Rudy's Diamond Strategies

This complementary Blog to the Chinese Challenge Blog is presenting studies to a mathematical theory of Diamonds. Diamond theory is studying for the first time, tabular categories as an interaction of categories and saltatories.

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Friday, June 22, 2007

The Diamond Book, Another Intro

The White Queen says to Alice:
"It's a poor sort of memory that only works backwards".

Diamond Strategies and Ancient Chinese thinking

"expanding categories", "mutual relations", "changing world"

To diamondize is to invent/discover new contextures.
"A good mathematician is one who is good at expanding categories or kinds (tong lei)."
"Chinese mathematical art aims to clarify practical problems by examining their relations; it puts problems and answers in a system of mutual relation—a yin-yang structure for all the things in a changing world. The mutual relations are determined by the lei (kind), which represents a group of associations, and the lei (kind) is determined by certain kinds of mutual relations."

"Chinese logicians in ancient times presupposed no fixed order in the world. Things are changing all the time. If this is true, then universal rules that aim to represent fixed order in the world for all time are not possible." (Jinmei Yuan)
Given those insights into the character of Ancient Chinese mathematical practice the question arises:
How can it be applied to the modern Western way of doing math?

The concept of composition is fundamental for category theory, thus we have to start our diamond deconstruction with it.
"... category theory is based upon one primitive notion – that of composition of morphisms." D. E. Rydeheard
If we agree, that the most fundamental operation in math and logic is to compose parts to a composed composition, as in category theory, then we have to ask:
How can the Chinese way of thinking being applied to this most fundamental operation of composition?

Tabular structure of the time "now"
"Chinese logical reasoning instead foregrounds the element of time as now. Time, then, plays a crucial role in the structure of Chinese logic." (Jinmei Yuan)
Because of the "mutual relations" and "bi-directional" structure of Chinese strategies I think the time mode of "now" is not the Western "now" appearing in the linear chain of "past–present–future". To understand "now" in a non-positivist sense of "here and now" it could be reasonable to engage into the adventure of reading Heidegger’s and Derrida’s contemplation about time. This seems to be confirmed by the term "happenstance" (Ereignis) which is crucial to understand the "now"-time structure."

Hence, the temporality of "now" is at least a complementarity of "past"- and "future"-oriented aspects. In other words, "now" as happenstance (Ereignis) is neither past nor future but also not present, but the interplay of these modi of temporality together.
"Deductive steps are not important for Chinese mathematicians; the important thing is to find harmonious relationships in a bidirectional order." (Jinmei Yuan)
There is no need to proclaim any kind of proof that the diamond strategies are the ultimate explication and formalization of Ancient Chinese mathematical thinking. What I intent is to elucidate both approaches; and especially to motivate the diamond way of thinking. Borrowing Ancient insights as metaphors and guidelines to understand the immanent formal stringency of the diamond approach.

Time-structure of mathematical operations
I’m in the mood, now, to belief that I just discovered a possibility to answer this crucial question, where and how to intervene into the fundamental concept of composition in mathematics and logic. The possibility to intervene discovered my readiness to perceive its lucidity to be written into the darkness of this text.

In a closed/open world things are purely functional (operational) and objectional, at once.
Western math is separating objects from morphisms. This happens even in the "object-free" interpretation of category theory.

My aim is not to regress to a state of mind, where we are not able to make such a difference like between objects and morphisms, but to go beyond of its fundamental restrictiveness.

Towards a diamond category theory

A morphism or arrow between two objects, morph(A, B), is always supposing, that A is first and B is second. That is, (A, B), is an ordered relation, called a tuple. It is also assumed that A and B are disjunct.

To mention such a triviality sounds tautological and unnecessarily. It would even be clumsy to write (A;first, B; second). Because we could iterate this game one step further: ((A;first;first, B; second;second) and so on.

The reason is simple. It is presumed that the order relation, written by the tuple, is established in advance. And where is it established? Somewhere in the axioms of whatever axiomatic theory, say set theory.

In a diamond world such pre-definitions cannot be accepted. They can be domesticated after some use, but not as a pre-established necessity.

