极限思想外文翻译pdf.docx
- 文档编号:11177726
- 上传时间:2023-02-25
- 格式:DOCX
- 页数:11
- 大小:26.08KB
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(11页珍藏版)》请在冰豆网上搜索。
极限思想外文翻译pdf
极限思想外文翻译pdf
BSHMBulletin,2014
DidWeierstrass'sdifferentialcalculushavealimit-avoiding
character?
His
,,definitionofalimitinstyle
MICHIYONAKANE
NihonUniversityResearchInstituteofScience&Technology,Japan
Inthe1820s,Cauchyfoundedhiscalculusonhisoriginallimitconceptand
,,developedhisthe-orybyusinginequalities,buthedidnotapplythese
inequalitiesconsistentlytoallpartsofhistheory.Incontrast,Weierstrassconsistentlydevelopedhis1861lecturesondifferentialcalculusintermsofepsilonics.HislectureswerenotbasedonCauchy'slimitandaredistin-guishedby
theirlimit-avoidingcharacter.Dugac'spartialpublicationofthe
1861lectures
makesthesedifferencesclear.Butintheunpublishedportionsofthelectures,
,,Weierstrassactu-allydefinedhislimitintermsofinequalities.Weierstrass's
limitwasaprototypeofthemodernlimitbutdidnotserveasafoundationofhiscalculustheory.Forthisreason,hedidnotprovide
thebasicstructureforthemodernedstyleanalysis.Thusitwas
Dini's1878text-bookthatintroducedthe
,,definitionofalimitintermsofinequalities.
Introduction
AugustinLouisCauchyandKarlWeierstrassweretwoofthemostimportantmathematiciansassociatedwiththeformalizationofanalysisonthebasisoftheeddoctrine.Inthe1820s,Cauchywasthefirsttogivecomprehensivestatementsofmathematicalanalysisthatwerebasedfromtheoutsetonareasonablycleardefinitionofthelimitconcept(Edwards1979,310).Heintroducedvariousdefinitionsandtheoriesthatinvolvedhislimitconcept.Hisexpressionsweremainlyverbal,buttheycouldbeunderstoodintermsofinequalities:
givenane,findnord(Grabiner1981,7).Asweshowlater,Cauchyactuallyparaphrasedhislimitconceptintermsofe,d,andn0inequalities,inhismorecomplicatedproofs.ButitwasWeierstrass's1861lectureswhichused
thetechniqueinallproofsandalsoinhisdefi-nition(Lutzen?
2003,
185-186).
Weierstrass'sadoptionoffullepsilonicarguments,however,didnotmeanthatheattainedaprototypeofthemoderntheory.Modernanalysistheoryisfoundedonlimitsdefinedintermsofedinequalities.HislectureswerenotfoundedonCauchy'slimitorhis
ownoriginaldefinitionoflimit(Dugac1973).Therefore,inordertoclarifytheformationofthemoderntheory,itwillbenecessaryto
identifywheretheeddefinitionoflimitwasintroducedandusedasafoundation.
Wedonotfindtheword‘limit'inthepublishedpartofthe1861
lectures.
Accord-ingly,Grattan-Guinness(1986,228)characterizesWeierstrass'sanalysisas
limit-avoid-ing.However,Weierstrassactuallydefinedhislimitintermsofepsilonicsintheunpublishedportionofhislectures.Histheoryinvolvedhislimitconcept,althoughtheconceptdidnotfunctionasthefoundationofhistheory.Basedonthisdiscovery,thispaperreexaminestheformationofedcalculustheory,notingmathematicianstreat-mentsoftheirlimits.Werestrictour
attentiontotheprocessofdefiningcontinuityandderivatives.Nonetheless,thisfocusprovidessufficientinformationforourpurposes.
First,weconfirmthatepsilonicsargumentscannotrepresentCauchy'slimit,
thoughtheycandescriberelationshipsthatinvolvedhislimitconcept.Next,weexaminehowWeierstrassconstructedanovelanalysistheorywhichwasnotbased
2013BritishSocietyfortheHistoryofMathematics
52BSHMBulletin
onCauchy'slimitsbutcouldhaveinvolvedCauchy'sresults.Then
weconfirm
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,Cauchydefinedthelimit
conceptasfol
lows:
‘Whenthesuccessivelyattributedvaluesofthesamevariableindefinitelyapproachafixedvalue,sothatfinallytheydifferfromitbyaslittleasdesired,thelastiscalledthelimitofalltheothers'(1821,19;Englishtranslationfrom
Grabiner1981,80).Startingfromthisconcept,Cauchydevelopedatheoryofcontinuousfunc-tions,infiniteseries,derivatives,andintegrals,constructingananalysisbasedonlim-its(Grabiner1981,77).
