Category Archives: Translation | 文字學

Visualizing higher dimensions

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The trick of visualizing higher dimension is: not to visualize it.

— Wikipedia

— Me@2011.08.19

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Besides trying to visualize, there are other methods to understand higher dimensions.

— Me@2018-10-28 04:28:01 PM

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What is the meaning of visualization?

— Me@2018-09-02 4:35 pm

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feel ~ receive all the data at once

(This definition is not totally correct, but is useful in the meantime.)

visual ~ feel at once through eyes

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you can visualize a 3D object ~ you can see all of a 3D object at once

you cannot visualize a 4D object ~ you cannot see all of a 4D object at once

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Actually, you can only visualize a 2D object, such as a square.

You cannot visualize a 3D object, such as a cube.

That’s why the screen of any computer monitor is 2 dimensional, not 3.

— Me@2018-10-28 04:32:41 PM

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2018.10.28 Sunday (c) All rights reserved by ACHK

Life, 3

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生命 3

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We exist in time because time is change.

Growing is part of the definition of life. Growing is a kind of change.

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Also, without time/change, there would be no thinking and no thoughts.

— Me@2017-12-26 11:42 am

— Me@2018-05-23 10:05:03 PM

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time ~ change

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Time is logically necessary if change is necessary.

— Me@2018-02-04 09:07:48 PM

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2018.05.23 Wednesday (c) All rights reserved by ACHK

The language of Change 1.2

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Energy conservation, 6.2 | Energy 5.2

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time ~ change

energy ~ the ability of _keeping_ changing

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constant velocity ~ the amount of an object’s change of position, measured with respect to its observer’s unit of change, is constant

s = \Delta x

v = \frac{s}{\Delta t} = \frac{\Delta x}{\Delta t}

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kinetic energy ~ the amount of the ability of keeping changing an object’s position

\frac{1}{2} m v^2 ~ the square of (the amount of change of position, relative to the observer’s unit of change)

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Energy difference is _not_ exactly a measurement of the amount of change, time interval is.

— Me@2018-02-20 09:39:30 AM

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2018.02.20 Tuesday (c) All rights reserved by ACHK

The language of Change

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Energy conservation, 6 | Energy 5

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time ~ change

energy ~ the ability of causing change

Assuming

1. a system of one single particle

2. has only kinetic energy

3. and that kinetic energy is conserved.

conservation of energy ~ an object’s potential amount of change of position, measured with respect to its observer’s unit of change, is constant

s = \Delta x

v = \frac{s}{\Delta t} = \frac{\Delta x}{\Delta t}

— Me@2018-02-15 02:21:20 PM

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Note:

The above argument has a bug:

If the mass m is constant, the kinetic energy E_K should be proportional to velocity squared v^2, instead of velocity v.

E_K = \frac{1}{2} m v^2

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However, the above argument is still technically correct:

When E_K is constant, v^2 is constant. In turn, the magnitude of v also remains unchanged.

— Me@2018-02-19 09:37:24 PM

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2018.02.15 Thursday (c) All rights reserved by ACHK

The meanings of ONE

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鑽石棉花 2

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One bag of apples, one apple, one slice of apple — which of these is one unit? Explore the basic unit of math (explained by a trip to the grocery store!) and discover the many meanings of one.

— Lesson by Christopher Danielson, animation by TED-Ed

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A unit ~ a definition of one

(cf. One is one … or is it? — TED-Ed)

— Me@2017-02-13 8:48 AM

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One is not a number, in the following sense:

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Primality of one

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be a prime number. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer’s list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to arrive at the consensus that 1 is not a prime number, but rather forms its own special category as a “unit”.

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid’s fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler’s totient function or the sum of divisors function.

— Wikipedia on Prime number

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As long as something exists, it is possible to define one.

One as the basis for counting (number); one itself is not a number, in the sense that one is for existence, not for counting.

When counting, we have to know count with respect to what. That “what” is a “unit”, aka one.

That is why

x times 1 = x

— Me@2017-02-13 8:48 AM

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\displaystyle{2 \times 1 = 2}

there are 2 units of apple == there are 2 apples

— Me@2021-08-22 07:13:46 AM

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\displaystyle{2 \times 1 = 2}

2 units = 2

— Me@2021-08-22 10:27:52 AM

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You define one in a context.

But you cannot define two without defining one.

— Me@2017-02-14 07:10:51 AM

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You define a unit in a context.

But you cannot define a number without defining a unit.

