這段改編自 2010 年 4 月 3 日的對話。
Young man, in mathematics you don’t understand things. You just get used to them.
— John von Neumann
數學之中，越基礎的內容，有時會越難解釋。那些基礎內容，我們有時會以為自己明白。那只是因為，我們已經習慣了那些東西。「習慣」冒充了「明白」的感覺。例如，「3 乘以 2」 是什麼意思？
（安：「3 乘以 2」即是有兩個 3 加在一起，所以是 6。
3 x 2 = 3 + 3 = 6
）
無錯。那「3 乘以 2.1」 呢？何謂「有 2.1 個 3 加在一起」？
3 x 2.1 = ?
（安：我們可以把「3 乘以 2.1」，看成有三個 2.1 加在一起：
3 x 2.1
= 2.1 x 3
= 2.1 + 2.1 + 2.1
= 6.3
）
都可以。那樣，「3.1 乘以 2.1」呢？
— Me@2013.01.30
2013.01.31 Thursday (c) All rights reserved by ACHK
西瓜 8.3 | Copy Me, 6.3 | Verification principle, 4.3
這段改編自 2010 年 4 月 3 日的對話。
The difference that makes no difference makes no difference.
無關痛癢的分別，毋須理會。
你這個問題，其實是哲學裡的 problem of identity，即是「同一問題」，或者「身份問題」。「身份問題」追究的是，何謂「同一個人」？
我們在上幾次，討論「時間定義」和「記憶」時，已經探討過這個問題。正如剛才所講，是否「同一個自我」，要視乎你自己的定義。「定義」即是「用法」，並沒有「對錯」可言，只有「恰不恰當」。
一個比較恰當的定義是，根據洛克（John Locke）的標準 —— 有同一個記憶，就為之「同一個自我」。
— Me@2013.01.28
2013.01.28 Monday (c) All rights reserved by ACHK
西瓜 8.2 | Copy Me, 6.2 | Verification principle, 4.2
這段改編自 2010 年 4 月 3 日的對話。
我們可以把「同一個宇宙」定義為，一個有「因果關係」的時空區域。「平行宇宙」這個詞語中的「平行」，是指老死不相往來，互不相干。如果有兩個時空區域，完全沒有任何形式的因果連繫，我們就應該把那兩個區域，標籤為「兩個宇宙」。
（安：但是，我問題的前提是，我有一部時光機。我有可能由一個平行宇宙「甲」，走到另一個平行宇宙「乙」。那樣，「宇宙甲」就可以透過我和我的時光機，去影響「宇宙乙」。甲乙就開始有因果關係。原本的兩個「平行宇宙」，亦都不再「平行」。那樣，難道甲乙宇宙，就為之「二合為一」嗎？
另外，即使現實世界只有一個宇宙，由於宇宙膨脹的速度比光速還高，而光速本身，又是宇宙中訊息傳遞的終極最高速度，宇宙中的很多區域之間，自盤古初開，就永世不相往來，不可能有任何「因果關係」。那樣，難道那眾多區域，就為之「多重宇宙」嗎？）
你帶出了問題的重點。那是言辭之爭。
是否「同一個宇宙」，要視乎你自己的定義。「定義」即是「用法」，並沒有「對錯」可言，只有「恰不恰當」。
如果是一個常用的詞語，社會有一個慣常的用法，而你卻在沒有任何理據，和沒有事先聲明的情況下，用了一個極端不同的定義，那就為之「不恰當」。例如，「電腦」的一般意思是解作「運算機器」。如果你卻用這個詞語來指「蘋果」，就會引起很大的誤會。例如，當你說「我每天也會吃一個電腦」時，大家也會覺得你是瘋子。
但是，「同一個宇宙」並不是常用詞語，並沒有一個約定俗成、相對客觀的用法。既然沒有「先天的恰當定義」，那就應該「用者自付」。既然是你帶頭使用這個罕見詞彙，你就應該自己先給予一個清晰的定義。如果大家也接受，我們就可以用它，來繼續討論問題。
換句話說，你應該由自己回答，你原本的問題 —— 怎樣分辨「兩個平行宇宙」和「同一個宇宙的兩個不同區域」？
你要講得出，至少在原則上，「去了另一個平行宇宙」和「去了同一個宇宙的另一個區域」，會有什麼觀察結果上的分別。
如果你講得出有什麼分別 —— 例如「在同一個宇宙，我的薪金不會變；但是在另一個宇宙，我的薪金會加倍」—— 你就毋須問我如何分辨，因為，你只要看看自己的薪金有沒有變，就可以知道，究竟自己是「去了另一個平行宇宙」，還是「去了同一個宇宙的另一個區域」。如果你講不出有什麼分別，你問我如何分辨也沒有意思，因為那代表了，根據你的用法，「兩個平行宇宙」和「同一個宇宙的兩個不同區域」，根本是同義詞。
The difference that makes no difference makes no difference.
