Laws of Form and the Logic of Non-Duality Louis H. Kauffman, UIC

Reﬂexivity - Laws of Form and the Logic of Non-Duality Louis H. Kauffman, UIC www.math.uic.edu/~kauffman

The Mind-Body Problem

A Mobius Strip

Marginal Note: The previous slide shows a Mobius brought to you from the fourth dimension. We take three dimensional sphere as the points at unit distance from the origin in four dimensional space.

(These are complex numbers. Two planes of complex numbers make four dimensional space.) The Mobius band is given by the equations below, and “seen” via stereographic projection.

Klein Bottle Available from ACME KLEIN BOTTLE, Oakland California Topological Thoughts

Torus + Mobius Klein + Mobius Topological Thoughts TOPO - LOGICAL THOUGHTS Epimenides Paradox of the Liar The ﬁctional speaker Epimenides, a Cretan, reportedly stated: The Cretans are always liars. L = Not(L) From the point of view of logic, the Liar is in an imaginary state that is neither true nor false. From the point of view of topology, the Liar has the shape of a Mobius band.

The boundary of the Mobius band is One, and yet it is Two. APPLIED MOBIUS INC. Problem: Design a switching circuit (with an economical simplicity of design) that can control a single light from an arbitrary number of locations.

This problem can be analysed by Boolean algebra. The following non-dual solution is the invention/discovery of cyberneticist Ricardo Uribe. Non-Dual Engineering Solutions Inc. Applications - The Mobius Belt The Mobius Shoe Mobius Fence - The inside is the outside is the inside ...

Fence Mobius Fence DISTINCTION

Laws of Form G. Spencer-Brown “We take as given the idea of distinction and the idea of indication, and that one cannot make an indication without drawing a distinction.”

“We take, therefore, the form of distinction for the form.” The circle “makes” a distinction in the plane. We make a distinction in the plane by drawing a circle. Circle and observer arise together in the act of perceiving. That circle, this observer and the distinction that arises are one. The Form We take to exist Arises From Framing Nothing. G. Spencer-Brown We could stop now. But the purpose/play of this talk is to look at how, by starting in unity we make imaginary complexity and how that is related to the original unity. Every discrimination is inherently a process, and the structure of our world as a whole comes from the relationships whose exploration constitutes that world.

It is a reﬂexive domain.

There is no place to hide in a reflexive domain, no fundamental particle, no irreducible object or building block. Any given entity acquires its properties through its relationships with everything else. A reflexive domain is like a conversation or an improvisation, held up and moving in its own momentum, creating and lifting sound and meaning in the process of its own exchange. Conversations create spaces and events, and these events create further conversations. The worlds appearing from reflexivity are worlds nevertheless with those properties of temporality, emergence of patterns,emergence of laws, that we have come to associate with seemingly objective reality. This talk will trace how a mathematics of distinction arises directly from the process of discrimination and how that language, understood rightly as an opportunity to join as well as to divide, can aid in the movement between duality and non-duality that is our heritage as human beings on this planet. The purpose of this talk is to express this language and invite your participation in it and to present the possiblity that all our resources physical, scientific, logical, intellectual, empathic are our allies in the journey to transcend separation. Here is how multiplicity arises in set theory.

0: Empty Set.

1: Set whose member is the empty set.

2: Set whose members are 0 and 1. TWO SETS ARE EQUAL IF AND ONLY IF THEY HAVE THE SAME MEMBERS.

Theorem. There is only one empty set. Proof. Suppose U and V are both empty. By the above principle, they must be equal. They have the same members, namely none! Q.E.D.

Theorem: 0 is not equal to 1. Proof. 0 has no members, while 1 has a member, namely 0. 3: Set whose members are 0,1,2.

In Set Theory multiplicities arise from nothing but the act of collection and the deﬁnition of equality of sets. Laws of Form The initial act of distinction. For the distinction to be (distinct) there must be a difference between the sides. Let us call one side Marked. The other side is Unmarked.

