QuantumGravityandTaoistCosmology:

ExploringtheAncientOriginsofPhenomenologicalStringTheory

StevenM.Rosen

[ThisarticleisslatedforProgressinBiophysicsandMolecularBiology,

Vol.131(2017),pp.34−60.]

ABSTRACTIntheauthor’spreviouscontributiontothisjournal(Rosen2015),aphenomenologicalstringtheorywasproposedbasedonqualitativetopologyandhypercomplexnumbers.ThecurrentpapertakesthisfurtherbydelvingintotheancientChineseoriginofphenomenologicalstringtheory.First,wediscoveraconnectionbetweentheKleinbottle,whichiscrucialtothetheory,andtheHo-t’u,aChinesenumberarchetypecentraltoTaoistcosmology.Thetwostructuresareseentomirroreachotherinexpressingthepsychophysical(phenomenological)actionpatternattheheartofmicrophysics.Buttacklingthequestionofquantumgravityrequiresthatawholefamilyoftopologicaldimensionsbebroughtintoplay.WhatwefindinengagingwiththesestructuresisacloselyrelatedfamilyofTaoistforebearsthat,inconcertwiththeirsuccessors,provideablueprintforcosmicevolution.Whereasconventionalstringtheoryaccountsforthegenerationofnature’sfundamentalforcesviaanotionofsymmetrybreakingthatisessentiallystaticandthusunabletoexplaincosmogonysuccessfully,phenomenological/Taoiststringtheoryentailsthedialecticalinterplayofsymmetryandasymmetryinherentintheprincipleofsynsymmetry.Thisdynamicconceptofcosmicchangeiselaboratedoninthethreeconcludingsectionsofthepaper.Here,adetailedanalysisofcosmogonyisoffered,firstintermsofthetheoryofdimensionaldevelopmentanditsTaoist(yin-yang)counterpart,thenintermsoftheevolutionoftheelementalforceparticlesthroughcyclesofexpansionandcontractioninaspiralinguniverse.Thepaperclosesbyconsideringtheroleoftheanalystperseinthefurtherevolutionofthecosmos.KEYWORDS:quantumgravity;Taoism;yinandyang;topology;phenomenologicalphilosophy;physics;cosmogony;stringtheory;symmetry;dimension1.CONTEMPORARYPHYSICS,ARCHETYPALNUMBERS,ANDTHEHO-T’UInrecentdecades,therehasbeenlittlemeaningfulprogressinmainstreamphysicstowardatheorythatwouldeffectivelyunifyalltheforcesofnature—thequantum

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mechanicalforcesaswellasgravitation.Musingironicallyonthis,physicistLeeSmolin(2006)observedthat,“formorethantwocenturies…ourunderstandingofthelawsofnatureexpandedrapidly….[yet]today,despiteourbestefforts,whatweknowforcertainabouttheselawsisnomorethanwhatweknewbackinthe1970s”(viii).WhatIdemonstratedinmypreviouscontributiontothisjournal(Rosen2015)isthatthecruxoftheproblemlayinquantummechanics’mostbasicterm:!,thequantumofaction.Althoughthisindivisibleatomofprocessfoundatthesub-microscopicPlancklengthof10–35mconstitutesthecornerstoneofmorethanacenturyofsuccessfulexperimentationinphysics,italsomarkstheendoftheroadforclassicalanalysis,sinceheretheanalyticalcontinuityofspacegiveswaytoa“graininess”ordiscretenessthatadmitsofnofurtherquantitativedetermination.Moreover,theindivisibilityofquantizedactioniscloselytiedtoanindivisibilityofobserverandobserved,andthisimplicitinseparabilityofsubjectandobjectisanathematotheprogramofobjectiveanalysisthathadgoneunquestionedforcenturies.Whilephysicistshavebeenabletouseprobabilisticmethodstoworkaroundtheunderlyingdifficultywhentheirinvestigationsarelimitedtothequantummechanicalforces,theproblemhasprovenintractablewhengravitationmustbeincludedinordertorealizeacomprehensiveaccountofnature.

Reflectingonthepersistentfailureofphysicstoarriveataworkabletheoryofquantumgravity,Smolincalledforadifferentstyleofdoingphysics,a“morereflective,risky,andphilosophicalstyle”(2006,294)thatconfronts“thedeepphilosophicalandfoundationalissuesinphysics”(290).Iapplaudthiscallforamorephilosophicallyorientedphysics,andIproposethatthestalemateinphysicssuggestsitwillnolongerbepossibleforustorelyontheoldphilosophicalfoundation.Thatisbecausethefusionofsubjectandobjectencounteredattheheartofquantumphysicsprofoundlyconflictswiththecurrentlydominantphilosophyofscience,presupposingasitdoesasharpdivisionofsubjectandobject.Inmy2015essay,Idescribedindetailanewfoundationforcontemporaryphysics:phenomenologicalphilosophy,anapproachinwhichtheintimateinteractionofsubjectandobjectisacceptedasfundamental.Althoughphenomenologyremainsimportantinthepresentpaper,itsharesthespotlightwithamucholderphilosophicaltradition,onethatinfactinfluencedMartinHeideggerandMauriceMerleau-Ponty,twooftheleadingfiguresinthephenomenologicalmovement.AsphilosopherStephenPriestputit:

MuchofwhatwehavethoughtofasHeidegger’soriginalityisessentiallyTaoistandZenandtheramificationsofthisthroughHeidegger’sinfluenceonMerleau-Ponty’scorpushaveyettobeidentified….Heideggerian‘ModernContinentalPhilosophy’shouldnolongerbestudiedinabstractionfromitsEastAsianground.(2003,243)1ThepictureofnatureadvancedbyTaoismsometwenty-fivehundredyears

agowasofacosmosinconstantcyclicalflux.Itsebbandflowofenergieswas1Inmakingthisassertion,PriestwascommentingonReinhardMay’sbook,Heidegger’sHiddenSources:EastAsianInfluencesonHisWork(1996).

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expressedbytheparadoxicaldynamicsoftwoopposingbutinterpenetratingforces:yangistheactiveforcerelatedtobrightness,positivity,expansionandmasculinity;yinisthepassivemediumassociatedwithdarkness,negativity,contraction,andfemininity.Operatingincloseharmony,yinandyanggeneratepatternsofchangethatarespecifiedbythetrigramsandhexagramsdepictedintheoldesttextofclassicalChina:theIChingorBookofChanges.AndthesystemoftransformationssetforthintheIChingiscloselyrelatedtoaprimalpatternofactionknownastheHo-t’u.Inthenextsection,wewillseethesignificanceoftheHo-t’uforcontemporaryphysics,butbeforeIproceedtoexaminethisarchetypalconfiguration,letmeclarifymyuseoftheterm“archetype,”anddiscussthegeneralmeaningofarchetypalnumbers.

InthepsychologyofC.G.Jung,archetypesarestructuresofthe“collectiveunconscious”imprintedinnatureatthedeepestlevel.ThiswasJung'swayofspeakingofthatwhichgoesbeyondtheegoicrealmofarbitrarilyfashioned,finiteformstoparticipateinsomethingmoretimelessanduniversal. Theprefixarch-comesfromtheGreekarchos,aruler,andappearsinsuchwordsas"archbishop"and"architect"(masterbuilder),signifyingwhatischieforprincipal.Archaeo-means"ancient"andderivesfromtheGreekarché,thebeginning,thefirst.Thusarchedenoteswhatcomesfirst,bothintimeandimportance.

Whatisthemeaningoftype?ThewordstemsfromtheGreektypos,"ablow,themarkofablow,figure,outline...fromtyptein,tobeat,strike"(Webster'sUnabridgedDictionary).Theetymologyof“type”thereforesuggestsaction.Puttingthistogetherwitharche,wecansaythatanarchetypeisaprimaryactionpattern,onethatisbothfundamentalandoriginary.WhatIamemphasizinghereisthedynamiccharacterofthearchetype.Therefore,unlikethetraditionalPlatonicsenseofarchetypeaseidos,anabsolute,changelessideaorform,Igiveprecedencetoviewingthearchetypeasanelementalprocess.Whileitistruethattheword“type”hasassumedapredominantlystaticmeaningforus—"asymbol;anemblem;atoken...[a]generalform,structure,plan...amodel"(Webster'sUnabridgedDictionary)—myownuseofthetermarchetyperetainsthedecidedlymoreactivesenseoftheoriginalGreektypos:“fromtyptein,tobeat,strike."

Now,beforehisdeath,Jungsuggestedtohiscolleague,Marie-LouisevonFranz,thattheprofoundestexpressionofarchetypaltruthmaybeembodiedincertainprimordialnumberpatterns.ThisprovidedtheimpetusforNumberandTime(1974),vonFranz'saccountofnumberasprimalquality,astheunderlying,energeticorganizingprincipleoftheunusmundus,thecosmicfieldfromwhichlessfundamentalarchetypesbecomemanifest.

DrawingfromawealthofancientandmedievalWesternandnon-Westernsources,vonFranzdemonstratesthatthecontemporaryWesternnotionofnumberasemptyquantityinfactgrewoutofaverydifferentmeaningofnumber.InbuildingonJung'scharacterizationofnumberasthemostbasicandprimitiveelementofmatterandpsychealike,vonFranzportraysnumbersas"psychophysicalpatternsofmotion"(1974,74)."Wearedealinghere,"saysvonFranz,

withtheprincipleaccordingtowhichnumber,takenqualitatively,isunderstoodtofunctionasapreconscious…principleofactivity;each

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numbermustbethoughtofascontainingaspecificactivitythatstreamsforthlikeafieldofforce.Fromthisstandpointnumberssignifydifferentrhythmicconfigurationsofthe[cosmicground].(1974,74–75)

Herethemeaningofnumberdifferssignificantlyfromthecommonunderstandingprevalenttoday.Accordingtothelatter,numbersaregenerallytakenasmathematicalvaluesrepresentingparticularquantitiesusedforcountingandmakingcalculations.Whilenumbersinthissensearecertainlyindispensableinmeasuringextrinsicmotions,theythemselvesarecompletelyinert.Incontrast,vonFranz'sarchetypalnumbershavetheirowninternalrhythmsorpulsations,eachoneconstitutingadistinctive"phaseoftransformation"intime(1974,42).Moreover,assembledintocertainfieldconfigurationsormatrices,thesenumbersconstitutenothinglessthan"cosmicplans"(22)."IntheancientChineseview,"saysvonFranz,"theuniversewas'organized'accordingtonumericalpatternsofthiskind"(24).EchoingvonFranz,letmeemphasizethatthenumbersinquestionarenotmereabstractionsbutare“psychophysical”innature.Ratherthanjustbeingobjective,theyexpresstheintimateinterplayofobjectandsubjectalsofoundattheheartofmodernphysics,withitsquantumofaction.

VonFranzdevotesconsiderableattentiontoaparticularnumericalarray,onethatisamongthemostprimordialnumberconfigurationsofarchaicChina:the"sequenceofolderheaven"orHo-t'u(Fig.1).2ThismodelprovidesablueprintforthegenerationofthetrigramsfeaturedinTaoism'sIChing.

Figure1.TheHo-t’unumberarchetypedepictedasanarrayofblackandwhitecircles(left)andasasequenceofArabicnumerals(right).(AdaptedfromvonFranz1974,23.Source:Granet,Penséechinoise,©AlbinMichel1936,1968,130.)

2InconsideringtheHo-t'u,VonFranzactuallyincludesanaccompanyingnumberarray,theLo-shuor"sequenceofyoungerheaven."ThisstructureislessprimordialthantheHo-t'uandthoughttoderivefromit(see235–241).Forourpurposes,itwillsufficetofocusourattentionexclusivelyontheolderconfiguration.

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TheHo-t’usequenceconsistsessentiallyofadoublecycleofnumbers,eachofwhichconstitutesarhythmicactionuntoitself.Inthefirstcycle,therearefoursuchpulsationsmakingupaninnernumberorbit:1,2,3,4.Thegesturalpatternsuggestedbythisiscross-like,andcanbetakenasfollowingthefourcardinalpointsofthecompass—North,South,East,andWest,respectively(contraWesternconvention,theoriginalChinesediagramsplacedNorthbelowandSouthabove).Cycle1issucceededbyamovementatthecenter,5,afterwhichweshifttotheouternumberorbitforthesecondcycleofactions:6,7,8,9.TheselatterpulsationscanbeseentocorrespondtotheintercardinalcompasspointsNorthwest,Southeast,Northeast,andSouthwest,respectively.Thesequenceendswithasecondcentralpulsation,10.

TherelationshipoftheHo-t'usequencetotheTaoisttrigramsisindicatedbyvonFranzasshowninFigure2.

Figure2.Ho-t’unumberarchetype(right)andassociatedtrigrams(left),withtrigramnamesdisplayedincenter.(AdaptedfromvonFranz1974,25.Source:Granet,Penséechinoise, © Albin

Michel1936,1968,148.)ThemovementfromK'un(N)toCh'ien(S)toLi(E)toK'an(W)correspondstothefirstHo-t'ucycle(1–4).VonFranz'sdiagramimpliesthatanactionintheunmarkedpositionatthecenterofthetrigramwheelwouldcomenext,thisbeingequivalenttopositionnumber5ontheHo-t'ucross.Thenthereisthesecondsetoftrigramactions—Ken(NW)àTui(SE)àChen(NE)àSun(SW)—correspondingtoHo-t'ucycle2.Thetrigramsequenceconcludeswithafinalpulsationatthecenterofthewheel,matchingHo-t'unumber10(seeFig.1,right).3

SpeakingoftheHo-t'uasinvolvingan"internaldoublemovement"(236),VonFranzcitesWilhelm'sexplanationofapassagefromTaoism'sIChing:

"Whenthetrigrams[...]areinmotion,adoublemovementisobservable:first,theusualclockwisemovement,cumulativeandexpandingastimegoes

3CommentatorshavenotalwaysbeenunanimousonthequestionofhowtoassigntrigramsandtheirassociatedcompassdirectionstothenumbersoftheHo-t’u.Butpursuingdebateonthismatterisbeyondthescopeofthepresentpaper.

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on,anddeterminingtheeventsthatarepassing;second,anopposite,backwardmovement,foldingupandcontractingastimegoesby,throughwhichtheseedsofthefuturetakeform[...]Ifweunderstandhowatreeiscontractedintoaseed,weunderstandthefutureunfoldingoftheseedintoatree."(WilhelmquotedinvonFranz,1974,236–237)

Alittlelaterinherbook,referringtothe"Pre-Worldheavenlyorder"

(anothertermassociatedwiththeHo-t'u)ascomprisinga"doubleinternalpatternofmotion,"vonFranzagaincitesWilhelm'stranslationoftheIChing:

“Thunderbringsaboutmovement,windbringsaboutdispersion,rainbringsaboutmoisture,thesunbringsaboutwarmth."[...]Thetextcontinues:"KeepingStillbringsaboutstandstill,theJoyousbringsaboutpleasure,theCreativebringsaboutrulership,theReceptivebringsaboutshelter."AsWilhelmpointsout,this[latter]passagedescribesaretrogrademovement.Init,theforcesofthecomingyearareunrolled.Pursuanceofthisline,saysWilhelm,leadstoknowledgeofthefuture,theeffectofwhichispreparedinadvancebyitscauses(i.e.,seeds),whichtakeshapebycontracting.This"retrograde"movementisanalogoustotheretrogradenumberstepswhich,Ihavepostulated,liebehindthequalitativenumberseries.(VonFranz1974,243–244)

VonFranzsummarizestheHo-t'u-related“Pre-World”sequenceoftrigramsascomprisinga"rhythmicalexchangeofpowers"inwhichthe"firstfourinnerrhythmsarephysical(movement,dissolution,moistening,heating)andthesecond[retrograde]fourarepsychic(maintenance,rejoicing,mastery,recovery)"(footnote16,243–244).ForvonFranz,thisdouble-cycle"representsanattempttovisualizethewholeofpsychologicalandphysicalexistence,theunusmundus,byantitheticaltimemovementsandinternalrhythms"(245).

Whatrelevancedoesthisancientcosmologyhaveforthecurrentcrisisinmoderncosmology,withitsproblemofquantumgravity?Inmy2015contributiontothisjournal,IoutlinedtheapproachtoquantumgravityofferedinTheSelf-EvolvingCosmos(Rosen2008a).Presently,IamgoingtobuildonwhatIsaidin2015,eventuallyexpandingittobringoutmorefullythediachronicaspect,thatofcosmicevolutionperse.Inthecourseofdoingthis,wewillseehowtheaccountofquantumgravityIhaveproposedisalignedwiththe“cosmicplans”(vonFranz1974,22)implicitintheHo-t’uandrelatedarchetypalstructuresofTaoistphilosophy.2.THETOPOLOGYOFPSYCHOPHYSICALACTIONImentionedattheoutsetoftheprevioussectionthatthesub-microscopicdynamismfundamentaltoquantummechanicsis!,thequantumofaction.Letusnowconsiderthisconceptingreaterdetail.Theformulafor!tellsussomethingaboutitsnature:!=h/2Π.Inquantumtheory,histheconstantofproportionality

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thatrelatestheenergy(E)ofaquantumofradiationtothefrequency(v)oftheoscillationthatproducedit:E=hv.Now,ifwerewritethisequationbyreplacingfrequency(v)withitsinverse,namely,time,wethenhaveE=h/Torh=ET,and,inphysics,energymultipliedbytimeisameasureofaction.Thush,Planck’sconstant,givesusquantizedaction.Theangularityofsaidaction,itsinternal“spin,”isexpressedbytheapplicationofphase,asweseeintheformula!=h/2Π.Herehisoperateduponbyaphaseof2Πradians,equivalenttoaturnof360°. Ialsomentionedearlierthatinquantummechanics,!isregardedasanindivisible“atomofprocess,”for!isnotreducibletosmallerunitsthatcouldbeappliedinitsquantitativeanalysis.Thisindivisibilityofthequantumdomainiscloselyrelatedtoitsbasicindeterminacyoruncertainty.AccordingtoHeisenberg’suncertaintyprinciple,thereisabuilt-inlimittotheinformationwecanobtainaboutthephysicalpropertiesofquantumsystems.ThislimitationcanbestatedintermsofPlanck’sconstant:ΔpΔq≈!,wherepandqarevariablessuchaspositionandmomentum,ortimeandenergy(variablesthatarepairedorconjugatedsoastobeessentiallyindivisiblefromeachother).Theformulasaysthattheproductoftheuncertainties(Δs)ofsuchpairedtermsapproximatelyequals(cannotbelessthan)thevalueofPlanck’sconstant.Clearlythen,thephasicindivisibility(h/2Π)ofPlanck-levelactionisequivalenttoitsuncertainty(ΔpΔq). Thereisanotherwaytolookatthequantumuncertainty.Nearingthesub-microscopicPlancklengthof10–35m,itappearsthatpreciseobjectivemeasurementisthwartedbythefactthattheenergythatmustbetransferredtoasysteminordertoobserveitdisturbsthatsystemsignificantly.Thiswell-known“problemofmeasurement”inquantummechanicsexpressesquantumindivisibilityintermsoftheindivisibilityoftheobserverandtheobserved.ItseemsthatinQM,theobservernolongercanmaintaintheclassicalpostureofdetachedobjectivity;unavoidably,s/hewillbeanactiveparticipant.Evidentlythismeansthatquantummechanicalactioncannotberegardedmerelyasobjectivebutmustbeseenasentailinganintimatemergingofobjectandsubject,ofmatterandpsyche,thatdefiesNewtonianorder.Andthepsychophysicalcharacterof!bringsbacktomindthekindofactioncharacteristicoftheHo-t’unumberarchetype.

Considerfurtherthequantizedactionof!.IttakestheformofanoddspinningthatWolfgangPaulimodeledbyusingcomplexnumbers.Musès(1976)suggestedthatPauli’sspinmatricesfortheelectronareactuallybasedonakindofcomplexnumberor“hypernumber”thatgoesbeyondPauli’simaginaryi:thehypernumberε(definedasε2=+1,butε≠±1).WhatIdemonstrateinCosmosisthatthegeometriccounterpartofεisatopologicalcuriosityknownastheKleinbottle.ThisstructureplaysahighlysignificantroleinCosmosandinmy2015paper,andIwouldnowliketoponderitoncemoreinthepresentcontext,bringingoutitsinherentlypsychophysicalnature.

LetusbeginwiththesimpleillustrationIofferedin2015.

