Alexander R Pruss Infinity, Causation, And Paradox Oxford University Press, USA ( 2018)

217 Pages·1999·3.2 MB·english-handwritten

by  Alexander Pruss

OUPCORRECTEDPROOF – FINAL,19/6/2018,SPi
Infinity, Causation,
and Paradox
Alexander R. Pruss
1
OUPCORRECTEDPROOF – FINAL,19/6/2018,SPi
3
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OUPCORRECTEDPROOF – FINAL,19/6/2018,SPi
Contents
ListofFigures xi
Acknowledgments xiii
1. Infinity,Paradox,andMathematics 1
1. ParadoxandCausalFinitism 1
2. SomeMathematicalandLogicalNotes 4
3. Modality 7
3.1 Metaphysicalpossibilityandnecessity 7
3.2 Rearrangementprinciples 7
3.2.1 Defeasibility 7
3.2.2 Causalpowers 9
4. Finitism:AnAlternateHypothesis 10
4.1 Timeandfinitism 10
4.2 Non-causalparadoxes:Anadvantage? 11
4.3 Mathematics:Adisadvantage 13
4.3.1 Infinitelymanyprimes 13
4.3.2 Potentialinfinity 15
4.3.3 ∗If-thenism 15
4.4 Futureinfinities 17
5. ∗DefiningtheFiniteandtheCountable 18
5.1 Thefinite 18
5.2 Acceptablemodelsfortheaxiomsofarithmetic 20
6. Evaluation 23
Appendix:∗CountingFutureThings 23
2. InfiniteRegresses 25
1. HowtoViolateCausalFinitism 25
2. InfiniteCausalRegresses 26
3. Type(i):UncausedRegresses 27
3.1 Viciousness 27
3.2 ViciousregressesandtheHume–EdwardsPrinciple 29
3.3 Regressesandexplanatoryloops 30
4. Type(ii):CausationPassingthroughInfinitelyManySteps 32
5. Type(iii):OutsideCauseDirectlyCausingEachItem 33
5.1 Options 33
5.2 Regresseswithoutsideoverdetermination 35
6. ∗AnalogywithAxiomofRegularity 36
7. Evaluation 37
Appendix:∗TwoKindsofViolationsofCausalFinitism 37
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vi contents
3. SupertasksandDeterministicParadoxes 40
1. Introduction 40
2. Thomson’sLampRevisited 40
2.1 Introduction 40
2.2 Causalfinitism 40
2.3 Non-standardanalysis 41
2.4 SpecialRelativity 42
2.5 Benacerraf’ssolutionandthePrincipleofSufficientReason 43
2.6 Twocounterfactuals 44
2.7 Evaluation 46
3. GrimReapers 46
3.1 Introduction 46
3.2 Causalfinitism 47
3.3 Theabsurdconclusionobjection 48
3.4 Arearrangementobjection 49
3.5 Themereologicalobjection 50
3.5.1 Fusion 50
3.5.2 Necessaryemergenceoforganicwholes 52
3.6 Uncausedlighting 52
3.6.1 Objection 52
3.6.2 TheCausalPrincipleistrue 53
3.6.3 Isthelamplightingreallyuncaused? 54
3.6.4 Amysteriouscorrelation 55
3.7 Discretetime 55
3.8 Evaluation 56
4. InfiniteNewtonianUniverses 56
4.1 Anargumentagainstcausalfinitismandariposte 56
4.2 Smullyan’srod 58
4.3 Theconditional 60
5. AnotherEternalLife 60
6. TimeTravelandCausalLoops 61
6.1 Grandfathersandtogglers 61
6.2 Timetravelandbackwardscausationwithoutcausalloops 63
7. Evaluation 63
4. ParadoxicalLotteries 64
1. Introduction 64
2. CountablyInfiniteFairLotteries 64
2.1 Background 64
2.2 Expectedsurprise 65
2.3 Aguessinggame 66
2.4 Symmetry 66
2.4.1 Symmetryandlotteries 66
2.4.2 ∗Symmetryandexpectedutility 68
2.5 Bayesianmanipulation 71
2.5.1 Theparadox 71
2.5.2 ∗Aswitchoverpoint? 74
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contents vii
2.5.3 ∗Countableadditivityandconglomerability 75
2.6 Improvingeveryone’schances 77
3. ConstructingParadoxicalLotteries 79
3.1 Fairnessandparadoxicality 79
3.2 Luckycoin-flipsequences 79
