Elementary Mathematics from a Higher Standpoint: Volume I: Arithmetic, Algebra, Analysis

325 Pages·2016·4.45 MB·English

by  Felix Klein (auth.)

Felix Klein
Elementary
Mathematics
from a Higher
Standpoint
Volume I: Arithmetic, Algebra, Analysis
Elementary Mathematics from a Higher Standpoint
Felix Klein
Elementary Mathematics
from a Higher Standpoint
Volume I: Arithmetic, Algebra, Analysis
Translated by Gert Schubring
FelixKlein
Translatedby:GertSchubring
ISBN978-3-662-49440-0 ISBN978-3-662-49442-4(eBook)
DOI10.1007/978-3-662-49442-4
LibraryofCongressControlNumber:2016943431
Translationofthe4thGermanedition„ElementarmathematikvomhöherenStandpunkteaus“,vol.1by
FelixKlein,GrundlehrenderMathematischenWissenschafteninEinzeldarstellungen,Band14,Verlag
vonJuliusSpringer,Berlin1933.ApreviousEnglishlanguageedition,FelixKlein“ElementaryMathe-
maticsfromanAdvancedStandpoint–Arithmetic,Algebra,Analysis”,translatedbyE.R.Hedrickand
C.A.Noble,NewYork1931,wasbasedonthe3rdGermaneditionandpublishedbyDoverPublications.
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Preface to the 2016 Edition
TheNotion ofElementary Mathematics
This is the first volume of the three-volume-series of Felix Klein’s “Elementar-
mathematik vom höheren Standpunkteaus”, the first two volumes nowin revised
and completed editions and the third volume the first time in English translation.
ThistranslationisbasedonthelastGermaneditionofvolumeI,thefourthof1933;
theEnglish translation of 1931, which was used for this newrevised version, had
thethirdeditionof1924asitsmastercopy.ThethirdandthefourthGermanedition
arebasicallyidentical, exceptthethreeparagraphsonpp.296–297,whichreplace
pp.297–303ofthe3rdedition.
The volumes are lectures notes of courses, which Klein offeredoften to future
secondaryschoolmathematicsteachersatGöttingenuniversity,andpublishedbe-
tween1902and1908,proposingandrealizinganewformofteachertraining,which
becameamodelformanymathematiciansandwhichremainedvalidandeffective
untiltoday.JeremyKilpatrickemphasizedtheimportanceofthesevolumesthus:
In print for a century, the volumes of Klein’s textbook have been used in
countless courses for prospectiveand practicing teachers. They provideex-
cellent early examplesof whattoday istermed mathematicalknowledgefor
teaching. Klein’s courses for teachers were part of his reform efforts to
improve secondary mathematics by improving the preparation of teachers.
Despitethemanysetbacksheencountered,nomathematicianhashadamore
profound influence on mathematics education as a field of scholarship and
practice(Kilpatrick2008,p.27).1
It was Kilpatrick, too, who as the first called attention to the misleading trans-
lation of the term “höher” in that English translation of the 1930’s. While all the
1IamquotingfromKilpatrick’slectureatICME11inMexico.Unfortunately,theProceedingsof
thisCongresswereneverpublished. TheICMIExecutiveCommitteedecidedtherefore,tomake
thelecturesdeliveredthereaccessibleonlineattheICMIsite.
v
vi Prefacetothe2016Edition
othertranslationshadgiventhe“höher”correctly,itbecame“advanced”insteadof
“higher”–andwasreceivedthusoversomanydecadesastheEnglishversion:
WhenitcametimefortheAmericantranslatorsofKlein’sElementarmathe-
matik to render the title in English, they chose to translate vom höheren
Standpunkte aus as from an advanced standpoint. The term higher is not
only a more literal translation of höheren than advanced is, but it also cap-
tures better the image Klein had for his work. Advanced can mean higher,
but its connotationis morelike ‘moredeveloped’or ‘further along in space
andtime’.Kleinwantedtoemphasizethathiscourseswouldgiveprospective
teachersabetter,morepanoramicviewofthelandscapeofmathematics. As
noted above, hewanted thoseteachersto ‘stand above’their subject. (ibid.,
p.40)
Infact,theterm“advanced”impliesafundamentalmisunderstandingofKlein’s
notion of elementary and of Elementarmathematik. The term “advanced”implies
thatelementarymathematicsis somewhatdelayed, laggingbehind, ofanotherna-
ture. It means exactly the contrary of what Klein was intending. By contrasting
twopoles,“elementary”versus“advanced”,onewouldadmitjustthatdiscontinu-
itybetweenschoolmathematicsandacademicmathematics,whichKleinwantedto
eliminate.
