dissemination_ introducing the proemial relationship
DESCRIPTION
artículo-proTRANSCRIPT
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7/7/2015 Dissemination:Introducingtheproemialrelationship
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ThinkArtLabHogmanay2004
ThinkArtLabAnimation:A.T.Kelemen
November12,1998Dr.RudolfKaehr
Dissemination:Introducingtheproemialrelationship
Therearemanywaysofcombiningabstractobjectsorinstitutions.Forexample,giventwoinstitutionsINS1andINS2which,intuitively,areindependentwecanformtheirproduct.ThisproductinstitutionhasallpairsofsignaturesfromINS1andINS2,respectively,asmodels,andsentenceswhichareeithersentencesfromINS1orfromINS2withtheobvioussatisfactionrelation."Cat.,p.357
Itisshown,thatthecategoryofinstitutionsiscomplete.
Theideaofdisseminationtriestoexplicateandformalizeaquitedifferentintuitionofcombininginstitutionswhichisnotproducingdiversityandmultiplicitybycombiningabasicsystemasaproductorsumorwhateverconstructionbutintroducesmultipledifferencesintheveryconceptofthebasicsystemitself.Afterthisconstructionapolylogicalorpolycontexturalsystemcanbecombinedinmanyways.Thisideaofmultitudesofbasicdifferencesinthenotionofformality,takenseriously,isinfundamentalcontrasttotheexistingconceptsofformalityinmathematics.Obviosly,thesemultitudesaremorefundamentalthanalltypesofmanysortedtheories,ortypedlogics,orpluralitiesofregionalontologies,domainsandcontexts.
1TheideaofproemialityAveryfirststepinthisdirectionwasmadebythephilosopher
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GotthardGuntherwithhisideaofaproemialrelationship"introducedinhispaperCognitionandVolition"(1970)aboutaCyberneticTheoryofSubjectivity.
Inordertoobtainageneralformulafortheconnectionbetweencognitionandvolitionwewillhavetoaskafinalquestion.Itis:Howcouldthedistinctionbetweenformandcontentbereflectedinanysortoflogicalalgorithmiftheclassictraditionoflogicinsiststhatinalllogicalrelationsthatareusedinabstractcalculithedivisionbetweenformandcontentisabsolute?Theansweris:wehavetointroduceanoperator(notadmissibleinclassiclogic)whichexchangesformandcontent.Inordertodosowehavetodistinguishclearlybetweenthreebasicconcepts.Wemustnotconfuse
arelation
arelationship(therelator)
therelatum.
Therelataaretheentitieswhichareconnectedbyarelationship,therelator,andthetotalofarelationshipandtherelataformsarelation.Thelatterconsequentlyincludesboth,arelatorandtherelata.
However,ifwelettherelatorassumetheplaceofarelatumtheexchangeisnotmutual.Therelatormaybecomearelatum,notintherelationforwhichitformerlyestablishedtherelationship,butonlyrelativetoarelationshipofhigherorder.Andviceversatherelatummaybecomearelator,notwithintherelationinwhichithasfiguredasarelationalmemberorrelatumbutonlyrelativetorelataoflowerorder.
If:
Ri+1(xi,yi)isgivenandtherelaturn(xory)becomesarelator,weobtain
Ri(xi1,yi1)whereRi=xioryi.Butiftherelatorbecomesarelatum,weobtain
Ri+2(xi+1,yi+1)whereRi+1=xi+1oryi+1.Thesubscriptisignifieshigheror
lowerlogicalorders.
Weshallcallthisconnectionbetweenrelatorandrelatumthe'proemial'relationship,forit'prefaces'thesymmetricalexchangerelationandtheorderedrelationandforms,asweshallsee,their
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commonbasis."
Neitherexchangenororderedrelationwouldbeconceivabletousunlessoursubjectivitycouldestablisharelationshipbetweenarelatoringeneralandanindividualrelatum.Thustheproemialrelationshipprovidesadeeperfoundationoflogicasanabstractpotentialfromwhichtheclassicrelationsofsymmetricalexchangeandproportionedorderemerge.
Itdoesso,becausetheproemialrelationshipconstitutesrelationassuchitdefinesthedifferencebetweenrelationandunityor,whichisthesamebetweenadistinctionandwhatisdistinguished,whichisagainthesameasthedifferencebetweensubjectandobject.
Itshouldbeclearfromwhathasbeensaidthattheproemialrelationshipcrossesthedistinctionbetweenformandmatter,itrelativizestheirdifferencewhatismatter(content)maybecomeform,andwhatisformmaybereducedtothestatusofmeremateriality"."
Westatedthattheproemialrelationshippresentsitselfasaninterlockingmechanismofexchangeandorder.Thisgaveustheopportunitytolookatitinadoubleway.Wecaneithersaythatproemialityisanexchangefoundedonorderbutsincetheorderisonlyconstitutedbythefactthattheexchangeeithertransportsarelator(asrelatum)toacontextofhigherlogicalcomplexitiesordemotesarelatumtoalowerlevel,wecanalsodefineproemialityasanorderedrelationonthebaseofanexchange.Ifweapplythattotherelationwhichasystemofsubjectivityhaswithitsenvironmentwemaysaythatcognitionandvolitionareforasubjectexchangeableattitudestoestablishcontactbutalsokeepdistancefromtheworldintowhichitisborn.Buttheexchangeisnotadirectone.
Ifweswitchinthesummerfromoursnowskistowaterskisandinthenextwinterbacktosnowskis,thisisadirectexchange.Buttheswitchintheproemialrelationshipalwaysinvolvesnottworelatabutfour!"Gunther
1.1Someexplanationsoftheideaofproemiality
Theproemialrelationshipisthereforeatfirstaninterlockingmechanismofthetwoconceptsofexchangeandorderorsymmetryandasymmetry.
