Small big bang 4

小宇宙大爆炸 4

Arthur Schopenhauer claimed that phenomena have no free will, but the will as noumenon is free.

— Wikipedia on Free will

Man can do what he wills but he cannot will what he wills.

On The Freedom Of The Will (1839)

— Arthur Schopenhauer

A will itself cannot be willed because it is the first cause of a causal chain. A first cause is a starting point. Anything can be willed would not be a first cause.

— Me@2012.11.29

2012.11.30 Friday (c) All rights reserved by ACHK

電流電壓 2

這段改編自 2010 年 6 月 8 日的對話。

(CPK:那樣,這一點的 voltage 數值是什麼?)

你要小心一點。Voltage(電壓)在這裡是指 potential difference(電勢差)。有兩點的 potential 數值,才會有所謂「difference」。所以,你問「這一點」voltage 數值是什麼,是沒有意思的,除非在事前已經設定好參考點,即是 ground(接地點)。

你應該問,甲點和乙點之間的 voltage 是什麼?又或者,甲乙兩點之中,哪一點的 potential 高一點?

— Me@2012.11.30

2012.11.30 Friday (c) All rights reserved by ACHK

Godel 4.2

Paradox 8.2

Godel’s Incompleteness Theorems are anti-self-reference:

For any formal system strong enough to include the system of arithmetic, to prove its consistency, you need a stronger system.

— Me@2012-04-02 9:35:17 AM

2012.11.30 Friday (c) All rights reserved by ACHK

Monty Hall problem 1.5.2

機會率哲學 4.1.2

這段改編自 2010 年 4 月 3 日的對話。

這個講法的好處是,既容易理解,又剛巧可以得出正確答案。可惜,這個講法的推論過程是錯的,即是詭辯。

推論過程的錯處在於,它忽略了剛才所講,機會率的數值,除了描述客觀的物理系統外,還會反映觀察者的主觀知識狀態。當主持人做了一些動作,而導致遊戲參加者知多了一些資料時,各道門的中獎機會自然有變。例如,假設參賽者的原本選擇是第一道門。當主持人打開第三道門,令到參賽者知道「門後是山羊」時,相對於參賽者來說,第三道門的中獎機會,就立刻變成了零。

同理,當主持人打開第三道門,令到參賽者知道「門後是山羊」時,相對於參賽者來說,另外兩道門的中獎機會,一般而言,都立刻有變。至於會變成什麼新的數值,則要重新運算。

剛才「淺白解釋」的其中一句是:「那樣,在主持人打開另外的其中一道門後,如果你維持原本的選擇,你中獎的機會就仍然只有三分之一。」這一句雖然答案正確,但是跳過了中間幾個必須的運算步驟,所以十分誤導。那個「仍然」,並不是必然的。

第一道門的中獎機會率剛巧不變,並不是必然的,而是有其他特定的原因。換句話說,我們不可以在沒有任何理據的情況下,貿貿然假設,在主持人開了一道門之後,原本選擇的中獎機會率,和之前一樣。同理,我亦不可以妄自宣稱,第三道門一打開了,第二道門就會自動繼承了它的中獎機會,除非有正確的運算支持。

而正確的運算是,使用「條件機率」(conditional probability)。「條件機率」的圖像版,叫做「樹形圖」(tree diagram)。

Tree showing the probability of every possible outcome if the player initially picks Door 1

— Wikipedia on Monty Hall problem

(安:但是,這個樹形圖,好像都是不太容易明白。可不可以再解釋一下?)

— Me@2012.11.28

2012.11.29 Thursday (c) All rights reserved by ACHK

Superdeterminism 2.2

Paradox 9.3 | Bell’s theorem, 4.2

The meaning of the phrase “counterfactual definiteness” in quantum mechanics or Bell’s theorem is not the same as that in the superdeterminism theory. They are two different concepts.