Hence, we have to reunite at each place the operational and the objectional character of our inscriptions.

As we know from mathematics, especially from category theory, a morphism at its own is not doing the job. We have to compose morphisms to composed morphisms. At this point, the clumsy notation starts to make some sense.

The conditions of compositions are expressed, even in classic theories, as a coincidence of the codomain of the first morphism with the domain of the second morphism. Hence, the composition takes the form:

When we met, it wasn't that you and me met each other, it was our togetherness which brought us together without our knowledge of what is happening with us together.

And now, a full complementation towards a Diamond category.

Your brightness didn't blend me to see this minutious difference in the composition of actions. What confused me, and still is shaking me, is this coincidence and synchronicity of our encounter and what I started to write without understanding what I was writing and how I could write you to understand our togetherness.

Which could be the words left which could be chosen to write you my wordlessness?

We are together in our differentness. Our differentness is what brought us together. We will never come together without the differentness of our togetherness.
Our togetherness is our differentness; and our differentness is our togetherness.

You have given me the warmth I needed to open my eyes.

Together we are different; in our differentness we are close.

Our closeness is disclosing us futures which aren't enclosing our past.

Was it coincidence, parallelism and synchronicity or simply the diamond way of life which brought us together, not only you and me, but us together into our togetherness and into the work which has overtaken me?

What I couldn't see before, that always was in front of me, was illuminated by the brightness of your feelings.

I was walking on the pavement, thinking about all this beautiful coincidences and the scientific problems of the temporal structure of synchronicity. And just at this moment I heard a voice calling my name. It was you on your bike. I had been stuck in my thoughts, you in a hurry and the dangers of the traffic. But down to earth and the street, doing what made me happy. A différence minutieuse. Giving me a hug and a kiss.

"Bump, is a meeting of coincidence!
", you text me.

Then I started to write this text as another approach to an Intro for The Book of Diamonds, to be written.

What are our diagrams telling us?
First of all, the way the arrows are connected is not straight forwards. There is additionally, a mutual counter-direction of the morphisms involved. Because of this split, the diagram is mediating two procedures, called the acceptional and the rejectional. Thus, an interaction between these two parts of the diagram happens. Such an interaction is not future-oriented but happens in the now, the happenstance, of its interactivity.

All the goodies of the classical orientation, the unrestricted iterativity of composition, is included in the diamond diagram. Nothing is lost. Morphisms in categories are not only composed, but have to realize the conditions of associativity for compositions.

Complementarity of composition and hetero-morphism

The composition is legitimate if its hetero-morphism is established. If the hetero-morphism is established the composition is legitimate. The hetero-morphism is legitimating the composition of morphisms.

Only if the hetero-morphism of the composition is established, the composition is legitimate.

Only if the composition of the morphism is realized, the hetero-morphism is legitimate.

connectivity and jumps

I didn’t look for you; you didn’t look for me. We didn’t look for each other. Neither was there anything to look.

It happened in the happenstance of our togetherness.

We jumped together; we bridged the abyss.
You bridged the abyss; I bridged the abyss.
In a balancing act we bridged the abyss together.
The abyss bridged me and you.
The bridge abyssed us together into our differentness, again.

Une quadrille burlesque indécidable.

Now I can see, I always was looking for you.
But I couldn't see in the darkness of my thoughts that you had been there for all the time.

We learned to live with the deepness of our differentness. Discovered guiding rules to compose our journeys.

The time structure of synchronicity is antidromic, parallel, both at once forwards and backwards. Not in chronological time but in lived time of encounters and togetherness.

You have given me the warmth I needed to open my eyes.

Associativity of saltatories
With the associativity of categories new insights in to the functionality of diamonds are shown.
Diamonds may be thematized as 2-categories where two mutual antidromic categories are in an interplay. Hence possibiliy, not exactly in the classic sense of 2-category theory neither in the sense of the polycontexturality of mediated categories.
Another notation is separating the acceptional from the rejectional morphisms of the diamond. A diamond consists on a simultaneity of a category and a jumpoid , also called a saltatory. If the category is involving m arrows, its antidromic saltatory is involving m-1 inverse arrows. Some simplification in the notation of saltatories is achieved if we adopt the category method of connecting arrows. This can be considered as a kind of a double strategies of thematization, one for compositions and one for saltos.

