.Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo The orthodox be acontin to it - approach states time to The Growing Blockand theProblem TEMPORAL EXTENSION TelUniversity Aviv Shira YechimovitzShira insights; and two the anonymous to constructive comments for their and referees suggestions. and group lab the Miller forvided; Centre and the at Kristie ofTime, Sydney and for University the helpful remarks conversations* Iwould and for Belkind fruitful thank Ori guidance; like for support the Cohn the Institute pro they - on account of being it ahybrid A-B-theory. Tension lies Finally, discuss Iwill possible consequences. uum. show to paper Inthis Iaim growing the that block slices of present.slices First, Ishow acontinuous that grow- ing block necessitates a present with zero temporal apresent with ing block necessitates acontinuous that in fact the block isbuilt B-theoretical model poses aunique problem continuity the to of time, duration; second, Ishow such that notion of present rules out some widely accepted B-theoretical solutions outrules some B-theoretical accepted widely BECOMING to the B-theory rules out B-theory the rules to some of the ones.A-theoretical problem the to of continuum, the commitment its while through the becomingA-theoretical of instantaneous of theContinuum CONTINUUM * theory. temporal order and direction in the a hybrid A-B theory, in particular focuses on growing the block as research currently Her University. of Science and Ideas, Tel Aviv andfor History the Philosophy Student Cohn the at Institute Shira Yechimovitz isan M.A. TEMPORAL DIRECTION GROWING BLOCK 85  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo A-series. Theyentailaspace-liketimeandmetaphysical pictureoftheuniverse as A-series (McTaggart 1908). A-categories, inrespecttowhichtimechanges. The standardconceptionisthattimeandspacearecontinuous.Sincecom- What changedisthefactthatanewentitycame intobeing,thusforming newrela- of timeisthatitappearstoinvolve somesortofflow orchange.Itseemsasifevery comes totime. er point;anditisinfinitelyindefinitelydivisible: any divisionofthecontinuumwill ed of point-like durationless instants. The continuum is gap free: it persists without cess iscalled becoming(Broad1923, 65-68). one by one,thusforming afour-dimensional growing block. Thefuturedoesnotex- est son.Achangehasoccurred,yet Tom himselfdidnotundergoany intrinsicchange. counts ofcontinuity.Finally,thegrowing blockviewisahybrid ofthetwo. Itiscom- the relations between its terms will be referred to as B-relations. The second feature does notonlyhave anorder,italsohas apreferred direction:itrunsfromearlierto dimensional slicesofreality keepcomingintoexistence,joiningthe world history tions withthepre-existingmembersoffamily. Suchisthechangeoftime.Three- dynamic metaphysical viewandaretherefore moreinclined to embracedifferentac- ther inthepast.Theserieswhosetermsarepast,present,andfutureiscalled than hisfather.Thechangeofthingsintimeis relative totime.Thechangeofamo- ter isthechangeTom Smithundergoesashegrows tallerthanhisfatherJohn.There I. a staticfour-dimensional spacetimeblock.Suchviews arecompatiblewiththecur- a longlastingphilosophicalandmathematical inquiry. The tensionincreaseswhen it always resultinpartsthatcanturnbefurtherdivided,withoutever resultingin any breaksorinterruption;itisdense:between eachtwo pointthereisalways anoth- so untilthebirthofanewsibling.Afterwards, Tom willnolongerbeJohn’s young- as anewchildisbornintothefamily.IfTom isJohn’syoungest son,hewillremain notion of time, since it lacks two important unique features. The first is that time rently accepted Cantorian account of continuity. A-theories argue that the A-series monly consideredstructurallysimilartoaone-dimensionalspace,thetimecontinu- ment intimeisdifferent,andcanbecompared tothechange Tom Smithundergoes mitted bothtotheviewofuniverse asa four-dimensional spacetimeblockthat put inthesametermsaschangethingsrespecttotime.Anillustrationoflat- um isoftenportrayed astheEuclidian straightgeometricalline.Justasthe is objective andthings in timedochangerespecttoitsterms.A-theoriesentaila instant intimechangesfrombeingfuture,topresent,furtherandfur- its beingconstitutedby pointsontheother,raisechallengesthatwere thesubjectof indivisibles (Bell2019, viii).Thesepropertiesofthecontinuumononehand,and is added,itbecomesthe newpresentandtheprecedentslicebecomes past. Thispro- ist. Thepresentisthemost recentslicethatcameintobeing,andonce anotherslice is atimetinwhichthesonshorterthanhisfatherand timet’inwhichheistaller is eternallyfixed inB-relations,andtothe claimthatthereareobjectively distinct later. Thetemporalseriesthatrunsfromearliertolaterisknown astheB-series,and line ismadeupofextensionlesspoints,thetimecontinuumtakentobeconstruct- A geometricallineisnotafulldescriptionoftheordinary(common-sense) According tothegrowing blocktheory,thechangeoftimeitselfcannotbe The growing block’sdualnature makesthepuzzle of thecontinuum Introduction B-theories oftimeembracetheB-seriesandrejectobjectivity

