Infinity,Causation, andParadox
AlexanderR.Pruss
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2.5.3
6.TheAxiomofChoiceMachine
2. ∗ TheAxiomofChoiceforCountableCollectionsofReals
3. ∗ ParadoxesofACCR
3.3Banach–Tarskiparadox
4. ∗
5.
5.1Strangemathematicsandparadox
5.2Coin-flipsandDutchBooks
5.3.2.1Makingthemachine
5.3.2.2Usingthemachine
5.3.2.3Causalinfinitismandverifyingthemachine’smatch
5.3.3Athree-dimensionalmachine
5.3.4
3.4Noinfiniteintensivemagnitudes
3.4.1Thebasictheory
3.4.2Someinfiniteintensivemagnitudes
3.4.2.1Centerofmassandmomentsofinertia
3.4.2.2Mentallife
3.4.2.3Blackholes
3.4.2.4Particles
3.4.3Huemer’sintensivemagnitudes
3.4.3.1Speed,Thomson’sLamp,andHilbert’sHotel
3.4.3.2 ∗ Smullyan’srod
3.4.3.3Immaterialminds
3.4.4Evaluation
4.3.1Somebackground
ListofFigures
Allillustrationsinthisvolumehavebeencompiledbytheauthor.
1.1Thomson’sLamp1
1.2Correspondencebetweennaturalnumbersandevennaturalnumbers5
1.3Intervalnotation6
2.1Thetwowaysofviolatingcausalfinitism:regress(left)andinfinite cooperation(right)25
2.2Thetestimonyofunicornexperts28
2.3Atheisticnon-viciousregress?33
2.4Here, I
3.1SomerepresentativeGrimReaperactivations47
3.2SomerepresentativereversedGrimReaperactivations48
3.3Smullyan’srodwithexponentiallydecreasingdensityandhence exponentiallydecreasingquasi-gravitationalpull58
4.1Aluckycasewherethelotteryworks,withthewinnerbeingthenumber279
4.2Atraverseofatwo-dimensionalarray80
6.1The(i)–(ii)bettingportfoliothatyoushouldbehappytopayadollar for.Thevolumeofeachsphereis1/100thofthatofthecube126
6.2The(i )–(ii )bettingportfoliothatyoushouldbehappytopayadollar foriftheargumentworks127
6.3The(i )–(ii )bettingportfolioyoushouldbehappytoacceptforfree128
6.4AsliceofaChoiceMachine132
7.1Benardete’sBoards147
7.2Fourparadigmaticviolationsof(8)163
Acknowledgments
IamespeciallygratefultoIanSlorachwhogavememanyveryinsightfulandhelpful commentsandcriticismsbothonmybloggedargumentsbeforeIstartedwritingthis bookandwhileIwascommittingmaterialtothebookdraft’sGitHubrepository.Iam alsoparticularlygratefultoMiguelBerasategui,BlaiseBlain,TrentDougherty,Kenny Easwaran,RichardGale,AlanHájek,JamesHawthorne,RobertKoons,Jonathan Kvanvig,ArthurPaulPederson,PhilipSwenson,andJoshRasmussen.Iamvery gratefultootherreadersofmyblogaswellastomyaudiencesatBaylorUniversity, CatholicUniversityofAmerica,UniversityofOklahoma,andthe“NewTheists” workshopfortheirpatienceasItriedoutversionsofthesearguments,andfortheir criticalcommentary.Moreover,Iamgreatlyinthedebtofanumberofanonymous readersofthismanuscriptwhosecarefulreadinghasresultedinmuchimprovement ofthebook.Theremainingobscuritiesaremyownaccomplishment.Finally,Iamvery gratefultoChristopherTomaszewskiforhiscarefulworkonindexingthisvolume.
Infinity,Paradox,andMathematics
1.ParadoxandCausalFinitism
Alampisonat10:00.Itsswitchistoggledinfinitelyoftenbetween10:00and11:00, sayat10:30,10:45,10:52.5,andsoon.Noothercauseaffectsthelamp’sstatebesides theswitch.Thus,afteranoddnumberoftogglingsthelightisoffandafteraneven numberit’son.Whatstatedoesthelamphaveat11:00?Thereseemstobenoanswer tothisquestion.Yetthelampiseitheronoroffthen(Fig.1.1).
ThisisknownastheThomson’sLampparadox(Thomson1954).Potentialanswers toaparadoxlikethisfallintothreegeneralcamps:logicallyrevisionary,metaphysical, andconservative.Logicallyrevisionaryanswersresolvetheparadoxbyinvokinga non-classicallogic,sayoneinwhichthelampcanbebothonandoffatthesametime, andcanusetheparadoxassupportforsuchrevision.Metaphysicalanswersresolvethe paradoxbyarguingforasubstantiveandgeneralmetaphysicalthesis,suchasthattime isdiscrete,thattherearenoactualinfinities,orthatitismetaphysicallyimpossible tomoveanything(say,aswitch)atspeedswhoselimitisinfinity(cf.Huemer2016, 12.10.3),athesisthatexplainswhythestoryisimpossible.