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Figure3.PartsoftheKleinbottle(afterRyan1993,98)

Figure3ismyadaptationofcommunicationtheoristPaulRyan’slinear

schematafortheKleinbottle(1993,98).AccordingtoRyan,thethreebasicfeaturesoftheKleinbottleare“partcontained,”“partuncontained,”and“partcontaining.”Hereweseehowthepartcontainedopensout(atthebottomofthefigure)toformtheperimeterofthecontainer,andhowthis,inturn,passesoverintotheuncontainedaspect(intheupperportionofFig.3).Thethreepartsofthisstructurethusflowintooneanotherinacontinuous,self-containingmovementthatfliesinthefaceoftheCartesiandivisionofcontained,containing,anduncontained—symbolically,ofobject,space,andsubject.Wecanalsoseeanaspectofdiscontinuityinthediagram.Atthejuncturewherethepartuncontainedpassesintothepartcontained,thestructuremustintersectitself.Wouldthisnotbreakthefigureopen,renderingitsimplydiscontinuous?WhilethisisindeedthecaseforaKleinbottleconceivedasanobjectinordinaryspace,properlyseentheKleinbottleactuallyenactsadialecticofcontinuityanddiscontinuity,aswillbecomeclearerinfurtherexploringthispeculiarstructure.Wecansaythenthat,initshighlyschematicway,theone-dimensionaldiagramlaysoutsymbolicallythebasictermsofa“continuouslydiscontinuous”dialecticinvolvingnotonlysubjectandobject(psycheandmatter),butthemediatingspaceaswell.Depictedhereistheprocessbywhichthethree-dimensionalobjectofthesub-microscopicrealm,intheactofcontainingitself,istransformedintothesubject.Thisblueprintforpsychophysicalinterrelatednessgivesusagraphicindicationofhowthemutuallyexclusivecategoriesofclassicalthoughtaresurpassedbyarelationofmutualinclusion.

TheparadoxicalnatureoftheKleinbottlefindsexpressioninitssurprisingpropertyofone-sidedness.Morecommonplacetopologicalfiguressuchasthesphereandthetorusaretwo-sided;theiropposingsidescanbeidentifiedinastraightforward,unambiguousfashion.Therefore,theymeettheclassicalexpectationofbeingclosedstructures,structureswhoseinteriorregions(“partscontained”)remaininterior.InthecontrastingcaseoftheKleinbottle,insideandoutsidearefreelyreversible.Thus,whiletheKleinbottleisnotsimplyanopenstructure,neitherisitsimplyclosed,asarethesphereandthetorus.InstudyingthepropertiesoftheKleinbottle,weareledtoaconclusionthatisparadoxicalfromtheclassicalviewpoint:thisstructureisbothopenandclosed.TheKleinbottlethereforehelpstoconveytousthefluidrelationshipsbetweeninsideandoutside,

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closureandopenness,continuityanddiscontinuitythatarelosttousinCartesianexperience.

However,musttheself-containingone-sidednessoftheKleinbottlebeseenasinvolvingthespatialcontainer?GrantingtheKleinbottle’ssymbolicvalue,couldwenotviewitsinside-outflowfrom“partcontained”to“partcontaining”merelyasacharacteristicofanobjectthatitselfissimply“inside”ofspace,withspacecontinuingtoplaytheclassicalroleofthatwhichcontainswithoutbeingcontained?Inotherwords,despiteitssuggestivequality,doestheKleinbottlenotlenditselftoclassicalidealizationasamereobjectinspacejustasmuchasanyotherstructure?

TheschematicrepresentationoftheKleinbottleprovidedbyFigure3showsthatitpossessesthecuriouspropertyofpassingthroughitself.WhenweconsidertheactualconstructionofaKleinbottleinthree-dimensionalspace(byjoiningoneboundarycircleofacylindertotheotherfromtheinside),weareconfrontedwiththefactthatnostructurecanpenetrateitselfwithoutcuttingaholeinitssurface,anactthatwouldrenderthemodeltopologicallyimperfect(simplydiscontinuous).SotheKleinbottlecannotbeassembledeffectivelywhenoneislimitedtothreedimensions.

Mathematiciansobservethataformthatpenetratesitselfinagivennumberofdimensionscanbeproducedwithoutcuttingaholeifanaddeddimensionisavailable.ThepointisimaginativelyillustratedbyRudolfRucker(1977).Heasksustopictureaspeciesof“Flatlanders”attemptingtoassembleaMoebiusstrip,whichisalower-dimensionalanalogueoftheKleinbottle.Ruckershowsthat,sincetherealityofthesecreatureswouldbelimitedtotwodimensions,whentheywouldtrytomakeanactualmodeloftheMoebius,theywouldbeforcedtocutaholeinit.Ofcourse,nosuchproblemwithMoebiusconstructionarisesforushumanbeings,whohavefullaccesstothreeexternaldimensions.ItisthemakingoftheKleinbottlethatisproblematicforus,requiringasitwouldafourthdimension.Tryaswemightwefindnofourthdimensioninwhichtoexecutethisoperation.

Incontemporarymathematicshowever,thefactthatwecannotcreateapropermodeloftheKleinbottleinthree-dimensionalspaceisnotseenasanobstacle.Themodernmathematiciandoesnotlimithim-orherselftotheconcreterealityofspacebutfeelsfreetoinvokeanynumberofhigherdimensions.Noticethough,thatinsummoningintobeingtheseextradimensions,themathematicianisextrapolatingfromtheknownthree-dimensionalityoftheconcreteworld.Thisprocedureofdimensionalproliferationisanactofabstractionthatpresupposesthatthenatureofdimensionalityitselfisnotaltered.InthecaseoftheKleinbottle,the“fourthdimension”requiredtocompleteitsformationremainsanextensivecontinuum:aninfinitelydense,structurelesssubstratethatitselfdoesnotchangebutthatmediateschangesinthestructuredobjectsitcontains(thepropertiesofclassicalspaceareexaminedindepthinRosen2008aand2015).Tobesure,the“fourthdimension”isacknowledgedasbutaformalconstructandtheKleinbottleisregardedasanabstractmathematicalobjectsimplycontainedinthishyperspace(whereasthesphere,torus,andMoebiusstriparerelativelyconcretemathematicalobjects,sincetangiblyperceptiblemodelsofthemmaybesuccessfullyfashionedinthreedimensions).WeseeherehowtheconventionalanalysisoftheKleinbottleunswervinglyadherestotheclassicalformulationofobjectsinspace.Moreover,

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whetheramathematicalobjectmustbeapproachedthroughhyperdimensionalabstractionoritisconcretizable,themathematician’sattentionisalwaysdirectedoutwardtowardanobject,towardthatwhichiscastbeforehisorhersubjectivity.Thisistheaspectoftheclassicalstancethattakessubjectivityasthedetachedpositionfromwhichallobjectsareviewed;here,neverissubjectivityassuchopenedtoview.ThusthepostureofcontemporarymathematicsisfaithfullyalignedwiththatofDescartesandNewtoninwhatevertopicitmaybeaddressing.Always,thereisthemathematicalobject(ageometricformoralgebraicfunction),theanalyticallycontinuousspaceinwhichtheobjectiscontained,andtheseldom-acknowledgeduncontainedsubjectivityofthemathematicianwhoiscarryingouttheanalysis.Andthereisneveranydoubtaboutthestrictseparationofthesethreebasiccategories.

Now,inhisstudyoftopology,Barradvisedthatweshouldnotbeintimidatedbythe“highermathematician….Wemustnotbeputoffbecauseheisinterestedonlyinthehigherabstractions:wehaveanequalrighttobeinterestedinthetangible”(1964,20).ThetangiblefactabouttheKleinbottlethatisglossedoverinthehigherabstractionsofmodernmathematicsisitshole.Becausethestandardapproachhasalwayspresupposedextensivecontinuity,itcannotcometotermswiththeinherentdiscontinuityoftheKleinbottlecreatedbyitsself-intersection.Therefore,alltooquickly,“higher”mathematicscircumventsthisconcreteholebyanactofabstractioninwhichtheKleinbottleistreatedasaproperlyclosedobjectembeddedinahyper-dimensionalcontinuum.Alsoimplicitinthemainstreamapproachisthedetachedsubjectivityofthemathematicianbeforewhomtheobjectiscast.Isuggestthat,bystayingwiththehole,wemaybringintoquestiontheclassicalintuitionofobject-in-space-before-subject.

LetuslookmorecloselyattheholeintheKleinbottle.Whatkindofholeisit?Thislossofcontinuityisnecessary.OnecertainlycouldmakeaholeintheMoebiusstrip,torus,oranyotherobjectinthree-dimensionalspace,butsuchdiscontinuitieswouldnotbenecessaryinasmuchastheseobjectscanbeproperlyassembledinspacewithoutrupturingthem.Itisclearthatwhethersuchobjectsarecutopenorleftintact,theclosureofthespacecontainingthemwillnotbebroughtintoquestion;inrenderingtheseobjectsdiscontinuous,wedonotaffecttheassumptionthatthespaceinwhichtheyareembeddedissimplycontinuous.WiththeKleinbottleitisdifferent.Itsdiscontinuitydoesspeaktothesupposedcontinuityofthree-dimensionalspaceitself,forthenecessityoftheholeinthebottleindicatesthatspaceisunabletocontainthebottlethewayordinaryobjectsappearcontainable.WeknowthatiftheKleinian“object”isproperlytobeclosed,assembledwithoutmerelytearingaholeinit,an“addeddimension”isneeded.Thus,fortheKleinbottletobeaccommodated,itseemsthethree-dimensionalcontinuumitselfmustinsomewaybeopenedup,itscontinuityopenedtochallenge.Ofcourse,wecouldattempttosidestepthechallengebyacontinuity-maintainingactofabstraction,asinthestandardmathematicalanalysisoftheKleinbottle.Assumingwedonotemploythisstratagem,whatconclusionareweledtoregardingthe“higher”dimensionthatisrequiredforcompletingtheKleinbottle?Ifitisnotanextensivecontinuum,whatsortofspaceordimensionisit?

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ThequestioncanbeaddressedbylookingagainatFigure3.Atfirstglance,weseeaschematiclinedrawingofaKleinbottlecontainedwithinthetwo-dimensionalframeoftheillustration.Butwhenweconsidertheactualcontentofthediagram,weseethat,insteadofshowingtheKleinbottleasanobjectenclosedinspace,thediagramportraysspace(“partcontaining”)asanaspectoftheKleinbottle!WhatissymbolizedschematicallyinFigure3isadimensionofaKleiniankind,oneinwhichspaceitselfisbutoneofthreeontologicaltermsallofwhichflowtogetherratherthanbeingsplitofffromoneanotherashappenswiththeCartesianparadigmofobject-in-space-before-subject.ItisclearfromthisthattheKleiniandimensionisnotphysicalbutpsychophysical,forheresubjectandobject,psycheandphysicalmatter,aremediatedbyaspaceinwhichtheypermeateeachother.SoIamproposingthat,whentheKleinbottleisregardedinconcretetermsratherthanasamathematicalabstraction,weactuallycannotcompleteitsconstructionasanobjectinaspatialcontinuumbutareobligedtoreformulatedimensionalityitselfalongKleinianlines.Notethat,inthisreformulation,theholeintheKleinbottleisneitherlightlydismissedintheinterestofmaintainingcontinuity,nordoescontinuitygivewaytosheerdiscontinuity.Instead,theverysplittingofcontinuityanddiscontinuityiscalledintoquestionandweentertainthepropositionthat,incompletingtheKleinbottle,inmakingitwhole,itdoesnotloseitshole.Graspedinthisdialecticalway,wemaysayinfactthattheKleinbottleconstitutesa“(w)hole,”aparadoxicalstructureinwhichcontinuityanddiscontinuityareentwined. Now,recallourassociationoftheKleinbottlewiththemicrophysicalspinningactionthatliesattheheartofquantummechanics.ThispointstothedynamiccharacteroftheKleinianconfigurationwhenitisunderstoodinthemicroworldcontext.Here,morethanafixedstructure,itisanaction,apsychophysicalatomofprocess.AndwhenwerememberthecorrelationofKleiniantopologywiththehypernumberε,whatcomestomindarethearchetypalnumbersoftheHo-t’udescribedbyvonFranz.4For,asnotedearlier,suchnumbersare“psychophysicalpatternsofmotion”(1974,74)andeachnumberis“thoughtofascontainingaspecificactivitythatstreamsforthlikeafieldofforce”(75).ButhowspecificisthecorrespondencebetweenthedynamicstructureoftheHo-t’unumberfieldandthatoftheKleinbottle? IntheprevioussectionwefoundthattheHo-t’uconsistsofadoublecycleofnumbers,witheachnumberembodyingarhythmicactionuntoitself,aninternal“phaseoftransformation,”asvonFranztermedit(1974,42).Thefirstcyclecontainsfoursuchphases.Thisisfollowedbyareturntocenter(markedbythenumber5inFig.1),afterwhichweenterthesecondcycleoffourphases.Heretheforwardor“clockwise”orientationofthenumberrhythmsincycle1hasbeenreversed,withphasesnowbeingcharacterizedashavingabackwardorretrogradeaction.And,whereasthe"firstfourinnerrhythmsarephysical…thesecondfourarepsychic”(footnote16,243–244).Arewetotakethesetwocyclesasrelatedinamerelysequentialfashion?Whenthesecondcycleisenactedisthefirstcyclesimply4WeknowthattheHo-t’uplaysacrucialroleintheIChing.JungiantheoristNathanSchwartz-Salant(2017)relatestheIChingtotheKleinbottlebysuggestingthatthelatteristhestructurethatbestexpressesthedynamicbalancebetweenorderanddisorderevidencedintheformer.

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leftbehind,whichwouldsuggestthatthecyclesareessentiallyexternaltoeachother?ItseemsthatvonFranzcannotbesuggestingthisinviewofheremphasisonthethoroughlypsychophysicalnatureoftheHo-t’unumberfield.Atonepointsheassertsthat“numberisboundupwiththelatentmaterialaspectofthepsycheandwiththelatentpsychicaspectofmatter”(157).Somehowthen,latentwithinthephysicalcirculationoftheHo-t’u(cycle1)isanaspectofthepsychicalcirculation(cycle2),andviceversa.VonFranzoffersanenigmaticcluefortheinterconnectionofpsycheandmatterinherdiscussionofarelatedmandala,thedoublewheelofEzekiel,whereintwowheelsaredepictedasintersectingeachotheratrightangles(seeFig.4).

Figure4.Ezekiel’swheelsThewheels,saysvonFranz,“donotworkinunisonbutarecontiguousatthecenter,whichisatechnicalimpossibility.Thetwosystemsareincommensurable”(262).Commentingfurther,shenotesthatthe“mysteriouspointofcontactbetweenthetwosystemsappearstobethecenterorasortofpivotwherepsycheandmattermeet”(263).ButtheiconicimageofEzekiel’swheelsdoesnotinitselfeffectivelyindicatethemutuallatencyofpsycheandmatter,sincethewheelsareshownasseparateexceptfortheaforementionedpointofcontact.IftheimagesoftheHo-t’u(Fig.1)andEzekiel’swheels(Fig.4)expresstheintimaterelationshipbetweenpsycheandmatterinmerelyabstractsymbolicterms,isthereadeeper,moreconcretelyembodiedwayofdeliveringthisconjunction?IsubmitthatthehiddentopologyoftheHo-t’uisnoneotherthanthatoftheKleinbottle.Toconfirmthis,webeginbyworkingwiththelower-dimensionalcounterpartoftheKleinbottle:theMoebiusstrip. LetuscomparethecuriouslyconfiguredMoebiuswithitsmoreconventionalanalogue,acylindricalring.

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Figure5.Cylindricalring(a)andMoebiusstrip(b)

Theordinaryring(Fig.5a)isconstructedbycuttingoutanarrowstripofpaperandjoiningtheends.ToproducethesurfaceofMoebius(Fig.5b),giveoneendofthepaperstripahalftwist(throughanangleof180°)beforelinkingitwiththeother.Now,imagineyourselfcirclingaboutthecylindricalring.Positionedontheinnersurfaceofthistwo-sidedstructure,youmove360°aroundtocompleteonerevolution,returningtoyourpointofdeparture.Naturally,yourpassagearoundtheinsideoftheringnevertakesyoutotheoutside.Throughoutthejourney,youremainonthesideonwhichyoubegan.

Itisnotlikethiswithmovementaroundtheone-sidedMoebiussurface.While360°ofrevolutiondoseemtobringyoubacktoyourpointoforigin,atthesametime,itisnotyourpointoforigin,sinceyouarenowontheothersideofthesurface.Noticethewayordinarycircularrevolutionmaintainsthesimpledichotomybetweenpointoforiginanddisplacementfromthatpoint:by180°ofmovementonthecylindricalring,youarefurthestremovedfromwhereyoubegan,andby360°,youarebackwhereyoustarted,thedisplacementbeingsimplyandcompletelyreversed.Ontheotherhand,with360°ofMoebiusrevolution,thereisacirclingbacktothepointoforiginthatisatoncethepointmostremotefromsaidorigin,since—whensidednessistakenintoaccount—itisthepointontheothersideofthesurfacethatisdiametricallyopposedtothestartingpoint.Whathappensifyoucontinuerevolvingbeyondthis360°pointofquasi-return?Afteranadditional360°,youfindthatyouhavenowtrulyreturned,sinceyouhavecomebacktoyourpointofdepartureonthesamesideofthesurfacefromwhichyoufirstsetout.

ConsiderthegeneralresemblanceofMoebiusactiontotheactionpatternoftheHo-t'u.Botharedoublecyclesinvolvinganinitialreturntoorigin(thephase-5returntocenter,inthecaseoftheHo-t'u;seeFig.1),followedbyasecondcircuitthatrepeatstheprocess,nowcompletingitinearnestwithasecondreturntoorigin(Ho-t'uphase10).However,insteadofshowingtheHo-t'uasconsistingsimplyofseparatecyclesconnectedbyaseparatethusexternalinter-cyclicallinkingphase(phase5),theone-sidednessoftheMoebiussuperimposesthecyclesuponeachother,concretelyconveyingtheparadoxthatthesetwocirculations—thephysicalandthepsychical—alsoareone.WecansaythatthecyclesoftheMoebiusaresointimatelyentwinedthatthetransitiontocycle2,thenewcycle,isjustasmuchaliteralrepetitionofthesamecycle.

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ArelatedfeatureoftheMoebiusstripfurtherconfirmsitsintimatecorrespondencewiththeHo-t'u.Forthelatter,wehavefoundthat,whereascycle1entailsclockwisemovement,cycle2iscounterclockwise.AshiftinclocksenseisjustwhattakesplaceinrevolvingaroundtheMoebius,thoughthemanneroftransformationismoresubtlethanasimpleassociationofonecyclewithclockwiseactionandtheotherwithcounterclockwiseaction.Todemonstratethisproperty,Iincorporateatestbodyintothemodel:afaceinprofile(Fig.6).

Figure6.Revolutionofleft-facingprofileoncylindricalring(a)andonMoebiusstrip(b) Figure6ashowsaleft-facingprofilerevolvinginacounterclockwisedirectionaroundacylindricalring(seearrows).Itisclearthatactiononthesimpleringwillcontinueindefinitelyinthismanner,withtheorientationoftheprofileneverchanging(thoughthehalf-faceisturnedupsidedown).InFigure6b,weseetheprofilemovingcounterclockwiseabouttheMoebiusstrip.Enteringthetwist,theleft-facingformischangedintoitsright-facingcounterpart,thetransformationbeingcompletedafter360°havebeentraversed.ThetransformationofleftintorightthereforecoincideswiththeoccurrenceofonefullcycleofMoebiusaction.Thischangeinorientationcanbeseentoreflectachangeinclocksense,for,whatiscounterclockwisetoaleft-facingprofilewillbeclockwisetoonethatfacesright.Thus,liketheactionoftheHo-t'u,Moebiusactioninvolvesareversalofclocksense.

However,thestructureoftheMoebiusrevealsaprocessoforientationreversalthatismorenuancedthanwhatvonFranz’stextindicatesfortheHo-t’u.WiththeMoebius,itisobviousthatcycle1actionisnotuniformlyclockwisefrombeginningtoend,thisbeingfollowedbyashiftingearsthatgivesasecond,purely

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counterclockwisecycle.Rather,theclockwiseorientationofcycle1isfullyrealizedonlyattheendofthecycleastheculminationofanongoingtransformationfromaninitialcounterclockwiseorientation.Then,inenteringcycle2,thedirectionofthegear-shifthasitselfshiftedsothatthemomentumisnowfromclockwisetocounterclockwise.ThefactthatchangeinclocksenseisinherenttoMoebiusactionmeansthatnoseparateswitchingofgearsisrequiredtobringitabout.Thegearsareshiftingthroughouteachcycle,withthequasi-returntooriginmarkingthecompletionoftheshiftinonedirection,andthereadinesstobeginshiftingintheother.Thebroaderimplicationsofthesetransformationsofclocksensewillsoonbediscussed.