With such a separation of different types of morphisms, diagram chasing might be supported.

What went together, too, is the fact that I changed to a PPC, hence, this text written here, is written on the fly. For you and me.

In our togetherness we are separated.

In our separateness we are associated.

Together, nous some un ensemble très fort.

Diamond rules

On the other side, I was aware that something special will happen this year. I told this my son. It is an odd year. I love odd numbers. But as we know there are about the same amount of even numbers. And there is something more.
Our society told me all the time, that, in my age, it will be
time for the very end of the game.

Hence, I had to make a difference and to start a new round in this interplay of neither-nor. And that's what's going on, now.

It is this difference you made , I was blind before.
After the difference made myself, I can see, how to meet you, again.

To play this game of sameness and differentness as the interplay of our relatedness.

I remember, you said: "Later!".

What’s new?
Hence, what is new with the diamond approach to mathematical thinking is the fact, that, after 30 years of distributing and mediating formal systems over the kenomic grid with the mechanism of proemiality and tetradic chiasms, which goes far beyond "translations, embeddings, fibring, combining logics", I discovered finally the hetero-morphisms, and thus, the diamond structure, inside, i.e. immanently and intrinsically, of the very notion of category itself.

First steps, where to go

Following the arrows of our diagram some primary steps towards a formalization of the structure of our cognitive journeys may be proposed.

As written above, diamonds don't fall from the blue sky, we have to bring them together, for a first trial, to borrow methods, with the well known formalizations of arrows in category theory.

After the entry steps, the nice properties of associativity for morphisms and hetero-morphisms are notified.

The definition of units has to interplay with identity and difference.

To not to lose ground, a smallness definition is accepted, at first.

As in category theory, many other approaches are accessible to formalize categories. The same will happen with diamonds; later.

Tuesday, June 19, 2007

The Book of Diamonds, Intro

Pour Lorna Duffy Blue, qui ma poussé, à tout hasard,

dans une quadrille burlesque indécidable.
Printemps 2007, Glasgow

A book I didn’t write

This is not the book I wanted to write. Nor did I want to read the book I didn’t write. What you are reading now is the book which has written me into the book of diamonds I never owned. I never wanted to write you such a book. Nor that you are reading the book I didn’t write.

It happened in a situation where I lost connection to what I have just written and what I had written before, again and again. While I was writing what I wanted to write I was writing what I never thought to write. A book of Diamonds. Or even The Book of Diamonds.

I haven’t written this book. After I have written some parts I started to read it. I think what happened is the most radical departure from Occidental thinking and writing I ever have read before.

I remember vaguely what I was writing all those years before. I tried to read it and had the feeling to discover a way of thinking which has become a dark continent of what I always wanted to think but never succeeded. This is because this darkness wasn’t illuminated enough to let discover this tiny but most fundamental difference in the way we are thinking and doing mathematics.
What jumped into my eyes, or was writing itself automatically into my formula editor, was the resistance of a difference to be levelled by the common approach of thinking.

The brightness of the new (in)sight is still troubling me.

It isn’t my aim to write this book. I never wanted to write a book.
Nevertheless, I don’t see a chance not to write this text as The Book of Diamonds wether or not I’m in the possession of diamonds. Nor do I want to be the author of a book I didn’t write myself.

What troubles me, is that, as a matter of course, I don’t understand what I have written in this book yet to be written.

The most self-evident situation, which is leading our thinking in whatever had been thought before, has become obsolete in its ridiculous restrictiveness.

Before I was overtaken by this tetra-lemmatic trance sans dance, I tried to overcome and surpass this boring narrowness of our common thinking by wild constructions of disseminated, i.e., distributed and mediated, formal systems. Like symbolic logic, formal arithmetic, programming languages and even category theory. This was a big step beyond the established way of thinking. And it still is.