Diagramm1
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7/7/2015 Dissemination:Introducingtheproemialrelationship
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cascadicrepresentation
Afurtherexplicationoftheintuitionofproemialityisachievedifweconsiderthefactthattheobjects,therelatorandtherelataoftherelations,havetofittogetherinacategoricalsense.Thereisasimilarityoftherelatorsofdifferentlevelsaswellasfortherelataofdifferentlevelsinthesensethatthedifferentrelatorsarerelatorsandnotsomethingelse.Andtherelataoneachlevelarerelataandnotrelators.ForthatIintroducethecoincidencerelation,whichdesignatescategoricalsameness(likeness,similtude).
TofinishthepictureIintroducetheexchangerelationbetweenthefirst"andthelast"elementoftheinterlockingmechanismoforderandexchangerelations.AsalaststepImentiontheposition,thelogicallocus,oftheorderrelationsaccordingtothehigherorlowerlogicalorders".
PrObj=(ObjOrd,Exch,Coin,Pos)
Diagramm2
Butthisexplanationstillexcludesthethirdtermofthedefinitionofarelation,therelationitself.Remember:Wemustnotconfusearelation,arelationship(therelator),therelatum.
AndfinallyIconsiderthefactthatthereisoneandonlyoneconceptofrelationandrelationalityunderconsideration.thereforetheconceptofrelationisbasedonunicity(uniqueness),representedby1.Thisissurelynotaharmlessstatement,itsupposesomethinglikeacommonintuitionofrelationalityoroperativitywhichfindsitselfexplainedandformalizedinsomemathematicalconstructionswhichareacceptedbythescientificcommunity.Therefore,Guntherschain"arelation,arelationship(therelator),therelatum"hastobecompletedbytheveryconceptofrelation,thatis,relationalitybasedin
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7/7/2015 Dissemination:Introducingtheproemialrelationship
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unicity.
Thefullfledgedexplanation,withoutthearrow"relation>relationality",oftheproemialrelationovertwolociisgivenbyitsconceptualgraph.Thescenarioisthesameforthedistributionandmediationofotherconcepts,likeoperations,functions,categories,institutionsetc.
Thusthedefinitionhastobeexpandedto:
PrObj=(ObjOrd,Exch,Coin,Pos)
withObj={relator,relatum,relation,relationality,unicity}
Inthiscontextitisnotmytasktodefendthisconstructionagainstthemanyattemptstoreduceittosomethingelse.TogofurtherinthegameImaketheoptionthatitwillbeusefulfordevelopingsomenewmechanismsofcombiningabstractobjectslikeinstitutions,logics,arithmetics,categorytheoriesandmore.Inexercisingthisgamethenewintuitionwillshapeitselfintoamoreacademicform.
Afterhavingintroducedtheideaofproemialityitwouldbepossibletoformalizeitfurtherandtodevelopapreliminarytheoryofproemiality,alsosometimescalledchiasticsortheoryofmediation.Themainthesis,therefore,isthatproemialityoffersamechanismofcombininginstitutionswhichdoesn'tbelongtotheuniverseofcombiningcategories.Thismechanismofcombininginstitutions,e.g.distributionandmediation,isfundamentallydifferentfromtheclassicalones.Despiteofthisdifferencethisstrategyisinnocontradictionoroppositiontotheknownprinciplesofcombiningsystemsoflogics.
Itissimplysomethingdifferentandtheclouwouldbetoexplainthisdifferenceinfull.Dontconfusetheexchangeofrelatorandrelatumofarelationinthemechanismoftheproemialrelationshipwiththesuperpositionofrelatorandrelationinrelationallogics.Thereisnoproblemtoapplyarelator,oraoperatororafunctortotheresultofarelationoroperationorfunctionase.g.inrecursiontheoryorinmetalevelhierarchies.
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7/7/2015 Dissemination:Introducingtheproemialrelationship
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Metaphor
Ifweproemializethelinguisticsubjectobjectrelationofasentenceweshouldn'thesitatetobestrictlystructural.TheexampleisborrowedfromHeinzvonFoerster.
"Thehorseisgallopping"(DasPferdgallopiert),theinterchangedsentencecanonlybe"Thegallopishorsing"(DerGalloppferdet).
Nobodysupposedthatwearedoinganalyticphilosophy.
1.2ProemialityandArchitectonics
1.2.1Abouttheascategoryinproemiality
WhatIhavedevelopedsofarisonlythehalfofthestory.Alsoitmightbeobviousthatthewordingofe.g."theoperator(ofonesystem)becomesanoperator(ofanothersystem)"isinstrongconflictwiththeidentityofitstermsthereforethissituationneedsamorepreciseexplication.Itshouldbeclearthatatermwhichisinonesystemanoperatorandsimultaneouslyanoperandinanothersystemsissplitinitsownidentity.Itisatonceitselfandsomethingelse.Thistermhasatoncetwofunctions,tobeanoperatorandtobeanoperand.Therefore,fromthepointofviewofidentityanditslogic,thistermisinitselfneitheranoperatornoranoperand.
Whatthenisit?Howcanwedefineitmoreaccurate?Itispartofanchiasticinterplayandwehavetobemoreexplicitwithourwording.Insteadofspeakingofan"operator"orofan"operand",weshouldusetheascategoryandusethewording"anobjectXasanobjectYisanobjectZ".Thus,anoperatorasanoperatorissimultaneouslyanoperand.
Anoperatorasanoperandisanoperand(ofanotheroperator)
1.2.2Aboutthearchitectonicsoftheascategory
TomakethiswordingmorepreciseIintroduceadiagramwhichiswellknownfromthetableauxmethodofformalizedpolycontexturallogic.
