The Map of Mathematics

Show to your schooling children why they need to study Maths – the Queen of all Sciences – which pushes the frontier of human evolution in last 3,000 years. Maths is always invented few centuries or decades before it becomes useful. For examples:  Complex numbers invented accidentally by the 16th century Italian Mathematicians for solving polynomial equation of 3rd degree, became useful in Physics Electrical and Magnetic Fields (19 CE) ; Invention of Analytic Geometry (17 CE) allowed Newton to trace the earth-sun orbit; Calculus propelled Physics and Physical Chemistry; Leibniz’s Binary Math (18 CE) discovery applied in Computing (20 CE)…

Latest Examples

1. Topology was invented in 1900 by French PolyMath Henri Poincaré, today applied in Big Data, AI…

2. His PhD student invented “Derivatives” Partial Differentiation, today applied in Commodity Trading, Stock Trading, Financial Derivatives… with Black-Sholes formula. 1998 USA Sub-Prime Crisis due to the misuse and lack of understanding of its limitation (“fat tail” ).

3. Mathematician SS Chern 陈省身and Nobel Physicist Yang Zhen-Ning 杨振宁were working independently in the USA for 40 years, Chern on Differential Geometry, Yang on Yang-Mill Equation (one the 7 unsolved Math Problems in 21st century). Through a common friend the hedge fund billionaire James Simons – Chern’s former PhD Math student and university colleague of Yang – they realised that the Math “Fiber Bundles” (纤维丛) invented by Chern 30 years earlier could apply in Yang’s Physics (Gauge Theory).

Russian & World Math Education

​In the world of Math education there are 3 big schools (门派) — in which the author had the good fortune to study under 3 different Math pedagogies:

“武当派” French (German) ,  “少林派” Russian  (China) ,  “华山派” UK (USA).

( ) : derivative of its parent school. eg. China derived from Russian school in 1960s by Hua Luogeng.

武当派 : 内功, 以柔尅刚, 四两拨千斤 <=> “Soft” Math, Abstract, Theoretical, Generalized.

少林派: 拳脚硬功夫 <=> “Hard” Math, Algorithmic.

华山派: 剑法轻灵 <=> “Applied” Math, Astute, Computer-aided.

The 3 schools’ pioneering grand masters (掌门人) since 16th century till 21st century, in between the 19th century (during the French Revolution) Modern Math (近代数学) is the critical milestone, the other (现代数学) is WW2 : –

France: Descartes / Fermat / Pascal  (17 CE : Analytical Geometry, Number Theory, Probability), Cauchy / Lagrange / Fourier /Galois  (19 CE, Modern Math : Analysis, Abstract Algebra), Poincaré (20CE Polymath, Topology), André Weil (WW2,  Bourbaki school), Alexander Grothendieck (21 CE, Differential Geometry)…

Note: 1/3 of Fields Medals won by French.

(Germany): Leibniz (18CE, Calculus, Binary Algebra). Felix Klein (20 CE, learnt from French master Camile Jordan) established the Gottingen school (World Center of Math before WW2, destroyed by Hitler). Successors: Gauss ( Polymath), David Hilbert , Riemann (Prime Number) , Cantor  (Infinity Math), Emmy Noether  (Axiomatic Algebra, Ring Theory) …

Note: Swiss-German branch – Bernouille father & 3 sons and student Euler (Polymath).

RussiaА. Н. Колмогоров (20 CE, Polymath), Grigori Perelman (21 CE, Topology, who rejected Fields medal).

(China): 华罗庚 (Hua Luogeng, Number Theory), 陈省身 (SS Chern, China/USA, Differential Geometry),  陈景润 (“Chen Theorem”, Number Theory), 丘成桐 (ST Yau, HK/USA, Differential Geometry), 吴文俊 (Wu Wenjun, Machine Proof of Geometry )。

UK: Isaac Newton  (18 CE,  Calculus), Hardy / Littlewood / Ramanujian (Number Theory), Bertrand Russell  (Logic), Andrew Wiles (21 CE, proved 380-year-old Fermat’s Last Theorem).

(USA): 20CE Gödel (Austrian / USA), Paul Erdös (Hungary / USA), Eilenberg /MacLane (WW2, Category Theory).

Note: Terence Tao (陶轩哲 Australia / USA, 21 CE, Number Theory)

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

The Pros & Cons of the French Elitist Grandes Ecoles

The French Grandes Écoles System is characterized by one unique Ultra-Selective Exams – “Concours” (pronounced as Kongu) or 科举 (pronounced in Chinese dialect Fujian as “Kogu”) a la the 1,300- year Chinese Imperial Exams dated since 600AD till 1905. Napoléon Bonaparte had great admiration of the Chinese mandarin meritocractic selection system, he was influenced by the Jesuit priests who were mostly working in China coastal province Fujian, decided to implement “Concours” for his newly established military engineering college  “École Polytechnique” (aka ‘X’).

Like any system, there are always two sides : the pros & the cons. Kogu served China well for 1300 years, producing top mandarins who ruled China with the most intelligent scholars through layers of selective exams from county (乡试选拔”秀才”) to province (省试选拔 “举人”) to the capital (京都 殿试选拔 “进士” – 前三名状元 /榜眼 / 探花). The Cons came from its Implementation “devils” – too focus on one syllabus ( literature), privileged family / political class with unfair inner-circle advantage, corruption, cheating, etc.Those real talents who did not play well within the rules were excluded outside the gate (李白, 杜甫, 吴承恩, 蒲松龄 李时珍, 曹雪芹 …). The Kogu was the key reason of China’s decline after the 18th century, having missed the European Industrial Revolution, due to the Qing Empire’s inward looking closed-door policy.