Conservativeanswers,ontheotherhand,refusetoreviselogicorpositsubstantive metaphysicaltheses,andcomeintwovarieties. Particularist conservativeanswers maintainthattheparticularstory(andminorvariantsonit)isimpossible,e.g.,preciselybecauseitisparadoxical. Defusing conservativeanswersmaintainthatthestory asgivenispossibleandthereisnoparadoxinit.
AparticularistanswertoThomson’sLampparadoxissimplythatthestoryasgiven isimpossible,sinceifthestorywerepossibleacontradictionwouldresult:thelamp wouldbebothonandoff.Adefusinganswer,ontheotherhand,asgivenbyBenacerraf (1962),notessimplythatthereisnocontradictioninsayingthatthelampison(or off,forthatmatter)at11am:wejustcan’tpredictthestatethelampwillhavefrom theinformationgiven. 10:00 10:30 10:45 10:52.5 11:00 ?
Fig.1.1 Thomson’sLamp.
Otherthingsbeingequal,conservativeanswerstoaparadoxarepreferableto metaphysicalones,whilemetaphysicalanswersarepreferabletologicallyrevisionary ones.Nonetheless,otherthingsneednotbeequal.Forinstance,whileagivenconservativeanswermaynotinvokeametaphysicalthesis,itmayunexpectedlycommit onetosuchathesis,andthenthebenefitsofconservativenessarelost.Forinstance, Benacerraf’ssolutionisintensionwiththePrincipleofSufficientReason.Foreven ifthereisnocontradictioninthelamp’sbeingonat11am,thereseemstobeno explanationastowhyit’sonthen(andifit’soff,thereisnoexplanationforthat).
Furthermore,ifanumberofparadoxesaregivenandeachcanberesolvedby meansofadifferentconservativeresponse,nonethelessitcouldbepreferableto resolvethemallinonefellswoopbyasingleelegantmetaphysicalhypothesisthat explainswhy none oftheparadoxicalstoriesarepossible.Foritisreasonabletoprefer unifiedexplanationsofphenomena.
Inthisvolume,Iwillpresentanumberofparadoxesofinfinity,someoldlike Thomson’sLampandsomenew,andofferaunifiedmetaphysicalresponsetoallof thembymeansofthehypothesisofcausalfinitism,whichroughlysaysthatnothing canbeaffectedbyinfinitelymanycauses.Inparticular,Thomson’sLampstoryis ruledoutsincethefinalstateofthelampwouldbeaffectedbyinfinitelymanyswitch togglings.Andinadditiontoarguingforthehypothesisasthebestunifiedresolution totheparadoxesIshalloffersomedirectargumentsagainstinfiniteregresses.Itis notthepurposeofthisbooktoconsider all paradoxesofinfinity—thatwouldbean infinitetask—orevenalltheonesthathavebeendiscoveredsofar.Rather,Iconsider asufficientnumbertomotivatecausalfinitism.1
Theavailabilityofanelegantmetaphysicalsolutionobviatestheneedforresorting tologicalrevisionism.Butwewillneedtobeconstantlyonthelookoutforconservativesolutionstotheparadoxes.Nonethelessonbalancecausalfinitismwillprovide asuperiorresolution.Furthermorewewillneedtoconsidercompetingmetaphysical hypothesesthatresolvesomeoralloftheparadoxes.However,itwillturnoutthat eachofthecompetinghypothesessuffersfromoneofthefollowingshortcomings:it isbroaderthanitshouldbe,itfailstoresolvealltheparadoxesthatcausalfinitism resolves,oritsuffersfrombeing adhoc.
Onecandistinguishtwowaysofresolvingaparadox:onecan solve itby showinghowanapparentlyincompatiblesetofclaimsisactuallycompatibleor byshowinghowanapparentlyplausibleassumptionisnolongerplausibleafter examination,oronecan kill itbyarguingthattheparadoxicalsituationcannotoccur.2 Insomecases,killingaparadoxisnotatenableoption.Forinstance,Zeno’sparadoxes ofmotioncanbesolved,saybyshowingthattheymakeassumptionsabouttimeor motionthatwecanreject,ortheycanbekilledbyholdingthatmotionisimpossible. Zeno,ofcourse,wantedtokilltheparadoxes,butsincethenmostphilosophershave preferredtosolvethem.
