Category Archives: mathematical logic

nCr

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乘法意思 6

這段改編自 2014 年 3 月 22 日的對話。

假設有 7 蘋果，你要選 3 個出來，總共有多少個選法？

答案是 7C3（「7 選 3」），即是總共有，35 個可能的組合。

（A：那我可不可以把題目看成，分兩次抽 3 個蘋果出來？

首先，我由那 7 個蘋果中，抽兩個出來，即是 7C2。然後，我由餘下的 5 個蘋果中，再抽 1 個出來，即是 5C1。所以，答案應該可以寫成「7C2 乘以 5C1」。

但是，「7C2 x 5C1」卻是 105，不是 35 。錯在那裡呢？）

「7C3」和「7C2 x 5C1」，所表達的情況不同。

「7C3」是指由 7 個蘋果之中，任意選 3 個出來，總共有多少個可能。

而「7C2 x 5C1」則是指，由一箱 7 個蘋果之中，任意選 2 個出來；然後，再由另一箱 5 個蘋果之中，抽一個出來，即是 5C1，總共有多少抽法。

留意，「7C2 x 5C1」根本不是你所指，代表「首先由 7 個蘋果中，抽兩個出來；然後，再由同一箱餘下的 5 個蘋果中，抽 1 個出來」。

（A：但是我仍然不太明白，「7 選 3」和「7 選完 2 後再選 1」，為何有所不同。）

互相獨立的因素，才會用乘法。你記不記得，在學習「機會率」時，學過這一點？

其實，歸根究底，「互相獨立的因素，如果一併考慮，總共有多少個組合」，就是乘法的根本意思，即是定義。

例如，一個長方形的長度增減，並不會影響闊度的大小，反之亦然。所以，長方形面積等於「長乘闊」的其中一個原因是，長和闊，是互相獨立的因素。

如果你把「7C2 x 5C1」看成，「由第一箱 7 個蘋果之中，任意選兩個出來；然後，再由另外箱 5 個蘋果中，抽 1 個出來，即是 5C1，總共有多少個抽法」，那就正確，因為，你由第一箱 7 個蘋果之中，抽了哪兩個出來，並不會影響到，你由第二箱 5 個蘋果之中，抽了 1 個出來時，會抽到哪 1 個。

但是，如果你把「7C2 x 5C1」看成，「由 7 個蘋果中，抽兩個出來；然後，再由同一箱餘下的 5 個蘋果中，抽一個出來」，那就不正確，因為，這兩個步驟，並不是互相獨立。第一個步驟結果，會影響到第二個步驟的結果。

你在「由 7 個蘋果中，抽兩個出來」時，抽了哪兩個，會影響到那箱中，將會餘下哪 5 個蘋果，給你第二個步驟去選。

— Me@2014.04.01

2014.04.01 Tuesday (c) All rights reserved by ACHK

Powers of Ten

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The Powers of Ten films are two short American documentary films written and directed by Charles and Ray Eames. Both works depict the relative scale of the Universe according to an order of magnitude (or logarithmic scale) based on a factor of ten, first expanding out from the Earth until the entire universe is surveyed, then reducing inward until a single atom and its quarks are observed.

Related films

Cosmic Voyage (1996), a loose remake of Powers of Ten for the National Air and Space Museum of the Smithsonian Institution.

— Wikipedia on Powers of Ten

2014.03.22 Saturday ACHK

流動電視

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「流動電視牌照」本身講明，不受廣播條例規管。

邏輯上，原本受廣播條例規管的牌照，才有機會「觸犯」廣播條例。

但是，通訊局現在的講法是，因為 hktv 的流動電視，「觸犯」了廣播條例，所以現在又要受廣播條例規管。

那是不是顛倒邏輯？

— Me@2014-03-15 01:01:04 PM

2014.03.16 Sunday (c) All rights reserved by ACHK

Existence, 10

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exist

~ is

~ has

存在

~ 有

~ 擁有

An apple exists in there

~ There exists an apple

~ There is an apple

~ There [has] an apple

一個蘋果（儲）存在那裡

~ 那裡存在一個蘋果

~ 那裡有一個蘋果

~ 那裡（擁）有一個蘋果

— Me@2012-10-19 09:20:39 AM

— Me@2014-03-13 11:25:58 AM

2014.03.16 Sunday (c) All rights reserved by ACHK

Antichamber, 3

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Gameplay/Plot

In Antichamber, the player controls the unnamed protagonist from a first-person perspective as they wander through non-Euclidean levels. Regarding typical notions of Euclidean space, Bruce has stated that “breaking down all those expectations and then remaking them is essentially the core mechanic of the game”.

— Wikipedia on Antichamber

2014.03.15 Saturday ACHK

Why does the universe exist? 6

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Why is there something instead of nothing?

— Me@2012-10-15 08:33:01 AM

If you use “universe exists” to mean “something exists” and use “why does the universe exist” to represent “why does something exist“, you have to specify which thing the “something” refers to.