86 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo A-categories inthemodel,whichentailsthatpresentmomentcannothave any This distinctionisderived fromwhatMillercalls«thegrowing blockontologicalthe- The A-theoreticalcommitmentsofthegrowing blockentailanobjective presentmo- The onlyintelligibleoptionfor thegrowing blocktheorististoadmitthatthetempo- When holdingtoacontinuous growing blockview,apoint-likepresentisinevitable. (Miller 2013, 348). Thepresentmomentistopologicallydistinctfromtherestof of theproblemspoint-likepresentbringsaboutandtheiracceptedsolution.Ithen of point-like instants. In the following section I show that the present slice cannot clude whatMillercalls«thedynamicthesis»,accordingtowhichtheanswer tothe ordered, directedblock, butisalsothelatest additiontoreality.But these two defin- entails enumeration, or consecutiveness, meaningtheinstantiationoftemporal entails notonlytheorderingrelationofearlier and laterthan,orsuccession–italso of onetemporalsliceatatime.Asnewcomes intoexistenceontheedgeofre- change ispossibleduetotheprocessofbecoming; namely,thecomingintoexistence this isnotthefullpicture;growing block’sA-theoreticalcommitmentsalsoin- than zerotobecomeallatonce. two pillarnotionsofthegrowing blocktheory.Thefirstisthe very definitionofthe der two restrictions:(a)thepresentcannothave any non-zerotemporalextension; features: one,ithas no successors;andtwo, itisnotjustthelast temporalsliceinthe question aboutwhichmomentisthepresent ever-changing (Miller2013, 348). This show why theydonotapplytothegrowing block.InthefinalsectionIdiscusspossi- seemingly impossibletoresolve, astheblockgrows throughthebecomingofthree-di - sence ofapresentevent isnotthatitprecedesfutureevents, butthatthereisquite sis»; namely, that the pastand the present exist, while the future is a non-existence stantaneous slice of reality at a time. These restrictions are a direct consequence of and (b)thegrowth oftheblockcanonlyoccurthroughbecomingonesuchin- II. as moreslicesaccumulateoneby oneinagradual process.Theprocessofbecoming ality, theslicethatwas oncepresentbecomes thepastandgoesdeeperinto Hence the very definition ofthe present inthe growing block consists oftwo distinct mensional sliceswithzerotemporalextension.Orifwe thinkoftimeasastraight ment thatisdistinctfromthetwo othertemporalcategories,thepastandfuture. non-zero duration;thesecondpertainstoforming oftheB-relationsthrough ral dimensionoftheblockisconstructedthroughadditionpresentslicesun- ration greater thanzero.Eachnew slicewould thenhave sometemporal extension, process ofbecoming,whichcannotpossiblyallow for any durationoftimegreater particulars oneby oneandtherefore one before theothers(Craig2000,234-235). ing features ofthepresent cannotbeconsistentwiththenotionof extendedpresent. have any non-zerotemporalextension,andthattheblockcanonlygrow throughthe line, timeinthegrowing blockuniverse literallyisformed throughtheaccumulation literally nothingtowhichithastherelationofprecedence»(Broad1923, 66). But ble consequencesandconclusions. becoming of one such durationless slice at atime. In the third sectionIlay outsome block by virtueofbeingtheonlysliceexistencethathasnosuccessor:«thees- II.1. The ProblemoftheDefining Features ofthePresent The Growing BlockandtheProblemofExtendedPresent Let usassume acontinuousgrowing blockinwhich the presenthasdu-

87 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo 191). Inthecontextof thegrowing block,pastgenesisrefers tothingsintimecom- As a result we will end up with an entity that is contradictory by its very definition: 2002) or «the now-now problem» (Braddon-Mitchell 2004) might be summarised ways have infinitelymany successors,andnoamountofdivisionwillever leadtoa even ifvery small.For convenience, considersuchanextendedtemporal sliceasa well. Indeed, the objectors could say that it is easy to see why the of thepastthatwas borndeadismorethan justgrim;pastgenesisdisqualifiesthe of thepresent,whileavoiding theproblemsofpoint-likepresentandcontinu- end-point; namely,theboundarybetween thisnewtemporalinterval andnothing- good reasonsnotto.For one,itstrengthensanotherobjectiontothegrowing block grow inextendedtemporalintervals mademostlyofpastwithoneuniquepresent force restriction(a),itdoesnotnecessarilyimplythat(b)hastoholdas the latestadditiontoreality,eachofthemwould alsobeconsideredapresentslice. that inturnarealsoindefinitelydivisible.Sincethosesub-intervals areallpartsof temporal interval onthegrowing block.Any suchtemporalinterval ofthegrow- this very instantcansustainactivityorprocesses andconsciousness.Themisfortune the now-now problem.(Forrest 2004,Forbes 2016) According todeadpastgrowing that we ever were trulyinthepresentisfaint.Pastgenesisalsoweakens thedead do we have noway ofknowing whetherourtimeisthetruepresent,buthope the edgeofrealityandpastismerelyanything thatprecedesthepresent– the following discussionofthe way theblock’sB-relations areformed. dead pastgrowing block’slineofdefence againsttheepistemicobjection, aswell as single presentslicethathasnosuccessors. a continuousinterval isbeingindefinitelydivisible,any such“present”slicecouldal- a “present”slicethathasoneormoresuccessors.Andsinceoftheproperties aphysic, we aremuchlikeliertobe inthepast.Ifwe embrace(a)butnot(b),only as follows: we believe thatourtimeisthepresent,butgrantedagrowing blockmet- supporters of past genesis might say –there is no reason why the block should not straightforward, conclusive way toprove pastgenesis tobeimpossibleinagrow- suggested by Forbes (2016), thatreliesonthefactthingsinpastusedtobe against theproblemofpasttruth-makers,inparticular