DOI: 10.4018/IJSVR.2018010101

International Journal of Semiotics and Visual RhetoricVolume 2 • Issue 1 • January-June 2018

Copyright©2018,IGIGlobal.CopyingordistributinginprintorelectronicformswithoutwrittenpermissionofIGIGlobalisprohibited.

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Fractal Entanglement Between Observer and ObservedTerry Marks-Tarlow, Insight Center, Los Angeles, USA & Italian Universita Niccolo Cusano, London, UK

ABSTRACT

Thisarticleexploresparadoxicaldynamicsembeddedinthefabricofexistence.ThelogicofSpencer-Brown,asextendedbyFranciscoVarela,isplacedwithinthehistoricalcontextofcybernetics.Varela’sconceptofreentryprovidesabridgetofractalgeometrythroughrecursiveiterationonthecomplexnumberplane.Fractalsaredynamicprocess-structuresthatnegotiatespatialandtemporalinterfacesbetweenvariousforcesanddimensionsofexistence.Thisarticlearguesthatfractalreentrydynamicsprovidecontradictoryfoundationsbywhichdifferentlevelsofnature,includingmindandmatter,bothseparateandconnect.Finally,naturalfractalsrevealanewlevelofreflexivityinscience,bywhichtheobserverisparadoxicallyembodiedwithintheobserved.

KEywORdSFractal Geometry, Imaginary Numbers, Jung, Paradox, Recursive Dynamics, Reentry, Reflexivity, Second-Order Cybernetics, Spencer-Brown, Varela

INTROdUCTION

Do what you will, this life’s a fiction/And is made up of contradiction. -William Blake

Weeasilytakeforgrantedtheabilitytodistinguishbetweenourselvesasobserversandwhatweobserveintheworld.Outwardlyourskinseemsvisibleproofofaclearboundarythatencasesandprotectsourorgans.Inwardlyoursenseofself,whenintact,alsofeelslikearelativelyclearboundary,attimeseventothepointofisolationfromothers.Yetwhetherweconsiderourbodiesorminds,thesubjectiveexperienceofclosedboundariesrestspreciselyontheoppositestateofaffairs–wide-openportalsthatcontinuallyallowtransactionbetweeninsideandoutside,bodyandworld,selfandnot-self.

Openportalsareevidentinour“posthuman”existence(seeHayles,1999),wheretheinterfacebetween human being and machine presents boundaries which have grown ever more complexovertime,witheachtechnologicaladvance.Weplugourconsciousnessintovirtualrealities,asweaugment,eveninvadeourbodieswiththepresenceofmachines.Thisintenseexchangebetweenfleshandmechanismdemandsnothingshortofaredefinitionofhumansubjectivity.

Mysticalpoets,likeWilliamBlakeintheaboveepigraph,alludetolifeasfictioninherentlymadeupofcontradiction.HinduismofferstheconceptofMayatodescribethefalseperceptualveilbywhichweshieldourselvesfromanultimatelymysteriousreality.Athigherlevelsofcognitiveorganization,

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psychologistsstudyrelatedphenomena.Forinstance,ShellyTaylor(1989)identifiesself-deceptionintheformof“positiveillusions,” thoseoverlyoptimisticattitudesandexpectationstowardsthefuturethatmaybeentirelyunrealistic,butneverthelesswhichcanhelpustobeatthemedicalodds.

Despitethecomplexitiesofourallegedposthumanexistence,mostofusliveasifconsistency,certainty, predictability and clear boundaries, especially between truth and falsity, reside at thebaseofthings,fromtheworkingsofourbodiesandmindstotheworkingsoftheuniverseatlarge.Boundariesareeverywhere,yetmostarepermeable.Byfocusingonthishighlycontradictorystateofaffairsthatextendsinvisiblyunderthesurface,IfollowBlake’sleadtoexploretheparadoxicaldynamicsembeddedintheveryfabricofexistence.

Thispapertracesalineofsemioticsandlogic,begunbyGeorgeSpencer-BrownandcontinuedbyFranciscoVarela,whichputsparadoxattheheartandseamofthings.IplaceVarela’sideasaboutreentrywithinthecontextofabranchofcontemporarymathematicscalledfractalgeometry.Iarguethatadeepunderstandingoffractalshelpstoilluminatetheprofoundyetinvisibleparadoxesthatpermeateordinarylife.

Tosetthestagehistorically,Ibeginbybrieflydescribingthecyberneticsrevolutionandhowreflexivityfirstenteredsocialsciences.TheprimitivelogicofGeorgeSpencer-Brown,plusextensionsaddedbyFranciscoVarela,arefittedintothishistoricalcontext.Varela’sdynamicsofreentryarticulateparadoxicalfoundationsnotonlyforlogicandbutalsoforthecreationofallstructure.Next,Iconnecttheselogicalassertionswithmathematicsofthecomplexplane,whereimaginarynumbersareusedtomodelextraorhiddendimensionality.Imaginarynumbersprovidethebridgetofractalgeometry,whosemathematicsinvolvesrecursiveiterationofsimpleformulasonthecomplexplane.

Fractalsaredynamicprocess-structuresthatetchtimeintospace.Theyareboundarykeepersthatnegotiatespatialandtemporalinterfacesbetweendifferentforcesanddimensionsofbeing.Mythesisisthatfractalsprovidetheparadoxicalfoundationbywhichdifferentlevelsofnaturebothconnectandseparate.Everyboundarybecomesadoor,everyborderaportal.Becausethesamedynamicsholdinsideaswellasoutsidethepsyche,fractalgeometryprovidesabridgeandlanguageforlinkinginsideandoutsideworlds.Whethertheyoccurinnature,ourbodiesorminds,fractalseparatricesorboundariesrevealinfinite,hiddenfrontiersinthespacebetweenordinary,Euclideandimensions.

Iconcludethispaperbyexaminingthemechanicsoffractalproductiontorevealanewtwistinthereflexivemarchofscience.Inaworldfilledwithfractals,notonlyistheobserverdetectableintheobserved,buttheobserverisalsoembodiedthere,inaprimordial,concreteway.Naturalfractals,likeshorelines,revealhowtheembodimentoftheobserverintheobservedparadoxicallyprecedesthepresenceofconsciousobservers.

