量子力學 1.14

因果律 1.19 | Verification principle, 5.19 | 西瓜 9.19

這段改編自 2010 年 4 月 3 日的對話。

(安:言歸正傳,剛才所講,「量子決定論」的難處在於:

理論上,『量子決定論』相當可信。但是,實際上,『量子決定論』並不可用。而這個『實際上』,實際上是『理論上』或者『原則上』,因為,即使只在原則上而言,任何觀察者也沒有可能,知道整個宇宙狀態的所有數據。

但是,再之前,討論「Laplace 因果律的宇宙版」(經典物理決定論)時,你又指出相同的難處:

即使假設在原則上,我們只要掌握某一個時刻,宇宙狀態的所有資料,我們就可以推斷到,宇宙在任何其他時刻的狀態;我們即使在原則上,也沒有可能,掌握某一個時刻,宇宙狀態的所有資料。

那樣,「量子物理決定論」和「經典物理決定論」,又有何分別呢?

核心分別在於,當兩者都遇上「宇宙版無從驗證」時,「經典決定論」可以由「可信而不可用」的「Laplace 因果律的宇宙版」,修減成「可信又可用」的「Laplace 因果律的局部版」 :

對於同一個物理系統而言,同一個設定(輸入),就每次也會得到,同一個對應的後果(輸出)。

例如,液態的清水,處於地球正常大氣壓力之下,會在攝氏零度開始結冰。我們所考慮的物理系統,就是『處於地球正常大氣壓力下的液態清水』。如果輸入是『溫度攝氏零度』,輸出就一定是『開始結冰』,又名『凝固』。那就為之『世事有常』。

」;

但是,「量子決定論」並沒有所謂的「局部版」,因為考慮「局部版」的話,就一定帶有隨機性,再也不成「一因一果」的「決定論」。

而在微觀粒子的世界,正正是那麼奇幻 —— 同一個情境之下,會有超過一個可能的結果。

用「量子力學」去預測,一個「微觀物理系統」的演化結果時,即是只在原則上而言,我們至多也只可以,預測有那些可能的結果,和各個結果的對應機會率;而大部分情況下,也不可以明確指出,結果一定是哪一個。

— Me@2013.09.30

2013.09.30 Monday (c) All rights reserved by ACHK

Block spacetime, 6

Clojure 4

fleitz 17 hours ago

Great article on the fundamental problems associated with mutable state. The fundamental problem is that the idea of an object with a set of state that is the same to all observers violates pretty much the whole of information theory. It’s not a problem that will ever be fixed [without] changing the fundamental laws of the universe.

Ditch the mutable data and you can stop asking questions like what do we do if 10 becomes 10.5 before it becomes 11 and start storing values which never change.

jerf 13 hours ago

Information theory does not have a problem, only, as you say, our universe. Mathematicians and their various hangers-on like programming language researchers often prefer to deal with models that have no concept of time, in which the very concept of “observer” is extraneous since there isn’t really anything like a “point of view”. Everything just… is.

— Hacker News

2013.09.29 Sunday ACHK

機器 3

原因 6 | 機會 3

機 = 因

因 = 先決條件

機器

= 全部先決條件之容器

= 充份條件之容器

故 = 因

故障

= 缺乏了起碼其中一個先決條件

= 因果網絡有障礙

— Me@2013.09.27 19.30.21

2013.09.29 Sunday (c) All rights reserved by ACHK

量子力學 1.13.2 (外傳)

這段改編自 2010 年 4 月 3 日的對話。

這個問題,可以用剛才討論的那個例子來理解:

那彷彿就好像,你中了彩票,獲得了一千億元的獎金,存入了你的銀行戶口。但是,銀行的職員跟你說,你每日最多只能從那個戶口中,提款一百元。你就立刻發覺,窮你一生,甚至是十世的時間,都不能用盡那一千億元。如果你的朋友問你:「你是否擁有一千億元?」

你就唯有無奈地答:「那要視乎你『擁有』的定義。擁有而不能用,還算不算是『擁有』呢?」

(安:但是,你又真的可以,從那一千億元的戶口之中,每天提取一百元去用。因為那些一百元,始終是來自那一千億元的,你不能說,那一千億元完全沒有用,完全不屬於你。)

無錯。

可信而不可用,還有資格叫做「可信」嗎?