1 Foramorethoroughsurvey,seeOppy(2006).
2 Iamgratefultoananonymousreaderforthisdistinction.
Whetherkillingorsolvingthemembersofafamilyofparadoxesisintellectually preferabledependsonthedetailsofthesituation.Forinstance,whentheparadoxes occurinsituationsthatwehaveapparentempiricalobservationsof—arrowsflying andfasterrunnerscatchingupwithslowerones,asinZeno’scase—killingtheparadox byrejectingtheactualityofthesituationsisapttoleadtoanunacceptableskepticism, pace Zeno.Ontheotherhand,whentheparadoxesoccurinsituationswhichwe merelyintuitivelythinkaremetaphysicallypossible,killingtheparadoxesbyrejecting themetaphysicalpossibilityofthesituationsmaybemuchmoretenable,sinceour intuitionsaboutmetaphysicalpossibilityareunlikelytobeasreliableasourempirical observations.
Wemayhaveacertainintuitivepreferenceforsolvingaparadoxratherthan killingit.Butunlesstheparadoxesarebasedonlogicallyinvalidreasoning,itwill beintellectuallypreferabletokillallthemembersofafamilyofparadoxesina unifiedwayratherthansolvetheminavarietyofdifferentways.Onereasonfor thisisthesimplefactthatto solve aparadoxbasedonlogicallyvalidreasoningwe havetorejectaplausiblepremise,andhencetosolveanumberofsuchparadoxes wehavetorejectanumberofplausiblepremises.Butitistypicallypreferableto makeasingleassumption—especiallyifthereissomeindependentreasontomake theassumptionbeyondtheneedtoresolveparadoxes—thantorejectanumberof plausiblepremises.
Themainstrategyofthebook,then,willbelikethatofZeno:ratherthanoptfor anumberofdifferentsolutionstodifferentparadoxes,theywillbeallkilledthrough thesingleassumptionofcausalfinitism.Butwhereastheno-motionthesisthatZeno defendsisonewehaveverystrongempiricalreasonstoreject,thethesisofcausal finitismiscompatiblewithourobservations(thougharguingforthiswilltakesome workininterpretingmodernphysics).
Formostoftherestofthepresentchapter,aftersomeimportantbackground notesbothtechnicalandphilosophical,Iwillconsideroneprominentalternate hypothesis—fullfinitism—andarguethatinordertogetoutoftheparadoxes,itneeds tobemarriedtoaparticulartheoryoftime,thegrowingblocktheory,andthatin anycaseitcausesseriousdifficultiesforthephilosophyofmathematics.Whileon thesubjectofphilosophyofmathematics,Iwillalsoofferanintriguingapplication ofcausalfinitism(andoffinitismaswell)totheproblemofdefiningthefiniteand thecountable.
InChapter2,Iwillconsiderinfiniteregresses,whichwillgiveussomereason toacceptcausalfinitismindependentlyoftheparadoxesitcankill.Then,inthe succeedingchapterswewilldiscussseveraldifferentkindsofcausalparadoxes: non-probabilisticparadoxes,paradoxicallotteries,otherprobabilisticanddecisiontheoreticparadoxes,andparadoxesboundupwiththeAxiomofChoicefromset theory.Attimeswewillalsoconsiderwhatwillbeseentobeananalogousquestion: whethertimetravelandbackwardscausationarepossible.Iwillthenofferwaysto refinetheroughthesisofcausalfinitisminlightofthedataadduced,andarguethat variousalternativestocausalfinitismareunsatisfactory.
Finally,Iwillconsidertwopotentialconsequencesofcausalfinitism.Thatatheory hasconsequencesbeyondwhatitwasintendedtoexplaingivessomereasontothink thetheoryisnot adhoc.Atthesametime,suchconsequencesmakethetheorymore vulnerabletorefutation,sincetheremightbeargumentsagainsttheconsequences. Thefirstapparentconsequenceisthattime,andperhapsspaceaswell,isdiscrete. Ifthisdoesindeedfollow,thatisintrinsicallyinteresting,butalsodamagingtocausal finitisminthatitappearstoconflictwithmuchofphysicssinceNewton.Weshall considerwhetherthediscretenessoftimeactuallyfollowsandwhetherthekindof discretenessthatissupportedbycausalfinitismisinfactinconflictwithphysics,and arguethatcausalfinitismcancoherewithmodernphysics.
Thesecondconsequenceisclearer.Ifcausalfinitismistrue,thentherecannotbe backwards-infinitecausalsequences,andhencetheremustbeatleastoneuncaused cause.Thereisalsosomereasontotakethisuncausedcausetobeanecessarybeing. Nowthemostprominenttheoryonwhichthereisacausallyefficaciousnecessary beingistheism.Thus,causalfinitismlendssomesupporttotheism.Interestingly,this willforceustoconsiderwhethertheismdoesn’tinturnundercutcausalfinitism.