THE CyBERNETIC REVOLUTION

TheperiodfollowingWorldWarIIwasatimeoftremendousintellectualgrowthinAmerica.Emergingfromtechnologydevelopedduringthewar,severaltrendsconvergedtolegitimatethescientificmeritofpsychology,includingthebirthofcybernetics,thescienceofinformation.Thisnewfield,spearheadedbythemathematicianNorbertWiener,mushroomedoutoftheinterdisciplinaryMacyConferencesheldyearlybetween1946and1953(SeeHeims,1991).Cyberneticsbroughtanewvariantofthemetaphorofthemindasmechanism.RootsofthisideaextendatleastasfarbackasRenaissancetimes,whenthenaturalsciences,onebyone,splitofffromphilosophy.Asmoreempiricalstudiesbegan,theheartresembledapump,thebodyamachine,andthewholeuniverseranlikeclockwork.

Mechanistic metaphors are evident within the psychoanalytic theories of Freud, where, forexample,theprimitiveinstinctsofdrivetheoryresemblehydraulicsofthesteamengine(seeDraaisma,1995/2000).Thecyberneticassociationbetweenmindandmachinemadeinthemid-20thcenturyprovedaboosttotheneurosciences,whenneuralloopsinthebrainweremodeledaslogicalchains.Mechanisticmetaphorsalsousheredinthecognitivistrevolution,asactivityinthepsychewaslikenedtoinformationprocessingincomputers.

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Ashumanistic thinkersragedagainst the‘cold’,mechanical,andat timesreductionistviewsbeingespoused,withintheMacyConferencesprotestsofadifferentkindbegantosurface.LawrenceKubie,apsychoanalystandrecentretreadfromthe“harder”fieldofneuroscience,stimulatedheateddiscussion among his colleagues by pointing to the problem of reflexivity (See Heims, 1991).Reflexivity,bywhichanassertionpointsself-referentiallytoitself,e.g.,“WhatIsaynowisfalse,”involvesaconfluenceormeldingbetweenobserverandobserver.Reflexivityisinherentintheverysubjectmatterofpsychology. Itoccurs, forexample,whenever researchersuseconsciousness tounderstandthenatureofconsciousness,narrativestostudythenarrativesofothers,orbehavioralrepertoirestoexaminebehavioralresponsesinothers.ResearchinpsychologyislikethemythicalUroborus,asnakeeatingitsowntale/tail.Despiteeveryattempttoremainobjectivebysidesteppingsubjectivity,evenbehavioristscanfindlittlerelieffromtheUroboricbeastofreflexivity.

During the Macy conferences, Kubie objected to the early cybernetic agenda of separatinginformationfullyfromitsmaterial,embodiedsources.Thepsychoanalystprotestedthatwithinanytheory,eveninsidethe“hardest”ofsciences,reflexivitylurks,andtheobserverlieshiddenwithintheobserved.Kubieclaimedthatalltheoriesabouttheoutsideworldsayasmuchabouttheunconsciousofthesubjectwhoespousesthemastheydoabouttheoutsideuniverseasconsciouslyperceived.Whenitcomestotheoryandmeaningmaking,nomatterwhatisobserved,theobserverwindsupimplicatedintheobserved.AlthoughKubie’sprotestsweredismissedbymostofhisfellowscientists,hisideasaboutreflexivitylaterbecameingrainedwithinthehistoryofpsychoanalysis.RobertStolorowandhiscolleagues(Atwood&Stolorow,1979/1993),cofoundersofintersubjectivitytheory,arguethateverytheoryofpersonalityisself-reflexiveinthatituniversalizesthetherapist’spersonalsolutiontothecrisesofhisorherownlifehistory.

DuringtheearlyyearsoftheMacyconferences,thenotionofsciencestillresteduponthehithertobedrockfoundationofobjectivity.Byrequiringaclearseparationofsubjectsfromobjects,objectivitywasapositionthatrancontrarytoreflexivity.BecauseearlymembersoftheMacyconferenceswereinterestedinmaintainingscienceasanexplicitlyobjectiveenterprise,theychosetoignoreKubieratherthantorevisetheirownideas.Insteadofincludingreflexivitywithintherubricofscience,theydismissedpsychoanalysisasscience.

Generally,duringthisfirstwaveofcyberneticstheory,theproblemofreflexivitywassuccessfullyavoidedbyisolatingpatternasaseparaterealmfromwhichallothersemanate.Whenthepatternofinformationreignssupreme,itsmaterialsubstratecanbefirstignoredandtheneliminatedfromconsiderationaltogether.Accordingtothisview,evenwithoutmatterthepatternstillmatters.Byremovinginformationentirelyfromitsmaterialsources,theneedforobserverswasalsoeliminated.Weareleftwithpatternasavirtualrealitywithneitherobservednorobserver.

Thisstrategyworkedtemporarily,butonlyuntilthewholeenterpriseofsciencebegantakingareflexivedive.Atthecosmiclevelofgrand-scaleevents,Einstein’searlierdiscoveriesinphysicsdestroyedthepreviouslyimmutableframeworkofspaceandtime.Thenotionofobjectiveobservationstretchedanddeformed,asrelativitytheoryandthesubjectivestanceofobserverstookcenterstage.Meanwhile, at the subatomic level of tiny, quantum events, another field spawned by Einstein’swork,consciousnessbeganpushingitswayself-reflexivelyintothemiddle.ThestillcontroversialCopenhageninterpretationassertedthatatthequantumlevel,theveryactofobservationisnecessarytomaterializethatwhichisobserved.

Evenmathematicswasnotimmunefromareflexivefall.Inthe1930s,anAustrianmathematiciannamed Kurt Gödel used recursive methods in order to code numbers and then talk about themreflexivelyatahigher,meta-levelofabstraction.Intheprocess,Gödelprovedthatnosingletheorycouldeverprovideasimultaneouslyconsistentandcompletefoundationtologic.Gödel’sassertionsannihilatedanyresidualhopesforperfectobjectivitywithinthemathematicalunderpinningsofscience.

As reflexivity was seeping into the physical and mathematical sciences, a second wave ofcybernetics arose between 1960 and 1985. Spearheaded by Francisco Varela, among others,informationscientistsbecamebetterprepared toembrace reflexivity (seeHayles,1999). In fact,

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theverynameofthisnewtrend,“second-ordercybernetics,”amountedtotherecursivestudyofobserversstudyingthehigherorderprocessesofobservation: theobserversobservedthemselvesobservingthemselves.