M M

Economy. The distinction is a circle. Let the circle itself stand for the marked state. M

Now Circle has a name-tag in her own form. The tag and the name of the tag can be confused. =

The name tag is not needed to identify the outside of our mark of distinction (in this representation). The principle that “the value of a call (of a name) made again is the value of the call” is quite general. Spencer-Brown: “The value of a call made again is the value of the call.” So far we have focused on the distinction as the locations of its sides and their names. Let the mark/circle/distinction be seen as a TRANSFORMATION from the state indicated on its inside to the state indicated on the outside.

Inside Outside Inside = Outside

UnMarked = Marked

Marked = UnMarked =

Cross from The marked the unmarked state. state. “The value of a crossing made again is not the value of the crossing.”

Cross from the marked state. Summary of Calling and Crossing

= Example

= = We have constructed an arithmetic of forms (patterns of distinction) that is a language speaking about a single distinction. Arithmetics have algebras, and the ﬁrst algebra associated with this arithmetic is Boolean algebra, the algebra of classical Aristotelian logic. Using

a = Not a.

So we see that the classical logic with all its dualities comes from and returns to a source that is the production and dissolution of imaginary distinctions in a world where there are no discriminations in the ﬁrst place. For example, we can return to the Liar paradox like this.

L = L

L = Marked L = = Unmarked

L = Unmarked L = = Marked But this equation

L = L suggests A form L that reenters its own indicational space. A form of self-reference, or self -observation. A ﬁxed point. An invariance. A recursion. An object produced from a process. A process indicated by a transformation. phase-shifted from the original one by one half-period. The

juxtaposition of the these two waveforms yields a marked state.

= ......

With this interpretation we would like to keep p o s i t i o n as a rule

about the reentering mark. But we also note, that as a waveform

the reentering mark, taken all by itself, is indistinguishable from its

crossed form. A Form Re-enters its Own Indicational Space.

= ......

= (all by itself)

One way to get partially out of this dilemma is to make two

imaginary values i and j, one for each waveform and to have the

following waveform arithmetic:

... i = ...

j = ......

ij = i = j = , ,

i = i , j = j

The waveform arithmetic satisfies occultation and transposition, but

not position. It is similar to the three-values Calculus for Self-

Reference, and has a completeness theorem using these values. This

rich structure is directly related to a class of multiple valued logics

An entity might be deﬁned in terms of itself.