Iwilloccasionallyusetheconvenientphrase“causalinfinitism”forthenegationof causalfinitism.Roughly,thus,causalinfinitismholdsthatitispossibleforsomething tohaveaninfinitecausalhistory.(Notethatcausalinfinitismdoesnotsaythatthere actually is anyinfinitecausalhistory.)Thusthepointofthebookistoargueforcausal finitismor,equivalently,toargueagainstcausalinfinitism.
LetmeendthissectionbynotingthatIdonottakeThomson’sLamptobea particularlycompellingversionofaparadoxmotivatingcausalfinitism.Therewill bemorediscussionofitinChapter3,Section2.Butitisahelpfulstand-informany ofthemorecomplicatedparadoxeswewillconsider.
2.SomeMathematicalandLogicalNotes
Wewillneedsometechnicalterminologyandsymbolismasgeneralbackgroundfor thebook,andthiswillbeintroducedinthissection.Additionally,thebookcontains sometechnicalsectionsmarkedwith“∗ ”andverytechnicalsectionswith“∗∗ ”.These canbeskippedwithoutlossofcontinuity.Notethatanysubsectionsofsomething markedwithoneofthesemarkerscanbepresumedtohaveatleastthatlevelof technicality.NotealsothatChapter6istechnicalorverytechnicalasawholeapart fromalesstechnicalintroductionandsummary.
Startwiththenotionofsetsascollectionsofabstractorconcreteobjects.The statement x ∈ A meansthat x isamemberof A.Wesaythataset A isa subset of aset B providedthateverymemberof A isamemberof B,andthat A isa proper subsetof B ifitisasubsetof B thatdoesnotincludeallthemembersof B.Forany set B andanypredicate F (x) wewrite {x ∈ B : F (x)} forthesubsetof B consistingof allandonlythe xssuchthat F (x) (sometimeswhenthecontextmakes B clear,wejust write {x : F (x)}).
Wecancomparethecardinalsizesofsetsasfollows.Ifthereisawayofassigninga differentmemberof B toeverydifferentmemberofaset A (i.e.,ifthereisaone-to-one functionfrom A toasubsetof B),thenwesaythat A ≤ B ,i.e.,thecardinalityof A islessthanorequaltothatof B.Forinstance,if B isthesetofrealnumbersbetween 0and1inclusive,and A isthesetofpositiveintegers,thentoeverymember n of A we canassignthemember1/n of B (notethatif n and m aredifferentmembersof A,then 1/n and1/m aredifferentmembersof B).
Wesaythat A hasfewermembersthan B,andwewrite A < B ,providedthat A ≤ B butnot B ≤ A .Wesaythatsets A and B havethesamecardinality when A ≤ B and B ≤ A .ThefamousSchröder–BernsteinTheorem(Lang 2002,p.885)saysthatunderthoseconditionsthereisaone-to-onepairingofallthe membersof A withallthemembersof B
Somesetsarefiniteandsomeareinfinite.Asetisfiniteifitisemptyorhasthe samesizeassomesetoftheform {1, ... , n} forapositiveinteger n.Otherwise,theset isinfinite.
Infinitesetsareinsomewayslikefinitesetsandinothersunlikethem.Boththeir likenessandtheirunlikenesstofinitesetsarecounterintuitivetomanypeople.
Awayinwhichinfinitesetsdifferfromfiniteonesisthatif A and B arefinitesets with A apropersubsetof B,then A alwayshasfewermembersthan B.Butinfinite setshavepropersubsetsofthesamesizeasthemselves.3 Forinstance,if B isthe set {0,1,2, ...} ofnaturalnumbers,thenthepropersubset A ={0,2,4, ...} ofeven naturalshasthesamesizeas B,ascanbeseenbyjoiningthemuponebyoneasin Fig.1.2.
Ontheotherhand,justasthefinitesetsdifferamongeachotherinsize,Georg Cantordiscoveredthatsodotheinfiniteones,ifsizeisdefinedasabove.Thedifference insizebetweeninfinitesetsishardertogenerate,however.Simplyaddinganew membertoaninfinitesetdoesn’tmakeforalargerinfiniteset.Butgivenaset A, wecanalsoformthe powerset P A ofallthesubsetsof A.Anditturnsoutthat P A alwayshasstrictlymoremembersthan A—thisisnowknownasCantor’sTheorem.4
Fig.1.2 Correspondencebetweennaturalnumbersandevennaturalnumbers.
3 ∗∗ Tobeprecise,thisisonlytrueforallDedekind-infinitesets.IftheCountableAxiomofChoiceis false,thentheremaybeinfinitesetsthataren’tDedekind-infinite(Jech1973,p.81).Butstandardexamples ofinfinitesets,suchasthenaturalorrealnumbers,are stillDedekind-infinite.Forsimplicity,Iwillwrite asifallinfinitesetswereDedekind-infinite.