theresolutiontoproblem reply totheobjectionsproblemsofcontinuummay give raiseto,butthereare ness. Thus,theproponentsofpastgenesisattempttoholdondefining features point-like, butthenask:musttheblockgrow throughthebecomingofoneduration- past growing block:avariety ofthegrowing blockthatwas conceived inresponseto present; itisnotonlythe lastmoment,butalsothelatest. Thiswillbecomeclearer in present. um (tobediscussedinsectionIII).Itmightseemtemptingtoturnpastgenesisasa ing blockwould thenhave tobeindefinitelydivisibleintosmallertemporalintervals ing intoexistencedirectlyaspast,withouthaving beenpresentfirst.Iftheis ing block’spresentby appealingtowhatForbes calls«pastgenesis»(Forbes 2015, ing blockuniverse, as pastgenesisignoresthesecondfeature of thegrowing block’s known astheepistemicproblem. less sliceat atime?Inlinewiththisobjection,somemighttrytoreconciletheconflict blockers, we can know withcertainty that we areinthe present moment,sinceonly between theextensionoftemporalslicesanddefining features ofthegrow- A possibleobjection might be that while this line ofargument might en- The epistemicproblem,alsoknown as«thepresentproblem»(Bourne Setting asidetheepistemic objectionandthedeadpastsolution,there isa present has to be

88 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo As proponentsoftheA-theoryobserved, theprocessofbecominginvolves notonly The derivation ofsuccessionfromconsecutiveness rendersboththe extendedpres- The successionoftemporalslicesdependsontheconsecutiveness oftheirbecoming, after Salreadyexisted. of timeorinourconsciousness,ratherthanfeatures oftimeitself.Butthegrowing widely heldthatincontrastwithastaticB-theory,objective temporalbecominghas will takethemtobeprimitive; others,astheresultofasymmetriesincontents event, atemporalsliceoraninterval of timeisimmaterial–itwould have hadtobe ent tohave any non-zeroduration.TheB-relationsofearlierthanandlater in ent andpast genesisasimpossiblefor the growing block. curred consecutively, orelsetheywould have beenoneandthesameslicethere - consecutiveness, meaningtheinstantiationoftemporalparticularsoneby one which makesitimpossiblefor asingletemporalslicetohave internalsuccession.Like growing blockmodel,theB-seriesstemsfromA-series.Whenanewsliceofreal- greater thanzero;whenever thereismorethan oneslice,itmeanstheirbecomingoc- the abilitytoaccountfor temporaldirection(Craig2000,256-258), andthegrowing t is laterthan slice S’ at time t’ is the direct consequence of S’ coming into existence fore itcameintobeingthatslicewas nothingatall(Broad1923, 66).That includesthe different strategiestoaccount for theorderand directionofthetimeseries.Some the B-theories,growing blockiscommittedtothethesisthateverything thatex- therefore ittakesmorethanoneslicetoform atemporalsegmentwhosedurationis tween itsearlierandlaterparts.Butaspreviously shown, anything thatcomesinto the growing blockareformed throughbecoming.Asatemporalslicecomesintoex- fore have a singleB-locationandzeroduration.Thus,there canonlybeonenewdu- the notions of temporal succession and consecutiveness are intimately connected. able, successive parts.Thecontentsofeach newslicearealways simultaneous and spread acrossseveral B-locations,andthere would have hadtobeadistinction be- anything in the spatiotemporal block to have duration – whether we refer to it as an a slicenecessarilyoccupiessingletemporallocationintheB-series.Obviously, for successor oftheslicethatwas formerly theedgeofbeing.Thus,any amountofex- and therefore onebefore theother(Craig2000,234-235). Inthegrowing blockview, succession, i.e.,theB-relationsofearlierthanandlaterthan,butalsoenumeration,or relations of temporal succession. To put it moresimply,thefactthatsliceSattime reality comeintoexistencethatdidnotandcouldhave existedbefore, sincebe- neous inrespecttothetemporaldimension,whichmeansallofcontentsuch rationless sliceatatime, andonlyonesuchslicecanbethelatestaddition toreality. ity isaddedthroughbecoming,newrelationsbetween thissliceandthesumtotalof ists intimedoessoeternallyfixed B-relations.DifferentB-theorists may employ istence thatjoinstheuniverse throughasingleactofbecomingcanonlybesimulta- istence throughtheprocessofbecoming,itconcurrentlybecomesimmediate block makesuseofthisnotionbecomingfor thisvery purpose. block are not arbitrary or illusory; rather they are objective features of time itself. It is block alsoadherestothecommitmentthattemporalorderanddirectionof being together is B-simultaneous, and therefore cannot be divided into distinguish - II.2. The ProblemofConsecutiveness inBecomingandtheForming of B-Relations In otherwords, very muchlikeinMcTaggart’s originalargument,inthe It now becomesclearthatthegrowing blockdoesnotallow for thepres-