Now,whiletheMoebiusstripdoesgenerallyprovideuswithanappropriatelydynamicandparadoxicalmodeloftheHo-t'udoublecycle,thespecificphasestructureoftheMoebiusdoesnotmeettherequirementsoftheHo-t'u.ThisobservationtiesinwiththepropositionIofferedabove:itistheKleinbottlethatisrequiredforthefullembodimentoftheHo-t'uarchetypalfield.Whatwewilleventuallyseeisthatthelower-dimensionalMoebiusstructureproperlyembodiesalower-orderarchetypalfield.

ThelimitationoftheMoebiuswithrespecttotheHo-t'ucanreadilybedemonstrated.Again,aMoebiusstripisproducedsimplybytwistingoneendofanarrowbandofpaperbeforejoiningittotheotherend.Ifthecontinuouslycurvedstructurethuscreated(showninFigs.5band6b)isflattenedout,weobtainthetriangularversionoftheMoebiusillustratedinFigure7.

Figure7.FlattenedMoebiusstrip TheflatteningoftheMoebiusbandgivesitsquantizedstructure.Insteadofeachcycleconsistingofanunbrokenrevolutionthrough360°,noweachiscomposedofdistinctphasessetofffromoneanotherbythecreationofedges.ForfullcorrespondencetotheHo-t'u,werequiretwocyclesoffourphaseseach.Whatwehaveinsteadisastructurewhosetwocycleseachconsistofonlythreephases. TobringoutthephasestructureoftheKleinbottle,webeginwithacomparison(seeFig.8andFig.9).

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Figure8.Constructionoftorus(upperrow)andKleinbottle(lowerrow)BothrowsofFigure8depicttheprogressiveclosingofatubularsurfacethatinitiallyisopen.Intheupperrow,theendcirclesofthetubearejoinedintheconventionalway,broughttogetherthroughthethree-dimensionalspaceoutsidethebodyofthetubetoproduceatorus.Bycontrast,theendcirclesinthelowerrowaresuperimposedfrominsidethebodyofthetube,anoperationrequiringthetubetopassthroughitself.ThisresultsintheformationoftheKleinbottle. BycomparingFigure8withFigure9,westarttoseehowaquantizedversionoftheKleinbottlecanbeproduced.

Figure9.Constructionofflattorus(afterBarr1964,23)(©CourtesyofDoverPublications)

InFigure9,Barrshowshowalongpapercylinder(picturedatthetop)canbeflattenedoutalongthedashedline,thenfoldedoversothattheendsarefacingeachother.Theendscanthenbegluedtogethertoproduceaflatversionofthetorus.

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Figure10.ConstructionofflatKleinbottle(adaptedfromBarr1964,23)

Figure10illustratesthecreationofaflatKleinbottle.Onceagainthepaper

cylinderisflattenedbut,thistime,wecutaslitneartheleftendofthecylinder,indicatedinthediagrambythedashedline(seemiddlestepofFig.10).Tocompletetheconstruction,weinserttherightendintotheslit,drawingitthroughsothatitcanbebroughttogetherwiththeleftendfrominsidethebodyofthetube,similartowhathappenswiththeunflattenedKleinbottleofFigure8whenitsendcirclesaresuperimposedviaself-intersection.(Theendtobeinsertedintheslitcanbenarrowedtoallowittofit.)

Byflatteningandfoldingthepapercylinderaswehave,eightdistinctsurfaceshavebeencreated.Thesesurfaces,distinguishedfromoneanotherbytheedgescreatedinthefolding,givethephasesofthequantizedKleinbottle,whichcorrespondintheirnumbertothetrigramphasesoftheHo-t’u.Buttheone-sidedKleinianconfigurationisstructuredmoresubtlythanitstwo-sidedtoroidalcounterpart,whosefourinsidesurfacesarecategoricallyseparatedfromthefoursurfacesontheoutside.LetusattempttoclarifytheKleinianstructurebyopeningupthefoldedpapermodel(Fig.11).

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Figure 11. Opened Klein bottle. The folded paper model is opened up to expose its eight sub-surfaces. Dashed lines indicate the placement of the slit connecting the two sides of the bottle.Accompanying theHo-t’unumbervaluesare thecorrelatedcompassdirectionsand trigramnamesgivenintheIChing(SeeFig.2).

Themovementfromonesurfacetoanotherincycle1marksthepassage

frominsidetooutside.ThesequantizedtransitionsareshownasfollowingthepatternoftheHo-t’ucross.WiththeHo-t’u(Fig.2),actionisdirectedfrombelow(North;K’un)toabove(South;Ch’ien),therebyestablishingthecross’verticalaxis.Then,aftermovingdownandtotheleft(East;Li),thereisatransitiontotheright(West;K’an)thatlaysdownthehorizontalaxis,amovementthatintersectsthevertical.Theself-intersectionoftheHo-t’ucrossthatclimaxesinpassingfromthethirdtofourthstagesofactionismirroredontheKleinbottle.Itsfirstthreephasesarefollowedbyafourththatbringsitsself-intersection,asindicatedbythedashedlinesignifyingtheslitwherethesurfacepassesthroughitself.5Andinmovingthroughtheslit,we“returntocenter”(phase5),whereuponwefindourselvesreadytobeginagainincycle2withphase6,thoughwearenowbeginningontheoutsideoftheKleinbottle.Notethat,unliketheotherphasesofcycle1,phase5isassociatedneitherwithacompassdirectionnoratrigram.Thefifthphasemerelymarksthecompletionofcycle1byreturningustoourpointofdeparture,thoughnowontheoppositesideofthesurface,anditisthisquasi-returnthatpreparesusforinitiatingthephaseactionsofcycle2.

IwanttoemphasizethattheKleinbottleisindeedone-sided.Thismeansthatinsideandoutsidearenotstrictlyseparatedfromeachotheraswiththetwo-sided

5Alongwiththecross,thepelicancametosymbolizeChristianityinmedievalEurope.Likethecross,thepelicanwasseenasasymbolofself-sacrifice,forthemotherofthisspecieswasbelievedcapableoffeedingheryoungwithherownbloodbypressingherbillintoherchest.ElsewhereIexploredthesymbolicrelationshipbetweentheself-intersectingpelicanandtheKleinbottle(seeRosen1995,2014).

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torusbutflowintoeachotherinaccordancewiththebottle’sowninternalstructure.Socycle1doesnotconsistoftransitionsbetweensub-surfacesallofwhichsimplylieontheoutsideoftheKleinbottle.Instead,movementfromsurfacetosurfaceentailsaprogressivepassagefrominsidetooutsidethattakesplacequitenaturally,reachingitsculminationwiththephase-5completionofcycle1.Itisperhapseasiertoseetheeffectofone-sidednesswiththecontinuousactionontheMoebiusstripshowninFigures5band6b.StartingfromanypointontheMoebius,arevolutionof360°willbringusbacktotheplacewherewebegan(we“returntocenter”)butontheoppositesideofthestrip.

WehaveobservedaswellthatMoebiusrevolutionchangesclocksense,transformingcounterclockwiseactionintoclockwiseactionandviceversa(left-rightorientationiscorrespondinglytransformed;seeFig.6b).ThesameappliestoactionontheKleinbottle.However,eventhoughclocksensechangesincycle1,wecanstillassociatetheoveralldirectionofmovementinthiscyclewithclockwise,giventhatthemomentumisshiftingthroughoutfromaninitialcounterclockwisespinningtoaclockwiseone.Cycle1canthusbesaidtoinvolvethegenerationofclockwiseaction.InrelatingthequantizedKleinbottletotheIChing,weidentifythefourspecificphasesofthecycle-1shifttoclockwiseasthecardinalcompassdirectionsoftheHo-t’u:North,South,East,andWest. Cycle2reversesthemomentumestablishedincycle1.Beginningwithphase6,ourtrajectorynowtakesusfromoutsidetoinsideandfromclockwisetocounterclockwise.Inviewofthewayclocksenseshiftsoverthecourseofthiscycle,wecansaythatthecycleinvolvesthegenerationofcounterclockwiseaction.RelatingthistotheIChing,thefourphasesofcycle2areidentifiedastheintercardinalHo-t’udirections:Northwest,Southeast,Northeast,andSouthwest.(Note,however,that,insection8,analternativeinterpretationofthephasesofcycle2isofferedthatdepartsfromIChingtradition.) Beforegoinganyfurther,Iwanttoclarifythenatureoftherotationalactionwearedealingwith.Firstofall,Figure6showstherelativesimplicityofcylindricalaction(6a)whencomparedwithactionontheMoebius(6b).Intheformer,theleft-facingprofilemerelyrevolvesaroundthelengthoftheringwhereas,inthelatter,italsorotatesaboutitsownaxis.Thesamecombinationofrotationalcomponents(longitudinalandtransverse)isfoundwithKleinianrotation.Itisthetransversecomponentofthistopologicalactionthattransformsleft-rightorientationandclocksense.InthecaseoftheactionportrayedinFigure11,jumpingfromonenumberedphasesurfacetoanotheraroundthecrossisthequantizedcounterpartofcontinuouslongitudinalrevolutionontheKleinbottle.ThecorrespondingHo-t’udiagramshownonlyintermsofitsnumericalsequence(Fig.2,right)showsonlythislongitudinalcomponent.Butwhenweconsidertheaccompanyingtrigramsandtheircompassdirections(Fig.2,left),thetransversecomponentofKleinianactionbecomesimplicated,withthefourdirectionssymbolizingfourdiscretestatesinthetransitionfromcounterclockwisetoclockwise.Thus,comingtounderstandthehiddentopologyoftheHo-t’uhelpsusseemoreclearlyhowtheHo-t’udiagramperseandthepulsatingtrigramsoftheIChingareaspectsofthesamedynamicstructure.

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Whatarethebroaderimplicationsofallthis?Inthepresentsection,IhaveemployedtopologytorelatethemicroworldactiondescribedincontemporaryphysicstotheactiongiveninancientChina’sHo-t’unumberarchetype.AlthoughIhavepresentedtangiblepapermodelstodetailthemannerinwhichthisactiontakesplace,theactionitselfinfactcannotberealizedasanobjectiveeventinphysicalspacebecauseitisessentiallypsychophysical.Andasanarchetypalaction(theprefixarchaeo-derivesfromtheGreekarché,thebeginning,thefirst),wemaytakeitasthepre-objectiveantecedentfromwhichtheexperienceofobjectivephysicalrealityfirstarises.ThisisconsistentwiththeviewpointexpressedbyvonFranz. AboveInotedvonFranz’scommentsthatnumberarchetypesliketheHo-t’uare“cosmicplans”(22),andthat“theuniversewas'organized'accordingtonumericalpatternsofthiskind"(24).Consideringalsothatthe“firstfourinnerrhythms[oftheHo-t’u]arephysical”(footnote16,243–244),theKleiniangenerationofclockwiseactionincycle1canberelatedtothecreationoftheobjectivematerialworld.InterestinginthisregardisthatthefirstcycleoftheHo-t’uisseenas“cumulativeandexpandingastimegoeson”(236).Incosmogonicterms,thiscanbeassociatedwiththeexpansionofthephysicaluniverse.Herewehave“theusualclockwisemovement”(236),thatinwhichtimeflowsforwardinitsnaturalsequence(Smithetal1990,118)followingtime’sarrowofincreasingentropyassociatedwithcosmicdilation.

AfterthegenesisofclockwiseactionfromaninitialcounterclockwiseorientationinthefirstHo-t’ucycle,themomentumshiftsbacktocounterclockwiseinthesecond.Interpretedcosmogonically,thisisthebackwardorretrogradecyclewhereinpreoccupationwiththephysicaluniverseprojected“outthere”incycle1isnowovercomeandaninwardawarenessofthepsycheisregained—“thesecondfour[innerrhythms]arepsychic,”saysvonFranz.Butitisnotthatpsycheisfavoredone-sidedlyinthiscycle.Instead,thepsychophysicalnatureofthecosmosisrealizedinfull.Whathadbeenprojectedaspurelyphysicalorobjectiveisthuspresentlydrawnbackintodisclosethearchetypalsourceofsaidprojection,asourcethatisitselfmoreprimordialthanthesplittingofmatterandpsyche,objectandsubject.Indrawinginward,theretrogradeactionofcycle2reversestheoutthrustoftheexpandinguniversetobring“anopposite,backwardmovement,foldingupandcontractingastimegoesby”(WilhelmquotedinvonFranz,236–237).Smithetal,citingapassagefromtheShuo-KuaoftheIChing,putitthisway:“‘goingalongwith…[time’sordinary]sequence’indicatesthatonecomestoknowthingsintheorderthattheyoccurnaturally…[whereas]‘goingagainstthedirectionoftime,’meansreachingwiththemindintothefuture,againsttheflow,andknowingthingsbeforethestreamoftimehasbroughtthemtous”(1990,118).

Iacknowledgethespeculativenatureofthistakeoncosmicevolution.Inthepagestofollow,Iwillelaborateonmyapproach,clarifyingit,andprovidingfurthersubstantiation.ThiswillinvolvereprisingmydemonstrationinTheSelf-EvolvingCosmos(2008a)thataneffectiverenderingofcosmogonyandtherelatedissueofquantumgravityrequiresustorecognizethattheKleinbottleisactuallybutonememberofawholefamilyofinterrelatedtopologicalstructures.Inbringingthistopologicalfamilyintoplay,wewillengagewithacorrespondingfamilyofTaoist

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structuresthatgoesbeyondthesystemoftrigrams.Inthisway,thelinkbetweenmodernWesterncosmologyandancientEasterncosmologywillbebroughtintosharperfocus.3.QUANTUMGRAVITYANDSTRINGTHEORYIopenedthispaperbynotingthelackofprogressintheoreticalphysicstowardanaccountthatwouldunifythefourfundamentalforcesofnature.Wefoundthatreconcilingthequantummechanicalforceswithgravitationrequiresatheorythatcanworkeffectivelyaroundthesub-microscopicPlancklengthof10–35m.Indispensabletosuchatheoryis!,thequantumofaction.Thisindivisibleatomofprocessconstitutesthecoreofquantummechanicsandcanbesuccessfullyemployedinrelativelylowenergyandlowmagnificationregimes,butdealingwithitbelowthePlancklengthinvolvesenergiessotumultuousthatthequantumpropertiesofdiscontinuityandobserver-observedinteractioncannolongerbemanagedviaprobabilisticanalysis. Itwasinthe1970s,followingtheprogressachievedinunifyingthequantummechanicalforces(strong,weak,andelectromagnetic),thatworkonatheoryofquantumgravitybeganinearnest.AndthisiswhenconfrontationwiththechaosofthePlanckrealmcouldnolongerbeavoided.TheequationsthatwouldunifythefourforceswerenowcompletelyunabletocontainthewildlyfluctuatingPlanckianenergies,asmanifestedbytheinfiniteprobabilitiesthatturneduptorenderthoseequationsuseless.Movementtowardaneffectivetheoryofquantumgravityhasthuscometoastandstilloverthepastfortyyears.Commentingontheimpasse,Smolinnotesthatadvanceshavebeenblocked“despiteourbestefforts”(2006,viii).What“bestefforts”ishereferringto? Sincethe1970s,thequestforamathematicalunificationofnaturehaslargelybeendominatedbyanapproachknownasstringtheory.Inthisendeavor,theattemptismadetoavoidprobingbelowthePlanckthresholdsimplybyassumingthatthesmallestconstituentsofnaturearenotindefinitelyminisculepoint-particlesasprevioustheoryhadassumed,butstring-likevibratingelementsoffiniteextensionconvenientlyscaledatthePlancklength.Itisbecausethisstratagemhasbeenabletoeliminateinfinitetermsfromquantumgravitationalequationsthatithasbecomethepreferredapproach.Butthepricepaidforthispositivisticployhascometobeacknowledged(Smolin2006,Woit2006).Inmyownexplorationsofthematter(Rosen2004,2008a,2008b,2015),Ihaveidentifiedtwoproblemswithstringtheory.

First,whileitistruethatstringtheoryservestheclassicalontologybysidesteppingsub-Planckianambiguity,anepistemicambiguitytakesitsplace.Stringtheory’sgeneralequationsmaybefreeofunmanageableinfinities,buttheoristsmustbeabletosolvethesehighlyabstractequationsinamannerthatproducesaspecificdescriptionoftheworldasweknowit.Asthingsnowstand,theequationsyieldavastarrayofpossiblesolutionswithnoguidingprinciplebymeansofwhichthefieldcanbenarrowedinuniquecorrespondencewithknownphysicalreality.Asecondlimitationofthetheoryisthatitseemstocontradictitselfinitsassumption

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offundamentalparticleswithfiniteextension.“Stringsaretrulyfundamental,”saysphysicistBrianGreene,“theyare‘atoms,’uncuttableconstituents”ofnature.So,“eventhoughstringshavespatialextent,thequestionoftheircompositioniswithoutanycontent”(1999,141).Butisn’tthisacontradiction?For—atleastaccordingtotheclassicalconceptofthecontinuumnotexplicitlychallengedbystringtheory,tobespatiallyextendedistobecuttable,infact,infinitelydivisible.Howthencouldastringbeafundamentalparticle,anatomicorindivisibleingredientofnature,whenitisspatiallyextended?Insum,stringtheoryisbothambiguousanditappearstocontradictitselfwhenseeninclassicalterms.

ButIsuggestthatdespitethelimitationsofconventionalstringtheory,ifwetakethevibratorypatternofthefundamentalstringsasessentiallyKleinianinnature—withKleinianspinnotobjectifiedbutunderstoodphenomenologically,inpsychophysicaltermsthatbridgethegapbetweensubjectandobject—stringtheorycangaingreatercoherence.Infact,IdemonstrateinCosmosthatbyreformulatingthetheoryinthecontextofthepsychophysicaldisciplineIcalltopologicalphenomenology,itcanbecastinaformthatprovidesadefinitive(albeitqualitative)accountofquantumgravity,onethatunambiguouslyyieldsthefundamentalparticlesofthestandardmodel.Wewillexplorethebasisforthisconclusionandreviewitsdetailsincomingsections.Inthecourseofdoingthis,theworkofCosmoswillbecarriedforwardandtheancientChineserootsofphenomenologicalstringtheorywillbebroughttolight.4.THETOPODIMENSIONALFAMILYWeknowthattheprimaryelementofactioninmicrophysicsis!,thequantumofactionassociatedwiththeemissionofradiantenergy.Wehavefoundaswellthatthecuriousspinningaccompanyingthisquantizeddynamicismodeledbythehypernumberε,andthatthegeometriccounterpartofεistheKleinbottle.Intheformofε!,theKleinbottleisthusseentoimplicitlyembodytheangularactionthatliesattheheartofquantummechanics.BearinginmindthattheKleinbottlecannotbeproperlyunderstoodasaCartesianobjectinphysicalspacebutmustbegraspedasinherentlypsychophysical,IproposeweviewKleinianspinasthebasicbuildingblockofapsychophysicalaccountofquantumgravity.

InPauli’smatrices,!/2istakenasthefundamentalunitofelectronspin.Infact,!/2isthebasisfordeterminingthespinofallsubatomicparticles,fermionsandbosonsalike.GiventheessentialroleplayedbyspininquantummechanicsandtheunderlyingsignificanceoftheKleinbottleinsaidspin,IsuggestedinCosmosthatallmicroworlddynamicsarisefromspinoftheKleiniankind:ε!/2. Now,inhisfurtherexplorationofthehypernumberε,Musèsindicateda“higherepsilon-algebra”wherein“√εninvolvesin,thesubscriptsofcoursereferringtothe(n+1)thdimensionsincei≡i1alreadyreferstoD2”(1968,42).InviewoftheintimaterelationshipbetweenεandtheKleinbottle,canMusès’implicationofadimensionalhierarchyofhypernumbervaluesbegiventopologicalexpression?TheKleinbottledoeslenditselftosuchageneralization.

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Mathematicianshaveinvestigatedthetransformationsthatresultfrombisectingtopologicalsurfaces.IftheKleinbottleisbisected,cutdownthemiddle,itwillfallintoapairofoppositely-orientedMoebiusstrips.Next,bisectingtheone-sidedMoebiusstrip,atwo-sidedlemniscatorysurfacewillbeproduced,itssidesbeingrelatedenantiomorphically(i.e.,asmirroropposites).Finally,cuttingthelemniscatedownthemiddleyieldsinterlockinglemniscates.Thetransformationbroughtaboutbythisbisectionisclearlythelastoneofanysignificance,sinceadditionalbisections—beingbisectionsoflemniscates,canonlyproducethesameresult:interlockinglemniscates.Thebisectionseriesiscompletedthenwhenweobtaininterlockinglemniscates,astructuretermedthesub-lemniscate.ByexperimentingwiththebisectionoftheKleinbottleinthisway,afamilyoffournestedtopologicalstructuresisdiscovered(Fig.12).