4 Hereisaproof.It’sclearthat P A hasatleastasmanymembersas A does,sinceforeachmember x of A,thesingleton {x} isamemberof P A.Soallweneedtoshowisthat A doesnothaveatleastasmany membersas P A.Fora reductio,supposethereisafunction f thatassignsadifferentmember f (B) of A
If A isfiniteandhas n members,then P A willhave2n members(forwecangenerate allthemembersof P A byconsideringthe2n possiblecombinationsofyes/noanswers tothequestions“DoIinclude a inthesubset?”as a rangesoverthemembers of A),and n < 2n .ButtheCantorianclaimappliesalsotoinfinitesets:ingeneral, A < P A
Inparticular,thereisnolargestset.Forif A werethelargestset,then P A wouldbe yetlarger,whichwouldbeacontradiction.
Setswhicharefiniteorthesamesizeasthesetofnaturalnumbers {0,1,2, ...}, whichwillbedenoted N,arecalled countable.Anexampleofanuncountableset is P N.Anotheristheset R ofrealnumbers,whichinfacthasthesizeas P N.
Ausefulnotationforcertainsetsofrealnumbersisgivenby[a, b], (a, b),[a, b), and (a, b](seeFig.1.3).Eachofthesedenotesanintervalfrom a to b,withthesquare bracketsindicatingthattheendpointisincludedintheintervalandtheparenthesis indicatingthatit’snot.Thus[a, b]isthesetofallrealnumbers x suchthat a ≤ x ≤ b, (a, b) isthesetofallreals x suchthat a < x < b,[a, b) isthesetofallreals x suchthat a ≤ x < b and (a, b]isthesetofallreals x suchthat a < x ≤ b.Aslongas a < b,all thefourintervalsareuncountablyinfiniteandofthesamesizeas R
Finally,itwillsometimesbeconvenienttotalkintermsofpluralities.WhenIsay (1)ThemembersofmyDepartmentgetalongwitheachother thegrammaticalsubjectof(1)isaplurality,themembersofmyDepartment.Theverb formthatagreeswiththatplurality,“get”,isinapluralconjugation.Thesubjectofthe sentenceisnotasingularobjectlikethesetofthemembersofmyDepartmentor somesortofamereologicalsumorfusionofthemembers,forthatwouldcallfora singularverb,anditwouldmakenosensetosaythatthatsingularobject“getalong witheachother”.
Wecanquantifyoverpluralities.Wecan,forinstance,saythatforanypluralityof membersofmyDepartment,the xs,thereisaplurality,the ys,ofpeopleinanother Departmentsuchthateachofthe xsisfriendswithatleasttwoofthe ys.Plural quantificationiswidelythoughttoavoidontologicalcommitmenttosets.Italso avoidstechnicaldifficultieswithobjectsthatdonotformaset.Thereisnosetofall sets,butitmakessensetoplurallyquantifyandsaythat allthesets areabstractobjects.
toeverydifferentmember B of P A.Let D bethesetofallmembers x of A thatareassignedby f tosome set B suchthat x isnotamemberof B,i.e., D ={x ∈ A : ∃B(x = f (B) & x / ∈ B)}.Let x = f (D).Note that x isamemberof D ifandonlyifthereisasubset B of A suchthat f (B) = x and x isnotamemberof B.Theonlypossiblecandidateforasubset B of A suchthat f (B) = x is D,since f assigns x to D andwill assignsomethingdifferentfrom x toa B differentfrom D.Thus, x isamemberof D ifandonlyif x isnota memberof D,whichisacontradiction.
Fig.1.3 Intervalnotation.
Iwillassumethatpluralitieshavetheirelementsrigidly.Thatis,if x0 isoneofthe xs, theninanypossibleworldwherethe xsexist, x0 existsandisoneofthe xs.
3.Modality
3.1Metaphysicalpossibilityandnecessity
Fairiesandwatermadeofcarbonatomshavesomethingincommon:theydon’texist. Buttherethesimilarityends.Foralthoughneitherisactual,thefairyispossible,while thewatermadeofcarbonatomsisnot.
Thekindofpossibilityatissuehereisnotmerelylogical.Nocontradictioncanbe provedfromtheexistenceofafairy,butalsonocontradictioncanbeprovedfromthe existenceofcarbon-basedwater.Ineachcase,empiricalworkisneededtoknowthe itemdoesnotexist.
Therearemanytheoriesofthenatureofmodality.5 Theargumentsofthisbook arenottiedtoanyparticularsuchtheory,butrathertointuitivejudgmentsabout cases.Theseintuitivejudgmentsaboutcasesmaythemselvesputconstraintsonwhich theoryofmodalityisplausible,thoughofcoursethereaderwillalsofindhertheoryof modalityaffectingwhattothinkaboutthecases.ThatisapartofwhyIgivesomany paradoxicalcasesinthisbook:somecasesmayappealtosomereaderswhileothers toothers.
