Loke Hagberg
Existence is indeed evidence of immortalityifthe past is infinite
Or,onproofs of eternal recurrenceusing transition systems 2025-02
Existenceisindeed evidence of immortalityifthe past is infinite
Or,onproofsofeternal recurrence usingtransitionsystems
2024-06-01 (withadjustments 2024-11)
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 infinitepastand afinite stateset,where time canbediscretelysubdivided, eternal recurrence is thecasefor everyobserverina worlddescribed by such asystem. Finally, the reasoning, potential andactualcounter arguments, consequences,and future research are 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 ochenändlig tillståndsmängd, där tidenkan delasupp diskret, är evig återkomstfalletför varjeobservatöri en världsom beskrivs av ett sådant system.Slutligen behandlasresonemangen, faktiskaoch potentiella motargument, konsekvenser, ochframtida forskning.
Acknowledgements
Iwould like to thankmywifeYuliyaHagberg forsupport throughout theprocess. 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 (6)),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 amount of times, whichissomething that hasbeen stated by Hinduscholars, Seneca,Nietzsche,and other thinkers (Teresi, 2010,p.174;Huemer, 2021,p.4). However, Nietzschemight only have used eternal recurrence as ametaphor(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’, andtherefore argued that 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 observerstakemultiplebranches, then 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”thatisthe same forevery observer (Taylor& Wheeler, 1992,p.1-20).Thismeans that thetheoryofrelativity is compatiblewiththe worldbeing subdivided into states whereone state passestoanother as time changes. Thereasonwhy measured time is relative forobservers canbe interpretedtobethatthere is alimited computation that is spentdifferentlydepending on pathdistance throughspacetime (Wolfram,2020, p.22-42).A cyclic or recurrentworld is onewhere events occurinloops,theycycle,and time is infiniteintothe past andintothe future.There are currentlyvarious models in cosmologythatare cyclic andhaveinfinite pasts(Huemer,2021, p.2-3).
Perhapsitcould be possibletoprove 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 amount of times– they reincarnate(Huemer,2021, p.1)
Huemer points outthatitismathematicallyconsistentthatthe worldhas afirststate,but it seems that we couldask what came before it just like we askwhatliesbeyonda possibleboundary of space (Huemer, 2021,p.1-2). LeonardMlodinowand StephenHawking pointedout that asking the 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 itslow entropyand that it couldjustaswellhavebeenthe case that theworld hada first statestarting in the1950’sinstead, with ahigherentropy,and that humansatthat 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 amount of times. It is notlogically necessary that theuniversehas aboundedand conservedphase space however(Huemer,2021, p.4).Huemeralsopointsout that this happensgiven aboundedand conservedphase spaceeventhoughthere is atendencyfor entropytoincrease(Huemer,2021, p.1011).
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 circumstance of abodyhavinga soul is repeated it couldindeedhavethe same soul (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 Bayesian proofofHuemeralsoassumes,for example, that thepriorsare non-zero.Jäger points out 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 infiniteamountof time-chunksthe probabilityofbeing aliveatany given time is 0 if theobserverisnot infinitelyrecurring.Ifanobserverhas non-zero priors forexisting a finitenumberof timesand an infiniteamountof 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 amount of 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 adeterministicworld,itrequiresfewer assumptions, anditisnot 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 partsofthisthesisare 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 andisalsocalled“countableinfinity”(allfinite sets as well as countable infinitesets, thosewithcardinality ℵ0,are countablesets).Two sets areofthe same cardinalityif they have abijection,thatisa one-to-one correspondence betweentheir elements,and twosets with acountably infiniteamountofelementshavethe same cardinality(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,and it is said that this statemapsthe states in the set).Eachapplication of �� on astate is a time-stepchangingwhatthe current stateis.