Mathematical concepts in second-order cybernetics

of the world in which we operate. They attain below we have indicated these relationships actions that I can perform. The wall is quite their stability through the limiting process with respect to the eigenform of nested boxes. transparent to a neutrino, and will not even be that goes outside the immediate worldMathematical of NoMathematicalte that the “It” is illustrated as a finite an eigenform for that neutrino. individual actions. We make an imaginaticovne ceapprptso iximationn seco (tnod the-o rinfinitder ec limit)yber thatne tisics This example shows quite sharply how the leap to complete such objects to become sufficientconc teo pallotsw iann obsersecvoern tdo -infer/perorde-r cynaturbeer ofn eant objectics is entailed in the proper- tokens for eigenbehaviors. It is impossible to ceive the generating process that underlies it. ties of its observer. make an infinite nest of boxes. We do not The eigenform model can be expressed in of the world in which we operate. They attain below we have indicated these relationships actions that I can perform. The wall is quite make it. We imagine it. And in imagining that The It quite abstract and general terms. Suppose their stability ofthr theoug who therld inlimiting whic hpr woce essoperatwe.ith The respecty attain to the eigenforbelowm w ofe nesthavede indicatboxes. edt rtheseanspar rentelationships to a neutrino, andactions will not that even I becan perform. The wall is quite that goes outside the immediate world of Note that the “It” is illustrated as a finite an eigenform for that neutrino. infinite nest of boxes, we arrive at the eigen- their stability through the limiting process with respectthat wtoe the ar eeigenfor given ma r ofecursion nested bo (notxes. nectressaransparilyent to a neutrino, and will not even be form. individual actions. We make an imaginative approximation (to the infinitneumer limit)ical) that iswith theThis equation example shows quite sharply how the leap to compthatlete sgoesuch ooutsidebjects to thebec oimmediatme sufficiente world to alloof w anN oobserte thatver tothe infer/per “It” is- illustnaturraete ofd anas objecta finit ise entailedan einig theenf oprroperm fo-r that neutrino. The leap of imagination to the infinite tokens for eigenbehaindividualviors. actions.It is impossible We mak to e ance ivimage theinati genervaeting prapprocessoximation that underlies (to it.theX (infinittt i+es o1)fe i t=limit)s o Fbs(eXr vthat(etr)).. is This example shows quite sharply how the eigenform is a model of the human abilitmakye anto infinitleae pnest to ofco bompxes.let eW es udoch notobjects to become sufficient to allow an obserThever eigenforto infer/perm model- cannatur be eex professed an object in is entailed in the proper- create signs and symbols. In the casemak of ethe it. We imagtokineens it. for And eigenbeha in imaginingviors. that It is impossible to Tcheei vIte the generHeraeting X( tpr) denotocessquit thatees abst the underliesra ctco nditionand it.gener ofalt ie tobserse roms.f it s-S oupposebserver. eigenform X with X = F(X), X can be reinfinitgarde dnest ofmak boxees, an we infinitarrive ate nestthe eigen- of boxes. We do not vation at time t.that X( wt)e caroulde give nbe a rasecursion simple (notThe as nec eigenforaessarilym model can be expressed in form. make it. We imagine it. And in imagining that The Itnumerical) with the equationquite abstract and general terms. Suppose as the name of the process itself or as the nameThe leap of imagination to the infinite set of nested boxes, or as complex as the entire infinite nest of boxes, we arrive at the eigen- L = L X(t + 1) = F(Xthat(t)). we are given a recursion (not necessarily of the limiting process. Note that if yeigenforou arem is a model of the human ability to configuration of your body in relation to the form. numerical) with the equation told that create signs and symbols. In the case of the known universe Hate retime X(t) denot t. esT htheen c onditionF(X(t)) of obser- eigenform X with TheX = F (leapX), X ofcan imagbe reginationarded to the infinite vation at time t. X(t) could be as simple as a The Process leading to It denotes the result of applying the operations X(t + 1) = F(X(t)). X = F(X), as the name of eigenforthe processm itself is a modelor as the of name the human ability to set of nested boxes, or as complex as the entire of the limitingcr preatoceess. sig Nnso tande that sy ifmbols. you ar eIn the case of the symbolized by Fc toonfigur the cationondition of your at bod timey in Htr.e Ylationeroeu X (tto) thedenotes the condition of obser- then, substituting F(X) for X, you can wtoldrit thate eigenform X with X = F(X), X can be regarded could, for simplicitknowyn, assumeuniverse atthat time F vationist. inde-The nat Ftime(X(t ))t. X(t) could be as simple as a The Process leading to It denotes the result of applying the operations as theX = name F(X), of the process itself or as the name pendent