But that isn’t the real thing to write.

The striking news is the discovery of a new way of writing. Writing, until now, was the composition of letters, words and sentences to a composite, called text or book. The composition operation is no different from the composition of journeys. Let’s have a look at how journeys are composed together to form a nice trip. We will be confronted with some surprising experiences in the middle of safe commodities.

Different times?

What is well known in time-related arts, that the temporality of a piece can be an intertwined movement of different futures and different pasts, is a thing of absolute impossibility in science and mathematics.

Time in science is uni-directional. It may be linear, branched or even cyclical, it remains oriented in one and only one direction. It is the direction of the next step into the future. But what we also know quite well is the fact that this is not the time of life, it is the time of chronology. Chronology is connecting time with numbers, forgetting the liveliness of lived time. Watchmakers know it the best.
Can you imagine a Swiss watch running forwards and backwards at once? Or our natural numbers, being disseminated and interwoven into counter-dynamic patterns? Utter nonsense!

Today, everything has to be linearized to be compatible with our scientific world-view and to be computed by our computerized technology and be measured by our chronology. No cash--point is working without the acceptance of global linearization.

We need this simple structure to compose our actions in a reasonable way. Reason is reduced to the ability to compose. To compose actions is the most elementary activity in life as well in science and maths. Hence, it is exactly the place to be analyzed and de-constructed in the search for a new way of composing complexity.

Well developed in time-based arts are patterns of poly-rhythms, poly-phony, multi-temporality of narratives, interwoven and fractal structures of stories, tempi developing in different directions, even the magic I’m interested in this book to be written, the simultaneous developments of tempi in contra-movements, at once forwards and backwards, and neither in the one nor in the other direction, and all that at once in a well balanced "harmony". This is not placed in the world of imagination and fantasy, only, but becoming a reality in our life, technology and science.

What’s for?

As we know, time-related arts can be of intriguing temporal complexity. And the fact, that it happens in a limited and measurable time at a well-defined place for a calculable price is not interfering with its artistic and aesthetic complexity.
In terms of a theatre play we can imagine, and realizing it much more distinctively as it has been done before, a development of the drama at once forwards, future-oriented, and backwards, past-oriented. Both, simultaneously interplaying together.

This is not really new in drama, music or dance, nor in film, video and other time-related arts. But there is no theory, no instrumental support for it, thus based entirely on intuition, and therefore highly vulnerable and badly restricted in its possible complexity. At the same time, the paradigm of linearized and calculable time is intruding all parts of our life. It becomes more and more impossible for the arts to resist this way of thinking and organizing life.

The aim of the diamond approach is to reverse this historic situation. Complex temporal structures have to be implemented into the very basic notions and techniques of mathematics itself. With the diamond approach we will be able to design, calculate and program the complex qualities of interplaying time structures.
To achieve and realize this vision of a complex temporality, we have, paradoxically, to subvert the hegemony of time and time-related thinking. Different time movements can be interwoven only if there is some space offered for their interactions. Hence, a new kind of spatiality, obviously beyond space and time, has to be uncovered, able to open up an arena to localize the game of interacting time lines.

How to travel from Dublin to London via Glasgow?

Metaphorically, things are as trivial as possible. If you are travelling from Dublin to Glasgow you are doing a complementarity of two moves: you are leaving Dublin, mile by mile, and at the same time you are approaching Glasgow, mile by mile. What we learned to do, until now, is to travel from Dublin to Glasgow and to arrive more or less at the time we calculated to arrive.

To practice the complementarity of the movement is not as simple as it sounds. You have to have one eye in the driving mirror and the other eye directed to the front window and, surely, you have to mediate, i.e., to understand together, what you are perceiving: leaving and approaching at once. And the place you are thinking these two counter-movements which happens at once is neither the forward nor the backward direction of your journey. It’s your awareness of both. Both together at once and, at the same time, neither the one nor the other. It is your arena where you are playing the play of leaving and arriving.