AdeterministicTSisa TS where �� does notmap anystate to more than onestate,and an indeterministicTSisa TS where �� does mapatleast onestate to at leasttwo states
Thedetermination 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).The transition probabilities arefixed, theMarkovpropertyisobeyed, andthe transition probabilities are timehomogeneous(theprocess is memoryless anddoesnot depend on time)(Häggström, 2002,p.8-10). If astate canhavecertain transition probabilities at one time andothersatanother then thesecases couldbetreated as differentstateswithfixedtransition probabilities instead. Note that astochastic matrix,alsocalled“transition matrix”, candescribethe transition probabilities entirely in thecaseof afinite stateset –withanimportant labelofsomestatesthatmap back to themselves with a probabilityof 1 butare actually final states.
An irreducibleMarkovchain is onewhere everystate canreach everyother statethrough some numberoftransitions. Theanalysisofa reducibleMarkovchain’s long-termbehaviorcan be found outbyanalyzing oneormoreMarkovchainswithsmaller statespaceswhich eventually are irreducibleMarkovchains, andirreducible Markov chains (Häggström,2002, p.23-25).A Markov chainisaperiodic if allits states have thepropertythatthe greatest commondivisor of theset of timesthatthe chaincan return to agiven stategiven that onestartsatthatstate is 1 (Häggström, 2002, p.25).
Afirststate anda final stateare states that thetransition function cannot be appliedoninthe backward (applyingthe inverseofthe transition function)orforward direction respectively.
Note that thefollowing setofoutcomesofapplying �� onto thefirststate to yieldthe second andso on so thesequence (��0, ��1, ��0, ��2) is notpossiblefor adeterministicTSbecause of “memorybeing encodedinthe state”,which meansthata stateina deterministicsystemcan only yieldanother state if “somevariablehas changed”,but atransition function is thestate setand thetransition function, whichmeans that thestatesmustcontain that change making it so that ��0 cannot lead to different states in adeterministicTS. To make this even more clear: acomputer contains itsentire memory andifa staterepresentsthe entire innerconfiguration of acomputer,thenthe memory of the 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 setis‘aconservative TS’. Theset of states that cannolongerappear at future timesrelative to some reference time is ‘the wandering set’.Bythe definition of aconservative TS thewandering setmustbeempty fora conservative TS.If thewhole statespace hasbeenpassedina deterministicTS, then thewandering setcan notgrow.
Discussing or usingthe statespace andthe wanderingset is somethingthatisdoneinvarious recurrence theorems (Wallace,2015)
2.2 Proofs abouttransition systemswithobservers
Axiom(A):there is acountable number of states in thestate set.
Axiom(B):the stateset is finite.
Theorem(1):a deterministicfinite stateset TS with no first statehas aconserved statespace
Proofoftheorem (1)using axiom(A):somefinite sequence of theform (��0,… , ����) with no repetition of states or (��0, ��0) must have occurred andincludesevery statebecause thestate setisfinite,then forthe first sequencesomestate must have mapped to ��0 andbecause everystate except ���� already maps to anotherstate then ���� must mapto ��0,which meansthatsucha TS hasa conservedstate space. Q.E.D.
Theorem(2):anindeterministicfinite stateset TS with no first state converges to aconserved state space.
Proof of theorem(2) usingaxiom (A): by definition an indeterministicfinite stateset TS stateisa time-homogeneous discrete-time Markov chain. Every time-homogeneous discrete-time finitestate setMarkovchain’s long-termbehaviorcan be analyzed by analyzingsuchMarkovchain’s that are irreducible, andiftheyare aperiodictheywill converge to auniquestationarydistribution (see Häggström(2002,p.28))and if they areperiodictheywill oscillate, that is in thelimit as thenumber of transitionsgotowardinfinity (which is reasonable as thereisnofirststate). Therefore, an indeterministicfinite stateTSwithnofirststate converges to aconserved statespace.Q.E.D
Theorem(3):aninfinite stateset TS with no first stateonlypossiblyhas or converges to aconserved statespace
Proof of theorem(3) usingaxiom (A): Fora deterministicTS, if �� =(−∞, ��] wherethe states are labeledbythe time-steps andevery state ���� maps to state ����+1 except forstate ���� that is afinal state, then thestate spaceisnever conserved. Forthe indeterministiccase, asimilar TS as the deterministicone canbeconsideredwhere everytransition hasa probabilityof 1 except that from state ����−1 it mighteithermap to ����,1 or ����,2 whichare both final states.Itishowever also possible that thefinalstatesbothfor thedeterministicTSand theindeterministicTSwas astate that only mapped to itself.Q.E.D.
