Digital physics, 6

Many people interested in physics keep on believing all kinds of evidently incorrect mystifications related to the notion of a “minimal length” and its logical relationships with the Lorentz invariance. Let’s look at them.

— Myths about the minimal length

— Lubos Motl

2012.09.30 Sunday ACHK

Universal Grammar, 2

Predicate logics may be viewed syntactically as Chomsky grammars. As such, predicate logics (as well as modal logics and mixed modal predicate logics) may be viewed as context-sensitive, or more typically as context-free, grammars. As each one of the four Chomsky-type grammars have equivalent automata, these logics can be viewed as automata just as well.

— Wikipedia on Predicate logic

2012.09.29 Saturday ACHK

Memory 4

這段改編自 2010 年 4 月 3 日的對話。

正常健康的腦袋,不會記下所有東西,而會把經驗過濾,把握重點記下,把其他遺忘。所以,記憶既不是事實的全部,亦未必是事實的真相。記憶時常會不完整,甚至不準確。

一個簡單的實驗,就可以證實,「記憶」和「事實」,並非百份百相乎。你可以試試再看,一部以前看過的電影。你會發現,電影中的很多細節,和你的記憶不一致,甚至相反。例如,你記得某輛車明明是在畫面的左邊,重看時,卻發現它竟然在右邊。

— Me@2012.09.29

2012.09.29 Saturday (c) All rights reserved by ACHK

西瓜 5

[physical geometry]

In so far as the statements of geometry speak about reality, they are not certain;

[mathematical geometry]

and in so far as they are certain, they do not speak about reality.

— Albert Einstein

Analytic statements are about the languages.

Synthetic statements are about the world.

Choosing the best language describing the world is itself a synthetic problem. 

— Me@2012-03-24 12:02:44 AM 

“Is logic empirical?” is not a valid question, because it does not specify the meaning of “logic”.

“Is logic empirical?” is due to the confusion of two different concepts. 

If you have no such confusion, the answer to the question is trivial.

As systems of analytical statements, the different theories of logic are not empirical.

But choosing the best among the logic systems to describe the real world is itself empirical.

— Me@2012-09-23 05:10:23 PM

2012.09.28 Friday (c) All rights reserved by ACHK

Completeness theorem, 4

Why truth table is not used in logic?

— JiminP

… the difference between semantics and syntax. A syntactic proof is a finite formal derivation of a sentence from the axioms of a theory using the logical axioms and the rules of inference of a logic. A proof by a truth table is a semantic proof; in allowing truth tables you are tacitly assuming the completeness theorem of propositional logic. Essentially, a priori, we don’t know that everything we can prove by studying the models of a theory (i.e. truth tables, in the case of propositional logic) can be proven syntactically, or even for that matter vice versa. It’s a non-trivial result in logic, …

— Zhen Lin

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

— Mathematics – Stack Exchange

2012.09.27 Thursday ACHK

Digital physics, 5

Part of our disagreement is a misunderstanding. I am not questioning that the usual notions of geometry break down at the Planck scale (or earlier).

But the reason in string theory is that it does not make sense to talk about shorter distances because the physics at “shorter” distances is not just normal geometry plus something else, but a stringy generalized fuzzy blah blah structure.

Loop quantum gravity, on the other hand, says that geometry is a good variable at all distance scales, and the areas etc. have discrete spectrum, which contradicts Lorentz invariance in any theory with local excitations.

— Lubos Motl

2012.09.26 Wednesday ACHK

考試美術 1.2

這段改編自 2010 年 6 月 9 日的對話。

正確的方法是,你先把整幅畫(人、樹、屋)的大致輪廓,先畫出來。因為你只是令「人、樹、屋」成形,所花的的時間一定不多。接著,你才為整幅畫,畫第一重的細節。之後,還有時間剩餘的話,你就加上第二重的細節。下一步,再有時間的話,你再增添第三重的細節,如此類推。

這個策略的好處是,無論考官在什麼時刻宣佈停筆,你都可以宣稱,你的畫作經已經完成。你不可能不合格。

— Me@2012.09.26

2012.09.26 Wednesday (c) All rights reserved by ACHK

Principia Mathematica, 3

Principia is nowadays only of historical interest, since the subject has developed in quite different directions from those initiated by Russell and Whitehead. The idea of basing mathematics (including the development of the usual integers, reals, function spaces) purely on “logic” has largely been abandoned in favour of set-theory based formulations. And Principia does not have a clear separation between syntax and semantics. Such a separation is essential to the development of Model Theory in the past 80 years.