In the superdeterminism theory, no non-local wave function collapse effect is needed, as all are pre-programmed, including the experimenters’ choices of measurement axes. Superdeterminism assumes that the world is classical and classically deterministic.

In such a situation, it is counterfactual definite in a sense that there is no quantum superposition. A system has a definite classical state even before any measurements.

However, it is not counterfactual definite in sense that there are no alternatives. It is not meaningful to ask, “What if the experimenter had chosen another measurement axis?”

Bell’s theorem assumes that the types of measurements performed at each detector can be chosen independently of each other and of the hidden variable being measured. In order for the argument for Bell’s inequality to follow, it is necessary to be able to speak meaningfully of what the result of the experiment would have been, had different choices been made. This assumption is called counterfactual definiteness.

But in a deterministic theory, the measurements the experimenters choose at each detector are predetermined by the laws of physics. It can therefore be argued that it is erroneous to speak of what would have happened had different measurements been chosen; no other measurement choices were physically possible.

— Wikipedia on Superdeterminism

 
Superdeterminism is cheating.

— Me@2012-11-24 11:21:01 AM

2012.11.27 Tuesday (c) All rights reserved by ACHK

Monty Hall problem 1.5.1

機會率哲學 4.1.1

這段改編自 2010 年 4 月 3 日的對話。

「蒙提霍爾問題」(Monty Hall problem)有一個比較平易近人的解答。

This is a public domain image.
Player’s pick has a 1/3 chance 
while the other two doors have 1/3 chance each, for a combined 2/3 chance.
— Wikipedia on Monty Hall problem

This is a public domain image.
With the usual assumptions player’s pick remains a 1/3 chance, 
while the other two doors have a combined 2/3 chance: 
2/3 for the still unopened one and 0 for the one the host opened.
— Wikipedia on Monty Hall problem

「蒙提霍爾問題」假設了,在遊戲開始時,三扇門「門後有房車」的機會均等。所以,你選定了一道門後,你中獎的機會就是三分之一,而其他門中獎機會率的總和,有三分之二。那樣,在主持人打開另外的其中一道門後,如果你維持原本的選擇,你中獎的機會就仍然只有三分之一。主持人打開了一道沒有車的門,而又容許你改變選擇,就相當於給予你,一次過選擇其他全部門的機會。因此,如果你肯改變選擇,你中獎的機會率,就會由三分之一,躍升至三分二。

如果你仍然不相信,你可以先假想這個遊戲的一個極端版本。假設這個「開門抽獎遊戲」改為有一千道門。其中只有一扇門的後面,有名貴房車。其餘的門後面,都是山羊。跟原本的版本一樣,在遊戲開始時,所有門的中獎機會均等。換句話說,無論那位參賽者選擇哪一扇門,中獎的機會,同是千分之一。

參賽者選了一道門後,主持人就會打開其餘 999 道門中的其中 998 道。那 998 扇門的後面,都各自有一隻山羊。然後,主持人又會問你,要不要更換選擇。你不更換的話,就相當預計了,自己在第一次選擇時一擊即中。那只有千分之一的機會。如果你意會到這一點,你就一定想放棄原選。

這個講法的好處是,既容易理解,又剛巧可以得出正確答案。可惜,這個講法的推論過程是錯的,即是詭辯。

— Me@2012.11.26

2012.11.27 Tuesday (c) All rights reserved by ACHK

Quantum teleportation

The no-communication theorem thus says shared entanglement alone can not be used to transmit any information. Compare this with the no teleportation theorem, which states a classical information channel can not transmit quantum information. (By transmit, we mean transmission with full fidelity.) However, quantum teleportation schemes utilize both resources to achieve what is impossible for either alone.

— Wikipedia on No-communication theorem

2012.11.26 Monday ACHK

Fixed stars

.

