The Mathematics of Niels Henrik Abel: Continuation and New Approaches in Mathematics During ...

470 Pages·2012·4.74 MB·English

RePoSS: Research Publications on Science Studies
#11:
RePoSS
The Mathematics of Niels Henrik
Abel: Continuation and New
Approaches in Mathematics
During the 1820s
Henrik Kragh Sørensen
October 2010
Centre for Science Studies, University of Aarhus, Denmark
Research group: History and philosophy of science
Please cite this work as:
Henrik Kragh Sørensen (Oct. 2010). The Mathematics of Niels
Henrik Abel: Continuation and New Approaches in Mathemat-
ics During the 1820s. RePoSS: Research Publications on Science
Studies 11. Aarhus: Centre for Science Studies, University of
Aarhus. url: http://www.css.au.dk/reposs.
Copyright c Henrik Kragh Sørensen, 2010
(cid:13)
The Mathematics of
N H A
IELS ENRIK BEL
Continuation and New Approaches
in Mathematics During the 1820s
H K S
ENRIK RAGH ØRENSEN
For Mom and Dad
who were always there for me
when I abandoned
all good manners,
good friends,
and common sense
to pursue my dreams.
The Mathematics of
N H A
IELS ENRIK BEL
Continuation and New Approaches
in Mathematics During the 1820s
H K S
ENRIK RAGH ØRENSEN
PhD dissertation
March 2002
Electronic edition, October 2010
History of Science Department
The Faculty of Science
University of Aarhus, Denmark
This dissertation was submitted to the Faculty of Science,
University of Aarhus in March 2002 for the purpose of ob-
tainingthescientificPhDdegree. Itwasdefendedinapublic
PhDdefenseonMay3,2002. Asecond,onlyslightlyrevised
editionwasprintedOctober,2004.
The PhD program was supervised by associate professor
KIRSTI ANDERSEN, History of Science Department, Univer-
sityofAarhus.
Professors UMBERTO BOTTAZZINI (University of Palermo,
Italy), JEREMY J. GRAY (Open University, UK), and OLE
KNUDSEN (History of Science Department, Aarhus) served
onthecommitteeforthedefense.
c Henrik Kragh Sørensen and the History of Science De-
(cid:13)
partment (Department of Science Studies), University of
Aarhus,2002–2010.
ThedissertationwastypesetinPalatinousingpdfLATEX.
First edition (March 2002, 7 copies) was copied and bound
bythePrintshopattheFacultyofScience,Aarhus.
Second edition (October 2004, 5 copies) was printed and
bound by the Printshop at Agder University College, Kris-
tiansand.
ThiselectroniceditionwascompiledonOctober28,2010.
For further information, additions, corrections, and
contact to the author, please refer to the website
http://www.henrikkragh.dk/phd/.
Thepictureonthefrontpageisapaintingof NIELS HENRIK
ABEL performed by the Norwegian painter JOHAN GØRB-
ITZduringABEL’stimeinParis1826. Itistheonlyauthentic
depictionof ABEL andisreproducedfrom(Ore,1957).
The picture on the reverse shows a curlicue frequently used
by ABELinhisnotebookstomarktheendofmanuscripts. It
isreproducedfrom(Stubhaug,1996).