那要視乎你「可信」的定義。

(安:等一等。其實這個例子很有趣,彷彿正在講述「個人潛能」似的。

每人出世時也有無限的希望,帶著價值一千億元的潛能。但是,每天可以提取的潛能不多,而一生人的生命卻又十分短促。平均而言,一生人大概只有,三萬至四萬日的時間。那樣,一生人就只能把自己,約略萬分之一的潛能,化成現實。)

無錯,那實在十分可惜。

不過,人類發展各門知識和發明各種科技,本身既是「化潛能為實在」的過程,同時又是逐步解決「人生苦短」這問題的手段。知識為人,節省很多無謂時間。機器為人,承受很多沉悶工序。

另外,釋放潛能,雖則極為重要,但也只是人生的其中一個意義,並非全部。

— Me@2013.09.28

You can do anything, but not everything.

– David Allen

2013.09.29 Sunday (c) All rights reserved by ACHK

Godel 12

Consistency and Completeness. We say a formal theory is consistent if you cannot prove both P and ¬P in the theory for some sentence P. In fact, because from P and ¬P you can prove anything using classical logic, it is equivalent that a theory is consistent if and only if there is at least one sentence Q such that there is no proof of Q in the theory.

[]

By contrast, a theory is said to be complete if given any sentence P, either [it] has a proof of P or a proof of ¬P. (Note that an inconsistent theory is necessarily complete).

[]

Hilbert proposed to find a consistent and complete axiomatization of arithmetic, together with a proof (using only the basic mathematics that both camps agreed on) that it was both complete and consistent, and that it would remain so even if some of the tools that his camp used (which the other found unpalatable and doubtful) were used with it.

— This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

— Mathematics – Stack Exchange

— Jan 5 ’11 at 3:21

— Arturo Magidin

2013.09.27 Friday ACHK

Always make new mistakes, 7.4

這段改編自 2010 年 7 月 27 日的對話。

重點是,你並沒有可能,可以「溫完書」。你只可以抽客觀上最重要,和主觀上自己最需要的部分來溫習。透過做 past paper(以往試題),你就可以釐清,要先溫哪些部分。然後,你又透過溫習,來提升每次做 past paper 時的成績。「溫習」和「做試題」應該是那樣相輔相成的。

你其他科都要這樣做,盡快開始啟動,做 past paper 的機器,因為沒有太多時間。試想想,現在是暑假,而約 30 日後就開學。中七開學時,距離你的高考公開試,就只有百幾日。

你感不感受到,「百幾日」很快就過?

(TK:感受到。)

(CYW:???)

(CYW:開學時,不是還有五個月嗎?)

「五個月」,不就是「百多日」嗎?

— Me@2013.09.26

2013.09.26 Thursday (c) All rights reserved by ACHK

Logical arrow of time, 5

Otherwise your games with the “definition” of initial and final states and with the sign of t are completely immaterial. “Initial” and “final” states are, according to logic, qualitatively different things, and the usual convention for the sign of t is that t_{initial} < t_{final}. But I have never even used this convention.

Even if I had, it wouldn’t matter. One can easily rewrite all proofs to the opposite convention by replacing t with −t; all those things are physically vacuous. The non-vacuous claim is that the future and past don’t play symmetric roles in logic.

