《数学与人类文明》Mathematics & Civilizations


1. 哲学:希腊 Euclidean Geometry

2. 艺术 : Arabic 文艺复兴, 达芬奇, Golden Ratio

3. 工业革命:Descartes Analytic Geometry, Newton Calculus

4. 抽象: Gauss “Non-Euclidean Geometry” , Paradox in Set Theory, Godel “Incomplete Theorem” .


大衍求一术 秦九韶





1. 秦汉:张苍&耿寿昌 编 《九章算术》, 3AD 东汉 刘徽 (注解)

2. 4AD 南北朝:祖冲之父子 圆周率π

3. 13-16世纪 元/明朝:珠算盘


只是Applied 应用, 没有 希腊Deductive Theory 理论。


China made 18 q-bit QUANTUM Computer

Intel had made 17-qbit Quantum Computer, recently China just made 18 q-bit but with only 6 photons (vs Intel’s 17).

叠加 :superposition
纠缠 :entanglement
并行计算 :parallel computing

Application: Decypher a 300-digits password would take 150, 000 years on a traditional computer, but only 1 sec on a Quantum Computer.



IMO 2020 Solutions

IMO 2020 : China World Team Champion ( 5 Golds + 1 Silver), the only one in the World with Perfect Score.

This Chinese IMO coach comments :

IMO 2020 is the easiest in the past 10 years, compared to 2015 (tougher) & 2018 (toughest).


李彦宏Baidu CEO : Internet 3 Episodes : PC->Mobile ->AI



李彦宏 Baidu CEO Cambridge Speech 剑桥大学演讲
《3 waves of Internet》:
1) PC- based (1997-)

  • Search Webpages
  • 6-month software update cycle

2) Mobile-based (2010 -)

  • “APP” is born
  • Eco-System : eg. Apple Appstore, Google PlayStore
  • O2O (Online to Offline) : Same day Hotel booking/Restaurant /…
  • SW Update everyday few times

3) AI-based (2017 – now)

  • Voice recognition sans keyboard input
  • Image recognition (eg. Customer ePayment :McDonald’s )
  • Natural language Pattern NLP (Salesman Virtual Assistant)

Chinese 3AD Arithmetics 东晋. 刘徽 《九章算术》更相减损术

Please explain the Number Theory behind this trick :\boxed{\frac {a } {b}= \frac {\frac {a}{b-a}}{\frac {b}{b-a}}}

Example: 246 - 205 = 41

\boxed {\frac {205} {246}= \frac {\frac {205}{41}}{\frac {246}{41}}=\frac{5}{6}}


27759 – 10227 = 17532 = 2 x 8766 = 2 x (2 x 4383) = 2 x 2 x (3 x 1461) = 2 x 2 x 3 x (3 x 487 )

\boxed {\frac {10227} {27759}= \frac {\frac {10227}{1461}}{\frac {27759}{1461}}=\frac{7}{19}}

Explanation:This method is from《九章算术》295AD 刘徽(曹魏/东晋),he invented the “Limit” 割圆法 method with 95-polygons to get the world’s best pi = 3.1416



Bézout’s Theorem :

For a, b CO-PRIME, ie gcd (a, b) = 1
There exist integers x and y such that ax + by = 1

书法”九宫格” 的”均” =黄金分割

书法是”字如人品” 。

大自然的美表现在 “黄金比率” (Golden Ratio) = , 暗藏在 唐初 欧阳询 发明的楷书”九宫格” : 一个比率 “均” (即:均称,对称 Symmetry) 。

1, 1, 2, 3, 5, 8,… (Fibonacci Series)

黄金分割= 0.618 ~ 1:2 ¦ 2:3 ¦ 3:5 ¦ 5:8

Note : Golden Ratio 黄金比率 =1.618 (长:短)

欧阳询 楷书 《九成宫》

晋/隋/唐的行书, 草书是在汉朝的楷书 (aka 真书) 的”“基础上”速度化” : “” (角度), (笔画突) ,尚保存”均” 的美。

东晋. 王羲之 《兰亭序》行书

唐.张旭 草书

毛泽东 草书沁园春· 雪




初等数学 (Elementary Math) : 高中 / 初级学院 JC (A-level / Baccalaureate) 毕业前的12年数学。

高等数学 (Superior Math, “Math Sup” in French) : 大学本科 (undergraduate) 的数学。

[高深数学 (Advanced Math) = Graduate PhD Math]

