Loke Hagberg
Existence is indeed evidence of immortalityifthe past is infinite
Or,onproofs of eternal recurrenceusing transition systems 2024-06-01
Existenceisindeed evidence of immortalityifthe past is infinite
Or,onproofsofeternal recurrence usingtransitionsystems
2024-06-01
B-level studyinphilosophy, SödertörnUniversityVT24
Author:LokeHagberg
Supervisor:NicholasSmith
Abstract
This studyisconcerned with thequestion if existenceisevidenceofeternal recurrence,thata currentobserveriswithina cyclic world, if thepastisinfinite.Michael Huemer’s proposed proofof existencebeing evidence of immortalityusing aBayesianapproachisdiscussed,aswellasvarious counterarguments.Thisstudy then uses transition systems, anon-Bayesianapproach, to prove variousresults aboutworldsthatcan be describedbythem. It is proved that in transition systems with an infinitepast, where time canbediscretelysubdivided, eternalrecurrenceisthe case for everyobserverina worlddescribed by such asystem. Finally, thereasoning,potential andactual counterarguments,consequences, andfutureresearchare considered.
Sammanfattning
Dennastudiebehandlar frågan om existensen är ett bevispåevigåterkomst,att en nuvarande observatör befinnersig iencykliskvärld,omdet förflutna är oändligt.Michael Huemersföreslagna bevisför att existensen är ett bevispåodödlighet, somanvänderett bayesianskt tillvägagångssätt, diskuteras,liksomolika motargument. Idenna studie användsövergångssystem,ett icke-bayesianskt tillvägagångssätt,för att bevisa olikaresultatomvärldar somkan beskrivasmed hjälpavdem.Det bevisasatt iövergångssystem medett oändligt förflutet,där tidenkan delasupp diskret, är evig återkomstfalletför varjeobservatöri en världsom beskrivs av ett sådant system.Slutligen behandlas resonemangen,faktiskaoch potentiella motargument, konsekvenser,och framtida forskning.
Acknowledgements
Iwould like to thankmywifeYuliyaHagberg forsupport throughthe process. Iwould also like to thankDavid Madsen forthe many yearsofphilosophicaldiscussion concerning this andother topics, my supervisor Nicholas Smithfor supportand feedback,and my fellowstudent Adrian Blivik for feedback.I first wroteabout this topicina book “Enhetsmallen” (2018) andoutlined theproofsthat arecarried outhere(except fortheorem (5)),where Ialsofor examplediscussed multiple dimensions of time.Later,I wroteabout theabsurdity of ‘the next state’ of theworld to notbe defined in “Collected papers on finitist mathematics andphenomenalism”(2023).
1Introduction
Time within metaphysics, in general, hasbeendevoted alot of attention over theyears (Gale, 2016).
Oneinquiry about time is whethereverythinginthe world, that is theworld we live in,happens again. Eternalrecurrenceisthe idea that everything in theworld repeatsitselfaninfinite number of times, whichissomething that hasbeen stated by Hinduscholars, Seneca,Nietzsche,and other thinkers (Teresi, 2010,p.174;Huemer, 2021,p.4). Nietzschemight only have used eternalrecurrence as ametaphororsuchhowever (Anderson, 2024).