of time. Time independencsete o off nestthe ed boxes, or as complex as the entire X = F(F(X)). of the limiting process.Space Note that if you are symbolized by F to the conditionconfigur at timeation t. Yo ofu your body in relation to the then, substituting F(X) for X, you can write recursion F will gcould,ive us for simple simplicit answy, assumeers and that w Fe is inde- told that known universe at time t. Then F(X(t)) Substituting again and again, you have can later Timediscusspen dwhatent of twimille. Thappenime inde peifn dtheence of the X = F(F(X)). The Process leading to It denotes the result of applying the operations X = F(X), actions depend ruponecursio nthe F w itime.ll give us I nsimple the answtime-ers and we X = F(F(F(X))) = F(F(F(F(X)))) = Substituting again and again, you have can later discuss what wislly mhappenbolize dif b they F to the condition at time t. You then, substituting F(X) for X, you can write independent caseactions we can depend wr itupone the time.could, In forthe simplicittime- y, assume that F is inde- F(F(F(F(F(X))))) = … X = F(F(F(X))) = F(F(F(F(X)))) = independent case we can wrpiteendent of time. Time independence of the F(F(F(F(F(X))))) = …X = F(F(X)). J = F(F(F(…))) The process arises from the symbolic J = F(F(F(…)))recursion F will give us simple answers and we expression of its eigenform. In this viewThe, the process arSubstitutingises from the again symbolic and again, you have – the infinite concatenation of F uponcan itself. later discuss what will happen if the expression of its eigenform. In this view, the – the infinite concatenationactions of F upon depend itself. upon the time. In the time- eigenform is an implicate order for the eprigeocessnform is an implicateX = F or(Fder(F for(X )the)) =pr ocessF(F(F(F(X)))) = Then Then independent case we can write that generates it. (Here we refer to imthatpli cgeanerte ates it. (Here weF r(eFfe(rF t(oF i(mF(pXlic)a)t)e)) = … F(J) = J F(J) = J order in the sense of David Bohm 1980).order in the sense ofJus Dat vsido ,Bohm an o 1980).bject in the wJousrldt so (c, aon gonbjeitcitv ine, the world (cognitive, J = F(F(F(…))) Sometimes oneThe styliz presoc theess starruisesctur e frbomy phtheysical, symbolic ideal, etc.) provides a conceptual since adding one more F to the concatenation Sometimes one stylizes the structurindicatinge by wherephxep theysical,res eigenforsion oideal,fm it sX e rieetgeenc.)ntefro sr primts .o Ivnidesc ethnties r v afio ercw ot,h ncthee eeptualxploration ofsinc a skei nadding of rela- onec hangesmore nothingF to the. concat–enation the infinite concatenation of F upon itself. indicating where the eigenform X reenotwenrs indicational its eciege nspacnfortere mwfoith isr antanh are implicate reoxwp orlo otherra toriodern tionshipsoforf athe sk pre irocessnelat oedf rteol aits- contcehangesxt and t onothing the . Thus J, the infinite concatThenenation of the graphical devicthate. See ge thener picturates it.e belo (Hwer fore w thee refeprr octoesses imp thatlica tgenere ate it. An object can have operation upon itself leads to a fixed point for own indicational space with an arrow or other tionships related to its context and to the Thus J, the infinite concatenation of the F(J) = J graphical device. See the picture belowcase for ofthe the nestorpredderoc boesses xines. the that sense gener of Daavtide it.Bohm Avarn yobject1980).ing deg rcanees of ha realitveyJ, usjustt oper saso ,an a nationeigenfor obje cupontm in th Fitselfe. Jw iso saidr leadsld (ctoo be gton the itai fixv eigenfore,ed pointm for for the recur- Sometimes one stylizes thedoes. stru Icturf wee tak bye the phsuggestionysical, ideal,to hear ett thatc.) prosvioidesn F . aW ce oseenc eptualthat every sincrecursione adding has onean more F to the concatenation case of the nested boxes. varying degrees of reality, justobjects as an are eigenfor tokens form eigenbehaF. vJiors, is said then tano bee theigen feigenfororm. Everym r eforcur stheion hraescur an -(imagi- indicating where the eigenform X reenters its center for the exploration of a skein of rela- changes nothing. does. If we take the suggestionobject tino itselfhear ist anthat entity, parsioticipatingn F. W ine asee narthaty) fixeveder point.y recursion has an own indicational space with an arnetwrowor ork of other interactions,tionships taking onrelat itsed appar to -its contWeext end and this tsectiono the with oneT hmorus eJ ,e xam-the infinite concatenation of the objects= are tokens for eigenbehaviors, then an eigenform. Every recursion has an (imagi- … graphical device. See the pictureent belo soliditw for ythe and prstabilitocessesy thatfrom gener theseate it.ple. An This object is the can eigenfor have m ofoper the ationKoch fruponactal itself leads to a fixed point for caseobject of thein nestitselfed isbo anxes. entity, intpareractions.ticipating invar a yingnar degy)rees fix ofed realit point.y, (Mjustandelbr as an eigenforot 1982). Imn this Fcase. J isone said can t wor beite the eigenform for the recur- network of interactions, takingAn onobject its isappar an amphibiandoes.