Frege’s theorem

322 Pages·2014·2.238 MB·English

by  Frege, Gottlob, Heck, Richard G.

Frege’s Theorem
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Frege’s Theorem
Richard G. Heck, Jr
CLARENDON PRESS • OXFORD
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Firstpublished2011
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ISBN978–0–19–969564–5
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Contents
Preface ix
EditorialNotes xi
OriginoftheChapters xii
1 Frege’sTheorem: AnOverview 1
1.1 FregeonFrege’sTheorem 9
1.2 TheCaesarProblem 13
1.3 WhatDoesHPHaveToDoWithArithmetic? 27
1.4 LogicismandNeo-Logicism 37
2 TheDevelopmentofArithmeticinFrege’s
GrundgesetzederArithmetik 40
2.1 BasicLawVinGrundgesetze 41
2.2 HPandFregeanArithmetic 45
2.3 Frege’sDerivationoftheAxiomsofArithmetic 47
2.4 Frege’sDerivationoftheAxiomsofArithmetic,
continued 51
2.5 AnElegantProofthatEveryNumberhasaSuccessor 53
2.6 Frege’sAxiomatizationofArithmetic 63
2.7 Closing 64
Postscript 66
3 DieGrundlagenderArithmetik§§82–83 69
Appendix: CounterpartsinGrundgesetzeofSome
PropositionsofDieGrundlagen 86
Postscript 87
4 Frege’sPrinciple 90
4.1 NumbersasExtensionsofConcepts 92
4.2 TheImportanceofHPinFrege’sPhilosophyof
Arithmetic 95
4.3 TheRoleofBasicLawVinFrege’sDerivationof
Arithmetic 97
4.4 Frege’sDerivationsofHP 99
4.5 HPversusFrege’sPrinciple 103
4.6 Frege’sPrincipleandtheExplicitDefinition 104
4.7 TheCaesarProblemRevisited 105
4.8 Closing 107
Postscript 108
vi Contents
5 JuliusCaesarandBasicLawV 111
5.1 TheCaesarProblem 113
5.2 TheCaesarProbleminGrundgesetze 115
5.3 TheCaesarProblemandtheApprehensionofLogical
Objects 118
5.4 Closing 125
6 TheJuliusCaesarObjection 127
6.1 WhytheCaesarObjectionHasToBeTakenSeriously 131
6.2 TheCaesarObjectionandtheFeasibilityofthe
LogicistProject 136
6.3 AvoidingtheCaesarObjection 147
6.4 Closing 152
7 Cardinality,Counting,andEquinumerosity 156
7.1 TechnicalPreliminaries 161
7.2 FregeandHusserl 163
7.3 CountingandCardinality 168
7.4 CountingandAscriptionsofNumber 172
7.5 Closing 176
8 SyntacticReductionism 180
8.1 MotivatingNominalism 182
8.2 TakingReductionismSeriously 187
8.3 TheIneliminabilityofNamesofAbstractObjects 193
8.4 WhereDoWeGoFromHere? 199
9 TheExistence(andNon-existence)ofAbstractObjects 200
9.1 TwoProblems 200
9.2 SemanticReductionismandProjectiblePredicates 206
9.3 Ideology,Existence,andAbstractObjects 216
9.4 TheJuliusCaesarProblem 223
10 OntheConsistencyofSecond-orderContextual
Definitions 227
Postscript 230
11 FinitudeandHume’sPrinciple 237
11.1 TheSystems 241
11.2 OnthePhilosophicalSignificanceofTheseResults 243
11.3 TheRelativeStrengthsoftheSystems 249
11.4 PAFisequivalenttoFAF 250
11.5 Closing 259
Postscript 260
Contents vii
12 ALogicforFrege’sTheorem 267
12.1 Predecession 271
12.2 AncestralLogic 274
12.3 SchematainSchematicLogic: ADigression 280
12.4 ArchéLogic 282
12.5 Frege’sTheorem 287
12.6 PhilosophicalConsiderations 290
Appendix: ProofofBegriffsschrift,Proposition124 296
Bibliography 297
Index 306
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Preface
Thisbookhasbeenalongtimecoming. Thepaperscollectedhererepre-
sentthe fruits ofatwo decadeinvestigationofphilosophical, historical,
andtechnicalquestionsrelatedtoFrege’sTheorem. WhileIcontinueto
think about these questions from time to time, my work on Frege has
movedinotherdirections,andIfeelasifIhavesaidmostofwhatIhave
tosayaboutFrege’sphilosophyofarithmetic. Thatmakesitagoodtime
tocollectthesepapers. Nonetheless,theprocessofrevisingthemforpub-
licationhasrenewedmypassionforproblemsI’dalmostforgottenlosing
sleepover. Ihadasurprsingamountofpurefunwritingtheoverviewthat
appearsasChapter1,aswellasthepostscriptsthatrecountchangesof
mindorrespondtopublishedcriticismsofmywork. Thankstothosewho
havepaidattentiontoit.
When I re-read these papers at the start of this project, I was sur-
prised by how well they fit together. The earliest of them was drafted
before I had my first job; the latest was finished only quite recently. So
there are plenty of changes of direction, emphasis, and view. But there
are,atthesametime,dominantthemesthatre-appearindifferentforms
indifferentplaces,andChapter1isintendedtogivethereaderasense
for what some of these are. The whole, I believe, is certainly greater
than the sum of the parts, and that is what makes it worth reprinting
thepaperstogether. Ihopethateventhosewhohavealreadyreadthem
separatelywillfindthatreadingthemtogetherrevealssomethingnew.
While working on these topics, I have been privileged to have the
support and encouragement of many friends, students, and colleagues,
and I have benefitted greatly from their comments and criticisms. So a
bigthankstoLawrenceAbrams,AndrewBoucher,SylvainBromberger,
TylerBurge,JohnBurgess,EmilyCarson,DickCartwright,KarinCase,
PeterClark,RoyCook,TonyCorsentino,AnnetteDemby,BillDemopou-
los, Mic Detlefsen, Burt Dreben, Philip Ebert, Kate Elgin, Delia Graff
Fara,FernandoFerreira,JanetFolina,MichaelGlanzberg,WarrenGold-
farb,AliceGraham-Brown,DaveGrishaw-Jones,StevenGross,BobHale,
JimHigginbotham,ChrisHill,DavidHunter,DarrylJung,JinhoKang,
KathrinKoslicki,MichaelKremer,ThomasKuhn,ØysteinLinnebo,Mary
Luti, Josep Macià-Fabrega, Lisa Marino, Mathieu Marion, Robert May,
Ute Molitor, Charles Parsons, Paul Pietroski, Carl Posy, Ian Proops,
HilaryPutnam,AgustínRayo,OfraRechter,MichaelRescorla,Thomas
Ricketts,GideonRosen,MarcusRossberg,TimScanlon,JoshSchechter,
RichardSchwartz,SallySedgwick,LisaSereno,BrettSherman,Stewart
Shapiro,OriSimchen,AlisonSimmons,DanSmith,GiselaStriker,Bob
Stalnaker, Zoltan Gendler Szabó, Bill Tait, Jamie Tappenden, Gabriel