Contents
Contents i
ListofTables vii
ListofFigures ix
ListofBoxes xi
ListofTheoremsetc. xiii
Summary xv
Prefacetothe2004edition xvii
Abel’smathematicsinthecontextoftraditionsandchanges . . . . . . . . . . xvii
Recentliterature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
Prefacetothe2002edition xxi
Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
I Introduction 1
1 Introduction 3
1.1 Thehistoricalandgeographicalsettingof ABEL’slife . . . . . . . . . . . 4
1.2 Themathematicaltopicsinvolved . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Themesfromearlynineteenth-centurymathematics . . . . . . . . . . . . 7
1.4 Reflectionsonmethodology . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Biographyof NIELS HENRIK ABEL 17
2.1 Childhoodandeducation . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 “Studythemasters” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 TheEuropeantour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 BackinNorway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
i
3 Historicalbackground 39
3.1 Mathematicalinstitutionsandnetworks . . . . . . . . . . . . . . . . . . . 39
3.2 ABEL’spositioninmathematicaltraditions . . . . . . . . . . . . . . . . . 41
3.3 Thestateofmathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 ABEL’slegacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
II “My favorite subject is algebra” 47
4 Thepositionandroleof ABEL’sworkswithinthedisciplineofalgebra 49
4.1 Outlineof ABEL’sresultsandtheirstructuralposition . . . . . . . . . . . 50
4.2 Mathematicalchangeasahistoryofnewquestions . . . . . . . . . . . . . 53
5 Towardsunsolvableequations 57
5.1 Algebraicsolubilitybefore LAGRANGE . . . . . . . . . . . . . . . . . . . . 59
5.2 LAGRANGE’stheoryofequations . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Solubilityofcyclotomicequations . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 Beliefinalgebraicsolubilityshaken . . . . . . . . . . . . . . . . . . . . . . 80
5.5 RUFFINI’sproofsoftheinsolubilityofthequintic . . . . . . . . . . . . . . 84
5.6 CAUCHY’theoryofpermutationsandanewproofof RUFFINI’stheorem 90
5.7 Somealgebraictoolsusedby GAUSS . . . . . . . . . . . . . . . . . . . . . 95
6 Algebraicinsolubilityofthequintic 97
6.1 Thefirstbreakwithtradition . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Outlineof ABEL’sproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 Classificationofalgebraicexpressions . . . . . . . . . . . . . . . . . . . . 101
6.4 ABEL andthetheoryofpermutations . . . . . . . . . . . . . . . . . . . . . 108
6.5 Permutationslinkedtorootextractions . . . . . . . . . . . . . . . . . . . . 110
6.6 Combinationintoanimpossibilityproof . . . . . . . . . . . . . . . . . . . 112
6.7 ABEL and RUFFINI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.8 Limitingtheclassofsolvableequations . . . . . . . . . . . . . . . . . . . 124
6.9 Receptionof ABEL’sworkonthequintic . . . . . . . . . . . . . . . . . . . 125
6.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7 Particularclassesofsolvableequations 141
7.1 SolubilityofAbelianequations . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2 Ellipticfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Theconceptofirreducibilityatwork . . . . . . . . . . . . . . . . . . . . . 157
7.4 Enlargingtheclassofsolvableequations . . . . . . . . . . . . . . . . . . . 160
8 Agrandtheoryinspe 163
8.1 Invertingtheapproachonceagain . . . . . . . . . . . . . . . . . . . . . . 163
8.2 Constructionoftheirreducibleequation . . . . . . . . . . . . . . . . . . . 165
ii
8.3 Refocusingontheequation . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.4 Furtherideasonthetheoryofequations . . . . . . . . . . . . . . . . . . . 176
8.5 Generalresolutionoftheproblemby E. GALOIS . . . . . . . . . . . . . . 181
IIIInterlude: ABEL and the ‘new rigor’ 189
9 Thenineteenth-centurychangeinepistemictechniques 191
10 Towardrigorizationofanalysis 193
10.1 EULER’svisionofanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.2 LAGRANGE’snewfocusonrigor . . . . . . . . . . . . . . . . . . . . . . . 197
10.3 Earlyrigorizationoftheoryofseries . . . . . . . . . . . . . . . . . . . . . 198
