Mathematics, 2

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Approximately speaking, mathematics is whatever language with the maximum precision.

— Me@2012-04-16

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2018.09.03 Monday (c) All rights reserved by ACHK

Is logic empirical?

Each of the logics is analytic.

Which logic is the best for describing the world” is synthetic.

— Me@2012-04-14 11:32:36 AM

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2018.04.22 Sunday (c) All rights reserved by ACHK

機遇再生論 1.6

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（而這個意思，亦在「機遇再生論」的原文中，用作其理據。）

$P(A) = \frac{1}{N}$

$P(\text{not} A) = 1 - \frac{1}{N}$

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$P(A) = \frac{1}{N}$

$P(\text{not} A) = 1 - \frac{1}{N}$

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(問：那樣，為什麼要問多一次呢？）

「如果洗牌兩次，起碼一次洗到原本排列 A 的機會率是多少？」

$A_2$ = 兩次洗牌的結果，起碼一次洗到原本排列 A

$A_2$ 的互補事件為「不是 $A_2$」：

= 兩次洗牌的結果，不是起碼一次洗到原本排列 A

= 兩次洗牌的結果，都不是排列 A

$P(\text{not} A_2) = (1 - \frac{1}{N})^2$

$P(A_2)$
$= 1 - P(\text{not} A_2)$
$= 1 - (1 - \frac{1}{N})^2$

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$P(A_m)= 1 - (1 - \frac{1}{N})^m$

$P(A_m)$
$= 1 - (1 - \frac{1}{N})^m$
$= 1 - (1 - \frac{1}{52!})^{10,000,000}$

$1.239799930857148592 \times 10^{-61}$

— Me@2018-01-25 12:38:39 PM

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2018.02.13 Tuesday (c) All rights reserved by ACHK

Riemann Surfaces

Imaginary Numbers Are Real [Part 1: Introduction]

Imaginary Numbers Are Real [Part 2: A Little History]

Imaginary Numbers Are Real [Part 3: Cardan’s Problem]

Imaginary Numbers Are Real [Part 4: Bombelli’s Solution]

Imaginary Numbers Are Real [Part 5: Numbers are Two Dimensional]

Imaginary Numbers Are Real [Part 6: The Complex Plane]

Imaginary Numbers Are Real [Part 7: Complex Multiplication]

Imaginary Numbers Are Real [Part 8: Math Wizardry]

Imaginary Numbers Are Real [Part 9: Closure]

Imaginary Numbers Are Real [Part 10: Complex Functions]

Imaginary Numbers Are Real [Part 11: Wandering in 4 Dimensions]

Imaginary Numbers Are Real [Part 12: Riemann’s Solution]

Imaginary Numbers Are Real [Part 13: Riemann Surfaces]

— Welch Labs

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In case the original videos are lost, please use the Internet Archive link:

https://web.archive.org/web/20170714105446/https://www.youtube.com/watch?v=T647CGsuOVU

— Me@2018-02-12 02:14:51 PM

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2018.02.12 Monday (c) All rights reserved by ACHK

機遇再生論 1.5

（請參閱本網誌，有關「重言句」、「經驗句」和「印證原則」的文章。）

「同情地理解」的意思是，有些理論，雖然在第一層次的分析之後，有明顯的漏洞，但是，我們可以試試，代入作者發表該理論時的，心理狀態和時空情境；研究作者發表該理論的，緣起和動機；從而看看，該理論不行的原因，會不會只是因為，作者的語文或思考不夠清晰，表達不佳而已？

（而這個意思，亦在「機遇再生論」的原文中，用作其理據。）

$P(A) = \frac{1}{N}$

$P($not $A) = 1 - \frac{1}{N}$

— Me@2017-12-18 02:51:11 PM

2017.12.18 Monday (c) All rights reserved by ACHK

Mathematics

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

A possible explanation of the physicist’s use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection.

— The Unreasonable Effectiveness of Mathematics in the Natural Sciences

— E. P. Wigner

2017.10.07 Saturday ACHK

The meanings of ONE

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.

A unit ~ a definition of one

(cf. One is one … or is it? — TED-Ed)

— Me@2017-02-13 8:48 AM

One is not a number, in the following sense:

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

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

2017.03.26 Sunday (c) All rights reserved by ACHK

Euler Formula

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

Self-information

The information entropy of a random event is the expected value of its self-information.

In information theory, self-information or surprisal is a measure of the information content [clarification needed] associated with an event in a probability space or with the value of a discrete random variable.

By definition, the amount of self-information contained in a probabilistic event depends only on the probability of that event: the smaller its probability, the larger the self-information associated with receiving the information that the event indeed occurred.

As a quick illustration, the information content associated with an outcome of 4 heads (or any specific outcome) in 4 consecutive tosses of a coin would be 4 bits (probability 1/16), and the information content associated with getting a result other than the one specified would be 0.09 bits (probability 15/16).

— Wikipedia on Self-information

2015.12.31 Thursday ACHK

Euler problem 27.2

Haskell

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 problem_27 = -(2*a-1)*(a^2-a+41)    where n = 1000          m = head \$ filter (\x -> x^2 - x + 41 > n) [1..]          a = m - 1

This is the “official” Haskell solution to Euler problem 27.

This solution is incomprehensible. The following hints are as far as I can get.

