Alfred Tarski, 3

The undefinability theorem shows that this encoding cannot be done for semantic concepts such as truth. It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any metalanguage capable of expressing the semantics of some object language must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language.

— Wikipedia on Tarski’s undefinability theorem

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Tarski’s 1969 “Truth and proof” considered both Gödel’s incompleteness theorems and Tarski’s undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.

— Wikipedia on Alfred Tarski

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2019.07.20 Saturday ACHK

PhD, 3.7.2

碩士 4.7.2 | On Keeping Your Soul, 2.2.7.2

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(問:根據你的講法,好像大部分情況下,都不應該讀研究院似的。)

在理想的情況下,你可能應該讀研究院。

(問:那樣,你心目中的理想情況是什麼?)

假設你已經有財政自由,你就有可能,適合讀研究院;因為,以前講有關讀研究院的一堆大問題,將會細很多,例如:

  1. 如果你是自資,就即是不拿學校的資助。那樣,你就不是僱員。研究以外的工作,例如做助教等,可以一概不理。

  2. 同理,你的博士導師再不會是,你工作上的上司。反而,你是消費者,他是你的僱員。

(問:怎樣為之「有財政自由」呢?)

即是你當時的積蓄,已足夠你一生人的使用。

(問:那就是即是「退了休」?)

不太一樣。

一來,「退休」通常是指,年紀大時才發生。追求財政自由的人,通常不會計劃,在年老時才財政自由;因為,追求財政自由的主要目的是,不用再每天上班,從而,有足夠的時間,去追求自己的理想。

二來,大有部分人「退休」時的積蓄,其實不夠餘生的使用。亦即是話,他們的財政,其實未有自由。

(問:那就即是話,要在還年青時,就賺到一生夠用的金錢?

為什麼要選擇,那麼難的目標?)

假設,你的理想是要,做一個物理學家。如果沒有財政自由,即使你做了物理學家,你也做不到物理學家。

(問:那麼玄?什麼意思?)

如果要做物理學家,通常的方法是,做物理教授。但是,做了教授後,你會發覺,教授的主要工作,其實是為其研究團隊,爭取研究撥款。

那樣,你只會剩下極少時間,給自己研究物理。

— Me@2019-07-06 10:57:22 PM

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2019.07.10 Wednesday (c) All rights reserved by ACHK

Ex 1.7 Properties of $\delta$

Let \displaystyle{F} be a path-independent function and \displaystyle{g} be a path-dependent function; then

\displaystyle{\delta_\eta h[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]~~~~~\text{with}~~~~~h[q] = F \circ g[q].~~~~~(1.26)}

— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics

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Prove that

\displaystyle{\delta_\eta F \circ g[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]}

~~~

\displaystyle{RHS = \lim_{\Delta t \to 0} \left( \frac{F \circ g[q](t+\Delta t) - F \circ g[q](t)}{\Delta t} \right) \lim_{\epsilon \to 0} \left( \frac{g[q+\epsilon \eta]-g[q]}{\epsilon} \right)}

\displaystyle{ \begin{aligned} LHS &= \delta_\eta F \circ g[q] \\   &=  \lim_{\epsilon \to 0} \left( \frac{F \circ g[q+\epsilon \eta]-F \circ g[q]}{\epsilon} \right) \\    &=  \lim_{\epsilon \to 0} \left( \frac{F \left[ g[q+\epsilon \eta] \right] - F \left[ g[q] \right]}{\epsilon} \right) \\   \end{aligned}}

Since \displaystyle{F} is path-independent,

\displaystyle{ \begin{aligned} LHS   &= \lim_{\epsilon \to 0} \left(  \frac{F \left( g[q+\epsilon \eta ] \right) - F \left( g[q] \right)}{\epsilon} \right) \\   \end{aligned}}

Let \displaystyle{ g[q+\epsilon \eta] = g + \Delta g}.

\displaystyle{ \begin{aligned} LHS   &= \lim_{\epsilon \to 0} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\epsilon} \right) \\   &= \lim_{\epsilon \to 0} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\   \end{aligned}}

When \displaystyle{ \epsilon \to 0}, \displaystyle{ \Delta g \to 0 }.

\displaystyle{ \begin{aligned} LHS   &= \lim_{\substack{\epsilon \to 0 \\ \Delta g \to 0}} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\   &= \lim_{\Delta g \to 0} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]} \lim_{\epsilon \to 0} \frac{g[q + \epsilon \eta] - g[q]}{\epsilon} \right) \\   &= DF \left( g[q] \right) \delta_\eta g[q] \\   &= RHS \\  \end{aligned}}

— Me@2019-06-24 10:55:28 PM

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2019.06.25 Tuesday (c) All rights reserved by ACHK

From classical to quantum

From this viewpoint, the move from a classical to a quantum mechanical system is not a move from a comutative to a non-commutative algebra \displaystyle{\mathcal{A}} of a real-valued observables, but, instead, a move from a commutative algebra to a partial commutative algebra of observables.

Of course, every non-commutative algebra determines an underlying partial commutative algebra and also its diagram of commutative subalgebras.

That fact that assuming the structure of a non-commutative algebra is the wrong assumption has already been observed in the literature (see, for example, [19]),

but it is often replaced by another wrong assumption, namely that of assuming the structure of a Jordan algebra.

These differing assumptions on the structure of \displaystyle{\mathcal A} affect the size of its automorphisum group and, hence, of the allowable symmetries of the system (the weaker the assumed structure on \displaystyle{\mathcal A}, the larger is its automorphism group).

— The Mathematical Foundations of Quantum Mechanics

— David A. Edwards

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2019.06.18 Tuesday ACHK

(反對)開夜車 4.1

本文章並(!)不(!)可作為醫學建議。如需醫學意見,請諮詢專業人士。

(問:但是,在現今社會,無論是上班,或是讀書,完全不「開夜車」,又好像不切實際。)

其實,主要是講讀書時代。如果在工作時代,你的職位需要,你時常「開夜車」的話,你根本就應該另謀高就。

試問,世間上,有什麼工作,竟然值得你冒生命危險,去時常「開夜車」呢?

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言歸正傳,讀書時代,如果時間管理得宜,需要「開夜車」的情況,其實是很少。

(問:那樣說有意思嗎?我正正是問你,在時間管理失宜,需要「開夜車」時,該如何自處?)

一定「開夜車」的話,你至少要做到以下幾點,去保障自己的安全:

  • 只可以間中,不可以經常。

  • 日間中途要有小睡。

  • 平均而言,你仍必須要有,充足的睡眠。亦即是話,某一天睡少了,必須於在當個星期,還回「睡債」。

    • 例如,如果你的充足睡眠是,每天七小時,而你在某一天只睡了六小時的話,你就有義務,在當個星期的另一天,睡多一小時。

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另外,有時,只要跳出框框,破格思考,或者,只要你的時間表稍改一點,就根本毋須「開夜車」。

本文章並(!)不(!)可作為醫學建議。如需醫學意見,請諮詢專業人士。

— Me@2019-06-06 08:23:56 PM

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2019.06.08 Saturday (c) All rights reserved by ACHK

Quick Calculation 15.1

A First Course in String Theory

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Recall that a group is a set which is closed under an associative multiplication; it contains an identity element, and each element has a multiplicative inverse. Verify that \displaystyle{U(1)} and \displaystyle{U(N)}, as described above, are groups.

~~~

What is \displaystyle{U(1)}?

— Me@2019-05-24 11:25:41 PM

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The set of all \displaystyle{1 \times 1} unitary matrices clearly coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1. Therefore, the circle group is canonically isomorphic to \displaystyle{U(1)}, the first unitary group.

— Wikipedia on Circle group

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In mathematics, a complex square matrix \displaystyle{U} is unitary if its conjugate transpose \displaystyle{U^*} is also its inverse—that is, if

\displaystyle{U^{*}U=UU^{*}=I,}

where \displaystyle{I} is the identity matrix.

In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger (\displaystyle{\dagger}) and the equation above becomes

\displaystyle{U^{\dagger }U=UU^{\dagger }=I.}

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

— Wikipedia on Unitary matrix

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2019.05.25 Saturday ACHK

Unitarity (physics)

Unitarity means that if a future state, F, of a system is unique, the corresponding past point, P,  is also unique, provided that there is no information lost on the transition from P to F.

— Me@2019-05-22 11:06:48 PM

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In quantum physics, unitarity means that the future point is unique, and the past point is unique. If no information gets lost on the transition from one configuration to another[,] it is unique. If a law exists on how to go forward, one can find a reverse law to it.[1] It is a restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1.

Since unitarity of a theory is necessary for its consistency (it is a very natural assumption, although recently questioned[2]), the term is sometimes also used as a synonym for consistency, and is sometimes used for other necessary conditions for consistency, especially the condition that the Hamiltonian is bounded from below. This means that there is a state of minimal energy (called the ground state or vacuum state). This is needed for the third law of thermodynamics to hold.