This complementarity of movements is just one part of the metaphor.
Because life is complex, it has to be composed by parts. Or it has to be de-composed into parts. We may drive from Dublin to Glasgow and then from Glasgow to London to realize our trip from Dublin to London. This, of course, is again something extremely simple to think and even to realize.

But again, there is a difference to discover which may change the way we are thinking for ever.

To arrive and to depart are two activities, i.e., two functions, two operations. Dublin, Glasgow and London as cities have nothing to do with arrivals and departures per se. They are three distinct cities. We can arrive and we can depart from these cities. But cities are not activities but entities, at least in this metaphor of traveling.

Things come into the swing if applied to the quadrille.

Obviously, Glasgow, in this case, is involved in the double activity of arrival and departure. It also seems to be clear, that the city Glasgow as the arrival city and Glasgow as the departure city are the same or even identical. It wouldn’t make sense for our exercise if the arrival city would be Glasgow in Scotland and the departure city Glasgow would be Glasgow in the USA.

But what does that mean exactly? If we stay for a while in Glasgow before we move on to London, Glasgow could have changed. Is it then still the same Glasgow we arrived in? And the same from which we want to depart? It could even happen that the city is changing its name in between!

On the other hand, it doesn’t matter how much Glasgow is changing, the activity of arrival and the activity of departure are independent of a possible change of Glasgow.

It seems also quite clear, that the activity of arrival and the activity of departure are not only different but building an opposition. They are opposite activities.
It is also not of special interest for our consideration if the way of arriving and the way of departing is changing. Instead of taking a bus to leave Glasgow we could take a train or an airplane. Nothing would change the functionality of departing and arriving as such.

Thus, we can distinguish two notions in the movement or even two separated movements playing together the movement of the journey:
1. Dublin––> Glasgow ––> London, and
2. departure ––> arrival/departure ––> arrival.
The classic analysis of the situation would naturally suppose that there is a kind of an equivalence or coincidence between Glasgow as arrival city and Glasgow as departure city, hence not making a big deal about the two distinctions just separated. Thus:
City-oriented travel diagram

A closer look at the place where the connection of both parts of the travel happens shows a more intricate structure than we are used to knowing. If we zoom into the connection of both journeys we discover an interesting interplay between the function of arrivals and the function of departures.

Activity-oriented diagram

The activity-oriented diagram is thematizing what really happens at the place of "arrival(Glasgow)=departure(Glasgow)". That is, the internal logical structure of the simple or simplifying equality, "arrival(Glasgow)" and "departure(Glasgow)", is analyzed and has to be studied in its 2-leveled structure and its complementary dynamics.

Obviously, the travel from Dublin to Glasgow, and from Glasgow to London is a composition of two sub-travels. Thus, the composition "o" in the first diagram is working only if the coincidence of both, Glasgow(arrival) and Glasgow(departure) is established.

If this coincidence is not given, the composition of the journeys cannot happen. Maybe something else will happen but not the connection of both journeys we wanted to happen. If we wanted to model what happened if it didn’t happen we would have to draw a new diagram with its own arrows and it wouldn’t be bad to find a connection from the old diagram to the new one.

What is the zoom telling us?

First, we observe the composition of the part-travels "o" aiming forwards to the aim.
Second, we discover a counter-movement in this activity of connecting parts, aiming into the opposite direction of the composition operation.

It may not be easy to understand why we have to deal with complicating such simple things. But we remember, even a single journey, without any connections, is a double movement. It is always simultaneously a dynamic of away and anear, to and fro, an intriguing mêlée of both. Not a toggle between one and the other, no flip-flop at all, but happening simultaneously both at once, coming and going.

Hence, it comes without surprise, that this mêlée happens for compositions too. In fact, it becomes inevitable in light of compositions. We simply have to zoom into it. We could forget about this complications if we would be on one and only one travel for ever. Then the backsight or retrospect would become obsolete. And only the foresight or prospect would count. Or in a further turn, only the journey per se without origin nor aim could become the leading metaphor.

Funnily enough, that is the way life is organized in Occidental cultures, modern and post-modern.

