The CIE (Ψ)

“CIE” is pronouced /saɪ/ or /si/ whichever rhymes with the Greek letter “Ψ”.

While the three components of the Trinion (○, ↑, =) show the objective and qualitative structure of the Universe, the Trinion itself observes the Universe via the subjective and quantitative observer-operator(obop) of the FoC. The obop being (◉) works with the following three components:

• Content (c): This is the object to be observed and measured, the measuree, the domain of operation, the unknown amount of material.
• Intent (i): This is the subject that measures the object, the measurer, the unit of measurement, the modulus of division, the operator, the known amount of material which act as the container of content. In perception, it's the “eye's self”, the projector that projects the material object onto the mental world. The operation of measuring or projecting the unknown (c) against the known (i) is illustrated by the division c/i.
• Extent (e): This is the number resulting from the measurement, the rage of the operation, the image of projection. This number is non-material and purely a mental construction by the intent.

There, “CIE” stands for “content/intent = extent” or “content = intent × extent”, where “extent” is the image of the object's “content”, and “intent” gives meaning to that image.

In dualistic view, these three components are 2-1 divided rotationally in three partitions:

• Sustent (c, i) - Extent (e): In the mapping i: c → e, the source c and the mapper i are the ones that determine and maintain the image e which, in its turn, extends the intent i into the extension i⋅e. Thus, the construct used to extend the intent is called “extent” (e). The extent is the extended part of the content, denoting the size, magnitude, amount of content. And the substances used to sustain the extension are called “sustents” (c, i). Sustent is the invisible underlying substance of all things that are expressed in the mind of the subject as visible extents which are just mental images. In this way, the partition “sustent - extent” is also called the “Physical (c, i) - Informational (e)” or “material - mental” partition. While the extent cannot exist without the sustent, the sustent is self-sustained because it's a closed circle comprising two complements of form (intent i and extension i⋅e) and content (c).
• Form (i, e) - Content (c): In the projection c →_i e, the internal form i of the obop's eye defines the format for the content c, so that c appears to the obop as the external form e. While in philosophy and science, “form” usually refers to internal forms like format, formular, structure, in natural language, “form” usually refers to external forms like shape, appearance. Here, the two forms are united to contrast with the content. While the external form e is “abstract”, i.e. empty of content, the internal form i is “concrete” with crystalized content from its sustents. Another concrete form is the “concrete extension” i⋅e denoting concrete objects in the material world of the obop. In contrast, the extent e is an “abstract extension” denoting abstract objects in the mental world of the obop. Most of the maths, logic, and information technology work with abstract extensions only, which manifests as numbers, symbols, signs, sets, etc. Because the internal form of the eye is especially important to the subject (obop) and the pronunciation of letter “i” is the same as the word “eye”, “i” is usually used to denote the eye of projection, where it can also means “I, the subject of projection”. Because the internal form i represents the self of the subject, and the external form represents the image projected in the mind of the subject, this partition is also called the “M(i, e) - W(c)” (“Mine vs Wild”, or “Me vs World”) partition, or the “Mirror partition”. Here, the form (concrete extension i⋅e) reflects the content: c = i⋅e.
• Extensive (e, c) - Intensive (i): In the observation function e = ob_i(c) = c/i and the operation function c = op_i(e) = i⋅e, the variables e & c are extensive properties, while the parameter i is an intensive property, as in the categorization of physical properties. That means, in the subjective view of the obop, its intent i is kept invariant, while the extent e varies directly with the conent c. Otherwise, in the objective view, the total content of the Universe (domain of discourse) is considered fixed c = 1, hence the inverse variation of the two forms, intension i and abstract extension e: 1/i = e. Here, the intent i gives intension, meaning, to the extent e, “e means e times i”, so that the abstract extension e can be used in place of the concrete extension i⋅e. As the obop i ranges over the content c to yield corresponding number e, i is known as the index, the indicator of the “current position” (the “here”), also the interator of the “current time” (the “now”).

Actually, the name of the three components “content - intent - extent” are taken from the fewer side of each partition. Othwerwise, the triple (c, i, e) can be called “content - context - form”.

