

InCoursd’analyseCauchyatfirstgaveaverbaldefinitionofacontinuousfunc-tion.Then,herewroteitintermsofinfinitesimals:

[Inotherwords,]thefunctionfðxÞwillremaincontinuousrelativetoxinagivenintervalif（inthisinterval）aninfinitesimalincrementinthevariablealwayspro-ducesaninfinitesimalincrementinthefunctionitself.（1821,43;Englishtransla-tionfromBirkhoffandMerzbach1973,2）.

Heintroducedtheinfinitesimal-involvingdefinitionandadoptedamodifiedversionofitinResume（1823,19–20）andLecons?

（1829,278）.

FollowingCauchy’sdefinitionofinfinitesimals,acontinuousfunctioncanbedefinedasafunctionfðxÞinwhich‘thevariablefðxþaÞfðxÞisaninfinitely

smallquantity（aspreviouslydefined）wheneverthevariableais,thatis,thatfðxþaÞfðxÞapproachestozeroasadoes’,asnotedbyEdwards（1979,311）.Thus,

thedefinitioncanbetranslatedintothelanguageofedinequalitiesfromamodernviewpoint.Cauchy’sinfinitesimalsarevariables,andwecanalsotake

suchaninterpretation.

Volume29（2014）53

Cauchyhimselftranslatedhislimitconceptintermsofedinequalities.Hechanged‘Ifthedifferencefðxþ1ÞfðxÞconvergestowardsacertainlimitk,forincreasingvaluesofx,（...）’to‘Firstsupposethatthequantitykhasafinite

value,anddenotebyeanumberassmallaswewish....wecangivethenumberhavaluelargeenoughthat,whenxisequaltoorgreaterthanh,thedifferenceinquestionisalwayscontainedbetweenthelimitske;kþe’（1821,54;English

translationfromBradleyandSandifer2009,35）.

InResume,Cauchygaveadefinitionofaderivative:

‘iffðxÞiscontinuous,then

itsderivativeisthelimitofthedifferencequotient,

yf（x，i）,f（x）

,xi

asitendsto0’（1823,22–23）.Healsotranslatedtheconceptofderivativeas

follows:

‘Designatebydandetwoverysmallnumbers;thefirstbeingchoseninsuchawaythat,fornumericalvaluesofilessthand,[...],theratiofðxþiÞf

ðxÞ=ialwaysremainsgreaterthanf’ðxÞeandlessthanf’ðxÞþe’（1823,

44–45;Englishtransla-tionfromGrabiner1981,115）.

TheseexamplesshowthatCauchynotedthatrelationshipsinvolvinglimitsorinfinitesimalscouldberewrittenintermofinequalities.Cauchy’sarguments

aboutinfiniteseriesinCoursd’analyse,whichdealtwiththerelationshipbetween

increasingnumbersandinfinitesimals,hadsuchacharacter.Laugwitz（1987,264;1999,58）andLutzen?

（2003,167）havenotedCauchy’sstrictuseoftheeN

characterizationofconvergenceinseveralofhisproofs.BorovickandKatz（2012）indicatethatthereisroomtoquestionwhetherornotourrepresentationusingedinequalitiesconveysmessagesdifferentfromCauchy’soriginalintention.But

thispaperacceptstheinter-pretationsofEdwards,Laugwitz,andLutzen?

.

Cauchy’slecturesmainlydiscussedpropertiesofseriesandfunctionsinthelimitprocess,whichwererepresentedasrelationshipsbetweenhislimitsorhisinfinitesi-mals,orbetweenincreasingnumbersandinfinitesimals.Hiscontemporariespresum-ablyrecognizedthepossibilityofdevelopinganalysistheoryintermsofonlye,d,andn0inequalities.Withafewnotableexceptions,allofCauchy’s

lecturescouldberewrit-tenintermsofedinequalities.Cauchy’slimitsandhis

infinitesimalswerenotfunc-tionalrelationships,1sotheywerenotrepresentableintermsofedinequalities.

Cauchy’slimitconceptwasthefoundationofhistheory.Thus,Weierstrass’sfull

epsilonicanalysistheoryhasadifferentfoundationfromthatofCauchy.

Weierstrass’s1861lectures

Weierstrass’sconsistentuseofedarguments

Weierstrassdeliveredhislectures‘Onthedifferentialcalculus’attheGewerbe

Insti-tutBerlin2inthesummersemesterof1861.Notesoftheselecturesweretakenby

1Edwards（1979,310）,L