More profane, everything in the modern world-view is conceived as a problem to be solved, i.e. life appears as problem solving.
Soon, happily enough, machines will overtake this destin sinistre.

Diamonds are not involved into the paradigm of problem solving and its time structure but are opening up playful games of the joie de vivre, spacing possibilities where problems can find their re-solution.

Lets go on! Keep it real!

This intriguing situation we are discovering with our zoom, happens for all stations of our travel. We started at Dublin and ended in London. And these two stations are looking simple and harmless. But this is only the case because we have taken a snapshot out of the dynamics of traveling. That is, in some way we arrived before in Dublin and at some time we will leave London. Hence, Dublin and London have to be seen in the same light of dynamics between the categories of arrival and departure as it is the case for Glasgow as the connecting interstation to London.

Coming to terms

In mathematics, the study of such composed arrows is called category theory. Category theory is studying arrows (morphisms), diamond theory is studying composition of morphisms as the primary topic. The activity is not in the arrows but in the composition of the arrows. Hence, the complementary movement of the rejectional arrows (morphisms). At the cross-point of compositions the magic complementarity of encounters happens.

There is nothing similar happening with morphisms alone and their objects. Category theory, without doubt, is dealing with compositions, too. But the focus is not on the intrinsic structure and dynamics of the composition itself but on the construction of new arrows based on the composition of arrows (morphisms).

Without such a magic of complementarity there is no realm for rendez-vous.
Departure is always the opposite of arrival. But this simple fact is also always doubled. The departure is the double opposite of arrival, the past arrival and the arrival in the future. Thus, the duplicity has to be realized at once.

Let’s read the diagram!

We can change terms now to introduce a more general approach to our intellectual journey. We replace for departure "alpha" and for arrival "omega" and omit the names of the cities. We get the first diagram. Then we stretch it to a nicer form. This is the diamond diagram of the arrows. Connected with a known terminology we get into the diamond of (proposition, opposition, acceptance, rejectance).

Further wordings
The class of departures can be taken as the position of proposition.
The class of arrivals can be taken as the position of the opposition.
The class of compositions can be taken as the position of the acceptance.
The class of complements can be taken as the position of the rejectance.

Acceptance means: both at once, proposition and opposition.
Rejectance means: neither-nor, neither proposition nor opposition.

Putting things together again, cities and activities, we get a final diagram

We learned to deal with identities, Glasgow is Glasgow. But our diagram is teaching us a difference. Glasgow as arrival city and Glasgow as departure city are not the same. As the location of arrival and departure of our journey, they are different.
More insights into the game are accessible if we go one step further with our journey.

Category theory as the study of arrows is studying the rules of the connectedness of arrows. The diagram above, with its 3 arrows f, g, h and its compositions (fg), (gh) and (fgh), shows clearly one of the main rules for arrows: associativity.

In a formula,
for all arrows f, g, h: ( f o g ) o h = f o ( g o h ).

Applying associativity to our journey analogy we have to add one more destination.
Hence, if we travel from (Dublin to Glasgow and from Glasgow to London) and then from (London to Brighton), we are realizing the same trip as if we travel first from (Dublin to Glasgow) and then from (Glasgow to London and from London to Brighton).

In contrast, within Diamond theory, for the very first time, additional to the category theory and in an interplay with it, the gaps and jumps involved are complementary to the connectedness of compositions. The counter-movements of compositions are generating jumps.

In our diagram: between the red arrows l and k there is no connectedness but a gap which needs a jump. We can bridge the separated arrows by the red arrow (kl), which is a balancing act over the gap, called spagat. If we want to compromise, we can build a risky bridge: (lgk), which is involving acceptional and the rejectional arrows. Both together, connectedness and jumps, are forming the diamond structure of any journey.

Positioning Diamonds

The part of the book I have written myself is the part of localizing or positioning diamonds into the kenomic grid of polycontexturality without knowing exactly their internal structure. Diamonds are not falling from the blue sky, they have to be positioned. This happens on different levels in the tectonics of the graphematic system. The logical structure of distributed diamonds, especially, is enlightening this brand new experience and is producing further insights into the diamond paradigm of writing.

