Monthly Archives: May 2021

一萬個小時 3.4

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重點副作用 6 | The non-side-effect-ness of side-effects, 6

這段改編自 2010 年 4 月 24 日的對話。

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現在，有三個概念混淆在一起。我們先釐清一下。

1. 幾乎

任何學問，或者才藝，如果要訓練自己，純熟到足以用來維生賺錢，或者娛樂滿足，需要的時間幅度是，一日幾小時，持續一兩年，才會開竅；然後，再持續一兩年，才會知道自己，在該門學問才藝，為何會開竅，開了什麼竅；又再過一兩年，才會有能力教人，如何在該門學問才藝開竅。

2. 過程中的一個重要環節是，會領悟到一些洞見。

3. 雖然要「自己領悟」第二點中的洞見，來自第一點中的「幾年加幾年」過程，但是，如果只是「吸收別人已知的」知識本身，絶不需要花「幾年加幾年」。有時，甚至只是，花數個小時就可以。

「學習」的意思正正就是，自己毋須親身經歷（例如）四年，也可以獲取本來要四年，才發掘得到的成果。

4. (1.) 但是，「知道」本身，並不代表自己有足夠功力去「運用」，而「運用」則是，第一點的水平。不過，如果中途有第三點，即是別人情報資料上的提點，你就可以走少很多冤往路。那樣，原本是（例如）十年的過程，可以壓縮到（例如）三年。

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剛才你提到：「

你的講法是，有些知識，你只需要幾個小時，就可以傳授給我，不成課程，因為時間所需太短；而另一些知識，我就需要花「幾年加幾年」，才可以領會得到，不成課程，因為時間所需太長。

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其實，我們可以考慮改變方案。雖然有些課題，可能幾個小時，就可以完成，但是，我們可以每個星期，也講不同的課題。

留意，雖然個別課題的成果，只會花幾小時，但是那是指，事先已有課題的情況下。那些有趣而深刻的課題，簡稱「勁題目」，並不會從天而降。那些勁題目本身，大部分情況下，只會在你開始研究，將會花「幾年加幾年」的苦功知識時，才會引發得到。

— Me@2021-05-23 04:39:54 PM

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2021.05.25 Tuesday (c) All rights reserved by ACHK

2.10 A spacetime orbifold in two dimensions, 2

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A First Course in String Theory

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(a) Use the result of Problem 2.2, part (a), to recast (1) as

\displaystyle{(x^+, x^-) \sim \left( e^{-\lambda} x^+, e^{\lambda} x^- \right)}, where \displaystyle{e^\lambda \equiv \sqrt{\frac{1+\beta}{1-\beta}}}.

What is the range of \lambda? What is the orbifold fixed point? Assume now that \beta > 0, and thus \lambda > 0.

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Range of \displaystyle{\lambda}:

\displaystyle{ \begin{aligned} 0 &< \beta < \infty \\ 1 &< \frac{1 + \beta}{1 - \beta} < \infty \\ 0 &< \ln \frac{1 + \beta}{1 - \beta} < \infty \\ 0 &< \frac{1}{2} \ln \frac{1 + \beta}{1 - \beta} < \infty \\ 0 &< \lambda < \infty \\ \end{aligned}}

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Fixed points:

\displaystyle{ \begin{aligned} \begin{bmatrix} (x^+)' \\ (x^-)' \end{bmatrix} &= \begin{bmatrix} e^{- \lambda} x^+ \\ e^{\lambda} x^- \\ \end{bmatrix} \\ \end{aligned}}

\displaystyle{ \begin{aligned} (x^+, x^-) &= (0, 0) \\ \end{aligned}}

— Me@2021-05-16 06:31:12 PM

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2021.05.17 Monday (c) All rights reserved by ACHK

Genius 4

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High school teachers are by nature non-intellectuals; students are by nature intellectuals.