Figure12.Topologicalbisectionseries.Fromtoptobottom:Kleinbottle,Moebiusstrip,lemniscate,sub-lemniscate

InCosmos,dimensionaldifferencesamongthefourmembersofthebisection

seriesarestudiedphenomenologically.Whiletoordinaryobservationeachmemberappearsasbutatwo-dimensionalsurfaceinthree-dimensionalspace,phenomenologicalreflectionleadstotheinsightthateachactuallyconstitutesadimensionallifeworlduntoitself(“lifeworld”isaphenomenologicaltermforthenon-classical,intrinsicallypsychophysicalrealmwhereinsubjectandobjectareintimatelyentwined).WhereastheKleinbottleisthree-dimensional,itsnestedcorrelatesareofprogressivelylowerdimension:theMoebiusistwo-dimensional,thelemniscateisone-dimensional,andthesub-lemniscateiszero-dimensional.And,liketheKleinbottle,eachoftheselower-dimensionalworldsisapsychophysicalrealmofactionthatsurpassestheCartesiandivisionofpsycheandmatter,subject

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andobject.Thisaccountofseveraldifferenttopodimensionallifeworldsembeddedwithineachotherisconsistentwiththehierarchyofε-likespinstructuressuggestedbyMusès.

εD0 εD0/εD1 εD0/εD2 εD0/εD3εD1/εD0 εD1 εD1/εD2 εD1/εD3εD2/εD0 εD2/εD1 εD2 εD2/εD3εD3/εD0 εD3/εD1 εD3/εD2 εD3

Table1.Interrelationalmatrixoftopodimensionalspinstructures

Table1,thetopodimensionalspinmatrix,givestheε-basedcounterpartof

thetopologicalbisectionseries.Thethree-dimensionalKleinianspinoriswrittenεD3,withlower-dimensionalmembersofthetightlyknitspinfamilydesignatedεD2,εD1,andεD0(correspondingtotheMoebial,lemniscatory,andsub-lemniscatorycirculations,respectively).Thesetermsarearrayedontheprincipaldiagonalofthematrix(extendingfromupperlefttolowerright).Theinterrelationshipsamongthefourprincipalmatrixelements,takentwoatatime,arereflectedintheelementsappearingoffthemaindiagonal.

Generallyspeaking,Table1unpacksthedialecticalstructureoftopodimensionalinterrelations.Inkeepingwiththe“musical”implicationsofstringtheory,wemayregardtopodimensionalactionasinherentlyvibratoryinnature.TheprincipaldiagonaloftheTablecontainsadimensionalseriesoffundamentalvibrationsortones,andthesefourprincipaltermsarecoupledtoeachothertwoatatimebysixpairsofovertone-undertoneintervalsrelatedtoeachotherinthemirror-opposedfashionofenantiomorphs.Thedimensionalovertoneratiosarethevaluesextendingbelowthefundamentaltones,whereastheundertoneratiosarethevaluesappearingtotherightofthefundamentals.(InCosmos,thetopodimensionalactionmatrixisseenasanalogoustotheoldPythagoreantable,whichisportrayedasanexpandingseriesofmusicalintervals,withfundamentaltonesontheprincipaldiagonal,flankedbyovertonesandundertones.Formoreonthis,seesection5.) ConsiderinTable1thetwoprincipaltonesofhighestdimensionality:εD2andεD3.Thesematrixelementsarelinkedbytheovertoneandundertonegiveninthetwocorrespondingnon-principalcells,εD3/εD2andεD2/εD3(respectively).Theenantiomorphically-relatedcouplingcellsinquestionarethehypernumbercounterpartsoftheconcretelyobservable,oppositelyorientedMoebiusstripswhich,whengluedtogether,formtheKleinbottle.Takenstrictlyasaprincipalmatrixelement,thehypernumberMoebiusvibrationisthespinstructurethatconstitutesthetwo-dimensionallifeworld(εD2).ButwhenweshiftourviewoftheMoebius,consideritinrelationtohigher,Kleiniandimensionality,akindof“doubling”takesplaceinwhichtheεD2singularMoebiusspinstructurebecomesapairofasymmetric,mirroropposedtwins,εD3/εD2andεD2/εD3.ItisthroughthefusionofthesedimensionalenantiomorphsthatKleiniandimensionalityiscrystallized.SincetheTable-1matrixindicatesthatallfourprincipaldimensionalitiesorfundamentaltonesareinterrelatedbyaccompanyingoff-

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diagonalovertone-undertonepairs,wecandrawthegeneralconclusionthathigherdimensionsemergethroughprocessesofenantiomorphicfusion(thiswillbespelledoutcompletelyinduecourse). Theprocessofdimensionalgenerationcanbeclarifiedinbroadtermsbyrelatingittoareversemovementthroughthebisectionserieswhereintopologicalstructuresarenotdividedbutgluedtogether.Tobegin,weimaginethefusionofinterlockinglemniscatesthatyieldsthesinglelemniscate.Thiscorrespondstothegenerationoftheone-dimensionallifeworld(εD1).Next,wepicturetheenantiomorphically-relatedsidesofthetwo-sidedlemniscatemergingtoformtheone-sidedMoebiusstructure,thisbeingassociatedwiththegenesisofthetwo-dimensionallifeworld(εD2).Finally,weimagineMoebiusenantiomorphsfusingtoproducetheKleinbottle,whichcorrespondstotheevolutionofourthree-dimensionallifeworld(εD3).Witheachfusion,alower-dimensionallifeworldisabsorbedbyaworldofhigherdimension,takenintoitinsuchawaythatthelowerdimensionisconcealed.Intheend,wehavethreelower-dimensionalvibratorystructuresconcealedwithinthethree-dimensionalKleinianvibration,muchaslowerdimensionsarehiddenbybecoming“curledup”withinvisible3+1-dimensionalspace-timeintheconventionalstringtheoreticaccountofdimensionalcosmogony.Itturnsout,infact,thatthephenomenologicalapproacharrivesatthesametotalnumberofdimensionsasdoestheconventionaltheory. WhatIdemonstrateinCosmosisthattheKleinianspinor,εD3,isnotitselfanextendedthree-dimensionalspace,butisaquantizedthree-dimensionalblendofspaceandtimethatfirstgivesbirthtoourfamiliar3+1-dimensionalspace-time.6Inlikemanner,thetwo-dimensionalMoebiusspinor(εD2)spinsouta2+1-dimensionalspace-time,thelemniscatoryspinor(εD1)sendsfortha1+1-dimensionalspace-time,andthesub-lemniscatoryspinor(εD0)projectsa0+1-dimensionalspace-time.Asimplesummationofprojectedspace-timedimensionsgivesusatotaloften,withthesixlowerdimensions—(2+1)+(1+1)+(0+1)—beinghiddenwithinthelarger3+1-dimensionalspace-time.Thispictureofoverallten-dimensionality,withsixdimensionsconcealed,accordswiththebasicaccountprovidedbyorthodoxstringtheory.Thuswemaysaythatourfourphenomenologicalspinorsspinoutthetenspace-timedimensionsofstringtheory.7 Yetdespitethegeneralagreementonthenumberofspace-timedimensions,importantdifferencesexistbetweenconventionalandphenomenologicalinterpretationsofstringtheory.Mainstreamtheoristshaveapproachedcosmogonybyadoptingtheconceptofsymmetrybreaking.Inthisnarrative,thefourforcesofnatureareconceivedasvibratingstringsthatinitiallyexistedinapurelysymmetricten-dimensionalspacescaledaroundthePlancklength.Subsequently,theperfectprimordialsymmetrywasspontaneouslybrokenbyadimensionalbifurcationinwhichfouroftheoriginaldimensionsexpandedtoproducethevisibleuniversewe

6TheKleinianspinorisa“natalspace,”a“matrix[for…]existingspace,”touseMerleau-Ponty’swayofdescribingthegenerativeroleplayedbyphenomenologicaldimensionality(1964,176).7WiththeextensionofstringtheoryknownasM-theory,elevendimensionsareactuallyentailed,thoughtheeleventhdimensionisnotliketheotherten.This“extra”dimensioninfactmaybeinterpretedinphenomenologicalterms.SeeTheSelf-EvolvingCosmos.

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knowtoday,withtheotherdimensionsremaininghidden.Coupledwiththiswasthebreakingofforce-fieldsymmetrytocreatetheappearanceofirreconcilabledifferencesamongtheforces. However,whiletheforegoingaccountofcosmogonyincorporatesbothdimensionalandforce-fieldsymmetrybreaking,thetwoarenotpreciselyalignedwitheachotherinthetheoreticalreckoning.Thisreflectsthefactthatcontemporarytheoristshavebeenunabletoarticulateacompletegeometricrenderingofcosmicevolution.Forthegeometricprogramfullytoberealized,thephysicaleventsdescribedinthestandardandinflationarymodelsofcosmicdevelopmentwouldneedtobespecificallyexpressibleasdimensionalevents.WhatHeinzPagelsnotedtwentyyearsagoindiscussingtheextra-dimensional(Kaluza-Klein)interpretationofcosmogonyremainstruetoday:“NoonehasyetbeenabletofindarealisticKaluza-Kleintheorywhichyieldsthestandardmodel”(1985,328).Inthestring-theoreticapplicationofKaluza-Kleintheory,oneobviousreasonforthislimitationistheabsenceofaconceptualprinciplethatcouldguidetheanalysttounambiguoussolutionsoftheten-dimensionalgeneralequations,solutionsspecifyingtheexactshapesofthehiddendimensionsthatwouldcorrespondtothephysicalfactsofthestandardmodel.Ofcourse,iftheprevailingtheorycannottelluswhatthedimensionalstructuresarethatcorrespondtophysicalreality,itcanhardlyinformusonhowthesedimensionsdevelop.Inpointoffact,thereisreallynopositivefeatureintrinsictothetheorythatprovidesfortheevolutionofdimensions.FromwhatIcantell,theonlyreasondimensionalbifurcationisassumedtohavetakenplaceatallisthattheoristsmustsomehowaccountforthepresentinabilitytoobservesixofthetendimensionsneededforaconsistentrenderingofquantumgravity(onethatavoidsuntenableprobabilityvalues). Smolinseemstoputhisfingerontheunderlyingproblemincallingattentiontothe“wrongassumption”physicists“areallmaking”whentheypresentthe“wholehistoryofconstantmotionandchange…assomethingstaticandunchanging”(2006,256–57).Whenauthenticchangeisthusdenied,itisnotsurprisingthatnonatural,parsimoniouswayofaccountingforcosmogonyisforthcoming.Conventionalstringtheorywellexemplifiesthisadherencetotheclassicalintuitionofchangelessnessintheprimacyitgivestothenotionofsymmetry.Itisinassuminganinitialstateof“perfectsymmetry”thattheoristsmustresorttotheartificeof“spontaneoussymmetrybreaking,”anallegedeventthat—farfrombeinganaturalconsequenceofthepurelysymmetrictheory—isgratuitouslyinvokedwithoutacompellingexplanationofitsbasis. Theinherentdynamismofphenomenologicalstringtheoryaffordsawayoutoftheimpasse.Insteadofartificiallyappendingasymmetrytoaprimordiallyperfectsymmetry,adialecticofsymmetryandasymmetryisofferedthatpermitsanunequivocal,intrinsicallymeaningfulaccountoftheevolvingforcesofnature.Thisprincipleof“synsymmetry”(Rosen1975,1994,2006,2008a)isimplicitinthetopologicalbisectionseriesanditsassociatedtopodimensionalspinmatrix(Table1). Forasimpleillustration,considertheMoebiusstrip.Itarisesfromthefusionofmirror-opposed,asymmetrically-relatedsidesofthelemniscate.Wecansaythat,throughthisunionofopposites,theasymmetryoflemniscatorysidesisrendered

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symmetric.However,whiletheMoebiuscanbedeemedsymmetricvis-à-visthefusedlemniscatorysidesthatconstituteit,atthesametimeitisitselfamemberofanenantiomorphicallyasymmetricpairwhoseownfusionproducestheKleinbottle.Wemaygenerallyconcludethatthemembersofourtopodimensionalfamilyareneithersimplyasymmetricnorsimplysymmetric,butsynsymmetric:agivenmembercombinessymmetryandasymmetryinsuchawaythatitissymmetricinrelationtoitslower-dimensionalcounterpartandasymmetricinrelationtoitshigherone(thesub-lemniscateisanexceptiontothis,sinceithasnolower-dimensionalcounterpart).Iproposethatthesynsymmetryconcept,vieweddynamicallyintermsofenantiomorphicfusionevents,constitutesaguidingprincipleforcosmogony.Theforcesandparticlesofnatureevolvebyageneralprocesswhereinasymmetricdimensionalenantiomorphsfusetocreateadimensionalsymmetrythatatonceinherentlygiveswaytonewasymmetry.Mytopo-phenomenologicalinterpretationofcosmogonyiselaboratedinsections6–9,wherealotmoredetailisgivenondimensionaldevelopment,cosmogonybyenantiomorphicfusion,andquantumgravity.Indeed,alotmoredetailwillbeneeded.

Recallthat,inthecourseofexaminingtherelationshipbetweentheKleinbottleandtheHo-t’unumberarchetypeinsection2,Iofferedapreliminaryglimpseofthecosmogonicimplicationsofmyaccount.Sincethetopodimensionalexplanationofcosmogonyjustgivendoesnotaddressthoseimplicationscompletely,itwillbenecessaryformetoexpandonwhatIhavesaidhere.Onlywhenfurtherdetailhasbeendisclosedonthespecificnatureofcosmicevolutionwillwebeabletofullyappreciatetheconnectionbetweenwhatwasintimatedinsection2andwhathasbeensetforthinthepresentsection.Andthis,inturn,willallowustograspthefullrelationshipbetweentopodimensionalandTaoistapproachestocosmogony.However,beforecarryingouttherequisiteelaboration,IwanttodemonstratethatthetopologicalfamilyformingthecoreofphenomenologicalstringtheoryhasitscounterpartinancientChina.5.THEWORLD’SOLDESTSTRINGTHEORY

Musichaslong…providedthemetaphorsofchoiceforthosepuzzlingoverquestionsofcosmicconcern.FromtheancientPythagorean“musicofthespheres”tothe“harmoniesofnature”thathaveguidedinquirythroughtheages,wehavecollectivelysoughtthesongofnatureinthegentlewanderingsofcelestialbodiesandtheriotousfulminationsofsubatomicparticles.Withthediscoveryofsuperstringtheory,musicalmetaphorstakeonastartlingreality,forthetheorysuggeststhatthemicroscopiclandscapeissuffusedwithtinystringswhosevibrationalpatternsorchestratetheevolutionofthecosmos.(Greene1999,135)

Idemonstratedinsection2thatthehiddentopologyofancientChina’sHo-t’unumberarchetypeisKleinianinnature.Intheprevioussection,wefoundthattheKleinbottledoesnotstandalonebutisamemberofaclosely-knitfamilyoffourtopologicalstructuresconstitutingthecoreofphenomenologicalstringtheory.

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WhatwearepresentlygoingtoseeisthattheHo-t’uanditsassociatedtrigramsystemalsoparticipateinalargerfourfoldgrouping,onethatcomprisesthe“stringtheory”fromwhichphenomenologicalstringtheorycanbesaidtohaveoriginated.Inotedearlierthatthevibrationsortonesofthetopodimensionalactionmatrix(Table1)beararesemblancetothetonalsystemsetforthintheoldPythagoreantable.ThistableisalsoknownastheLambdoma(sinceitisoftenshowninaconfigurationsuggestiveoftheGreekletterlambda),anditisconnectedtotheideaofthe“musicofthespheres.”ThefirstfourintervalsofthePythagoreantablearedisplayedinTable2.

Table2.SectionofthePythagoreantable

Thetableisusuallyportrayedasanindefinitelyexpandingseriesofmusical

intervals,buttherearemanyvariationsofitintheliterature,withsomeexcludingthe0/0intervalentirelyorplacingitoutsidethebodyofthetable(see,forexample,Godwin1989,95).WhatisshownintherepresentationofthetableIhaveemployedisasetofrelationshipsthatmirrorsourtopodimensionalactionmatrix(Table1).Thereisaprincipaldiagonalthefirsttoneofwhichis0/0,seenintheupperleft-handcorner;thiscorrespondstoεD0,thezero-dimensionaltermofTable1.Iwillhavemoretosayabout0/0below.The0/0fundamentaltoneisfollowedonthemaindiagonalbyaseriesofnon-zerofundamentalsconstitutingreflexive,self-divisive“formsofunity”(Musès,1968,32).Thatis,eachnon-zerofundamentalcanbetakenasaself-divisionthatbyordinaryarithmeticwouldequal1butthatisneverthelessdistinctiveinthePythagoreancontext:1/1≠2/2≠3/3.Thisseriesofself-divisionsmayberelatedtothetopologicalseriesofself-intersectingformspicturedinFigure12,anditisimplicitaswellinthevaluesofTable1.Notealsothat,inbothTables1and2,thefourprincipaltermsarecoupledtoeachothertwoatatimebysixenantiomorphicallyrelatedpairsofovertone-undertonevalues.WemaythereforeentertainthepropositionthatthevaluesprovidedinTable1arethetopodimensionalcounterpartsofthePythagoreanmusicalintervalsappearinginTable2.Accordingly,theserhythmicalrelationshipscanbesaidtogiveusthe“musicofthedimensionalspheres.”

0/0 0/1 0/2 0/3

1/0 1/1 1/2 1/3

2/0 2/1 2/2 2/3

3/0 3/1 3/2 3/3

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MusicologistJoscelynGodwinassertsthatthePythagoreantableis"animageoftheuniverse"(1992,191)and"ameanstowardsymbolicexplanationandpossibleilluminationconcerningcosmicandmetaphysicalrealities"(190).MusicologistRudolfHaasesimilarlyemphasizesthespiritualandcosmogonicimplicationsoftheLambdoma,intimatingthatitmaybenothinglessthananarchetypalgroundplanforthecreationofallthatis:

Itiswellknownthattheconstructionoftheworldonthebasisoftwoantitheticalprinciplesisanage-oldconcept:itisbestknownastheyangandyinofChinesetradition,butalsoformsanimportantelementofPythagoreanphilosophy,whichiswhyKayser[Haase'steacher]revivedthisdualisminreferringtothesimilarlydualstructureoftheLambdoma,regardingitasthefoundationofthelatter.(Haase,1989,103)

Forherpart,vonFranzobservesthatthe“ancientChinese…dividedtheir

wholepsychophysicalcosmosintoaperiodictwofoldrhythm,areciprocal…Yin-Yangmotion.YinandYangare…asymphonyofalternatingrhythmsinwhichspatialelements(infront–behind)andtemporalelements(before–after)arenotseparable.”VonFranzgoesontociteMarcelGranet’smusicalsimilewherein“YinandYangplayinconcert(tiao)andharmonize(ho).‘Thewholeuniversehasarhythmicalbasicstructure’”(1974,95).Inafootnoteonthesamepage,vonFranzsuggeststhat“Homeanstheharmonyofapieceofmusic.”Andinthatsamecontext,shespeaksofthe“rhythmicalinternalmovement”foundinthe“olderheavenlyorder,”theorderrelatedtotheHo-t’u.Later,afterreferringtofield-likeChinesenumbermatrices“whichservedasrhythmicallyorganized…imagesoftheaspectsofthecosmicwhole,”shenotesthepossibilitythat“thefamousPythagoreanlambdoma…alsowasoriginallyafieldarrangementincircularform”(1974,147). TheforegoingreflectionsonthePythagoreantableandtheharmoniesofyinandyangofferanintimationoftheprimordial“stringtheory”onwhichourtopo-phenomenological/psychophysicalstringtheoryisbased.TheHo-t’unumberarchetypeisofcoursepartofthisancientsymphony.ButwehaveseenthatthetrigramsystemassociatedwiththeHo-t’umapsontotheKleinbottleandthelatterisjustonememberofawholetopologicalfamily.Wearenowpreparedtolookatthecorrespondingyin-yangfamilyofwhichtheHo-t’utrigramsystemisbutonemember.Insection1,IintroducedthecosmologyofTaoismbydescribingitsvisionofauniverseinconstantflux.PhilosopherJeaneaneFowlerdevelopsthisthemeatlength,beginningwiththeobservationthat“thatwhichinformedallthetransformationsinthecosmoswaspositedasTao”(2005,46).CitingtheIChing,FowlerfurtherassertsthattheTaois

“theimmutable,eternallawatworkinallchange…theprincipleoftheoneinthemany.”…[Tao]isthatwhichgeneratesthetensionbetweenopposites,thatwhichmakeschangesandtransformationspossible,andthepowerthatrenewsthattensionfrommomenttomoment.Itisthequiet,spontaneous

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powerthateternallygivesenergytothecosmos,totherhythmiccompositionofthestarsandplanetsasmuchastotheenergythatatinyseedneedsforgermination.(46)Laterinhertext,FowlercorrectstheearlierimpressionthattheTaois

associatedexclusivelywiththeOne,for“TaoisbeyondeventheOne”(76).Indeed,wereallycannotsaywhattheTaois.ClassicalTaoisttextssuchastheIChingandtheTaoTeChingcharacterizetheTaoasunnamable,incapableofbeingexpressedinlanguage.TheTaohasthustypicallybeenlinkedtonegativeconceptssuchasemptiness,limitlessness,ornothingness(Ozaki2001,ChiaandWei2009,Grigg1994).IftheTaoisindeedassociatedwithnothingness,andifTaoismformsthefoundationofthePythagoreanLambdomaasHaasesuggests,wemightexpectthezero-dimensionalintervaloftheLambdoma,0/0,toreflectnothingness.Inconfirmingthis,Godwinfirstunderscoresthespecialstatusof0/0asthatwhich“soundsnotonebutisthesilencetowardwhichalltonestend"(1992,192).Then,workingwiththeanalogybetweensilenceandNon-Being,Godwinsuggeststhat,justasthepositivetonesarerootedinthesilenceof0/0,"allBeingculminatesinaNothingnessbeyondBeingthat…isparadoxicallyitssolesupportandpositiveorigin"(193).Wemaythustake0/0asthePythagoreanexpressionoftheTao.