— Me@2021-08-21 10:08:58 PM

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2017.03.26 Sunday (c) All rights reserved by ACHK

Euler Formula

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Exponential, 2
 

a^x

general exponential increase ~ the effects are cumulative
 
e^x

natural exponential increase ~ every step has immediate and cumulative effects

— Me@2014-10-29 04:44:51 PM
 

exponent growth

e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n

~ compound interest effects with infinitesimal time intervals
 

multiply -1

~ rotate to the opposite direction

(rotate the position vector of a number on the real number line to the opposite direction)

~ rotate 180 degrees
 

multiply i

~ rotate to the perpendicular direction

~ rotate 90 degrees
 

For example, the complex number (3, 0) times i equals (0, 3):

3 \times i = 3 i
(3, 0) (0, 1) = (0, 3)
 

multiplying i

~ change the direction to the one perpendicular to the current moving direction

(current moving direction ~ the direction of a number’s position vector)
 

exponential growth with an imaginary amount

e^{i \theta} = \lim_{n \to \infty} \left( 1 + \frac{i \theta}{n} \right)^n

~ change the direction to the one perpendicular to the current moving direction continuously

~ rotate \theta radians

— Me@2016-06-05 04:04:13 PM
 
 
 
2016.06.08 Wednesday (c) All rights reserved by ACHK

注定外傳 2.6

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Can it be Otherwise? 2.6 | The Beginning of Time, 7.3

還有，「宇宙」這個詞語，其實分析下去，是不合法的，因為「宇宙」的意思，就是「所有事物」。

而「所有」這個詞語的意思，是相對的，因為「所有」，即是「百分之一百」。

在沒有一個基數時，講「百分之一百」，其實不會知道，是指多少數量。同理，在沒有上文下理時，講「所有」，其實不太知道，是指什麼意思。例如，「所有人」即是有「多少人」呢？

沒有明確的上文下理，「所有人」自然沒有明確的意思。

詳情請參閱，我以往有關「所有」的文章，例如：

「

相反，如果有明確的上文下理，就自然有明確的意思。例如，『三十元中的百分之一百』，就很明顯是指，那三十元。

又例如，『這間屋的所有人』，都有明確的意思，因為有明確的範圍；有範圍，就可點人數：

凡是在這間屋內遇到的人，包括你自己，你都記下名字，直到在這間屋，再不找到新的人為止。那樣，你就可以得到，有齊『這間屋所有人』的名單。

『所有』，就是『場所之有』。

沒有明確的場所，就不知所「有」何物。

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— Me@2016-05-18 11:40:31 AM

2016.05.18 Wednesday (c) All rights reserved by ACHK

Quantum Indeterminacy

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注定外外傳 1

Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics.

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Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that errors in measurement may lead to indeterminate outcomes. However, by the later half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.

— Wikipedia on Quantum indeterminacy

Quantum indeterminacy is the inability to predict the behaviour of the system with 100% accuracy, even in principle.

If everything is connected , quantum indeterminacy is due to the logical fact that, by definition, a “part” cannot contain (all the information of) the “whole”.

An observer (A) cannot separate itself from the system (B) that it wants to observe, because an observation is an interaction between the observer and the observed .  

In order to get a perfect prediction of a measurement result, observer (A) must have all the information of the present state of the whole system (A+B). However, there are two logical difficulties.

First, observer A cannot have all the information about (A+B).

Second, observer A cannot observe itself to get (all of) its present state information, since an observation is an interaction between two entities. Logically, it is impossible for something to interact with itself directly. Just as logically, it is impossible for your right hand to hold your right hand itself. 

So the information observer A can get (to the greatest extent) is all the information about B, which is only part of the system (A+B) it (A) needs to know in order to get a prefect prediction for the evolution of the system B.

— Me@2015-09-14 08:12:32 PM

2015.09.15 Tuesday (c) All rights reserved by ACHK

機遇再生論 1.3

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「

『機遇再生論』的大概意思是，所有可能發生的事情，例如重生，在無限長的未來時間中，必會發生。

」

機遇再生論原始版本，有問題的字眼中，除了「所有」之外，還有「無限」。「無限」通常都是一個違法詞語。「無限」引起的問題，以前論述過，現不再詳談。請參閱「無限」系列的文章。

你可以嘗試移除「無限」這個詞語，只把「無限」的意思中，有意義的部分保留：

「

『機遇再生論』的大概意思是，所有可能發生的事情，例如重生，只要等足夠長的時間，總會發生。

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但是，即使避開了「無限」，用了「足夠長」，仍然會有其他問題。「足夠長」這個詞語雖然不算違法，但是十分空泛，空泛到近乎沒有意義。

試想想，怎樣才為之「足夠長」呢？

— Me@2015.04.08

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2015.04.09 Thursday (c) All rights reserved by ACHK