3.3 Whatitistoconstructacountablyinfinitefairlottery 81
3.4 ∗Coin-flipsandtheAxiomofChoice 83
3.5 Randomwalks 85
4. Objections 86
4.1 Infinitelotteriesanduniformdistributions 86
4.1.1 Theproblem 86
4.1.2 ResponseI:Nocontinuousdistributions 87
4.1.3 ResponseII:Measurementofinfiniteprecisiondata 88
4.1.4 ResponseIII:TheuseoftheAxiomofChoice 88
4.2 ∗Anon-normalizablequantumstate 89
4.3 Limitationsonourreasoning 90
5. Evaluation 91
5. ProbabilityandDecisionTheory 93
1. Introduction 93
2. GuessingwithFinitelyManyErrors 93
2.1 Doingalittlebetterthanonecan 93
2.2 Acontradiction 95
2.3 Doingmuchbetterthanonecan 96
2.4 ∗Constructionofstrategyguaranteeingatmostfinitelymanyerrors 98
2.5 Amultipersonalsynchronicversion 98
2.5.1 Anangelicannouncement 98
2.5.2 Anobjectionandatweak 100
2.5.3 ∗Makingtheparadoxrobust 101
2.6 Aparody? 102
2.6.1 Thestory 102
2.6.2 Evaluatingtheparody 103
3. Satan’sApple 106
3.1 Thestory 106
3.2 Synchronicversion 107
3.3 Diachronicversion 108
3.4 Objection:Scores,desires,andpromises 108
3.5 Evaluation 110
4. Beam’sParadox 111
4.1 ∗Themathematicalformulation 111
4.2 ∗Synchronicversion 112
4.3 ∗Diachronicinfinitefutureversion 113
4.4 ∗Diachronicsupertaskversion 114
4.5 EvaluationofBeam’sparadox 114
5. EvaluationofDecision-TheoreticParadoxes 114
Appendix:∗ProofoftheTheoremfromSection2.1 115
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viii contents
6. TheAxiomofChoiceMachine 117
1. LessTechnicalIntroduction 117
2. ∗TheAxiomofChoiceforCountableCollectionsofReals 119
3. ∗ParadoxesofACCR 120
3.1 Die-guessinggames 120
3.2 Non-measurablesets 120
3.3 Banach–Tarskiparadox 121
4. ∗AnArgumentforACCR 122
5. ∗AChoiceMachine 125
5.1 Strangemathematicsandparadox 125
5.2 Coin-flipsandDutchBooks 129
5.3 HowtoconstructaChoiceMachine 130
5.3.1 Angels 130
5.3.2 Afour-dimensionalmachine 131
5.3.2.1 Makingthemachine 131
5.3.2.2 Usingthemachine 133
5.3.2.3 Causalinfinitismandverifyingthemachine’smatch 134
5.3.3 Athree-dimensionalmachine 136
5.3.4 ∗∗IsACneeded? 136
5.3.5 Luck 137
6. Evaluation 137
Appendix:∗∗DetailsofCoin-TossRearrangement 138
7. Refinement,Alternatives,andExtensions 140
1. Introduction 140
2. Refinement 140
2.1 Eventandtropeindividuation 140
2.2 Historiesgeneratedbypartialcausalrelations 142
2.3 AcloserlookatGrimReapers 143
2.4 Objectionstocausalfinitisminvolvingpartialcausation 147
2.5 Absencesandomissions 148
3. SomeCompetitorstoCausalFinitism 150
3.1 Finitism 150
3.2 Noinfiniteregresses 151
3.3 Nopastinfinities 151
3.4 Noinfiniteintensivemagnitudes 153
3.4.1 Thebasictheory 153
3.4.2 Someinfiniteintensivemagnitudes 154
3.4.2.1 Centerofmassandmomentsofinertia 154
3.4.2.2 Mentallife 154
3.4.2.3 Blackholes 155
3.4.2.4 Particles 155
3.4.3 Huemer’sintensivemagnitudes 156
3.4.3.1 Speed,Thomson’sLamp,andHilbert’sHotel 156
3.4.3.2 ∗Smullyan’srod 157
3.4.3.3 Immaterialminds 158
3.4.4 Evaluation 159
3.5 Noroom 159
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contents ix
4. WhyisCausalFinitismTrue? 161
4.1 Thequestion 161
4.2 Someexplanatorysuggestions 161
5. FurtherExtensions 162
5.1 Causalloops 162
5.2 Explanatoryrelations 164
6. OverallEvaluation 165
8. DiscreteTimeandSpace 167
