[按：下文是群邮件的内容，最近的"代表笔记"，标题是另拟的。原标题 "The hub"，意为"枢纽"。]

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.

Not competition, but illumination.

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

"Hub" plays a role of "wormhole" of the universe of knowledge.

---- It serves as a direct access to anywhere in mathematics and beyond.

---- Yet, it is more universal, "specific" and easy to handle, compared to philosophy.

---- I decide "Hub" is a system of four concepts.

---- That is, "mode", "world", "understanding" and "name".

---- This answers the four questions of theory of mathematics (TOM).

---- Is not "Hub" just another name for TOM ?

---- I leave this question not answered at the moment.

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The deployment of "Hub" ——

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u        n     ♖       ♘

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m       w     ♔      ♗

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I give a primary description to organize and deliver the "raw" thoughts ——

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A "mode" is a combination of symbols or any other thing (e.g. phenomenon, relation, etc), specific or abstract, showing itself for many times.

---- Important modes are all named.

---- Creation (or finding) is nothing but of new modes.

---- In mathematics or other disciplines (or realms), one just deals with modes in one way or another.

---- One may judge or identify a mode by counting its occurrences.

---- "mode" is of invariance, with unique nature.

---- If one holds the view of mode, one is already connected to the whole knowledge, including the un-invented.

---- That is, all the "walls" in front of the viewer are collapsed in principle.

---- In this sense, one is able to learn "everything" in one day or just a few minutes.

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Note: "pattern" is formed by arranging one or more modes, optionally with their variants, in a mode manner.

---- In my view, "mode" is more simple than "pattern".

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Credit: a few years ago, I encountered "Fourier modes" somewhere (Wikipedia ?), yet no explanation found.

---- At some later point, by chance, I saw the function called "mode" in Matlab (R), which was illuminating for me.

---- I leave deeper checks to later time.

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A "world" is a collection of objects subjected to interactions yielding consequences that do not exceed itself (i.e. the world).

---- One will agree that world is an instance of mode, yet not informative.

---- This concept of "world" was addressed at some earlier point^.

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Credit: I noticed the term of "world" by chance at some earlier point* (2018-8-25), yet not well remembered, actually totally forgot ——

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In Grothendieck it is an extreme form of Cantor’s freedom of mathematics. It is not only the freedom to build a world of set theory for mathematics but to build an entire world — specifically a “topos”, as large as the universe of all sets — adapted to any single problem such as a single polynomial equation on a finite field. (Credit to Wikipedia as the first source to me).

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Comment: I do not know to what extend it has been defined or fulfilled, but I believe similar thoughts propagate from an individual to another.

---- "world" is just one of the component of "hub" under TOM talking here.

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An "understanding" is a map from one world to another one.

---- "understanding" is the hub of "Hub".

---- "artificial understanding" is a special instance^.

---- One needs to see the difference from the core^.

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The "name" here refers to a system of names, or the spirit to give names.

---- Say, "main theorem", "vice theorem", "transfer theorem", "executing theorem" and "technical theorem", etc.

---- Say, "main set", "main map", "main interaction".

---- Say, homemade names for specific modes.

---- One may find use of name in context^.

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Comment: I prefer to use "vice theorem", instead of "second main theorem" (as used by Prokhorov and Shokurov for a 2009 paper).

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H       E

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C       V

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* It's not yet clear what H is about, but I feel there should be one.(Now, it's fulfilled to some extend).