因果律 1.1

Posted on August 3, 2013 by rpflee

語意互相推卸責任論 1.1

這段改編自 2010 年 4 月 3 日的對話。

（安：在《心通識六講》之中，李教授提過，依照 Pierre-Simon Laplace（拉普拉斯）的看法，如果我們了解全部的物理定律，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，無論是過去或者將來。我簡稱這個理論，為之「因果律」。

而對於「因果律」，我可以有兩個截然不同的看法。

第一個看法是，視之為「經驗句子」，即是「科學理論」。意思是，我們需要靠多次的觀察，和無數的實驗，才能判斷「因果律」正確與否。

另一個看法是，我們不把「因果律」視為「科學句子」，而視之為「重言句子」。意思是，我們視它為正常人「思考架構」和「語言系統」的一部分。李教授用的比喻是，這樣對待「因果律」，就好像我們對待「度量衡系統」一樣。例如，無論我們對這個世界有什麼觀測結果，「1 米等於 100 厘米」必然正確。

但是，我就是正正不明白第二個看法。「因果律」何以視為「重言句」呢？）

我都不是十分明白。我猜想，大概的意思是，如果抱著第一個看法，當我們觀察到，涉嫌違反「因果律」的事件發生時，我們就自然會質問，「因果律」的可信性；但是，如果採取第二個看法，當我們遇到貌似違反「因果律」的實驗結果時，我們就反而會懷疑，那個實驗本身，做得不夠精確。

比喻說，你和我都是一間水果公司的員工。你負責把每 10 個蘋果放在 1 個紙皮箱之中；而我就負責在每個紙皮箱的表面，寫個「10」字，以標明箱中蘋果的數目。有一次，你只放了 9 個蘋果落一箱之中，但我仍然標示「10」個，導致訊息錯誤，被顧客投訴。那是誰人的錯呢？

我可以質疑你，為何不放 10 個；你又可以反駁我，為何不寫「9」字。

（安：那樣，哪一個看法才算是正確的呢？）

因為「因果律」並不是常用的字眼，所以沒有一個精確的意思。學術討論以外，你大概不會想像得到，在日常生活中，你何來會提及「因果律」這個詞語。

— Me@2013.08.02

無限旅程 4.4

Posted on July 28, 2013 by rpflee

Meaningful 12.4 | 惜此際 4 | A. J. Ayer, 2.2

這段改編自 2010 年 4 月 3 日的對話。

（安：那我自己的存在呢？為什麼「自己存在」，好過「自己不存在」？）

那是一個奇怪的問題。那亦是一個「不成問題」的問題，因為，「自己」並沒有所謂「存在不存在」。意思是，「你的存在」並不是「你的性質」。

「性質」的意思是，你可以利用它，來把人群分成兩組。例如，「輕盈」是一個性質。所以，你可以把人群分成兩組：一組是「輕盈」的人，另一組是「不輕盈」的人。

但是，你可不可以假設，「存在」是一個「性質」呢？亦即是話，你可不可以把人群分成兩組：一組是「存在」的人，而另一組是「不存在」的人呢？

「你的存在」，並不是「你的性質」，而是其他東西的性質。準確一點講，「你的存在」，是「你環境的性質」。例如，你在學校參加課外活動，加入了足球學會，但沒有加入籃球學會。那樣，「你的存在」就是足球學會的性質，而不是籃球學會性質。

「性質」的意思是，你可以利用它，來把人群分成兩組。因為你不可以用「存在」和「不存在」，把人群分成兩組，所以一個人的存在，並不是那個人自己的性質。

但是，某個人的存在與否，可以把學校中的課外活動學會，分成兩組。例如，我們可以把所有學會分成兩組：一組是「有你」加入的學會，另一組是「沒有你」的學會。在這個上文下理之下，「你的存在」就是課外活動學會的性質。