POSTMOdERNISM

Second-ordercyberneticsarosewithinabroad,societalseachangeknownaspostmodernism.Overtheyearstheuseofthistermhasbeenstretchedsofarastoencompasspracticallyeverything,whilebeingdeconstructedsothoroughlyastomeanalmostnothing.Forthisreason,Ibegtodismissitsbroaderdefinitioninordertofocusuponasinglefacet,itsinherentreflexivity.Tosymbolizethepostmodernimagination,RichardKearney(1988)offerstherecursivesymboloftwomirrorsreflectingoneanother.Hecontraststhiswiththepremodernimagination,symbolizedbyasinglemirrorinwhichhumancreativityreflectsGod’screation,aswellaswiththemodernimagination,symbolizedbyalampinwhichhumancreativityisilluminatedfromwithin.

Becauseofitsreflexivity,theposthumanimaginationbecomeslostinsideaninfiniteregressofimitations,copiesandsimulacra.Withoriginsdeconstructedintodust,thepostmodernbeingisoftenportrayedasrootless,wanderinginsideamechanical,artificialdesertofre-production.Withinthisbleakfrontier,ontheonehand,thedemiseofhumancreativityandoriginalityisdecried.Ontheotherhand,thereloomsthecyberneticthreatofmachinesusurpingtheveryautonomy,indeedexistence,oftheirhumanistcreators.

InHow We Became Posthuman,EnglishprofessorKatherineHayles(1999)detailsthethreateneddestruction of embodied existence. She analyzes cyberpunk novels with heroes that evaporateintovirtualrealityastheirconsciousnessbecomesthoroughlyenmeshedandencapsulatedwithinmachines.Theflipsideofthisfuturisticnightmareportraysmachinessophisticatedenoughtotakeovertheevolutionoflifeitself.Asartificialintelligencebecomesincreasinglyequippedwithemotion,creativity, the capacity to learn, self-repair and self-generate, this sci-fi genre depicts humanoidmachinesthatthreatentoextinguishcarbon-basedevolutionasweknowit,replacingitwiththefar-superior,silicon-basedlifeforms.

Asposthumanboundarieshavebecomemoreblurredandhumanbeingsself-reflexivelyentangledwithfacetsoftheirowntechnologicalproduction,linesbetweenobserversandobservedcontinuetogrowmorecomplex.Asreflexivityisintegratedmoreandmoreconsciouslyintocybernetics,onepositiveoutcomeisthatthedooristhrownopenforthescientificstudyofsubjectivity.Sincesubjectscannowstudytheirownsubjectivity,consciousnessitselfhasrecentlyregainedstatusasalegitimateandseriousobjectofscientificstudy.Eversophisticatedcapacityforfunctionalbrainscansnowblendsfirstandthirdpersonperspectives(Northoff&Heinzel,2006),whilethenewfieldofhyperscanning(e.g.,Babiloni&Astolfi,2014;Dumasetal.,2011)allowssimultaneousbrainmeasurementoftwosubjectivitiesininteractionwithoneanother.

In the postmodern view, reflexivity is often viewed as a by-product of modern technologyconstructedinthecontextofparticularsocialandeconomictrends.Contemporarymethodssuchasneuralfeedbackevenallowustobecomeactiveobserversofourownbrainprocesses.Inarguably,computer-driven,cyberneticextensionsofourperceptualandconceptualapparatusesdohelpustodetect,directandevencreatereflexivitywithgreaterease.YetIbelievethattherootsofreflexivityaremuchdeeperandmoreorganicthansocialandhistoricaltrendssuggestedbypostmodernism.Imaintainthatthedisciplineoffractalgeometryprovidesevidencethatreflexivityisintrinsicnotonlytohuman-madeproductions,butalsotonatureatlarge.

Fractalshelpus toadvancebeyond thecyberneticmetaphorofpsycheasmechanismto themoreorganiconeofnature,includinghumannature,asfractal.Heremechanisticmeansofcomputersimulationreflexivelyguideusbeyondmechanism,aswecirclebacktoadifferentkindoforigins,forbothhumanandmachine,infractalbasesofnature.Beforeturningtofractalgeometryitself,the

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sectiontocomepresentsSpencer-BrownandVarela’slogicalunderpinningsforreentrydynamicsastheyareembeddedintheveryfabricofcreation.

CONTRAdICTION BUILT INTO THE FABRIC

A great truth is a truth whose opposite is also a great truth. -Neils Bohr

Whendevelopinghis“LawsofForm,”mathematicianandlogicianGeorgeSpencer-Brown(1969;1979)triedtospecifyhowwecreate“some-thingfromno-thing”inconsciousness(SeeRobertson,1999).Spencer-Brownuseda2-valuedsystemthatconsistedonlyof“marked”and“unmarked”statesplustwoaxioms.Fromthesesimplebases,hederivedacalculusoffirstdistinctions.AlthoughitiscommonlybelievedthatGeorgeBoole(1958)developedthemostbasicformoflogic,Spencer-Browndisagreed,claiminghisowncalculusissoprimordialastoprovideacradlenotonlyforlogicitself,butalsoforthebasicstructureofanyuniverse.

WithinSpencer-Brown’ssystem,onedistinguishesmarkedfromunmarkedstatesbyattributingvaluetoonestateovertheother.Thisactofmarkingormakingadistinctionrequiresanobserver.Wecanreadilyunderstandthisrequirementforlogic:tomakeamark,applyasetofaxioms,ordistinguishtruthfromfalsity,aconsciousobservermustbepresent.Buthowdoesthisprocessofvaluationapplytomoreprimitivelevelsofasystemthatsupposedlyprecedelogicandevenpeople?Isanobserverimplicatedalongwiththeobservedtheretoo?Iwillreturntothisissueinmysubsequentdiscussionoffractalgeometry.

AsSpencer-Brownprogressedwithhiswork,heusedbasicaxioms toderivehigherdegreeequations.Butthensomethingstrangebegantohappen:anomaliesappeared;reentryofequationsback into themselves sometimes resulted inparadox.Thisoccurredwhenmarked statesbecameequatedwithunmarkedones.Spencer-Brownofferedaninterestinginterpretation.Ratherthantoviewthisasthesimultaneouspresenceofcontradictorystates,hesuggestedanalternative.Maybethesystemwasoscillatingbetweenoppositestatesintime.Ifso,thenself-reflexiveactsofreentry,orself-indication,wouldaddthedimensionoftimetothatofspacealreadyimpliedbyfirstdistinctions.Givenenoughtime,bothmarkedandunmarkedcanexistinthesamespace.

NeuroscientistandresearcherFranciscoVarelawasintriguedbySpencer-Brown’sideas,especiallybyhisexplanationforthedynamicsofreentry.Varela(1975;1979)developed“ACalculusofSelf-Reference”toextendSpencer-Brown’swork.Insodoing,hetookabold,ifnotradicalleap.Ratherthantoconceptualizereentryascharacterizinghigherdegreeequationsonly,Varelaproposedthatreentrybeaddedatthegroundfloor,asitsownterm,alongwiththeothertwomarkedandunmarkedstates.

