A Tale of Fishes, Birds and Diamonds in Second-Order Epistemology

Why it is useless to write about the mono-contexturality of alphabetism and digitalism

Rudolf Kaehr Dr.^@

ThinkArt Lab Glasgow

Abstract

In a closed world, which consists of many worlds, there is no narrowness. In such a world, which is open and closed at once, there is profoundness of reflection and broadness of interaction. In such a world, it is reasonable to conceive any movement as coupled with its counter-movement. Endness has to be connected with endless rhythms instead of linear or non-linear progressions.

1.  The Endness of Events

The endness or finiteness of events in a open/closed world are not simply ending in an unqualified way. Endness has to be connected with rhythms instead of linear or non-linear progressions.

A rhythm has a beginning and an end; endlessly. An open/closed world is poly-rhythmic. Scientific linear time structures of whatever complexity are without rhythms. Western science beliefs in a 1-rhythm world: from the big bang to the wee crash.

In many papers I emphasized the importance of linearity for the Western way of thinking and its mathematics and mathematically based technology.

In-between I have the feeling that I always experienced a strange lack of response to my argumentations. In a metaphor, I feel like a fish telling his female fish friend: "Honey, do you know, we are living in water?" And getting the harsh response: Shut up you wanker, I don’t fancy you!.

OK, not everybody can be mesmerized like Monsieur Jourdain after he learned that he is speaking all his life prose. And not everybody thinks that this is trivial anyway.

For good reasons we can belief that there is no reason to think that the fish girl was stubborn or even stupid. She easily could have pointed to the un-denial fact that there is no such thing like water in the water to perceive. What is in the water are all these different plants, stones, animals, and surely, other fishes. But no water at all. This is more than clear. There might be some areas where it is harder to swim or where other stuff is moving very fast or areas where nothing is moving at all. The stuff might also move in all direction, at once. And as far as she can swim there is no limit and no reason to stop her swimming. What can be perceived and sensed in her world as a fish are objects of all sorts but no water. All that is crystal clear.

And without doubt, if the water in which everything is swimming becomes itself an object of the water with all its objects, something quite confusing will happen to the whole world of swimming forever.

Another approach, which has not to struggle with the problems of the abstractness of the arguments for linearity of alphabetism with its atomicity and ideality of signs, could be the more generally acknowledged fact of the endless repeatability of (sign) events. This concept of iterability can be thought independent of dimensionality, parallelisms, circularities, interactions and other seemingly non-linear complex and pictorial or sonic processes and structures. Hence, for now, we will free the notion of iterability and repeatability from all restricting connections to alphabetism and digitalism.

As for the swimming moves of our fish girl, which are not restricted by any obstacle, to each move there is a next move, and so on. There is no last move in such a world of movements. Swimming is producing swimming; only a swimmer is swimming, and no swimming is leaving the category of swimming. Outside of swimming there is no swimming. Swimming adds to swimming, and remains swimming; endlessly. No swimming transforms into flying; no swimming permutes into walking. And so on.

OK, in real-world conditions, the fish girl will stop to swim because of physical limitations of her life-span. The same happens, evidently, to the chalk and blackboards of the high priests of formal systems. The endless iterativity of their sign systems will have, in real-world conditions, unavoidably, some natural ends. This is in sharp contradiction to the abstractness of the definition of signs and Obs in formal systems.

Nevertheless, repeatability is open and endless. The iterability of repeatability is stable.

The other fact, we could agree to some degree, is given by the identity of the repeated objects. It may not be a too big challenge to see and perceive, clara et distincta, that this concept of identity is best realized, as Hegel pointed out, by the Western alpha-numeric sign systems. A number or a letter is as a number or as a letter strictly identical with itself. Take the inscription on your bank note: 5 USDollar. There is nothing to interpret, 5 is 5 and USDollar is USDollar. And nothing else.

Hence, endless repeatability is realized within the realm of identical entities. Or: identities are realized in the realm of iterability. There is no identity without iterability and no iterability without identity. This, again, happens in the ideal world of sign systems, i.e., in the mind of semioticians and mathematicians; and not at the blackboard, nor in citations or plagiarism¹.