- If w ebetwW take eeenden the the this suggestion sectionsymbolically to w hearith the tone eigenfor that morms eequationi oenxam- F. We see that every recursion has an … = symbolic and imaginary world of the mind ent solidity and stability from theseobjects ple.are t oThiskens isfor the eigenbeha eigenforviors,m of thenK =the K an {K Ko Kc he} i Kgfrenactalform. Every recursion has an (imagi- and the complex woobjectrld of personalin itself isex peran ientit- y, participating in a nary) fixed point. interactions. ence. The object, when viewed(M asandelbr a process,ot is 1982).to indicat In thise that case the Koneoch canFractal w rriteente ers its network of interactions, taking on its appar- We end this section with one more exam- Does the infinitAe nnest object… of box esis e xist?an= Ceramphibian- a dialo guebetw beteweneen the these wosyrldmbolicallys. The object, theo eigenforwn indicationalm equation space four times (that is, it is tainly it does notsymbolic exist on andthis page imag orinar any-y wwhenorld seen of asthe a sig mindn forent itself, solidit or in andy ofand itself, stabilitmadey upfr omof four these copies ofple. itself, This eac ish theone- eigenform of the Koch fractal interactions. K = K { K K } K (Mandelbrot 1982). In this case one can write where in the phandysical the world c owmpleith whicx hw woe rldare ofis personalimaginary. experi- third the size of the original. The curly brack- familiar. The infinite nest of boxes exists in the Why are objects apparAn entlyobject solid? is an Of amphibianets in the betw centeeren of the this equationsymbolically refer to the eigenform equation imagination. Itenc is a esy. mbolicThe object, entity. when viewcourseed as y oua pr cannotocess, walksy ismbolic througto handindicat a br imagick walle inarthaty factthewo rldthat K oof ctheh the Ftwr actalmindo middle reent copiesers witsithin the K = K { K K } K Does the infinite nest of boxes exist? TheCer eigenfor- a md iais lothegue imag binedetw eboundaren theyse weovernld ifs y. oTuh thei nokb ajebocandutt, it dtheiffoe wrceonnmplet lindicationaly. I dxo wnootrld frof actalspac personal ear foure inclined e timesxperi - w(thatith r espectis, it is to one tainly it does not exist on this page orin thean y-reciprowhencal relationship seen as aof sig then forobject itself,mean or in appar andent of in itself, theenc sensee. The ofmade thoug object, htup alone. when of four vianotherew edco piesas anda pr of ocw ithess,itself, r espectis eac ttooh indicat theone- twoe outthater the Koch Fractal reenters its (the “It”) and the prDoesocess the leading infinit to thee nest object of boxI esmean exist? appar Cerent- in thea d iasenselogue of appearbetweeancn te.hese cwopies.orld sIn. Tthehe figurobjeec belot, w owwe nsho indicationalw the geo- space four times (that is, it is where in the physical world with which(the we praroceess leadingis imag to inar“It”).y I.n the diagram The wall appears solid to methir becaused the of siz thee of metther oric cigonfigurinal.ation The ofcurly the r ebrentack-ry. familiar. The infinite nest of boxes exists in the tainlyW hity does are not objects exist on appar this pageently or solid?any- Ofwhen seenets asin a the sign c forent itself,er of or this in and equation of itself, refermade to theup of four copies of itself, each one- imagination. It is a symbolic entity. whercoursee in y theou ph cannotysical w walkorld w thrithoug whichh a w bre aricke wallis imagfactinar y.that the two middle copies withinthird thethe size of the original. The curly brack- familiar. The infinite nest of boxes exists in the Why are objects apparently solid? Of ets in the center of this equation refer to the Volume 4 · Number 3 · July 2009 125 The eigenform is the imagined boundary imagevenination. if you tIht isin ak sy ambolicbout it entit diffey.rently. I do ncoourset fryouactal cannot are walk inclined throug h wa ithbrick r espectwall factto thatone the two middle copies within the in the reciprocal relationship of the object meanThe appareigenforentm in is thethe imagsenseined of thougboundarhty alone.even if anotheryou think andabou tw itith dif freespectrently. I tdoo ntheot twfroactal out earr e inclined with respect to one (the “It”) and the process leading to the object inI mean the re apparciprocalent re inlationship the sense of ofthe appear object ancmeane. apparcopies.ent Iinn thethe sense figur ofe thougbelowht w alone.e show theanother geo- and with respect to the two outer (the process leading to “It”). In the diagram (theThe “I wallt”) and appears the proc solidess leading to me to because the object of theI meanmet apparricent co innfigur the senseation of of appear the reancente.ry. copies. In the figure below we show the geo- (the process leading to “It”). In the diagram The wall appears solid to me because of the metric configuration of the reentry.