Anotherinquiryabout time is aboutits topology,whether it is infiniteornot (and in what directions), if it branches or not, etc. Aristotleansweredthe first question by defininga moment of time as that whichhas amomentof time before it anda moment of time after it,and thereforearguedthat time is infinitebothintothe past andintothe future becausethere cannot be a“first moment of time” andhence afirststate of theworld with no statebeforeit(astate is aslice of space-time at agiven time containing everything in theworld at that time). This argument hasgenerally notbeenaccepted becauseitencodes thetopologyof time within thedefinition of time throughwhich relationsa moment of time is allowedtohave(Emery, Markosian& Sullivan,2020).Itisworth to mention that with branching time,observers still take onesinglepath, if observerstakemultiplebranchesthen thosebranchescould be treatedasa “superbranch”instead whichthencorresponds to theirsingle path
In Newtonianphysics spaceand time areabsolute, andthe time measured betweentwo events is thesamequantity fordifferentobservers (which areassumed to measurecorrectly). However, in Einstein’s theory of relativity,the time betweentwo events that observersmeasure candiffer (Emery,Markosian &Sullivan, 2020). Is that reconcilablewith“afirststate of theworld”? Wouldthat stateberelative,could differentobservers have different“first states of theworld”? In thespecial theory of relativity,itisstill thecasethatthere is an observer-independent type of time called proper time that is thesamefor everyobserver(Taylor &Wheeler,1992, p.1-20). This meansthatthe theory of relativity is compatiblewiththe worldbeing subdivided into states whereone statepassesto anotheras time changes. Thereasonwhy measured time is relative forobservers is that thereisa limitedcomputation that goes slower when theobservermoves faster and“spends”more computation when moving faster (witha possiblelimit at thespeed of light)(Wolfram, 2020, p.2242). Acyclic or recurrentworld is onewhere events occurinloops,theycycle,and time is infiniteinto thepastand into thefuture. Thereare currentlyvarious models in cosmologythatare cyclic and have infinitepasts (Huemer, 2021,p.2-3).
Perhaps it couldbepossibletoprove that theworld must be cyclic andthatanobserverinitmust recurifthatworld hasa first state. MichaelHuemertried to carryout such aproof (Huemer, 2021, p.1).
1.1Backgroundand previous studies
MichaelHuemerpublished apaper called “Existence is evidence of immortality” (2021).Huemer states that if thereisnofirststate of theworld,the worldhavinganinfinite past,thenpersons recur, they areimmortaland liveaninfinite number of times– they reincarnate(Huemer,2021, p.1)
Huemer points outthatitismathematicallyconsistentthatthe worlddoesnot have afirststate,but it seemsthatwecould askwhatcamebeforeitjustlikeweask what lies beyond apossibleboundary of space(Huemer,2021, p.1-2).Leonard Mlodinow andStephen Hawkingpointed outthataskingthe question “whatcamebeforethe bigbang”,ifthe bigbangwas thefirststate of theworld,isjustlike asking “whatissouth of thesouth pole”(forwhich theansweris“nothing” or thequestion canbe considered to be meaningless) (Mlodinow& Hawking, 2010,p.135)
Huemer points outthatsomephilosophersmisguidedly object to an infinitepast, arguingthatthere cannot be an actual infinityand sometimesmention Zeno’s paradox, that some physical movement betweentwo points must pass halfway, halfwayofhalfway,halfway of halfwayofhalfway,etc.and that such amovementtherefore is problematic, but Huemer argues that becauseobjects do in fact finishsuchmovements that is nota problem(Huemer,2021, p.2).Huemeralsomentionsthe big bang andarguesthatitwas ahighlyimprobablestarting pointbecause of itshighentropy andthatit couldjustaswellhavebeenthe case that theworld hada first statestarting in the1950’sinstead, with alower entropy, andthathumansatthat time came into existencewithfalse memories. Becausethe 1950’sbeing thestarting pointseems implausible, so should thebig bang beingthe starting pointaccording to Huemer (Huemer, 2021,p.3).
Huemer also mentionsthe Poincaré recurrence theorem, stating that physicalsystems with bounded andconserved phase spacewillreturnto“arbitrarilyclose”tothe starting state, andwithinfinite time thephysicalsystemwillreturn“arbitrarilyclose”tothe starting stateaninfinite numberof times. It is notlogically necessary that theuniversehas aboundedand conservedphase space however(Huemer,2021, p.4).Huemeralsopointsout that this happenseventhoughthere is a tendency forentropy to increase (Huemer, 2021,p.10-11).
Huemer uses Bayesian probabilityand argues that thecorrect theory of personsisone that an arbitrarilysimilar individual is thesamepersonand argues probabilisticallythatweare to accept that if we existnow then thereisnofirststate andobservers usingthe Bayesian updating are reincarnating or thereisa first state(Huemer,2021, p.5-10). Huemer also uses theprinciple of indifference to arguefor this (Huemer, 2021,p.12-15).