Prerequisite considerations:

b must be a prime number, since x^2 + a*x + b must be a prime number when n=0.

 a = m - 1  is a choice of the prime number b (in x^2 + a*x + b) just(?) smaller than 1000.

a == 32

(\x -> x^2 – x + 41) 31 == 971

Why not the prime number 997?

— Me@2015.06.14 08:53 AM

It is because 997 is not a prime number generated by the formula x^2 - x + 41.

— Me@2015-06-30 11:07:53 AM

map (\x -> x^2 - x + 41) [0..40]

is for generating 41 primes with the greatest Euler’s lucky number 41.

— Me@2015.06.14 08:34 AM

2015.07.01 Wednesday (c) All rights reserved by ACHK

Ideal clock 1.3

An ideal clock is a clock (i.e., recurrent process) that makes the most other recurrent processes periodic.

— Wikipedia on Clock

The ideal clock itself is a meta-clock — a property of a set of clocks.

— Me@2015-04-16 9:35 AM

If you compare the accuracies of two clocks (A and B) by comparing each of them with a third clock (C), then I can ask, “How can you know that C is more accurate than both A and B?”

By defining the ideal clock as a meta-clock, we avoid infinite regress.

— Me@2015-04-27 11:36:59 AM

2015.04.29 Wednesday (c) All rights reserved by ACHK

Ideal clock 1.2

A clock is a recurrent process and a counter.

A good clock is one which, when used to measure other recurrent processes, finds many of them to be periodic.

An ideal clock is a clock (i.e., recurrent process) that makes the most other recurrent processes periodic.

— Wikipedia on Clock

The ideal clock itself is a meta-clock — a property of a set of clocks.

— Me@2015-04-16 9:35 AM

2015.04.27 Monday (c) All rights reserved by ACHK

Godel 19

… But it so happens that you just used second-order logic, because you talked about groups or collections of entities, whereas first-order logic only talks about individual entities.

— Standard and Nonstandard Numbers

— Eliezer Yudkowsky

— Less Wrong

2015.04.17 Friday ACHK

機遇再生論 1.4

『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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— Me@2015.04.08

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2015.04.15 Wednesday (c) All rights reserved by ACHK

The Beginning of Time, 6

B is to the north of A

~ B is between A and the North Pole

B is to the north of North Pole

~ B is between the North Pole and the North Pole

Therefore, the phrase “north of North Pole” is meaningless.

— Me@2015-04-06 12:48 AM

2015.04.10 Friday (c) All rights reserved by ACHK

What Is it Like to Be a Bat?

Feeling is a relationship between a particular observer and a particular observed.

So the question of “whether the color red I see is the same as the color red you see” is logically meaningless.

— Me@2015-04-06 1:13 PM

If observer B can get the memory of observer A, it is logically possible to feel another mind’s feelings (to a certain extend).

In that situation, the question of “whether B’s feeling of seeing the color red is the same as A’s” is meaningful.

— Me@2015-04-07 03:58:46 PM

2015.04.09 Thursday (c) All rights reserved by ACHK

機遇再生論 1.3

『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』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。

— Me@2015.04.08

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2015.04.09 Thursday (c) All rights reserved by ACHK

Learn Physics by Programming in Haskell

Learning functional programming and partially applying functions to other functions and such helped me understand tensors a lot better, since that’s basically what contraction is doing. It’s nice to see that the approach can be taken further.

— Snuggly_Person

I also think Haskell and some similar languages (especially Idris) have a great conceptual synergy with physics.

In physics too we strive to express things in ways that strip out extraneous details as much as possible. Haskell really embraces this concept in the sense that you write functions essentially by writing equations. You don’t describe all the mechanical steps to produce an output, you just write down the ‘invariant content’ of the function.

— BlackBrane

2015.03.29 Sunday ACHK

機遇再生論 1.2.2

「所有」，就是「場所之有」。

「機會再生論」原始版本的邏輯矛盾來源，在於「所有」。論述中，運用「所有」這個詞語時，並沒有講清楚情境，導致它不自覺地，包括了元層次的事物。「機會再生論」原始版本的邏輯矛盾，來自「本層次」和「元層次」（meta level）的矛盾。

『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。

— Me@2015-03-21 10:07:51 PM

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2015.03.28 Saturday (c) All rights reserved by ACHK

Logical Fatalism

Logical Fatalism and the Argument from Bivalence

Another famous argument for fatalism that goes back to antiquity is one that depends not on causation or physical circumstances but rather is based on presumed logical truths.

The key idea of logical fatalism is that there is a body of true propositions (statements) about what is going to happen, and these are true regardless of when they are made. So, for example, if it is true today that tomorrow there will be a sea battle, then there cannot fail to be a sea battle tomorrow, since otherwise it would not be true today that such a battle will take place tomorrow.

The argument relies heavily on the principle of bivalence: the idea that any proposition is either true or false. As a result of this principle, if it is not false that there will be a sea battle, then it is true; there is no in-between. However, rejecting the principle of bivalence—perhaps by saying that the truth of a proposition regarding the future is indeterminate—is a controversial view since the principle is an accepted part of classical logic.

— Wikipedia on Fatalism

Quantum superposition can solve logical fatalism:

Macroscopic time is due to quantum decoherence.

The future is a coherent (constant phase difference) superposition of eigenstates.

That’s why classical probability can be regarded as part of quantum theory.

Quantum decoherence gives classically consistent histories.

— Me@2012.04.08

— Me@2015.03.26

2015.03.27 Friday (c) All rights reserved by ACHK