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Thistypeofdiagramswasfirstintroducedtodealinaproperwaywiththetableauxmethodinpolycontexturallogic.Especiallytounderstandthefunctioning,andthisgivesprobablyalsosomelightonitsmeaning,ofthesocalledtransjunctions,Iintroducedthistabulationofthestepwisedecompositionofsignedformulasintableauxproofs.Transjunctionshavereachedindifferentscientificandartisticareassomedegreeofacceptanceandarewidelyusedasimportantmechanismofsubversivethinkingandmodeling.Alsothenumberofoccurrenceofthisterminliteratureisquiteimpressivethereisnotmuchscientificunderstandingtofind.
Transjunctionsarelogicalfunctionsoroperatorswhichareinvolvedinsomesortsofbifurcationsandaresplitintodifferentpartsbelongingatoncetodifferentlogicalsystems.Theyarethereforecomposedofpartialfunctionsincontrasttothetotalfunctionsofclassicallogicaljunctionslikeconjunction,disjunction,implicationandsoon.
Thischangeoflogicalsystembybifurcationwhichpresupposethedifferenceofaninsideandanoutsideofalogicalsystemisruledbytheproemialrelationbetweenthepartsofthetransjunctionandthedifferentlogicalsystemsinvolved.Tothestepwisedecompositionofatransjunctionalformulacorrespondsanorderrelation,tothebifucationtoothersystemstheexchangerelationbecauseofitsinside/outsidedifference,andtothecomponentsandthestepsofdecompositionofthetransjunctionalformulaasawholetherelationofcoincidence.Therefore,theoperationoftransjunctioncanbeunderstoodasaproemialobject.
Thisdiagramwhichgivessomefirststepsinthedesignofpolycontexturalarchitectonicscannowbeusedforfurtherexplicationsofthemechanismofproemiality.
Theexchangebetweenoperatorandoperandhastobedescribedsimultaneouslyfrombothpositions.Thatiswhywehavetorealizeadoubledescription,adoublegestureofinscribingtheproemialityoftheconstellation.Tovisualizethisprozedurewehavetorealizeadoubledescriptionofthediagram
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Thefirstdiagramsarecorrectinsofarastheydescribethestructureofproemiality.Butatthesametimetheyareabbreviationsinsofarastheprocessofreadingthem,thatistoreadthematoncefrombothsides,isnotinscribed.Thisprocessofreadinghastobedonebyareader.Butwehavetomakeitexplicitandtovisualizeit.Therefore,evenifitseemstobeobvious,ithastoberealizedandnotonlybementioned.Thenewdiagramisfocussingmoretheprocessofproemialitythanonitsgeneralstructure.TonottooverloadtheschemeIreducedittothedistributionoftheIF/THENrelation.Maybewithallthatinmindwearenowreachingslowlythefamousproemialcube.
Diagramm3
Theproemialcube
Again,thegreendoublearrowrepresentstheexchangerelation,theredlinethecoincidencerelation,theblackarrowtheorderrelation,and,new,thebluelinerepresentsthedistributionofthetwoproemialrelationsinacommonarchitecture.
Idontcommentthefullcombinatoricsbetweenallknotsofthediagram.Also,Iwouldliketoleavethestudyoffurtherdimensionsofvisualizationsandtheirexplanationsasaninterestingjobtotheprogrammers.InthistextDERRIDASMACHINESIwillreducemypresentationtothegraphicallymoresimplecaseofthevisualizationofthestructureoftheconceptofproemialityanditsapplications,thatis,tothetwodimensionaldiamonddiagraminsteadofthecube.
1.3ProemialityandHeterarchyinaUMLFramework
TogiveamoretransparentmodelingoftheproemialrelationshipitmaybehelpfultosetthewholeconstructionandwordingintoanUMLdiagramandtousethemodelingofheterarchyworkedoutbyEdwardLeeasahelpfultooltoexplicateproemialityintermsofUMLmodeling.
AlsotheproemialrelationshipisnotrestrictedtoontologyandthedistributionofhierarchicalontologiesinaheterarchicframeworkanddespitethefactthatUMLhasnomechanismsofcategorychange,metamorphosisandmediationitseemstobeahelpfulexercisetofindacorrespondencebetweentheUMLheterarchydiagramandthe
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constructionofproemialitywhichismorebasedonelementarytermsofrelationality.Theheterarchydiagramisaclassdiagramwhichmodelsthestaticstructureofthesystem.Proemialityhas,alsoitisfundamentallydynamic,itsstaticaspects.ItisthisstaticaspectwecanmodelwiththehelpoftheUMLheterarchydiagram.
AfurtherstepofUMLmodelingofproemialitywillhavetoinvolvemoredynamicmodelslikeinteractionandactivitydiagrams.
Diagramm4
UMLdiagramofheterarchyin:EdwardA.Lee,OrthogonalizingtheIssues,UCBerkeley
WhatisthedifferenceinmodellingbetweenconceptualgraphsandUMLdiagrams?
Aconceptualgraphof
theUMLheterarchy
diagram.
1.4Complementarityofdisseminationandtogetherness
Complementarytothenotionandprocedureofdissemination,whichismotivatedbythenecessityofconstructingcomplexandpolycontexturalsystemsoutofsimpleones,thatis,monocontextural
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systems,wehavetoconsiderthepolycontexturalityofthecomplexsystemassuch.Onefirstcategoryweobserveisthecategoryoftogethernessofthelocalsystemsinthecomplexandinteractingwholeness.
Anothercategorythatemergesnaturallyoutofthedisseminatedsystemsisthecategoryofwholenessormoreprecisethecategoryofsuperadditivityofdisseminatedsystems.
Inthissense,disseminationisaprocessofdisseminatingsinglesystemsandatthesametimeitisthewholeness,thetogethernessofthedisseminatedsystems.Thisisalsoincludedinthenotionofdisseminationasaprocessofdistributionandmediationofsystems.Disseminationisalwaysboth:multitudeandwholeness.