記住，「你的存在」，並不是「你的性質」。

剛才提到，「你的體重」是「你的性質」。所以，你可以考慮一下，究竟「輕盈」一點會好一點，還是「豐滿」一點會更健康？

但是，因為「你的存在」，並不是「你的性質」，所以，「究竟『自己存在』對我來說好一點，還是『自己不存在』可以為我刪除痛苦」這個問題，根本沒有任何形式的意思。

例如，你可以問，一般來說，「輕盈」的人還是「豐滿」的人，會快樂一點？

但是，如果你問：

「

平均而言，究竟『存在的人』，還是『不存在的人』，會快樂一點？

」，

我就根本完全不知你在說什麼。「不存在的人」，還算是「人」嗎？「不存在的東西」，還算是「東西」嗎？既然「不存在」，又何來「快不快樂」呢？

自殺的人，至少犯了兩大錯誤。而這個邏輯錯誤，就是第一個。

他們以為，自殺以後，自己就不再存在，所以可以減輕痛苦，增加快樂。但是，如果真的「不再存在」，就沒有所謂的「增加快樂」。換而言之，自殺以後，即使假設自己不再存在，也沒有「減輕痛苦」的功效。只有「存在的人」或者「存在的意識體」，才有能力和有機會，減輕自己的痛苦。

第二個錯誤是，自殺的人以為，自殺以後，自己就不再存在。那是不一定的。

有時，他們又會以為，即使意識仍然繼續存在，他們也可以成功逃避到現世的痛苦。但是，那亦是沒有根據的。如果現世是痛苦的，你憑什麼可以假設，下一個世界是快樂的呢？

或者，「下一世」比「現世」，更加光怪陸離。

比喻說，逃學的年青人，以為輟學可以刪除讀書的痛苦。但是，他們不知道的是，「輟學」的後果是，要「上班」養活自己。對於一個年青人來說，「上班的痛苦」，往往大於「上學的痛苦」。

相反，如果那位年青人正常地完成學業，他反而可以大大減輕，未來上班的痛苦。

— Me@2013.07.28

The Beginning of Time, 2

Posted on July 25, 2013 by rpflee

Brian Greene continues with all the delusions and delusions, infrequently spiced with a correct proposition. The Big Bang created the arrow of time (the latter has nothing to do with the laws of physics), holy cow. “We don’t know why the Universe started in a low-entropy state,” holy cow. We perfectly know why it did. If it started with a state of a high entropy, we could always ask “what was before that”. The only thing that prevents us from going before a moment is that the moment has the minimal mathematically possible value of the entropy, namely zero.

— The Fabric of the Cosmos II

— Lubos Motl

2013.07.25 Thursday ACHK

無限旅程 4.3

Posted on July 25, 2013 by rpflee

Meaningful 12.3

這段改編自 2010 年 4 月 3 日的對話。

（安：但是，我又可以追問，為什麼「存在」好過「不存在」？）

那不是一定的。那要視乎上文下理而定。

你只可以說，「好東西存在」，好過「好東西不存在」；而「壞東西不存在」，好過「壞東西存在」。那都是重言句。

例如，銀包（錢包）中的金錢，對你來說是好東西。所以，「金錢存在於你的銀包之內」，相對於你而言，好過「金錢不存在於你的銀包之中」。

又例如，銀包中的蟑螂，對你來說是壞東西。所以，「蟑螂不存在於你的銀包之內」，相對於你而言，好過「蟑螂存在於你的銀包之中」。

（安：那我自己的存在呢？為什麼「自己存在」，好過「自己不存在」？）

那是一個奇怪的問題。那亦是一個「不成問題」的問題，因為，「自己」並沒有所謂「存在不存在」。意思是，「你的存在」並不是「你的性質」。

但是，你可不可以假設，「存在」是一個「性質」呢？亦即是話，你可不可以把人群分成兩組：一組是「存在」的人，而另一組是「不存在」的人呢？

— Me@2013.07.24

Godel 11

Posted on July 21, 2013 by rpflee

1931: Publication of Gödel’s incompleteness theorems, showing that essential aspects of Hilbert’s program could not be attained.