Thissimpledifferencemadeallthedifference,asGregoryBatesonmighthavesaid.ItsignaledVarela’sdeparture fromAristotelian logic,whichhadheldan irongriparoundphilosophersandlogiciansformillennia.VarelaabandonedAristotle’sdichotomoussystem,whereallpropositionsareeitheronlytrueorfalse;itslawofidentity,whereAcanneverequalnot-A;aswellasitslawoftheexcludedmiddle,wherethespacebetweentruthandfalsityispristinelyempty.

Byaddingreentryasathirdterm,Varelaopenedupaninfinitelydeep,Pandora’sboxofmiddlegroundfilledwithfuzzygrays,lostidentity,andunfathomablecomplexity.Herenotonlycansomethingbetrueandfalsesimultaneously,butevenmore,Varelaactuallybelievedthattheexistenceofautonomyinnaturedependsuponthiscontradictorystateofaffairs.Varelaandhismentor,HumbertoMaturana,coinedtheterm“autopoeisis”toexplainhowbiologicalsystemsself-organize(Varela,MaturanaandUtribe,1974).Withreentrydynamicsatthecore,autopoeiticsystemsembodyparadoxattheirboundaries, expressing their autonomous functioning through remaining functionally closed, yetstructurallyopen.

Byassertingreentryasathirdvalueinitsownright,VarelaagreedwithSpencer-Brownthatself-referentialdynamicsestablishthepresenceoftime.Buthewentevenfurther,toassertthatparadox

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becomesembodiedatthemostbasiclevel,in the very form itself.Whetherinorganicorinorganicforms,autonomoussystemsappearsupportedbyinherentlycontradictoryunderpinnings.

IMAGINARy NUMBERS ANd HIddEN dIMENSIONS

The shortest path between two truths in the real domain passes through the complex domain. -Jacques Hadamard

Tovisualizehowparadoxbecomesembodiedintoform,ithelpstounderstandimaginarynumbers.WhenSpencer-Browninterpretedcontradictionasoscillation,helikenedhis‘discovery’oftimetothatofthecomplexnumberplane:

When Spencer-Brown introduces reentry and arrives at an expression equivalent to its content, f = f¦, what we call a self-cross, he notes its disconnection with his arithmetic and thus chooses to interpret it as an imaginary state in the form seen in time as an alteration of the two states of the form. This interpretation is, in my opinion, one of his most outstanding contributions. He succeeds in linking time and description in a most natural fashion. (Varela,1975, p. 20)

Thislinkagemaybeevendeeperthanmereanalogy.Imaginarynumbers,discoveredinthe17thcentury,bytheItalianmathematicianJeromeCardan,involvetwoorthogonalaxes,onerealandoneimaginary,andtheseeminglyimpossiblesquarerootof–1.Astheirnamesuggests,thesenumbersoriginallywereconsideredentirelyfanciful.Althoughtheykeptcroppingupinequationsoftheform,X2=-1,theywereoriginallybelievedtohavenopracticalapplications.

Yetasoftenoccurswitheventhemostfar-outseemingmathematicalabstractions,unexpectedpracticalusesareeventuallyfound,sometimesinthestrangestplaces.Forexample,non-Euclideangeometrywasoriginallythoughtimpracticalbecauseitseemstocontradicteverydayperception.Yetitsratherstrangegeometryperfectlymodelsthecurvedspace-timeofEinstein’sequations.

Suchhasbeenthecaseforimaginarynumbersaswell.Generally,withinmathematicalequations,thesquarerootof-1representsinvisible,extraorhiddendimensionality.Forexample,inelectricalengineering it models reversed polarities of alternating currents. In Einstein’s famous equation,E=mc2,thesquarerootof-1representsthe4thdimension–time–addedtothethreecustomarydimensionsofspace–length,width,depth.Imaginarynumbersalsocropupinquantummechanics,wheretheycapture(orfailtodoso)immeasurable,nonlocalaspectsofelectronbehaviorknownas“wavefunctions.”Generally,itappearsthatthoseartificiallyconstructed“fictions”calledimaginarynumberskeepemergingintheverysame,reflexiveplaceswheretheobserverisincreasinglydetectablewithintheobserved.

WhenEinsteinrevolutionizedphysicswithhisworkonrelativityintheearly20thcentury,mostlaypeoplefelthistheoriestobebeyondtheircomprehension.YetEinstein’sideasdidfindawaytofilterdowntoasegmentofthepopulation.Ratherthantointerpretthe4thdimensionnarrowlyastime,itbecamefashionabletoviewthisextradimensionmorebroadly,astheinteriorspace-timeofimaginationorevenconsciousnessitself.

Asmentioned,ataboutthesamepointinhistory,consciousnessbeganoccupyingcenterstageinquantummechanics,wherewhatwasfoundatthemicroscopiclevelseemedtodependentirelyonmethodsofobservation.TheCopenhageninterpretationassertsthatinordertocollapsethequantumwaveformfromastateofinfinitepotentiality,whereanythingispossible,tothatofmaterialreality,whereonlyonepossibilityisactualized,anobserverisrequired.

Herewehaveobserverspostulatingthenecessityoftheirownobservationstobringintoexistencethephenomenaunderobservation.Quantummechanicswasclosingalooponitself,by self-referentially implicating consciousness into the phenomenon under study. While an

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objectivist’snightmare, thezeitgeist seems tohavearrived for the recursive inclusionof theobserverwithintheobserved.

Insidetherealmofpsychology,aneophytefielddevotedentirelytothereflexiveenterpriseofconsciousnesscirclingbackaroundtoexamineitself,imaginarynumbersalsowerecomingintoplay.CarlJungconsideredtheconceptofnumberhighlysignificanttothepsyche(seeRobertson,2000).Infact,theSwisspsychiatristcametoviewnumberasthemostbasicarchetypeofall.Althoughunabletounfoldmanydetails,Jungconceivedofnumberasprovidingthebasisforaunifiedstratum,whichhetermedapsychoidlevel,whereinnerandouter,mindandmattermeetasone.

Mistakenly, Jung called imaginary numbers transcendent ones and named a correspondingpsychologicalfunction–thetranscendentfunction–afterthem.Junginterpretedrealandimaginaryaspectsofthecomplexplaneintermsoftheunionofconsciousandunconsciouscontentsofthepsyche.Thisunion,forJung,constitutestheessenceofindividuation,theprocessbywhichweeachcomemostfullyintoourselves.Duringindividuation,the“littleself”oftheconsciousegoalignsandbalanceswiththe“bigSelf”oftheunconsciouswhole:theknowncoincideswiththeunknown.

