[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.
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"Understanding" is composed of four elements, that is integrity (or identification), map, formation and learning-circle ——
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"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.
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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.
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"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".
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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.
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"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?)
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Comment: one will appreciate the formation of these lines.
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"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.
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Reference: The hub.(https://www.researchgate.net/project/Theory-of-Mathematics-TOM/update/5ff8371be35e2b000103ec30)
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