* Writers may be classified as meteors, planets, and fixed stars. A meteor makes a striking effect for a moment. You look up and cry “There!” and it is gone forever. Planets and wandering stars last a much longer time. They often outshine the fixed stars and are confounded by them by the inexperienced; but this only because they are near. It is not long before they must yield their place; nay, the light they give is reflected only, and the sphere of their influence is confined to their orbit — their contemporaries. Their path is one of change and movement, and with the circuit of a few years their tale is told. Fixed stars are the only ones that are constant; their position in the firmament is secure; they shine with a light of their own; their effect today is the same as it was yesterday, because, having no parallax, their appearance does not alter with a difference in our standpoint. They belong not to one system, one nation only, but to the universe. And just because they are so very far away, it is usually many years before their light is visible to the inhabitants of this earth.

o Vol. 2 “The Art of Literature” as translated in Essays and Aphorisms (1970), as translated by R. J. Hollingdale

– Arthur Schopenhauer

.

.

.

2010.06.06 Sunday ACHK

Monty Hall problem 1.4.3

Frequency probability and Bayesian probability, 3.2.3

機會率哲學 3.2.3

這段改編自 2010 年 4 月 3 日的對話。

既然是兩個不同的概念,我們何不索性賦予它們,兩個不同的名字。那就可以避免再混淆。我們可以把「頻率學派」的「機會率」,叫做「頻率機會率」、「客觀機會率」,或者「物理機會率」。然後,我們把「貝葉斯學派」的「機會率」,叫做「貝葉斯機會率」、「主觀機會率」,或者「知識機會率」。

(安:「頻率機會率」和「貝葉斯機會率」的分別作於,「頻率學派」認為,機會率的數值是客觀的,反映著被觀察系統的物理性質;而「貝葉斯學派」則認為,機會率的數值是主觀的,反映著觀察者對一個物理系統的知識多寡。

但是,既然「頻率機會率」和「貝葉斯機會率」都叫做「機會率」,即是它們有著密切的關係,共通的地方。正如,既然「紅蘋果」和「青蘋果」都是「蘋果」,即是它們有著很多相同的性質。那樣,我想問,「頻率機會率」和「貝葉斯機會率」的關係是什麼?有沒有可能把兩者統一起來?或者說,可不可以用同一個架構來處理它們呢?)

你的意思是,可不可以把「頻率機會率」和「貝葉斯機會率」視為,同一樣東西的兩個方面?

對於「機會率真義」這個問題,我的第二個解答,正正是企圖處理這個點。

「頻率機會率」和「貝葉斯機會率」衝突的來源,在於雙方也假設了,對於一個機會率問題,我們可以有把「觀察者」和「被觀察者」,百分百地截然分開。在這裡,「觀察者」即是「機會率使用者」;「被觀察者」即是「外在物理系統」。

如果視「觀察者」和「被觀察者」為兩個互不相干的系統,我們就可以爭論,究竟「機會率」是在描述「被觀察者」的性質,即是「物理性質」;還是在表達「觀察者」的性質,即是「知識多寡」。

但是,如果「可以把『觀察者』和『被觀察系統』完全分開」這個假設,根本不成立,我們就應該把「觀察者」,視為對應機會率系統的一部分。那樣,「觀察者的性質」就化成「系統性質」的一部分。

「觀察者」加「被觀察者」作為一個整體,作為一個單一系統的話,「機會率」就是那個大系統的性質。

This work is in the public domain in the United States, and those countries with a copyright term of life of the author plus 100 years or less.

(安:假設梵高(Van Gogh)的一幅名畫,給一位有錢人甲,用一億元買下來。然後我們問,究竟那幅名畫的「價值」,是那幅畫本身的性質,還是有錢人甲的性質?

