白马非马

韓非子是战国法家, 荀子的高徒, 秦始皇宰相李斯的同学。他说"白马非马", 即白马不是马, 可以用集合論(Set Theory) 证明:

Let 马 = H = {w, b, r, y ...}
w : 白马
b : 黑马
r ：红马
y：黄马

Let 白马 = W = {w}

To prove:
H = W
We must prove:
H ⊂ W and H ⊃ W

From definition we know:
$latex w \in H \supset W $
$latex H \nsubseteq W $
$latex \implies H \neq W $

白马≠马
白马非马
[QED]

其他例子:
木魚非鱼

More on Linguistic "Half Life"

Proto Indo-European and Chinese in the Late Neolithic Age
后新石器时代的原欧-印语与汉语

Tsung-tung Chang[張聰東] 1988:
“Indo-European vocabulary in Old Chinese: A new thesis on the emergence of Chinese language and civilization in the Late Neolithic Age”, Sino-Platonic Papers 7, Philadelphia.

This Chinese scholar wrote the 1988 paper on the Chinese language origin with the proto-Indo-European (proto IE).

Interestingly very similar 'coincidence' occurs in 1500 words between Chinese and proto IE:

Take -> 得 tek (ancient Chinese sound as in Fujian dialect today)
Mort -> 殁 mo
See -> 视 see
Cow -> 牛 gu
...

http://www.sino-platonic.org/complete/spp007_old_chinese.pdf


After the Tower of Babel, God confused the human into different languages, but by the linguistic 'archaeology' 'Half Life' Theory, we can deduce ~ 4,900 years ago the Chinese and the Germanic (English, Denmark, German ...) shared the same common linguistic root.

The ancient Chinese scholar Xu Shen许慎(东汉 : 58 CE -147 CE) who compiled the first dictionary on Chinese characters and its origins "说文解字", wrote that the character :
白 (White) is "从入合二" : from the origin character 入 together with 2 (二).


For 2,000 years nobody understood what Xu meant or he had made a mistake for the character 白 (white)。

Now from the "Half-Life Theory", we trace to the Germanic common root, Chinese scholars confirm Xu was right.

The character '入' pronounced in ancient sound (still remains in southern dialect like Fujian in Singapore / Taiwan) is *njub, '入' is the ancient character for 乳头( nipple). *njub sounds like 'nipple' in Germanic languages.

白 means use 2 fingers or 2 lips (合二) to squeeze out white milk from nipple (入).

In European languages, 'White' / 'Milk' are also connected in etymology by 'b' 'mb':
English: bleach / milk
Denmark: bleg / maek
Sweden: blek / mjök
German: bleich / milch
Dutch: bleek / melk
Ancient Irish: mlicht/blicht (bleacht)
Polish: biały / mleko

Conclusion :

白 => milk => white (color)

-----------------------------------------
Language 'Half-Life' Formula

$latex \boxed{T = \frac{\log N} {2.\log 85\%}}$
(N= % of vocabulary with common root)

Modern English & German:
N=60%
=> T =1.561 (K-Yrs)
=> 1,561 yrs ago the 2 languages shared the common root @ 449 CE. [confirmed by historians]






Genesis 11:1-9 Tower of Babel

God confused Noah's descendants from their common language root 'Proto-Canaanite' to different languages:
Phoenician
Greek
Aramaic -> Arabic / Jewish
Chinese
...
1. Chinese / 日耳曼語系 Germanic (English) have common root T = 4,951 yrs ago [尧/舜]

2. Chinese / Latin (French)
T= 3,700 yrs ago [夏/商]

3. Chinese / Persian (Iran) T= 3,100 yrs ago [商/周]

Proof: [Chinese / Germanic / Latin Common Roots]

背bei -> Back

爸ba -> papa

鳳feng -> [Phoin] = Phoenix [Greek]

殁mo (=死) -> Mort (English) / Mort (French)

民min -> man (eng) / mann (German)

核he /ker -> Heart / kerd (Kernel)

酸suan -> sour

QuYuan 屈原 Symmetry

屈原 QuYuan (343–278 BCE) Symmetry:
http://en.wikipedia.org/wiki/Qu_Yuan

离騷《天问》
1. "九天之际, 安放安属,
隅隈多有, 谁知其数 ?"
=> 天 (Sky) 和 地 (Earth) must be 2 symmetric spheres.