It showed how to construct, for any sufficiently powerful and consistent recursively axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic on the (infinite) set of natural numbers – a statement that formally expresses its own unprovability, which he then proved equivalent to the claim of consistency of the theory; so that (assuming the consistency as true), the system is not powerful enough for proving its own consistency, let alone that a simpler system could do the job.

It thus became clear that the notion of mathematical truth [cannot] be completely determined and reduced to a purely formal system as envisaged in Hilbert’s program. This dealt a final blow to the heart of Hilbert’s program, the hope that consistency could be established by finitistic means (it was never made clear exactly what axioms were the “finitistic” ones, but whatever axiomatic system was being referred to, it was a ‘weaker’ system than the system whose consistency it was supposed to prove).

— Wikipedia on Foundations of mathematics

2013.07.21 Sunday ACHK

Godel 10

Posted on July 19, 2013 by rpflee

An important consequence of the completeness theorem is that it is possible to enumerate the logical consequences of any effective first-order theory, by enumerating all the correct deductions using axioms from the theory.

Gödel’s incompleteness theorem, referring to a different meaning of completeness, shows that if any sufficiently strong effective theory of arithmetic is consistent[,] then there is a formula (depending on the theory) which can neither be proven nor disproven within the theory. Nevertheless the completeness theorem applies to these theories, showing that any logical consequence of such a theory is provable from the theory.

— 14 February 2012

— Wikipedia on Gödel’s completeness theorem

2013.07.19 Friday ACHK

Godel 9

Posted on July 17, 2013 by rpflee

For example, the statement S = “There exists an a with a*a = 2″ is true in R, but false in Q since the square root of 2 is irrational. Similarly, the statement T = “There exists an a with a*a = -1″ is true in C, but false in R (the imaginary unit i satisfies the statement in C).

[]

Hence, if you believe the Soundness Theorem, we should not expect to be able to prove either S or (not S) from F because there is one model of F in which S is true, and one where S is false. Similarly, we should not expect to be able to prove either T or (not T). Thus, our system F is incomplete, i.e. there are statements X such that we can neither prove X nor (not X).

[]

However, there are [] good reasons why it is incomplete [;] there are statements which can be either true or false depending on which model of F you are currently working.

The Completeness Theorem basically says that this is the only way a system can be incomplete. In other words, the above converse question is true, which implies that we can prove absolutely everything that is not ruled out for the above basic reason.

— Godel’s Completeness Theorem

— Joe Mileti

2013.07.17 Wednesday ACHK

Meta-time 5

Posted on July 15, 2013 by rpflee

Two dimensional time 5 | 二次元時間 5

You are the main author of your future.

In a special sense, the real-present is the meta-time of the potential future timelines, selecting the future among multiple possible timelines.

— Me@2013-07-11 3:48 PM

Looper, 5.4

Posted on July 11, 2013 by rpflee

Paradox 5.5 | Meta-time 4.5 | Cumulative concept of time, 13.5 | Two dimensional time 4.4 | 二次元時間 4.4

To be logically consistent WITHIN the movie’s story, Young Joe (in the year 2044) should not be able to influence Old Joe, who had time-travelled to the year 2044 from the year 2074，because that Old Joe is from another timeline. The proof is that Young Joe’s experience in the year 2044 is different from Old Joe’s experience in the year 2044 when he was young.

They are not the same person, nor the same person at different ages within the same timeline. At most, they are different versions of the “same” person from two different timelines (aka “parallel universes” or “histories”).

Young Joe’s changes should affect the same-timeline-Old-Joe, but not any Old Joe’s from any other timelines. So the Old Joe within the movie should not have been affected when Young Joe hurt himself.

Also, the changes of the same-timeline-Old-Joe due to the actions of Young Joe should be seen only by the author (meta-time), but not by Young Joe until he has become that Old Joe 30 years later.

The author unintentionally, or intentionally, has confused two story timelines. Moreover, the author unintentionally, or intentionally, has confused the story-time and its meta-time.