2Combinatoricsofchiasticchangesofcategories
2.1ConservativemappingsorCategorytheoreticcombinations
Ifthecontexturaldifferencesbetweentwoobjectsaredeniedwecanmodeltherelationshipbetweenthemintermsofmorphismsinthecategorytheoreticsense.Thesemorphismsarethestructurepreservingmappingsofnamestonames,sortstosorts,operationstooperations,equalitiestoequalities,andunitytounity.,etc.oftheabstractobjects.Butagain,inthiscaseweareneglectingthefact,thattheybelongtodifferentlogicalcontextures.
Ontheotherhand,ifwetaketheircontexturaldifferencesintoconsideration,thesemappingsarepreservingthetectonicalstructureofthesystems,despitetheirlogicalincompatibility.Intermsofproemialitythesemappingsarenotofthesortoforderrelations,likemorphisms,butofthesortofcoincindencerelation.Inacategorytheoreticalmodeltheywouldbesomeidentitymorphismsorisomorphisms.
2.2MetamorphosisorProemialcombinationsinabstractobjects
1Chiasmofsortsandnames:CHI(sorts,names)
Thisissimilartothechiasmofsortsandtheuniverse(ofsorts)inamanysortedlogic.
Itseemsnottobeunnaturalthatasortcanchangeintoanameofanewobjectandontheothersideanameasbeinghierarchicallysuperiortothesortscanchangeintoalowerlevelobjectasasortinanothercontexture.
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Butthisseemstobeanordinaryprocedureforinteractingsystems.Theconceptualizingprocessofdifferentagentscandifferexactlyinthesensethatforoneagentthesetofsortsorofoneofthesortsoftheotheragentcorrespondstothename,thatis,thewholeorcontextureofhisownsystem.Incontrast,whatisthewholescopeofoneagentcanbeasortwithmanyothersortsforanotheragent.Thereisnothingmagicwiththat.Andthereisalsonoreasonforunsolvableconflictsifbothareawareaboutthissituationandunderstandthemechanismofchangebetweeneachother.Thiscommonunderstandingcanbemodelledorrealizedinafurthersystem,withoutbeingforcedtonegatethedifferencesbetweenthetwoagents.
Sortsandnamesoccursondifferentlevelsoftheconceptualhierarchy.Themechanismisgeneralizationandreductionorspecializationofconcepts.
2Chiasmofsortsandoperations:CHI(sorts,opns)
3Chiasmofoperationsandequations:CHI(opns,eqns)
4Chiasmofnamesandoperations:CHI(names,opns)
5Chiasmofnamesandequations:CHI(names,eqns)
6Chiasmofunicityandnames:CHI(unicity,names)
Unicitycanbeunderstoodasthecontextureofthelocalabstractalgebra.Classicaltheorieshavenottobeconcernedwiththeircontextureanduniquenessbecausetheyareuniqueperse,thatistheyaremonocontextural.Becauseoftheiruniquenessthereisnoreasontonotifyitbyaspecialtermlike1.
Becausetheunicity(unity)isabsolute,everypossiblechangeofithasfundamentalconsequencesforthewholeframeworkofreasoning.Thechiasmbetweentheabsoluteunicity(uniqueness)andtherelativityofthenamesdeniesasimplemappingofthelociofthedifferentsystemsontothelinearityofnaturalnumbers.Thechiasmbetweenunicityandtheotherhasnobeginningandnoend.
Thechiasmisthemechanismofchange.Toconnectthedifferentunitizeswithnumberswehavetoabandontheideaofaninitialobject,astartingpointofthenumberseries.Naturalnumbers,asweunderstandthem,areconstructedbyalgebras,inductionandinitiality.Asafirststep,wecantrytomodelthechiasticsituationinthecontextofcoalgebras,coinductivityandfinality.Thischiasticwayofthinkingisclosertothemetaphorsofstreamsandflows,andthelackofultimatebeginningsandendingsasoriginsandtelos.
Moreprecisely,weshouldthinkofthechiasticparadigmasaninterlockingplayofalgebraicandcoalgebraicstrategiesandmethods.
Withthisinmind,allattemptstoformalizepolycontexturalsystems,logicsandarithmetics,withthemethodsofcategorytheoryalonehave
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toberelativized.Itisneverthelessofgreatimportancetostarttheprocessofformalizationofpolycontexturalitywiththemethodswhichareaccessible.Oneverystrongmethod,whichiswellaccessible,isthemethodoffiberingorindexing(Pfalzgraf,Gabbay).Inotherterms,themethodofmappinglocalsystemstoanindexsetasavehicleofdistributionofformalsystems.Butthisprocedureinvolvesthewholeapparatusofthealgebraicparadigm:equality,identity,linearity,initiality,inductivity,etc.Which,asItriedtomakeclear,isinstrongconflicttotheveryideaofproemialityanditschiasticmechanisms.
Thechiasmbetweennamesandcontextures(unicity)isofgreatimportanceforaseriousmodelingofreflectionalcomputationbecauseitopensupthepossibilityofadistributedselfreferentialitybetweensystemsaswholes.Furthermore,namesinacontexturecanbeinterpretedasthereflectionalmappingofothercontexturesintothereflectingcontexture.
2.3Chiasms,metamorphosisandsuperoperators
ThesuperoperationCHIcanbeinterpretedastheoperatorofchangesofcategoricalperspectives,contextsorcontexturesandpointsofview.
Thesepossibilitiesofchangingthecategoricaltermsisexactlywhatmakesthedifferencebetweenchiasmsandcategorytheoreticmorphismswhicharepreservingtheconceptualstructuresofthesystemintheprocessofmappingitintoanothersystem.
Proemialityincorporatesboth,categorytheoreticalandchiasticmorphisms.
ChiasticmorphismsarenotconservativeinthesensethattheyarepreservingthetectonicalorconceptualstructureofasystembutmoresubversiveinthesensequiteanalogtothecatastrophesinThomsCatastropheTheorythattheyarechangingandnotpreservingtheconceptualorder.Thesemorphismsareinastrictsensenotonlyforgetfulmappingsbutrulesofmetamorphosis.
###ChaoticLogicsChaoticlogicsarenotthelogicsofchaosbutthelogicsofchange.Changeinchaoticsystemsisnotacontinuosprocessbutthe
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switchfromonemodetoanothermodeofasystembysomechangesofthestatesofthesystem.Chaoticlogicsarethelogicsofinteractinglogicalsystems.Changesinchaoticlogicsaremodeledbytranscontexturaljumpsfromonesystemtoanothersystemandaredefinedinsharpcontrasttotheintracontexturalstepsoftheexpansionruleinasingularsystem.Transjunctionaljumpsdon'texcludethepossibilitytostayintheprimarysystematthesametimeofthejump.CyberneticOntologyOrderfromNoise.####
Asaconsequenceofthesefirstinsights,inthischiasticpartoftheproemialrelationship,thecategorytheoreticlawsofidentityandassociativityarelost,oratleastfundamentallytransformed.
Thepossibilityofmetamorphosisisgivenbytheinterlockingmechanismofthechiasm.Alsothesuperoperatorshadbeenintroducedprimarilytodealwithcontexturesassuchthereisnoreasontonottoapplytheseoperatorstotheinternalstructureofthecontexture,thatishere,totheinternalstructureoftheabstractobjects.Thereforethegeneraloperatorofmetamorphosisiscomposed,atfirst,bythesuperoperators{ID,PERM,RED,BIF).
Thisallows,thattheremaybeanidentityrelationIDbetweentocontexturesandchangesintheirinternalstructurewithe.g.sort1incontexture1becomessort2incontexture2producedbythesuperoperatorPERM.Or,thecontexturesandthesortsarestable,buttheinternaloperationsofthecontexturesmaychange.
Itisnotexcludedinthischiasticconceptofarchitecturesofdifferentsystems,thatforonesystemallthedifferencesoftheothersystemboilsdowntoonenotion.Thiswouldbeafurtherstepinmappingthearchitectureofonesystemintoanothersystem.MaybethattheinterlockingmechanismbetweenthesystemswouldbereducedtoastrongreductionproducedbytheextensiveapplicationofthesuperoperatorREDtoallcategoriesofthesysteminconsideration.
Fromthepointofviewofproemiality,metamorphosisisnotasimpleconfusionofthecategoricalframeworkbutawellruledoratleastruleguidedchangeofcategoriesintheprocessofchange,emanationandevolutionorothertypesoftransformations.Thistypeofmetamorphosisisnotwildinthesenseoftheabsolutenovum,becauseitsscenarioisfoundedontheknowncategories(names,sorts,operations,etc.)ofthesystemsintransformation.Ifwewouldchooseanothersettinginsteadofalgebras,wewouldhaveasimilarscenarioofchangewithintheframeworkofthedefiningconcepts.Anothertypeofchangecouldbethoughtforthecasewherethetransformationchangestocategoriesunknownbefore.Forthiscasewewouldbeforcedtoadtoourframeworkofproemialchangebetweencategoriessomethinglikeanemptyboxfortheunknown.
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Whynot?
Again,theprocessoftransformationruledbytheproemialrelationshiphasnottohappenonlybetweenobjectsofthesamearchitecture,likealgebrastoalgebras.Italsocanhappenbetweenobjectsofdifferentarchitectures.Aninterestingcasecouldbethechangebetweenalgebrasandcoalgebras.Thesamesituationistoobservebetweendistributedcategorysystems.Morphisminonesystemcanchangetoobjectsinanothersystemofcategories.Oreventheveryconceptofcategoryofonesystemcanbetransformedinamereobjectofanothersystem.Andsoon.
Usualmathematicalpractice?
Computerscientistshavefarmoreflexibleviewofformalismandsematicsthantraditionallogicians.Whatisregardedasasemanticdomainatonemomentmaylaterberegardedasaformalisminneedofsemantics."
M.P.Fourman,TheoriesasCategories,in:CategoryTheoryandComputerpogramming,SpringerLNCS240,p.435,1986
Idon'tsaythatthisisnotthewaymathematiciansareanywayworking.Butitseemstobeobviousthattheyarenotreflectingorevenformalizingthisprocess,thisuseoftermsandmethods,thatistheiractualpracticeofdoingcreativelymathematics.Withoutevermentioningwhatthismeansandhowitisformalized,theas".
Maybecomputerscientisthaveamoreflexibleuseofformalismsthanlogicians.Butlogicianshavenotonlyproducedmostoftheseformalismslongbeforebutalsoknowverywellthattheyaredealingwithhighlyidealizedsituationsgovernedbytheprincipleofidentity.
Ontheotherside,philosophersandphilosophicallogicianshavedevelopedmuchworkinexplainingtheascategoryofthinkingandbeing(analogy).Butwhatiscalled,especiallyinEuropeanphilosophy,hermeneutics,deniesanypossibilityofformalizationoftheascategory.Wealsoshouldn'tconfusetheascategorywiththemorepopularasifcategoryoffictionalism(HansVaihinger)andconstructivism.
Itwouldbeveryinterestingtostartsomecasestudiesofthispracticeofcomputerscientistsandmathematicians.Averyinterestingcasewouldbethewayorworkingwithswingingtypes,thatistheswitchfromalgebrastocoalgebrasandback,inthesenseofPeterPadawitz.
Ormoretraditional:InthesummertermyougetLogicsasalgebras,inthewintertermtheyofferyouAlgebrasaslogics.Andinbetweenyouenjoythesummerholidaystoforgetanypossibleconflicts.
Translations,GoguensSemioticAlgebras
Itturnsoutthatcorrecttranslationsareconservativemetamorphosis.