If 地 (Earth) were flat, then there would be (隅隈) edges and angles at the 天 (Sky) & 地 (Earth) boundary (九天之际).

2. "东西南北, 其修孰多,
南北顺橢, 其衍几何。"
=> 南北顺橢 = The Earth is ellipse (橢), with north-south (南北) slightly flatten.

几何 = Geometry

3. How did QuYuan know this advanced astronomy & geometry in ~ 300 BCE?

[caption id="attachment_2393" align="alignnone" width="201"] 屈原[/caption]

墨子 Mozi & Force

[caption id="attachment_2380" align="alignnone" width="166"] Mozi[/caption]

墨子 Mozi (468 BCE~ 376 BCE), 2000 years earlier than Newton

"墨子" :  “力, 行之所以奋也。”

行: Moving
奋: Acceleration
力: Force is due to acceleration by the moving object.

F ∝ a
F = m.a

杨振宁欣赏数学

1938 杨振宁高二（JC1) 时，父亲杨武之（华罗庚，陈省身的老师 ）借他二本书：
Hardy's "Pure Mathematics",
E.T. Bell's "Men of Mathematics".
并讨论 Set Theory, The Continuum Hypothesis.

40年后杨振宁说：
"我的物理学界同事们大多对数学采取功利主义态度。也许受父亲的影响，我较为欣赏数学，其优美和力量：它有战术上的机巧与灵活，又有战略上的雄才远虑。而且，奇迹的奇迹，它的一些美妙概念竟是支配物理世界的基本结构。＂

陈省身: 数学=美

陈省身 数学=美

“宇宙世界万象纷呈, 都暗含着一种伟大的秩序, 這种秩序, 可以被数学家发现, 用数学公式表示出来; 而公式的演算, 应该做到由繁就简, 最后的形式最简洁, 才最漂亮。”

Win Win Problem

Yoyo lost $3, Pierre found it and returned to Yoyo.
Yoyo wanted to give all $3 to Pierre as gift, but Pierre refused.

Joe is the judge. How to solve it ?

Solution:
Joe donates $1 to make total $4,
divide between Yoyo and Pierre，

=> Yoyo and Pierre each gets $2
Also Yoyo, Pierre and Joe all lose $1.

Note: "三方各损一两" Win-Win Problem

无用之用

周培源教授： 数学是＂无用之用＂

"Useless" Mathematics is "Useful".

Chicken & Rabbits

This is an ancient Math Puzzle before Algebra was invented, so the ancient intelligent Chinese used this funny "Method of Elimination" below:

Puzzle: In a cage housing some chicken and rabbits. There are total of  35 heads, and 94 legs. Find how many chicken and rabbits ?

Solution:

Imagine letting each animal raises up 2 legs, then there would be 35x2 = 70 legs raised.

94 - 70 = 24 legs

Note that all chicken raising 2 legs would have fallen on ground (no leg how to stand ?), so the 24 remaining legs still standing on the ground MUST belong ONLY to the rabbits standing on 2 legs (the other 2 legs raised up)

24 / 2 = 12 rabbits (Ans)

35 - 12 = 23 chicken (Ans)

Abstract Algebra compulsory

Prof S.S. Chern 陈省身 (the “Gauss No. 2" in Differential Geometry) retired from Berkeley University , he went back to his Chinese Alma mata Nankai University 南开大学 in Tianjing city. (Nankai produced the first prime minister Zhou Enlai 周恩来 and the recent Prime Minister Wen Jiabao 溫家宝).

Knowing the important role of Abstract Algebra in linking all branches of Math and all sciences, Prof Chern  made "Abstract Algebra" compulsory with effect 2001 for all students of Maths, Science, Engineering, IT, Finance:

19 weeks of 3 hrs per week = 57 hrs.

