# 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

# Constructive competition

Habit 4: Think Win-Win or No-Deal

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Constructive competition means that, whoever wins, the whole society wins.

— Me@2011.10.11

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# PhD, 3.7.2

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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。）

（問：那樣，你心目中的理想情況是什麼？）

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？）

（問：那就是即是「退了休」？）

（問：那就即是話，要在還年青時，就賺到一生夠用的金錢？

（問：那麼玄？什麼意思？）

— Me@2019-07-06 10:57:22 PM

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# Physical laws are low-energy approximations to reality, 1.5

… difficult, as you have to heat up [the system] …

… messenger …

… collider particle …

… cool it down to discover new physics …

— Me@2019-06-27 11:23:18 PM

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# 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

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

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

\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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# 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

# Free will, 6

The factors are so many and so complex that no single observer can ever predict my actions with 100% accuracy, even in principle.

$\displaystyle{\equiv}$

I have free will.

— Me@2018-05-06 12:21:36 AM

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# （反對）開夜車 4.1

（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。）

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（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？）

• 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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— Me@2019-06-06 08:23:56 PM

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# Physical laws are low-energy approximations to reality, 1.4

$\displaystyle{\vdots}$

$\displaystyle{\uparrow}$

quark

$\displaystyle{\uparrow}$

plasma

$\displaystyle{\uparrow}$

vapour

$\displaystyle{\uparrow}$

water

$\displaystyle{\uparrow}$

ice

$\displaystyle{\downarrow}$

f-magnetism

$\displaystyle{\downarrow}$

QCD

$\displaystyle{\vdots}$

— Me@2019-06-04 08:42:39 PM

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# 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.

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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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# Free

So long as a man can look into the eyes of his oppressor, he is free.

— Me and the Big Guy

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2019.05.18 Saturday ACHK

# PhD, 3.7.1

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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。）

（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？）

（問：但是，現實可能同理想，相差「十萬八千里」。）

（問：那麼小的進步，又有什麼用呢？）

（問：那樣，你心目中的理想情況是什麼？）

— Me@2019-05-18 03:14:02 PM

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# The Principle of Stationary Action

Varying the action

$\displaystyle{S[q](t_1, t_2) = \int_{t_1}^{t_2} L \circ \Gamma[q]}$

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The Principle of Stationary Action

$\displaystyle{\delta_\eta S[q] (t_1, t_2) = 0}$

$\delta_\eta S[q] (t_1, t_2) = \int_{t_1}^{t_2} \delta_\eta L \circ \Gamma[q]$

— Structure and Interpretation of Classical Mechanics

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2019.05.13 Monday 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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# In Search of Lost Time, 4

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You can’t change the beginning, but you can change the ending.

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You can’t go back and change the beginning, but you can start where you are and change the ending.

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2019.05.11 Saturday 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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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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— Me@2019-05-09 10:04:55 PM

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# Physical laws are low-energy approximations to reality, 1.3.2

QCD, Maxwell, Dirac equation, spin wave excitation, superconductivity, …

~ low energy physics

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symmetry breaking

$\displaystyle{\downarrow}$

local minimum

$\displaystyle{\downarrow}$

simple physics

— Me@2019-05-06 11:12:02 PM

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# 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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