— Me@2013-07-05 10:32 PM

Looper, 5.3

Posted on July 9, 2013 by rpflee

Paradox 5.4 | Meta-time 4.4 | Cumulative concept of time, 13.4 | Two dimensional time 4.3 | 二次元時間 4.3

In a single-mutable-timeline time travel story, the two dimensional time is not due to the internal causal structure of the story. Instead, it is due to the author’s timeline (aka meta-time). The author’s timeline is the second time dimension (aka independent direction).

The single-mutable-timeline model of time travel is not logically consistent within the story. If it is “mutable”, it is not “single”.

The single-mutable-timeline model of time travel is logically consistent only outside the story, from the perspective of the story’s author.

— Me@2013-07-02 3:47 PM

Looper, 5.2

Posted on July 8, 2013 by rpflee

Paradox 5.3 | Meta-time 4.3 | Cumulative concept of time, 13.3 | Two dimensional time 4.2 | 二次元時間 4.2

In the movie Looper, Young Joe (in the year 2044) influences Old Joe (in the year 2074) in the sense that Young Joe’s every action affects the state of Old Joe, because Old Joe is Young Joe’s future self.

For example, after Young Joe had hurt his own arm, the corresponding wound also appeared on Old Joe’s arm, even though Old Joe had already time-travelled back to the year 2044.

All of Young Joe’s actions are the causes of Old Joe’s state. Young Joe is in the past of Old Joe.

Old Joe (2074-Joe) = [ …, Young Joe (2044-Joe), … ]

B = [ …, A, … ]

However, Old Joe (2074-Joe) had time-travelled back to the year 2044, meeting the Young Joe.

So, some of Old Joe’s actions would affect Young Joe’s decisions on his own actions. In this sense, Old Joe also influences Young Joe indirectly. Some of Old Joe’s actions are the causes of Young Joe’s state. Part of Old Joe is also in the past of Young Joe.

Young Joe (2044-Joe) = [ …, Old Joe (2074-Joe), … ]

A = [ …, B, … ]

However, it is logically impossible to have both

B is in the past of A

and

A is in the past of B

just as it is logically almost impossible to have both

D is a part of C

and

C is a part of D

If you insist that it is the case, the only possibility is that

C = D

In this analogy, neither C nor D is really a “part” of another. In the time travel case, neither A nor B is really in the past of another. In other words, A (Young Joe) and B (Old Joe) have no time relationship. Neither’s actions are the causes of the state of another.

The real causes of Young Joe or Old Joe’s states are actually not within the movie story’s timeline. The real causes are the decisions of the author of the story.

— Me@2013-07-03 6:19 PM

Looper, 5

Posted on July 5, 2013 by rpflee

Paradox 5.2 | Meta-time 4.2 | Cumulative concept of time, 13.2 | Two dimensional time 4.1 | 二次元時間 4.1

In a “if and only if” case, there is no time.

If A is a necessary condition of B, we say “A is a cause of B“. In other words, A is in the past of B.

However, in some time travel story, it is “possible” to have both

A is a cause of B

(A is a necessary condition of B)

(B -> A)

and

B is a cause of A

(B is also a necessary condition of A)

(A -> B)

In this case, A and B are just equivalent.

(A B)

Neither is in the past of another. A and B have no causal relationship. In this sense, there is no time.

— Me@2013-07-03 6:19 PM

The P = NP problem

Posted on July 4, 2013 by rpflee

In essence, the P = NP problem can be restated as the following question:

Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be computed quickly?

— Wikipedia on P versus NP problem

— 17:14, 31 July 2011

If the question of whether P=NP were to be answered affirmatively[,] it would trivialize the rest of the Millennium Prize Problems (and indeed all but the unprovable propositions in mathematics) because they would all have direct solutions easily solvable by a formal system.

—

2013.07.04 Thursday ACHK

Conscious time

Posted on June 29, 2013 by rpflee

Cumulative concept of time, 15

In 1895, in his novel, The Time Machine, H.G. Wells wrote, “There is no difference between time and any of the three dimensions of space except that our consciousness moves along it.”