Therefore, if we accept iterability, we have not to struggle with the strangeness of the challenge to be aware of swimming in strange waters. Identity, at least to some degree of fuzziness, and the endlessness of repeatability in all its mathematical forms, seems to be accessible to everyone and understood universally without getting involved with the paradox of the medium we are living in.

Things are getting less natural and universal if we stipulate a pluri-verse instead of a classic universe as the ultimate condition. But this is a story to come!

It seems that nobody wants to share my linearity thesis. It is said, all over again: The world is hyper-complex, fractal, undecidable and the World Wide Web decentralized and chaotic. Old alphabetism is loosing its dominance in the Western world to images, graphics, pictograms, videos and sound. More theoretical motivated guys are talking about cellular automata, parallelism, actor communities, grid computing, multi-located λ-calculi, etc.

Therefore, there is no such thing to observe like a dominance of linearity and identity in a post-modern world full of paradoxes, parallaxes, ruptures and abysses.

A.A. Markov’s linearity thesis - equivalent to the Church-Turing Thesis - is not only unknown by media scientists but put under the carpet by computer scientists as old foundational fundamentalism (FOL) and bad reductionism.

What to do against such a poverty of thinking?

Simply, change topic! Give it up! Ask our fish!

Hence, forget linearity!
Forget alphabetism and its digitalism!

Enjoy endless repeatability! The world is rich and complex, and you too.

And there is also conceptual space enough available to defend this situation of repeatability before we end up in the annoyance of paradoxical self-defence.

2.  The Obs of Paradoxes and Second-Order Cybernetics

2.1.  The fish’s parallax

Our poor fish had a fight with his mate, made a big jump; and ended outside of his aquarium and discovered, for a short time in his life before he died, that there must really be something like an outside of his medium. He was right to tell his bird that they are both living in water! But then it was too late.

A bird from Raymond Smullyan’s bird zoo has taken him into other dimensions, not being aware of what an epistemological mystery happened to his easy prey. As we know, birds don’t swim in waters.

This was his moment of self-referentiality.

What he couldn’t know, at this exquisite time of his death, that he was an observer observing himself as an observer. Living in water and at once not living in water. Living and dying at once. The high-noon of observability: simultaneously, internal and external observer; at once! A real double-bind! What a paradox! The parallax killed him.

This double blindness of Western civilization is on the train to experience the same misery.

Time has gone on, and Uncle Heinz, the Magician from Urbana/Vienna brought this paradoxical experiences to the point. And this long before Niklas Luhmann released a wave of re-entry beasts into the still waters of sleeping West Germany.

2.2.  The misery of re-entry figures

Heinz domesticated, what the Swiss psychologist, Jean Piaget, described in great precision and concreteness, into his calculus of recursive Eigen-values. For that, Heinz transformed the observers into the magic object Obs. Obs are not only abstract Obs, they are objects obs, observers Obs, observables obs and more. And obviously, they come in the form of fishes. And sometimes they look like ancient birds.

The epistemological results are well known:
1. Cognition is the computation of a computation.
2. Objects are tokens for eigenbehaviour.

In the case of two observers observing an observation, Heinz von Foerster writes:

“Under which conditions, then, do objects assume ‘objectivity'?
Apparently, only when a subject, S, not unlike himself, who, in turn, stipulates the existence of still another subject, not unlike himself, who may well be S₁."

The observational double play, defined by Heinz, is not closed with a couple of internal observers S₁ and S₂, a new observer is entering the game, enjoying to observe from a cosy external position, those observers observations. In an actualized version, deployed from medieval mysticism, this new external observer of this devilish mismatching re-entry match, distinctively drawn from the neutral position of Australia, is nourishing himself with a non-observational drink on his own. Meanwhile, the combatants have lost all their magicks, reduced to snakes and trapped in the Sisyphus play of eating their own re-entry tails. In real-world time measures, the magic of the game is over and on the way of being re-entering into a non-circular harmony.

Nonetheless, the magic is more profound. Our Obs are in fact birds, too. More precisely, Heinz’ fishes and snakes are profoundly, i.e., in their very deep-structure, birds, and nothing else than birds.

Adepts of Heinz’ constructivism are mostly mesmerized by re-entry forms of indefinite recursions. What they miss to read is the hint to the real trick, the magician is giving clearly: Contemplate!

