The Diamond Book, Another Intro
The White Queen says to Alice:
"It's a poor sort of memory that only works backwards".
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."Given those insights into the character of Ancient Chinese mathematical practice the question arises:
"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)
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. RydeheardIf 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?
"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.
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.
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.
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.
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.
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.
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.
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!".
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.