Philosophical perspectives on infinity

336 Pages·2006·1.815 MB·English

by  Oppy, Graham

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PhilosophicalPerspectivesonInfinity
Thisbookisanexplorationofphilosophicalquestionsaboutinfinity.
Graham Oppy examines how the infinite lurks everywhere, both in
scienceandinourordinarythoughtsabouttheworld.Healsoana-
lyzesthemanypuzzlesandparadoxesthatfollowinthetrainofthe
infinite. Even simple notions, such as counting, adding, and maxi-
mizing, present serious difficulties. Other topics examined include
thenatureofspaceandtime,infinitiesinphysicalscience,infinities
intheoriesofprobabilityanddecision,thenatureofpart/wholerela-
tions, mathematical theories of the infinite, and infinite regression
andprinciplesofsufficientreason.
GrahamOppyisAssociateDeanofResearchintheFacultyofArtsat
MonashUniversityinAustralia.HeistheauthorofOntologicalArgu-
mentsandBeliefinGodandtheforthcomingArgumentsaboutGods,and
he has received fellowships from the Australian Research Council.
HeisanassociateeditoroftheAustralasianJournalofPhilosophyand
sitsontheeditorialboardsofPhilo,ReligiousStudies,andSophia.
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Philosophical Perspectives on Infinity
GRAHAM OPPY
MonashUniversity
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cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo
Cambridge University Press
TheEdinburghBuilding,Cambridgecb22ru,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521860673
©GrahamOppy2006
Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof
relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace
withoutthewrittenpermissionofCambridgeUniversityPress.
Firstpublishedinprintformat 2006
isbn-13 978-0-511-19110-7 eBook(EBL)
isbn-10 0-511-19110-3
eBook(EBL)
isbn-13 978-0-521-86067-3 hardback
isbn-10 0-521-86067-9 hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurls
forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot
guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
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Contents
Preface page ix
Acknowledgments xv
Introduction 1
1. BeginningsandPuzzles 7
2. MathematicalPreliminaries 20
2.1 SetTheory 20
2.2 Numbers 23
2.3 Cantor’sParadise 24
2.4 StandardAnalysis 31
2.5 Non-StandardNumbers 34
3. SomeCasesDiscussed 48
3.1 Al-Ghazali’sProblem 49
3.2 Hilbert’sHotel 51
3.3 Craig’sLibrary 53
3.4 TristramShandy 56
3.5 CountingfromInfinity 59
3.6 InfiniteParalysis 63
3.7 Stick 66
3.8 Spaceship 69
3.9 Thomson’sLamp 71
3.10 Black’sMarbleShifter 72
3.11 PiMachine 74
3.12 GoldbachMachine 77
3.13 RossUrn 79
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vi Contents
3.14 DeafeningPeals 81
3.15 Invisibility 83
3.16 InfinityMob 85
3.17 String 86
3.18 SomeConcludingRemarks 88
4. Space,Time,andSpacetime 90
4.1 Zeno’sParadoxes 91
4.2 Gru¨nbaum’sMetricalPuzzle 99
4.3 Skyrms’sMeasurePuzzle 106
4.4 Points,Regions,andFiniteLattices 110
4.5 FirstKantianAntinomy 115
4.6 InfinityMachinesinRelativisticSpacetimes 123
4.7 Singularities 128
4.8 ConcludingRemarks 131
5. PhysicalInfinities 133
5.1 HotterthanInfiniteTemperatures 134
5.2 ExtensiveMagnitudes 136
5.3 InfiniteExtensiveMagnitudes? 138
5.4 Renormalisation 141
5.5 TheDarkNightSky 146
5.6 SomeMoreGeneralConsiderations 150
5.7 Possibilities 153
5.8 PhysicalPossibilities 157
6. ProbabilityandDecisionTheory 160
6.1 Probabilities 160
6.2 AdditivityPrinciples 165
6.3 DecisionTheory 170
6.4 ApproachingInfiniteDecisionTheory 175
6.5 InfiniteUtilityStreams 177
6.6 InfiniteDecisionRules 181
6.7 TwoEnvelopes 185
6.8 St.PetersburgGame 194
6.9 HeavenandHell 196
6.10 ConcludingRemarks 199
7. Mereology 201
7.1 SecondKantianAntinomy 202
7.2 SomePostulates 204
7.3 Forrest’sChallenge 208
7.4 TakingStock 213
7.5 AtomsandIndiscernibleParticles 215
7.6 VaguenessandInfiniteDivisibility 219
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Contents vii
7.7 Continuity 224
7.8 OurUniverse 226
8. SomePhilosophicalConsiderations 231
8.1 Distinctions 232
8.2 PhilosophiesofPureMathematics 242
8.3 KnowingtheInfinite 247
8.4 PuttingClassicalMathematicsFirst 250
8.5 ExtrapolationfromFiniteMathematics 253
8.6 SomeNotesaboutSetTheory 256
8.7 PhilosophiesofAppliedMathematics 259
8.8 Skolem’sParadox 266
8.9 Infinitesimals 270
8.10 ConcludingRemarks 274
9. InfiniteRegressandSufficientReason 275
9.1 StrongPrinciplesofSufficientReason 275
9.2 Self-ExplanationandSufficientReason 277
9.3 NecessitationandSufficientReason 278
9.4 ProblemsforStrongPrinciplesofSufficientReason 280
9.5 WeakerPrinciplesofSufficientReason 281
9.6 InfiniteRegresses 285
9.7 ConcludingRemarks 290
Conclusion 291
References 297
Index 311
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