The search for mathematical roots, 1870-1940: logics, set theories and the foundations of mathematics from Cantor through Russell to Godel

705 Pages·2000·3.35 MB·English

by  I. Grattan-Guinness

The Search for Mathematical Roots,
1870(cid:1)1940
Τηισ παγε ιντεντιοναλλψ λεφτ blank
The Search for
Mathematical Roots,
1870(cid:1)1940
LOGICS, SET THEORIES AND
THE FOUNDATIONS OF MATHEMATICS
FROM CANTOR THROUGH
¨
RUSSELL TO GODEL
I. GRATTAN-GUINNESS
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright(cid:1)2000byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,3MarketPlace,
Woodstock,Oxfordshire,OX201SY
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
Grattan-Guinness,I.
Thesearchformathematicalroots,1870(cid:1)1940:logics,settheoriesandthefoundations
ofmathematicsfromCantorthroughRusselltoG¨odel(cid:1)I.Grattan-Guinness.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN0-691-05857-1Žalk.paper. (cid:2)ISBN0-691-05858-XŽpbk.:alk.paper.
1.Arithmetic(cid:2)Foundations(cid:2)History(cid:2)19thcentury.2.
Arithmetic(cid:2)Foundations(cid:2)History(cid:2)20thcentury.3.Settheory(cid:2)History(cid:2)19thcentury.4.
Settheory(cid:2)History(cid:2)20thcentury.5.Logic,Symbolicandmathematical(cid:2)History(cid:2)19th
century.6.Logic,Symbolicandmathematical(cid:2)History(cid:2)20thcentury.I.Title.
QA248.G6842000
510--dc21 00-036694
ThisbookhasbeencomposedinTimesRoman
Thepaperusedinthispublicationmeetstheminimumrequirementsof
ANSI(cid:1)NISOZ39.48-1992ŽR1997.ŽPermanenceofPaper.
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PrintedintheUnitedStatesofAmerica
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C O N T E N T S
CHAPTER1
Explanations
1.1 Sallies 3
1.2 Scopeandlimitsofthebook 3
1.2.1 Anoutlinehistory 3
1.2.2 Mathematicalaspects 4
1.2.3 Historicalpresentation 6
1.2.4 Otherlogics,mathematicsandphilosophies 7
1.3 Citations,terminologyandnotations 9
1.3.1 Referencesandthebibliography 9
1.3.2 Translations,quotationsandnotations 10
1.4 Permissionsandacknowledgements 11
CHAPTER2
Preludes: Algebraic Logic and Mathematical Analysis up to 1870
2.1 Planofthechapter 14
2.2 ‘Logique’andalgebrasinFrenchmathematics 14
2.2.1 The‘logique’andclarityof‘id´eologie’ 14
2.2.2 Lagrange’salgebraicphilosophy 15
2.2.3 Themanysensesof‘analysis’ 17
2.2.4 Two Lagrangianalgebras:functionalequations
anddifferentialoperators 17
2.2.5 Autonomyforthenewalgebras 19
2.3 SomeEnglishalgebraistsandlogicians 20
2.3.1 ACambridgere(cid:2)i(cid:2)al:the‘AnalyticalSociety’,Lacroix,
andtheprofessingofalgebras 20
2.3.2 Thead(cid:2)ocacyofalgebrasbyBabbage,HerschelandPeacock 20
2.3.3 AnOxfordmo(cid:2)ement:Whatelyandtheprofessingoflogic 22
2.4 ALondonpioneer:DeMorganonalgebrasandlogic 25
2.4.1 Summaryofhislife 25
2.4.2 DeMorgan’sphilosophiesofalgebra 25
2.4.3 DeMorgan’slogicalcareer 26
2.4.4 DeMorgan’scontributionstothefoundationsoflogic 27
2.4.5 Beyondthesyllogism 29
2.4.6 Contretempso(cid:2)er‘thequantificationofthepredicate’ 30
2.4.7 Thelogicoftwo-placerelations,1860 32
2.4.8 Analogiesbetweenlogicandmathematics 35
2.4.9 DeMorgan’stheoryofcollections 36