89 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo The factthatthegrowing blockbuildsthroughtheaccumulationofdurationlessslic- 231a21-b10). Thecontinuumhasthepotential tobeinfinitelydivided,notintopoints, or elsetheblockcouldnotbegap-free.Inthatcase,how canB-locationsbedistin- ed throughbecoming,itmustalsocometobeexactlyonthevery edgeoftheblock, es posesatwofold problem.Ontheonehandthereisproblemofaddition;clear- challenge tothetimecontinuumingrowing blockmodel;thefirstisprob- ent is point-like, the temporal location of each and every new slice would be the same equally toany numberbetween nandn+1,nomatterhow small.Given thatthepres- come allatonce,whichIshowed tobeimpossibleonthegrowing block.Thisapplies es. We cannotsay thatthe timecontinuumispriortothesetemporalslices, because curs intheprocessofbecoming.Given agrowing blockmetaphysics, therehastobe cannot bedividedeverywhere atthesametime.Aristotle’sargumentisasfollows: guished onefromtheother?Letedgeofrealitybeintimet growing blockmetaphysics allows for thetimeseriestobedense. growing blockaccountpreviouslygiven, itbecomesevidentwhy proponentsofthe tions; thepresentsliceliesatvery edgeoftheblock,andwhenanewsliceisadd- tially divisibletoinfinityinthesensethattheymay bedividedanywhere, thoughthey first introducedinAristotle’sdiscussionofZeno’sparadoxes andlaterembraced by them tobe actual entities, thecomponents oftheblockand notjustthepotential these slices. Thus,ifwe are committed tothegrowing blockview we mustadmit tinuum is prior to its parts and points exist not in actuality, but as limitsoflines(Phy. first, sincepointshave noparts,theycannot form acontinuum,andsecond,the con- zero duration,andeven infinitelymany suchslicescouldnever amounttoany du- are actualizedonthatvery location,therecanbenofurtherlocations;inorderfor the as ofitspredecessor,andconsequently,theentireblock. such athingasanactual instantaneoussliceoftime,andasIshowed theblockcan- are the boundaries of the segments that are the parts of the continuum, but any such III. In thissection I will explore two ways inwhichthepoint-likepresentmay posea Brentano, Peirceandothers.(Bell2019, 157, 163). Continuousmagnitudes arepoten- ration greaterthanzero.Ontheotherhandthereisproblemoftemporalloca- model mightwant tohold ontosuchanassumption. mon: theydiscredittheunderlyingassumptionthatlengthofaninterval onaline nothing would have existedintimeifitwere notfor theconstantaccumulationof not existpriortothese slices andcanonlygrow throughthebecoming ofsuchslic- parts onlycomeintobeingasthewholesaredivided. up andform temporalduration(zerotimesinfinitystillequalszero);andthesecond is thatoneofthepropertiescontinuumdensity,yet itisunclearwhethera is generatedby addingupthelengthsofitssmallestcontinuants.Lookingbackto ly theblockitselfhastemporalduration,butitconsistsofindividualslicesthathave lem famouslyknown asZeno’sparadox ofplurality:how candurationlessslicesadd location but intosegmentsthatcanalsobeinfinitelydivided, andsoon,on.Thepoints III.1. t The firstsolutionistoviewthelineaspriorpoints.Thisapproach was The solutionsIamabouttodiscussinthissectionhave onethingincom- The Point-LikePresentandtheContinuum n+1 But thiscannotbesaidofthegrowing block, seeingthatadditioninfactoc- tocomeintoexistence, a temporalinterval fromt The ProblemwithPoint-LikePresentsForming Duration n tot n . Aslongastheslices n+1 willhave tobe-

90 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo 2010, 277-280). Therefore, theextensionofacontinuousline doesnotdependonthe of t es constitutingtheblocktobeactual.OurbestsciencetakesCantorianapproach whatsoever onitslength.Instead,theline must beassociatedwithametricordis- val’s endpoints.Sincelengthisnotanintrinsic feature ofasetpoints,thepoints count, and then show how it resolves the paradox of plurality. Then I will show why was abletoprove thatany finitesegmentontherealnumberline–regardlessofhow would berighttothinkofthesegment’slengthassumlengthsthose correspondence to any other proper subset on the real number line,and to the real of realnumberscannotbeputinaone-to-onecorrespondencewiththeintegers. ence with the set of positive integers, which is neither gap free nor dense. Cantor each ofitsmembershasfinitelymany predecessors.Anexampleofadenumerable was abletoestablishaone-to-onecorrespondencebetween thesetofintegersand tn consistsofslicesS1+S2+…+Sn.Letussay thetemporaldurationofuniverse the growing block,we willrunintoaproblem. Thesumtotalofrealityatany time tance function (see Dainton 2010, 281-283). But if we try to apply this answer to themselves donotmatter.Theadditionof a singlepointtoaninterval hasnoeffect to thecontinuum,whichmodelscontinuumafterrealnumberline.Cantor they cannotexistpriortothebecomingoftheircontent. the bigbang, therewillbenotemporal durationatall.Another thingisthataccording treat everyday objects. denumerable. Aninfinitesetisdenumerableifitcanbeorderedinsucha way that ferent sizesofinfinity.Aninfinitesetcanbeeitherdenumerable(countable)ornon the Cantoriansolutioncannotbeappliedtogrowing block. there are more rational numbers than integers on finite segments of the same length, the setofrationalnumbers,whichisdense,butnotgapfree.Sodespitefactthat as oftn+1begreaterthantheduration oftimeastn?Even ifwe assumethatas a thenewtimetn+1,whicharealsodurationless. Sohow canthedurationoftime as oftimetnisnotthesumduration the temporalslicesforming it(which sion, we cannotsay thattheaddingofSn+1toblockmakesitgrow inrespectto amount ofpointsfor ittohave length. Thelengthisthedistancebetween theinter- short orlong–containsthesamenumberofpointsasany othersegmentonthereal size oftheirsmallerparts.Ifasegmentthesidewalk consistsofpaving stones,we short, containsthesamenumberofpointsasentirerealline(Dainton set istheofintegers.Alldenumerablesetscanbeputinaone-to-onecorrespond- Furthermore, any propersubsetontherealnumberlinecanbeputinaone-to-one ry”. According tomeasuretheory,aninterval mustcontainnon-denumerablyinfinite