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Maybethemainproblemofmachinetranslationisjustthisdecision,tostartwithconservativetranslationsandtotrytomodelcommonsensetexts,whicharefullofgamesofviolatingthisconservativity,withthisrestrictedapproach.Inotherword,conservativetranslationsarebasedondisambiguatedandcontextfreesemantics.Acasewhichisveryartificialanddoesn'tmatchnaturallanguageatall.
Aconservativeexample:conflictsinthetreeofdataobjects
Allprogramminglanguagesarebasedonverystrictandstableconceptualstructures.Ifthedataobjectsareintroducedasanorderedsystemlikethetreeofdataobjects",thisstructurewillneverbechangedintheprocessorexecutionofaprogram(Programmablauf).Ifsomethingwouldbechangedinthisorderitwouldautomaticallyproduceseriousconflicts.
Becauseofthefact,thatclassicalprogramsareessentiallymonologic,thereisnospaceforconflictsinapositivesense.Butrealsystems,thatisinteractingsystemsastodaycomputing,arepermanentlyconfrontedwithconflicts.Whynotintroducingdialogsintheverystructureofprogramminglanguagesandsystems?I'mnotwritinghereaboutspecialinteractiveprograms,e.g.,butonthearchitectureandfundamentalconceptualityordefinitionofprogramminglanguagesassuchandnotofspecialapplicationsoftheselanguages.Likeinteractiveproofsystemsorinteractivegames.
Thereisaneasywayofproducingconflictsinadialogicalsystem,ife.g.L1declaresAasasimpleobjectandL2declaressimultaneouslyAasacomplexobject,thatisastructure.Obviouslyitispossible,inthepolycontexturalapproach,tomodelthisconflictandtoresolveitinanotherlogicalsystem,sayL3,thiswithoutproducingametasystemsubordinatingL1andL2.
Diagramm5
Treeofdataobjects
Furthermore,theconflicthasaclearstructure,itisametamorphosisofthetermssimpleobject"inL1andstructure"inL2.Thismetamorphosisisasimplepermutationbetweensortsovertwodifferentcontexturesbasedonthechiasticstructureofthemediationofthesystems.Butitrespectsthesimultaneouscorrectnessofbothpointsofviewinrespectofbeingasimpleobject"andbeinga
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structure".Inthissenseitcanbecalledasymmetricalmetamorphosis.
Todaycomputingisoftencharacterizedbyitsinteractivity.Buttheprogramminglanguageshavenotchangedtorespondtothissituation.Theyarestill,inprinciple,monologic.
Afurtherexampleofaninterchangebetweenprogramminglanguageswouldbethechiasmbetweendataobjectsandcontrolstructures.
Averyshyimplementationofthisinterlockingmechanism,withfarreachingconsequences,isatthebasisofallartificialintelligenceattempts,theinternaldifferenceandpossibleambiguityinLISPbetweendataandprogramsruledbytheQUOTE/EVALfunction.
Theseexamplesshouldnotbeconfusedwithcontradictionsarisingbyaconflictinattributesbetweendifferentinformations.Thisimpliesalogicalandlinguisticlevelofcommunicationanddoesn'ttouchthecategoricalframeworkofinteraction.
AfterWegner,interactionsareparaconsistent,oratleastbelongtoaparaconsitenttypeoflogic.Thismaybetrueonalinguisticlogicallevel,butitisnotincorrespondencewithamoreachitectonicandchiasticviewofinteractivity.
blindspots
Strategiesofdetectingtheontological,logical,computational,epistemological,reflectional,andwhatever,blindspotofaninteractingagent.
2.4Asimpletypologyofchiasms
Tostudysomeaspectsofchiasmswecanrestrictourselftothestudyoftheinterplaybetweenrelatorsandrelata,neglectingthefullfledgedexpositionofthechiasmwithitsconceptrelationandunicity(uniqueness).
Inpracticeitiseasytodiscoverthatmanyvariantsofrealizationsofchiasmareintheepistemologicalplay.Mostly,chiasmarenotfullydesigned,reductionsareusedandsometimestheuseisoverdeterminated.
Wecanclassifythesinglechiasmsasbalanced,underandoverbalanced.Asdistributedandembeddedchiasmswecandistinguishtwomodiofdistribution,iterationandaccretionanditscombinations.
2.4.1Iterationsofchiasms
2.4.2Accretionsofchiasm
2.4.3Mediationofiterationandaccretionofchiasms
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2.4.4Overdeterminationofchiasms
2.4.5Examplesofunderbalancedchiasms
Diagramm6
Examplesofchiasms
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3Proemialitybetweenstructuralandprocessualunderstanding
3.1FormalLogic,TotalityandTheSuperadditivePrinciple
in:GotthardGntherBeitrgezurGrundlegungeineroperationsfhigenDialektik,Band1,MeinerVerlag,Hamburg,1976,p.329351,firstpubl.:BCLReport,1966
Wehavegiventhemainreasonabove:iftherelationbetweenthoughtanditsobjectisbasicallyunderstoodasasymmetricexchangerelationthephenomenonofsubjectivitydisappears.Buta"totality"inwhicheverythingisreducedtoobjectivitycanneverbetotalbecausesomethingismissing.Atotalityis,inHegelsterminology:1)aniteratedselfreflectionof2)anoniteratedselfreflection,and3)aheteroreflection.
Ifwepermit,forthedescriptionofthisstructure,onlylogicaloperationswhichleadtoreflectionsymmetrythen1)iseliminated,and2)and3)turnouttobeindistinguishableandlogicallyidentical...because1)isnothingelsebutthecapacityofkeeping2)and3)apart.
(...)