Volume 4 · Number 3 · July 2009 125 Volume 4 · Number 3 · July 2009 125 Mathematical concepts in second-order cybernetics

Volume 4 · Number 3 · July 2009 125 On encountering reentering and reﬂexive structures we leave simple dualities for a complex world. Once this sort of pattern sets it is a challenge to go back to the beginning.

One can be aware of one’s own thoughts. An organism produces itself through its own productions. A market is composed of individuals whose actions inﬂuence the market just as the actions of the market inﬂuence these individuals. The participant is an observer but not an objective observer. There is no objective observer. There is no objective observer, and yet objects, repeatablity, a whole world of actions, and a reality to be explored arise in the relexive domain. Describing Describing Describing Describing Consider the consequences of describing and then describing that description.

We begin with one entity:

* And the language of the numbers: 1,2,3.

Yes, just ONE,TWO,THREE. * Description: “One star.” 1* Description: “One one, one star.”

111* Description: “Three ones, one star.”

311* Description: “One three, two ones, one star.”

13211* Describing Describing * 1* 111* 311* 13211* 111312211* 311311222111* 1321132132311* 11131221131211131213211* K 71377—21/12/2004—RAK 71377VICHA—21/12/2004—RANDRAN—127841VICHANDRAN—127841

beings6.1 AlicAude,io-activBob andity andCarolthewhosociaeachl contexttake on the task to describe another, with Bob EigenForm K I kept thinking about that question, and wondering about finding a good mathematical describing Alice, Carol describing Bob and Alice describing Carol. In the mutual round 34,1/2 example. Then I remembered learning about the “audio-active sequence” of numbers of their descriptions they may converge on a mutual agreement as do the triplet of from Conway (1985). This is a number sequence that begins as: audio-active sequences (in the limit). Yet, it may take some coaxing to bring forth the agreement and1; som11;e 21crea; tivity1211;as11122well.1More; 31221complex1; 13112socia221l ;situatio11132ns13211will;be. .beyo. nd calculation, and yet, the principles of the interaction, the possibility of eigenforms will 140 applyCan. Theyouconceptfind theis nextpowerfunumberl and impoin thertantsequenceto consider? If y,oparu readticularlythemwheoutn oloud,ne is the 139 facedgenerawithtingthe ideaincalculablbecomeesnatureapparentof complex interaction. one; one one; two ones; one two; one one; . . . 7. Generation of objects TheEachtrue termquestioinntheaboutsequencean objeiscta descriptiois: How isnitofgenerathe digted?its in the previous member of the Thesequefalsence.questionThe recursionabout angoesobjectbackis: andWhatforthis itsbetwclassificeen numberation? and description of Takenumbaer.mathemWhat aticalhappenscaseasinthispoint.recursionLet R begoestheonsetanofdallon?sets that are not members of themHereselveiss.a(Russbit moreell’s fofamousit: paradoxical set.) We symbolize R as follows. Let AB denote the condition that B is a member of A. 1 Define R by the equation 11 21 RX ,XX ¼ which 1211says X is a member of R means that it is not the case that X is a member of X. From this11122we1 reach the paradox at once. Substitute R for X you obtain: 312211 RR ,RR 13112221 ¼ R is a 11132member13211of R means that it is not the case that R is a member of R. Someth31131ing21113122curious1has happened. We attempt to classify R by finding if it was or was no13211t a membe31112311r of31122itself11and we are led into a round robin that oscillates between membe11131rship22113311and nonme21321mbership.13212221Classification creates trouble. Ask31132how22123211R is genera21113ted.12211312113211 We13211start 33211121with som31221e sets12311311we know2221131. For examp11221131le, the221empty set is not a member of itself,Nowneithyouercanis thebeginsettoofseeall thatcats.thereSo aisfirsa tapproapproachximato tiona tripleto Rofcoulinfinited besequences, each describing the next, with theR1first{ des{ };cribingCats};the last. This triple is the limiting ¼ wherconditioe Catsndenoteof the authedio-actset ofiveallsequencecats (Cats. Inisonnote sensea cat.).theNowaudio-acwe notivetesequencethat R1 ioscillates also s notamonga membertheseof itself.three seqSo weuenceshave(intotheaddlimit),R1 toandget ayetbetterin anothapproerximationsense thisR2.triplet of infinite sequences is the eigenform in back of the audio-activity! R2 { { }; Cats; { { }; Cats} }: A ¼11131221131211132221. . . But R2 is also not a member¼of itself and so we would have to add R2 and keep on with this as well as throwing inBother311sets31122211that come31112311along and are332no.rmal.. . A set is normal if it is not a member of itself.¼ C 132113213221133112132123. . . ¼ The triple of infinite sequences are built by continually cycling the self-description through the three sequences. This leads to a definite and highly unpredictable buildup of the three infinite sequences, A, B, and C such that B describes A, C describes B and A describes C! (Figure 5). This triplication is the eigenform for the recursion of the audio-active sequence. The triplicate mutual description is the “fixed point” of this recursion. With this example, we begin to see the subtlety of the concept of an eigenform, and how it may apply to Figure 5. diverse human situations. For indeed imagine the plight of three individual human

