陈省身:数学之美 SS Chern : The Math Beauty

There are 5 great Geometry Masters in history: 欧高黎嘉陈

Euclid (300 BCE, Greece), Gauss (18CE, Germany), Riemann (19CE, Germany), Cartan (20CE, France), Chern (21CE, China).

Jim Simons (Hedge Fund Billionaire, Chern’s PhD Student) quoted Chern had said to him:

“If you do One Thing that is really good, that’s all you could really expect in a life time.” 一生作好一件事, 此生足矣!


1. Video below @82:00 mins, SS Chern criticised on Hardy’s famous statement: “Great Math is only  discovered by young mathematicians before 30.” Chern’s response: “Don’t believe it ! 不要相信它”.

2. Chern’s Conjecture :“21世纪中国将是数学大国。 ” China will be a Math Kingdom in 21st century.


北京大学:数学是什么 ?


第1讲 数学的思维方式 

3000 年前 希腊,巴比伦,中国,印度, 10世纪阿拉伯, 16世纪欧洲文艺复兴 数学 => [经典数学 Classical Math]

1830 年 数学的革命 – 法国天才少年 伽罗瓦 (Évariste Galois 1811 – 1832) => [近代数学 Modern Math]

观察 (Observe): 客观现象
抽象 (Abstraction) : 概念, 建立 模型 (Model)
探索 (Explore): 自觉 (Intuition), 解剖 , 类比(Analogy), 归纳 (Induction), 联想, 推理 (Deduction) 等…
猜测 (Conjecture) : eg. Riemann Conjecture (unsolved)
论证 (Prove): 只能用公理 (Axioms)(已知的共识), 定义 (概念), 已经证明的定理 (Theorems), 进行逻辑推理并计算.
揭示 (Reveal): 事物的内在规律 (井然有序)

第二讲: 例子 – 微积分 (Calculus) 的诞生, 演变, 严谨化


15 世纪 天体运动的观察: 哥白尼, 开普勒 三大定律 (天文数据结论, 非数学证明)

17 世纪 理论化: [英]牛顿,[德] Leibniz (非严密的数学)

19 世纪 严密数学: [法] Cauchy 柯西, [德] Wierstrass => “epsilon-delta” 极限 (Limit) => 柯西 数列 (Series).

实数 (R Real Numbers) 的 Complete (完备性 ) : [德国中学数学 老师] Dedekind (戴德金)’s Cut

有理数 (Q Rational Numbers): 稠密 但 不 Complete , 即 有漏洞, 穿插进 无理数 (irrational like \pi, \sqrt{2}  ) 

定理:  如果 数列是 柯西数列 => 一定有极限, 且此 极限一定是 实数

例子: Series S = {1.4 , 1.41, 1.414 … }

S has no limit in \mathbb {Q}, but limit = \sqrt{2} \in  \mathbb{R}

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).

Trump’s Speaking Math Formula

The lower in the score the better : Trump (4.1) beats Hillary (7.7) who beats Sanders (10.1)

Trump defied most expectation from the world to win the 2017 President of the USA. His victory over the much highly educated Ivy-league Yale lawyer-trained Hilary Clinton who speaks sophisticated English is “SIMPLE English“: seldom more than 2-syllable words.

1-syllable words mostly: eg.dead, die, point, harm,…

2-syllable words to emphasize: eg. pro-blem, ser-vice, bed-lam (疯人院), root cause, …

3-syllable words to repeat (seldom): eg. tre-men-dous

His speech is of Grade-4 level, reaching out to most lower-class blue-collar workers who can resonate with him. That is a powerful political skill of reaching to the mass. Hilary Clinton’s strength of posh English is her ‘fatal’ weakness vis-a-vis connecting to the mass.

In election time, it is common to see candidates who win the heart of voters by using the local dialects of the mass, never mind they are discouraged in schools or TV: Hokkien, Teochew, etc.

Math Education Evolution: From Function to Set to Category

Interesting Math education evolves since 19th century.

“Elementary Math from An Advanced Standpoint” (3 volumes) was proposed by German Göttingen School Felix Klein (19th century) :
1)  Math teaching based on Function (graph) which is visible to students. This has influenced  all Secondary school Math worldwide.

2) Geometry = Group

After WW1, French felt being  behind the German school, the “Bourbaki” Ecole Normale Supérieure students rewrote all Math teachings – aka “Abstract Math” – based on the structure “Set” as the foundation to build further algebraic structures (group, ring, field, vector space…) and all Math.

After WW2, the American prof MacLane & Eilenburg summarised all these Bourbaki structures into one super-structure: “Category” (范畴) with a “morphism” (aka ‘relation’) between them.

Grothendieck proposed rewriting the Bourbaki Abstract Math from ‘Set’ to ‘Category’, but was rejected by the jealous Bourbaki founder Andre Weil.

Category is still a graduate syllabus Math,  also called “Abstract Nonsense”! It is very useful in IT  Functional Programming for “Artificial Intelligence” – the next revolution in “Our Human Brain” !