"Contemplate [emph, rk] the above expression (6) and note:
(i)  that the independent variable obsthe “primary argument" [of the recursive function, rk],
     has disappeared [...].
(ii)  that, because obs_∞ expresses an indefinite recursion of operators CORD onto CORD,
     any indefinite recursion within that expression can be replaced by obs_∞.
(iii) Hence:

Note that while in this form the horror infinitaris of expression (6) has disappeared (all expressions in CORD are finite), a new feature emerged, namely, that the dependent variable obs_∞ is, so to say, “self-depending" (or “self-defining” or “self-reflecting,” etc., through the operator CORD)." (Heinz von Foerster, Observing Systems², p.277)

With that, and the following contemplations on constructions, Heinz succeed to suggest that contemplation is equal computation. And obviously, it goes on and on. Even if it might become circular by contemplation and meditation, the circularity of the indefinite recursion is constructed in one and only one direction. Its recursion is trapped into linear iterability. The “primary argument” as the entry step of recursive game, might have disappeared in such a contemplation, but it still dictates the start as a singularity. That is, there is one and only one beginning, even in the case we are eliminating it. Only one “primary argument” is disappearing in the contemplation. There is no colored multitude involved. Colored figures are not allowed on this stage. Contemplate on that and you will find it boring! Given into other hands, it will be, i.e. it is becoming dangerous.

2.3.  Curry’s ornithology

Haskell Curry, one of the most passionate ornitologues, invented the radically naked objects of thinking ever appeared in the Western world: the Obs of Formal Systems.

The philosopher Emil Lask risked it before Curry and Schönfinkel but didn't succeed, except in some philosophical circles in Japan, to entertain the European academics with his Urform³ of logical nakedness.

The job of the Obs was to save the struggling fundaments of mathematical thinking.

This Snob of an English aristocratic logician, Bertrand, got the trick to show that the whole work of brave Gottlieb was paradoxical. Frege was destroyed. He didn’t know Heinz and his magic tool box. Bertrand became famous: the Russell Paradox.

Today, we are used to enjoy bursting bubbles of all sorts. Van Benthem⁴ has written a kind of a manifesto for the domestication and integration of paradoxes: Four Paradoxes. Similar proclamation appeared by Heinz and Richard Howe, written to promote Francisco Varela’s Calculus for Self-reference.

Curry’s Obs had a much too hard burden to carry, and they failed too. But Haskell was tricky enough to keep his job running. The paradox of the Obs, he claimed, is a natural quality of the world of Obs. The faulty worm is not in the Obs but in logic. The Obs are flying free, with and without paradoxes.

Who cares about logic?

Heinz Second-Order cybernetics had a similar short life as our thinking fish.

But those bubbles hadn’t been useless or without influence.

Another magician, Raymond Smullyan, had the real smile.

Curry's operators had been hidden by the high priests of formalistic mathematics. Only the free city of Amsterdam got and offered some insights into the secret life of the Obs and their derivatives.

When I visited the institute in the search for papers about Ultra-intuitionism I got a very friendly and open-minded welcome. But what was most striking was the set of beautiful colonial chairs in the foyer of the Centrum voor Wiskunde. In Westberlin, at this time, they wouldn’t have had a chance to survive.

Smullyan, himself a Magician, and one of the most ingenious logician ever, has freed these Obs by transforming them into birds.

Obviously Raymond wasn't aware or more probably, he might kept it as a special secret of his logical riddle books, the Urbird of all combinatorial birds, was a “schöner Fink”⁵, called Moses⁶, hidden, and nearly not recognized, in Moscow of the 20s. He was the Urvogel of the Urlogik. Unfortunately, he didn't make it over the ocean. Today, the game is called by Strachey “currying”, nobody can guess the pleasure of “schönfinkeling”. Nobody wanted to walk or rest in the Finken of Moses. Neither there is a Fink or a Moses in Ratheman’s bird zoo.

Now, there is a zoo of free birds, Mockingbirds and ordinary birds, similar to the one in Singapore, accessible for free or with ticket to everyone.

One of the most fanciest bird is the paradoxical bird Why. He sings, logically, exactly when he/she doesn’t sing. He/she must, therefore, be beyond such differences. That must be the reason why he is called Sage Bird.⁷ His derivation is impressive but nevertheless well controllable.