— Wikipedia on Unitarity (physics)

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2019.05.23 Thursday (c) All rights reserved by ACHK

PhD, 3.7.1

碩士 4.7.1 | On Keeping Your Soul, 2.2.7.1

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(問:根據你的講法,好像大部分情況下,都不應該讀研究院似的。)

在理想的情況下,你可能應該讀研究院。

(問:為什麼不講實際情況,而要講理想情況。講理想情況,即是脫離現實、執行不到。那不是浪費時間嗎?)

不是。

一來,「理想不能達到」,只是通常,並非必然。

二來,即使在「理想不能達到」的情況下,探討理想,都可以有益處;因為,那會令你知道,如何接近它。

(問:但是,現實可能同理想,相差「十萬八千里」。)

那樣,你仍然可以研究,如何把那「十萬八千里」的距離,縮減成「十萬七千九百九十九里」。

(問:那麼小的進步,又有什麼用呢?)

多麼小的進步,都可能帶來,鉅大的後果。例如,考試僅僅合格,和僅僅不合格,相差多少分?

還有,你並不是進步一點後,就原地踏步。

反而,進了一小步後,現實就立刻稍有不同,令你很有機會,可以繼而,進下一小步,甚至是下一大步。

簡言之,「見步行步,行步見步」,是一個不斷循環的過程。

(問:那樣,你心目中的理想情況是什麼?)

假設你已經有財政自由 …

— Me@2019-05-18 03:14:02 PM

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2019.05.18 Saturday (c) All rights reserved by ACHK

Multiverse

A physics statement is meaningful only if it is with respect to an observer. So the many-world theory is meaningless.

— Me@2018-08-31 12:55:54 PM

— Me@2019-05-11 09:41:55 PM

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Answer me the following yes/no question:

In your multi-universe theory, is it possible, at least in principle, for an observer in one universe to interact with any of the other universes?

If no, then it is equivalent to say that those other universes do not exist.

If yes, then those other universes are not “other” universes at all, but actually just other parts of the same universe.

— Me@2019-05-11 09:43:40 PM

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2019.05.11 Saturday (c) All rights reserved by ACHK

追憶逝水年華, 3

In Search of Lost Time, 3 | (反對)開夜車 3.1 | 止蝕 4

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其實,我中五那年,試過企圖「開夜車」。詳細過程怎樣,已不太記得。但是,總體的感受,仍然是深刻。

那時,我有很多天真的想法,例如:

1. 只要每晚睡少四小時,就每晚可以多四小時溫習。

2. 只要每晚可以多四小時溫習,就可以追回之前,落後了的進度。

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實情是,「貪心」帶來「貧窮」:

很多時,我也是覺得自己,溫習的進度落後了,來不及準備那年的公開試。所以,想透過凌晨(例如)三時起牀,來追回之前,落後了的進度。但是,我卻幾乎每天凌晨,也起不到牀,導致溫習大計失敗,遺撼非常。繼而,因為失落了,更多的時間,我就覺得,更加需要在下一天,凌晨起牀。

惡性循環,持續了很久,浪費了我大量的時間。其實,只要我睡眠適量,作息定時,反而有不少機會,有上佳的成績。

現在回想起來,仍然驚嘆著,為何年輕時,可以那麼愚蠢。

只可以說,人在惶恐時,多麼糊塗的事,都可以發生。

— Me@2019-05-09 10:04:55 PM

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2019.05.10 Friday (c) All rights reserved by ACHK

Ex 1.8 Implementation of $\delta$

\displaystyle{ \begin{aligned} \delta_\eta f[q] &= \lim_{\epsilon \to 0} \left( \frac{f[q+\epsilon \eta]-f[q]}{\epsilon} \right) \\ \end{aligned}}

The variation may be represented in terms of a derivative.

— Structure and Interpretation of Classical Mechanics

\displaystyle{ \begin{aligned} g( \epsilon ) &= f[q + \epsilon \eta] \\ \delta_\eta f[q] &= \lim_{\epsilon \to 0} \left( \frac{g(\epsilon) - g(0)}{\epsilon} \right) \\ &= D g(0) \\ \end{aligned}}

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A lambda expression evaluates to a procedure. The environment in effect when the lambda expression is evaluated is remembered as part of the procedure; it is called the closing environment.

— Structure and Interpretation of Classical Mechanics

(define (((delta eta) f) q)
  (let ((g (lambda (epsilon) (f (+ q (* epsilon eta))))))
    ((D g) 0))) 

— Me@2019-05-05 10:47:46 PM

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2019.05.05 Sunday (c) All rights reserved by ACHK