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The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel PDF

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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. www.pup.princeton.edu PrintedintheUnitedStatesofAmerica 10987654321 Disclaimer: Some images in the original version of this book are not available for inclusion in the eBook. 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

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