2.5 ALincolnoutsider:Booleonlogicasappliedmathematics 37
2.5.1 Summaryofhiscareer 37
2.5.2 Boole’s‘generalmethodinanalysis’,1844 39
2.5.3 Themathematicalanalysisoflogic,1847:‘electi(cid:2)esymbols’andlaws 40
2.5.4 ‘Nothing’andthe‘Uni(cid:2)erse’ 42
2.5.5 Propositions,expansiontheorems,andsolutions 43
vi
CONTENTS
2.5.6 Thelawsofthought, 1854:modifiedprinciplesandextendedmethods 46
2.5.7 Boole’snewtheoryofpropositions 49
2.5.8 ThecharacterofBoole’ssystem 50
2.5.9 Boole’ssearchformathematicalroots 53
2.6 Thesemi-followersofBoole 54
2.6.1 SomeinitialreactionstoBoole’stheory 54
2.6.2 ThereformulationbyJe(cid:2)ons 56
2.6.3 Je(cid:2)ons(cid:2)ersusBoole 59
2.6.4 FollowersofBooleand(cid:1)orJe(cid:2)ons 60
2.7 Cauchy,Weierstrassandtheriseofmathematicalanalysis 63
2.7.1 Differenttraditionsinthecalculus 63
2.7.2 Cauchyandthe EcolePolytechnique 64
2.7.3 ThegradualadoptionandadaptationofCauchy’snewtradition 67
2.7.4 TherefinementsofWeierstrassandhisfollowers 68
2.8 Judgementandsupplement 70
2.8.1 Mathematicalanalysis(cid:2)ersusalgebraiclogic 70
2.8.2 TheplacesofKantandBolzano 71
CHAPTER3
Cantor: Mathematics as Mengenlehre
3.1 Prefaces 75
3.1.1 Planofthechapter 75
3.1.2 Cantor’scareer 75
3.2 Thelaunchingofthe Mengenlehre,1870(cid:1)1883 79
3.2.1 Riemann’sthesis:therealmofdiscontinuousfunctions 79
3.2.2 Heineontrigonometricseriesandtherealline,1870(cid:1)1872 81
3.2.3 Cantor’sextensionofHeine’sfindings,1870(cid:1)1872 83
3.2.4 Dedekindonirrationalnumbers,1872 85
3.2.5 Cantoronlineandplane,1874(cid:1)1877 88
3.2.6 Infinitenumbersandthetopologyoflinearsets,1878(cid:1)1883 89
3.2.7 TheGrundlagen, 1883:theconstructionofnumber-classes 92
3.2.8 TheGrundlagen:thedefinitionofcontinuity 95
3.2.9 ThesuccessortotheGrundlagen, 1884 96
3.3 Cantor’s Actamathematicaphase,1883(cid:1)1885 97
3.3.1 Mittag-LefflerandtheFrenchtranslations,1883 97
3.3.2 Unpublishedandpublished‘communications’,1884(cid:1)1885 98
3.3.3 Order-typesandpartialderi(cid:2)ati(cid:2)esinthe‘communications’ 100
3.3.4 CommentatorsonCantor,1883(cid:1)1885 102
3.4 Theextensionofthe Mengenlehre,1886(cid:1)1897 103
3.4.1 Dedekind’sde(cid:2)elopingsettheory,1888 103
3.4.2 Dedekind’schainsofintegers 105
3.4.3 Dedekind’sphilosophyofarithmetic 107
3.4.4 Cantor’sphilosophyoftheinfinite,1886(cid:1)1888 109
3.4.5 Cantor’snewdefinitionsofnumbers 110
3.4.6 Cardinalexponentiation:Cantor’sdiagonalargument,1891 110
3.4.7 Transfinitecardinalarithmeticandsimplyorderedsets,1895 112
3.4.8 Transfiniteordinalarithmeticandwell-orderedsets,1897 114
3.5 OpenandhiddenquestionsinCantor’s Mengenlehre 114
3.5.1 Well-orderingandtheaxiomsofchoice 114
vii
CONTENTS
3.5.2 WhatwasCantor’s‘Cantor’scontinuumproblem’? 116
3.5.3 ‘‘Paradoxes’’andtheabsoluteinfinite 117
3.6 Cantor’sphilosophyofmathematics 119
3.6.1 Amixedposition 119
3.6.2 (No)logicandmetamathematics 120
3.6.3 Thesupposedimpossibilityofinfinitesimals 121
3.6.4 AcontrastwithKronecker 122
3.7 Concludingcomments:thecharacterofCantor’sachievements 124
CHAPTER4
Parallel Processes in Set Theory, Logics and Axiomatics, 1870s(cid:1)1900s
4.1 Plansforthechapter 126
4.2 ThesplittingandsellingofCantor’s Mengenlehre 126