無關痛癢的分別，毋須理會。
— Me@2013.01.26
2013.01.26 Saturday (c) All rights reserved by ACHK
多重小宇宙 1.2 | 二次元時間 2.6 | Dimension 1.3.6 | Two dimensional time 2.6 | A little bit of yourself, 2 | 心靈互聯網 2 | Mind Internet 2
這段改編自 2010 年 4 月 3 日的對話。
但是，記住，那只能作比喻，而並不是實情，因為所有的「主觀時間線」，都會同時影響和受制於同一條「客觀時間線」。任何兩個人，即使從不相遇，兩條「主觀時間線」永不相交，他們的人生歷程，也不可能百分百互不相干。任何一條「主觀時間線」，都不如我所講的「平行宇宙」一般，有機會獨立存在。
不過，你這個講法雖然不是鉅細無遺，但是極度有用，因為它帶出了一個超級重點。現實世界的時間，雖然只有一個次元，但那一個次元，就已經足夠難明了。
剛才我把「一次元時間」講成一條「時間線」或者「因果鏈」，只是為了方便簡化。實情是，「時間」是一個「因果網絡」。意思是，「因」和「果」並不是一一對應。一個「因」，可以引發多個「果」；而一個「果」，又可以來自多個「因」。比喻說，一個學生，會有很多老師；而一個老師，又會有很多學生。「一因多果」和「一果多因」，可以統稱為「多重對應」。
你「現實版二次元主觀時間」的講法，雖然不是分毫不差，但是可信可用，因為，現實世界的「因」和「果」，是「多重對應」的。
— Me@2013.01.21
時間者
因果網絡也
— Me@2007.09.17
2013.01.21 Monday (c) All rights reserved by ACHK
無我 3
You cannot see yourself directly from your own point of view. Instead, you can only see your mirror, photo, and video images.
You exist only in other people’s world. You do not exist in your own world.
— Me@2013-01-20 01:08:34 AM
2013.01.21 Monday (c) All rights reserved by ACHK
Limitations of Godel’s theorems
The conclusions of Godel’s theorems are only proven for the formal theories that satisfy the necessary hypotheses. Not all axiom systems satisfy these hypotheses, even when these systems have models that include the natural numbers as a subset. For example, there are first-order axiomatizations of Euclidean geometry, of real closed fields, and of arithmetic in which multiplication is not provably total; none of these meet the hypotheses of Godel’s theorems. The key fact is that these axiomatizations are not expressive enough to define the set of natural numbers or develop basic properties of the natural numbers. Regarding the third example, Dan E. Willard (Willard 2001) has studied many weak systems of arithmetic which do not satisfy the hypotheses of the second incompleteness theorem, and which are consistent and capable of proving their own consistency (see self-verifying theories).
Godel’s theorems only apply to effectively generated (that is, recursively enumerable) theories. If all true statements about natural numbers are taken as axioms for a theory, then this theory is a consistent, complete extension of Peano arithmetic (called true arithmetic) for which none of Godel’s theorems apply in a meaningful way, because this theory is not recursively enumerable.
The second incompleteness theorem only shows that the consistency of certain theories cannot be proved from the axioms of those theories themselves. It does not show that the consistency cannot be proved from other (consistent) axioms. For example, the consistency of the Peano arithmetic can be proved in Zermelo–Fraenkel set theory (ZFC), or in theories of arithmetic augmented with transfinite induction, as in Gentzen’s consistency proof.
— Wikipedia on Godel’s incompleteness theorems
2013.01.20 Sunday ACHK
Looper, 2.2 | 二次元時間 2.4 | Dimension 1.3.4 | Two dimensional time 2.4 | 孖生宇宙 2.4
這段改編自 2010 年 4 月 3 日的對話。
而《Looper》作者自己的時間線，則可以視為《Looper》故事本身的「第三個時間次元」。一般而言，由構思故事到完成劇本，通常也不會一筆過，而會反覆修改。換而言之，那是一個演變的過程：
《Looper》故事版本一 –>（影響）《Looper》故事版本二 –>（影響）《Looper》故事版本三 –> … …
製在《Looper》這部電影時，作者很多時會和製作人員討論劇情。指清故事中的事件時，作者就需要講明，他所討論的那個事件，發生在「哪一個版本」中的「哪一個平行宇宙」中的「哪一點時間」，例如：
（故事版本二，宇宙三，2017 年 5 月 10 日）
亦即是話，作者需要有三個時間坐標數字，才可以「設置」，或者「定位」一個事件。
— Me@2013.01.18
2013.01.18 Friday (c) All rights reserved by ACHK
In ordinary quantum mechanics, observers or measuring devices are macroscopic. So they are classical, in the sense that each of them is always in a macroscopic-eigenstate, aka “a macrostate“. A classical object would not be in a macroscopic superposition, in the sense that there would not be in a superposition of macroscopic-eigenstates. Macroscopic reality is always definite, unless you are talking about future events.