number line,andastheentirerealline.IwillbrieflypresentCantorianac- number lineitself.Hence,any segmenton acontinuousline,nomatterhow longor paving stones.Butsinceany segmentonacontinuouslinecontainsthesamenum- points itconsistsof.Ineveryday life, we takethesize of objectstobethesum its temporal duration.Andifwe assumethereisastarting pointtorealitysuch as has tobezero).Reality thenincreasesthroughthebecomingofanewsliceSn+1at boundaries of its potential parts. Moreover, the problem of locations still remains as by addingupitspoints?Oneway available inmathematicsiscalled“measuretheo- ber of points regardless of its length, we would be wrong to treat it the same way we both infinitesetshave thesamenumberofmembers.ButasCantorproved, theset n therealreadywas a pre-existingfour-dimensional blockwith temporalexten- Cantor proved somethingthatseemscounter-intuitive: thattherearedif- Let usmove ontoanapproachthatmightallow for theinstantaneousslic- But whatotherways aretheretodetermine thelengthofaninterval ifnot

91 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo As thisdefinitionmakes very clear,aconsequenceofdensityisthatmember As previouslymentioned,theB-theoriesoftime neednotbethreatenedby thenotion The truenatureoftime isgiven tousby ourintuition.Thosewho holdtheseviews cessor comesintobeing,thereisnoway ofcomingupwithinfinitelymany slices in of reality.Theserelations,onceformed, areeternallyfixed (Broad 1923, 69).Soonce ed to the sum total of reality through becoming, it comes into relations with the rest one ofthekeycommitmentsgrowing blockview.Onceatemporalsliceisadd- of thepoint-likeinstant,andmorespecifically thepoint-likepresent.B-theoretical endorsing differentsolutions. ThinkerslikeBergson,Brouwer, Wyle, andothersassert of thegrowing blocktoo? Cantorian continuumorthenotionthatpoints arepotentialorideal.Soperhapsthe dense seriescannothave animmediate successororpredecessor.Thiscontradicts formed throughbecoming. tive A-theoreticalbecomingconflictswiththepropertyofdensity.TheB-theoristcan the B-theoryandthoseofA-theorythatkeepsgrowing blocktheoristfrom the growing blocktimeseries,therecanonlybeafiniteorzeronumberofelements. to theproblemofcontinuum.Thebindisthat thedoctrineofinstantaneous time isspace-likeandsoB-theoristsarenotcommitted eithertotheobjectivityof that thenumbered,mathematical notionofthecontinuumisnothow timeisinitself. time andaccept aprimitive pre-metrical notionofthepresent. Couldtheproponents form adenseseries: to measuretheory,aninterval mustcontainanon-denumerablyinfinitenumberof are a non-denumerably infinite number of points composing the continuum. But this still acceptthattheindefinitedivisibilitywillnever leadtoindivisiblesorthatthere achieving resolution. The fact that the B-series of time is formed through consecu- sor and every single a new slice becomes on the edge of reality, a relation of succession is formed between answer tothegrowing block’sproblem canbefound intheA-theoreticalsolutions are freetoassertthatall thatexistsisacompletelyunifiedandindivisible durationof Euclid defined thestraight line asa lengthwithout breadth,and ifthe line means givingupcompletelyonany A-theoretical notions. Suchablockcannever be present isincompatiblewithbecoming(Craig2000, 236).ApureA-theoryisopento present momentortoitsdynamicity.Thus,the B-theoristsarefreetoembracethe points between every two points on the line. Namely, that instances in time points inorderfor ittohave measure,whichleadsustotheproblemofdensity. it andthepreviousslices.Thus,every pointonthetimeserieshasonedirectpredeces- is composedofpoints,we takethattomeanthereareinfinitely many between themasdensityrequires.Inotherwords: between two differentelements of (Huntington 2003,34) finity of other elements; so that no element has a successor, and no element a predecessor Between every two elementsofadenseseriestherewillbeatleastoneandtherefore anin- III.3. III.2. Once again,itisthecombinationofmetaphysical commitmentsof The ProblemofDensityandBecoming Pure A-TheoreticalSolutions past slice has one direct successor. Once a slice’s immediate suc-

92 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo 79-84). Broad declares that it is misleading to call becoming “a change” (Broad 1923, Another istoreconceive thegrowing blockinsuchaway thatcantranscendthecon- The Cantorianviewofthecontinuumisnotwithout itsweaknesses, whichgave rise This is a direct result of its being a hybrid A-B-theory. The fact that the growth of the 246). Byacceptingaprimitive pre-metricalnotionofthepresent,A-theoreticmeta- 68), whichisonereasonitcannotfitwiththeideaexpressedasBergson’sdurée come welded togetherintoaunified continuum.