李善兰 (晚清 19CE):继续翻译 晚明 Ricci Mateo / 徐光启 《几何原理》(《The Elements》13 books)没翻译的最后4本。

李善兰引进中国数学 "Function" (法国Descartes “Relationship” 概念 / 德国 Leibniz’s 引进 / 瑞士 Euler 正式化 y = f(x) ), Felix Klein 带入中学数学) 。他翻译成 "函数", 为什么?因为他用信函举例子: Function 好比 一封信 (f) ,写信人集合 Set {A} ,收信人集合 Set {B} :

f(A) – > B


每个B人 可以收到 0 1 封信。

不可能 同一封信 发给 > 1 个 B人。

Key Points (Read below link) :

高等数学的三个"挡路虎"(Road Blocks) ,排除他们三个障碍, 才能登堂入室”高等数学” ,游刃自如。

  1. 函数 Function,
  2. 连续 Continuity ,
  3. 极限 Limit



中三化学刘老师教我们背这个金属活动性顺序表, 排前面的元素比排后面的活跃,写化学反应方程式时,非常有用。基本上整个 Physcial Chemistry 就是浓缩在这表里,朗诵上口,终身难忘,50年后还记得!- 化学靠记忆,用中文学化学最方便 。(我在法国精英大学 Classe Preparatoire 念化学,打败全班法国人得第一名,中学用中文读化学的基础, 功不可没。)

铅 (氢)

K Na Ca Mg Al Zn Fe Sn Pb (H)

[嫁 倩 (轻)]

Cu Hg Ag Pt Au

[统 金]

应试教育 :学多悟少

应试教育 = Exams-centric Education

放任教育 = Laissez-faire Education


美国[#] 放任教育: 学悟多 “Teach Less Learn More

中国[*] 应试教育 :学悟少 “Teach More Learn Less

Note :

[#] : 包括 以色列,芬兰,…

[*] : 包括新加坡,法国,韩国,…

The american education before university is considered inferieur to most asian education, because the curriculum is more relaxed, book knowledge gained in school is behind by a year relative to the Asian counterparts. [Note §]

Yet the american universities produce the most inventors, scientists, mathematicians, entrepreneurs. Why? That is due to the “Laissez-faire” education for american kids who have more relaxed time to explore the nature, with keen curriosity for finding scientific truth on their own.

[Note §]: Singaporeans after Junior College “JC” Year-1 (Pre-University 1) can enter directly to good American universities (eg. John Hopkins, Harvard), without the GCE A-level exams certificate. Because Singapore JC Year 2 covers the same curriculum in the USA University Year 1.

The disparity is due to the 1960s UK University Reform which pushed the first year university to pre-University in High schools, thereby shortens the more expensive university years from 4 to 3, and increases the much cheaper pre-University by one more year. Singapore follows the UK GCE system of 2-year Pre-U, without shortening the general Batchelor (non-Honors) degree academic years from 4 to 3 (eg. Engineering).

Japan is an exception from the typical stringent rigor of asian universities. Japan produces a Japanese Nobel Prize each year. Why? Notice these Japanese Nobel scientists do not come from top universities (where exams are very competitive) but the “second-rank” universities.

France is declining from the 19CE scientific grandeur also due to the Grandes Ecoles (Elite Universities) Concours competitive Entrance Exams – introduced from the ancient China Imperial Exams (科举) by Napoleon Bonaparte for his military Ecole Polytechnique (“X” ) – since then it has spread to entire France in last 200 years. Each year the top 1% of high school French talented kids are sent to these “cramped” schools (Classes Préparatoires) to study like “Taupes” (Moles) for 2 to 3 years, “no-day-no-night” to prepare for the competitive exams in Advanced Math. After the Concours, these “successful” but totally exhausted students no longer are keen to study as hard but their prestigious career guaranteed after being “X Diplômé” . President Emanuel Macron failed twice in such Concours, as did the “Father of Modern Math” Evariste Galois in the French Revolution.

Similarly the 应试教育 in China 科举 was first introduced since 600AD 隋朝 till 1905. Many top Imperial scholars (进士,状元,榜眼,探花) are unknown in history today, but the many failed scholars made tremendous contributions in science, math, literature (曹雪芹,) , medicine (李时珍)and military (左宗棠,曾国藩) … also some revolted against the ruling dynasties (黄巢, 洪秀全) . Today the Chinese “Gaokao” (高考) University Entrance Exams since 1980s has produced millions of graduates each year, but there is still ZERO Nobel Prize or Math Fields Medal in the past 40 years, compared to those world-class scientists (Nobel Physicists 杨振宁, 李政道, Nobel Medicine 屠呦呦, 钱学森, 钱三强, 钱伟长,… ) & mathematicians (陈省身, 吴文俊, 华罗庚, 苏步青,… ) educated before 1949. Why? This question has been the “Paradox” asked by 钱学森 to the Chiness Primier 温家宝,also known as the “陈省身 Conjecture“. [The answer is in this blog title ! ]

Other competitive education countries fall into the same “Paradox” are Korea & Singapore – Any of the Oxf-Bridge or Ivy League Universities trained, among the 200+ President scjolars or 400+ SAF / Police scholars or few thousand Colombo / PSC scholars since 1965, has won a Nobel Prize or Fields Medal or any reputed international scientific award? NONE !