Y Why Bird (aka Sage Bird) SLL  (((SS)K)((S(K((SS)(S((SS)K)))))K)

Alas, Smullyan’s birds might smile, they might iterate and clone their existence in great ingeniousity, they are even able to produce a mass invasion of new birds. But those newcomers are only combinations of existing birds. And even worse, there is a kind of an oligarchy, others are calling this a family dictatorship, of a very small group of original birds. The committee is called: (IKS).

Proof that the Amsterdamer have no clue about the radicality of this ultra-paradoxical SKIer, Gotthard. A real hard guy in SKIing and thinking.

Smullyan’s birds might sing but they don’t mate. Olivier,⁸the Messiaen of living birds, Traité de rythme, de couleur et d'ornithologie (1949-1992), is feeding them, and they are feeding him, with all kind of fishes and rhythms. He is the only one who knows the life of birds, their melodies, rhythms, figures of flights. He knows the fishes and the cats, too.

To tell people what our fish couldn’t tell and what Curry disguised in complex formalistic calculi, which, today, appear in computer programming, and Heinz in his observing systems, which are observing only themselves, Olivier Messiaen’s⁹generosity to the Lord let us hear them all together in his musical work. He even changed the shape of the most classical instrument, the piano, to stay in the tempi of his birds, to an oval. His wife Yvonne Loriot¹⁰, the pianist, was his impressive bird of inspiration.

Also Olivier and Haskell probably didn’t met in Paris, there are some interesting connections between HASKELL and music composition: Haskore¹¹, runs on NeXT computers, written by Paul Hudak.

2.4.  Diamonds

Now, what can we do?

• Our fish got a knock back from his fish girl, and died.
• Curry’s Obs disappeared into computer programming languages, like HASKELL.¹²
• Heinz Second-Order Cybernetics got occupied and watered down by Niklas.
• Smullyan’s birds are singing everywhere but never mate.
• Oliver’s rhythms are too complex for the digital age based on Curry’s Obs.
• And, the real birds on my balcony at Garnethill are all dying out...

Obviously, we need something more stable.

Diamonds!

Diamonds are making birds happy.

Therefore, I have to fight against the power of my own star sign: Pisces. Or was it Aquarius?

But first, listen to Fish Fish Fish,¹³and relax.¹⁴

Watch Saul Steinberg’s fisherman!

        Saul Steinberg: The Labyrinth, N.Y. 1954

Diamonds are not unknown in logic. Nathaniel introduced the “heller Stein” as his main player of his “Diamond, A Paradox Logic”¹⁵. But again, the hidden story is not yet told. It was the “schonfinkeling” Fink who has offered “schon” before the diamond at the very beginning of the “paradox avoiding” strategy game of formal systems players. Well disguised, until now, as the bird-stone “Finkelstein".

The eyes of the fish had a glance like a diamond. But he couldn’t see the glance of his own eyes. Heinz always said, ‘you cannot see that you see’. Or, was it: We don’t see that we don’t see.

Does it matter?

Without doubt, our fish had a knowledge that he was living in water and not a perception, neither clara nor distincta; there was clearly nothing to see at all. What he could see was his complex world full of strange stuff, and this funny fish girl. But not the water. What the fish girl didn’t know, neither Heinz, was that he properly acted according to Heinz’ Ethical Imperative:

"Always act to augment the possibilities of the others."

But how could such a meta-message augment the possibilities of actions for the fish girl? Isn’t it simply an unnecessarily luxury to know that she is living in water?

But he, our fish, not Heinz, didn’t accept the fish girl’s ignorance to try to reduce the necessities of his insights. Therefore, intuitively, his dual imperative of Heinz’s altruistic maxim came into force.

Never contemplate to reduce the amount of necessities of yours!

2.5.  Skiing with IKS

Gotthard’s SKIing was on the right path of illumination. His path was profoundly chiastic. With his 2 skiers and his 2 staves he mapped with his shadow the living chiasm into the morning snow. But again, his great sophistication worked only down hill.

He was very much into Aliens, too. Gotthard told his decade long German friend, Helmut Schelski, I’m not a human being, I’m only looking like one. But with skiing he was down hill terrestrial.