1. Introduction 167
2. CausalFinitismandDiscreteness 167
2.1 Thebasicargument 167
2.2 Fromdiscretetimetodiscretespace? 167
3. TwoKindsofDiscreteness 168
3.1 Subdivisibilityandfixeity 168
3.2 RefiningtheAristotelianpicture 169
3.2.1 AnobjectiontoAristoteliandiscreteness 169
3.2.2 Internalandexternaldiscreteness 170
4. Physics 172
4.1 Anobjectiontocausalfinitism 172
4.2 Causationandphysics 172
4.3 Quantumcollapse 174
4.3.1 Somebackground 174
4.3.2 Causation 176
4.3.3 Backtodiscretetime 177
5. FieldsandDiscreteSpace 178
6. Evaluation 180
9. AFirstCause 181
1. Introduction 181
2. AnUncausedCause 181
2.1 Thequickargument 181
2.2 Towardsanecessarybeing 182
2.3 SupportfortheCausalPrinciple 183
2.4 TheKala¯margument 184
3. CompatibilitywithTheism? 184
3.1 Theism 184
3.2 Divinemotivation 184
3.3 Divineknowledge 186
3.4 Divineaction 188
3.5 Limitsonmetaphysicalpossibility 191
4. Evaluation 192
10. Conclusions 193
References 195
Index 201
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List of Figures
Allillustrationsinthisvolumehavebeencompiledbytheauthor.
1.1 Thomson’sLamp 1
1.2 Correspondencebetweennaturalnumbersandevennaturalnumbers 5
1.3 Intervalnotation 6
2.1 Thetwowaysofviolatingcausalfinitism:regress(left)andinfinite
cooperation(right) 25
2.2 Thetestimonyofunicornexperts 28
2.3 Atheisticnon-viciousregress? 33
2.4 Here,I ={a ,a ,a },L ={e,a ,a ,a },andM ={a ,a ,...} 38
1 1 2 3 1 1 2 3 2 4 5
3.1 SomerepresentativeGrimReaperactivations 47
3.2 SomerepresentativereversedGrimReaperactivations 48
3.3 Smullyan’srodwithexponentiallydecreasingdensityandhence
exponentiallydecreasingquasi-gravitationalpull 58
4.1 Aluckycasewherethelotteryworks,withthewinnerbeingthenumber2 79
4.2 Atraverseofatwo-dimensionalarray 80
6.1 The(i)–(ii)bettingportfoliothatyoushouldbehappytopayadollar
for.Thevolumeofeachsphereis1/100thofthatofthecube 126
(cid:3) (cid:3)
6.2 The(i)–(ii)bettingportfoliothatyoushouldbehappytopayadollar
foriftheargumentworks 127
(cid:3)(cid:3) (cid:3)(cid:3)
6.3 The(i )–(ii )bettingportfolioyoushouldbehappytoacceptforfree 128
6.4 AsliceofaChoiceMachine 132
7.1 Benardete’sBoards 147
7.2 Fourparadigmaticviolationsof(8) 163
OUPCORRECTEDPROOF – FINAL,19/6/2018,SPi
Acknowledgments
IamespeciallygratefultoIanSlorachwhogavememanyveryinsightfulandhelpful
commentsandcriticismsbothonmybloggedargumentsbeforeIstartedwritingthis
bookandwhileIwascommittingmaterialtothebookdraft’sGitHubrepository.Iam
alsoparticularlygratefultoMiguelBerasategui,BlaiseBlain,TrentDougherty,Kenny
Easwaran, Richard Gale, Alan Hájek, James Hawthorne, Robert Koons, Jonathan
Kvanvig, Arthur Paul Pederson, Philip Swenson, and Josh Rasmussen. I am very
gratefultootherreadersofmyblogaswellastomyaudiencesatBaylorUniversity,
Catholic University of America, University of Oklahoma, and the “New Theists”
workshopfortheirpatienceasItriedoutversionsofthesearguments,andfortheir
criticalcommentary.Moreover,Iamgreatlyinthedebtofanumberofanonymous
readersofthismanuscriptwhosecarefulreadinghasresultedinmuchimprovement
ofthebook.Theremainingobscuritiesaremyownaccomplishment.Finally,Iamvery
gratefultoChristopherTomaszewskiforhiscarefulworkonindexingthisvolume.