3.2Rearrangementprinciples
..defeasibility
Ifitismetaphysicallypossibletohaveonehorseandtwodonkeysinaroom, intuitivelyit’spossibletohavetwohorsesandonedonkeyinaroomaswell.Lewis (1986,Section1.8)attemptedtoformulatea“rearrangementprinciple”thatjustifies inferencessuchasthis(seeKoons2014forsomemorerigorousformulations).The basicideabehindrearrangementprinciplesisthat:
(2)Givenapossibleworldwithacertainarrangementofnon-overlappingspatiotemporalitems,any“rearrangement”oftheseitemsthatchangesthequantities,positions,andorientationstosomeothercombinationofquantities, positions,andorientationthatisgeometricallycoherentandnon-overlapping isalsometaphysicallypossible.
Unrestrictedrearrangementprinciplessitpoorlywithcausalfinitism.Justmultiply thenumberofbuttonflippingsandyougofromanordinarybedsidelampbeing turnedoffatnightandoninthemorningtoThomson’sparadoxicallamp.Andeven ifwerestrictchangesinquantitytobefinite,aninnocentforwards-infinitecausal sequencecanbetransformedintoabackwards-infiniteone.
5 InPruss(2011)Idefendacausalpowersaccountofmodality,butnothinginthepresentbookdepends onthatdefense.
Nonetheless,manyofourargumentsforcausalfinitismwilldependonrearrangementconsiderations.Isn’tthatcheating?
Toseeourwaytoanegativeanswer,observethatunrestrictedrearrangementprinciplescarrymanyheavymetaphysicalcommitments.Theyruleoutclassicaltheism, sinceonclassicaltheismGodisanecessarybeing,andasituationthatcouldcoexist withGodcouldberearranged,saybygreatlymultiplyingevilsandremovinggoods, intoasituationthatcouldn’tcoexistwithGod(cf.Gulesarian1983).Theygivean argumentforthepossibilityofauniverseconsistingofasinglewalnutthatcomes intoexistenceatsometime,andhenceforthepossibilityofsomethingcomingfrom nothing.TheyruleoutAristoteliantheoriesoflawsandcausationonwhichthe exercisesofcausalpowersnecessitatetheireffectsintheabsenceofcounteracting causes.Theysitpoorlywiththeessentialityofevolutionaryoriginsforbiological naturalkindsandwiththeessentialityoforiginsforindividuals.Andtheyevenrule outcertaincolocationisttheoriesofmaterialobjects.Forcolocationistswillsaythat whereveryouhaveaclaystatueyoualsohavealumpofclay,butanunrestricted rearrangementprincipleshouldletyouhavetheclaystatuewithoutthelump!
EvenphilosophersofaHumeanbentwhoareuncomfortablewiththeismand essentialityoforiginsandhavenoproblemswiththingscomingintobeing exnihilo needtorestrictrearrangementonmetaphysicalgrounds.Forinstance,Lewis(1986, p.89)saidthatallrearrangementsarepossible“sizeandshapepermitting”.Theworry isthatitmightturnoutthatmaterialobjectscannotinterpenetrate,soonecannot rearrangeaworldwithahorsestandingbesideacowintoaworldwheretheyoccupy thesamelocation.Thisseeminglypurelygeometricalconstraintinfactneedsto dependonthemetaphysicsofthematerialobjectsinquestion.Perhapsindeedahorse andacowcouldnotoccupythesamelocation,butwehavegoodreasontothinkthat multiplebosonslikephotonscanoccupythesamelocation,sincetwobosonscanhave thesamequantumstate(Dirac1987,p.210).Sowhatthe“sizeandshapepermitting” constraintcomestodependsonthemetaphysicsoftheobjects,namelywhetherthey canbecolocated.
Thus,rearrangementprincipleslike(2)shouldbecurtailedinsomewaylestthey rideroughshodovertoomuchmetaphysics.OnewaytodothisistoextendLewis’s strategybygivingalistofspecificmetaphysicalconstraintslikehis“sizeandshape permitting”.Butitisdifficulttoseehowwecouldeverbejustifiedinthinkingthat ourlistofconstraintsiscomplete.
Abetterwayistostipulatethattherearrangementprinciplesaredefeasible,withthe understandingthatitisbestifdefeatersforrearrangementprinciplesareprincipled ratherthan adhoc.Theism,Aristotelianviewsofcausation,essentialityoforigins, orcausalfinitismcouldeachprovideprincipleddefeaterstoparticularcasesof rearrangement.ButifoneruledoutThomson’sLampbysayingthatthisparticular rearrangementoftheordinarybedsidelampsituationisimpossible,thatwouldbe adhoc.If,instead,wecouldruleoutThomson’sLampaswellasanumberofother paradoxesbymeansofasinglegeneralprinciple,namelycausalfinitism,thatwould behighlypreferable.Anditisthisthatisthestrategyofthepresentbook.