For Klein, therewas noseparation between anelementarymathematics andan
academic mathematics. His conception for training teachers in higher education
departed from a holistic vision of mathematics: mathematics, steadily developing
andreformingitselfwithinthisprocess,leadingtoevernewrestructuredelements,
provides therefore new accesses to the elements. There is a widespread under-
standingoftheterm“elementary”,meaningitsomething“simple”andnotloaded
with conceptualdimension – even somehowapproaching“trivial”. Connected, in
contrast, with thenotion of element, “elementary”meansfor Klein to unravelthe
fundamentalconception. Whatisatstake,hence,isthenotionofelements.
Beyond mere factual information, with his lecture notes Klein leads the stu-
dents to gain a more comprehensive and methodological point of view on school
mathematics. ThethreevolumesthusenableustounderstandKlein’sfar-reaching
conceptionofelementarisation,ofthe“elementaryfromahigherstandpoint”,inits
implementationforschoolmathematics: Theelementsareunderstoodasthefunda-
mentalconceptsof mathematics, relatedto thewholeofmathematics– according
toitsrestructuredarchitecture.
This notion of elements corresponds neatly to the first reflections on the na-
tureofelementsundertakeninthewakeofEnlightenmenthowtomakeknowledge
teachable and how to disseminate knowledge thus in society to ensure its general
understanding. One has to name in particular Jean le Rond d’Alembert (1717–
1783)whoconceptualizedinaprofoundmannerwhathecalledto“elementarise”
thesciences. Itwashisseminalandextensiveentry“élémensdessciences”inthe
Encyclopédie,thekeyworkoftheEnlightenment,wherehegavethisanalysisand
reflection howto elementariseascience, thatis howto connecttheelementswith
thewholeofthatscience. Thisprocedureisto beableto identifytheelementsof
TheNotionofElementaryMathematics vii
a science, or in other words, have rebuilt it in a new coherent way all parts of a
sciencethatmayhaveaccumulatedindependentlyandnotmethodically:
Onappelleengénéralélémensd’untout,lespartiesprimitives&originaires
dontonpeutsupposerquecetoutestformé(d’Alembert1755,491l).
Inthissense,thereisnoqualitativedifferencebetweentheelementarypartsand
thehigherparts. Theelementsareconsideredasthe“germs”ofthehigherparts:
Ces propositions réunies en un corps, formeront, à proprement parler, les
élémens de la science, puisque ces élémens seront comme un germe qu’il
suffiroit de développer pour connoitre les objets de la science fort en détail
(d’Alembert1755,491r).
Anextensivepartoftheentryisdedicatedtothereflectiononelementarybooks,
suchasschoolbooks,whichareessential, ontheonehand,to disseminatethesci-
encesand,second,tomakeprogressinthesciences,thatis,toobtainnewtruths. In
hisreflectiononelementarybooks,d’Alembertemphasisedanotheraspectofgreat
importance regarding the relationship between the elementary and the higher: he
underlined that the key issue for the composition of good elementary books con-
sistsininvestigatingthe“metaphysics”ofpropositions–orintermsoftoday: the
epistemologyofscience.
Infact,Klein’sworkcanbeunderstoodexactlyasprovidingsuchanepistemo-
logical, or methodological access to mathematics. It was not to provide factual
knowledge:
Ishallbynomeansaddressmyselftobeginners,butIshalltakeforgranted
thatyouareallacquaintedwiththemainfeaturesofthemostimportantdis-
ciplinesofmathematics(Klein,thisvolume,p.[1]etseq.).
Whereasheoutlinedashisgoal:
AnditispreciselyinsuchsummarisinglecturecoursesasIamabouttode-
livertoyouthatIseeoneofthemostimportanttools(ibid.,p.[1]).
Indeed,Kleinexplicitlyexposedtheepistemologicalaspectofhiswork:explain-
ing the connections, the connections between subdisciplines, which normally are
treatedseparately andpointingoutthelinksofparticularmathematicalissues and
questionswithasyntheticviewofthewholeofmathematics. Thus,futureteachers
wouldachievetodeepentheirunderstandingofthebasicconceptsofmathematics
andappreciatethenatureofmathematicalconcepts:
My task will always be to show you the mutual connection between prob-
lems in the various disciplines, these connections use not to be sufficiently
considered in the specialised lecture courses, and I want more especially to
emphasizetherelationoftheseproblemstothoseofschoolmathematics. In
thiswayIhopetomakeiteasierforyoutoacquirethatabilitywhichIlook
upon as the real goalof your academic study: the ability to draw (in ample
viii Prefacetothe2016Edition
measure)fromthegreatbodyofknowledgetaughttoyouherevividstimuli
foryourteaching(ibid.,p.[2]).