Huemer argues that reincarnation is compatiblewithphysicalism andevenCartesian dualism, if a physical object is recreatedafter having been destroyed, we thinkofitasthe same object,and if the circumstancesofa body having asoulisrepeateditcould indeed have thesamesoul(Huemer,2021, p.11-12).
Jens Jägerin“Immortal Beauty:DoesExistence Confirm Reincarnation?” (2022) argues that Huemer’s reasoningwithBayesianprobability is unsound, becauseofthe observer carrying outthe probabilitycalculation is “privileged” by observingattheir current time,and that observerscould possiblyhavenon-qualitative evidence making them notaccept theBayesianreasoning (p.3-4). The
Bayesianproof of Huemer also assumesfor examplethatthe priors arenon-zero. Jägerpointsout that Huemer’s proofisvalid:if time is dividedupinchunks, beingalive at afinite numberofchunks of time (for examplecenturies)has thesameprobability to allother configurationsofchunks(by indifference), then with an infinitenumberof time-chunksthe probabilityofbeing aliveatany given time is 0 if theobserverisnot infinitelyrecurring.Ifanobserverhas non-zero priors forexisting a finitenumberof timesand an infinitenumberof times, thesecondshouldbecomelargerthanthe first by theevidenceofknowing they existina particular time-chunk (Jäger,2022, p.5).
JägerarguesthatHuemer’sargumentfails becauseofthe waythe evidence is calculated:ifan observer is observingat time �� anditisthe only time-chunk they arealive in,thenthe probabilityof �� beingequaltothe currentlyobserved time is 1,total certainty, if theobserverobservesaninfinite numberof times, then theprobability of �� beingequaltothe currentlyobserved time fallsto 0.The de re anddeseevidencetakeeachother out, this de se evidence (not actually beingabletopickout aspecific nowwithinfinite uncertainty)cancels Huemer’s de re evidence forimmortality andfor its opposite such that thereisnoBayesianupdating at allfor theobserver(Jäger, 2022,p.6-9). Jägeris inclined to thinkthatitisunjustifiedtoinfer immortalityfromexistence forany arbitraryobserver but argues that therecan be such Bayesian evidence in certaincases with no first state(Jäger, 2022, p.17).
Jägerfurther argues that thetheoriesofconsciousness areindependent to reincarnation, consciousnesscould be connectedtobodiesinvarious ways (Jäger,2022, p.17).
RandallMcCutcheonin“Existenceisnot Evidence forImmortality” (2020) also argues againstthe Bayesian evidence argument of Huemer (p.1). McCutcheon argues that Huemer changesindexical expressionsintotheir secondary intension, so forexample “thatpersonexists1950” where“that person”issomeone specificintoits secondary intension(thesecondary intensionpicks that person in everypossibleworld instead),which is incompatiblewithanthropic reasoning. Usingthe second intensionchanges theBayesianevidenceinproblematicwaysinthiscase(McCutcheon,2020, p.2-3) McCutcheon argues that if reincarnation is to have evidence it wouldrequire some newargument (McCutcheon, 2020,p.7).
1.2Purpose andresearchquestion
Thepurpose of thestudy is to examineifexistence is evidence forimmortality by usingtransition systems. Thelogic used in thereasoning is onethataccepts thelawsofthought:propositionsare identicaltothemselves, thereare no contradictions, anda proposition is exclusivelytrueorfalse (which allows proof by contradiction forexample), forexample first-order logic(Russell, 2001,p.35) What will be considered is theworld that Ilivein(referred to as “the world”), but no weight is put into that,and theargumentation should applytoany possibleworld.Thismotivatesthe research question,which is:
Is existenceevidenceofimmortality if thereisnofirststate of aworld?
Theresearchquestion asks if an observer that is consciousatsomepoint in time recurorreincarnate infinitelyoften.