Atthesametime,thereisalwaysacosttointroducinganothermetaphysical principlelikecausalfinitismthatdefeatsparticularinstancesofrearrangement. Butthecostissurmountable.
Idonotwanttheargumentsofthisbooktobehostagetoaparticularrearrangementprinciple.Rather,Iwanttorelyontheintuitiveplausibilityoftheparticular rearrangementsthatIwillmakeuseof.
..causalpowers
Onecrucialquestioninformulatingarearrangementprincipleiswhatpropertiesare carriedalongwiththeobjectsastheyarerearranged.Icanrearrangearoomwith abrayingdonkeyintoaroomwithtwobrayingdonkeys.ButIcannotrearrange aroomwithasolitarydonkeyintoaroomwithtwosolitarydonkeys.Astandard thingtosayisthatthepropertiesthatcanbecarriedaroundbytheitemsbeing rearrangedarethe intrinsic properties:solitarinessisnotintrinsic,butbrayingmight be.Butitisnotoriouslydifficulttodefineanintrinsicproperty(seeWeathersonand Marshall2014).
Thereis,however,onecontroversialchoicethatmanyofourargumentswillrequire, andthisisapictureofobjectsandtheiractivitiesashavingacausalnaturethatis carriedalongwiththeirrearrangement.Whenonerearrangesalampswitchfrom onelocationinspacetimetoanother,therearrangedswitchcontinuestohavethe samecausalpowers,andwhenputinthesamerelevantcontext(say,alamp)these causalpowerswillhavethesameeffects.Ifintrinsicpropertiesarewhatcanbe carriedalongwithrearrangements,thenIamtakingcausalpowerstobeintrinsicproperties.
Thisisaveryintuitivepictureofcausalpowers.Itis,nonetheless,inconflictwith widelyheldHumeanviewsonwhichcausalfactssuperveneontheglobalarrangement ofmatterintheuniverse.Itakethisconflicttoprovideanargumentagainstthe Humeanview.Thepossibilityofrearrangingthingsintheworldwhilekeepingfixed thecausalpowersofthingsisintuitivelymoresecurethantheHumeantheories ofcausation.
Onestrengthofmakingacase-by-casejudgmentaboutthepossibilitiesofrearrangementsofpowerfulobjectsratherthanpositingasinglegeneralprincipleis thateachsuchjudgmentcanbeseparatelyevaluatedbyaHumeanreader.Thereader maydecideoneofthreethingsaboutaparticularapplicationofrearrangement:
(i)theapplicationisincompatiblewithHumeanismandplausibleenoughto providesignificantevidenceagainstHumeanism;or
(ii)theapplicationisincompatiblewithHumeanismbutnotveryplausibleandso insteadHumeanismprovidessignificantevidenceagainstthisapplication;or
(iii)theapplicationcanbemadecoherentwithHumeanism,forinstance,by supposingmanyanalogousbackgroundeventssufficienttogroundcausallaws thatapplyalsototherearrangedcase.
4.Finitism:AnAlternateHypothesis
4.1Timeandfinitism
Finitism holdsthattherecanonlybefinitelymanythings(includingbothsubstances andevents).Finitism,however,allowsforpotentialinfinities.Thusacollectionoftoy soldierstowhichanewtoysoldierwillbeaddedeverydaywouldbepotentially infiniteasforanynumber n,itwouldeventuallyhavemorethan n elements.But accordingtofinitismtherearenoactualinfinities.Therearealwaysonlyfinitely manythings.
Finitismhasanimpressivephilosophicalhistory,goingbackatleasttoAristotle’s responsestoZeno’sparadoxes,6 andbeingthegenerallyacceptedphilosophical orthodoxyintheMiddleAges.
Theexactupshotoffinitismdependsonwhichtheoryoftimeitiscombinedwith.
The eternalist thinksofpast,present,andfuturethingsasallontologicallyon par,andbelievesthat(barringsomecatastrophe)ourgreat-great-great-grandchildren existandAlexander’sgreatwarhorseBucephalusalsoexists.Ofcourse,thegreatgreat-great-grandchildrenandBucephalusdon’t presently exist.Buttheynonetheless reallydoexist.The growingblocktheorist takesrealitynottoextendtothefuture,but toincludethepastandpresent.7 Thusourgreat-great-great-grandchildrendon’texist (thoughitmightbetruethattheywillexist),butBucephalusdoes.The presentist,on theotherhand,onlyacceptspresentlyexistingentitiesasexisting.Iwill,further,take allofthesethesesabouttimetoclaimtobenecessarilytrue.