(Bell 2019, 163; Dainton2010, 306). of continuity.Oneapproachistocontinuelookingfor otheraccountsofcontinuity. continuum that I tested so far conflict with the A-theoretical elements of the grow- events intimeareorderedandrespecttowhichrealitykeepsexpanding.Suchor- with suchconceptionsoftemporalcontinuityliesinthefactthatgrowing block or ashe puts it:«thebasicrealityis things acting»(Priorunpublished, in Craig2000, growing block. réele, orPrior’sideathatthereisnotransitionofinstantsfrombeingfutureto the analogyofreal number line,asthetruecontinuumrequiresan even greater tinuum, thesepointsmust bewelded together,whichcannotbeconceptualized using taking extension as fundamental. Perhaps one of them could be a better fit for the to (roughly)two groupsofopposingviews: thehyper-dense Peirceancontinuumand flicting traits.Oritcouldbethatthegrowing blockspacetimeisinherentlydiscrete. dered homogenousspace-liketimeisincompleteoppositiontothesenotionsofdu- tion suchas“whatisthepresenttime?”or“how longisthepresenttime?”vary de- the viewthatthereisnosuchthingaspresentsimpliciter.Theanswers toques- sis thattimeisacontinuum.AlltheB-theoreticalsolutionsfor theproblemsof IV. accept suchnotion.Perhapsthemostevidentincompatibilityofgrowing block stantly, but once something enters existence, it remains unchanged (Broad 1923, 66, allows for durationtobetakenaspriormetrics. second, the present hour or the present decade (Loizou 1986 in Craig 2000, 248). This I showed thatinacontinuousgrowing blockuniverse, thepresenthasnoduration. number of points: nothing less than the maximal possible number of points. When not merelybeacollection ofinfinitelymany points.Inordertogenuinely form acon- packed inthis hyper-dense manner, thepointslosetheir individualidentity andbe- ration andcontinuity. present tobeingpast:thetimeisallthatexistsandchangeinthings, ment, but is also, in a way, committed to a static block; new things come to be con- raising Zeno’sparadoxes ofpluralityandmotion.Onepossiblesolutionisadheringto physics canavoid Zeno’sparadoxes (Craig2000,248). Butthegrowing blockcannot pending onthecontextinwhichtheyareasked.“Presenttime”canmeanpresent ically appealing theory,so it would be beneficial tofinda way tosettletheproblems ing block,andviceversa. Butthegrowing blockisstillavery intuitively andmetaphys- is committedtotheclaimthatthereatemporaldimensionalongwhichallof block canonlyoccurthroughtemporalslicesofzeroextensionconflictswiththe- IV.1. The idea underlying the hyper-dense continuum is that the continuum can- A-theoretical becomingcannotendorseaninstantaneouspresentwithout The growing blockmay holdtoadynamic thesisregardingthepresentmo- Conclusion Revising theOrthodox Continuum

93 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo 310). Taking extensionasfundamentalcanprovide asolutiontotheparadox ofplu- T-instantaneous, buthas Θ While this view was proven useful in solvingsome of thepuzzlescontinuum (Broad 1959, 769-772) (see Dainton2010, 307-309), itishardtosee how hyper-density couldbeappliedto of theblockcannotoccurthroughbecomingdurationlessinstants,yet at extension asfundamentalistoabandontheideathatpointsaremostfun- conceived ashaving indivisible,extensionless parts –i.e.thepointsthatconstitutea of thecontinuumhave extension,no matterhow small. True, theorthodox contin- of smaller lines, and so on without ever reaching a bottom level (Dainton 2010, 309- ordinary time.AtemporalsliceshallberepresentedasapointtontheT-axis.Butif can beextended.ThinkofthingsintimeasorderedalongaT-axis,whichstandsfor ous presentandsuggeststhattimecouldhave anotherdimensionalongwhichslices we adda theses asthegrowing block.Butthegrowing events theorydeparts fromthegrow- finite regressionorcircularity. the sametimeitmustthusoccur.Onabove suggestion,thetemporalsliceis damental constituentsofaline.Instead,thepartslinearelines,whichmade the growing blockviewgiven theproblemofregulardensityandbecomingdiscussed the blockisconstructed. abstraction form existingevents, andsoaretheinstantaneousslices: accepting asecondtemporaldimensionstillseemslikeslipperyslopeleadingtoin- slice would be represented by a straight line of finite extension, parallel to the straight line–anditisthisthatthedifferencebetween theviews hingeson. To take III. Moreover, whilethereappearstobenoinconsistencyonthisparticularaccount, In “AReply toMyCritics”,Broadhimself deals withtheproblemofinstantane- II – there must be such a bottom level in the form of instantaneous slices from which poses atheoryofgrowing events thatstill holdstothesameontologicalanddynamic rality, but itisincompatible with thegrowing block because –asIshowed insection uents ofreality(Perović 2019, 19). Sothecontinuous four-dimensional blockisbutan uum alsohasthepropertyofbeinginfinitelyandindefinitelydivisible,butitis ing blockinthattheformer, events arethemostfundamentalontological constit- instantaneous andextended.However, Idonotseeany way how incorporating is hardtoseehow theycanever reachsuchmaximal quantity. in sectionIII.2.Moreover, sincethetemporalslicescomeintoexistenceoneby one,it by adheringtoatheoryofgrowing events, ratherthangrowing block.Perović pro- -duration canhelpthegrowing blockovercome theproblemsdescribedinsection events take some time to unfold and such temporal extendedness of an event is difficult(if events] theorist’sevents; rather, events aremetaphysically prior and instantaneousslices and theirpropertiesareabstractions fromevents. Thisisjustanother way ofsaying that ties. (Perović 2019, 20) not impossible) torecover frominstantaneoustemporal slicesofobjectsandtheir proper- […] itisnotsuchsliceswith their instantaneouspropertiesthatbuilduptheGE[growing IV.2. There mightstillbeaway toloosentheclutchofsomecommitments Recall thattheproblemthispaperisconcerned withisthatthegrowth Extension asfundamentalisthenotionthatmostparts Θ -axis, whichstandsfor anadditionaltemporaldimension,the The Growing BlockRevised Θ -duration, soonemightsay itsucceeds inbeingboth Θ -axis.