— Me@2011.08.23

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As a mathematician there is a story I hear a lot. It tends to come up whenever I tell someone what I do for the first time, and they admit that they don’t really like, or aren’t very good at, mathematics. In almost every case, if I bother to ask (and these days I usually do), I find that the person, once upon a time, was good at and liked mathematics, but somewhere along the way they had a bad teacher, or struck a subject they couldn’t grasp at first, and fell a bit behind. From that point on their experiences of mathematics is a tale of woe: because mathematics piles layer upon layer, if you fall behind then you find yourself in a never ending game of catch-up, chasing a horizon that you never seem to reach; that can be very dispiriting and depressing.

— Zen and the Art of Mathematics

— The Narrow Road

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All children are born geniuses; 9999 out of every 10000 are swiftly, inadvertently degeniusized by grownups.

— Buckminster Fuller

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2021.05.15 Saturday (c) All rights reserved by ACHK

千里共嬋娟

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尋覓 2.2.3.5 | Passion Test 2.3 | 二百萬 4

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如果你覺得你女朋友，會阻礙你工作進度，或者是你時間負擔的話，你倆大概由始至終，就不應在一起；你倆根本不是，對方的另一半。

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第三，太陽會發光，月亮不會。但是，月亮單單是存在本身，就已經透過反射，為地球黑夜一面，帶來不止的光芒。

發現「質量守恆定律」的，是一位化學家，名叫拉瓦節（Antoine-Laurent de Lavoisier）。他的正識是一位稅務官。因為高薪厚職，他有充足的金錢，去購買很多高級精準的化學儀器，以作化學研究之用。他的其中一個科學功績是，透過他的一些實驗，印證了「質量守恆定律」。

拉瓦節的太太 Marie-Anne Pierette Paulze，同時是他的助手 —— 為拉瓦節把文章翻譯成外語，從而把拉瓦節的化學研究成果，發揚光大。

拉瓦節太太的化學造詣，不及丈夫；而拉瓦節的文字功力，亦不及太太。但是，他們卻是夢幻組合，合力發表了，很多改變歷史的作品。

化學方面，拉瓦節是太陽，太太是月亮；文字方面，太太是太陽，拉瓦節是月亮。

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那是直接的合作。但是，有時，另一半單單是存在本身，就令你可以創作到，一些原本不可能存在的作品，又名「神作」。

例如，作家如果沒有（現在或未來的）知己知音，其實不會有心神動機（心機），去開始創作。即使有因為金錢的引誘，而夾硬創作，結果也不會是神作，只會是鬼作，連自己也不想閱讀。

相反，作家如果有（現在或未來的）知己知音的話，創作的過程就如，穿梭時空，探索宇宙；創作的結果就成，神來之筆，扣人心弦。

— Me@2021-05-14 11:02:51 PM

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2021.05.15 Saturday (c) All rights reserved by ACHK

Ex 1.21 The dumbbell, 2.2.2

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[guess]


(define (KE-particle m v)
  (* 1/2 m (square v)))

(define ((L-free-constrained m0 m1 l) local)
  (let* ((extract (extract-particle 2))
	 (p0 (extract local 0))
	 (q0 (coordinate p0))
	 (qdot0 (velocity p0))
 
	 (p1 (extract local 1))
	 (q1 (coordinate p1))
	 (qdot1 (velocity p1))
 
	 (F (ref (coordinate local) 4)))

    (- (+ (KE-particle m0 qdot0)
	      (KE-particle m1 qdot1))
          (U-constraint q0 q1 F l))))

(let ((L (L-free-constrained 'm_0 'm_1 'l)))
  (show-expression
   ((compose L (Gamma q-rect)) 't)))

\displaystyle{ \frac{1}{2} m_0 \left( \dot x_0^2 + \dot y_0^2 \right) + \frac{1}{2} m_1 \left( \dot x_1^2 + \dot y_1^2 \right) + \frac{F}{2 l} \left( l^2 - y_1^2 + 2 y_0 y_1 - x_1^2 + 2 x_0 x_1 - y_0^2 - x_0^2 \right) }

