The problem of induction 2

Posted on December 5, 2012 by rpflee

Occam’s razor 2

If induction has worked in the past, it is a simpler theory that it will continue to work, than it will stop working at precisely midnight, because the theory that it will stop working has to contain additional information specifying when it will stop working.

Marcus Hutter, in 2005, demonstrated that simpler theories are ([given that] all other things being equal) more likely to be true.

This gives us the bootstrap we need to have as least a minimal reason to trust induction that doesn’t require inductive reasoning.

— 31 August 2012 11:20:20AM

— Douglas_Reay

— Less Wrong

2012.12.05 Wednesday ACHK

Godel 4.2

Posted on November 30, 2012 by rpflee

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

Superdeterminism

Posted on November 23, 2012 by rpflee

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

Godel 4

Posted on November 22, 2012 by rpflee

Paradox 8

Any formal system strong enough to include arithmetic unavoidably has self-references.

— Me@2012-04-02 9:32:06 AM

機會率哲學 2.7.2

Posted on November 17, 2012 by rpflee

The problem of induction 1.7.2 | Paradox 7.2

這段改編自 2010 年 4 月 3 日的對話。

（安：但是，你不能不考慮「機會率」。

例如，你發覺一粒骰子，一百次之中，全部一百次都是擲到「一」。那樣，你會認為，那一粒骰子不正常，不是公平的。你歸納到的規律是，那粒骰子次次也會擲到「一」。所以，你會預測，下次擲骰子的結果都是「一」。正如，因為以往的每天，太陽都由東邊升起，你自然會預期，明天都是那樣。這是一個「完全有規律」的例子。

又例如，你發覺一粒骰子，一百次之中，有九十次都是擲到「一」。你會覺得，「擲到一」的機會率，遠高於其他五個數字。你歸納到的規律是，那粒骰子傾向擲到「一」。所以，你會預測，下次擲骰子的結果都是「一」。這是一個「既不是完全沒有規律，亦不是完全有規律」的例子。一方面，這個事件並不是「完全不可預測」的，因為你相當有信心，骰子會擲到「一」。另一方面，這個事件亦不是「完全可預測」的，因為你的信心並不至於大到，願意用整副身家到擔保。

再例如，你發覺一粒骰子，一百次之中，有大概十六次是擲到「一」，而其他數字的出現次數，也是差不多。那樣，你會認為，那一粒骰子是正常公平的。這是一個「完全沒有規律」的例子。因為骰子對那六個數字，無所偏好，導致你「完全不可預測」，下一次會擲到哪一個數字。

但是，從另一層次看，「無所偏好」即是「隨機」。那樣，你就可以使用「機會率法則」。雖然你不可以預測，下一次會擲到哪一個數字，但是你可以宣稱，下一次擲到任何一個數字的機會率，都是六分之一。換句話說，如果你把骰子擲很多次，每個數字出現的次數，都會佔全部次數的大概六分之一。）

你的意思是，個別事件「完全不可預測」的話，即是在「個別事件層次」，完全沒有規律。那樣，如果提高一個層次，改為觀察「大量個別事件」，就反而會有明顯的規律。完全沒有規律的個別事件，即是「隨機事件」。既為「隨機事件」，就可以用「機會率」去處理。

又或者說，如果個別事件完全沒有「必然定律」，集體事件就會遵守「概然定律」。「概然」即是「大概而然」，亦即「集體而言」。「概然定律」，亦稱「機會率法則」。

— Me@2012.11.17

機會率哲學 2.7.1

Posted on November 15, 2012 by rpflee

The problem of induction 1.7.1 | Paradox 7.1

這段改編自 2010 年 4 月 3 日的對話。

（安：Hans Reichenbach 講法的下半段是，如果自然現象根本沒有規律可言，就不會有任何方法，可以找到任何規律。那樣，即使找不到規律，也不是「歸納法」有錯。

這裡我不明白的地方是，何謂「完全沒有規律」呢？

我彷彿想像不到，有一個情況，可以真的「完全沒有規律」。）

你的意思是，「完全沒有規律」本身，都可以視為一個「規律」？

（安：大概是。）

那樣不行。有沒有「規律」，不純粹是感覺。你不可以隨便標籤，因為我們正在討論的「規律」，有明確意思。「有規律」是指「可預測」。如果一件事件是「不可預測的」，你就不可以宣稱它「有規律」。

（安：但是，你不能不考慮「機會率」。）

— Me@2012.11.15

Paradox 6

Posted on November 9, 2012 by rpflee

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

2012.11.09 Friday ACHK

Paradox 5

Posted on October 24, 2012 by rpflee

Meta-time 4 | Cumulative concept of time, 13

The grandfather paradox has this structure:

Your-2012-self = [ …, your-2002-self, … ]

B = [ …, A, … ]

If your-2012-self can go back to, such as, year 2001,

Your-2002-self = [ …, your-2012-self, … ]

A = [ …, B, … ]

Unless A = B, it is logical impossible to have both “A is part of B” and “B is part of A“.