However,iftheconceptoftheuniversalsubject,i.e.ofBewussteinberhaupt(Kant),iseliminatedthelogicalconstrainttoreduceeverythingtoultimateparityrelationsdisappears.WewillstillhavereflectionsymmetrybetweenSSandSObutnotlongerbetweenSandOingeneral.Inotherwords:itwillturnoutthatthefoundingrelationbetweensubjectandobjectorbetweenThoughtandBeingisnotasymmetricalexchangerelationbutsomethingelse.ThisisthepointwherethetransitionismadefromformalclassiclogicofAristoteliantypetoatheoryoftransclassic,nonAristotelianRationality.WebeginbyredrawingFigure1omittingSUandhavingthephalanxoftheSOreplacedbyasingleSwiththeindexO.WeindicatetherelationsbetweenSS,SOandObyarrowsoffourdifferentshapes.Accordingtothelogicalcharacteroftherelationanarrowwilleitherbedoublepointedoritwillhaveoneshaftorbedoubleshaftedhavingeithercontinuousordottedlines.Figure5willthenshowthefollowingconfiguration:
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IfSSdesignatesathinkingsubjectandOitsobjectingeneral(i.e.theUniverse)therelationbetweenSSandOisundoubtedlyanorderedonebecauseOmustbeconsideredthecontentofthereflectiveprocessofSS.Ontheotherhand,seenfromtheviewpointofSSanyothersubject(theThou)isanobservedsubjectanditisobservedashavingitsplaceintheUniverse.ButifSSis(partof)thecontentoftheUniverseweobtainagainanorderedrelation,thistimebetweenOandSO.ThereremainsthedirectrelationbetweenSSandSO.Thisisobviouslyofadifferenttype.SOisnotonlythepassiveobjectofthereflectiveprocessofSS.Itisinitsturnitselfanactivesubjectwhichmayviewthefirstsubject(andeverythingelse)fromitsvantagepoint.InotherwordsSOmayassumetheroleofSSthusrelegatingtheoriginalsubject,theSelf,tothepositionoftheThou.AndthereisneitheronearthnorinheaventheslightestindicationthatweshouldpreferonesubjectivevantagepointforviewingtheUniversetoanother.Inshort,therelationbetweenSSandSOisnotanorderedrelation.Itisacompletelysymmetricalexchangerelation,like"left"and"right".Anorderedrelationbetweendifferentcentersofsubjectivereflectioncomesintoplayonlyifwereintroducetheconceptofauniversalsubjectwhichcontainsallhuman"souls"ascomputingsubcenters.Ofthetworelationswehavesofarconsidered,theexchangerelationissymmetricalandtheorderedrelationrepresentsnonsymmetry.
ThisinvestigationintendsonlytoshowthattheconceptofTotalityorGanzheitiscloselylinkedtotheproblemofsubjectivityandtransclassiclogicandthatitisbasedonthreebasicstructuralrelations:
anexchangerelationbetweenlogicalpositionsanorderedrelationbetweenlogicalpositionsafoundingrelationwhichholdsbetweenthememberofarelationandarelationitself.
Itmaybesaidthatthehierarchyoflogicalthemesasindicatedintable(II)representsanhierarchyofimplicationalpower.Allthemeshaveincommonthattheyareselfimplicationstheyimplythemselves.Howeverthefirsttheme(objectiveexistence)impliesonlyitselfandnothingelse.Inthisrespectit
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differsfromanysucceedingthemewhichimpliesitselfaswellasallsubordinatedthemes.Forthisreasonitispropertocalltheinitialtheme"irreflexive"andallthefollowing"reflexive".Irreflexivitymeansthatsomethingwethinkofisonlyanimplicatebutnotanimplicandforsomethingelse.Ontheotherhandifwereferlogicallytoreflexivitywemeanthatour(pseudo)objectofthoughtisanimplicandrelativetoalowerorderandaswellanimplicaterelativetoathemethatfollowsitinthehierarchyoftable(II).
Wearenowabletoestablishthefundamentallawthatgovernstheconnectionsbetweenexchange,orderedandfoundingrelation.Wediscoverfirstinclassictwovaluedlogicthataffirmationandnegationformanorderedrelation.Thepositivevalueimpliesitselfandonlyitself.Thenegativevalueimpliesitselfandthepositive.Inotherwords:affirmationisneveranythingbutimplicateandnegationisalwaysimplication.Thisiswhywespeakhereofanorderedrelationbetweentheimplicateandtheimplicand.Thenameofthisrelationinclassictwovaluedlogicisinference.
Itisnownecessarytorememberthatthepossibilityofcoexistenceoftwoindependentsubjects(IandThou)intheUniverseisbasedonanexchangerelationbetweenequipollentcentersofreflection.Moreover,thesesubjectsareallcapableofbeingimplicands.Moreobjectsdonotoperateinferentially.Thatmeanstheydonotimplyanythingelse.
IfwenowconsiderthefoundingrelationinwhichasubjectconstitutesitselfasdiametricallyposedrelativetoallobjectsandthetotalobjectiveconceptoftheUniversewewilldiscoverthatthisrelationrepresentsaninterestingsynthesisofanexchangerelationbetweenlogicalpositionsanorderedrelationbetweenlogicalpositionsafoundingrelationwhichholdsbetweenthememberofarelationandarelationitself.10exchangeandorder.Thefoundingrelationisinitselfanexchangerelationinsofarasthelinkingsubject(SS)mayassumethelogicalpositionoftheothersubjectwhichisthoughtof(SO).SOmayinitsturnassumetherankofSS.Anytwocentersofsubjectivereflectionofthesameordermutuallyimplyeachother.ButsuchanexchangedoesnotoperatebetweenSandO.Aswepointedoutbefore:thebonafideobjectcannotinferthesubjectandbydoingsousurptheroleofasubject.Ifitcoulditwouldimplythatsubjectsareirreflexiveentitieswhichforasubjectisacontradictioinadjecto.Itfollowsthattherelationbetweenimplicateandimplicandhastwodifferentaspects:betweentwosubjectsthisrelationassumestheroleofasymmetricalexchange.Betweensubjectandobjectitappearshoweverasanorderedrelation.Thefoundingrelationisthereforealsoanorderedrelation.Ortoputitdifferently:thefoundingrelationisacombinationof
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exchangeandorder.Whatistheimplicand(SS)maybecometheimplicatenotrelativetoObuttoourimpartialobserverSSS.Wemightsaythatthefoundingrelationisaconcatenationofsequencesofexchangeandsequencesoforderedrelations.