一方面,我們不可以說,那一億元的「價值」,純粹是那幅畫的性質,因為對於一個三餐不計的平民來說,「衣食豐足」遠勝過「一幅梵高大作」。甚至,他可能連「梵高」是誰,也不太知道。

另一方面,我們亦不可以說,那一億元的「價值」,純粹是那位有錢人甲的性質,因為對於甲來說,並不是任何一位畫家的任何一幅畫,都價值一億大元。)

— Me@2012.11.25

2012.11.25 Sunday (c) All rights reserved by ACHK

Superdeterminism 2.1

Paradox 9.2 | Bell’s theorem, 4.1

Bell’s theorem states that if Bell’s inequality is violated by experimental results, then the original quantum mechanics is correct in a sense that no local hidden variable theory is possible to replace it. Nature is either non-local or non-counterfactual-definite (or both).

1. The principle of locality:

There are two possible meanings of “locality” here.

1.1 The principle is correct in a sense that no causal influence can be faster than light.

1.2 The principle is incorrect in a sense that distant particles can be entangled. Correlation without causation can be instantaneous.

Assume that a pair of particles are entangled. Measuring one particle will collapse the wave function, which governs both particles, instantaneously.

2. Counterfactual definiteness:

2.1 It is correct in a sense that an object has a definite quantum state.

2.2 It is incorrect in a sense that, more often than not, the definite quantum state is not corresponding to a definite classical state (aka eigenstate). Instead, that quantum state is a superposition of different eigenstates. 

The meaning of the phrase “counterfactual definiteness” in quantum mechanics or Bell’s theorem is not the same as that in the superdeterminism theory. They are two different concepts.

— Me@2012-11-24 11:21:01 AM

2012.11.25 Sunday (c) All rights reserved by ACHK

A Word to the Wise

It turns out there is, and the key to the mystery is the old adage “a word to the wise is sufficient.” Because this phrase is not only overused, but overused in an indirect way (by prepending the subject to some advice), most people who’ve heard it don’t know what it means. What it means is that if someone is wise, all you have to do is say one word to them, and they’ll understand immediately. You don’t have to explain in detail; they’ll chase down all the implications.

— A Word to the Resourceful

— January 2012

— Paul Graham

2012.11.24 Saturday ACHK

Monty Hall problem 1.4.2

Frequency probability and Bayesian probability, 3.2.2

機會率哲學 3.2.2

這段改編自 2010 年 4 月 3 日的對話。

「頻率學派」和「貝葉斯學派」之中,究竟哪一個正確呢?又或者說,哪一個較為有用呢?

「機會率」既有客觀的成份,又有主觀的成份。留意,這裡「主觀」的意思,並不代表「不正確」。一方面,我們真的使用「機會率」,來描述外在事件;另一方面,「機會率」又真的會反映,一個人對一件外在事件的結果,無知程度的深淺。

「機會率」處理「未知」,但不一定處理「未來」,因為「未知」不一定代表「未來」。例如,剛才的例子中,骰子已擲,而我亦知道了結果,所以不是「未來」。但是,因為你還未看結果,所以相對你來說,那仍然是「未知」。你仍然需要使用「機會率」,來估計結果。大概而言,所有人也不知的,為之「未來」;精確而言,尚未發生的,才為之「未來」。

對於「機會率真義」這個問題,我的第一個解答是,其實「頻率學派」所講的「機會率」,和「貝葉斯學派」所講的「機會率」,根本是兩個不同的概念,有著不同的意思,雖然兩個意思十分相關,相關到會用同一個名字「機會率」,甚至很多時會有相同的數值。

既然是兩個不同的概念,我們何不索性賦予它們,兩個不同的名字。那就可以避免再混淆。我們可以把「頻率學派」的「機會率」,叫做「頻率機會率」、「客觀機會率」,或者「物理機會率」。然後,我們把「貝葉斯學派」的「機會率」,叫做「貝葉斯機會率」、「主觀機會率」,或者「知識機會率」。

— Me@2012.11.23

2012.11.23 Friday (c) All rights reserved by ACHK

Superdeterminism

Paradox 9

In the context of quantum mechanics, superdeterminism is a term that has been used to describe a hypothetical class of theories which evade Bell’s theorem by virtue of being completely deterministic. Bell’s theorem depends on the assumption of [non-] counterfactual definiteness, which technically does not apply to deterministic theories. It is conceivable, but arguably unlikely, that someone could exploit this loophole to construct a local hidden variable theory that reproduces the predictions of quantum mechanics.