ThroughlensesofVarela’srecursivedynamics,theindividuationprocessitselfcanbeseenself-reflexivelyasalifelongcycleofreentry.Iandmycolleagues(Marks-Tarlow,Robertson&Combs,2002)havedescribedtheevolutionofconsciousnessintermsofsuchcycles.Psychologicalbirthisprecededbytheparadoxicalunionofoppositeswithintheunconscious,aswebeginourmentallifewithgood/bad,you/me,inside/outsidemeldedtogether.Inordertomakedistinctions,nascentconsciousnessmustseparateeachpolefromitsopposite.Thisincludesconsciousfromunconsciouselements.Andyet,completepsychologicalevolutionrequiresthatwebringtogetherorbalanceallopposites,includingconsciousandunconsciousrealmsintoacoincidentia oppositorium, a coincidence of opposites.

I(Marks-Tarlow,2008,2010)suggestthatallpsychologicaldevelopmentpossessesparadoxicalunderpinnings.Forexample,spiritualadvancementcanbecharacterizedastheuseofself-reflectiontoachieveincreasingobjectivity.Yettofollowapathtowards“objectivity,”istoattempttostepoutsideourselvesandundertakeanendeavorinwhich,bydefinition,wecan’tsucceed.Forwecanneverstepfullyoutsideourselves.Whatismore,mostspiritualpathsaimedtowardsobjectivityinvolveparadoxicallymovingintheexactoppositedirection,e.g.,usingmethodsofmeditation,todelveyetfurtherinsideourownsubjectivity.

Intermsofmathematicalmodeling,followersofJungandEinsteinplayedwiththeideathatimaginarynumberscanservetounifyinnerandouterworlds.Physicist,psychologistArnoldMindell(2000)makesasimilarassertion,believingallmathematicstobeacodeforlinkingobserverswiththeobserved.MetaphoricallyMindellbelievesimaginarynumbersrepresentqualitative,subjectiveaspectsofexperience,orwhathecalls“nonconsensualreality.”HisthinkingisinfluencedbyWolfgangPauli,aphysicistwhosedreamsinspiredJungtowritesomeofhismostimportantwork.PerhapsPauli’smostfamousdreaminvolvesaninnermusicteacherwhoconceptualizesimaginarynumbersasthekeytounifyingphysicsandpsychology(SeeWolf,1994).

I also speculate within the arena of imaginary numbers as a method for invisibly unifyingpreviously separated levels (Marks-Tarlow, 2004, 2012, in press). Fractals are cycles of reentryrecursivelyiteratedonthecomplexnumberplane.Theydescribequalitativeratherthanquantitativeaspectsofnature,suchastheruggednessofacoastlineorthejaggednessofamountainchain.Sinceimaginarynumbersmodelhiddendimensionality, in thecaseof fractals, thisconsistsof infiniteexpanseslurkinginthespacesbetweenordinary,Euclideandimensions.

Fractalsappeartoconnectaswellasseparateinnerandouterlevelsinatleasttwoways.Oneisbyofferingalexiconofcomplexity,whichappliesequallyaswelltonaturearoundusasitdoestohumannatureinsideus.TheabilityoffractalstodescribeandunifyinnerandouterworldshelpstofleshoutGregoryBateson’slargelyunformulatedintuition(1979)concerning‘anecessaryunity’ofmindandnature.Secondly,fractalseparatricescomprisecomplexboundaryzonesbetweenvariousrealmsofexistence,spanninglevelsfromthe“lowest”materialtothe“highest”spiritualrealms.

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FRACTAL GEOMETRy

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. -Benoit Mandelbrot

Computer-generatedfractalsprovidethemostsuccessfultooleverfoundforsimulatingnature.Tocreateamathematicalfractal,averysimpleformula,suchastheoriginalonediscoveredbytheFrenchengineerBenoitMandelbrot(1977),z→x2+c,isiteratedrecursivelyforeverypointofthecomplexplane.Thismeansthatthesameformulaispluggedinoverandover,withtheendproductsfedbackinasnewstartingpoints.Intheory,thiscontinuesindefinitely.Inpractice,itcontinuesuntileithertheequationsettlesdowntoastableoutcomeoranarbitrarypointisreachedindicatingacleardirection(seeFigure1).

Whenobservingafractalimagerenderedbycomputeriteration,threedistinctzonesareapparent.Thereistheinsideofafractal,whichinthecaseoftheMandelbrotset,isusuallyindicatedbyasolidblackzonerepresentingstability.Hereonthecomplexplane,theequationconvergestoorderintheformofperiodicity,bydisplayingoneormorestableend-points.Bycontrastthezoneoutsideofthefractal,symbolizedbyvariouscolorgradations,representsthepartofthecomplexplanewheretheverysameequationfliesoutofcontrol,movingtowardsinfinityatvariousspeeds.Thethirdandmostinterestingzoneconsistsoftheedgebetweenthesetwo.Thisistherealfractal,whichisinfinitelydeepwhenthecomputerisusedasamicroscopetozoominatsmallerandsmallerscales.Here,iterationneithersettlesdowntostabilitynorfliesofftoinfinity,butinsteadexhibitsunceasingcomplexityshiftingdynamicallywiththeperspectiveoftheobserver.

Whetherrenderedonacomputerorfoundinnature,thehallmarkofafractalisself-similarity.Self-similarityisanewlydiscovered,self-reflexivesymmetryinwhichpartofanynaturalfractalbearsresemblancetothewholething.Sometimesthisresemblanceisexact,asinthecaseoflinearfractals,liketheKochsnowflakeorCantordust.Othertimes,theresemblanceisapproximateorstatistical,asinthecaseoftheMandelbrotsetorothernonlinearfractals.Whereaslinearfractalstendtobeabstractandidealizedmathematicalforms,nonlineartendtoberealizedinnature.Clouds,mountains,coastlines,andotherbranchingshapesareallpartofthenaturallexiconoffractals.Withfractals,insteadofthewholebeinggreaterthanthesumofitsparts,thewholecanbedetectedwithinthepartatmultiplescalesofobservation.