— Physics Stack Exchange

— Jan 25 ’12 at 9:49

— Lubos Motl

2013.09.26 Thursday ACHK

受難曲 2

沒有攝影師,就沒有畫面。但是,有畫面,就一定沒有攝影師。

— Me@2013.09.22

… 因為,攝影師要離開畫面,才可以拍攝到該個畫面。

— Me@2013.09.23

2013.09.25 Wednesday (c) All rights reserved by ACHK

量子力學 1.13

因果律 1.18 | Verification principle, 5.18 | 西瓜 9.18

這段改編自 2010 年 4 月 3 日的對話。

理論上,「量子決定論」相當可信。但是,實際上,「量子決定論」並不可用。而這個「實際上」,實際上是「理論上」或者「原則上」,因為,即使只在原則上而言,任何觀察者也沒有可能,知道整個宇宙狀態的所有數據。

換而言之,無論智力有多高超、科技有多先進,也沒有任何 人類、生物、電腦 或者 神明,可以達到運用「量子決定論」的先決條件。邏輯上,沒有任何觀察者,可以百分之一百地,執行到「量子決定論」。

「量子決定論」可信而不可用。可信而不可用,還有資格叫做「可信」嗎?

這個問題,可以用剛才討論的那個例子來理解:

那彷彿就好像,你中了彩票,獲得了一千億元的獎金,存入了你的銀行戶口。但是,銀行的職員跟你說,你每日最多只能從那個戶口中,提款一百元。你就立刻發覺,窮你一生,甚至是十世的時間,都不能用盡那一千億元。如果你的朋友問你:「你是否擁有一千億元?」

你就唯有無奈地答:「那要視乎你『擁有』的定義。擁有而不能用,還算不算是『擁有』呢?」

(安:但是,你又真的可以,從那一千億元的戶口之中,每天提取一百元去用。因為那些一百元,始終是來自那一千億元的,你不能說,那一千億元完全沒有用,完全不屬於你。)

無錯。

可信而不可用,還有資格叫做「可信」嗎?

那要視乎你「可信」的定義。

— Me@2013.09.25

2013.09.25 Wednesday (c) All rights reserved by ACHK

Logical implication 3

The paradox ceases to exist the moment we replace informal logic with propositional logic. The Tortoise and Achilles don’t agree on any definition of logical implication. In propositional logic the logical implication is defined as follows:

P ⇒ Q if and only if the proposition P → Q is a tautology

[H]ence de modus ponens [P ∧ (P → Q)] ⇒ Q is a valid logical implication according to the definition of logical implication just stated. There is no need to recurse since the logical implication can be translated into symbols and propositional operators such as →. Demonstrating the logical implication simply translates into verifying that the compound truth table is producing a tautology.

— Wikipedia on What the Tortoise Said to Achilles

2013.09.25 Wednesday ACHK

Always make new mistakes, 7.3

最近五年 4.3

這段改編自 2010 年 7 月 27 日的對話。

(CYW:但是,我又真的很想溫完書才做。)

那豈不是成世人都不用做 past paper?

(CYW:但是,如果做 past paper 時,大部分題目也不懂的話,那不會好像是,浪費了那份 past paper 嗎?)

我上次講過,最近五年的 past paper(公開試試題)你要封起。任何情況下,你也不要看到它們,直至你下年公開試前,一個月的「試前休假」為止。換而言之,在「試前休假」開始之前,並沒有可能,會因為大部分題目不懂做,而「浪費」掉最重要的那五份 past paper。

至於其餘年份的 past paper,即使「大部分題目也不懂做」,其實也並不構成浪費。我有沒有講過,「always make new mistakes」呀?

你要記住,「做了也不懂」就正正是做 past paper 的目的。你要透過做 past paper,去發現自己有什麼不懂,從而盡早改善。如果你遇到極端的情況,例如碰見一題長題目時,整題要留空,連第一部分也不懂做的話,那就代表你要刻意地,重新溫習那一課。如果沒有那麼不幸,只是有一兩部分不懂的話,那你就只需要溫習,那一課中的那一兩個部分,和練習類似的題目。

重點是,你並沒有可能,可以「溫完書」。你只可以抽客觀上最重要,和主觀上自己最需要的部分來溫習。透過做 past paper,你就可以釐清,要先溫哪些部分。

— Me@2013.09.24

2013.09.24 Tuesday (c) All rights reserved by ACHK

Quantum Mechanics 4

Instead, it is a lesson that string theory has clearly taught us. If we want to talk about exact observables only, it’s simply not true that physics may be thought of as being composed of sharply localized objects in a predetermined classical geometric background, at least not in a uniquely specified way.

