[按：下文是群邮件的内容，最近的涂鸦笔记，标题是另拟的。原标题 "Hub-u"，附带看出"枢纽"位于"形成"的最初阶段。]

This is coming to you from Yiwei LI (PhD, Applied math), Taiyuan University of Science and Technology  (TYUST) Taiyuan, China

Scratch time is a topic under the column of Theory of Mathematics (TOM), an attempted framework to understand invented and un-invented mathematics from the core.

Soft thinking is to do almost nothing, yet covering everything.

---- Not criticizing anyone, just a reflection of the moment.

.

"Understanding" is composed of four elements, that is integrity (or identification), map, formation and learning-circle ——

.

f           l

.

i          m   

.

"integrity" refers to the completeness of the object or process (specific or abstract) to be understood.

---- Incompleted things can not exist on their own in anyone's mind.

----| When one says "half of an egg", one inevitably refers to the whole concept of "egg".

----| Also, one is able to identify an incompleted thing only because that thing is reconstructed as a whole in one's mind, automatically.

----| For example, given three points on the blackboard, one may view them as a triangle.

.

Credit: this comes from the residual memories from the psychology course in my univerity era.

---- I still remember the image of a "triangle" with three its "horns" removed.

---- Yet, everyone tends to identify that image as a triangle, though it only represents part of a triangle.

.

"map" is the action occurring in one's mind, to refer to one thing from another.

---- Cayley's theorem says, any group is mapped to a permutation group in some equivalent sense.

---- Why did Cayley think of mapping any group to a permutation group ?

---- Needs arose in Cayley to understand those other groups.

---- Guess Gayley was to understand a few groups at hand, yet surprised at finding a universal understanding of any group.

--*-- One might realize that permutation group serves as the "hub" in group theory.

---- If one enters a new world never known before, one has to enter that world through no where but its "hub".

---- So did Galois (or other genuine mathematicians).

---- Why do others find group theory dry and difficult to enter ?

---- Ordinary textbook writers tend to switch others from the "hub", by adding a "ladder" that never touches the genius' shoulder.

---- Fortunately, Harold Edwards is not among them.

---- Ordinary textbook writers are useful, of course, to show what should be "circulated".

.

Note: the thoughts of taking the map as the way to understand things arose around some moment when I encountered the "log resolution" in the paper I was reading.

---- It is surprising to see a map setup in a backward way, viewing the object as an image of something deeper.

---- Eventually, I realize that things in mathematics (and beyond) are understood through their images!

---- By the way*, if one finds the right "hub" in mathematics, one will realize that it also serves as a hub to any realm.

---- Just like one finds the "wormhole" in the universe of knowledge.

* The lines staring from "--*--" are added after this "by the way" complement.

.

"formation" is about the process through which the object is formed.

---- One cannot understand a thing thoroughly without revealing the formation of that thing.

---- Say, to understand the universe, one needs to figure out the formation of the universe.

---- However, it's usually hard to recover the process of something formed by itself or others...

---- If one formulates a conjecture or theorem, one must have touched the "hub" toward that conjecture or theorem.

---- So*, the hub is "located" at the beginning of the universe ! (Glad to figure out this point in this way).

---- To genuinely enter the proof of a theorem, one has to find out the hub in the proof.

---- The hint is that, hub is connected to "everything".

---- One may try to identify the mode amounting to the biggest occurrence, while counting out the false-hubs.

---- After that, one may resort to an artificial formation of concern.

* This is added after the last sentence —— a jump event occurred in my mind when I re-read these lines, likely at "everything".

** By the way, I just realize that I am the hub of this "universe" of flock emails.(Why me?)

.

Comment: one will appreciate the formation of these lines.

.

"learning-circle" refers to the complete process of learning.

---- Most people are less successful only because their learning-circles are not completed**.

---- So, every effort must be taken to fulfill the integrity of learning-circle.

---- There is no one else who concerns oneself the most. 

---- Resort to the latest No.1 on this planet, or no one.

---- Read their latest peak work, and read thoroughly.

---- Take residence in their work; retype and write comments sentence by sentence.

---- Make sure to finish the learning-circle, step by step, day after day.

---- Or, don't.

---- Do not let others know that; or let everyone know that.

---- Alles aufgeben. Verlass die Last.  مدين للآخرين.

.

.

Reference: The hub.(https://www.researchgate.net/project/Theory-of-Mathematics-TOM/update/5ff8371be35e2b000103ec30)