The down-hill approach and Gunther’s deep desire to find a Urform, similar to the one of Lask and Curry, but profundly philosophical, albeit not necessarily Western, intrigued him into the icy world of the Form of Form, the μορϕη of form, i.e., the morphogram and its grammatics, its morphogrammatics. Only week thinkers, trapped in their logocentrism, can come with the yellow card that it better should be called grammer instead of grammatics.

The operator Gotthard discovered in his God hard insistence to his path, was what he called the operator before the operator, the prelude to any relationship, i.e. the pro-ömiality of relationality and operativity as such. In a proemial system, the difference between applicator and applicand, e.g, combinator, becomes entangeld into a chiastic interplay of changing roles.

Im neither a skier nor a pilot. This story came to (my) mind when I was siting in a bus, commuting between Glasgow and Edinburgh.

SKI or IKS is, according to the prominent computer scientist Philip Wadler, "universal, ultimate and natural". Yes, in Switzerland we say: Petri heil!, for fishing. And Hals- und Beinbruch! for skiing. And in the city some girls are hugging each other: “Chum guet unders tram!”. I’m not sure what Wadler’s aliens would wish. But one thing is crystal clear, they are all counting on SKI.

Diamonds are the most flexible Obs which have ever occurred in history. They are neither fishes nor birds nor skiers. Not definable by IKS. In fact, they are not part of history, neither of this nor of another. And neither belong to the superpositions of the negations used in this explanation.

Currys’s paradoxical Obs needed an observer to observe their paradoxical behavior. In itself, this paradoxical Ob was not moved by any struggle of a paradox. Unlike to Mao TseTungs insight: Everything is in a struggle, hence dialectical. This burden belonged to the observing observer of the Ob, not to the real world objects. Curry kept the secret of the real behavior of his Obs in his mind. Ink and paper couldn’t catch it properly. What we got are some calculi, which need an initiation by a high priest, like Henk Barendregt, to be understood.

Diamonds, as hard as they are, are a living paradoxon.

So, what’s so special to diamonds? They are and they are not. That means nothing and/or everything, hence I should stop talking and try to begin to write.

The paradoxicality of the known paradoxes, is constructed always in one direction only. Or as Curry would insist, in no direction at all. It’s not about perception and observation but about thinking only. But readable in fact in one or the other direction, exclusively.

How are paradoxes constructed? There is a sophisticated tool-box on the mathematical market. Also out of fashion a lot of devices are still in offer. Tools offered are: re-entry, recursion, diagonalization, substitution, contradiction, flip-flop, etc.

Their movement or their temporality is always uni-directional. They need time. They are consuming time. Niklas even thought, they are producing time. Produced or consumed or both at once, at the same time, their time structure is uni-linear.

Everything else appeared as nice phantasies of the post-modern couple Deleuze/Guattari with their open chiastic multitudes. Or as Paul said before/during the Parisian Cultural Revolution: "Anything goes!".

Or look at Escher! Surely, you can read his figures up and down, but you will not read the figures at once up and down, nor neither up nor down. As the neuro-cyberneticists would say, today: “Our brain is not build for that”. Or the other fundamentalists: “We haven’t yet found the proper gene for that”.

If you need more stuff to swallow, read instead of Gödel Hofstadter’s “Gödel, Escher, Bach".

The temporality of diamonds is moving at once forwards and backwards and, at the same time, diamonds are in fact the most stable Obs ever, they are beyond any move at all.

I can see this is making you dizzy and vomiting. Sorry. That’s because you are still a fish or a bird or a skier or whatever terrestrial being.

Diamonds are interacting, and enjoying their interplay; independently.

2.6.  Uni-formity of self-reference

Symbols ...presuppose the difference between familiar and unfamiliar and...enable the re-entry of this difference into the familiar...They are forms of self-reference using the self-reference of form. (Niklas Luhmann)

Luhmann re-enters the unfamiliar of his familiar/unfamiliar difference into the familiar, i.e., familiar/(familiar/unfamiliar).

What are the obstacles or fears for the inverse scenario: the familiar re-enters into the unfamiliar, i.e., unfamiliar/(familiar/unfamiliar)? Hence, at least the double movement of re-entry would have a chance to be thought. This wouldn’t be achieved with a second application of the simple re-entry manoeuvre onto the inverse difference: putting the familiar into the unfamiliar of the familiar/unfamiliar difference. This manoeuvre would have to be realized at once for the familiar and the unfamiliar re-entry.