\displaystyle{ = \frac{1}{2} m_0 \left( \dot x_0^2 + \dot y_0^2 \right) + \frac{1}{2} m_1 \left( \dot x_1^2 + \dot y_1^2 \right) - \frac{F}{2 l} \left[ (y_1 - y_0)^2 + (x_1 - x_0)^2 - l^2 \right] }


(define ((local_ m0 m1 l) local)
  (let* ((extract (extract-particle 2))
	 (p0 (extract local 0))
	 (q0 (coordinate p0))
	 (qdot0 (velocity p0))
 
	 (p1 (extract local 1))
	 (q1 (coordinate p1))
	 (qdot1 (velocity p1))
 
	 (F (ref (coordinate local) 4)))
    local))

(show-expression
 ((compose (local_ 'm_0 'm_1 'l) (Gamma q-rect)) 't))




(define ((p0_ m0 m1 l) local)
  (let* ((extract (extract-particle 2))
	 (p0 (extract local 0))
	 (q0 (coordinate p0))
	 (qdot0 (velocity p0))
 
	 (p1 (extract local 1))
	 (q1 (coordinate p1))
	 (qdot1 (velocity p1))
 
	 (F (ref (coordinate local) 4)))
    p0))
 
(show-expression
 ((compose (p0_ 'm_0 'm_1 'l) (Gamma q-rect)) 't))




(define ((p1_ m0 m1 l) local)
  (let* ((extract (extract-particle 2))
	 (p0 (extract local 0))
	 (q0 (coordinate p0))
	 (qdot0 (velocity p0))
 
	 (p1 (extract local 1))
	 (q1 (coordinate p1))
	 (qdot1 (velocity p1))
 
	 (F (ref (coordinate local) 4)))
    p1))
 
(show-expression
 ((compose (p1_ 'm_0 'm_1 'l) (Gamma q-rect)) 't))

[guess]

— based on /sicmutils/sicm-exercises

— Me@2021-04-27 05:03:59 PM

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2021.05.11 Tuesday (c) All rights reserved by ACHK

Goodstein’s theorem

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[guess]

Goodstein’s theorem is an example that sometimes a finite result requires the existence of infinity in its proof.

— Me@2021-05-09 11:06:34 PM

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Goodstein’s theorem itself assumes that there is an infinite number of natural numbers, so it is not really a finite result.

— Me@2017-02-20 06:16:28 PM

[guess]

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2021.05.09 Sunday (c) All rights reserved by ACHK

卡拉OK, 4

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Facebook time does not change anything.

Doing things changes things.

If you (and other people) do nothing, the situation stays exactly where it is.

— Me@2016-12-12 03:18:13 PM

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2021.05.09 Sunday (c) All rights reserved by ACHK

To realize is to realize, 1.2.2

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So in theory, there is no free will, because the future is already fixed, by the physical laws.

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However, even if we knew the exact physical laws, it would be still logically impossible to get all the data of the present state of the whole universe, because it is logically impossible for any observer to observe itself, with 100% details, directly. For example, no camera can take a picture of itself directly.

So “in practice”, which is actually also “in principle”, there is free will, because logically, no one can predict, with 100% accuracy, your future actions.

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In one logical sense, the future is already fixed, so there is no free will. In another logical sense, the future is fixed, but no observer can know that “fixed future” with 100% accuracy, so there is free will.

As a result, whether you label your actions are due to “free-will” or “not-free-will” has no real consequence. In other words, whether there is free will or not has no meaningful difference.

The difference that makes no difference makes no difference.

So you can actually transcend the free will problem altogether. You can just ignore it and live your life.

Or, you can somehow capitalize on this freedom of labelling your (future) life as either fixed or free, depending which label is more beneficial for you in a particular situation.

For example, when you are highly under pressure, you know that everything is fixed by the physical laws, from god’s point of view. When you are highly above pressure, you know that you are partially responsible for creating your own reality, because the future is not fixed for any one observer, for there is no “god’s point of view”.

You have the flexibility to label it in one way or another.

— Me@2021-05-07 10:27:04 PM

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2021.05.09 Sunday (c) All rights reserved by ACHK

2.10 A spacetime orbifold in two dimensions

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A First Course in String Theory

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Consider a two-dimensional world with coordinates \displaystyle{x^0} and \displaystyle{x^1}.