Thereasonfor choosing transition systemsinthe proofstrategyisbecause such systems, or similar ones,are used forvarious recurrence theorems regardingdynamical systems(Wallace, 2015,p.1-2). Themethodofusing transition systemsalsodiffersfromHuemer’sapproachinthat, forexample,an observer candirectlyprove theirimmortalitygiven theassumptionswithout theuse of probabilityin
a deterministicworld,requiresfewer assumptions, andisnot open to exactlythe same setof counterarguments.Further,“evidence”isinterpreted in abroader sensethanonlyBayesian evidence.
Proofs from otherassumptionsleading to assumptionsusedtoprove that existenceisevidenceof immortalityare also demonstrated becausetheyare instrumental in thediscussion.
Proofs caningeneral have objectionsagainst theaxioms, thetypeofinference system,orthe validity of theproofs(or some combination thereof).One delimitation is that thetypeofinference system will notbecriticallyassessed(whichismotivatedbythe type used beinga standard type). Another delimitation is that partial or wholemetaphysicalsystems that maycomeintocontradiction with the proofs but arenot mentioned in thepreviousliteraturewillnot be discussed.
1.3Disposition
Thefollowing partsofthe paperare subdivided into:‘2Transition systemswithobservers’and ‘3 discussion’. ‘2 Transition’isfurther subdivided into 2.1and 2.2. ‘2.1 Transition systems’ contains an explanation of what transition systemsare andwhatproperties they have,these properties arethen used in theproofsabout thetransition systemsin‘2.2Proofsabout transition systemswith observers’.‘3Discussion’containsa discussion aboutthe relation betweenthe previous studiesand theresults,and then adiscussionabout some potential future research.
2Transition systemswithobservers
Theentities used arenon-numberelementsinsets, numbers, andsets. Thenotation that is used is written aboutbyJoanBagaria in “Set theory”(2023). Thecardinality of aset is itssize, forexample theset of twoelementsis 2,and thecardinality forall of thenatural numbersorall of thewhole numbers is denotedby ℵ0 andisalsocalledcountable infinity(allfinite sets arealsocalledcountable sets). Twosetsare of thesamecardinality if they have abijection,thatisa one-to-one correspondencebetween theirelements, andtwo sets with acountably infinitenumberofelements have thesamecardinality (Bagaria,2023)
2.1Transition systems
Atransition system (TS) is defined as astate set �� anda transition function �� that maps everystate to some setofstates(aset that is possiblyempty). Each application of �� on astate is a time-step changing what thecurrent stateis.
AdeterministicTSisa TS wherethe �� does notmap anystate to more than onestate,and an indeterministicTSisa TS wherethe �� does mapatleast onestate to at leasttwo states.The determination of what statecomes next in an indeterministicTSisprobabilistic, followingthe standard Kolmogorov axioms of probability: theprobability of allpossibleoccurrencesconsidered addupto 1,possibleoccurrenceshavenon-zeroprobabilities,and theprobability of multipledisjoint occurrencesisthe sumoftheir probabilities,see theaxiomsin“Foundationsofthe theory of probability” by Andrey Kolmogorov &AlbertTurnerBharucha-Reid (2018, p.1-2).A first andfinal
state arestatesthatthe transition function cannot be appliedoninthe backward (applyingthe inverseofthe transition function)orforward direction respectively.
Note that thefollowing setofoutcomesofapplying �� onto thefirststate to yieldthe second andso on {��0, ��1, ��0, ��2} is notpossiblefor adeterministicTSbecause of “memorybeing encodedinthe state”,which meansthata stateina deterministicsystemcan only yieldanother stateif“some variable haschanged”, buta transition function is thestate setand thetransition function,which meansthatthe states must containthatchangemakingitsothat ��0 cannot lead to differentstates in adeterministicTS. To make this even more clear: acomputer contains itsentire memory andifa staterepresentsthe entire innerconfiguration of acomputer, then thememoryofthe computer is encodedwithinthe state.