But this little antidromic challenge is not included in any of the recursive re-entry forms George Spencer-Brown is offering his admirers. It is also not done with any double or multiple recursion function - ”Only two can play this game” - as they are well known from recursive function theory. Not to mention the complementary situation of the manoeuvre, the neither-nor of the familiar/(familiar/unfamiliar) and the unfamiliar/(familiar/unfamiliar) re-entries.

There are two reasons why such a double re-entry is not in the range of constructivist thinking.
First, what is lacking is a polycontextural number theory and semiotics, able to disseminate recursive formulas, like Spencer Brown’s re-entries.
Second, there is no insight into an antidromic time structure. In other words, recursion is realized inside the paradigm of linearity of counting and time. Antidromic movements are asking for some more space and don’t wont to be pressed into a linear succession, only.

Antidromic strategies would confuse the innocence of linear repeatability. If something is moving forwards and backwards at once the whole concept of movement becomes obsolete. If movements are neither forwards- nor backwards-oriented, then movements need structural space to realize their parallax. Such a structural space is neither endless nor finite; it is not connected to any kind of iter-/alterability. Such a world is simultaneously, beyond time, open and closed.  

Heinz is more on pictorial scenarios: On his arena of self-reference, where the play of the Circulus Creativus is performed, the popular self-form, played by the serpent Uroboros is the crazy monster-form, which is eating his own tail. Well, that's funny. But it is also quite boring. I never had the opportunity to read something deviant to this construction, say the complementary wording or, at least, dual version of the play: The tail of the Uroboros is feeding his own mouth. Well, the paradox as a whole would then be read as “eating his own tail” and “feeding his own mouth”, at once. Hence, a queer double fun in a Kasper¹⁶puppet theater.

Again, to satisfy more sophisticated clients, the play would have to become an antidromic double play, like it was such often been risked in Paris and Yale under the directorship of the master of the double-play championships, Jacques Derrida. It got its seminal performances under the auspices of Péter Szondi and Samual Weber in the 60s/70s at the Institute of Comparatistics in Westberlin. There we enjoyed, despite the total lack of any deconstructive attempts towards mathematical logic, the double science of ”La Double Séance”.

2.7.  Double-faces in real-world constellations

Self-descriptions of double-faced experiences are not always easy to grasp.

The multi-cultured¹⁷artist

"An artist's split between cultures becomes a potential means of deautomatizing worn-out formal devices, a strategy of inserting and asserting, of uprooting and defamiliarizing formal contextures. In a true sense the process exhibits a polycontextural stylistic matrix and a distributed artistic identity  -  one runs across the terms heterarchy, contexture, polycontexturality, transjunction and kenogrammatics in Gotthard Guenther’s transclassic logic investigations. Polycontextural systems apply different codes of self-observation related to different observation positions.” (Arteni)

The ¹⁸baby girl

"A BABY born with two faces is doing well one month on from her birth.
Tot Lali was born in a northern Indian village with two noses, two pairs of lips and two pairs of eyes - but only two ears.”

Her full name is hinting to a solution: symmetric and at once, asymmetric. Tot is symmetrically inverse to toT, i.e., palindromic, and Lali is asymmetrically inverse to ilaL. This is reflected in the symmetry of two noses, two pairs of lips, two pairs of eyes - and the asymmetry of only two ears.
No wonder that the baby girl Tot Lali is conceived as an incarnation of a Buddhist God: Baby Girl Born With Two Faces Worshipped as Reincarnated God. (FoxNews)

3.  New birds in the coming

3.1.  Curtain up!

Back to our story about fishes and birds, in what ever disguise.
The scenario has changed. We are now in a world of multitudes of worlds. All interacting together in a harmonic togetherness.

Alas, nothing new happens here. Now surprise, no challenge, except of keeping the multitudes together in harmony. Cleanly distributed, well-balanced and mediated.
But only epistemological blindness can oversee what is happening behind the stage.

Birds are mating!

New unforeseen birds are emerging; never seen before. Neither confirming nor disturbing the brave harmony of polycontextural choreography. On stage, behind stage, without stage. Everywhere/nowhere.