Then, would the macroscopic reality actually be a superposition of microscopic eigenstates?
Yes, it is. That is a logical implication from quantum mechanics. However, that makes no experimental difference, since those microstates of a lot of particles constitute a single macrostate.
In conclusion, a macrostate is not a superposition of macroscopic eigenstates. And although it is a superposition of microscopic eigenstates, it makes only conceptual difference but no experimental difference even if we ignore this fact. So for a classical observer, we do not have to consider whether it is in a superposition or not.
How about the observed particle? Would it be in a superposition?
It can and probably is.
However, superposition is a logical implication only. It cannot be observed directly using a macroscopic measuring device. Also, by using a macroscopic measuring device, aka “a classical observer“, to measure or observe a microscopic event, we will always collapse the wave function of the observed system (due to the decoherence effect), yielding a definite macroscopic result (which is corresponding to one of the eigenstate components in the original microscopic superposition).
What if I have a microscopic measuring device as a “quantum observer”?
— Me@2013-01-16 10:53:06 AM
2013.01.16 Wednesday (c) All rights reserved by ACHK
Anatta 2 | 中道 4
In Buddhism, the term anatta or anatman refers to the notion of “not-self” or the illusion of “self”.
— Wikipedia on Anatta
Thus, in Theravada Buddhist soteriology, there is neither a permanent self nor complete annihilation of the ‘person’ at death; there is only the arising and ceasing of causally related phenomena.
— Wikipedia on Middle Way
The Buddha attacked all attempts to conceive of a fixed self, while stating that holding the view “I have no self” is also mistaken. This is an example of the middle way charted by the Buddha.
— Wikipedia on Philosophy of self
The following is a completely different point, but still relevant to the “not-self” topic:
The Buddha’s concept of “not-self” does not mean “there is no me“, just as “a camera cannot take a picture of itself” does not mean “the camera does not exist“.
A hand cannot hold itself, but it still exists.
— Me@2013.01.14
2013.01.15 Tuesday (c) All rights reserved by ACHK
Universal wave function, 19 | Reductionism 4
Impartial/All is the Noumenon, which is logically impossible for any single observer to observe directly, unless the observer is the whole of the universe. But “self-observation” is meaningless.
— Me@2012.04.07
Because “state” is expressed in RQM as the correlation between two systems, there can be no meaning to “self-measurement”.
— Wikipedia on Relational quantum mechanics
The Noumenon is a logical implication. It cannot be observed directly. It can be observed partially only, through senses, or phenomena. An observation is an interaction between the observer and the observed.
To really “observe” the Noumenon, all we can do is to observe as many phenomena as possible. In other words, we consider as many observer-observed pairs as possible.
— Me@2013.01.14
This is because this state would have to be ascribed to a correlation between the universe and some other physical observer, but this observer in turn would have to form part of the universe, and as was discussed above, it is impossible for an object to give a complete specification of itself. Following the idea of relational networks above, an RQM-oriented cosmology would have to account for the universe as a set of partial systems providing descriptions of one another. The exact nature of such a construction remains an open question.
— Wikipedia on Relational quantum mechanics
nomenon = all
phe- = part
noumenon = all aspects of the universe
phenomenon = part of the reality of the universe
— Me@2012.04.07
2013.01.14 Monday (c) All rights reserved by ACHK
Recursion 12
Self-reference may not be paradoxical, as long as there is a terminating condition / boundary case.
As long as there is a terminating condition, the self-reference is not really “hundred-percent-self”-reference. In other words, it is just self-similar, but not self-identical.
 |
| Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. |
Being self-similar is possible, but being self-identical is logically impossible since it creates infinite regress.
— Me@2013.01.12
2013.01.13 Sunday (c) All rights reserved by ACHK
In particular, arguably the greatest achievement of metamathematics and the philosophy of mathematics to date is Godel’s incompleteness theorem: proof that given any finite number of axioms for Peano arithmetic, there will be true statements about that arithmetic that cannot be proved from those axioms.
— Wikipedia on Metamathematics
2013.01.09 Wednesday ACHK
“Mutual Transformer” is an application of the Yin Yang principle.