Syllabus:

Group, Ring, Field, Galois Theory.

Note: The French Classe Preparatoire for Grande Ecoles already implemented Abstract Algebra for more than 100 years.

Game of Go

Chinese invented Go (圍棋 \weiqi) since antiquity. The Japanese called it 碁 (Go) and introduced it to the West.

Go is the game of wisdom of Universe and Life.

There are 19 x 19 grids = 361
Each dot can have 3 ways= black, white or empty
So total combinations = 3^361

Take a look at this number: 3^361
is the approximate between 3^360 and 3^365

In a Lunar calendar a year = 360 days.
In Solar calendar a year = 365 days

3 represents the Trinity: 天地人 (God, Earth, Man)
(In Christianity : God, Son and Holy Spirit)

Yijing & Orchestra Conductor

Problem:
The Orchestra conductor has been asked to conduct a series of concert for 2 consecutive months.
The Orchestra has to play few pieces of classical music with the following conditions requested by the organising committee:

1) Play at least 1 piece of music a day;
2) Avoid playing exactly the same music in 2 consecutive days (although partially is allowed)
3) Due to resource constraint, it is impossible to play all pieces of music within a single day.

Question:
At least how many (N) pieces of music the conductor has to prepare in order to fulfill these conditions?

Solution:
Using Yijing （易经）notation:
* a piece of music being played in a day is denoted by
__ (solid line);
while not being played is denoted by
- - (broken line)
* 2 consecutive months in any year have maximum 62 (= 31+31) days.
* A bagua (8 diagramme 八卦) has 64 gua（卦), starting from 6 solid lines stacked on each other (Qian 乾gua) to 6 broken lines stacked on each other (Kun 坤gua), each line (broken or solid line) is called a Yao (爻). 6 Yao makes a gua.
There are total of 64 ($latex = 2^6 $) gua.
* Condition 1) eliminates Kun gua (all broken lines, ie all N pieces unplayed) since at least 1 piece of music to be played in a day.
* condition 3) eliminates Qian gua (all solid lines, sice all pieces can't be played in a single day.
* 64 gua minus 2 above-eliminations left 62 gua.
* these 62 gua have mixture of 6 solid/broken lines, representing 6 played/unplayed combination pieces of
music, with each gua being unique.

=> Answer: The conductor needs to prepare MINIMUM 6 pieces of music.

Calculus = 微积分

Calculus = 微积分
出自: 荀况《荀子.大略》300 BCE [same time as Euclid]
“尽小者大, 积微者著”
=> 見微知著

数学家三素质

数学家需有三素质: 才,学,识

才: 能力
学: 学问
识: 見识 (世界/人生观, 对社会、 历史、 人生的感悟)

数学大师: 无法

我现正在上2010全国华乐比赛-大师班。
大师音乐评分标准：
1. 大法（基本功）
2. 小法（特色）
3. 无法（风格）

数学大师Math Masters 也是如此，
1. 大法（基本功 Classical Math）
2. 小法（特色 Math Olympiad techniques）
3. 无法（风格 Abstract Math -> French "Math Composition"）

ST Yao丘成桐

丘成桐 (ST Yao 1949~) Fields Medal @1982 [33岁] proved Calabi Conjecture

1. 读私立 培正中学, 高中遇 好数学老师. @香港中文大学, 发觉 Math Beauty, '豁然开朗'.
2. Best Math student not necessary Mathematician, only sufficient!
3. 一名数学科学家 都应对 文学,哲学 这类 学科有基本的涉猎. 好的数学 使你体验到庄子讲的
"天地与我并生, 万物与我为一" 的境界
4. 成功 = 要有数学热情.
Strategy:
a. 深入思考
b. 在心中或纸上仔细研究
c. Find clues from book, till get answer.
d. 出题目给自己

China 'Gauss': 秦九韶 Qin Jiushao

秦九韶 Qin Jiushao(1202-1261 AD): http://en.wikipedia.org/wiki/Qin_Jiushao

A Southern Song dynasty (南宋) officer. During his 3-yr leaves when his mother died, he generalised 孙子算经 (4th century)'s "Chinese Remainder Theorem" in '大衍求一术'. After leaves, he went back to chase money & women, produced no more Maths.