Relativity & Pseudo-absoluteness

• Intrinsic relativity, apparent form & measured content: All forms and contents are only visible, measurable via their extent against some eye i, and that extent shows the external form apparent to that eye, which is called “apparent form”. That means all forms and contents are intrinsically relative to each other, and their relation c/i, denoted by e, is all the information they can see and agree together. In other words, the “content” c is just the form that is subjected to measurement by the “intent” i, hence “content” = “object form” and “intent” = “subject form”. The extent e seen by i agrees with c in the sense that, as a subject form, c sees that i has extent 1/e. Considering intents as units, we have the measured content of c is i⋅e and the measured content of i is c⋅1/e. Note that the measured content i⋅e is just an aspect of the underlying content c which cannot be measured completely, i.e. the content itself is immeasurable even though measurable .
• Pseudo-absoluteness, crystal form & crystalized content: When there's a common intent i₀ which is so fine that can measure all forms in a world, that world has a sense of absoluteness where all forms have “pseudo-absolute extent e₀” defined by i₀. As long as i₀ is stable, the “pseudo-absolute extent e₀” is stable and can be used to express the “crystalized content” of all forms in this world, where the common unit i₀ is ignored. The finer i₀ is, the more stable the “crystalized content” is, as well as the “crystal forms” made up from this “crystalized content”. Note that while the notion of an “infinitely fine common intent” can define an “absolute world” with all forms absolutely defined exactly by an “absolute extent”, in reality, no matter how fine the common intent is, all extents are relative to that common intent. Hence, that absoluteness is just a pseudo-absoluteness. However, that pseudo-absolute extent is enough to show various hidden structures underlying the relative world, e.g. the triangle inequality.
• Triangle inequality:
  
  The the amount of crystalized content of a form must be greater, and usually much greater, than any measured content of that form, i.e. c > i⋅e and i > c⋅1/e. That crystalized content is called the “capacity” of the corresponding crystal form.

The CIE in Unithread and Knowledge Theory

View - proj = distortion = curve = force
rotation = flection (deflect, reflect, multiflect)
self-reflect: i = c/e

Apendix

A. About the symbol “Ψ”

This Greek letter, psi, is not only pronounced similar to “cie”, but also composed of all three capitalized letters “C, I, E”.

B. Vietnamese terms

• Extent - intent: số lượng - tính chất (lượng - chất), (ngoại) hình - (nội) thức
• Số lượng (extent) = quantity = số (number, phase) + lượng (amount): Here "number" means "whole number" and "amount" means abitrary number.
• Sustent = content + intent: Tính = cảm tính (qualia) + lý tính (quanta)
• Sustent - extent: Tính - lượng
• Tính chất (intent) = quality = tính (nature, property = thuộc tính = attribute) + chất (substance, phase): Nature is the link between subject's property and object's attribute. A substance is a wholesome collection of complementary properties.
  同性相連，補性相合
  Đồng tính tương liên, bổ tính tương hợp.
  同聲相應，同氣相求『易経・乾・文言・九五・孔子曰』(Khổng Tử viết văn ngôn cho hào thứ 5 của quẻ Càn trong Kinh Dịch)
• Nội dung = tính chất × số lượng
• Intension - extension: nội hàm - ngoại diên
• Extensive - intensive: phân tán (mở rộng) - tập trung
• Extensity - intensity: độ phân tán - độ tập trung (độ dãn - độ nén)
• Form - content: hình thức - nội dung
• Hình thức = (ngoại) hình + (nội) thức (mô thức)
• Nội hàm = nội dung (cảm tính) + hàm thức (lý tính)
• Thuộc tính mở rộng: nội dung, kích thước (ngoại diên)
• Function = hàm: The old meaning of “function as a container of variables and parameters” (Bernoulli 1718, Euler 1748), as captured by the Sino-Vietnamese term “hàm” [函], is retrieved here in the partition “extensive - intensive”. The modern meaning of “function as a mapping” was introduced by Dirichlet in 1837.