Finitismpluseternalismstraightforwardlyentailscausalfinitism:iftherecanonly befinitelymanythings,andthatincludespast,present,andfuture,thenofcourse nothingcanbeaffectedbyinfinitelymanycauses.Thusanyparadoxruledoutby causalfinitismwillberuledoutbyfinitismpluseternalism.Butunfortunatelyfinitism pluseternalismalsoentailsthatthefuturemustbefinite—thattherecannotbe infinitelymanyfutureevents.Butsurelyitispossibletohaveaninfinitefuturefull ofdifferenteventsorsubstances,saywithanewtoysoldierbeingproducedeveryday forever.Thus,finitismisimplausiblegiveneternalism.
Givenpresentism,ontheotherhand,finitismiscompatiblewithinfinitesequences ofcauses,aslongasatnoparticulartimearethereinfinitelymanycauses.Thus, finitismpluspresentismdoesnothingtoruleouttheinfinitelymanytogglingsofthe switchinThomson’sLamp.Whiletheotherparadoxeshavenotyetbeendiscussed, manyofthemwillalsohavethediachroniccharacterofThomson’sLampandhence willbeuntouchedbypresentistfinitism.
6 Ofcourse,inadegenerateway,Parmenideswasafinitist,ashethoughtthattherecouldbeonly onething.
7 ThereisalsoavariantbyDiekemper(2014)thatincludesonlythepast.Thatvariantwillnotbehelpful tothefinitist,andIwillsticktothecanonicalversionthatincludesthepresent.
Thisleavesgrowingblockplusfinitism.Ifit’snecessarilytruethatacauseisearlier thanorsimultaneouswithitseffect,thengrowingblockfinitismdoesentailcausal finitism,andhencecanruleoutalltheparadoxesthatcausalfinitismcan.Giventhe combinationof(a)growingblocktheory,(b)finitism,and(c)thethesisthatcausesare eithertemporallypriortoorsimultaneouswiththeireffects,wedogetcausalfinitism again,andhencewecanruleoutalltheparadoxesthatcausalfinitismcanruleout.
Thefinitist’sbestbetatparadoxremovalisthustoadoptgrowingblocktogether withthethesisthatcausesarepriortoorsimultaneouswiththeireffects.
Unfortunately,thereisapowerfulargumentagainstgrowingblocktheorydue toMerricks(2006).Manypeoplehavethoughtthoughtsaboutwhatdateortime itis,thoughtsexpressibleinsentenceslike:“Itisnow2012”or“Itisnownoon.” Ifgrowingblocktheoryistrue,manyofthesethoughtsareinthepast,andmost haveacontentthatisobjectivelyfalse.Onthegrowingblocktheory,the“now” istheleadingedgeofreality,theboundarybetweentherealandtheunreal.The thoughtexpressedby“Itisnow2012”istrueifandonlyif2012isattheleading edgeofreality.But2012isnotattheleadingedgeofreality.Moreover,mypresent thoughtthatitisnow2018hasnobetterevidencethanthe“Itisnow2012”thought thatwasin2012.Sincemostthoughtsofthissort,withtheusualsortofevidence forthem,arefalse,Ishouldbeskepticalaboutwhetheritisnow2018.Andthat’s absurd.Presentismescapesthisargumentbydenyingthatthepastthoughtsexist. Movingspotlightversionsofeternalism,onwhichthereissomethinglikeanobjective “movingspotlight”illuminatingthe“now”arealsosubjecttothisobjection:mostof the“Itisnow t ”thoughtsarenotilluminatedbythespotlight,andyet“now”implies such“illumination”.ButB-theoreticeternalism(e.g.,Mellor1998),whichclaimsthat the“now”isamereindexical,ratherthananexpressionofanobjectivechanging property(likebeingattheleadingedgeofrealityorbeing“illuminated”),isnotsubject totheobjection.Hence,thefinitist’sbestbetatparadoxremovalrequiresadoptinga particularlyvulnerabletheoryoftime.
Wenowconsidermoreadvantagesanddisadvantagesoffinitism vis-à-vis causal finitismasawayoutofparadoxes.
4.2Non-causalparadoxes:Anadvantage?
ImagineHilbert’sHotel—ahotelwithinfinitelymanyroomsnumbered1,2,3, ... . Youcanhavelotsoffunwiththat.Putapersonineveryroom,andthenhangupthe sign:“Novacancy.Alwaysroomformore.”8 Whenanewcustomerasksforaroom, justputtheminroom1,andtellthemtotellthepersonintheroomtomoveto thenextroom,andtopassonthesamerequest.Youcanevenhaveinfinitelymany peoplevacatethehotelandstillhaveitfull.Ifallthepeopleintheodd-numbered roomsleave,youcantelleachpersonintheeven-numberedroomstomovetoaroom whosenumberishalfoftheirroomnumber.
8 ThesignsuggestioncomesfromRichardGale.