4.2.1 Nationalandinternationalsupport 126
4.2.2 Frenchinitiati(cid:2)es,especiallyfromBorel 127
4.2.3 Couturatoutliningtheinfinite,1896 129
4.2.4 Germaninitiati(cid:2)esfromKlein 130
4.2.5 GermanproofsoftheSchro¨der-Bernsteintheorem 132
4.2.6 PublicityfromHilbert,1900 134
4.2.7 Integralequationsandfunctionalanalysis 135
4.2.8 Kempeon‘mathematicalform’ 137
4.2.9 Kempe(cid:2)who? 139
4.3 Americanalgebraiclogic:Peirceandhisfollowers 140
4.3.1 Peirce,publishedandunpublished 141
4.3.2 InfluencesonPeirce’slogic:father’salgebras 142
4.3.3 Peirce’sfirstphase:Booleanlogicandthecategories,1867(cid:1)1868 144
4.3.4 Peirce’s(cid:2)irtuosotheoryofrelati(cid:2)es,1870 145
4.3.5 Peirce’ssecondphase,1880:thepropositionalcalculus 147
4.3.6 Peirce’ssecondphase,1881:finiteandinfinite 149
4.3.7 Peirce’sstudents,1883:duality,and‘Quantifying’aproposition 150
4.3.8 Peirceon‘icons’andtheorderof‘quantifiers’,1885 153
4.3.9 ThePeirceansinthe1890s 154
4.4 Germanalgebraiclogic:fromtheGrassmannstoSchr¨oder 156
4.4.1 TheGrassmannsonduality 156
4.4.2 Schro¨der’sGrassmannianphase 159
4.4.3 Schro¨der’sPeircean‘lectures’onlogic 161
4.4.4 Schro¨der’sfirst(cid:2)olume,1890 161
4.4.5 Partofthesecond(cid:2)olume,1891 167
4.4.6 Schro¨der’sthird(cid:2)olume,1895:the‘logicofrelati(cid:2)es’ 170
4.4.7 PeirceonandagainstSchro¨derinThemonist,1896(cid:1)1897 172
4.4.8 Schro¨deronCantorianthemes,1898 174
4.4.9 ThereceptionandpublicationofSchro¨derinthe1900s 175
4.5 Frege:arithmeticaslogic 177
4.5.1 FregeandFrege(cid:3) 177
4.5.2 The‘concept-script’calculusofFrege’s‘purethought’,1879 179
4.5.3 Frege’sargumentsforlogicisingarithmetic,1884 183
4.5.4 Kerry’sconceptionofFregeanconceptsinthemid1880s 187
4.5.5 Importantnewdistinctionsintheearly1890s 187
4.5.6 The‘fundamentallaws’oflogicisedarithmetic,1893 191
viii
CONTENTS
4.5.7 Frege’sreactionstoothersinthelater1890s 194
4.5.8 More‘fundamentallaws’ofarithmetic,1903 195
4.5.9 Frege,KorseltandThomaeonthefoundationsofarithmetic 197
4.6 Husserl:logicasphenomenology 199
4.6.1 AfollowerofWeierstrassandCantor 199
4.6.2 Thephenomenological‘philosophyofarithmetic’,1891 201
4.6.3 Re(cid:2)iewsbyFregeandothers 203
4.6.4 Husserl’s‘logicalin(cid:2)estigations’,1900(cid:1)1901 204
4.6.5 Husserl’searlytalksinGo¨ttingen,1901 206
4.7 Hilbert:earlyproofandmodeltheory,1899(cid:1)1905 207
4.7.1 Hilbert’sgrowingconcernwithaxiomatics 207
4.7.2 Hilbert’sdifferentaxiomsystemsforEuclideangeometry,1899(cid:1)1902 208
4.7.3 FromGermancompletenesstoAmericanmodeltheory 209
4.7.4 Frege,HilbertandKorseltonthefoundationsofgeometries 212
4.7.5 Hilbert’slogicandprooftheory,1904(cid:1)1905 213
4.7.6 Zermelo’slogicandsettheory,1904(cid:1)1909 216
CHAPTER5
Peano: the Formulary of Mathematics
5.1 Prefaces 219
5.1.1 Planofthechapter 219
5.1.2 Peano’scareer 219
5.2 Formalisingmathematicalanalysis 221
5.2.1 Impro(cid:2)ingGenocchi,1884 221
5.2.2 De(cid:2)elopingGrassmann’s‘geometricalcalculus’,1888 223
5.2.3 Thelogisticofarithmetic,1889 225
5.2.4 Thelogisticofgeometry,1889 229
5.2.5 Thelogisticofanalysis,1890 230
5.2.6 Bettazzionmagnitudes,1890 232