94 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo 353). Sophysics gives usevidencecontra thegrowing block,beitcontinuousordis - The propertiesofthediscreteandcontinuousarediametrically crete. Evidenceinsupportofthegrowing block may turnupinthefuture(morelike- ents. Soperhapsifthegrowing blockisdiscrete,thereneedbenotension inthefact considering adiscreteaccountofthegrowing block,questionsfromthephysical of any discretedurationoftimeendsinindivisible atomicquantitiescalledchronons. opposed. Whilethecontinuoustemporalinterval isindefinitelydivisible,thedivision ory’s originalcommitmenttotheprivilegedpresent,avariant ofthegrowing event objective present,orthecommitmentofgrowing blocktotemporalorderand could not be farther from the growing block. They do not incorporate the notion of crete (Dainton2010, 300-301; Rovelli 2018, 54-56). Chronons are usually defined as a certain minimal physical quantity. Indeed,when On theotherhandthereisworry thatthesetwo theoriesdifferonsuchfunda- On thisaccounttheuniverse grows, butitisagrowing event ratherthanagrowing growing block,andcertainlytothenotionof theobjective present.(Miller2013, 352- they can have a metric and they can be seen as formed of their smallest constitu- growing event theoristhave an“edge of existence”andstillavoid theB-theoretical to overcome theproblems of additionandlocations mentionedinsection III.1.But the slicescould betemporallyextended, ratherthaninstantaneous, we willbe able that itstemporalduration isconstructedthroughthebecomingoftemporal slices.If tinuous spacetime)andthegrowing blockare farfromaperfect fit.Thespecialtheo- tum mechanicsmightsupportadiscretespacetime,andtherearestillsomevery the very edgeofbeing(Perović 2019, 22)andthequestionremains:how canthe they arethesourceoftheory’sstrongintuitive appeal,andtheblockslicesare tological and dynamic theses of the growing block, because it may very well be that the continuum.Perhapsthisisasacrifice worth makinginordertopreserve theon- fixed, objective temporalrelationsordirection,and«inthe vast universe thereis direction. AsRovelli putsit:«Time hasloosenedintoanetwork ofrelationsthatno sciences arise.Whilesomeofourbestsciencereliesonthecontinuitytime,quan- succession? meantime, thereisstillvalue ininvestigating otherconsequencesofadiscretegrow - ry ofrelativityposesachallengetoboththeontological anddynamicalthesesofthe needs tobeconstructedthatidentifiesthepresentnotwith“ongoingness”but rather itreplacesit.Perović alsonotesthatinordertokeepthegrowing blockthe- mental grounds,thatthegrowing events theoryisnotadefence ofthegrowing block, not indispensablefeatures, but–somemightsay –redundantandeven disqualifying. nothing thatwe canreasonably call“present”»(Rovelli 2018, 59). promising theories, such as quantum gravity, according to which time might be dis- ing blockpicture. ing event theoristmightbeabletoadoptadifferent,moreA-theoreticalaccountof ly intheform ofphilosophicalarguments,rather thanscientificobservations). Inthe longer holdstogetherasacoherentcanvas» (Rovelli 2018, 58).Reality possesses no block, anditconsistsofaccumulatingevents, ratherthanslices.Itappearsthegrow- block metaphysic. Theymightentailadynamicmetaphysical picture,butonethat IV.3. A clearadvantage of takingspaceandtimeasdiscreteisthat,onthis view, On theotherhand,specialtheoryofrelativity(whichsupportsacon- But itseemsthatthetheoriesinquestiondonotconcurwithagrowing Can theGrowing BlockbeDiscrete?