Taoistthinkers,drawingfromtheearlierYin-YangSchoolofphilosophy(theSchoolofNaturalists),positedtheideathat,fromthenothingnessoftheTaotherearisestheparadoxicaldualityofyinandyang,theopposedyetinterpenetratingforcesthatdrivethedynamicsofnature.ButtheIChinggoesfurther,identifyingtwoadditionallevelsofevolutionfromtheoriginalTao.AstheologianWhalenLaiputsit:“TheIChingsays:FromtheGreatUltimate[i.e.,theTao]comethetwopoles[yinandyang];fromthetwopolescomethefourforms[…and]fromthefourforms[bigrams]cometheeighttrigrams”(1980,254).Laiprovidesadiagramofthis(Fig.13).

Figure13.Theyin-yang“familytree”:fourlevelsofcosmogony(adaptedfromLai1980,254).Copyright©2008byJohnWiley&Sons,Inc.ReprintedbypermissionofJohnWiley&Sons,Inc.

Havingconsideredabovethepropositionthatthevaluesofthe

topodimensionalactionmatrix(Table1)arecorrelatedwiththevaluesofthePythagoreantable(Table2),Iwouldnowliketoextendthistoincludethefourmembersoftheyin-yangfamilydepictedinFigure13.Wehavealreadyseentheinvolvementoftwofamilymembers:(1)theTao,associatedwiththe0/0

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Pythagoreanintervalandthezero-dimensional,sub-lemniscatorymemberofthetopodimensionalfamily;and(2)thetrigrams,linkedtothe3/3Pythagoreanintervalandthethree-dimensional,Kleinianmemberofthetopodimensionalfamily.Induecourse,wewillseehowtheothertwobranchesoftheyin-yangfamily—theunigrammodeofyinandyangandthebigrams—arecorrelatedwiththeone-andtwo-dimensionalmembersofthetopodimensionalfamily,respectively.Forthemoment,Iwilljustobservethatthenumbersofyin-yanglinescorresponddirectlytoourtopodimensionalnumbers.TheTaoitselfisdevoidofyin-yanglinesandthismatchesthezero-dimensionaltermofTable1,εD0.Thesingleyin-yanglinesemergingfromtheTaoparalleltheone-dimensionalterm,εD1.Theyin-yangcounterpartsofthetwo-dimensionallifeworld(εD2)arethedoublelinesconstitutingthebigrams,andthetriplelinescomprisingthetrigramsrelatetothethree-dimensionalrealm(εD3).Evidentlythen,thenumberofyin-yanglinesbearsasimple,one-to-onerelationshiptodimensionnumber. Now,letusfocusagainonthetrigrams.Inthefirsttwosectionsofthispaper,weinvestigatedtheIChing’sHo-t’unumberarchetypeandfoundthatitsassociatedtrigramsareorganizedintotwocyclesoffourphaseseach,withphaseactionfollowingtheunderlyingtopologyoftheKleinbottle.Thecosmogonicimplicationsofthiswereexploredandwewereledtotheprovisionalconclusionthatcycle1involvesthegenerationofforwardorclockwiseactionentailinganexpansionoftheuniversethatprojectstheappearanceofanobjectivephysicalworld.Then,inKleiniancycle2,thegearsshifttobackwardandtheprojectioniswithdrawn,theuniversecontractingtobringtoawarenessthepsychophysicalnatureofreality.WhatwehavecometorealizeinsubsequentsectionsisthattheKleinbottleandtheHo-t’u,besidesbeingrelatedtooneanother,areeachmembersofwholefamiliesofinterrelatedstructures—onefamilytopodimensional,theotherTaoist.Forafullunderstandingofcosmogony,theentirefamilymustbetakenintoaccountineachcase,soweclearlywillneedtoworkouttherelationshipsamongallthemembersofbothcosmicfamilies.Moreover,whiletheoverviewofdimensionalgenerationofferedinsection4providesageneralideaofthepatternofcosmicevolutiongivingrisetothefundamentalforcesofnature,itaccountsonlyfortheexpansionoftheuniverse,notitssubsequentcontractionasadumbratedintheKleinianHo-t’u.Theaspectofcontractionsurelymustbeincluded. Inthenextsection,Iofferageneralinterpretationofdimensionaldevelopmentthataddressestheissueoflifeworldexpansionandcontraction.Then,insection7,thespecificstagesofcosmogonyarelaidoutforthetopodimensionalfamily,andthisisfollowedbyasectionshowingthatthefamilyofTaolendsitselftoessentiallythesameanalysis.WhatcomesthroughinthesesectionsisthedeepaffinityandunderlyingharmonyofWesternandEastern“stringtheories.”6.GENERALSTAGESOFDIMENSIONALDEVELOPMENTPhenomenologicalthinkingprovidesuswithaninsightintotheembodieddialecticalinterplayofobject,space,andsubjectlyingbehindtheCartesianfacadeofachangelessspacewhereinobjectsarecastbeforedetachedsubjects.Phenomenologically,spaceisnotastaticcontextforthemechanistictransformation

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ofexternallyrelatedobjects;rather,thelifeworlddimensionisa“spatio-sub-objective”being(whatMerleau-Pontycalled“an‘element’ofBeing”;1968,139)thattransformsitselforganically.Wecanseefromthisthatthephenomenologicalapproachlendsitselftotheideaofdimensionalgenerationinawaythatclassicalthinkingdoesnot.Thelifeworlddimensiondoesnotjustcontainearthlymatterbutisitselfoftheearth.Likeearthlymatter,thisdimensionisinprocess.Andweareabouttoseethat,likelivingmatter,itpassesthroughstagesofbiogenesis.Thusthephenomenologicaldimensionpossessesthecharacterofalivingorganism—thoughnotafiniteparticularorganism,tobesure.Insteadthedimensioninquestionisagenericorganicity,awholedimensionoflife.Andthatdimensiondevelops. Dimensionaldevelopmentdealsnotwithchangestakingplaceinanindividualorsubjectperse,butwithhowtheveryrelationshipofsubject,object,andspacechanges.IntheTheSelf-EvolvingCosmoswediscoverthattheCartesianframeworkofobject-in-space-before-subjectthatseemstoprecludeanorganismicbasisfordimensionalchangeinfactitselfarisesintheintermediarystageofadevelopmentalprocessthatisindeedorganicallydimensional.Intheopeningstage,object,space,andsubjectarelargelyundifferentiated.Ratherthanconstitutingwell-definedontologicalcategories,theycompriseonlyanincipientfluxofembryonicpossibilities.IntheearliestfragmentofWesternphilosophy,Anaximanderreferredtothisinchoateconditionastheapeiron(seeRosen2004).Literallymeaning“withoutmeasure,”theoldGreekwordwasvariouslyinterpretedas“limitless,”“boundless,”“indeterminate,”or“unintelligible”(Angeles1981,14).Intheproto-scientificdisciplineofalchemy,theinitialstateofaffairswastermedprimematter:“primamateria,whichistheoriginalchaosandthesea”(Jung1970,9).Then,fromtheprimordialflux,thefirstdifferentiationofsubjectandobjectemergeswithinthemediumofanascentlydifferentiatedspace.Thismarksthebeginningofstage2and,asitunfolds,distinctionsamongsubject,object,andthespacethatcontainsthemhardenintocategoricaldivisionsthatarenowassumedtohavebeentherefromthestart.Completelyovershadowedisthedialecticalprocessthatgaverisetothesedivisions.Inthusexpandingfromthe“blackhole”oftheprimevallifeworld,anextensiveuniverseisopenedupandtheoriginalconnectionwiththepsycheisobscured.Whatbeganasadensepsychophysical“soup”isnowprojectedasapurelyphysicalworldstretching“outthere”beforeus,anobjectiverealmfromwhichthelivedsubjecthasbeendropped.(EarlierIgaveanexampleofthisobjectivestanceindiscussingthepostureofconventionalmathematicswhereinattentionisalwaysfocusedonthemathematicalobject,withthesubjectivityofthemathematicianseldombeingacknowledged.) ThefirsttwogeneralstagesofdimensionaltransformationhavetheircounterpartsintheKleinianactionoftheHo-t’udescribedinsection2.Theinitialstagecorrespondstotheoriginal“counterclockwise”actionfromwhichclockwiseactionisgeneratedinHo-t’ucycle1.SincethepapermodeloftheKleinbottlegivingthetopologyoftheHo-t’uisitselfbutanobjectinthree-dimensionalspace,itcannotshowthattheopeningphaseofcycle1actuallyentailsneitherclockwisenorcounterclockwiseaction,forthisundifferentiatedstateofaffairsaffordsnodefiniteorientation.ThethreesubsequentphasesofHo-t’ucycle1parallelthesecondgeneralstageofdimensionaldevelopment:“theusualclockwisemovement,

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cumulativeandexpandingastimegoeson.”Itisherethatanobjectiveworldisprojectedandtimeflowsforwardinitsordinarysequenceaccordingtotime’sarrowofincreasingentropyinanexpandingphysicaluniverse.

Whatofthethirdgeneralstageofdimensionaldevelopment?ThiscoincideswiththesecondcycleoftheKleinianHo-t’u.Wehavealreadyconsideredthepropositionthat,inenteringcycle2,thegearsshiftfromclockwisetocounterclockwise,forwardtobackward,andtheuniversecontracts.Asthisprocessisenacted,theprojectionofapurelyphysicalthree-dimensionaluniverseiswithdrawn.Itisbyovercomingourfixationontheexternalworldofobjectsanddrawingourattentioninwardthatwebecomeawareofthepsychophysicalnatureofthecosmos.Wearegoingtoseethat,inthusmovingbackward,lower-dimensionallifeworldsconcealedinearlierstagesofcosmogonyarebroughttolight.TheIChingimpliesthat,throughthecosmiccontractionandreversaloftimethattakesplaceincycle2,“theseedsofthefuturetakeform.”Inthisregard,WilhelmsaysintheIChingcommentaryIcitedsection1:“Ifweunderstandhowatreeiscontractedintoaseed,weunderstandthefutureunfoldingoftheseedintoatree"(WilhelmquotedinvonFranz1974,236–37).Iwillshedlightonthemeaningofthisenigmaticpassageinthenextsection.7.THECOSMOGONICSPIRALThedescriptionofdimensionaldevelopmentjustgivendoesnottakeintoaccountthefactthateachmemberofourtopodimensionalfamilyhasitsowndistinctivepatternofchange,itsownuniquephasestructure.Wehavelearnedthatdimensionaltransformationgenerallyentailsstagesofexpansionandcontraction,forward-directedstagesinwhichaworldisprojectedandretrogradestagesinwhichtheprojectioniswithdrawn.Butwehavenotyetseenhowthisisspecificallyorchestratedforthefourfundamentallifeworld“tones”thatsoundthemusicofthedimensionalspheres.Sothetimehascometobringourpictureofcosmogonyintosharperfocus. ConsideragainTable1,theinterrelationalmatrixoftopodimensionalspinstructures.Takenbyitself,thistableaffordsbutastaticpictureofdimensionalassociations,onethatis“averagedover,”i.e.abstractedfrom,theactualfactsofdimensionalchange.Therefore,tofillintheconcretedetailsofhowtheseveraldimensionalspinorsevolveinrelationtooneanother,wemustsetthematrixinmotion.ThisisachievedinTable3.

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Table3adisplaysthefullcourseofdevelopmentofallordersof

topodimensionalaction.Ineffect,theTableexpandsTable1diachronically,nowshowingthestagesofcontractionaswellasthoseofexpansion.WhilethehypernumbervaluesgiveninthecellsofthesematricesarethesameasthoseexhibitedinTable1,withthenewtablewecanstudythespecificlinesofevolutionofthespinstructuresthatthehypernumbersrepresent.Employingtheprincipleofsynsymmetrysetforthinsection4,wewilltracktheseinterwovencosmogonicpathways.

Generallyspeaking,eachmatrixofTable3afeaturesonitsmaindiagonal(thatarrayedfromupperlefttolowerright)oneofourfourfundamentalspinors.Theseprincipalterms—εD0,εD1,εD2,andεD3—correspondtothefourreflexiveself-divisions(0/0,1/1,2/2,3/3)constitutingtheseriesoffundamentaltonesofthePythagoreantable(Table2),which,inturn,arealignedwiththetopologicalseriesofself-intersectingforms(Fig.12).Thehypernumberratiosappearingoffthemaindiagonalofeachmatrixaremembersofenantiomorphicpairingsthatpertaintodevelopmentalrelationshipsbetweenthedifferentdimensionalspinors(aswillsoonbediscussed). Table3aisbasicallytobereadinacircularfashion:webeginbyreadingitfrombottomtotop,thenreversecoursetoreaditfromthetopdown.Startingfromthematrixatthebottom,theupwardmovementthroughthefourmatricesgivestheclockwise,forward,projectivestagesfoundincycle1ofdimensionalgeneration,indexedbythestagevaluesappearingtotheleftofthematrices.Wethenswitchtothecounterclockwiseorretrogradestagesofcycle2whereinprojectionsarewithdrawn,nowreadingbackdownthroughthosesamematrices,withstagenumberspresentlydisplayedtotheright.Theparenthetichypernumbertermsaccompanyingeachstagenumberindicatethetopodimensionalspinororspinorstowhichthatstagenumberapplies;sincethezero-dimensionalspinordoesnotundergodevelopment,itdoesnotappearhere.Ifthesequenceofstagesforeachtopodimensionalspinorisconsideredseparatelyfromthatoftheotherspinors,weseethatTable3ainfactdoesnotdescribetheactionofasinglecircle(upwardthroughtheclockwisestages,downwardthroughthecounterclockwiseones)butofcirclesnestedwithincircles,sothattheoverallpatternisactuallythatofaspiral. Topreservethethoroughlyinterwoven,nonlinearcharacterofdimensionalinterrelatedness,Table3adisplaystheseveralwindingsofthedimensionalspiralasoverlappingoneanother.However,thismakestheTablesomewhatdifficulttoread.Tofacilitateunderstanding,IofferTable3basavisualaid.Herethecirculationsofthedimensionalspiralhavebeenparsed,teasedapartforeasieridentification.

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Thedrivingforcebehindthegenerationoflifeworlddimensionsdepictedin

Table3wasalreadyidentifiedinsection4:ann-dimensionallifeworldiscrystallizedthroughthefusionofn−1-dimensionalenantiomorphs.InvolvedhereisthedialecticalinterplayofsymmetryandasymmetrythatIhavetermedsynsymmetry.Priortotheirfusion,enantiomorphsarenon-identicalmirroropposites,thusasymmetricwithrespecttoeachother.Thisconditionofenantiomorphicasymmetryprevailsatthebeginningofawinding.Then,ascycle1ofthewindingprogresses,themergerofn−1-dimensionalenantiomorphsgivesriseton-dimensionalsymmetry(anexamplefromsection4isthefusionofopposingsidesofthelemniscatetoyieldtheMoebiusstrip).Thesynsymmetryconceptfurtherimpliesthattheestablishmentofsymmetrymustbefollowedbyaneworderofasymmetry.Thisisinitiatedincycle2ofthewinding.Hereenantiomorphsareincubatedforthen-dimensionalstructure(theMoebiusstrip,forexample)thatmeetthenecessaryconditionsforopeningupthenextwinding,thatwhereinn-dimensionalenantiomorphswillfusetogeneraten+1-dimensionalsymmetry(e.g.,thesymmetryoftheKleinbottle).Itisthissynsymmetricprocessofdimensionalfusionand“diffusion”(i.e.,thecreationofasymmetricenantiomorphs)thatfuelsthemovementfromonewindingofthecosmogonicspiraltoanother.Isuggest,moreover,thatthesynsymmetryideahelpsuscomprehendthemeaningoftheIChingpassagecitedearlier:“Ifweunderstandhowatreeiscontractedintoaseed,weunderstandthefutureunfoldingoftheseedintoatree."Inthecosmiccontractionandtimereversaloccurringincycle2ofagivenwinding,“theseedsofthefuturetakeform”bypropagatingtheenantiomorphsrequiredforgenerationofnewsymmetry(anew“tree”)inthenextwinding.

InTable3b,weseetheprogressiveincreaseinthenumberofstagesthroughwhichdimensionaldevelopmentoccursaswegofromlower-tohigher-dimensionalwindingsofthedimensionalspiralviatheprocessofsynsymmetry.Thefirstwindingofthespiralisthatofthesub-lemniscatorymatrix,whichconsistsofbutasinglecell.Ineffect,the“radius”ofthiscirculationis“zero,”foritentailsnotransformationwhatsoever.8NostagesofdevelopmentcanbefoundforεD0norareanynecessary,sincethiszero-dimensionalsphereiscomprisedonlyoftheseedsofhigher-dimensionalstructureswhosegenerationwillbefacilitatedinsubsequentwindingsbythesupporttheyreceivethroughεD0’senantiomorphicovertones(εD1/εD0,εD2/εD0,εD3/εD0)andundertones(εD0/εD1,εD0/εD2,εD0/εD3).

Advancingtothesecondwindingofthespiral,thematrixexpandstothe2×2structureassociatedwiththegenerationoftheone-dimensionallemniscatorylifeworld.Theopeningstagefindsaprimordialmatrixinwhichtheone-dimensionalspinorresidesinbutanembryonicform.Inourpsychophysicalstringtheory,wherespinningparticlescanbeseenasvibratingstringscreatingwavesthatpossess

8Ofcourse,the“radii”ofalldimensionalcirculationsare“zero”inthesensethatstagetransitionsarenotactuallydisplacementsinanextensivespatialcontinuum,thoughtheyappearassuchinTable3.TheseveraldimensionalcirculationsportrayedintheTablecannotreallyhavefiniteradiibecause—insteadoftakingplacewithinspace,theyconstitutethequantizedpre-spatialactionsfromwhichspacefirstarises.

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overtonesandundertones,wemaycharacterizethezero-dimensionalsub-lemniscatoryspinorasa“carrierwave”whoseεD1/εD0overtoneandεD0/εD1undertonecarrythefledglingone-dimensionalspinor(see“WavesCarryingWaves,”chapter7ofCosmos).One-dimensionalactionisbutnascentlyorientedhere;thedifferentiationofthelemniscatorylifeworldhasnotyettakenplace.

Movingupwardnowtothestage-2matrixofthelemniscatorywinding,weseethatεD1hasgainedmaturity.Thisentailsthefusionoftheasymmetricsub-lemniscatoryenantiomorphs,εD1/εD0andεD0/εD1,whereintheyare“annihilated,”beingabsorbedintotheemergentsymmetricstructureoftheexpandinglemniscatoryuniverse(tobemoreexact,εD1isthematurequantizedspinorthatprojectsanexpandeduniverse).However,withthetransitiontostage2,theprimordialstateofaffairsisnotsimplyleftbehind;rather,itisrelegatedtothebackground.AndwhiletheprimalpotencyofεD0isineclipse,adepotentiated,objectifiedversionofitisprojectedasapoint-likelowerdimensionintheone-dimensionalworld(concreteexamplesofthiswillbegiveninsection9whenweareworkingwithparticleevolution).TheattenuationofεD0isrepresentedinTable3bbyplacingthisvalueinparentheses.Atthesametimethatthesub-lemniscateisprojectedasazero-dimensionalobjectintheone-dimensionalworld,thelemniscatoryworldisitselfprojectedassomethingobjectively“outthere.”Soforward-orientedprojectiveactivityisoccurringinearnestinthissecondstageofcycle1.Whereasthelemniscate’sactionisbutincipientlyorientedinstage1,itpresentlyassumesadefiniteclockwisedirection. Thecompletionofstage2bringsthefirstcycleoflemniscatoryarchetypalactiontoaclose.Acriticalturningpointisnowreachedwherethegearsshiftfromforwardtobackward,clockwisetocounterclockwise.Enteringthesecondcycle,theexpansionoftheone-dimensionalworldisreversedanditsprojectioniswithdrawn.9

Letuslookmorecloselyatthenatureofthisactofwithdrawal,orwhatwemaycallretrojection.Itdoesnotimplythattheforwardmovementofcycle1simplyceasesincycle2.Projectiveactioncontinues,andyetitistransformedbybeingcounter-acted.Whatexactlydoesthismean?Supposeyouwerehandlingatexturedpieceoffabric,onewhosefiberswerearrangedinacertaindirection.Isitnotbyrunningyourfingersagainstthegrainofthematerialthatitsdirectionbecomesmoreclearlydiscernable?Similarly,inthecycle-twobackwardmovementagainstthegrainofcycleone,awarenessisgainedoftheveryprocessofforward-directedprojectionthattranspiresinthatfirstcycle.Itisinthiswaythattheprojectionisretracted,consciouslytakenback,inthemidstofitsongoingoccurrence.