… in a deterministic theory, the measurements the experimenters choose at each detector are predetermined by the laws of physics. It can therefore be argued that it is erroneous to speak of what would have happened had different measurements been chosen; no other measurement choices were physically possible. Since the chosen measurements can be determined in advance, the results at one detector can be affected by the type of measurement done at the other without any need for information to travel faster than the speed of light.

— Wikipedia on Superdeterminism

Even if there are no other physical possibilities for a measurement choice, there are other logical possibilities. The goal of quantum mechanics, or science in general, is to consider, for an identical system, what input results what output.

The problem of superdeterminism in quantum mechanics is not “claiming the observers’ action are deterministic”, but by claiming so, claiming also that there is no decoherence (wave function collapse).

When we say that the observer cannot be separated from the observed, we mean that we have to consider the whole (observed + observer), instead of shifting the system from the observed to the observer, and then ignoring the original observed itself.

— Me@2012-11-20 02:11:06 PM

2012.11.23 Friday (c) All rights reserved by ACHK

Monty Hall problem 1.4.1

Frequency probability and Bayesian probability, 3.2.1

機會率哲學 3.2.1

這段改編自 2010 年 4 月 3 日的對話。

「機會率」這門學問,有兩個主要的學派:「頻率學派」(frequentist probability)和「貝葉斯學派」(Bayesian probability)。兩派對「機會率」的詮釋,有相反的意見。

「頻率學派」認為,機會率的數值是客觀的,反映著被觀察系統的物理性質。例如,你把某一粒骰子擲了六千次。如果每一個數字出現的次數,都大概是一千次的話,你就可以宣稱,那一粒骰子是正常的。對於該粒骰子,你擲到任何一面的機會率,都是六分之一。「頻率學派」認為,「擲到任何一面機會都是六分之一」這句本身,其實間接地描述了,該粒骰子的物理性質,例如「骰子有六面」和「質量均勻分佈,沒有偏袒」等。

「貝葉斯學派」則認為,機會率的數值是主觀的,反映著觀察者對一個物理系統的知識多寡。即使是同一個系統的同一件事件的同一個可能結果,不同觀察者可以運算到,截然不同的機會率,而他們彼此都沒有錯。例如,我擲了一粒公平骰子後,立刻檢查一下結果,但又不給你看。我發現我擲到「三」。然後,我要你估一估計,結果是什麼。相對於我來說,我擲到「三」的機會率是 100%,而其他數字的機會則是 0%,因為我已經知道結果。但是,你卻仍然不知道結果。所以,相對於你來說,我擲到「三」的機會率只是六分之一,而不是 100%。

「頻率學派」和「貝葉斯學派」之中,究竟哪一個正確呢?又或者說,哪一個較為有用呢?

— Me@2012.11.22

2012.11.22 Thursday (c) All rights reserved by ACHK

Hero 4

Seen by itself the David’s body might be some unusually taut and vivid work of antiquity; it is only when we come to the head that we are aware of a spiritual force that the ancient world never knew. I suppose that this quality, which I may call heroic, is not a part of most people’s idea of civilisation. It involves a contempt for convenience and a sacrifice of all those pleasures that contribute to what we call civilised life. It is the enemy of happiness. And yet we recognise that to despise material obstacles, and even to defy the blind forces of fate, is man’s supreme achievement; and since, in the end, civilisation depends on man’s extending his powers of mind and spirit to the utmost, we must reckon the emergence of Michelangelo as one of the great events in the history of western man.

— Ch. 5: The Hero as Artist

— Civilisation (1969)

— Kenneth Clark

2012.11.21 Wednesday ACHK