— Me@2012.04.02

Paradox 4.3

Posted on October 15, 2012 by rpflee

According to the definition/usage of the word “true”:

A = “A” is true

“A” is true = A

e.g. The sentence “Snow is white” is true = Snow is white

Not-A = “A” is false

“A” is false = Not-A

e.g. The sentence “Snow is white” is false = Snow is not white

Paradox is due to the mixing of para-level (meta-level) and original level. As long as we do not allow mixing levels, there are no paradoxes. Every sentence should only be allowed to describe sentences which have lower levels. So the sentence “this sentence is false” should not be allowed.

this sentence is false = “this sentence is false” is (in ordinary level) true

this sentence is false = “this sentence is false” is (in meta level) false

In detail, according to the definition of the word “true”:

this sentence is false = “this sentence is false” is true

But according to the meaning of the sentence itself:

this sentence is false = “this sentence is false” is false

So there is a contradiction.

— Me@2012.04.01

— Me@2012.10.05

— Me@2012.10.15

Curry’s paradox 1.2

Posted on October 14, 2012 by rpflee

Paradox 4.2

The resolution of Curry’s paradox is a contentious issue because resolutions (apart from trivial ones such as disallowing X directly) are difficult and not intuitive. Logicians are undecided whether such sentences are somehow impermissible (and if so, how to banish them), or meaningless, or whether they are correct and reveal problems with the concept of truth itself (and if so, whether we should reject the concept of truth, or change it), or whether they can be rendered benign by a suitable account of their meanings.

— Wikipedia on Curry’s paradox

Regarding Curry’s paradox is an unresolved important question is silly.

— Me@2012-10-13 02:41:50 PM

Curry’s paradox 1.1

Posted on October 13, 2012 by rpflee

Paradox 4.1

When you equal the meta sentence level and the sentence level, it is nothing special about the paradox, for it is just a contradiction between the sentence and the meta-sentence.

In Curry’s paradox, “X -> Y” is what you want to prove. So you try “if X”, to see if you can get Y.

But by letting “X = X -> Y” and if X, and X = X -> Y, then Y. You are actually doing circular proof.

Effectively you use X to prove X, which is a useless proof.

— Me@2012.04.01

— Me@2012.10.05

— Me@2012.10.13

Paradox 3.4

Posted on October 6, 2012 by rpflee

Meta-time 3.4

Time travel is only possible if there is a meta-time. Time travel is only possible if our physical time is fake, in a sense that it is not the real causal chain; and the meta-time is real, in a sense that it is the real causal chain.

For example, if our so-called physical world is actually a computer video game simulation, then the “physical” laws and the “physical” time are fake, in a sense that they are illusions simulated by a computer. The world to which that computer belongs is our meta-time. That meta-world is the real physical world. That meta-time is the real causal chain.

— Me@2012-10-01 12:53:48 PM

— Me@2012-10-03 02:21:45 PM

— Me@2012-10-05 11:24:43 AM

Paradox 3.3

Posted on October 6, 2012 by rpflee

Meta-time 3.3

Paradox is due to the mixing of para-level (meta-level) and original level, including the time-travel paradoxes.

The grandpa paradox has the similar structure as

“This sentence is true.”

The ontological paradox has the similar structure as

“This sentence is false.”

— Me@2012-10-06 09:35:11 AM

Paradox 3.2

Posted on October 5, 2012 by rpflee

Meta-time 3.2

Paradox is due to the mixing of para-level (meta-level) and original level. For example, consider this sentence:

“This sentence is false. “

There are two problems for this sentence.

First, is this sentence true or false?

If it is true, according to itself, it is false.

But if it is false, then the assertion that “this sentence is false” is false, so it is true.

Second, what is the level of this sentence?

We don’t know, because it is referring to nothing, except itself. Let us just assume that it is an order-n sentence.