Theretrogradeorientationisestablishedinstage3ofthelemniscatorywinding.Whatwasprojectedasanobjective,“purelyphysical”one-dimensional

9Intheinterestofmanageableexposition,actioncyclesarenotshowntopologicallybutappearaslinearsequencesofmatrixcellsinTable3.Thus,inthelemniscatorywindingofTable3b,cycle1iscomprisedofstages1and2,andcycle2consistsofstages3and4.Butthesephasesofactionareindeedtopologicalandamoreaccuratewayofindicatingthedoublecyclicityofthelemniscatewouldbeviatheemblemofinfinitythattangiblydisplaysthelemniscate’stwocyclesofopposingclockorientations:∞.

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worldinstage2ispresentlyrecognizedasfirstarisingfromthesymmetry-creatingpsychophysicalactoffusingthesub-lemniscatoryenantiomorphsofstage1.ThisbasicallyaccordswithvonFranz’sassessmentofthecyclesoftheHo-t’u,wheresheidentifiesHo-t’ucycle1as“physical”andcycle2as“psychic”(seesection1).Itshouldbeclearhowever,thattheone-dimensionalrealmofthelemniscateisfarsimplerthanthethree-dimensionalworldoftheKleinianHo-t’uthatvonFranzwasdealingwith.ThegreaternumberofstagesintheKleinianwindingshowninTable3breflectsthefactthatdevelopmentgoesfurtherhere,withasharperdivisionemergingbetweensubjectandobject,psycheandmatter.Wecansayingeneralthattheprogressivelyincreasingnumberofstagesfoundinpassingfromonewindingofthecosmogonicspiraltoanotherindicatesthegrowingcomplexityofthelifeworldsgenerated.

Notethatinstage3ofthelemniscatorywinding,theawarenessgainedofthestage-1sourceofprojectionissomewhatlimitedinasmuchasthiscognizanceofthezero-dimensionalrealmitselfremainsstrictlyone-dimensional.Itislemniscatoryconsciousnessthatprevailsinstage3asitdidinstage2andthelemniscate’sabstractperceptionofthesub-lemniscatoryspheredoesnotreturnittothatprimevalsettinginaconcreteway.Althoughtheprojectionoftheone-dimensionalworldhasbeeneffectivelywithdrawninstage3,thatofthezero-dimensionalworldhasnot(intheTable,εD0remainsenclosedinparentheses,indicatingitscontinuingpresenceasadepotentiatedprojection).

Itisthroughtheretrojectionoccurringinstage4thattherepressionofthesub-lemniscateisliftedanditspotencyrestored.Here,ratherthanmerelyapprehendingthesub-lemniscatefromalemniscatoryperspective,thelemniscateiscarriedbacktoitsembryonicoriginandthezero-dimensionalstructureitselfgainsfullpresence.Thisreturntothebeginningisnomereregressioninwhichthematurelemniscateutterlyrevertstoitsincipientform.DespiteTable3b’slinearsequencingofstages,theyareinfactnotsimplyseparatedfromeachotherbutoverlap.Therefore,justasthezero-dimensionalstructureofstage1isnotleftbehindinpassingtostage2butcontinuesinthebackground,thematureone-dimensionalstructureofstage3maintainsitspresenceinstage4.10

Withtheretrojectionenactedinstage4,closureisbroughttothelemniscatorywindingofthecosmogonicspiral.Thefullydevelopedlemniscateandrevitalizedsub-lemniscatearenowbothonthescene.Theyarewhollypresenttoeachotherandcanenterintoresonancetosoundthe“musicofthedimensionalspheres.”Becausetherelationshipisnolongerthatofamaturespinorcarryingaspinorthatisstillundeveloped,becausebothdimensionalspinorsarenowmature,wecanreinterprettheenantiomorphicratiosasexpressingtheintimatereciprocityofarchetypalplayers.

ThesecondcycleofthelemniscatorywindingnotonlycompletesthedevelopmentofthelemniscatebutalsopavesthewayforentryintotheMoebiuswinding.Foritishere,bythesynsymmetricprocessofdimensionaldiffusion,that

10Itistrue,however,thatwhilestagesoverlaponeanotherinbothcyclesofdimensionaldevelopment,thequalityofoverlapincycle2differsfromthatincycle1.SeeCosmos,Chapter10,foradiscussionof“transparent”and“translucent”formsofstageoverlap.

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thelemniscatory“tree”contractsintoa“seed”requiredforthegrowthoftheMoebius“tree.”Theother“seed”necessaryforsubsequentMoebiusdevelopmentisofcoursethede-repressedsub-lemniscate.Theseedssynchronouslysownincycle2ofthelemniscatorywindingbecometheasymmetriccarrierwaveswhoseenantiomorphicfusionsgenerateMoebiussymmetryinthenexttopodimensionalepoch.

Whenthelemniscatorywindingcloses,thecosmogonicspiralopensoutintotheMoebius,consistingof3×3matricestransformedoversixstages.Thecompasshasshiftedsothatinitialactionisonceagainunoriented.Instage1ofthistwo-dimensionalcirculation,wefindthat—asaresultofthegerminalactivityinthesecondcycleofthelastepoch—thecarryingcapacityofthezero-dimensionalsub-lemniscatehasbeenenlarged.Throughitsnewpairofenantiomorphs,εD2/εD0andεD0/εD2,thesub-lemniscateservesascarrierwavefortheembryonicMoebiusspinor.ThefusionoftheseasymmetricenantiomorphsonbehalfofthedevelopingMoebiusthenbringsusintostage2.Withthesub-lemniscatorycarrierhavingbeenabsorbedintotheMoebiuswave,theoriginalzero-dimensionalspinoriseclipsedandanattenuatedversionofitisprojected(asindicatedbytheappearanceofεD0inparentheses).Foritspart,theMoebiusgainsitsmeasureofsymmetry,becomingmoremature.Aclockwiseorientationisassumedinthisstage,andwehavethefirstprojectionofanobjectivetwo-dimensionalworldstretchingbeforeanemergenttwo-dimensionalsubjectivity.However,whereastheone-dimensionallemniscaterequiredthefusionofbutasinglepairofenantiomorphstocompleteitscycle-1development,themorecomplexMoebiuspatterncallsfortwosuchfusions.

Thisiswherethelemniscatorycontractionofthepreviouswindingbearsfruit.Itseffectisthatthelemniscatecancurrentlyplaytheroleofacarrierwave,expressedastheenantiomorphicovertone-undertonecoupling,εD2/εD1andεD1/εD2.ItisthefusionoftheseenantiomorphsthatbringsusintothethirdstageofMoebiuscosmogony,theconclusivestageinitsforward-directedunfoldment.Withthemergeroflemniscatoryenantiomorphs,thelemniscatorycarrierwavedissolvesintoεD2andadepotentiatedformofitisprojected(denotedinstage3byitsenclosureinbrackets).Thisattenuatingprojectionisaccompaniedbyasecondandfinalprojectiveexpansionofthetwo-dimensionalworld.Thenow-matureMoebiusspinorthusreachesfullsymmetry.

Nextcomesthetransitionfromcycle1tocycle2oftheMoebiuswinding.Enteringstage4,thegearsareputintoreverse,expansionswitchestocontraction,andtwo-dimensionalspinisreoriented,becomingcounterclockwise.Inthisfirstretrogrademovement,thereisaninitialrecognitionthatthecycle-1preoccupationwithanobjectivephysicalrealityexternaltotheperceivingsubjecthasactuallyarisenfromsub-objective,psychophysicalactsofprojection.(Donotethatsubject-objectrelationsintheMoebiuswindingaresimpler,lessdifferentiated,thanthoseintheKleinianwinding;seeRosen2006and2008aforanintensiveexaminationofthisdistinction.)Butthestage-4withdrawalofthetwo-dimensionalprojectionleavesuntouchedthedepotentiatinglower-dimensionalprojections.Then,movingconcretelybackwardinstage5,two-dimensionalretrojectionistakenfurther.Theearlier(stage-2)projectionofεD2iscounteractedandtherepressionoftheone-dimensionallemniscatorycarrierwaveislifted.TheMoebiusspinornowentersinto

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harmonywithitslemniscatorycounterpart,andweinterprettheenantiomorphicratiosofstage5asexpressingthatharmony.

Goingstillfurtherbackinstage6,thezero-dimensionalsub-lemniscatorycarrieremergesfromobscuritytobecomeharmonicallyattunedtotheMoebius.Theprojectiveactionsofcycle1havenowbeenwhollycounteractedandtheMoebiusspinorhasachievedsynchronywithitslower-dimensionalrelativesinthetopodimensionalfamily.Andsincethesynchronyisrealizedamidsttheprocessofcontraction,thesethreespinorscometoconstitutetheseed-structuresmanifestedascarrierwavesinthenextepoch.

WiththecompletionoftheMoebiuswinding,thedimensionalspiraldilatesonceagainandwefindourselvesintheopeningstageoftheKleinianwindingof4×4matricesevolvingovereightstages.Beyonditsincipientlyorientedstage,thethree-dimensionalKleinianorganismgainssymmetryandmaturesthroughthreeclockwisephases.Thisfollowsthepatternoftheearlierwindings.Ineachstageadifferentfusionofenantiomorphsoccursentailingtheabsorptionofalower-dimensionalcarrierwave(sub-lemniscatory,lemniscatory,andMoebial)thatfacilitatesthegrowthoftheKleinianwave.Andwitheachnewfusioninthecycle-1expansionofKleiniandimensionality,thereisanewandmoredifferentiatedprojectionofanobjectivethree-dimensionalworld.

Subsequently,thegearsarereversedoncemoreandwepassintothesecondcycleofKleinianevolution,whereexpansionbecomescontractionandclockwiseactionbecomescounterclockwise.Stage5bringstheKleinianorganism’sfirstrealizationthatthe“objectivephysicaluniverse”isinfactapsychophysicalprojection.Inthethreeensuingstagesofcycle2,thecycle-1repressionsimposedonthelower-dimensionalcarrierwavesaresuccessivelyliftedandeachentersintoresonancewiththeKleiniandimension.Withallthemembersofthetopodimensionalfamilyresoundinginharmony,the“symphonyofdimensionalspheres”isheardinitsfullness. Butwhathappensnext?Doesthecosmogonicspiralnowcloseintoacirclethatbringsustotheendofcosmogony?Whatofthelogicofsynsymmetry,ofdimensionalfusionanddiffusion?Incycle2oftheKleinianwinding,musttheKleinian“tree”notcontractintoa“seed”foranew,meta-Kleinian“tree,”anewroundoftopodimensionaltransformationsinanovelwindingofthespiral?Thisimplicationisinescapableifwearetoavoidabandoningarbitrarilythesynsymmetryprinciplesocriticaltoouranalysis.Indeed,whentheanalysiswasrelatedtothePythagoreantableinsection5,Inotedthatthistableisusuallyportrayedasanindefinitelyexpandingseriesofmusicalintervalsthatgoesbeyondthe3/3intervalcorrelatedwithKleinianthree-dimensionality.Nevertheless,withtheprospectofsurpassingtheKleiniandimension,itappearswehavereachedawatershed,forouranalysisofcosmicevolutionhascomeupagainstthelimitsofourownanalyticalcapability.Whilewecanabstractlyanticipatehigher-dimensionalwindingsofthecosmogonicspiral,wecannotgainconcreteknowledgeofthembecausetheyliebeyondtheconceptualframeworkweemployasthree-dimensionalanalysts.

IdonotholdwiththeKantianviewthatthetangibleexperienceofadimensiontranscendingourcognitiveframeisforeverinaccessibletous.InsteadI

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suggestthatwecanindeedcometopalpablyrealizethehigherdimensionbutthatdoingsorequiresmorethanjustchangingthecontentofouranalysis.Ouranalyticframeworkitselfmustevolve,stretchingtoaccommodatethenewdimension.BeforeIamfinished,Iwillattempttoclarifywhatthismeansandwhatitmayrequire.Tosumupthissection,theforegoinganalysisbringstolightthestagesofdimensionaldevelopmentthatservetodistinguishonetopodimensionalspinstructurefromanother.Whiletheseveralwindingsofthedimensionalspiraloverlap,eachlifeworldcirculationrunsitsowncourse,withdistinctcirculationsbeingmarkedbydifferencesinthenumberofstagesthateachrequirestocarryoutitsclockwiseprojectionsandcounterclockwiseretrojections.Thesedifferencesreflect,inturn,differingdegreesoflifeworldcomplexity,whichshowupinthecapacityfordiscriminatingsubjectandobject,psycheandmatter.8.THECOSMOGONICSPIRALINLIGHTOFTHETAOIncompletingourstudyofhowdimensionsevolve,wehavereachedanothercrossroadbetweenWestandEast.Forwearenowpreparedtograsptherelationshipbetweenoursection-7treatmentoftheKleinbottleinthecontextofdimensionalgenerationandourapproachtoitinsection2,whereitistakeninconjunctionwithTaoism’sHo-t’uandassociatedtrigrams. Intheearliersection,thequantizedphasestructureoftheKleinbottlewasseentocorrespondtothatoftheHo-t’u:bothconsistoftwooppositelydirectedcyclesoffourphaseseach.ThisisofcoursethesamestructuremanifestedintheKleinianwindingofthecosmogonicspiral.Whatwecouldnotseeinsection2isthethoroughlyinterrelationalnatureofKleiniandevelopment.ThisonlycomestolightwhenwetakeintoaccounttheKleinbottle’smembershipinitstopologicalfamily.Table3showshoweachstageintheunfoldmentoftheKleinbottleisdefinedbyitsrelationshiptoanotherfamilymember.AndwhatweseeinTable4isnotonlythecorrespondenceoftheKleinianwindingofthecosmogonicspiraltothetrigramsoftheHo-t’u,butthecorrelationofallmembersofthetopodimensionalfamilywiththefamilyofTao.

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ForTable4,IhavereplacedthetopodimensionalvaluesofTable3bwiththeirintimatelyrelatedTaoistcounterparts.ThefourordersoftheTaoaregiveninthefourwindingsofthecosmogonicspiral.ThefirstwindingconsistsoftheTaoitself,shownintheupperleft-handquadrantoftheTableasthesinglevalue,0.Whereasallotherwindingscompriseextendeddevelopmentalmatrices,theTaostandsalone.Aswassaidofthezero-dimensionalsub-lemniscate,theTaopersedoesnotevolvethroughstages.Actingviaitsseedstructures,itfunctionssolelyasacatalystforthematricalgenerationofhigherordersofitself:theunigrams,consistingofthebasicmodesofyin(--)andyang(—);thebigrams,entailingthedoublingoftheyin-yanglines;andthetrigrams.Parallelingthegenerationofthetopodimensionalspinors,Taoistcosmogonyisseenasdrivenbythesynsymmetricprocessofenantiomorphicfusionanddiffusion.WithineachwindingofTable4,theunbracketedtermsonthemaindiagonalsofthematricesshowthestagesofdevelopmentofthefundamentalyin-yangspinors.Thecoupledenantiomorphslocatedoffthemaindiagonalsaretheyin-yangratiopairs,eachofwhichexpressesarelationshipbetweenmembersoftwodifferentordersoftheyin-yangfamily.Cosmicevolutionisfacilitatedbythesymmetry-generatingenantiomorphicfusionoccurringinthefirstcycleofagivenwinding,followedbythecycle-2diffusionthatsowstheseedsfortheasymmetricenantiomorphsofthenextwinding. EverywindingoftheTaoistcosmogonicspiralistobereadinthesamecircularmannerasitstopodimensionalcounterpartinTable3.Readingthroughthestagesofagivenwindingfromthebottommatrixtothetop,wehavetheclockwise,projectivestagesofcycle1.Thenswitchinggears,wereaddownwardfromthetopmatrix,andthisgivesthecounterclockwisestagesofretrojectionthatconstitutecycle2.

LetusbeginourreadingofTable4byreversingthesequenceinwhichthewindingswerestudiedinTable3b.WewillnotproceedfromtheTaoandtrackthecosmogonicspiral’sexpansionthroughtheunigrams,tothebigramsandthetrigrams.Insteadwewillstartwiththetrigramwindingsincethisiswherewecanbestarticulateaprinciplethatwillinvolveasignificantdeparturefromtheclassicaltreatmentoftheyin-yangfamily.

ThetrigramwindingofTable4displaysfour4×4matricesevolvingovertwocyclesoffourstageseach(àlatheKleinianwindingofTable3b).Inthefirstcycle,weseethecosmogonicprocessbywhichthepsychophysicaltrigramspinorprogressivelydevelopsfromaninitiallyembryonicorigin;asitmaturesandgainssymmetry,itspinsoutanobjectivephysicalworld.ThetrigramsequencefollowsthatoftheHo-t’ucross.Inthefirststage,theembryonicspinorisrepresentedbythetrigramK’un,consistingofthreebrokenlines;itscompassdirectionisNorth.WeobserveinthebottommatrixofthewindingtheenantiomorphicsupportK’unreceivesfromtheTaocarrierwave(denotedby0).Then,viathefusionoftheseenantiomorphs,wemoveupwardtothenextstage,wherethespinorisnowdirectedSouthandismanifestedasCh’ien,expressedbythreesolidlines.Notyetmature,Ch’ieniscarriedbytheenantiomorphsofthematureyin-yangspinororiginatingintheunigramwinding.Repeatingtheprocessoffusion,thetrigramspinoradvancestostage3,assumingthestructureoftheEasterly-directedLi:a

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brokenlinebetweentwosolidlines.Liiscarriedbytheenantiomorphsof“YoungYin,”thematurespinorofthebigramwinding.(Bearinmindthatacarrierwavealwaysinvolvesthematureformofalower-orderspinor,sinceaspinorcannotplaythesupportiveroleofcarrieruntilitsdevelopmentiscompletedinitsownwinding.)Thelastprojectivestageofthetrigramwindingarisesfromthefusionofbigramenantiomorphs.Inthisphase,thetrigramspinorreachesfullmaturityasK’an,asolidlinebetweentwobrokenlines.NotethatthedepotentiatingprojectionspreviouslydiscussedforKleinianevolutionaremirroredintrigramdevelopment:witheachenantiomorphicfusionofcycle1,thelower-orderspinorthathadsupportedthetrigram’sprojectivepotentiationisitselfsubjectedtoadepotentiatingprojectionthatnarrowsitdownandobjectifiesit,occludingitstruenature.InTable4,spinorsthusattenuatedareenclosedinbrackets.

Weknowwhathappensincycle2.Thegearsshiftfromforwardtobackward,projectiontoretrojection,andtheretrogradestagesofthecycleunfold.Theprojectionsofcycle1arepresentlywithdrawn,withlower-orderspinorsemergingfromeclipsetoenterintosynchronywiththetrigramspinor.Andinthecontractionofthetrigramcosmos,thesespinorsbecomeseedsforthenextwinding.Noticethat,inTable4,thestagesofcycle2areassociatedwithnonewtrigramsandaccompanyingcompassdirections,astheyareintheclassicalrenditionofthetrigrams(seeFigure11).WhyhaveIdroppedfourtrigrams?

InthetraditionalapproachtotheIChing,alleightpermutationsofthetrigramsareemployedandfromthis,64hexagramsarebuiltforthepurposesofpersonalguidanceanddivination,withphilosophicalandcosmologicalcommentaryadded.WhatIamproposingisanalternativewayofworkingwiththetrigramsthatusesonlytheonesspecifiedincycle1.Insteadofadheringtothecustomofintroducingadditionaltrigramsandcompassdirectionsforcycle2,Isuggestthattheverysametrigramsbeemployed,nowintheretrogradeorientation.