A time interval �� is aset of numbers, such as {0, 1, 2}.A time-stepisa difference of 1unitina time interval.If ��0 is agiven stateand ��∈ �� is thenumberof time-steps that have passedsince thestate ��0 wasthe currentstate,thenthe followingequality holds: �� ��(��0) = ��(��(… ��(��0))) such that there are ����’s on theright-handsideofthe equation
An observer setisa setofthe states in asystemfromsomereference time (possiblytosomeother time if theobserverset is bounded).The observer canbeconsideredtobea part of certainstates, theobserverisassumedtobea consciousness, this is somethingthatcan be questioned andwillbe treatedinthe discussion.
Thestate spaceset of aTSisthe setofstatesthatcan possiblyappearatfuture timesrelative to some reference time.A system with aconserved statespace setiscalleda conservative TS.The setof states that cannolongerappear at future timesrelative to some reference time is called the wanderingset.Bythe definition of aconservative TS thewandering setmustbeempty fora conservative TS.Ifthe wholestate spacehas been passed in adeterministicTSthenthe wandering setcan notgrow.
Discussing or usingthe statespace andthe wanderingset is somethingthatisdoneinvarious recurrence theorems (Wallace,2015)
2.2Proofsabout transition systemswithobservers
Axiom(A):there is acountable number of states in thestate set.
Theorem(1):a TS with no first statecan notreach afinalstate.
Proofoftheorem (1)using axiom(A):assume that (1a) �� is thenatural number time forthe last state in thestate setand (2a) that �� canbeyielded from applying �� some naturalnumberof times. Assumptions(1a)and (2a) together with axiom(A) leadstoa contradiction becausenonatural numbercan be addedtonegative infinitytoget anyarbitrary naturalnumber, for �� =[−∞, ��] no number of applicationsof �� caneveryield �� forany naturalnumber �� andtherefore (2a) cannot be true when (1a) andaxiom (A)are true.Q.E.D.
Theorem(2):a TS with no first state converges to aconserved statespace
Proof of theorem(2) usingaxiom (A)for adeterministicTS: thereisa bijection betweenthe stateset andthe setofevery statethathas alreadybeenpassed(becausebothare of cardinality ℵ0)(Bagaria, 2023).Thismeans that everypossibility of adding astate to thewandering sethas been passed and thereforethe resulting statespace is conserved, thewandering setcan notgrowwhenevery statein astate spacehas been passed in adeterministicTS. TheTSwill neverreach afinalstate accordingto theorem(1).Q.E.D
Proof of theorem(2) usingaxiom (A)for an indeterministicTS: if thereisa non-zero probabilityofthe wanderingset growing, over an infinite time it will happenuntilthere arenostatesoronlystates with zero probabilityofdoing so left.The probabilityofatleast onestate ending up in thewandering setis �� andthatnostatesend up in thewandering setis 1 – �� at agiven state, then as that stateis returned to over andoveragain,tending toward infinity, thelaw of largenumbers (which follows from theKolmogorovian probabilitytheory, see“Foundationsofthe theory of probability” by Andrey Kolmogorov &AlbertTurnerBharucha-Reid (201,p.57-69))statesthatthe distributionsofthe events will converge to theprobabilities,and forthattobethe case at leastone statemusthaveended up in thewandering set(whichmay then change theprobabilities). TheTSwillnever reacha final state accordingtotheorem (1). Q.E.D.
Theorem(3):‘existenceisevidenceofimmortality’, that is,anobserverrecursinfinitely if thereisno first state.
Proof of theorem(3).Itfollows from theorem(2) that anyobserverstarting from some statebeing thecurrent stateiswithina TS wherethe currentwandering setisempty,meaning that theobserver will recurifthere is no first state. Q.E.D.
Theorem(4):a TS whereevery stateisyielded by thetransition function from some stateisidentical to aTSnot having afirststate.
Proof of theorem(4):ifstate �� is yieldedby ��− 1 andthere is astate at time �� then theremusthave been astate forevery time preceding �� by induction.Q.E.D
Theorem(5):the next stateisalwaysdefinedthenthere cannot be afinalstate.
Proofoftheorem (5): if thenextstate is always defined then anystate that thetransition function reachesmustmap to at leastone state, possibly itself,toallowany number of time-steps.Q.E.D.