A late boomer, he invented Lithium battery (bought by SONY) at 90 years old:

克隆之父 Nobel (2012) Medicine : another “hopeless” student in the eyes of his professors.

。。。中国学生 早学迟悟


Relativistic Space Time, Fengshui, Isometry

风水 Fengshui (aka Geomancy) is Einstein Relativistic using Compass (东西南北上下 =Space) & Timing (吉凶日/时间 = auspicious date & time).

\Yu (Space) \Zhou (Time) = Space-Time
宇环 (universal surrounding)
宙时 (instant timing)


Axonometry = Isometry ( dengjiao toushi 等角透视,ie Angle Preserved Projection )

Chinese art is axonometric using isometry geometry, scrolling from right to left along time. Example : this digital animated 12-century painting

清明上河图 Qingming Festival River Scene(北宋)

“Why the world relies on a Chinese ‘perspective’” by Jan Krikke, China, AI and Quantum Physics




清. 李善兰 翻译 Function 为函数。函,信也。只能有一个收信人,所以 只有一个 f(x) 值。

The unique 1 single output of a function becomes very important for subsequent development in Math & IT:
functions are composable, associative, identify function,etc (distributive,… ) => it can be treated like vector => structure of a Vector Space “Vect”

Extended to..

“Vect” is a bigger structure “Category” in which “function of functions” is a
Functor” (函子)F:F(f)

Example : F(f) = fmap (in Haskell)

fmap (+1) {2,7,6,3}

=> {3,8,7,4}

here F = fmap, f = +1

The Math branch in the study of functions is called “functional” 泛函。

IT : Functional Programming in Lisp, Haskell, Scala, ensure safety of guaranteed output by math function property. Any unexpected exception (side effects: IO, errors) is handled by a special function called “Monad” (endo-Functor).

(想看更多合你口味的内容,马上下载 今日头条)



Princeton University Math Dean Prof Chang 张圣容


传统研究学问的方法: 万事俱备,等东风。

Princeton 研究学问的方法: "先上车,后补票"

All-rounded Education important for Math Research:

(想看更多合你口味的内容,马上下载 今日头条)


(想看更多合你口味的内容,马上下载 今日头条)

小算盘 大乾坤 Abacus

浙江临海市 国华珠算博物管主人 木匠 雷国华 收藏算盘20年,凭个人力量,保存中国算盘发源地的文化遗产,为此散尽钱财借债,免费开放给公众参观。说到家人因支持他而受苦,感慨激动不已。

中国的算盘上面2珠 (2×5),下面5珠(5), 共15,加1 = 16 进位。古代是16位 (Hexadecimal) 制, 半斤 八两 (= 1/2 斤 x 16 两 = 8 两)。

日本改良成10位制: 上面1珠 (1×5),下面4珠(4), 共9,加1 = 10 进位。

明朝万历年间的 “平民王子”朱载堉 (1536 AD – 1610 AD) 发明现代音乐的 十二平均律 (12-tone Equal Temperament) , 用81档的大算盘算出:先开立方根,后开平方根 2 次

\boxed { \displaystyle \sqrt[12] {2} = 2^{\frac {1}{12} }= 2^{{\frac {1}{3}}.{\frac {1}{2}}.{\frac {1}{2}}} = \sqrt {\sqrt {\sqrt[3]{2}}}}

Chinese Primary School Math

As shown in the picture below, find the height of the table?

(No Algebraic equation please for Primary School Math).

Answer: [hint = 130 cm]

Try yourself before scrolling down.

. .


The 2 sleeping cats and 2 sitting cats cancel out, left 260 (= 110+150) for the 2x Table Heights.

1 Table Height = 260/2 = 130 cm.

数学范畴化 Categorification


Key Points:

  • Category and Functor are above the underlying algebraic structures (Set, Group, Ring, Vector Spaces, etc), study the relations between these structures.
  • Early 19 CE mathematicians before “Category Theory” already knew there is 1:1 mapping between the Field Extension and Galois Group.
  • Treat Structures and Relation between them (Functors) on equal footing.