A boost with velocity parameter \displaystyle{\beta} along the \displaystyle{x^1} axis is described by the first two equations in (2.36). We want to understand the two-dimensional space that emerges if we identify

\displaystyle{(x^0, x^1) \sim ({x'}^0, {x'}^1)}.

We are identifying spacetime points whose coordinates are related by a boost!

(a) Use the result of Problem 2.2, part (a), to recast (1) as

\displaystyle{(x^+, x^-) \sim \left( e^{-\lambda} x^+, e^{\lambda} x^- \right)}, where \displaystyle{e^\lambda \equiv \sqrt{\frac{1+\beta}{1-\beta}}}.

~~~

\displaystyle{ \begin{aligned} (x')^0 &= \gamma (x^0 - \beta x^1) \\ (x')^1 &= \gamma (- \beta x^0 + x^1) \\ \end{aligned}}

\displaystyle{ \begin{aligned} \begin{bmatrix} (x^+)' \\ (x^-)' \end{bmatrix} &= \begin{bmatrix} \gamma (1-\beta) & 0 \\ 0 & \gamma (1+\beta) \\ \end{bmatrix} \begin{bmatrix} x^+ \\ x^- \end{bmatrix} \\ &= \begin{bmatrix} \frac{1}{\sqrt{1 - \beta^2}} (1-\beta) x^+ \\ \frac{1}{\sqrt{1 - \beta^2}} (1+\beta) x^- \\ \end{bmatrix} \\ &= \begin{bmatrix} e^{- \lambda} x^+ \\ e^{\lambda} x^- \\ \end{bmatrix} \\ \end{aligned}}

— Me@2021-05-04 10:48:28 PM

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2021.05.05 Wednesday (c) All rights reserved by ACHK

Logical arrow of time, 9.2

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To confirm or disconfirm a prediction, you cannot check record; you can only observe the system evolving.

To confirm or disconfirm a retrodiction, you can only check record (or the logical consequence of that retrodiction); you cannot observe that past event directly.

— Me@2013-08-10 08:00 PM

— Me@2021-05-03 12:28 PM

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2021.05.03 Monday (c) All rights reserved by ACHK

相逢恨晚 2

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尋覓 2.2.3.4 | Passion Test 2.2

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如果你覺得你女朋友，會阻礙你工作進度，或者是你時間負擔的話，你倆大概由始至終，就不應在一起；你倆根本不是，對方的另一半。

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第二，如果你喜歡一個人，她真的是你另一半的話：

初期相識時，甚至是第一次相見時，就已經有相逢恨晚、無所不談之感。　

We talked half the night, and in the middle of talk became lovers. — Bertrand Russell

你自然視她的意見，如珠如寶，尤其是有關工作上的；因為，你工作上的決定，直接影響著你倆，將來可否開花結果，化成夫妻。

中期相處時，就會心有靈犀。

心有靈犀者，心靈感應也。人的身體大腦，有如電腦硬件。人的思想靈魂，其實就是程式軟件。聊天等溝通，其實就是不斷地，複製自我於，對方的腦子之中。所以，相識不淺時，你倆其實就已經有了，對方的「自我」複本，在腦海心中。

深情者，即使不見也同在。

有感情 = 感受到相同的情境 — Me@2011.06.04

所以，你倆每天，毋須再 24 小時也，物理聊天相見。

每天再相見時，就只須談情說愛，毋須再談天說地；因為，你倆已能心靈感應，開聲前就大概知道，對方的心思。你再毋須事事也，問女何所思，問女何所憶。

後期相愛時，則會穿梭時空。

愛情三階段：

相識相逢恨晚，相處心有靈犀，相愛穿梭時空。

單身三境界：

青年工程師，中年物理學家，老年魔法師。

— Me@2021-04-05 06:14:46 PM

— Me@2021-05-01 04:46:18 PM

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2021.05.02 Sunday (c) All rights reserved by ACHK