95 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo one (orinthefirstsingularinstantaneoussliceofexistence,if you prefer) stepping commitments of the block on the one hand, and its B-theoretical commitments on change inthegrowing blockisreducibletobecoming(Broad1923, 67) doesnotcon- can onlybecomeonthevery edgeofreality.Andthefactthatevery othertypeof en duration?Perhapstheanswer isthatthefeatures ofthepresentsliceentailthat ent B-locations,andtherefore –by theway B-locationsareformed –consistsofmore ons doexist,theirsizewillbeuniform anddetermined.Inthewords ofLeeSmolin: which inevitablyconflicts withthedoctrinethattimeiscontinuous.The A-theoretical Of alltheoptions Iconsidered,the growing event theory andthediscrete growing from section IImust be rejectedfor discrete time, orelsewe end up backinsquare the durationofentireblockcouldbeadditive. from tion torealitycouldbeanatomicslice.Butwhatwillofourcommitmentthe the other, rule out any possibility to resolve this tension. It seems that the answer that becomingwilloccurthroughtheaccumulation ofinstantaneoustemporalslices, the continuityoftime.Thedefining features ofthegrowing block’spresentdemand the next.Theslicesjustbecomeandby theconsecutiveness ofthatbecomingthey the block,asitisnottraverse oftheblock’sedgefromonetemporallocationto duration of the slicesandhow isitdetermined? Will theystillbeuniform once giv- the ideaofdiscretegrowing block.Butsomequestionsstillremain.Whatisthe flict withthefact thatoncetheycome intoexistence, theslicesremainstatic.Any the extensionofasingleslicecanonlybeasgreatsmallest to beincoherentandirrelevant (seeCraig2000,240-242). Butitseems thatifchron- than asingleslice.Butwhatkeepsthemfrombeingshorterchronon?There sarily. The growth of the block is not a kind of motion and must not be mistaken for are stillvoices inthediscussionwhodoubtwholenotionofchronons,claimingit space; any duration greater than a chronon is spread between more than two differ - B-series? Cansuchanaccountgive risetoZeno’sparadoxes ofmotion?Notneces- In conclusion,asahybrid A-B-theorythegrowing blockposesauniqueproblemto B-theoretical way suchasRussell’s“at-at”theory. right intoZeno’sparadox ofplurality.Butmaybe by acceptingthedoctrinethattime motion (Broad 1959, 766-767), so there isnoneed to explainhow the block grows no paradoxical entitiessuchas“present”sliceswithsuccessors,andthelatestaddi- might bethatthegrowing blocktheorycannotholdontoallofit. Itsdefenders will nothing paradoxical abouttemporalslicesjustcomingintoexistenceontheedgeof motion ofobjectsacrossspaceinthegrowing blockcanbeanalysed inacompletely possible lapseoftime:thechronon.Theonlythingthatcanbeuneven isthequal- is discrete,thegrowing blocktheoristcansalvage theextendedpresent.Thatway, itative differencebetween two slices,namelydifferencesacrossthedimensionsof long asthepresentslicecannotbedividedintofurtherslices,therewill have togive upeither some ofthetheory’scommitments, orthecontinuity oftime. before we canarrive atthatconclusion,theargumentsagainstextendedpresent According toloopquantum gravity, spaceismadeofdiscreteatoms eachofwhichcarriesa (Smolin 2000a,106inDainton2010,300). which isarbitrarilybigorsmall–instead,thevolume mustbeoneofafinitesetnumbers tiny unitofvolume. Incontrasttoordinarygeometry,agiven regioncannothave avolume t n tot Thinking backtothedefining features ofthegrowing block’spresent,as I trustthatthislineofthoughthasthereaderspaperwarming upto n+1 withoutever traversing infinitely many locations inbetween. Thereis

96 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo on acontinuousgrowing blockview. on theotherhand,isabletopreserve alloftheelementsthatseemedcontradictory growing events theoristswillbeforced tokeeptheB-commitments,willtheystill tions: iftheysucceedindisposingoftheB-theoreticalcommitmentsentirely,would it to thegrowing block’sontologicalanddynamicaltheses,buttherearestillopenques- still beagrowing block?Andcouldthetheorystillsupportanobjective present?Ifthe block seemthemostpromising.Thegrowing events theoristsshouldbeabletohold be abletoadoptanA-theoreticalaccountfor continuity?Thediscretegrowing block,

97 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The  .Philosophy Kitchen #13 — Anno 7 — Settembre 2020 — ISSN: 2385-1945 — Il Tempo e il Continuo Aristotle (2008).Physics. (Waterfield, R.trans.).(Bostock,D.ed.)Oxford University Craig, W.L. (2000a).TheTensed TheoryofTime:ACriticalExamination. Springer Stillwell J.(2002).TheContinuum Problem.TheAmerican Mathematical Monthly. Id. (1959). A Reply to My Critics. In The Philosophy of C.D. Broad , ed. Paul Arthur Broad, C.D.(1923). Scientific Thought. NewYork: Harcourt,Brace&Company, Inc. Braddon-Mitchell, D.(2004)How dowe know itisnow now? Analysis, 64(3),199–203. Bourne, C.(2002).WhenamI?Atensetimefor sometensetheorists?Australasian Bell, J.L.(2019). TheContinuous, theDiscreteandInfinitesimalinPhilosophy and Rovelli, C.(2018). TheOrderofTime.(AllenLane). Perović, K.(2019). ThreeVarieties ofGrowing BlockTheory.Erkenntnis, 1-23. Miller, K. (2013). Presentism, eternalism, and the growing block. In H. Dyke & A. Bardon McTaggart, J.E.(1908). Theunrealityoftime.Mind17(68), 457–474 Huntington, E.V. (2003).TheContinuum andOtherTypes ofSerialOrder.Dover Forrest, P. (2004). The real but dead past: A reply to Braddon-Mitchell. Forbes, G.(2016). Thegrowing block’spastproblems.PhilosophicalStudies,173, Forbes, G.(2015). Accounting for ExperiencesasofPassage:Why Topology Isn’t Dainton, B.(2010). TimeandSpace.Durham:Acumen. Journal ofPhilosophy, 80(3),359–371. 358–362. Vol. 109, No.3(Mar.,2002),286-297. (Eds.), 699–709. Mathematics. SpringerVerlag. Schilpp, LibraryofLivingPhilosophers(NewYork: Tudor) Netherlands. London:Kluwer Academic Publisher. Press. Enough. Topoi 34, 187–194 publication, Inc. Bibliography A companiontothephilosophy oftime.Malden,MA:Wiley-Blackwell. Analysis, 64(4),

98 The Growing Block and the Problem of the Continuum — Shira Yechimovitz Shira — of the Continuum and the Problem Block Growing The