这位中国留英数学专业的小伙子有很好的口才: 数学之美, 他的数学遗憾。

华威大学(The University of Warwick),famous for mathematics in UK.

Key Points:

Einstein用 Riemann Geometry 数学救了Newton 物理。

中国数学的没落: 自从明朝科举废除数学考试

3位大师救近代中国数学: 华罗庚, 陈省身, 苏步青

数学光明的未来: AI, Big Data, Cloud Computing

天才的崎岖道路: “扫地僧”张益唐

时代的”指数” exponential快脚步, 不要追风, 要一以贯之 : Prof. Andrew Wiles proved 350-year-old “Fermat’s Last Theorem” (FLT).

[纠正]: Andrew Wiles 超过40岁, 没赶上Fields Medal, 只得个”奖励”。他看到椭圆(Ellipse)气球, 得到突破 FLT “工具”的灵感 – “Elliptic Curve” 。

数学是什么: 爱情, 艺术, 音乐, 科技

直观 数学 Intuition in Abstract Math

Can Abstract Math be intuitive, ie understood with concrete examples from daily life objects and phenomena ?

Yes! and Abstract Math should be taught by intuitive way!

1. 直观 线性空间 : Intuition in Linear Space

(Part I & II) 矩阵 (Matrix), 线性变换 (Linear Transformation)


(Part III)


Animation: English (Chinese subtitles)


2. 直观 群论 (Intuition in Group Theory)



UK Textbooks à la Chinese / Singapore Math Style

[Original Financial Times Article] Google: UK maths books fail DfE test

\boxed {\text {Ideal Math = (Chinese + English) * French }}

UK, USA, France are copying Chinese Math (from which the “Singapore Modeling Math” derived) in Primary schools, this proves my above-mentioned “Ideal Math” formula is correct. The 2 Asian countries were top in 2015 PISA Math Test for 15-year-old students, while UK was ranked 27th.

Note: Shanghai used to represent China to be the World Top PISA in math for decades till averaged down by other chinese cities.

It remains to see if China / Singapore reciprocate the French Math theoretical foundation rigor in High Schools (Junior Colleges) and in the first 2 years of undergraduates for STEM (Science, Technology, Engineering, Math) students.

Compared with Fields Medals (equivalent to Nobel Prize in Math), the picture is reversed – where USA, France and UK are top. The secret lies in the formula : the multiplying factor – (* French), ie the math theoretical foundation notably in French Math.

Note: 2 Chinese Fields Medalists are listed in the nationalities of “British / HK, now China” (ST Yau 丘成桐), Australia (Terence Tao 陶哲轩 ).

The World’s Best Mathematician 

Terence Tao 陶哲轩 : 

  • Research style: both Collaborative à la  “PolyMath” Project & “Lone Wolf” à la Andrew Wiles (“The Fermat’s Last Theorem”), G. Perelman (“The Poincaré Conjecture”) or Zhang Yitang 张益唐 (“70-million Gap in Twin Primes”)
  • Not attempting the Riemann Hypothese: tools not there yet.
  • His Weakness in Math: Algebraic Topology (拓扑代数) [*]

[*] Algebraic Topology: apply algebra (linear) in Topology: eg. Homology (同调), Co-homology (上同调), Homotopy (同伦) or Homological algebra, etc. A powerful applied mathematics used in Big Data Analytics.

3 Phases of Math Training:

  1. Pre-rigourous: intuition à la Ramanujian 
  2. Rigorous: formal proof
  3. Post-rigourous: 1 + 2 (intuitive bold “cheating” guess, followed by rigorous proof)

Prof ST Yau’s 丘成桐 Talk to Chinese Youth on Math Education 

Prof ST Yau 丘成桐 , Chinese/HK Harvard Math Dean, is the only 2 Mathematicians in history (the other person is Prof Pierre Deligne of Belgium) who won ALL 3 top math prizes: Fields Medal 1982 (at 27, proving Calabi Conjecture), Crafoord Prize (1994) , Wolf Prize (2010).

Key Takeaways :

1. On Math Education
◇ Compulsary Math training for reasoning skill applicable in Economy, Law, Medicine, etc.
◇ Study Math Tip: read the new topic notes 1 day before the lecture, then after lecture do the problems to enhance understanding.
◇ Read Math topics even though you do not understand in first round, re-read few more times,  then few days / months / years / decades later you will digest them. (做学问的程序).
◇ Do not consult students in WHAT to teach, because they don’t know what to learn.
◇ Love of Math beauty is the “pull-factor” for motivating  students’ interest in Math.
◇ Parental Pressure.