Theorem(4):‘existenceisevidenceofimmortality’, that is,anobserverrecursinfinitely if thereisno first stateand thestate setisfinite
Proof of theorem(4) usingaxiom (A)and axiom(B).Itfollows from theorems (1)and (2)thatany observer starting from some statebeing thecurrent stateiswithina TS wherenostate is addedto thewandering setfromthe time of thecurrent stateasthe reference time,meaning that the observer will recurifthere is no first stateand thestate setisfinite.Q.E.D.
Theorem(5):a TS whereevery stateisyielded by thetransition function from some stateisidentical to aTSnot having afirststate.
Proof of theorem(5):ifthe stateat time �� is yieldedbythe transition function of thestate at time ��− 1 andthere is astate at time �� then theremusthavebeena statefor every time preceding �� by induction.Q.E.D.
Theorem(6):the next stateisalwaysdefinedthenthere cannot be afinalstate
Proofoftheorem (6): 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
3Discussion
Theresearchquestion is:
Is existenceevidenceofimmortality if thereisnofirststate of aworld?
Existenceisevidenceofimmortality if thereisnofirststate of aworld,the question is answered in theaffirmative by theorem(4) giventwo axioms
Concerning axiom(A):evenif time is fundamentally continuous, an observer candiscretize time by dividing up continuous time into discrete chunks, whichallows axiom(A) to hold.Continuous time is associated with Zeno’s paradox, howeverthere arevarious proposed solutionsthatholdthat continuous time passing is notactuallyproblematic, andthe problematicpartisnot thechunkingof discrete intervalseitherway,see forexample Bertrand Russell in “Our knowledgeofthe external world” (2015, p.168-198).
Concerning axiom(B):itisthe case if theworld,exceptfor thedimension of time,isfinite (and bounded) in itspossibleconfigurations. In thecontext of aworld with no first stateaxiom (B)seems to be abit more reasonable to accept than outsidethatcontext,after allitmeans that thebig bang wasnot thebeginning.One argument for(B) in this contextisthatitfollows if onedoesnot accept that an infiniteamountofstatescan pass such that anew stateisreached.There is also aproposalof apossiblediscreteSchrödinger evolution by Carroll(2023)and he states that,“Theoverall evolution of this discretizedmodel is periodic in time,withprecisely �� distinctstatesrealized alongthe way. Theevolution of theuniverseiscyclic,repeating after �� timesteps.”(p.4).Notealsothataninfinite stateset in thesamecontext couldalsoallowfor acyclicworld
Theorems (1)-(4)are themaintheorems, asserting theantecedentintheorem (5)for agiven transition system then that transition system does nothavea first state– whichisanantecedentin theorems (1)and (2)which leadstothe conclusion that thegiven transition system cannot reacha final state. Theorem(6) is mostly an interesting finding in this work that hassomevalue in the discussion.For example, asserting that thereisnot an infiniteamountofdifferentstates, therebeing no first stateand theantecedentintheorem (6)for agiven transition system,thenthe transition system is either cyclic or will become cyclic at some pointin time becausethe next stateisalways defined andatsomepoint thewandering setwillstopgrowing andthe states that areleft will be repeated.
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 and afinite state set, where time can be discretely subdivided, eternal recurrence is the case forevery observer in aworld described by such a system.Finally,the reasoning,potential and actual counter arguments, consequences, and future research are considered.