Diamonds in Ancient thinking

Furthermore, a connection is risked between diamond thinking and ancient Greek, Pythagorean, and the ancient Chinese way of thinking. Diamonds are not necessarily connected with any phono-logocentric notions. That is, diamonds are inscribed beyond the conception of names, notions, sentences, propositions, numbers and advice. Diamonds are not about eternal logical truth but are opening up worlds to discover. Diamonds had been surviving in Western thinking as neglected creatures, reduced to logical entities, like Aristotle’s Square of Oppositions. To do the diamond, i.e., to diamondize is still the challenge we have to enjoy to risk.

We are proud to live our life in an open world, not restricted to any limitations, allowing all kind of infinities, endless progresses, and feeling open to unlimited futures.

This enthusiasm for an open, infinite and dynamic world-view can be summarized in the very concept of natural numbers. Their counting structure is open and limitless.

With such an achievement in thinking and technology we are proud to distinguish our culture from Ancient cultures which had been closed, limited and static, and often involved with cyclic time-structures and their endless repetition of the same.
At a time where this proudness has achieved its aims, we are wakening up from this dream of liberty.

The whole hallucination of the openness turns round into the nightmare of a sinister narrowness of endless iterativity and the shocking discovery of the endlessness of its resources.

It is time to acknowledge that the Ancient world-view wasn’t as closed as its critics propagated. In fact since Aristotle we simply have lost any understanding of a world-view which is neither open nor closed, neither finite nor infinite, and neither static nor dynamic, simply because these distinctions are not thought in the sense of the Ancient world-view but in the modern way of thinking. Its simple two-valuedness is automatically forcing this attitude of thinking to evaluate the binaries involved, i.e. open is good, closed is bad, dynamic is good, static is bad, infinite is good, finite is bad.

closed, static, temporal vs. open, dynamic, eternal worlds

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.

In an open world it wouldn’t make much sense to run numbers forwards and backwards at once. But in a closed world, which is open to a multitude of other worlds, numbers are situated and distributed over many places and running together in all directions possible. Each step in a open/closed world goes together with its counter-step. There is no move without its counter-move.

If we respect the situation for closed/open worlds, then we can omit the special status of an initial object. That is, there is no zero as the ultimate beginning or origin of natural numbers in a diamond world. Everything begins everywhere.

Thus, parallax structures of number series, where numbers are ambivalent and antidromic, are natural. It has to be shown, how such ambivalent and antidromic number systems are well founded in diamonds.

What’s new?

So, after all these journeys about journeys, what is new and interesting about at all?
To cite, what I might have written, I can answer this question with an interrogative first trial. But first, I have to write, what’s new is the fact that I’m writing without knowing what I’m writing. Until now, I was quite aware and in control of my writings.
"If everything is in itself in a contractional struggle, involved into the dynamics of its opposites, hence, what does it mean for the most fundamental mathematical action, the composition of objects (relations, functions, morphisms, etc.)? The main opposites of thinking are sameness and differentness (difference, distinctness, diversity). They have to be inscribed in their chiastic interplay. How can their struggle at the place of the most elementary mathematical operation be inscribed?"
The discovery of the realm of rejectionality, the "royaume sans roy et capital", which is inscribing the writer into his writing, is the new theme of writing to be risked and explored.

All this together could become a book I would like (you) to read. What is written now could be called a sketch, or a proposal of a book I would like to write. But such a book would remain, necessarily, an endless sketch. What I have done until now was to disseminate formal systems (logics, arithmetic, category theory, etc.) based on triadic structures, i.e., I diamondized triangles (triads).

Classical thinking is dealing with dyads, like (yes/no), (true/false), (good/bad).

Modern thinking tries to be involved with triads: (true/false/context) or (operator/operand/operation).

The brand new exciting event to enjoy is: Diamondization of diamonds!

How to play the game of tetrads of tetrads, diamonds of diamonds?

How to do it?
Let’s do it!

Read the book to be written: "The Book of Diamonds".