Note: '求一': solve a.X ≡ 1 (mod b); a < b

Baccalaureat Origin

1. Arabic Origin: Latin translation of Arabic word 'Bi-haqq al-ruwayeh'. Arabs learnt from China.

2. Chinese Origin: Chinese ‘秀才' (xiu-cai) scholar system (equivalent to Cambridge GCE A level).

3. First appeared as 'Baccalareus' in the University of Paris (1232 AD) by Pope Gregory IX, means 'The right (license) to teach on the authority of another'.

Chinese Remainder Theorem 中国剩余定理

中国剩余定理CRT (Chinese Remainder Theorem)
X ≡ 2 (mod 3)
X ≡ 3 (mod 5)
X ≡ 2 (mod 7)
Solve X?
明. 程大位 "算法统宗" (1593)
3人同行70稀
5树梅花21支
7子团圆半个月(15)
除百零五(105)便得知
Let remainders:
$Latex r_3=2, r_5=3, r_7=2$
$Latex r= r_3.70+ r_5.21 + r_7.15 (mod 3.5.7)$
r= 2.70 +3.21 +2.15 (mod 105)
r= 140 +63 +30 (mod 105)
r= 233 (mod 105)
$latex r= 23 = x_{min}$
or X= 23 +105Z (23 + multiples of 105)

-------------------------------------------
CRT: Why 3:70, 5:21, 7:15
X ≡ 2 (mod 3)
X ≡ 3 (mod 5)
X ≡ 2 (mod 7)

1) Find A such that
A ≡ 1 (mod 3)
A ≡ 0 (mod 5)
A ≡ 0 (mod 7)

=> 5|A, 7|A => 35 |A

A=35, 70 ...

70 ≡ 1 (mod 3)

=> 70x2 ≡ 2 (mod 3)

2) Find B s.t.:

B ≡ 0 (mod 3)

B ≡ 1 (mod 5)

B ≡ 0 (mod 7)

3|B, 7|B => 21|B

21 ≡ 1 (mod 5)

=> 21x3 ≡ 3 (mod 5)

3) Find C s.t. :

C ≡ 0 (mod 3)

C ≡ 0 (mod 5)

C ≡ 1 (mod 7)

=> 3|C, 5|C => 15|C

=> C=15≡ 1 (mod 7)

15x2 ≡ 2 (mod 7)

4)

X ≡ 70x2 +21x3+ 15x2 (mod 3x5x7)

X≡ 233 (mod 105)

X≡ 23 (mod 105)

X= 23+105Z

----------------Ring Theory -------------------------------
Commutative Ring CRT by Ring/ Ideal Theory:
If R is a commutative ring & A1,...An are pairwise coprime ideals,
Prove that if
$Latex r_1, r_2, \cdots, r_n$ belong to R
Then there exists 'a' belongs to R with
$Latex a + A_j= r_j + A_j$ ; j ∈[1,n]
Interpretation:
Let R= Z ring
X ≡ 2 (mod 3)
X ≡ 3 (mod 5)
X ≡ 2 (mod 7)
Or
X ≡ rj (mod mj)

Ideal = (m) = mZ
$Latex A_3=(3), A_5=(5), A_7=(7)$
$latex r_3=2, r_5=3, r_7=2$
There exists
a=23
$Latex a +A_3 = r_3 + A_3$
23+(3)= 2+ (3)
23 + 1x3 =2 + 8x3 =26

π

Memorize: 1 3 5 => 113355


=> ∏ = 355/113


Note: Chinese mathematician Zhu Chongzhi (祖冲之 429–500 AD) computed ∏ up to 7 decimals, accuracy 1,000 yrs earlier than the West.