There is a decisive differencebetween d’Alembert’s and Klein’s notion of ele-
mentarisation. D’Alembert’s notion basically was not a historical one; he did not
reflecttheeffectofscientificprogressontheelements. ButthiswasexactlyKlein’s
notion. Heemphasised:
Thenormalprocessofdevelopment[...]ofascienceisthefollowing:higher
and more complicated parts become gradually more elementary, due to the
increase in thecapacity to understandtheconceptsandto thesimplification
oftheirexposition(“lawofhistoricalshifting”). Itconstitutesthetaskofthe
school to verify, in view of the requirements of general education, whether
the introduction of elementarised concepts into the syllabus is necessary or
not(Klein&Schimmack1907,p.90).
The historical evolution of mathematics entails therefore a process od restruc-
turationofmathematicswherenewtheories,whichatfirstmighthaverangedsome-
what isolated and not well integrated, turn well connected to other branches of
mathematics andeffecta newarchitectureof mathematics, based onre-conceived
elements,thusonanewsetofelementarisedconcepts.
SettheorywasacaseforKleinwherethistheoreticaldevelopmentwastoofresh
andevennotyetaccomplishedandevenmorefarfromhavingmaturedinamanner
to having induced an intra-disciplinary process of integration and restructuration.
The concepts of set theory did not (yet) providenew elements for mathematics –
thereforeKlein’spolemicagainstFriedrichMeyer’sschoolbookof1885who’sin-
tentionhadbeen,infact,tousesettheoryasnewelementsforteachingarithmetic
and algebra (see the note on p. [289]). In Klein’s times, mathematics had not
achieved the level of the architecture established by Bourbaki– and hence not of
“modernmath”.
This volume I is devoted to what Klein calls the three big “A’s”: arithmetic,
algebra and analysis. They are presented and discussed always together with a
dimensionofgeometricinterpretationandvisualisation–givenhisepistemological
viewpointofmathematics beingbasedin spaceintuition. Aparticularlyrevealing
exampleforelementarisationishischapteronthetranscendenceofeand(cid:2),where
hesucceedsingivingaconcise,wellaccessibleproofforthetranscendenceofthese
twonumbers.
TheUse ofHistory ofMathematics
Aparticularlycharacteristic featureofKlein’slecturecoursesandofhisapproach
of “Elementary Mathematics from a Higher Standpoint” is the important role at-
tributedtothehistoryofmathematics. Kleinexplainsvarioustimeshisconviction
that exposing key features of the historical development of concepts will support
hismethodologicalorientationtoleadtoadeeperunderstandingofthefundaments
WhyaRevisedTranslationofVolumeI? ix
ofmathematics. Kleinthusrevealshimselfasaprobablyfirstandstaunchsupporter
oftheuseofhistoryfortheteachingofmathematics–whatthusbecamesince1976
theinternationalmovement,wellknownasHPM.2
Yet, one has to admit that at his time there was only one conception available
fortheuseofmathematicshistoryinteaching:theso-calledbiogeneticlaw,affirm-
ing a recapitulation of the historical development by the individual. There was a
widespreadconvictionofthevalidityofthis“law”forbiology;anditsapplicability
foreducationbelongedtothedominantmentalitiesoftheepoch,thoughwithsome
more reservations. As quickly as the biogenetic law had spread after Haeckel’s
propagation, as quickly it disappeared from publicdiscourses in education, in the
inter-war period, and seemingly completely. It was mentioned for the first time,
afterthisfallingintooblivion,in1962,inthememorandumof65mathematicians
against “new math”. Whatis the most astonishing, however, is the revival of this
conception–reputedtobedead,andthisexactlywithmoreworkdoneontheuse
ofthehistoryofmathematicsinteaching:sinceaboutthe1990s.Itseemsthatthere
arestill no other well-developedor known conceptionshowto relate history with
teaching(seeSchubring2006).Therefore,onecannotblameKleintohavereferred
oftentothisconception.
Afurtherissueinthisregardiswhichhistoriographyofmathematicsisadapted
to be used for the contextof teaching. Felix Klein was notonly highly interested
inthehistoryofmathematics; hepromotedstronglyresearchinto thehistory. For
instance, he initiated research into the manuscripts of Gauß and he organised the
publication of Collected Works of several mathematicians. Thus, he was com-
pletely aware of the results of historical research into the history of mathematics
as achieved until his times. And this knowledge was the basis for his historical
annotationsandaffirmationsinthethreevolumes. Clearly,ashistoriographicalre-
search has progressed since then, not all his information is today still the state of
theart.