10.4 Newtypesofseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
11 CAUCHY’snewfoundationforanalysis 207
11.1 Programmaticfocusonarithmeticalequality . . . . . . . . . . . . . . . . 207
11.2 CAUCHY’sconceptsoflimitsandinfinitesimals . . . . . . . . . . . . . . . 209
11.3 Divergentserieshavenosum . . . . . . . . . . . . . . . . . . . . . . . . . 210
11.4 Meansoftestingforconvergenceofseries . . . . . . . . . . . . . . . . . . 212
11.5 CAUCHY’sproofofthebinomialtheorem . . . . . . . . . . . . . . . . . . 214
11.6 Earlyreceptionof CAUCHY’snewrigor . . . . . . . . . . . . . . . . . . . 218
12 ABEL’sreadingof CAUCHY’snewrigorandthebinomialtheorem 221
12.1 ABEL’scriticalattitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.2 Infinitesimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
12.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
12.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
12.5 ABEL’s“exception” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
12.6 Acuriousreaction:LehrsatzV . . . . . . . . . . . . . . . . . . . . . . . . . 241
12.7 Frompowerseriestoabsoluteconvergence . . . . . . . . . . . . . . . . . 246
12.8 Producttheoremsofinfiniteseries . . . . . . . . . . . . . . . . . . . . . . 251
12.9 ABEL’sproofofthebinomialtheorem . . . . . . . . . . . . . . . . . . . . 254
12.10Aspectsof ABEL’sbinomialpaper . . . . . . . . . . . . . . . . . . . . . . . 260
13 ABEL and OLIVIER onconvergencetests 265
13.1 OLIVIER’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
13.2 ABEL’scounterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
13.3 ABEL’sgeneralrefutation . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
13.4 Morecharacterizationsandtestsofconvergence . . . . . . . . . . . . . . 272
14 Receptionof ABEL’scontributiontorigorization 277
14.1 Receptionof ABEL’srigorization . . . . . . . . . . . . . . . . . . . . . . . 277
iii
14.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
IVElliptic functions and the Paris mémoire 283
15 Ellipticintegralsandfunctions:Chronologyandtopics 285
15.1 Elliptictranscendentalsbeforethenineteenthcentury . . . . . . . . . . . 286
15.2 Thelemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
15.3 LEGENDRE’stheoryofellipticintegrals . . . . . . . . . . . . . . . . . . . . 292
15.4 Leftinthedrawer: GAUSS onellipticfunctions . . . . . . . . . . . . . . . 296
15.5 Chronologyof ABEL’sworkonelliptictranscendentals . . . . . . . . . . 297
16 Theideaofinvertingellipticintegrals 299
16.1 Theimportanceofthelemniscate . . . . . . . . . . . . . . . . . . . . . . . 299
16.2 InversionintheRecherches . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
16.3 Thedivisionproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
16.4 Perspectivesoninversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
17 Stepsintheprocessofcomingto“know”ellipticfunctions 321
17.1 Infiniterepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
17.2 Ellipticfunctionsasratiosofpowerseries . . . . . . . . . . . . . . . . . . 325
17.3 Characterizationof ABEL’srepresentations . . . . . . . . . . . . . . . . . 328
17.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
18 Toolsin ABEL’sresearchonelliptictranscendentals 331
18.1 Transformationtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
18.2 Integrationinlogarithmicterms . . . . . . . . . . . . . . . . . . . . . . . . 339
18.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
19 TheParismemoir 347
19.1 ABEL’sapproachtotheParismemoir . . . . . . . . . . . . . . . . . . . . . 347
19.2 Thecontentsof ABEL’sParisresultanditsproof . . . . . . . . . . . . . . 351
19.3 Additional,tentativeremarkson ABEL’stools . . . . . . . . . . . . . . . . 371
19.4 ThefateoftheParismemoir . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
19.5 ReceptionoftheParismemoir . . . . . . . . . . . . . . . . . . . . . . . . . 376
19.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
20 Generalapproachestoellipticfunctions 379
20.1 ABEL’sversionofageneraltheoryofellipticfunctions . . . . . . . . . . . 379
20.2 Otherwaysofintroducingellipticfunctionsinthenineteenthcentury . . 381
20.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
iv