Whendiscussingtheevolutionofthelimitconcept,Grabinerwrites:
‘Thiscon-
cept,translatedintothealgebraofinequalities,wasexactlywhat
Cauchyneededforhiscalculus'(1981,80).Fromthepresent-daypoint
ofview,Cauchydescribedratherthandefinedhiskineticconceptoflimits.Accordingtohis‘definition'—
whichhasthequalityofatranslationordescription—hecould
developanyaspect
ofthetheorybyreducingittothealgebraofinequalities.
Next,Cauchyintroducedinfinitelysmallquantitiesintohistheory.
Whenthesuc-
cessiveabsolutevaluesofavariabledecreaseindefinitely,insuchawayastobecomelessthananygivenquantity,thatvariablebecomeswhatiscalledaninfinitesimal.Suchavariablehaszeroforitslimit'(1821,19;Englishtranslation
fromBirkhoffandMerzbach1973,2).Thatistosay,inCauchy's
limitofvariablexisc
framework‘the
isintuitivelyunderstoodas
indefinitelyapproachescisinfinitesimal
Cauchy'sideaofdefininginfinitesimalsasvariablesofaspecialkindwasoriginal,becauseLeibnizandEuler,forexample,hadtreatedthemasconstants(Boyer1989,575;Lutzen?
2003,164).
InCoursd'analyseCauchyatfirstgaveaverbaldefinitionofacontinuousfunc-tion.Then,herewroteitintermsofinfinitesimals:
incontinuousrelative
[Inotherwords,]thefunctionfexTwillrematoxinagivenintervalif(inthisinterval)aninfinitesimalincrementinthevariablealwayspro-ducesaninfinitesimalincrementinthefunctionitself.(1821,43;Englishtransla-tionfromBirkhoffandMerzbach1973,2).
Heintroducedtheinfinitesimal-involvingdefinitionandadopteda
modifiedversionofitinResume(1823,19-20)andLecons?
(1829,278).
FollowingCauchy'sdefinitionofinfinitesimals,acontinuous
functioncanbedefinedasafunctionfexTinwhich‘thevariablefex
taTfexTisaninfinitely
smallquantity(aspreviouslydefined)wheneverthevariableais,
thatis,thatfextaTfexTapproachestozeroasadoes',asnoted
byEdwards(1979,311).Thus,
thedefinitioncanbetranslatedintothelanguageofed
inequalitiesfromamodernviewpoint.Cauchy'sinfinitesimalsarevariables,andwecanalsotake
suchaninterpretation.
Volume29(2014)53
Cauchyhimselftranslatedhislimitconceptintermsofed
inequalities.Hechanged‘Ifthedifferencefext1TfexTconverges
towardsacertainlimitk,forincreasingvaluesofx,(...)'to
‘Firstsupposethatthequantitykhasafinite
value,anddenotebyeanumberassmallaswewish....wecangivethenumberhavaluelargeenoughthat,whenxisequaltoor
greaterthanh,thedifferenceinquestionisalwayscontainedbetween
thelimitske;kte'(1821,54;English
translationfromBradleyandSandifer2009,35).
InResume,Cauchygaveadefinitionofaderivative:
‘iffexTis
continuous,then
itsderivativeisthelimitofthedifferencequotient,
yf(x,i),f(x)
,xi
asitendsto0'(1823,22-23).Healsotranslatedtheconceptof
derivativeas
follows:
‘Designatebydandetwoverysmallnumbers;thefirst
beingchoseninsuchawaythat,fornumericalvaluesofilessthand,[...],theratiofextiTf
exT=ialwaysremainsgreaterthanf'exTeandlessthanf'exTt
e'(1823,
44-45;Englishtransla-tionfromGrabiner1981,115).
TheseexamplesshowthatCauchynotedthatrelationshipsinvolving
limitsorinfinitesimalscouldberewrittenintermofinequalities.
Cauchy'sarguments
aboutinfiniteseriesinCoursd'analyse,whichdealtwiththe
relationshipbetween
increasingnumbersandinfinitesimals,hadsuchacharacter.
Laugwitz(1987,264;1999,58)andLutzen?
(2003,167)havenoted
Cauchy'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's
limitsandhis
infinitesimalswerenotfunc-tionalrelationships,1sotheywerenotrepresentableintermsofedinequalities.
Cauchy'slimitconceptwasthefoundationofhistheory.Thus,Weierstrass'sfull
epsilonicanalysistheoryhasadifferentfoundationfromthatofCauchy.
Weierstrass's1861lectures
Weierstrass'sconsistentuseofedarguments
Weierstrassdeliveredhislectures‘Onthedifferentialcalculus'
attheGewerbe
Insti-tutBerlin2inthesummersemesterof1861.Notesoftheselecturesweretakenby
lEdwards(1979,310),Laugwitz(1987,260-261,271-272),and
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 极限 思想 外文 翻译 pdf