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

— Mathematics – Stack Exchange

— Andre Nicolas

2012.09.25 Tuesday ACHK

三國演義

這段改編自 2010 年 4 月 3 日的對話。

(安:你網誌的部分文章,雖然是改編自你我的對話,即是我親身和你討論過那些話題,但是現在看回,感覺十分奇怪,而又帶點震撼。)

我聽自己的錄音紀錄時,我都覺得很奇怪,好像聆聽另一個人在說話。

真實版的「安」 和 文字改編版的「安」的分別,就好像《三國志》諸葛亮 和《三國演義》諸葛亮的分別。兩者差距可以很大。

— Me@2012.09.24

2012.09.24 Monday (c) All rights reserved by ACHK

Information lost, 4

Losing information is the same as generating entropy. And generating entropy means generating heat. The virtual black holes that Stephen had so blithely postulated would create heat in empty space.

— The Black Hole War, p.23

— Leonard Susskind

2012.09.23 Sunday ACHK

Teaching is useless

無足夠資料 9 | 數學教育 4 | What is statistics? 2 | 軟硬智力 10

Intellect is invisible to those have none.

– Arthur Schopenhauer

Teaching is useless. Information is useful. Although intelligence can seldom be increased, giving relevant information to a GOOD person so that he can fully realize his existing intelligence potential is always useful.

The intelligence-absorption principle should actually be called the information-absorption principle, because intelligence cannot be increased directly.

「智攝原則」應該改稱為「資攝原則」,因為你沒有可能,直接增加別人的智力。你可以做到的,就只有透過給予相關資料,去加快別人智力潛能的發展。換句話說,你只可以破除無知,而不可刪減愚蠢。

Information is just like technology: Intellectually challenging to create doesn’t mean intellectually challenging to use.

— Me@2011.11.10

You cannot teach a man anything, you can only help him to find it for himself.

– Galileo

2012.09.23 Sunday (c) All rights reserved by ACHK

考試美術 1.1

這段改編自 2010 年 6 月 9 日的對話。

當年我報考中學會考的美術科。該科有兩份卷。第一份是素描;第二份我選修了設計。每份卷的考試時間長度,都是三個小時。

美術老師教了我,一個至關重要的考試技巧。在一份卷的三個小時內,千萬不要力臻完美,十分詳細地畫任何一部分;因為,那樣會令你沒有足夠時間,去畫作品中的其他部分,而導致該卷不合格。

例如,試題的要求是,畫一個人、一棵樹 和 一間屋。如果你打算集中火力,先「完成」那間屋,才開始描繪其他的話,計劃就一定行不通。任何一部分,你都可以不斷迫近完美,無限詳細地畫下去,並沒有所謂「客觀的完成」。所以,「打算先『完成』那間屋」的真正意思是,你不企圖畫其他。

典型的劇情是,你花了頭兩個小時,都仍然在畫那間屋,不肯放手。結果,拖到二小時三十分,才開始用餘下的半小時,將「人」和「樹」草草了事。到頭來,三樣東西中,你只畫了一樣,是似模似樣的。你根本沒有完成試題的基本要求,成績自然是不合格,

— Me@2012.09.23

2012.09.23 Sunday (c) All rights reserved by ACHK

Existential Import, 1

Universal claims about empty sets are all true, because there are no falsifying instances.

NOTE: Claims about empty sets are trivially true. Sure, “all irrational prime numbers are odd” because there are no irrational prime numbers, but it is equally true that “all irrational prime numbers are even”.

— Tutorials, PL 120 Symbolic Logic I, Fall 2011

— Professor H. Hamner Hill

2012.09.22 Saturday ACHK