Thereisasenseinwhichthetrigramsofcycle1andtheiraffiliatedcompasspointscanbeconsideredmorefundamentalthantheircycle-2counterparts.Firstofall,weobservethatthecompasspointsconventionallyfeaturedincycle2donotconstitutedirectionsthatareuniquelydifferentfromthefourcardinaldirectionsarisingincycle1.Thecycle-2directionsaredeemed“intercardinal”inthattheycombinethealreadyestablishedprimarycompassdirectionswithoutintroducinganythingqualitativelynew.WeseefromFigure2thattheintercardinaldirections—Northwest,Southeast,Northeast,Southwest—areobtainedsimplybyrotatingtheHo-t’ucrossbyanangleof45°togivethecompassmarkingsthatliehalfwaybetweenthecardinals.

Perhapsmoreimportantly,whiletheclassicaltrigramsofcycle2—Ken,Tui,Chen,andSun—providenovelpermutationsofthetripleyin-yanglinesandthesearesubjecttonewinterpretations,thereisnochangeintheunderlyingcombinatorialstructureofthetrigrams.Thefourtrigramsofcycle1exhaustthepossiblecombinationsofbrokenandsolidlinestakenthreeatatime:allsolid;allbroken;twosolid,onebroken;twobroken,onesolid.Whatwehaveinthesecondcyclearebutrepetitionsofthoseprimarystructures.Thepermutationsaredifferentbutthemorebasiccombinationsstaythesame.So,justastheclassicalcompasspointsofcycle2providenofundamentallynewcompassdirections,thetrigramsof

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thiscycleoffernoinnovationsinbasiccombinatorialstructure.Whathappenswhenwelimitthetrigramstotheprimaryonesgivenincycle1andtakecycle2asabackwardmovementthroughthoseverysametrigrams?Wecanthenseeclearlytheisomorphicalignmentofthetrigramorderoftheyin-yangfamilywithitsKleinianequivalentinthetopodimensionalfamily.

MovingbackwardnowinthecosmogonicspiralofTable4,letusconsidertheyin-yangwindingthatcorrespondstotheMoebiuswindingofTable3b.ThecounterpartoftheMoebiusisthearchetypalfieldthatinvolvesthegenerationofthebigramspinstructureovertwocyclesofthreestageseach.Intheclassicalliterature,thesetofbigramsisknownastheFourEmblemsorFourSymbols(Hulse2002).ThetraditionalbigramsareexhibitedinFigure13,alongwiththeireightassociatedtrigrams,onlyfourofwhichweareemployinginourcombinatorialapproach.ComparingFigure13withTable4,weseetheconnectiongivenintheformerbetweenthestage-1trigramK’unwithitsthreebrokenlinesandthebigramcomposedoftwobrokenlines,called“OldYin”or“GreaterYin”(Hulse2002,379).Aligningbigramwithtrigramdevelopment,OldYinistakentosignifythefirststageofthebigrams.Bythesametoken,thelinkinginFigure13ofthestage-2trigramCh’ien(threesolidlines)withthe“OldYang”bigramcomposedoftwosolidlinessuggeststhatOldYangbeassociatedwiththesecondstageofthebigramwinding.InTable4,thethirdandfinalstageincycle1ofthiswindingcorrespondstothebigramdesignated“YoungYin,”consistingofabrokenlineatopasolidline.ButwhatofthefourthbigramshowninFigure13,madeupofasolidlineaboveabrokenline?Thisbigram,named“YoungYang,”isapermutationofthesamecombinationoflinesthatconstitutesYoungYin:asolidlineandabrokenline.Inkeepingwiththeprincipleofworkingonlywithbasiccombinations,oneofthesepermutationsmustbeexcluded.WhyhasYoungYangbeenchosenforeliminationandnotYoungYin?Figure13showsthattheformerisconnectedwiththestage-4trigramK’anwhereasthelatterislinkedtothestage-3trigramLi.SinceTable4maintainsconsistencyindevelopmentalsequencingbypresentingthebigramsinthesamestageorderasthetrigramswithwhichtheyarecoupled,YoungYinmustcorrespondtostage3ofthebigramwinding,leavingYoungYangastheextraneouspermutationtobedropped.Thecompassdirectionsforthethreestagesofbigramcycle1followthesequenceofthetrigrams—NorthàSouthàEast,withWestexcluded.

Aswiththetrigramwinding,thatofthebigramsdisplayedinTable4showsthegenerationofthespinorthroughtwocycles,thefirstexpandingprojectivelytospinoutanobjectiveworld(thoughonelessdifferentiatedandcomplexthanthetrigramworld),thesecondcontracting,movingbackward,andwithdrawingtheprojectionsofthefirst.Againweseetheprocessofdevelopmentfacilitatedincycle1bymaturelower-orderspinorsactingascarrierwavesthatgiveenantiomorphicsupporttotheevolvingspinor.Instage1,OldYin,theembryonicbigram,issupportedbytheenantiomorphsoftheTao.EnantiomorphicfusionthenbringsustoOldYanginstage2.Notyetfullydeveloped,thisbigramiscarriedbytheenantiomorphsoftheyin-yangunigram.Next,whenunigramenantiomorphsfuse,weencounterYoungYin,thematurebigramspinstructureofthethirdandfinalstageofprojection.Thisisfollowedofcoursebytheswitchingofgears.Passing

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backwardthroughthebigramstagesofcycle1,thecycle-2spinorwithdrawsitsprojectionsandentersintoharmonywiththelower-orderspinorsthathadbeenrepressedonitsbehalf.Inthiscontractionofthebigramuniverse,thespinorscondenseintogerminatingseeds,tobearfruitastheenantiomorphsthatfacilitatetrigramgeneration.

Wenowgostillfurtherbackonthecosmogonicspiraltotheyin-yangequivalentofthelemniscatorywinding.Thislower-orderarchetypalfieldcomprisesadoublecycleofspintransformationslinkedtotheunigrams.Cycle1involvesbuttwostages.Inthefirstofthese,wehavetheembryonicspinstructurerepresentedbyasinglebrokenline,designated“PrimalYin”(associatedwiththecompasspoint,North).ThisspinoriscarriedbytheenantiomorphsoftheTao.Justonefusionisneededtobringforththematurestructureofstage2,denotedbysingleyinandyanglines(andorientedtotheSouth).Withthemovementbackwardthatensuesincycle2,thereisinitialrecognitioninstage3thataworldhasbeenprojected,followedbythemoreconcreteretrojectionenactedinstage4throughwhichtheTaoemergesfromitsstage-2eclipsetoconvergeharmonicallywithyinandyang.Theconcomitantcosmiccontractionformstheseedsforthebigramwinding.

Letmeemphasizetherelativesimplicityofthecosmogonicprocessintheunigramwinding.Asalreadynoted,lower-orderspinstructuresprojectlesscomplexlifeworlds,withalessercapacityfordifferentiatingsubjectandobject.Wecansayaccordinglythatthe“objectivephysicalworld”projectedintheunigramwindingisbutaweaklydifferentiatedoneinvolvingaformofconsciousnessthatisnotassharplyfocusedanddiscriminatingastheawarenessachievedinhigherdimensionalwindings.

Continuingbackthroughthespiraltoitssource,wereach“thatwhichinform[s]allthetransformationsinthecosmos”(Fowler2005,46):theTaoitself.TheTao,saysFowler,iswhat“generatesthetensionbetweenopposites[i.e.yinandyang]….Itisthequiet,spontaneouspowerthateternallygivesenergytothecosmos,totherhythmiccompositionofthestarsandplanetsasmuchastotheenergythatatinyseedneedsforgermination”(46).TheTaoisaprimordialmatrix.Whilethistermisbroadlydefinedinmathematicsasanarrayofnumbers,itsgeneraldictionarydefinitionindicatesanenvironmentormaterialcontextinwhichsomethingdevelops.TaoappearsonitsowninTable4asthesingle-celledmatrixfromwhichalldevelops.ItisbecausetheTaoisassociatedwithsilence,nothingness,andzero-dimensionality(seesection5)thatwehaveassignedtoitthevalueof0.RecallingGodwin’scouplingofthePythagorean0/0withthatwhich“soundsnotonebutisthesilencetowardwhichalltonestend"(1992,192),wenowturnourattentionbacktothepsychophysical“tones”ofquantumgravityforastringtheoreticspecificationofthe“musicofthedimensionalspheres.”9.THESPIRALOFQUANTUMGRAVITYInthepasttwosections,theevolutionoftheprimaryspinorswasexplainedinhighlytheoreticalterms.Iwouldnowliketofleshthisoutbyshowingitscorrespondencewiththeelementaryforcesofstringtheory.TheSelf-EvolvingCosmosdescribesinconsiderabledetailthephenomenologicalapproachtothe

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standardmodelofparticlephysics.Iofferedasynopticsketchofthisinmy2015articleandmustlimitmyselftoanabbreviatedrenditioninthecurrentpaperaswell.Butthelatterdoesgofurtherthantheformer,and,insomeimportantrespects,thepresentanalysisadvancestheworkdoneinCosmositself.

WhatCosmossuggestsisthatafullaccountofthefundamentalparticlesofstringtheorymaybeprovidedbyembeddingthetheoryinthematrixofprimordialspinstructuresgiveninTable1.Thismatrixconstitutesaspecialapplicationofthehypernumberidea,onethatdeliversahighlyspecificreckoningofprimordialspinaction.Thetopodimensionalarrayoffourfundamentalspinors(shownontheprincipaldiagonalofthematrix)canbedirectlyassociatedwiththefourtypesofgaugebosonsfoundinnature.Thegauge-bosoncorrelatesofTable1aredisplayedinTable5.Whatisthebasisofthesecorrelations?

G G/g G/(W,Z) G/γg/G g g/(W,Z) g/γ

(W,Z)/G (W,Z)/g W,Z (W,Z)/γγ/G γ/g γ/(W,Z) γ

Table5.Spinmatrixofgaugebosons.Gisthegraviton;gisthestronggaugeboson;W,Zistheweak

gaugebosonparticlepair;andγisthephoton

WeknowthatTable1signifiesaprocessofgenerationinwhichhighertopologicaldimensionsevolvefromlowerones.Thefactsofparticleevolutionlendthemselvestostraightforward,one-to-onecorrelationwiththisprocess.Thefirstforceparticleto“freezeout”oftheBigBang’shotprimordialsoupisthehypothesizedgraviton,G.ThegravitonofTable5isassociatedwithεD0,thezero-dimensionalsub-lemniscatoryactionofTable1,whichcanbewrittenεD0(!/2)togiveexpressiontosubatomicparticlespin;thus,G≡εD0(!/2).Nexttoseparateitselffromtheprimordialchaosisthestronggaugeboson,g,andwerelateittoεD1lemniscatoryaction,writingg≡εD1(!/2).Thentheweakforceemerges,givenbythebosonpairWandZ,whichweidentifywithεD2(!/2).Whenthethreeordersoflower-dimensionalgaugebosonshave“frozenout,”whatremainsisγ,thephoton,topodimensionallyexpressedasεD3(!/2). Iftheprincipaltermsor“fundamentaltones”oftheTable-5matrixgivethefourgaugebosons,whatisthesignificanceofthe“overtone-undertone”couplingsappearingofftheprincipaldiagonal?InTable1,thesearethetopodimensionalenantiomorphswhosesynsymmetricfusionsdrivetheprocessofdimensionalgeneration.Theovertone-undertonecouplingsappearinTable5asenantiomorphicallyrelatedbosonratios.Itisfromtheirinteractionsthattheprimarygaugebosonsemerge.Inphenomenologicalstringtheory,boson-ratiointeractionnotonlyaccountsforthegenerationofthefourkindsofgaugebosons,butalsofortheproductionofthe12fermionsofthestandardmodel.Thesixpairsofratiosinvolvedindistillingthebosonslikewiseinteracttoyieldthesixpairsoffermions(threeleptonpairsandthreequarkpairs).Geometricallyspeaking,thefermionsfunctionas“dimensionalboundingelements,”localfeaturesofglobal

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bosonicdimensionality,withlocalandglobalaspectsintimatelyinterwoven.Needlesstosay,fermiongenerationrequiresclarification,butIwillnotelaboratefurtheronithere(seeCosmos).WhatIwillelaborateonisthespecificcourseofbosongenerationthattracestheevolutionofnature’sfundamentalforces.

WhileTable5canbereadasaprocessofdimensionalgeneration,whatweacknowledgedearlierforTable1alsoappliestoit:ifTable5istakenbyitselfwithoutreadingchangeintoit,theTableprovidesamerelystaticpictureofparticlerelationships,asnapshotthatmissesthekineticfactsofcosmogony.Therefore,toseemoreclearlythedevelopmentalbasisofthecorrelationssuggestedinTable5,wemustfillintheconcretedetailsofhowthefourgaugebosonsevolveinrelationtoeachother.ThisisdoneinTable6.

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Table6presentstheparticulatecounterpartsofthecosmogoniceventsstudiedinTables3and4.Theparticlesarenottoberegardedasmerephysicalobjectsappearinginspace.Eachspinningparticle(or,bytheprincipleofcomplementarity,eachwave)isinfacttobeunderstoodasadynamicdimensionuntoitself,anarchetypalactionthatcombinesmatterandpsycheinanintimateway.Phenomenologically,wecandescribetheparticlesasconstitutinglifeworlds,butitmaybeevenbettertoviewthemas“braneworlds.”Forweareemployingacertainkindofstringtheory,and,intheM-theoreticelaborationofstringtheory,“branes”enterthepictureashigher-dimensionalversionsofstrings.Therefore,ifstringsaretakenasone-dimensionalvibrations,branes(asin“membranes”)canbevibrationsofhigherdimension.AccordingtoGreene,ourwholeuniversemightbeconceivedasa“braneworld”(2004,386).Whatwearedealingwithpresentlyarefoursuchuniverses. WithTable6,wehaveanoveliterationofthecosmogonicspiral.Herethefourbraneworldsarerepresentedinthefourwindingsofthespiral.Insection4,wesawtheconsequenceofconsideringcosmogonyasaprocessofsymmetrybreaking:startingfromsymmetry,aneffectivedimensionalrenderingofnature’sevolvingforcescannotberealized.IntherenditionsignifiedbyTable6,theissueisresolvedbyregardingthedimensionalgenerationofparticles(waves,braneworlds)asresultingfromthedialecticalinterplayofsymmetryandasymmetrythatwehavecalledsynsymmetry:theenantiomorphicfusionthatcreatesbraneworldsymmetryinonewindingcarrieswithittheimplicationofanenantiomorphicdiffusiongivingrisetonewasymmetryforthecreationtotakesplaceinthenextwinding. WemaycomparetheoverallspiralpatternofcosmogonybroughtoutinTables3,4and6tomainstreamcosmologicaldepictions.Whilemostcosmologistsnowappeartoagreethatthethree-dimensionaluniversewascharacterizedearlyonbyphasesofexpansion,thereislessagreementonthefateoftheuniverse.Willitcontinuetoexpand,orwillitreverseitselfandbegintocontract?CosmologistsPaulSteinhardtandNeilTurok(2002)haveaddressedthisquestionbyofferinganewversionoftheoscillatinguniverseidea,onethathasattractedmuchattentionamongastrophysicists.Takingtheircuefromstring/M-theory,SteinhardtandTurokproposeacyclicmodelinwhichtheuniverseundergoescountlessroundsofexpansionandcontraction.

Ourtopo-phenomenologicalversionofstringtheorysuggestsanevolvingpsychophysicalcosmosthatisnotmerelycyclicalbutspiralic.InagreementwithSteinhardtandTurok,thenotionofsynsymmetryrequiresthatthecosmicexpansionwearecurrentlyexperiencinginthisthree-dimensionalKleinianbraneworldwillbefollowedbyacontraction,afterwhichanotherperiodofexpansionshallensue.Butthesubsequentexpansionwillnotjustrepeatthepreviousone.Itwillinvolvetheopeningupofawholenewdimension,includingnewformsofmatterandanewforceofnaturebeyondwhatappearsinTable6.IshallhavemoretosayaboutthisprospectbeforeIconclude.WhatIpresentlywanttoemphasizeisthatthegeneralpictureofcosmicdevelopmentIamproposingisneitherofasimplyopenuniversewhosegivendimensionsexpandindefinitely,norofacloseduniversefeaturingendlesscyclesofexpansionandcontraction.Wemust

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imagineinsteadanevolvingcosmoswhosecontractionsarethe“laborpains”thataccompanythebirthingofnewdimensionalorganisms.Eachcosmicorganismexpandsinitsturn,onlytoexperiencecosmiccontractionsthatpavethewayforthenextroundofcreativecosmicgrowth.

ThewindingsofTable6unfoldincloseparallelwiththoseofTables3band4,allowingustoseetheconnectionsamongtopodimensional,Taoist,andparticulatedescriptionsofcosmogony.Inthefirstwinding,wehavethegravitonorgravitationalwave,G.GivenitsequivalencetoεD0(!/2),wemaytakeittocomprisethesub-lemniscatoryzero-dimensionalbraneworldorvibratinguniverse.Ratherthanundergoingitsownevolution,Gfunctionspurelyasacarrierwaveforthegenerationofhigher-dimensionalwavesthatinitiallyresideasseedswithinitszero-dimensionalfield.

Thesecondwindingofthecosmogonicspiralconcernsthegenerationofthestrongnuclearforce:g≡εD1(!/2).Thestrongforcecorrespondstothelemniscatorybraneworldconsistingofonedimension.Instage1ofthiswinding,wefindaprimordialmatrixwhereingismanifestedinnascentform,anembryonicparticle-wavesupportedbytheovertoneandundertoneenantiomorphsoftheGcarrierwave:g/GandG/g,respectively.Theprojectionoftheone-dimensionalbraneworldhasnotyetoccurred.

Inadvancingtothesecondstageofthisprojectivecycle,Genantiomorphsfusetobringsymmetrytog.Thusgainingmaturity,thestrong-forceuniverseexpands,and,intheprocess,Gisabsorbedintotheemergentstructureofg.This“annihilation”ofGatthesametime“freezesitout.”Wecanseehowthisworksfromouranalysisinsections7and8.

Coincidentwiththeprojectivepotentiationofgastheone-dimensionalbraneworldprevailinginstage2,Gissubjectedtoadepotentiatingprojection.Withitsoriginalvitalitypresentlymasked,itismanifestedasanobjectifieddimensionembeddedwithinthelarger,ostensiblyobjectiveuniverseofg.ItisinthisformthatGis“frozenout”instage2(indicatedinTable6byitsenclosurewithinparentheses).Sotheprimordialgravitationalfieldthathadservedascarrierwavefortheinchoatestrongforceinstage1isnowineclipse,withanattenuatedremnantofitoperatingwithinthefieldofthematurestrongforce.TherewillbefurtherocclusionsofGinsubsequentwindingsandthiswilleventuateintheattenuatedformofgravitationknowntoustoday—theformthatconventionalanalysiscannoteffectivelyreconcilewiththeotherthreeforcesofnaturetoyieldaconsistenttheoryofquantumgravity.

Theeventsattendingthetransitiontocycle2comeaswemightexpectatthispointinourinvestigation.Instage3,thegearsshiftfromforwardtobackward,expansiontocontraction,andthestage-2projectionofapresumablyobjectivestrong-forceuniverseiswithdrawnviaretrograderecognitionofitssub-objectivesource.Then,instage4,theretrojectiongoesfurtherandGemergesfromeclipse,convergingharmonicallywithg,thussowingtheseedsforthenextwinding.

Withtheopeningofthethirdcosmogonicepoch,wehavethegenerationoftheweakbosonpair:WandZ≡εD2(!/2).Topodimensionally,thisentailstheevolutionofthetwo-dimensionalMoebialbraneworld.Inthefirststageofthis

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winding,wefindourselvesbackintheprimordialmatrixofthegravitationalforce,withtheembryonicweakforcecarriedbythenewenantiomorphsofG:(W,Z)/GandG/(W,Z).Then,instage2,theseenantiomorphsfuseandsymmetryisenhancedinafirstprojectiveexpansionoftheweak-forceuniverse,coupledwithadepotentiatingprojectionofthegravitationalforcethatfreezesitout.Atthesametime,thepotentialforfurtherweak-forcedevelopmentiscarriedbytheenantiomorphsofthestrongforce,(W,Z)/gandg/(W,Z).Thefinalstageofcycle1completesthematurationoftheweakforce.Herestrong-forceenantiomorphsfuse,thestrongforcefreezesout,andtheweakforcegainsfullsymmetrywiththeprojectionofanostensiblyobjectivetwo-dimensionalbraneworld.Needlesstosay,allthisiscounteractedinshiftingtotheretrogradecontractionsofcycle2:Theprojectionsoftheweakforcearewithdrawn,therepressionsofthestrongandgravitationalforcesarelifted,andthethreeforcesenterintoharmony,sowingtheseedsforthecosmogonicepochtocome.