But since it describes itself, it describes an order-n sentence. So it is an order-(n+1) sentence.

But since it describes itself, it describes an order-(n+1) sentence. So it is an order-(n+2) sentence.

Contradiction!

How can the same sentence have more than one order?

That is exactly the problem of mixing levels. The meaning of the sentence and the meaning of the meta-sentence may contradict.

“This sentence is false.” is with level n, (n+1), (n+2), … at the same time.

But if it is true at level n, it is false at level (n+1), and true at level (n+2), etc.

So it is true and false and true …

Then S is not allowed to describe any other level n (or higher than level n) sentences.

— Me@2012-10-05 02:00:04 PM

Paradox 3.1

Posted on October 5, 2012 by rpflee

Meta-time 3.1

Objects and events are of level zero.

Sentences about objects and events are of level one. They are called order-one sentences, e.g.

“Here is an apple.”

Sentences about sentences are of level two. They are called order-two sentences or meta-sentences, e.g.

“”Here is an apple.” has 4 words.”

Sentences about order-two sentences are of level three. They are called order-three sentences or meta-meta-sentences, e.g.

“”Here is an apple.” has 5 words.” is false.”

…

— Me@2012-10-05 12:00:04 PM

Paradox 2

Posted on October 3, 2012 by rpflee

Meta-time 2

Paradox is due to the mixing of para-level (meta-level) and original level.

— Me@2012-09-29 02:22:14 PM

… including the time-travel paradoxes.

As long as you put time-travel into a story, you mix the meta-time and the original time within that story. Thus time-travel paradoxes appear.

— Me@2012-10-01 10:33:05 AM

The two typical time-travel paradoxes are the grandfather paradox and the ontological paradox.

The grandfather paradox is that time-travel would create an inconsistent story. For example, if you time-travel back to 10 years ago and kill your younger self, you present-self cannot exist. So you could not have time-travelled back to 10 years ago and kill your younger self, you present-self can exist. But your present-self have time-travelled back to 10 years ago and kill your younger self, then you present-self cannot exist.

The ontological paradox is that information can come from nowhere and events can happen with no cause. For example, your future-self goes back in time to give you the solution of a homework problem. After copying it, you go back in time to give your past-self the solution of that homework problem. The question is, where does that homework solution come from?

The meta-time is the author’s time or the readers’ time, which is the real, in a sense that it is the real causal chain. The original-time is the time within that story, which is fake, in a sense that it is not the real causal chain. As long as we distinguish the meta-time (author’s time) and the original-time (story-time) clearly, the two paradoxes can be transcended.

To avoid the grandfather paradox, only the author should be allowed to go back into an earlier story-time. For example, after finishing the 10 chapters of a story, the author goes back to the first chapter to rewrite and polish it. The characters within that story should not be able to go back into an earlier story-time.

To transcend the ontological paradox, we should realize that the “information from nowhere” is actually from the meta-time; the “event with no cause” is actually caused by the author of that story.

— Me@2012-10-03 02:21:45 PM

Paradox

Posted on October 1, 2012 by rpflee

Meta-time

para- (“above, beyond; abnormal”)

— Wiktionary

Paradox is PARAdox.

Paradox is due to the mixing of para-level (meta-level) and original level.

— Me@2012-09-29 02:22:14 PM

… including the time-travel paradoxes.

As long as you put time-travel into a story, you mix the meta-time and the original time within that story. Thus time-travel paradoxes appear.

— Me@2012-10-01 10:33:05 AM

Logical implication

Posted on September 3, 2012 by rpflee

Entailment [logical implication] vs material implication

The difference between material implication and entailment is that they apply in different contexts. The first is a statement of logic, the second of metalogic. If p and q are two sentences then the difference between “p implies q” and “p is a proof of q” is that the first is a statement within formal logic, the second is a statement about it. Entailment is a concept of proof theory, whereas material implication is the mechanics of a proof.

— Wikipedia on Entailment

2012.09.03 Monday ACHK

Self aware, 2

Posted on August 15, 2012 by rpflee

Recursion 10

Consciousness is a self aware.

— Me@2012-03-09 10:44:35 PM

Chrono Trigger 3.2

Posted on August 3, 2012 by rpflee

Meta 4 | The Neverending Story 2

Each computer game is not never-ending, but the meta-game (the game of making and playing computer games) can be.

— Me@2012.08.02

Physics Town

continues

Category Archives: meta-dox