Figure 1. Computer-generated fractals

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PARAdOXICAL BOUNdARIES

Whilemountainsand riversappear tobestable“things” toourWesternminds, theyare insteadcontinuallymovingprocessesthatevolvedynamicallyonvarioustimescales.Thefictionofstabilityand“thingness”isreifiedbytheEnglishlanguage,asdominatedbynounsacteduponbyverbs.ThedynamismoffractalsinnatureseemsmuchmoreconsistentwithseveralAmericanIndianlanguages,wherefeaturessuchaslightningandcoastlinesaredescribedasprocessesusingverbs.

Asembodiedinnature,fractalsoccupythecomplexinterfacebetweenchaoticforces,suchasthatbetweenwind,waterandheatcomprisingtheweather.Howeverslowlytheymaymove,clouds,riverbeds andmountains are all dynamicprocesses.They involve recursive cyclesof reentrybywhichtimegetsetchedfractallyintoself-similarform.Cloudsarefractalswhoserapidmovementprovidesaminute-to-minuteaccountofwindandweatherconditions.Riversembodyminute-to-minutedynamics,butalsogloballyoccupyan intermediate time frameofevolution: thepresentriverbedisbutthelatestinaseriesofconfigurationswhoseevolutioncanbetracedonthecanyonwalls.Mountainsandhillshavetheentirehistoryofcreeks,rivers,windandotherweather-relatedforcesetchedintotheirrocksurfacesovereons.

Fractaldynamicsalsopervadeourbodies(e.g.,IannacconeandKhokha,1996).Elsewhere(Marks-Tarlow,2002,2008)Isuggest that fractalscomprisezonesofcommunicationand transportationbetweenvariousorgansandsubsystemsofthebody,aswellasbetweenthebodyandtheoutsideworld.Bloodcirculatesthroughoutthebodyinthefractalbranchingofarteriesandveins.Thelungscycleoxygeninandcarbondioxideoutthroughfractalbronchioles.Eventheneuralpathwaysinthebrain,ourmainorganforperceptionandcommunication,arefractal.

Skinpores,wrinklesandothermarkingsonthe“sacks”inwhichhumansandotheranimalsareenclosedarefractallydistributed.Self-similardynamicsalsopervadepsychophysics,bywhichphysical stimulioutsideourbodiesare transmuted toperceptionandsensation inside. Inhumanphysiologygenerally,asinnaturebroadly,fractalsserveparadoxicalfunctionsasboundarykeepers,bothtoseparateandconnectvarioussubsystemsandlevelsofbeing.

Elsewhere(Marks-Tarlow,1999;2002,2008,2011)Ialsodescribetheexistenceoffractalsatamoreabstract,symboliclevelinthepsyche.Intheparadoxicalspacebetweenselfandother,fractaldynamicsariseinhumanbehavior,aspersonalitytraitsmanifestinself-similarpatternsevidentatmultiplescalesofexistence.Forexample,aggressivepeoplemaypushothersasideverballywithovertalk,pushaheadofothersphysicallyinlinesatthebank,whilepushingtheirneedsaheadofothers’insignificantrelationships.Frommicrotomacrolevels,eachinstanceoccursself-similarlyatdifferentscalesofsocialobservation.Fromthisbriefdescriptionoffractalsastheyoccurnaturally,itbecomesclearthatself-similardynamicsspanthefullrangeofexistence,fromthemostconcrete,materiallevelstothemosthighlyabstractandpsychologicalones.

Attheabstractlevel,self-similardynamicsarealsoevidentinthelogicofCharlesPeirce(e.g.,1998).Withinhistriadicsystemandthesemioticsbaseduponit,thefirstistheobject;thesecondistheinterpretant;andthethirdisthesign.Thesignexpressesageneralorabstractrelationbetweenthefirstandsecond,andcarriesthemeaningofthisrelationship.Becauseeachsigncanitselfbethefirstforothersigns,withinunlimited semiosis,thissystemoperatesasaninfinitelyrecursivenetworkofre-entrantloops,eachreflectingbackupontheothersinself-similarfashion.

SørenBrier(2010)describedanontologicalframeworkofcybersemioticsinordertounderstandthe emergence of form, life, mind and consciousness. Nature is viewed in terms of multilevel,multidimensionalhierarchiesofinter-relatedclustersthatformametaheterarchy,orheterogeneousgeneralhierarchy.Asnewlevelsemergeviarelatedprocesses,thereisself-similarityacrosslevelsbywhichgeneral, translevelprinciplescanbeabstractedandplaced intoaPeirceanperspective.Potentialities(firstness)arecausallymanifestedasconstraintsandforces(secondness),whilemeaningiscreatedbyintegratingmanifestationwithpotentialityintoregularitiesandpatterns(thirdness).The

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processcontinuesrecursively,witheachnew,emergentlevelservingaspotentialforthedevelopmentofthenext.

Ingeneral,self-similardynamicsappearinthejoints,inthespacebetweenlevels,relationsanddimensions.Theysupplyboundariesthatareinfinitelydeepandparadoxical.IproposethatfractalscompriseVarela’s(1979)realmof“autonomousfunctioning,”whichoperateswithbordersthatarefunctionallyclosedwhileremainingstructurallyopen.

FRACTAL dIMENTIONALITy

The imaginary numbers are a wonderful flight of God’s spirit; they are almost an amphibian between being and not being. -Leibniz

Astheyoccuraroundandinsideourbodies,fractalsoccupythecomplicatedinterfacebetweenvariousforcesinnature.Astheyoccurinthepsyche,fractalsariseoutofendlessfeedbackloopsbetweenselfandotherinrelationships.Oursenseofself,onceagainparadoxicalinbeingbothopenandclosed,arisesdynamicallyinthetransitionalspace betweenpeople.

Itappearsthateverywheretheyarise,fractalsoccupytheboundaryzonebetween.Thiscanbeilluminated with a technical understanding of fractal dimensionality, as mathematically, fractalsresideintheinfinitelycomplexspacebetween ordinary Euclidean dimensions.Cloudsarecomposedofwatervapor,zerodimensionalpointsthatoccupythree-dimensionalspace.Acoastlineisaone-dimensionallinethatoccupiesatwo-dimensionalplane.Mountainsaretwo-dimensionalsurfacesdrapingathree-dimensionalworld.Quaternionsareproductsofthehypercomplexplaneconsistingofonerealandthreeimaginaryaxes.Becausetheyarethree-dimensionalshadowsoffour-dimensionalspace,ifthefourthdimensionreallydoesrelatetoconsciousness,quaternionsmayprovidesomecluestotheinternallandscapeofhigherdimensionalthought.