3 Discussion
Theresearchquestion is:
Is existenceevidenceofimmortality if thereisnofirststate of aworld?
Existenceisevidenceofimmortality if thereisnofirststate of aworld,the question is answered in affirmative by theorem(3) givena very reasonable axiom. Concerning axiom(A):evenif time is fundamentally continuous, an observer candiscretize time by dividing up continuous time into discrete chunks,which allows axiom(A) to hold.Continuous time is associated with Zeno’s paradox, howeverthere arevarious proposed solutionsthatholdthatcontinuous time passingisnot actually problematic, andthe problematicpartisnot thechunkingofdiscreteintervals either way, seefor exampleBertrandRussell in “Our knowledgeofthe external world” (2015, p.168-198)
Theorems (1)-(3)are themaintheorems, asserting theantecedentintheorem (4)for agiven transition system then that transition system does nothavea first state– whichisthe antecedent in theorem(1) whichleads to theconclusionthatthe giventransition system cannot reacha final state. Theorem(5) is mostly an interesting finding in this work that hassomevalue in thediscussion.For example, asserting that thereisnot an infinitenumberofdifferentstates, therebeing no first state andthe antecedent in theorem(5) fora giventransition system,thenthe transition system is either cyclic or will become cyclic at some pointin time becausethe next stateisalwaysdefinedand at some pointthe wanderingset will stop growingand thestatesthatare left will be repeated
Both Jägerand McCutcheon pointout thedere/de se issueinHuemer’sargumentregarding the Bayesian evidence,and it very well maybeflawed in that way, butthatflaw does notapply for theorems (1)-(3).
Theobservercorresponding to aconsciousness wasassumed,and this is similartowhatHuemer does (Huemer, 2021, p.11-12).Jäger essentially agrees with this treatmentand states that the workings of consciousnesscan be treatedseparately(Jäger, 2022,p.17).I wouldargue that the argumentsthatI have made is aboutthe worldbeing cyclic,and includedconsciousness as apartof theworld,itdoesnot matter if theworld is idealist andthatthere wouldbeonlymentalphenomena in it or if thereisalso, or exclusively, amaterialpartofthe world. Theargumentappliesaslongas thereissometypeoftransition system,and thetheoremsfollowasthe givenassumptionsfor a theoremholdtrueand thegiven inferencesystemisused(an inferencesystemwiththe laws of thoughtwas in this case,suchasfirst-orderlogic). Theassumptionsand theinference system canof course be questioned or criticized, andthe conclusionsmay notbecompatiblewithcertain metaphysical systems.
Thetheoremsand proofs areindependent to if theworld is deterministicornot.Itcan be notedthat theindeterministiccaseoftheorem (2)there is convergence to aconserved statespace,thisis similartothe so called “Infinite monkey theorem”,ifa monkey randomly hits random keys on a typewriter then over an infinite time thefulltextwritten on thetypewriterwould converge to atext containing everypossibletext, it is assigned probability 1 to happen (Gut,2006, p.100-101).Thisis typicallytaken to mean that it does happen,but it is slightly weaker than that
It is worthtomention that reincarnation of observersinwhathas been describedmeans that the observerswillrecur exactlyastheywereover time,which is stronger than therecurrenceinthe Poincaré recurrence theorem, anditalsomeans that this specifictypeofreincarnation is not compatiblewith‘memories from previous lives’. ‘Memoriesfrompreviouslives’require adifferent kind of reincarnation andhence requires otherevidenceand that meansthatthistypeof
Thisstudy is concernedwith the question if existence is evidence ofeternal recurrence, that acurrent observer is within acyclic world, if the past is infinite. Michael Huemer proposeda proof of existence being evidence of immortality using aBayesian approach, which is discussed, as well as various counter arguments. This study then uses transition systems, anon-Bayesian approach, to prove various results about worlds that can be described by them.Itisprovedthat in transition systems with an infinite past, where time can be discretely subdivided, eternal recurrence is the case for every observer in aworld described by such asystem.Finally,the reasoning, potential and actual counter arguments, consequences,and future research are considered.