Author Archives: rpflee

Godel 6

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This theorem implies that the only way a language can be incomplete is [that] there is a model of the language in which a particular statement is true, and another in which the statement is false.

For example, we can see that [for] the language [comprising] the symbols 0, 1, +, -, and , [] the statement []   is true [] if we take the structure to be C or R, but not if we choose Q. So it is clear that the formula a*a = 2 is not true in every model of the language and the thus the language is incomplete.

What the completeness theorem asserts is that this is the only way that a theory (set of formulas) can be incomplete and that every formula that satisfies every structure is provable in the language.

— Godel’s Completeness And Incompleteness Theorems

— Ben Chaiken

2012.12.04 Tuesday ACHK

第 N 減 1 步

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背誦製成品 10

這段改編自 2012 年 12 月 2 日的對話。

…

這一題用這個方法，就可以輕易完成。

（A：真的嗎？好像很難會想得到。）

你覺得很難，正正是因為你以為，需要由自己想得到。但是，現在我並不是要求你自己創作方法，而是要求你記得我教你的方法，然後在考試時見到同類題目時，可以高速拿出來運用。

情形就製作三文治前，我叫你準備材料，例如兩片麵包。如果你問：「真的嗎？好像很難才會弄到一片麵包。試想想，由種小麥開始，到製成一片麵包，有很多工序，很辛苦啊！」

我就會答：「但是，現在我並不是要求你，自己製作麵包，而是要求你，先買兩片麵包回來，然後用來製作三文治。」

記住，正常的考試，並不會要求你，由起點「第一步」開始，走到目的地「第 N 步」。大概而言，考試只會要求你由「第 N – 1 步」，走到「第 N 步」。只要你肯先行背誦，常用的高深方法，你在考試作答題目時，就可以由「第 N – 1 步」開始，輕易走到「第 N 步」。

— Me@2012.12.02

2012.12.03 Monday (c) All rights reserved by ACHK

Bell’s theorem, 5

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De Broglie–Bohm theory, 4

[guess]

The spirit of de Broglie–Bohm theory is non-local realism.

The particles can be regarded as being “actually” in eigenstates with respect to any variables, even before any measurements, as long as somehow, we take into account the fact that each particle is influenced by all other particles in the universe, creating the “illusion” of superposition of eigenstates.

— Me@2012.12.01

2012.12.03 Monday (c) All rights reserved by ACHK

Results

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Mathematicians won the war.

Mathematicians broke the Japanese codes and built the A-bomb.

Mathematicians… like you.

In medicine or economics,

in technology or in space,

battle lines are being drawn.

To triumph we need results — publishable, applicable results.

Now, who among you will be the next Morse,

the next Einstein?

Who among you will be the vanguard of democracy, freedom and discovery?

Today we bequeath America’s future into your able hands.

Welcome to Princeton, gentlemen.

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– A Beautiful Mind (2001)

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2009.09.11 Friday ACHK

Monty Hall problem 1.5.4

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機會率哲學 4.1.4

這段改編自 2010 年 4 月 3 日的對話。

主持人選擇先打開哪一扇門，其實受制於參賽者的原初選擇，是否房車。換句話說，「房車在第一扇門後面，而主持人打開第三扇」和「房車在第二扇門後面，而主持人打開第三扇」這兩個遊戲中途結果，是不對稱的，因為兩者有著不同的歷程。既然不等價，機會率就自然不同。

（安：我同意這個想法，會得到這個結論。但是，如果我用另一個想法，卻會得出另一個結論。

假設原本的參賽者叫做「甲」，而在遊戲中途，會加入另一位參賽者「乙」。主持人在打開第三扇門後，會改為叫乙，為甲繼續選擇。亦即是話，乙要為甲決定，究竟是選第一道門，還是第二道門。如果乙的選擇可為甲贏得房車，乙自己亦會獲得一千元的獎金。

那樣，對於乙來說，他見到的，就只是一個「二選其一」的「開門抽獎遊戲」。

如果主持人沒有任何提示，乙就只會知道「那裡有兩道門」、「一道有房車」和「另一道有山羊」，而不會掌握任何其他資料。換而言之，乙對第一第二道門，無所偏好。所以，兩道門的中獎機會率，理應相同，都是二分之一。

我這個想法有錯嗎？）

如果乙在兩門選其一時，不知道甲的原本選擇，你的講法就是正確的。兩門的中獎機會相同，都是二分之一。

但是，如果乙知道甲的原選，兩門的歷史，相對乙來說，就會不同。所以，機會率不會均等。

— Me@2012.12.02

2012.12.02 Sunday (c) All rights reserved by ACHK

Godel 5

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Now when you combine the Completeness and Incompleteness Theorems, you can get some really remarkable results. If you work with the axioms of number theory, call them N (which include many of the above axioms F along with axioms for < and axioms for mathematical induction), for example, we know by the Incompleteness Theorem that there is a statement X such that neither X nor (not X) is provable. Hence, by the Completeness Theorem, there is a model of N in which X is true and a model of N in which X is false.

It follows that there are mathematical universes which look and act very much like the regular natural numbers, but do in fact have some subtle differences. One of the most fascinating results I’ve seen is that there is a model of number theory which “thinks” (in a precise sense) that the axioms N are inconsistent, even though they are not (roughly, the “proof” of an inconsistency that it “sees” is infinitely long, and so is not a real proof).

— Godel’s Completeness Theorem

— Joe Mileti

2012.12.02 Sunday ACHK

Monty Hall problem 1.5.3

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機會率哲學 4.1.3

這段改編自 2010 年 4 月 3 日的對話。

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？）

參賽者觀察到的事件是，主持人打開了第三扇門打開了，而門後有山羊。對於參賽者來說，有兩個可能的事件歷史，可以導致這個暫時的結果。

第一個可能是，房車在第一扇門後面，而主持人打開第三扇。

那樣，「打開第三扇門」其實只是主持人的兩個可能選擇之一。因為第二扇後面的，都是山羊，主持人選擇「打開第二扇」都可以。

第二個可能是，房車在第二扇門後面，而主持人打開第三扇。

那樣，「打開第三扇門」就是主持人的唯一選擇。因為第二扇後面的是房車，主持人被迫選擇「打開第三扇門」。

主持人選擇先打開哪一扇門，其實受制於參賽者的原初選擇，是否房車。換句話說，「房車在第一扇門後面，而主持人打開第三扇」和「房車在第二扇門後面，而主持人打開第三扇」這兩個遊戲中途結果，是不對稱的，因為兩者有著不同的歷程。既然不等價，機會率就自然不同。

— Me@2012.12.01

2012.12.01 Saturday (c) All rights reserved by ACHK

Small big bang 4

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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

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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

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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

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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

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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

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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

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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

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* 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

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2010.06.06 Sunday ACHK