To calculate fractal or Hausdorf dimension (one of many possible ways to measure fractaldimensionality)alog/logrelationshipestimatestherateatwhichmoreinformationbecomesavailabletousasweshrinkthesizeofourmeasuringdevice.Logarithmsinvolveexponentsofthepowerstowhichabasenumbermustberaisedtoequalagivennumber.Forexample,thelogarithmof100inbase10is2.WecangetafeelforthenonlinearityoflogarithmsbyconsideringtheRichterscale,alogarithmicscalemeasuringthepowerofearthquakes.Onanylogarithmicscale,thedistancefromtheorigintoanymarkisproportionaltothelogarithmofthenumberattachedtothatmark.OntheRichterscale,anearthquakeofmagnitude8.0isnottwiceaspowerfulasonemeasuring4.0,whichwouldindicatealinearrelationshipbetweenmarks.Instead,amagnitudeof8.0is10x10x10x10timesmorepowerfulthanoneof4.0.

Thelog/loggraphusedtocalculatefractalHausdorfdimensionplotsonelogarithmicrelationshipagainstanother–theratebywhichmoreinformationbecomesavailableasweshrinkthesizeofourmeasuringdevice.Inthelog/logrelationshipsoffractaldimensionality,akindofmagicbecomesevident: an unexpected relationship exists between the observer and the observed, i.e., betweenthemeasuringdeviceweuseandthatwhichwemeasure.Withfractalobjects,becausethissamerelationshipholdsateveryscaleofmeasurement,wewindupwithself-similarpattern.

Withmathematicalfractals,thesmallerourmeasuringsticks,thelargerthemeasurement.Becausethequantityofmeasurecontinuallyaltersrelativetothesizeoftheyardstick,wecanseethatonadeeplevel,fractaldimensionisnotameasurementofquantityasweusuallyconceiveit.Instead,itmeasuresthequality of relations between observer and observed.Thisiswhatremainsconstantwithfractalsaswechangescales.Thisisalsowhyfractalpatternturnsouttobescaleinvariant.

Withfractals,stickingtoquantitativemeasurementonlyleadstoparadox,preciselybecausethesmallerourmeasuringdevice,thelongerthemeasurement.Atthelimitofaninfinitelysmalldevice,wewillobtainaninfinitelylargemeasurement.ThisledMandelbrot’s(1977)tohisfamousconclusion

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thatthelengthofanystretchofcoastlineisinfinitelylong.Asoftenhappensinscience,paradoxisavertedthroughnewdiscovery.Inthiscase,expandingthetraditionalnotionofdimensionalityintothefractionalembodiesparadoxasitredefinesnewrelationsbetweenobserversandtheobserved.

THE OBSERVER IN THE OBSERVEd

In the characterization of organizational closure, nothing prevents the observer himself from being part of the process of specifying the system, not only by describing it, but by being one link in the network of process that defines the system. -Francisco Varela

Whetherexistingabstractly,inthephasespaceunderneathchaos,inphysicalormaterialformorpsychologically,fractalsimplicatetheobserverintheobserved.Thisisapparentinthenotionoffractalseparatrices,whereinsideandoutside,selfandother,spiritandmatterareparadoxicallyseparatedthroughfractalboundaries,yethopelesslyentwined.Intheprevioussection,wealsosawhowtheobserverlieshiddenintheobservedasbuiltrightintotheveryconceptoffractaldimensionality.Thisgivesusaclueaboutreflexivityevenatthemostprimitivelevelsofmacroscopicexistence.

Oneverylevel,recursivecyclesofreentryseemtoetchtemporaldynamicsintospatialform.Varela(1975)suggestedthatself-referenceis“thehingeuponwhichlevelsofserialinclusivenessintercross.”

Itappearsasifdifferent,successivelylargerlevelsareconnectedandintercrossatthepointwheretheconstituentsofthenewlowerlevelrefertothemselves,whereantinomic[contradictory]formsappear,andtimesetsin.Werecognizethisfactinordinaryspeech.Whentryingtoconveyadescriptionofanewdomainweoftenconstructanapparentantinomytoinducethelistener’scognitioninawaysuchastocompelhisimaginationtowardstheconstructionofalargerdomainwheretheapparentoppositescanexistinunity.(Amoralexample:onceyouloseeverything,youhaveeverything;aphilosophicalone:abeingiswhenitceasestobe)(p.22).

Itiseasytoviewtheuniverseintermsofsuccessivelylargerlevelsofserialinclusiveness.Asacrudeexample,physicsisembeddedwithinchemistry,whichisembeddedwithinbiology,whichisembeddedwithinpsychology.Grim,MarandSt.Denis(1998)havepioneeredtheuseoffractalgeometrytoenvisionformalsystems,includingtherepresentationofparadoxandincompleteness.

I propose fractal dynamics to be one way to conceptualize how Varela’s levels of serialinclusiveness connect and intercross. Whether comprising attractors beneath chaotic forces innature,includinghumannature,whetherexistinginphysicalorsymbolicform,fractalsnegotiatetheboundaryzone,theplacewherelevelscontaintheantinomyofoppositesbothunifiedandseparated.Becausefractalsspanalllevels,separatingandconnectingthematboundarypoints,thesedynamicsmighthelptobridgetraditionallevelsofanalysis,fromthepurelyphysicallevelofPrigogine’sfar-from-equilibriumthermodynamics, throughbiologicallevelsofreentryinMaturanaandVarela’sautopoieticsystems, to thehigherordersocialandcultural levels impliedbyPeirce’sconceptofhistoricaldriftandpraxis.

Theironyisthis.Whilewemayuse‘unnatural’ormechanicalmeans,suchascomputers,todiscoverandilluminatetheworkingsoffractals,whatweilluminateextendsbeyondandbeneaththepostmodernmirrorsthatreflectinfiniteregressioninformsofhumanproductionandreproduction.Byilluminatingfractals,weself-reflexivelyilluminatetheobserverintheobservedinnothingshortofnatureherself.

This insightdovetailswith the frequent religiousvisionofallofcreationasaself-reflexiveexpressionofselfintoformtoknow,shareandlovinglyreunitewithSelf.Italsodovetailswiththeposition of neurological positivism espoused by philosopher and psychologist Larry Vandervert(1990).Vandervertclaims thatevolution itselfproceedsself-reflexively,culminating in themostcomplexobjectintheknownuniverse,thehumanbrain.Asweuseourbrainsself-reflexivelyto

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examineourselves,ourmodelsgrowmoresophisticatedandeverclosertothefractalstuffofwhichourminds,brainsandnatureatlargeiscomposed.