For every particular thing, you can explain how come it exists, at least in principle. For example, when you ask,

“How come there is an apple?”

I can answer,

“Because I have just bought it.”

Without specifying what “something” is, “why does something exist” is a meaningless question.

— Me@2012-10-18 12:47:32 PM

— Me@2014-03-08 08:22:11 AM

2014.03.10 Monday (c) All rights reserved by ACHK

Property

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Something is a “property” means that you can use that “property” to separate objects into two groups, such that all the members of one group have that “property”, while all the members of another group have not.

For example, “being big” is a property of some dogs, because you can separate all the dogs into two groups: group with big dogs and group with non-big dogs.

However, “existence” is not a property of any dogs, because you cannot separate all the dogs into two groups, such that one has existing dogs and one has non-existing dogs.

The existence of a dog is not a property of the dog itself. Instead, the existence of a dog is a property of, for example, rooms.

You can use “the existence of dogs” to separate all the rooms in this world into two groups: the room with dogs and the room without dogs. 

— Me@2012-10-18 12:47:32 PM

— Me@2014-03-03 02:46:35 PM

2014.03.03 Monday (c) All rights reserved by ACHK

Universe 7

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The universe as a whole is an un-observable for two reasons, one physical and one logical.

The physical reason is that the speed of light, while being the maximum possible signal transmission speed, is finite. However, the expansion of the universe, in a sense, is faster than the speed of light. So the light rays emitted by some objects can never reach your eyes, no matter how long you wait. You cannot observe everything at once at any particular moment of time.

The logical reason is that, for any observer, at least one thing in this universe it cannot observe: itself. You can never see yourself directly, just as a camera can never take a picture of itself directly.

— Me@2012-10-18 12:47:32 PM

— Me@2014-02-25 01:57:06 PM

2014.02.26 Wednesday (c) All rights reserved by ACHK

Universe | I

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Onion self 9 | 洋蔥自我 9 | Inner and outer, 7

Universe is not something you can observe directly, but a logical implication.

Whatever you can observe, it is only part of the universe, not the universe itself. 

“I” is also a logical implication or logical limit.

Whatever you can observe, belongs to your “I”, but is not your “I”.

For example, you can see your right hand.

It is part of you.

It is yours, but it is not you.

— Me@2012.10.18

— Me@2014.02.09

2014.02.10 Monday (c) All rights reserved by ACHK

Godel numbering

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Leibniz dreamt of a characteristica universalis: a universal language that could express any scientific concept. He didn’t get too far – nobody was interested. But it involved a strange idea. Basic concepts would be encoded as prime numbers. More complicated ones would be encoded by multiplying these numbers. So, you could see which basic concepts a complicated one contained, just by factoring it!

When I heard this I immediately thought of Gödel numbering. This is a much better-worked out idea which Gödel used to encode statements of arithmetic as numbers. It also uses primes, but in a smarter way.

Did Gödel get his inspiration from Leibniz? I haven’t been able to find out. But I discovered that he was obsessed with Leibniz’s characteristica universalis!

— John Baez

2014.02.08 Saturday ACHK

Limit vs Boundary

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Logically, it is impossible for space has boundary. If space has boundary, what is the other side?

However, logically, it is possible for space has limit. Whatever object reaches the limit, disappears.

… just as physical death is a limit of biological life, but not a boundary within your biological life.

In reality, the space has neither boundary nor limit, just as the surface of the Earth has neither boundary nor limit.

— Me@2012.10.17

— Me@2014.02.06

2014.02.06 Thursday (c) All rights reserved by ACHK

The Beginning of Time, 3.2

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Cumulative concept of time 1.3.2

The past is part of the future.

The first cause is the smallest part.

cause ~ component

The first cause is not a boundary, because there is no “before”.

The first cause is a physical limit, not a physical boundary.

The first cause is a logical boundary, not a physical one.

— Me@2012.10.17

— Me@2014.02.03

2014.02.03 Monday (c) All rights reserved by ACHK

Godel 18

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Note that whether or not a property can be expressed by a formula of arithmetic has to do exclusively with the power of the language, whereas the question of whether or not it can be captured has to do, in addition, with the power of the selected axiom system. Thus, for example, the formula

there-exists-a-number  z  such-that   x * z = y.

is a formula with two free variables which expresses the relation “y is divisible by x”; but if we choose a very weak axiom system – for example, one consisting only of the axiom “0 + 1 = 1” – the property “y is divisible by x” will not be captured by this (or any other) formula.

— Gödel’s Incompleteness Theorems – A Brief Introduction

— Malgosia Askanas

2014.02.01 Saturday ACHK

Universe 5

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Existence, 9

meaning ~ use

The sentence “the universe exists” is meaningless

~ The sentence “the universe exists” has no use

X exists

~ X is in a container

~ X can be found in that container

universe ~ the maximum possible container

So, it is bizarre to state, “The universe exists.“

the universe exists

~ the universe is in a container

~ the universe is within something bigger

— Me@2012.10.17

2014.01.30 Thursday (c) All rights reserved by ACHK