Whya RevisedTranslationofVolumeI?
Thisedition isthefirst completeEnglish translation ofKlein’s first volumeof the
Elementarmathematik. Infact,theoriginalvolumecontains,atitsend,twoappen-
dices, of 14 pages: one on the efforts to reform mathematics teaching, while the
othergivescomplementaryinformationonmathematicalandpedagogicalliterature
–thusrevealingsectionsforcomplementingtounderstandKlein’sviewsonteach-
ing mathematics. The translators were aware of these two appendices: they are
mentionedintheirversionof1931,inthefootnote1onp.1,addedbythemselves.
However,theyomittedthesetwosectionswithoutanycommentorjustification.
2The International Study Group on the relations between the HISTORY and PEDAGOGY of
MATHEMATICS,foundedin1976,anaffiliatedGroupofICMI.
x Prefacetothe2016Edition
Moreover,themisleadingtranslationofthetitlefortheentireserieshadalready
tobeoutlined. OneisthereforeledtoaskwhowerethetwoAmericantranslators.
Their biography shows them to have been well-qualified mathematicians. Both
translators, Earle Raymond Hedrick and Charles Albert Noble studied in Göttin-
gen, then the internationally leading centre of mathematics, with Klein and with
Hilbert. Both obtained their PhD as doctoral students of Hilbert, in 1901. After
theirreturntotheStates,theyplayedanimportantroleinbuildingupthecountry’s
mathematicalinstitutions. Hedrick,firstmathematicsprofessorattheUniversityof
Missouri, was called in 1924 to the University of California at Los Angeles. He
servedasfirstPresidentoftheMathematicalAssociation ofAmericain1916,and
later as President of the American Mathematical Society (1929–1930). Noble, at
firstamathematicsteacheratcolleges,becameamathematicsprofessorattheUni-
versity of Californiaat Berkeley(Parshall & Rowe1994, p. 410 and 440et seq.).
One is thereforestruck to remark their translation being marred by numerouster-
minologicalandtextualfaults.
Whatcatchestheeyeimmediately, besidesthe“advanced”issue, istheirtrans-
lation for Mengenlehre. Although Mengenlehre was a major issue of discussion
duringthetimeoftheirstudiesinGöttingen,theyarenotfamiliarwiththeEnglish
term. In the first part, they use persistently “theory of point sets”. In the spe-
cialsection onsettheory,theygiveastitle “Theoryofassemblages”andusehere
“assemblage”forset, butnotconsistently–theyalso use“aggregate”. Onemight
inferthatsettheoryhadnotyetreallyarrivedintheStatesbythe1930s.
Actually, the two translators have qualified, in their preface, their work as “a
rather free translation”. That would be admissible, but this is not the case. At
toomanyplaces,thetextgivesnottheintendedmeaning,butrathererroneousand
misleading translations, in particular with regard to mathematics, not only with
regardtothegeneralstyle.
Surely, a problem might have arisen by the character of Klein’s text as lecture
notes–takenduringhiscoursesbystudents. Thus,thetextrepresentsoralteaching,
andnotatextintentionallycomposedforprinting. Thelecturenotesaretherefore
writtenquiteofteninarathercolloquialstyle–andthespecifityoftranslatingcon-
sistsinrenderingwellthemeaningofsuchidioms. Notbeingawareofthistextual
style is already a first reason for misleading translations. Surely, both translators
did not acquire such an intimate knowledge of the German language during their
stayinGermany. Noble,forinstance, gavehispresentationsinKlein’sseminarin
English(Parshall&Rowe1994,p.257).
ButevenKlein’sfamousstatementaboutthedoublediscontinuitybetweensec-
ondaryschoolsanduniversitystudiesisrenderedinamisleadingway: whileKlein
iscomplainingaboutthisdiscontinuityas apersistentproblem, includinghisown
times, the translators transformed this into past tense – as if the problem had al-
readybeenovercome!Thisintroductorypassageofthebook,onp.[1],containsno
colloquialtermsatall.
AproblempervadingbothvolumesIandIIconsistsinnotbeingawareofoneof
themostbasicconceptsofKlein: ofAnschauung–forhimthefundamentofcon-
ceptualdevelopmentofmathematics. And,asitiswellknow,Anschauungpresents