Self-Mutuality and Fundamental Triplicity

Trefoil as self-mutuality. Loops about itself. Creates three loopings In the course of Closure. Patterned Integrity

The knot is information independent of the substrate that carries it. In the diagram above, we indicate two sets. The first (looking like the mark) is the empty set. The second, consisting of a mark crossing over another mark, is the set whose only member is the empty set.

We can continue this construction, building again the von Neumann construction of the natural numbers in this notation:

{}

{{}}

dimension. After that invention, it turned out that the diagrams

represented knotted and linked curves in space, a concept far

beyond the ken of those original flatlanders.

Set theory is about an asymmetric re{la ti{on} c a{ll{ed} m}em }bership.

We write a ε S to say that a is a member of the set S. And we are

loathe to allow a to belong to b, b to belong to a (although there is

really no law against it). In this section we shall diagram the

membership relation as follows: Knot Sets{ {} {{}} {{} {{}}} }

a b Crossing This notation allows us to also have sets that are members of a t h e m s e l v e s , as Relationship a εb

The entities a and b that are in the relation a ε b are diagrammed as

segments of lines or curves, with the a -curve passing underneath the

b -curve. Membership is representead by under-passage of curve segments. A curve or segment with no curves passing underneSelf-ath it is the empty set.

a ε a Membership

{ } a = {a} and sets can be members of each other. a

{ } Mutuality { { } } b

a={b}

b={a}

Mutuality is diagrammed as topological linking. This leads the question beyond flatland: Is there a topological interpretation for this way of looking at set-membership?

Consider the following example, modified from the previous one.

a = {}

a b = {a,a} b

topological

equivalence

a={}

a b={}

The link consisting of a and b in this example is not topologically linked. The two components slide over one another and come apart.

The set a remains empty, but the set b changes from b = {a,a} to empty. This example suggests the following interpretation. Architecture of Counting 0

3 Topological Russell (K)not Paradox

A A does not belongs to A. belong to A. This slide show has been only an introduction to certain mathematical and conceptual points of view about reflexivity. In the worlds of scientific, political and economic action these principles come into play in the way structures rise and fall in the play of realities that are created from (almost) nothing by the participants in their desire to profit, have power or even just to have clarity and understanding. Beneath the remarkable and unpredictable structures that arise from such interplay is a lambent simplicity to which we may return, as to the source of the world.