The French Concours also has done well for France since the Great Napoléon Empire till the glorious 30 years of the Fifth République after WW2. Like the Chinese Kogu, Concours has its many cons too : 1) biased in narrow syllabus (predominantly in the less applied but abstract French Math); 2) Privileged to the rich and self-perpetuating class of grande-ecole family tradition; 3) the real talents like Évariste  Galois (the greatest mathematician of the 19th century) failed École Polytechnique Concours  twice; 4) Locking the most talented Scientific youth in preparing for Concours, the excellent French tradition of scientific invention & discovery spirit since 16th to 19th century has been replaced by complacent, privileged, bureaucratic elitism. From the 1980s, France has been lacking behind the USA in the next revolution of Information Technology in the Internet & Mobile Phone Age.

System Flaws Biased Elitism Losses
Kogu 科举 Literature 八股文 Rich Officials / Merchants 官商 Failed talents 落第才子
Concours Abstract ‘Pure’ Math Educated / Upper-class French Entrepreneurs, Scientific talents

Two World’s of Higher Education

The Making of the French Ruling Elites From A Small Circle:

Self-Perpetuating Elitist Class : Imagine a large, potentially flourishing country (France) which is held back by its own selfish, self-perpetuating elite alumni…

Peter Gumbel: Elites d’Academie

Utter Elitism : Louis-Le-Grand (the Prépas to École Polytechnique & École Normale Supérieure)

Before Concours, Prépa life is like catching TGV Speedy Train; but after passing it, life is partying next 3 years …

« Tu bosses dur pendant deux ans, et après tu es tranquille en école jusqu’au diplôme », témoigne Louise R, 29 ans. Une fois la grande école intégrée, ces surentraînés de l’étude sont nombreux à partager une sensation de vide, le rythme de travail étant bien moins soutenu. Jérome D, 21 ans estime« perdre son temps » dans son école d’ingénieur parisienne : « Après tant d’efforts et de renoncements, quelle n’a pas été ma désillusion ! ». Également passé par une école d’ingénieur, Benoît M juge que durant les trois années qui suivent la prépa, « On n’apprend finalement pas grand-chose ».

Mathematics: The Next Generation

Historical Backgroud:

Math evolves since antiquity, from Babylon, Egypt 5,000 years ago, through Greek, China, India 3,000 years ago, then the Arabs in the 10th century taught the Renaissance Europeans the Hindu-Arabic numerals and Algebra, Math progressed at a condensed rapid pace ever since: complex numbers to solve cubic equations in 16th century Italy, followed by the 17 CE French Cartersian Analytical Geometry, Fermat’s Number Theory,…, finally by the 19 CE to solve quintic equations of degree 5 and above, a new type of Abstract Math was created by a French genius 19-year-old Evariste Galois in “Group Theory”. The “Modern Math” was born since, it quickly develops into over 4,000 sub-branches of Math, but the origin of Math is still the same eternal truth.

Math Education Flaw: 本末倒置 Put the cart before the horse.

Math has been taught wrongly since young, either is boring, or scary, or mechanically (calculating).

This lecture by Queen Mary College (U. London) Prof Cameron is one of the rare Mathematician changing that pedagogy. Math is a “Universal Language of Truths” with unambiguous, logical syntax which transcends over eternity.

I like the brilliant idea of making the rigorous Math foundation compulsory for all S.T.E.M. (Science, Technology, Engineering, Math) undergraduate students. Prof S.S. Chern 陈省身 (Wolf Prize) after retirement in Nankai University (南开大学, 天津, China) also made basic “Abstract Algebra” course compulsory for all Chinese S.T.E.M. undergraduates in 2000s.

The foundations Prof Cameron teaches are centered around 4 Math Objects:

1. SET 集合
– Set is the founding block of the 20th century Modern Math, hitherto introduced into the world’s university textbooks by the French “Bourbaki” school (André Weil et al) after WW1.

Note: The last “Bourbaki” grand master Grothendieck proposed to replace Set by Category. That will be the next century Math for future Artificial Intelligence Era, aka “The 4th Human Revolution”.

– A vision first proposed by the German Gottingen School’s greatest Math Educator Felix Klein, who said Functions can be visualised in graphs, so it is the best tool to learn mathematical abstractness.

– The German mathematician Leopold Kronecker, who once wrote that “God made the integers; all else is the work of man.”

– The universe is composed of numbers in “NZQRC” (ie Natural numbers, Integers, Rationals, Reals, Complex numbers). After C (Complex), no more further split of new numbers. Why?

4. Proofs
– reading and debugging proofs.

Example 1: Proof by Contradiction, aka Reductio ad Absurdum (Euclid’s Proof on Infinitely Many Prime Numbers)


Challenge the proof: Why ?
Induction intuitively by:


Example 2: Proof by Logic

By Reasoning (which is unconscious), most would get “2 & A” (wrong answer)

By Logic (using consciousness), then you can proof …
Correct Answer: 2 and B
Test on all 3 Truth cases below in Truth Table:
p = front side
q = back side


Recommended The Best Book on Abstract Algebra by Prof. Peter J. Cameron