CONCLUSION

With fractals as a new metaphor of mind, we no longer need to deify or defile mechanism asmetaphor.Insteadwecanusemechanismasatool.Aswecontinuetousetherecursiveformulasofchaos,complexity,fractalgeometryandcellularautomatatosimulatethenaturalworld,weseeacrossmultiple,nestedlevelsofserialinclusiveness,thatself-similarfractaldynamicsrecurasameta-pattern.Cyclesofreentrycontinuallyoscillatebetweencreatinganderasingtheseamwhereobserverandobserved,perceiverandperceived,innerandouter,selfandother,intersectandself-crossparadoxically.Atthisseam,selfandworldappearmutuallyco-determining.Thisiswheretheactofmakingadistinctioncreatestheworldasweperceiveit,andwheretheworldbringsdistinctiontotheconsciousnessoftheperceiver.

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REFERENCES

Atwood,G.,&Stolorow,R.(1979/1993).Faces in a cloud: Intersubjectivity in personality theory.Northvale,NewJersey:JasonAronson.

Babiloni,F.,&Astolfi,L.(2014).Socialneuroscienceandhyperscanningtechniques:Past,presentandfuture.Neuroscience and Biobehavioral Reviews,44,76–93.doi:10.1016/j.neubiorev.2012.07.006PMID:22917915

Bateson,G.(1979).Mind and nature: A necessary unity.NewYork:BantamBooks.

Brier,S.(2010).Cybersemiotics:Anevolutionaryworldviewgoingbeyondentropyandinformationintothequestionofmeaning.Entropy (Basel, Switzerland),12(8),1902–1920.doi:10.3390/e12081902

Draaisma,D.(1995/2000).Metaphors of memory: A history of ideas about the mind.Cambridge,UK:CambridgeUniversityPress.

Dumas,G.,Lacat,F.,Martinerie,J.,Nadel,J.,&George,N.(2011).Fromsocialbehaviortobrainsynchronization:Reviewandperspectivesinhyperscanning.IRBM,32(1),48–53.doi:10.1016/j.irbm.2011.01.002

Grim,P.,Mar,G.,&St.Denis,P.(1998).The philosophical computer: Exploratory essays in philosophical computer modeling.Cambridge,MA:MITPress.

Hayles, K. (1999). How we became posthuman: Virtual bodies in cybernetics, literature, and informatics.Chicago,IL:UniversityofChicagoPress.doi:10.7208/chicago/9780226321394.001.0001

Heims,S.(1991).The cybernetics group.Cambridge,MA:TheMITPress.

Iannaccone,P.,&Khokha,M.(Eds.).(1996).Fractal geometry in biological systems.NewYork:CRCPress.

Mandelbrot,B.(1977).The fractal geometry of nature.NewYork:W.H.Freeman.

Marks-Tarlow,T.(1999).Theselfasadynamicalsystem.Nonlinear Dynamics Psychology and Life Sciences,3(4),311–345.doi:10.1023/A:1021958829905

Marks-Tarlow, T. (2002). Fractal dynamics of the psyche. In B. Goertzel & A. Combs (Eds.), Dynamical Psychology, an International, Interdisciplinary E-Journal of Complex Mental Affairs.

Marks-Tarlow,T.(2004).Semioticseams:Fractaldynamicsofreentry.Cybernetics & Human Knowing,11(1),49–62.

Marks-Tarlow,T.(2008).Psyche’s veil: Psychotherapy, fractals & complexity.NewYork,NY:Routledge.

Marks-Tarlow,T.(2010).Thefractalselfatplay.American Journal of Play,3(1),31–62.

Marks-Tarlow,T.(2011).Mergingandemerging:Anonlinearportraitofintersubjectivityduringpsychotherapy.Psychoanalytic Dialogues,21(1),110–127.doi:10.1080/10481885.2011.545333

Marks-Tarlow,T.(2012).Fractalgeometryasabridgebetweenrealms.InF.Orsucci&N.Sala(Eds.),Complexity science, living systems, and reflexing interfaces: New models and perspectives(pp.25–43).Hauppauge,NY:NovaScience.

Marks-Tarlow, T. (in press). A fractal epistemology for transpersonal psychology. International Journal of Transpersonal Psychology.

Marks-Tarlow,T.,Robertson,R.,&Combs,A.(2002).VarelaandtheUroboros:Thepsychologicalsignificanceofreentry.Cybernetics & Human Knowing,9(2),31–47.

Mindell,A.(2000).Quantum mind: The edge between physics and psychology.Portland,OR:LaoTsePress.

Northoff,G.,&Heinzel,A.(2006).First-personneuroscience:Anewmethodologicalapproachfor linkingmentalandneuronalstates.Philosophy, Ethics, and Humanities in Medicine; PEHM,1(1),E3.doi:10.1186/1747-5341-1-3PMID:16759399

Peirce,C.S.(1998).The essential Peirce: Selected philosophical writings.Bloomington,IL:IndianaUniversityPress.

International Journal of Semiotics and Visual RhetoricVolume 2 • Issue 1 • January-June 2018

14

Prigogine,I.,&Stengers,I.(1984).Order out of chaos: Man’s new dialogue with nature.Toronto:BantamBooks.

Robertson,R. (1999).Some-thingfromno-thing:G.Spencer-brown’s lawsof form.Cybernetics & Human Knowing,6(4),43–55.

Robertson,R. (2000).Theevolutionof Jung’sarchetypal reality.Psychological Perspectives,41(1),66–80.doi:10.1080/00332920008403393

Schroeder,M.(1991).Fractals, chaos, power laws.NewYork:Freeman.

Spencer-Brown,G.(1969).Laws of form.London:AllenandUnwin.

Spencer-Brown,G.(1979).Laws of form(rev.ed.).NewYork:E.P.Dutton.

Taylor,S.(1989).Positive illusions: Creative self-deception and the healthy mind.NewYork:BasicBooks.

Vandervert,L.(1990).Systemsthinkingandneurologicalpositivism:Furtherelucidationsandimplications.Systems Research,7(1).doi:10.1002/sres.3850070101

Varela, F. (1975). A calculus for self-reference. International Journal of General Systems, 2(1), 5–24.doi:10.1080/03081077508960828

Varela,F.(1979).Principles of biological autonomy.NewYork:NorthHolland.

Varela, F., Maturana, H. R., & Uribe, R. (1974). Autopoiesis: The organization of living systems, itscharacterizationandamodel.Bio Systems,5(4),187–196.doi:10.1016/0303-2647(74)90031-8PMID:4407425

Wolf,F.(1994).The dreaming universe: A mind expanding journey into the realm where psyche and physics meet.NewYork:SimonandSchuster.