— Wikipedia on Spacetime

Consciousness “moves” from the past to the future because consciousness is a kind of reflection.

To be conscious, one has to access its own states. But only the past states are available. Accessing one’s own now-here state is logically impossible, because that creates a metadox (paradox).

— Me@2013-06-26 02:28:51 PM

We can remember the past but not the future because the past is part of the future; the whole contains its parts, but not vice versa.

— Me@2011.08.21

Past and Future

Posted on June 26, 2013 by rpflee

“The past cannot be changed” is a tautology, because “cannot be changed” is included in the meaning of the word “past“.

Similarly, “the future is not fixed” is also a tautology.

Anything you cannot change is within your past.

Anything you can change is within your future.

— Me@2013-06-23 3:40 PM

— Me@2013-06-26 10:40 AM

Creator

Posted on June 24, 2013 by rpflee

Assume that your existence is a property of X. Then in order to create you or annihilate you, X has to change the corresponding property.

For example, “having beard” is a property of me. In order to destroy that beard, I have to be willing to change my property of “having beard” to “having no beard“.

— Me@2013-06-11 2:30 AM

Consciousness 3

Posted on June 23, 2013 by rpflee

Copy Me, 4

Being conscious is being able to form memories.

forming memories

~ forming an identity

~ forming a causal chain of thoughts

To form memories, one needs to access and then store its own states.

But due to metadox (paradox), no one, or no single part of the brain, can access its own now-here state directly.

That is why different parts of the brain have to communicate and coordinate in order to be conscious.

— Me@2013-05-30 1:47 PM

Eigenstates 2.3

Posted on June 22, 2013 by rpflee

So, after all, what is the meaning of “a quantum eigenstate”?

One way to resolve the circular definition is to define

a definite state

a state whose measurement result can be predicted with 100% certainty provided that the initial condition is given with 100% accuracy

Another way to resolve the circular definition is to realize that

1. a classical state, as a macroscopic definite state, is experimental or observational;

2. a quantum eigenstate, as a microscopic definite state, is conceptual.

A classical state is what we, as macroscopic observers, can see directly.

A quantum eigenstate is what we cannot see. Moreover, it is not absolute. For the same system, there are more than one choice of state vector bases, in the sense that different sets of measurements can get different sets of eigenstates.

The concept of “quantum eigenstates” exists because we insist to express quantum states in terms of daily-life (classical (macroscopic) physics) language.

— Me@2013.06.22

Eigenstates 2.2

Posted on June 18, 2013 by rpflee

But there is a problem. The definition of “quantum eigenstate” seems to be circular:

an eigenstate = a definite state = a classical state

a quantum eigenstate = a microscopic state corresponding to a macroscopic (classical) state

The phrase “quantum eigenstates” is defined in terms of “classical states”. However, classical states exist only because of the decoherence of quantum states of a lot of particles. The universe is fundamentally quantum, not classical. The classical world exists only as an approximation to the quantum universe.

Also, we cannot define a quantum eigenstate as a collapsed quantum state, because in reality, there is no wave function collapse. Collapse is only an illusion due to quantum decoherence.

So, after all, what is the meaning of “a quantum eigenstate”?

— Me@2013.06.18

Eigenstates 2.1

Posted on June 17, 2013 by rpflee

A (macroscopic) classical state is due to the decoherence of quantum states of a lot of particles.

A quantum state is a quantum eigenstate or a superposition of quantum eigenstates.

a classical state = a macroscopic definite state

a quantum eigenstate = a microscopic definite state

a definite state = a state whose measurement result can be predicted with 100% certainty

— Me@2013.06.16

Physics Town

continues

Category Archives: mathematical logic

因果律 1.1

無限旅程 4.4

The Beginning of Time, 2

無限旅程 4.3

Godel 11

Godel 10

Godel 9

Meta-time 5

Looper, 5.4

Looper, 5.3

Looper, 5.2

Looper, 5

The P = NP problem

Conscious time

Past and Future

Creator

Consciousness 3

Eigenstates 2.3

Eigenstates 2.2

Eigenstates 2.1