The very assumption that there exists a black hole interior (and the strict event horizon that separates it from the rest of the Universe) is an approximation. Exact stringy rules of evolution don’t allow such a conclusion to be 100% certain, ever. The probability may converge to 100% for a star collapse but the remaining deviations from 100% are always necessary to restore the unitarity and guarantee other high-precision consistency conditions.

— String theory is a complete theory of quantum gravity

— Lubos Motl

2013.09.23 Monday ACHK

量子力學 1.12

因果律 1.17 | Verification principle, 5.17 | 西瓜 9.17

這段改編自 2010 年 4 月 3 日的對話。

簡而言之,

    宇宙隨著量子物理定律演化,一切事件皆是必然的。

但是,你要小心一點,這只是暫時的結論,而未是最終的結論。到這一步為止,我們只處理了「量子決定論」的第一個問題,而未處理第二個問題。我們暫時只知道,理論上,「量子決定論」相當可信。但是,我們還未確定,實際上,「量子決定論」可不可信。換而言之,我們只知其「可信」,而不知其「可用」與否。

「量子決定論」的第二個問題是,在任何一次的實驗之前,你都要知道整個宇宙狀態的所有數據,才可以百分百準確地,預測到該個實驗的結果。

但是,「宇宙」就是「所有東西」。邏輯上,任何觀察者也沒有可能,觀察到「所有東西」,因為任何觀察者本身,也必定是「宇宙」的一部分。正如,在拍大合照時,並沒有可能會拍到所有人,因為,總要有一個人,去做攝影師。

準確一點的比喻是,邏輯上,任何攝影機也沒有可能,拍攝到「所有東西」,因為至少有一樣東西,它一定拍不到;那就是它自己。

理論上,「量子決定論」相當可信。但是,實際上,「量子決定論」並不可用。而這個「實際上」,實際上是「理論上」或者「原則上」,因為,即使只在原則上而言,任何觀察者也沒有可能,知道整個宇宙狀態的所有數據。

換而言之,無論智力有多高超、科技有多先進,也沒有任何 人類、生物、電腦 或者 神明,可以達到運用「量子決定論」的先決條件。邏輯上,沒有任何觀察者,可以執行到「量子決定論」。

「量子決定論」可信而不可用。

— Me@2013.09.22

2013.09.23 Monday (c) All rights reserved by ACHK

Idea babies 4

On why failing in the startup world is normal

“In the startup world, ‘not working’ is normal … You might wonder why ships have [bilge] pumps [to remove water from below the deck] … Why don’t they just make one that’s waterproof, right? And the fact is, one way or another, all ships take on water … and one way or another, practically all startups internally are disasters. And they just hide this from the outside world.

— Paul Graham

— Failure: The F-Word Silicon Valley Loves And Hates

— NPR Staff

2013.09.22 Sunday ACHK

David Attenborough

In a December 2005 interview with Simon Mayo on BBC Radio Five Live, Attenborough stated that he considers himself an agnostic. When asked whether his observation of the natural world has given him faith in a creator, he generally responds with some version of this story, making reference to the Loa loa parasitic worm:

My response is that when Creationists talk about God creating every individual species as a separate act, they always instance hummingbirds, or orchids, sunflowers and beautiful things. But I tend to think instead of a parasitic worm that is boring through the eye of a boy sitting on the bank of a river in West Africa, [a worm] that’s going to make him blind. And [I ask them], ‘Are you telling me that the God you believe in, who you also say is an all-merciful God, who cares for each one of us individually, are you saying that God created this worm that can live in no other way than in an innocent child’s eyeball? Because that doesn’t seem to me to coincide with a God who’s full of mercy’. 