ThediagramofFig._6willillustratewhatwemean:
Fig._6indicatesasequenceofsinglepointedandasecondsequenceofdoublepointedarrowssuchthatasinglepointedarrowalwaysalternateswithadoublepointedone.Aconcreteexampleofwhatthefigureillustratesisthefathersonrelation.Thisisfirstanorderedrelation.Butthesoncanalsobecomeafather.Inthissensefathersonisalsoanexchangerelation.Butthesondoesnotacquirethestatusoffatherrelativetohisownfatherbutrelativetothegrandsonofhisfather.Inabstractterms:whatismember(orargument)oftheorderedrelationOSS,namelySS,maybecomeanargumentofanexchangerelationnotrelativetoObutrelativetoSSSwhichimpliesthisexchangeSSSO.
Thuswemaysay:thefoundingrelationisanexchangerelationbasedonanorderedrelation.Butsincetheexchangerelationscanestablishthemselvesonlybetweenorderedrelationswemightalsosay:thefoundingrelationisanorderedrelationbasedonthesuccessionofexchangerelations.Whenwestatedthatthefoundingrelationestablishessubjectivitywereferredtothefactthataselfreflectingsystemmustalwaysbe:selfreflectionof(selfandheteroreflection).AsHegelpointedoutinhisdialecticlogiconeandahalfcenturiesago,theoppositionofheteroandselfreflectionisnotaparityrelationbecauseitrequiresaniterationofselfreflectionincontrasttothenoniterativecharacterofheteroreflection.Itfollowsaswaspointedoutabove,thatonevalueissufficienttodesignateinheteroreflectionbuttwovaluesarerequiredapartfromthevalueSSSOOSSSSOSSSFig_6
forobjectdesignationtoseparateselfreflectionfromthe
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object.Thisisconfirmedbythecharacterofthefoundingrelation.Table(VI)clearlyshowsthatitrequiresaminimumofthreevaluesforitsownestablishment.Buttheintroductionofathirdvaluegeneratesanewprincipleofsuperadditivity.
3.2Irreflexivityastheultimatebeginning
Incontrasttomyworkinghypothesis"Thereisnoorigin,onlyamultitudeofbeginnings"irreflexivityinGunthersapproachtothefoundingrelationhasthevalueofanultimatebeginning,whichistheorigininitsunicity.Thisoriginischaracterizedasaselfimplication.
Itmaybesaidthatthehierarchyoflogicalthemesasindicatedintable(II)representsanhierarchyofimplicationalpower.Allthemeshaveincommonthattheyareselfimplicationstheyimplythemselves.Howeverthefirsttheme(objectiveexistence)impliesonlyitselfandnothingelse.Inthisrespectitdiffersfromanysucceedingthemewhichimpliesitselfaswellasallsubordinatedthemes.Forthisreasonitispropertocalltheinitialtheme"irreflexive"andallthefollowing"reflexive".Irreflexivitymeansthatsomethingwethinkofisonlyanimplicatebutnotanimplicandforsomethingelse.
Tostartproemiality(foundingrelation)withabeginninginthesenseofanoriginisnotincludedinthegeneraldefinitionofthefoundingrelation.Itisanadditionaldecision,basedonspecialontologicalinterests.
Neithertheabstractformulationnortheexamplegiven,fathersonrelationship,involvesanultimativebeginning.Otherwisethefathersonrelationshipconnotedwithanoriginwouldforceustoaccepta"Urfather".MaybeGod.Butthisisnotphilosophicalthinking.
Tointerpretproemialityashavingabeginningisguidedbytheprincipleofwellfoundedness.Thisprincipleisnecessaryforanalgebraicorconstructivistapproach.Incontrasttothisinterpretationofthefoundingrelationitisequallypossibletounderstandthismechanisminanonfoundedwayofcoalgebraiccoinduction.
AsanexamplewemaythinkofachainofalternatingXsandYswithaoutanoriginnoranend:
...XYXYXYXYX...
IsitreasonabletotakeXoralternativelyYasthestartelementofthechain?Obviouslynot.Itmaybe,insomespecialsituations,areasonabledecisiontotakeYasthestart.
Wemightsaythatthefoundingrelationisaconcatenationofsequencesofexchangeandsequencesoforderedrelations.
Thesameistruefortheconcatenationchainoforderandexchange
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relations.Butthisdecisionisarbitrarilyandnotpartofthemechanismofthefoundingrelation.
Tomakethesetwointerpretationsmoreclear,IintroducedinmyMaterialien197375thedistinctionbetweenopenandclosedproemialrelationship.
Evenifweacceptthattheenvironmentofalivingsystemhasincontrasttoitsmodelingofitanirreflexivecharacterforthemodelingsystem,itisimportanttoseethatthisirreflexivityisofrelativenature.Otherwiseitwillbeverydifficultforacognitivesystemtohavedifferentinterpretationsofitsenvironmentandtochangeitsontology.
Manyconstructivistshaveintroducedthedistinctionbetweenrealityandobjectivity(Maturana)todealwiththisdifficulty.Intheirapproachirreflexivityispurereality,whichassuchescapesanyknowledge.Ontheothersideobjectivityisaresultoftheprocessofinterpretation.ButsinceKantweshouldknowthatthistrickisnotproperlyworking.