ThethirdroundofparticlegenerationisgivenintheKleinianepochthatbringsintheelectromagneticforce,γ≡εD3(!/2).Bynowthereaderiswellequippedtotrackthestagesofprojectionshownforthiswinding.InTable6weseethedevelopmentofγfromitsembryonicbeginningsinthearchaicgravitationalmatrixthroughthreestagesofexpansionfacilitatedbyenantiomorphicfusionsofthelower-dimensionalforceparticles.Theprocessculminatesinstage4withtheprojectionofourvisiblethree-dimensionaluniverse—visiblebymeansoftheparticlepermittingustoviewit,thephoton.

Beforeconsideringthesecondcycleoftheelectromagneticwinding,IwouldliketodiscussafeatureofTable6thatbearsinterestingimplicationsforcosmogony.Onlytheelectromagneticwindingisindexedchronologicallybycosmogoniceras:thePlanckera(t<10–43sec),theGUTera(10–43to10–36sec),theElectroweakera(10–36to10–12sec),andourpresentera(t>10–12sec).Whyaretherenochronologicallydatederasgivenforthelower-dimensionalwindingsofthespiral?Itisbecausetheformsoftimeassociatedwiththemarenotchronological.Insection4,Iindicatedthateachtopodimensionalspinorisrelatedtoadifferentordernotonlyofspacebutoftimeaswell,thoughIdidnotelaborateonthesedifferences.Itdoesseemreasonabletohypothesizethatdifferentlifeworldsentaildifferentordersoftime,andthatworldsindependentofourthree-dimensionaluniversearegovernedbyformsoftimequalitativelydistinctfromthelineartimesofamiliartous.WhileIhaveexplorednonlineartemporalityelsewhere(Rosen2006),hereIwillonlyobservethatourfirstcosmogonicwindingsurelydefieschronologicaldatingsinceitisutterlywithouttime,expressingasitdoesthetimelessnessoftheTao(onthetimelessaspectoftheTao,seeFagg1985andButler-Bowdon2012).

Now,theKleinianwindingofthecosmogonicspiralhasspecialsignificanceforus.Itisourownwinding.Thedimensionalcontextdevelopedinitisthethree-dimensionalworldthatframesourimmediateperceptionsandservesastheconcretegroundoftheveryanalysiscarriedoutinthispaper.ThephotongeneratedintheKleinianepochisindeedtheprimarymeansbywhichwemakeobservations,conductexperiments,performmeasurements,andgenerallyinteractwithour

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environment.So,onthecosmogonicmap,thewords“youarehere”mightwellhavebeenmarkedforthisepoch.

Fromourthree-dimensionalvantagepoint,wecandescribetheretrogradestagesofthelowerdimensionsinadetachedfashion,butwedonothavethatluxurywhenitcomestodescribingthesecondcycleofourowndimension.Tounifythequantumgravitationalfield,thefrozen-out,depotentiatedbraneworldsmustberepotentiatedinconjunctionwiththeelectromagneticbraneworld.Sincetheelectromagneticworldistheanalyst’sownworld,andsincethisisapsychophysicalworldratherthanmerelybeingphysically“outthere,”theanalystcannotstandalooffromtheprocess,treatingitassomethingunfoldingobjectivelybeforeher.Theanalystishereonthecosmogonicmap,andhemustparticipateaccordingly.

Tobesure,inenteringcycle2,wehavealreadytakenthefirststepinthisdirectionwithourretrojectivereflectionsonthepsychophysicalnatureofouruniverse.Yet,inmakingthetransitiontostage5,ourperspectiveremainsstrictlythree-dimensional,asitwasinstage4.Whatweneedtodointhesubsequentstagesofthiscycleistangiblyengagewiththelowerdimensionalsub-objectivitiesastherepressionsofthemareliftedandtheiroriginalpotenciesarerestored.Butjustwhatwoulditmeanforthescientisttoforegohiscurrentlyobjectifyingpostureinfavorofonethatisdeeplyparticipatory?

10.TOWARDAPARTICIPATORYCOSMOGONY

ThephilosopherofscienceEvelynFox-Kellercallsforanewformofperceptioninscientificinquirythatshenames“dynamicobjectivity”(1985,115).Theoldapproach,shesays,involvesa“staticobjectivity”inwhich“thepursuitofknowledge...beginswiththeseveranceofsubjectfromobject”(117).Incontrast,

dynamicobjectivityaimsataformofknowledgethatgrantstotheworldaroundusitsindependentintegritybutdoessoinawaythatremainscognizantof,indeedrelieson,ourconnectivitywiththatworld.Inthis,dynamicobjectivityisnotunlikeempathy,aformofknowledgeofotherpersonsthatdrawsexplicitlyonthecommonalityoffeelingsandexperienceinordertoenrichone'sunderstandingofanotherinhisorherownright.(1985,117)Dynamicobjectivityemploysatypeofawarenessakintotheretrogradeact

ofwithdrawingprojections.ThisisevidencedinFox-Keller'scitationofPiaget:“‘Objectivityconsistsin…fullyrealizingthecountlessintrusionsoftheselfineverydaythoughtandthecountlessillusionswhichresult....Solongasthoughthasnotbecomeconsciousofself,itispreytoperpetualconfusionsbetweenobjectiveandsubjective’”(117).AccordingtoFox-Keller:

Dynamicobjectivityisthusapursuitofknowledgethatmakesuseofsubjectiveexperience(Piagetcallsitconsciousnessofself)intheinterestsofamoreeffectiveobjectivity.Premisedoncontinuity[ofselfandother],itrecognizesdifferencebetweenselfandotherasanopportunityforadeeper

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andmorearticulatedkinship.Thestruggletodisentangleselffromotherisitselfasourceofinsight—potentiallyintothenatureofbothselfandother....Tothisend,thescientistemploysaformofattentiontothenaturalworldthatislikeone'sidealattentiontothehumanworld:itisaformoflove.Thecapacityforsuchattention,likethecapacityforloveandempathy,requiresasenseofselfsecureenoughtotoleratebothdifferenceandcontinuity.(1985,117–18)Writinginthesamevein,Fox-KelleradducesErnestSchachtel'sdistinction

between“autocentric”and“allocentric”perception.Whereastheformeris“dominatedbyneedorself-interest,”thelatter“isperceptionintheserviceofalove‘whichwantstoaffirmothersintheirtotalanduniquebeing.’Itisanaffirmationofobjectsas‘partofthesameworldofwhichmanisapart,’”onewhich“permitsafuller,more‘global’understandingoftheobjectinitsownright”(119).AlthoughFox-Kellerpaysscantattentiontophenomenologicalphilosophyassuch,themainthrustofherpresentationismuchinkeepingwithphenomenology'scentralaim,asexpressedinitswell-knownslogan:“Tothethingsthemselves!”Anditseemsclearthattheworldsharedbythe“allocentric”observerandtheobjectsthats/heobservesisthelifeworldofphenomenology.

Fox-Kellerhelpsusgainabettergraspofthenewmodeofscientificinquirybyofferingaspecificexampleofoneofitspremierpractitioners:theNobelprize-winningbiologist,BarbaraMcClintock.Instarkcontrasttothedetached,dispassionateattitudeoftheCartesianscientist,McClintockspeaksofobtaininganintimatefeelingfortheplantssheworkswith:“‘Idon'tfeelIreallyknowthestoryifIdon'twatchtheplantallthewayalong.SoIknoweveryplantinthefield.Iknowthemintimately,andIfinditagreatpleasuretoknowthem’”(Fox-Keller1985,164).Inanotherplace,McClintock:

describesthestateofmindaccompanyingthecrucialshiftinorientationthatenabledhertoidentifychromosomesshehadearliernotbeenabletodistinguish:“IfoundthatthemoreIworkedwiththem,thebiggerandbigger[thechromosomes]got,andwhenIwasreallyworkingwiththemIwasn'toutside,Iwasdownthere.Iwaspartofthesystem....ItsurprisedmebecauseIactuallyfeltasifIwasrightdownthereandtheseweremyfriends....Asyoulookatthesethings,theybecomepartofyou.Andyouforgetyourself.”(McClintockquotedinFox-Keller1985,165)

Fox-KellerobservesthatMcClintock'svocabulary“isconsistentlyavocabularyofaffection,ofkinship,ofempathy,”anempathythatconstitutes“thehighestformoflove:lovethatallowsforintimacywithouttheannihilationofdifference”(164).Heretheword“love”isused“neitherlooselynorsentimentally,butoutoffidelitytothelanguageMcClintockherselfusestodescribeaformofattention,indeedaformofthought”(164).

Fox-Kellerarrivesattheseconclusions:

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ThecrucialpointforusisthatMcClintockcanriskthesuspensionofboundariesbetweensubjectandobjectwithoutjeopardytosciencepreciselybecause,toher,scienceisnotpremisedonthatdivision.Indeed,theintimacysheexperienceswiththeobjectsshestudies...isawellspringofherpowersasascientist....Inthisworldofdifference,divisionisrelinquishedwithoutgeneratingchaos.Selfandother,mindandnaturesurvivenotinmutualalienation,orinsymbioticfusion,butinstructuralintegrity.(1985,164–165)

Finally,afterrecountingthegoalofconventionalscience,Fox-Kellerobserves

that,“ToMcClintock,sciencehasadifferentgoal:notpredictionperse,butunderstanding;notthepowertomanipulate,butempowerment—thekindofpowerthatresultsfromanunderstandingoftheworldaroundus,thatsimultaneouslyreflectsandaffirmsourconnectiontothatworld”(166).

Inphenomenologicalterms,theworldtowhichMcClintockisconnectedinfeelingandembodiedempathyisthelifeworld.Itisaworldinwhichthedialecticofdifferenceandidentityisenactedthroughanintimateknowledgeofotherthatrequiresandisinseparablefromtheknowledgeofself(a“consciousnessofself”).McClintock's“revolutionthat‘willreorganize...thewaywedo[scientific]research’”(Fox-Keller1985,172)dependsupondescendingfromtheCartesianstratosphereandimmersingourselvesinthepsychophysicaldimensionwhereinobjectandsubject,symmetryandasymmetry,continuityanddiscontinuitymediateoneanotherinternallyinanencompassingcircularflow.Wehavenotforgottenthatthesourceofthisphenomenologicalcirculationliesintheancienteddiesofyinandyang.

Fox-Keller's“dynamicobjectivity”asexemplifiedbyMcClintockishardlytheonlyinstanceoftheburgeoningofanewdialecticalscience.Thephenomenologicalinitiative,begunearlyinthetwentiethcentury,hasbeenadvancedbythinkerslikeHeelan(1983)andGendlin(1991),whohaveproposedthattheworkofsciencenotproceedfrom“stratospheric”perception,butfromtheintricaciesofthelifeworldorlivedbody.AdialecticalapproachtosciencealsoisadvocatedbybiophysicistKoichiroMatsuno(1995).Matsunohascalledfora“dialogical”sciencethatwouldsupersedetheold“monologue”carriedonbythesolitaryCartesiansubjectlookingdownupontheworldfromabove.InMatsuno'svision,scientificactivitywouldinvolveacommunityofsubjectsconcretelyengagedwitheachotherindynamicandgenerativenegotiations.WhereastheCartesiansubjectisanonymous,absentfromtheeventsthattranspire,theparticipantsinthedialecticalcommunitywouldfunctionself-referentiallytoincludethemselvesintheprocess(Matsunoexemplifiesthisbyexplicitlyincludinghimselfasauthorinwhathewrites;1995,1998).OtherimportantcontributionscomefromPlamenSimeonov(2012),whohasemphasizedtheneedtodevisefirst-personmethodologiesforthenaturalsciences;fromArranGare(2013),withhisinsistencethatsciencebegroundedinawaythatincludeslivedsubjectivity;andfromLouisKauffman’s(2015)reflectionsonhowmathematicalself-referenceisrelatedtotopologyandphenomenologicalphilosophy.StillanothercontributiontoemergentdialecticalscienceisofferedbytheJungianpsychologistNathanSchwartz-Salant(2007).Operatingself-referentially,Schwartz-SalantemploysMerleau-PontyandtheKleinbottlein

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characterizingthedeeppsychodynamicsofhumanrelationships,andhelikensthefieldsoperativeintheseparadoxicalinteractionstofieldprocessesinfundamentalphysics(seealsoSchwartz-Salant2017).

Whatwerequireinthepresentcontextisadialecticalcosmogony.Inthisapproach,theanalysisofcosmogonyissituatedwithincosmogonyitself(“wearepartoftheworldwearetryingtoknow,”saysGare;2013,25).TheunquestionedobjectivestanceanalystshavetendedtotaketowardcosmicevolutionisinfactcharacteristicofthefourthandfinalstageofprojectionintheKleinianepochofdimensionalgeneration.Thisisthestageofdevelopmentinwhichweassumethatcosmogoniceventsare“objectivelyoutthere,”andthatweanalystsaredetachedfromthem,withourlivedsubjectivityplayingnoroleinwhatwesee.Inthisstage,thecommonsensenotionofanexternaluniversedevelopingonitsownissocompellingthatitseemsabsurdforustothinkotherwise.But,inadvancingtothestagesofcycle2,thepointcomeshometousthatweareindeedintimateparticipantsinthestoryofcosmiccreation.Thusenteringintocosmogonicprocess,theclassicalpostureofanalysisgiveswaytoaphenomenologicaloneinwhichourownprocessofdevelopmentplaysanintegralrole.Intheactofinwardlygraspingthetransformationofthecosmos,theanalystsurpassestheprojectiveconstructionofherselfasanisolatedonlookerandtakespartinthedramaofcreatingaworld.So,ifthecosmosisself-evolving,theselfoftheanalystfiguresessentiallyinthereflexiveenactmentofthisprocess.

Butletmetrytobecleareraboutwhattheinvolvementoftheanalystspecificallyentails.ThepropositionIventuretosuggestisthatafullyreflexiveanalysisofcosmogonyrequiresthat,ininvestigatingthestagesofcosmogonicretrojection,theanalystmustgainpalpableawarenessofhisorherownstagesofdevelopment.Onlythencanthelinktocosmicdevelopmentberealizedinitsexistentialimmediacy,since,onlythenwouldtheanalystrealizecosmictransformationasaself-transformation,notjustatransformationofwhatisother.Ofcourse,theanalyticalselfinquestioncannotmerelybethatofaparticularindividual.Theselfthatparticipatesinthearchetypalprocessesofcosmiccreationmustfunctionarchetypally.Yetitseemsweneedtobeginwiththeparticularpersoniftheprocessistobegroundedinexistentialreality.Presumably,inthecourseofdeeplyexploringhisorherownpast,theanalystwouldcrossathresholdandherpersonalbeingwouldshadeintothetranspersonal.ThetranspersonalpsychiatristStanislavGrofexpressedasimilarideaindescribingthetransformationofawarenessthatcanoccurintheactofre-experiencingthe“perinatal”stagesofdevelopment,thoseoccurringaroundthetimeofbirth:“Allwecansayisthatsomewhereintheprocessofconfrontationwiththeperinatallevelofthepsyche,astrangequalitativeMoebius-like[!]shiftseemstooccurinwhichdeepself-explorationoftheindividualunconsciousturnsintoaprocessofexperientialadventuresintheuniverse-at-large”(1985,36).

Intheprojectivemomentofcosmogony,itmaywellseemaflightoffancytolinkthestagesofhumandevelopmenttothoseofthecosmosasawhole.Thephenomenologicalresponsetothisincredulityextendsthebiologicaldictumthat“ontogenyrecapitulatesphylogeny”tothefieldofphysicsandsays,ontogenyrecapitulatescosmogony.For,ifitistruethatweparticipateinthestoryofcreation

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inafull-fledgedway,itwouldseemthatourownhistorywouldbeinseparablenotonlyfromthatofthebroaderbiologicalworldbutfromnatureatlarge.Evidentlythen,whenwemovebackwardthroughthestagesofcycle2togain“allocentric”awarenessofthebraneworldsbelongingtonature’sarchaicpast,itseemswemustworkthroughourownarchaicpastifwearetoapprehendthoseworldsinthemostconcrete,immediate,anddeeplyreflexiveway.

Embryologicalresearchcertainlyappearstosupporttheideathattheearlydevelopmentofthehumanindividualmirrorsthedevelopmentofthespeciesasawhole.Infact,myearlierworklinksontogenyandphylogenyexplicitly,andinadetailedway(Rosen2006).Presently,itisontogenyandcosmogonythatmustbelinked.Sometheoristshavebroadlyspeculatedthattheuniversefunctionsasagianthologram(Bohm1980,189).Suchacosmosshouldpossessafractalpatternofself-similarity,withthestructureanddevelopmentofthewholebeingmirroredrecursivelyoneveryscaleofmagnitudedowntothesmallestpart.Then—ifprobingtheearlyhistoryofanindividualmemberofthephylogeneticorderopensoutintophylogenyasawhole—itisperhapsnotunreasonabletohypothesizeadeeperstratumofself-similarityinvolvingthehistoryofthecosmostowhichwebelong.RelevantinthisregardisthevisionofphysicistLeeSmolin:

Livingthingsshareinsomeways,andextendinotherways,thebasicpropertiesofnon-equilibriumself-organizedsystemsthatseemtocharacterizetheuniverseoneveryscale,fromthecosmosasawholetothesurfaceofplanets….Iflife,orderandstructurearethenaturalstateofthecosmositself,thenourexistence,indeedourspirit,mightfinallybecomprehendedascreatednaturally,bytheworld,ratherthanunnaturallyandinoppositiontoit.(1997,160)

Inasimilarvein,biophysicaltheoristHectorSabelliassertsthat“thecontinuityofevolutionrequiresthatthesamefundamentalformsmustbeexpressedatthephysical,biological,andpsychologicallevelsoforganization”(2005,431).Thisisconsistent,ofcourse,withthepsychophysicalnatureofcosmogonicprocess. InTheSelf-EvolvingCosmos,Iattemptedtospelloutmorespecificallythemannerinwhichtheanalyst’sowndevelopmentcanbelinkedtothatofthecosmosatlarge.ButIcangonofurtherhereifthisalreadylengthypaperistostopshortofbecomingabook! Letmeconcludebyreturningtotheissueraisedneartheendofsection7.ThereInotedthattheguidingprincipleofsynsymmetryimpliesthebirthofahigher-dimensionallifeworldsurpassingtheKleinian.ThequestionIposedwashowwecancometoknowthisfour-dimensionalrealityinatangibleway.WhatIindicatedisthatgoingbeyondanabstractthree-dimensionalanalysisofthefourthdimensionnecessitatesthedimensionalevolutionofouranalyticframeworkitself.Atthatpoint,itwasalreadyimplicitthatthebackwardmovementthroughthesecondKleiniancyclebringsthisevolutionabout.Weknewthat,inthecourseofcycle2,lower-dimensionalorganismsrisefromobscurity,and—operatinginsynchronywiththeKleinianorganism—sowtheseedsforthemeta-Kleinian,four-dimensionalframework.Butweknewthisonlyabstractly,sinceourwayofknowing

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wasitselfrestrictedtothethree-dimensionalcontext.Sotheobstacletotangiblyapprehendingthefourthdimensionisourinabilitytograspitinamannertranscendingthelimitsofourthree-dimensionalframeofanalysis.ClearlythetheoreticalanticipationofcompletingthesecondKleiniancycleisnosubstituteforcompletingitinactuality.Howcanthelatterbeachieved?Isuggestitcanhappenbyadoptingthereflexivephenomenologicalpostureintimatedabove.Hereweparticipateallocentricallywiththelower-dimensionalorganismsofcycle2,investigatingthestagesofdevelopmentasstagesinourowndevelopment,experiencingthetransformationsoccurringasself-transformations.Itiswhenthisprocessisbroughttofruitionthatanewwindingofthecosmogonicspiralopensupandawholenewworldisintroduced—aworldwewillhavecometoknowsubstantivelybythedimensionalexpansionofourcapacityforknowing. Whilemyownepistemiccapacityremainsdistinctlythree-dimensional,letmeofferforwhatitisworthaprovisional,stillquiteabstractimpressionofwhatmaybeinstore.Inthewiderturningofthespiral,anewandmoredialecticallyintricate,four-dimensionalbraneworldwouldcomeintoplaybeyondtheKleinianworld,atopologicalactionpatternlaidoutin5×5matrices.Inthisnovelworld,anewforceofnaturepresumablywillemerge,alongwithnewformsofmatter.Afreshconfigurationofyinandyangshouldexpressthistransformation,symbolizedbyfourlinesinsteadofthree(the“quadrigram”wouldsurpassthetrigram).InthelanguageoftheIChing,the“forcesofthecomingyear”wouldthus“unroll.”Andascosmogonywouldadvanceoncemoreintheepochtocome,themusicofthedimensionalspheresshouldresoundinapreviouslyunheardregister,audibleonlytothosewiththecapacitytohearit.REFERENCESAngeles,PeterA.1981.DictionaryofPhilosophy.NewYork:BarnesandNoble.Barr,Stephen.1964.ExperimentsinTopology.NewYork:Dover.Bohm,David.1980.WholenessandtheImplicateOrder.London:Routledgeand

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