5.3 The Ri(cid:2)ista:Peanoandhisschool,1890(cid:1)1895 232
5.3.1 The‘societyofmathematicians’ 232
5.3.2 ‘Mathematicallogic’,1891 234
5.3.3 De(cid:2)elopingarithmetic,1891 235
5.3.4 Infinitesimalsandlimits,1892(cid:1)1895 236
5.3.5 Notationsandtheirrange,1894 237
5.3.6 Peanoondefinitionbyequi(cid:2)alenceclasses 239
5.3.7 Burali-Forti’stextbook,1894 240
5.3.8 Burali-Forti’sresearch,1896(cid:1)1897 241
5.4 The Formulaireandthe Ri(cid:2)ista,1895(cid:1)1900 242
5.4.1 Thefirsteditionofthe Formulaire,1895 242
5.4.2 Towardsthesecondeditionofthe Formulaire,1897 244
5.4.3 Peanoontheeliminabilityof‘the’ 246
5.4.4 Frege(cid:2)ersusPeanoonlogicanddefinitions 247
5.4.5 Schro¨der’ssteamships(cid:2)ersusPeano’ssailingboats 249
5.4.6 Newpresentationsofarithmetic,1898 251
5.4.7 Padoaonclasshood,1899 253
5.4.8 Peano’snewlogicalsummary,1900 254
5.5 PeanistsinParis,August1900 255
5.5.1 AnItalianFridaymorning 255
ix
CONTENTS
5.5.2 Peanoondefinitions 256
5.5.3 Burali-Fortiondefinitionsofnumbers 257
5.5.4 Padoaondefinabilityandindependence 259
5.5.5 Pierionthelogicofgeometry 261
5.6 Concludingcomments:thecharacterofPeano’sachievements 262
5.6.1 Peano’slittledictionary,1901 262
5.6.2 Partlygraspedopportunities 264
5.6.3 Logicwithoutrelations 266
CHAPTER6
Russell’s Way In: From Certainty to Paradoxes, 1895(cid:1)1903
6.1 Prefaces 268
6.1.1 Plansfortwochapters 268
6.1.2 Principalsources 269
6.1.3 RussellasaCambridgeundergraduate,1891(cid:1)1894 271
6.1.4 Cambridgephilosophyinthe1890s 273
6.2 Three philosophical phases in the foundation of mathematics,
1895(cid:1)1899 274
6.2.1 Russell’sidealistaxiomaticgeometries 275
6.2.2 Theimportanceofaxiomsandrelations 276
6.2.3 Apairof pasdedeux withParis:CouturatandPoincar´eongeometries 278
6.2.4 TheemergenceofWhitehead,1898 280
6.2.5 TheimpactofG.E.Moore,1899 282
6.2.6 Threeattemptedbooks,1898(cid:1)1899 283
6.2.7 Russell’sprogresswithCantor’sMengenlehre, 1896(cid:1)1899 285
6.3 Fromneo-Hegelianismtowards‘Principles’,1899(cid:1)1901 286
6.3.1 Changingrelations 286
6.3.2 Spaceandtime,absolutely 288
6.3.3 ‘PrinciplesofMathematics’,1899(cid:1)1900 288
6.4 ThefirstimpactofPeano 290
6.4.1 The ParisCongressofPhilosophy,August1900: Schro¨der
(cid:2)ersusPeanoon‘the’ 290
6.4.2 Annotatingandpopularisingintheautumn 291
6.4.3 DatingtheoriginsofRussell’slogicism 292
6.4.4 Draftingthelogicofrelations,October1900 296
6.4.5 Part3of Theprinciples, No(cid:2)ember1900:quantityandmagnitude 298
6.4.6 Part4,No(cid:2)ember1900:orderandordinals 299
6.4.7 Part5,No(cid:2)ember1900:thetransfiniteandthecontinuous 300
6.4.8 Part6,December1900:geometriesinspace 301
6.4.9 Whiteheadon‘thealgebraofsymboliclogic’,1900 302
6.5 Convolutingtowardslogicism,1900(cid:1)1901 303
6.5.1 Logicismasgeneralisedmetageometry,January1901 303
6.5.2 ThefirstpaperforPeano,February1901:relationsandnumbers 305
6.5.3 CardinalarithmeticwithWhiteheadandRussell,June1901 307
6.5.4 ThesecondpaperforPeano,March(cid:1)August1901:settheorywithseries 308
6.6 From‘fallacy’to‘contradiction’,1900(cid:1)1901 310
6.6.1 RussellonCantor’s‘fallacy’,No(cid:2)ember1900 310
6.6.2 Russell’sswitchtoa‘contradiction’ 311