He has explained that he feels the evidence all over the planet clearly shows evolution to be the best way to explain the diversity of life, and that “as far as [he’s] concerned, if there is a supreme being[,] then he chose organic evolution as a way of bringing into existence the natural world.”

— Wikipedia on David Attenborough

2013.09.21 Saturday ACHK

Always make new mistakes, 7.2

最近五年 4.2

這段改編自 2010 年 7 月 27 日的對話。

不如這樣,你訓練到自己,有能力去完成 past paper 中的所有題目,包括那些牽涉到,學校未教課題的題目。

你先行自修那些課題。你先自己閱讀課文、嘗試例題。看到不明白的地方,就把它們記下來,一次過問我。

(TK:更大的問題是,probability(機會率)課題之中,老師教了的部分,其實我仍然不明白。)

你做 past paper(過往試題)時,有沒有做 probability 的題目?

(TK:有呀。)

你先問我,你做了的 probability 題目之中,錯了而又不明白的地方。然後,就繼續做那些 past paper。無論是教了的課題,還是未教的,你都嘗試去做。亦即是話,你除了要先行自修未教的課題外,你還要一併「自修」,那些「教了不明」的章節。

重要的是,你在每星期一次的會面時,主動問我問題,把你在自修或者做題目時的疑難,一一定期清除。

你要立刻開始「做 past paper」的進度,無論已教的課程,有多少不明白,還是未教的章節,有多少不清楚。千萬不要等,千萬不要用一般人的處事方法。通常,我叫人「做 past paper」時,一般人都會答:「我還未溫好習。等我先溫完書再說。」

你知不知道,企圖先溫完書,才開始做 past paper,會有什麼後果?

(TK:來不及完成那麼多年的 past paper?)

不是「來不及」,而是「根本不會開始做」。

(TK:為什麼呢?)

你試想想,怎樣才為之「溫完書」呢?

根本沒有所謂的「溫完書」,因為,任何一個課題,你都可以無限深入、無限研究下去。莫講話這一科有十多個課題,即使只有一個,你窮一生的時間,也未必研究得「完」。

(CYW:但是,我又真的很想溫完書才做。)

那豈不是成世人都不用做 past paper?

(CYW:但是,如果做 past paper 時,大部分題目也不懂的話,那不會好像是,浪費了那份 past paper 嗎?)

你誤會了我的意思。我不是叫你,在完全不用溫習的情況下,就去開始做 past paper。我只是叫你,千萬不要企圖「溫完書」。換句話說,你溫到某個程度,或者指定時限,就要收手,改為立刻開始做 past paper 題目。簡而言之,要「溫書」,但不要「溫完書」。

— Me@2013.09.21

2013.09.21 Saturday (c) All rights reserved by ACHK

Inner and outer, 6

Onion self 8

In a sense, the fewer qualifiers you give to yourself, the smaller “you” are.

As you have fewer unnecessary qualifiers, “you” are purer. For example, when you label yourself as “a person” rather than “a thin person”, “you” are purer. Furthermore, if you label yourself as “an animal” rather than “a person”, “you” are even purer.

In the most extreme case, you label yourself just as “a thing”. You are just a being. You are just you.

In another sense, the fewer adjectives you give to yourself, the bigger “you” are.

As you have fewer unnecessary adjectives, “you” are more complex. For example, when you define yourself as “a human” rather than “a thin human”, “you” are greater, because there are more objects satisfying the condition of “being a human” than “begin a thin human”. There are more “humans” than “thin humans”.

Furthermore, when you define yourself as “an animal” rather than “a human animal”, “you” are even greater.

In the most extreme case, you define yourself just as “a thing”. In other words, as everything is “a thing”, you are “everything”.

The smallest possible you is “nothing”. The biggest possible you is “everything”, aka “the universe”.

— Me@2013-09-17 10:15 PM

2013.09.20 Friday (c) All rights reserved by ACHK