It was a hard work of sophisticated combinatorics to combine the basic birds of the SKI committee together to invent new birds on stage. A wise bird like LU(LU) was cleverly disguising the fact that she was nothing more than the old guys in a fancy disguise.

Combinator birds at Ratheman’s¹⁹experimental zoo - free access! - are giving the clue of LU(LU)'s complexity.

L  Lark   ((S((S(KS))K))(K((S((SK)K))((SK)K))))

U Turing ((S(K(S((SK)K))))((S((SK)K))((SK)K)))

A bird of a highly complex behavior is the Finch. He can’t do it in a simpler way.

  F Finch
((S(K((S((SK)K))(K((S(K(S((SK)K))))K)))))((S(K((S(K((S(KS))K)))((S(KS))K))))((S(K(S((SK) K))))K)))

Rathman’s bird zoo might be rich but there is no such birds to find like a Fink or another diamond bird.
There is also not much to bet on the game of disseminating LU(LU) into different shapes of colored contextures. Such a dissemination of LU(LU) over the grid of polycontexturality would be highly colorful and intriguing. But it wouldn’t leave the general scenario of repeatability.

This can be seen with the colorful behavior of a colleague of LU(LU), the distributed Y-operator WHY.²⁰His advantage is clear, he freely can dance his circular pirouettes and jumping his salti without being conceptually confused and logically trapped by Curry’s paradoxes and the Uroboros of Heinz’ observers pas-de-deux.

Things are much more surprising if we change, not only our view-points of observation, but if we jump out of the whole observational circus of observing systems.

What was not seen by the designers of the play was the fact, that with each combination of birds not only an anticipated new combinator was defined, fulfilling the conditions of constructive definitions, but also a new, not expexted surprising and not allowed counter-combinator, violating the rules of definitorial reductionism, is taking place, well hidden in the backstage. It is time to open the eyes to see what always was there to see.

Curtain up!

Each observation in this non-observational game is co-creating to its observables of observation new observables, which are creating new observers, observing, in an act of self-observing, their complementary play of being observers and observables at once. Hence, diamond observers are neither observables nor observers of any observation.

3.2.  Diamond category

Hence, a combination of two combinator has, according to the diamond rules²¹, a double result: horizontally, the superpositional composition of the two combinators and simultaneously, vertically, an antidromic hyperposition of the creation of a new combinator.

Thus, diamond combinations are not simply behaving like Heinz’ copulation of his Bio-Logic (1962), and are also not primarily producing superadditive logical systems, like it is the case, in Gunther’s setting, for the intriguingly complex polycontextural birds, but are enabeling by co-creation the emergence - ("rise from water and obscurity”) - of a new kind of combinators.

To use category theoretical terms, the composition of morphism f and g to (fg) is not only resulting in the composed morphism h, h=(fg), but at once, antidromically and enantiomorph into the hetero-morphism l. In other words, a diamond object Diam(Ob) of a diamond composition is a bi-object [X, x], consisting at once of categorical and saltatorical components. Bi-objects are complexions, unifying interactively, identity and diversity. Hence, Obs in diamonds or Diamonds in Obs, are always appearing in a double determination, at once being identical and different to themselves.

The consequences for the entire paradigm of composability, based, as we learned, again and again, on iterability and repeatability, linear or not linear, are enormous. Not only an absolute new kind of double-compositionability²²appears on stage, even more. Primary to all kind of composition, there is the difference, i.e. the differentness of identity and difference, between superpositional and antidromic combinations. Instead of dealing with superpositionality alone, interactionality and reflectionality between superpositional and antidromical movements, iter-/alterabilities, are taking place, well positioned in the kenomic grid of Diamond ²³Strategies.

The double strategie of moving at once forwards and backwards, of counting and combining formal term sequences, simultaneously in different direction, at the very basis of the definition of rationality, has lost all of its absurdity. Neither parallax nor paradox, nor any re-entry is able anymore to seduce to enter any trance.

This is really a great relief!

Forget debates about the monocontexturality of combinatorial logics, their fixation on alphabetism and its linearity and atomicity, as sine qua non of all their composability.
Forget the postmodern theater of disseminating colored contextures of repeatability.
Forget the phantasm of our hidden universal mockingbirds in whatever fibered forests.

Listen to the songs of free mating birds! Enjoy your Diamonds!

Notes