2. “3D” facial photo using Math

3. Pi-Music: 1 = “do”, 2 = “re”, 3 =”me”…
Pi =3.1415926…

4. Math Olympiad: Prof ST Yau had criticised publicly it as a bad Math training, not the “real” Math. 

An audience tested Prof ST Yau on a Math (Accounting) Puzzle which he couldn’t  solve on the spot. He said Mathematicians are poor in +-×÷ arithmetic. 

5. Chinese students in USA: China sends over 200,000 students to USA universities. They are good in secondary / high school Math with known solutions,  but poor in graduate PhD Math which requires “out-of-the-box” independent thinking skill for finding unknown solutions. Recent few years Chinese students (eg. Stanford Prof 李骏 : 1989 Harvard PhD)  in USA have improved standard in PhD research.

6. Research is not for fame. It takes many years to think through an interesting topic.


1. Prof ST Yau’s Best Seller Book 《The Shape of Inner Space》avail @ NLB (Ref #530.1) 11 copies in most NLB branches@ AMK, Bishan etc.

2. Interview Prof ST Yau by HK TV (Cantonese)

3. 丘成桐 (2008) 评中国 和 美 国的教育 : 中国学生不爱看课外书, 因为考试太重, 课余时间花在玩电脑游戏。

4.  丘成桐 (2016): 中国大学本科要注重基础教育, 才能培养世界级一流人才

Homology (同调 ) in Geometry & Topology





同胚” homomEophism (eg. Donnut 和茶壶), 可以扭捏泥土从前者变后者。

“同态” (同样形态homomOrphism), 就是Same-Shape-ism. eg. (相似) Similar Triangle.

如果是congruent (全等), 就是 Isomorphism (“同构“, 同样结构)。

所有新加坡人自己人批评自己人kiasu, 其实大家都kiasu, 因为是”自同态” (自己同样态度kiasu), 自=”Endo”
=> Endomorphism.

如果猪八戒照镜子, 看到镜子里面的丑八怪, 还是他猪八戒,
=> Automorphism “自同构” (镜里的影子和自己同样结构)

这些构造(structure)在WW1后被当时Structurism哲学思想影响, Bourbaki 法国师范大学一批学生 (犹太人 André Weil是领袖)把全部人类的数学重写, 以structures (Set, Group, Ring, Module, Field, Vector Space, Topology. .. )为基础 就是新(抽象)数学, 影响到今。
WW2 后, 美国人Sanders MacLane 更上一层楼, 把Set/Group/Ring…等structures 再归类成Category (范畴), 研究其共通的性质 (Morphism 动态), 能够 举一反十。应用在IT 里, 其 Category 就是Functional programming, Types…


[录音小声, 请用earphone耳机听更清楚。]

Key Points Take Away:

1. 身处逆境, 不是勇气, 是淡定。

2. 对目的要穷追不捨, 不要放弃。他从北大的Analytic Number Theory (解析数论)兴趣, 被”人为”的转道去搞博士论文Algebraic Geometry, 7年毕业却无业。从新回到” 解析数论”的跑道, 才得到大成就。

3. 如果2个不同领域的学问之间有些联系, 只要往里鑽, 必能发现新东西。

4. 人生低谷, 碰到3个贵人(2位北大校友, 一位美国系主任青睐)协助。

5. 太太不知他干何学问, 不给 他家庭经济压力, 才能安心于数学。

1. 对于天才儿童, 他劝家长不要 “压 “也不要”捧”, 只要多鼓励, 像Perleman 的(俄国数学家, 证明100年的Poincaré Conjecture)父母循循教导儿子

2. 希望能收PhD学生, 会对他们负责任, 不要有像他个人的悲剧发生 (指被教授利用做私人的项目, 误了学生的前途)。他手头有半’成品’和 3/4’成品’, 可让学生拿去参考, 继续完成当论文。

Interview With The Smartest Man In The World

Terence Tao 陶哲轩 :
♢ Born in 1975 @ Adelaide. Parents from Hong Kong immigrants to Australia.
♢ Chinese-american
♢ IQ 230-240
♢ Full professor of UCLA at 24
♢ Fields Medalist (2006), the 2nd youngest winner in history (after French J.P. Serre at 27)

The TV interviewer pretending to be an “idiot”, a mentality of most Americans who pride themselves of being poor in Math !