在本網誌中，「互化法」是「間書原理」的一個應用。
— Me@2012-03-09 11:31:45 AM
2013.01.09 Wednesday (c) All rights reserved by ACHK
A time to fear, 4 | Recursion 8
人害怕「不確定性」，簡稱「未知」、「未定」、或者「不安」。「未知」是知識狀態；「不安」則是情緒起伏。「未知」導致「不安」。
人腦的特定是，時刻都想確定，下一步應該怎樣做。「不確定性」導致人腦，不能立刻決定，下一步的行動是什麼。那樣，人腦就會處於，一個「不斷企圖做決定，但又做不到決定」的無限循環（infinite loop）、跳上跳落的躍動狀態：
要做決定 –> 資料未夠–> 再試 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> 要做決定 –> 不行 –> …
所以，「未知」導致「不安」。
對電腦程式來說，要打破一個「無限循環」，要麼到達「終止條件」（terminating condition / boundary case）；要麼乾脆把它，從程式碼中刪除。
對人腦心靈而言，要結束一個「煩惱不安」，要麼有新的資料，導致做到決定；要麼索性把它，從思考中忘記。
簡而言之，要麼「解結」，要麼「斬結」。
— Me@2012.03.08
— Me@2012.12.31
2012.12.31 Monday (c) All rights reserved by ACHK
exists = is in = belongs to = can be found
“Exists” is “存在” in Chinese. Literally,
exists = 存在 = is stored in
“X is stored in” is not a complete sentence because it lacks an object. X is stored in where?
存 = 儲存
嚴格來說，「甲存在」並不是一完整句子，因為它沒有指清，甲存在哪裡？
— Me@2012.10.16
— Me@2012.12.30
2012.12.30 Sunday (c) All rights reserved by ACHK
exists = can be found
Existence is a relation. In order words, the sentence “X exists” means:
X exists = X can be found by something else
If there is only one object, the word “existence” has no meaning, because there is no “something else”. For example, by the definition of the word “universe“, there is only one universe. So the sentence “universe exists” has no meaning.
uni- = one
universe = all the things = all as one
— Me@2012.10.16
2012.12.26 Wednesday (c) All rights reserved by ACHK
Universe 2.2
Defining the word “universe” as “all the things” does not totally make sense, since the meaning of “all” is relative to a place. Without a context such as “all things in this house“, the word “all” is meaningless. To be meaningful, you have to specify what the word “all” is with respect to.
Instead of “all the things“, we can define “universe” as
universe = all the things observable by an observer, directly or indirectly, in practice or in principle, plus the observer itself
— Me@2012.10.16
— Me@2012.12.24
2012.12.24 Monday (c) All rights reserved by ACHK
— Draft only. These few lines are not 100% correct. —
In other words, whatever list you have constructed, it contains only part of the universe, not all of the universe. The universe, as infinitesimal and infinity, is not a thing, but a process (of keeping collecting things). The universe is a logical implication, or a logical limit.
— Draft only. These few lines are not 100% correct. —
— Me@2012-12-20
2012.12.23 Sunday (c) All rights reserved by ACHK
Universe 2.1
“All” is “所有” in Chinese. Literally,
all = 所有 = the place has
The definition of word “universe” is “everything” or “all the things“.
universe = all the things
However, this definition is not 100% sharp, because the meaning of “all” is relative to a list or a place. In order to give a meaning to the word “all“, you have to provide a list or a place in the sentence it appears.
When referring to a list, for example, the meaning of the phrase “all of A, B, and C” is the same as “A, B, and C“.
all of A, B, and C = A, B, and C
However, the definition of “universe” does not specify a list, such as
universe = all of A, B, and C
When referring to a place, for example, the meaning of the phrase “all the people in this house” is:
You keep looking for people in this house. Whoever you can find, include him or her onto your list of people. Once you cannot find new people in the house anymore, your list will have “all the people in this house“.
In other words, in order to give the word “all” a meaning, you have to provide a range for searching. However, the definition of “universe” does not specify a place, such as
universe = all the things in X,
unless we define the word “universe” as
universe = all the things in universe
However, this is a circular definition.
— Me@2012.10.16
— Me@2012.12.11
— Me@2012.12.20
2012.12.21 Friday (c) All rights reserved by ACHK
Nothing is better than having a good wife.
Having a toothbrush is better than having nothing.
—
Therefore, having a toothbrush is better than having a good wife.
Nothing is better than having a good wife.
{} > {good wife} (wrong)
{x : x > good wife} = {} (correct)
Having a toothbrush is better than having nothing.
{toothbrush} > {}
The is‘s have two different senses here. The first “is better than” is for comparing objects. The second “is better than” is for comparing sets. An object and a meta-object should not be put together for comparison.
— Me@2012-04-02 10:46:22 AM
2012.12.15 Saturday (c) All rights reserved by ACHK
Physics Town 
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