《数学与人类文明》Mathematics & Civilizations

《数学与人类文明》数学与现代文明:

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” .

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数学武林的四大门派

数学武林的四大门派比喻:

[0. 希腊 = 达摩祖师]

  1. 法国 =少林派
  2. 德国 = 武当派(德国祖师 Lindemann 是法国宗师 Charles Hermite 的得意外国徒弟
  3. 俄国 = 峨眉派 (俄国祖师奶 Sofya Vasilyevna Kovalevskaya 是 德国大师 Karl Weierstrass 的得意女弟子)
  4. 中国 = 古墓派 (‘小龙女’ 的奇特功夫 自成一派:中国古代数学 + 融合 留 法/英/美/日 数学派 )

https://m.sohu.com/a/216801634_404328/

英/美/日 数学都是支派,非主流宗派 掌门:

英国 受法国影响 : Newton 说他是站在巨人 Descartes, Fermat 的肩膀. 近代 Bertrand Russell 也是读法国Camile Jordan 的 《Group Theory 》书。

美国 数学 : WW2 德国/东欧 犹太人移民: Noether, Godel, Courant, Artin,…

日本数学 : 唐朝留学 中国 (日本”和算” 鼻祖 关孝和 把中国 “行列式” 传给欧洲Leibniz : Determinant ) ,明治天皇 时期 日本留学欧洲。

Note:

1. 奇特数学: 东汉 3AD  Chinese Remainder Theorem (韩信点兵 Modular Arithmetics), Singapore Modelling Math (eg. 鸡兔问题) ,13CE 的 行列式 (Determinant, before “Matrix” was invented in 20th CE by JJ Sylvester )

2. Determinant was invented by the ancient Chinese Algebraists 李冶 / 朱世杰 /秦九韶 in 13th century (金 / 南宋 / 元) in《天元术》.The Japanese “和算” mathematician 关孝和 spread it further to Europe before the German mathematician Leibniz named it the “Determinant” in 18th century. The world, however, had to wait till the 19th century to discover the theory of Matrix 矩阵 by JJ Sylvester (Statistical Math private Tutor of Florence Nightingale, the world’s first nurse) closely linked to the application of Determinant.

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
eg.
宇环 (universal surrounding)
宙时 (instant timing)

https://www.asiatimes.com/2019/08/opinion/why-quantum-physics-needs-asian-philosophy/

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

https://link.medium.com/z5LaoL0tdZ

丘成桐:数理与人文(原创完整版)

【丘成桐:数理与人文(原创完整版)】
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Mind over Matter : Bruce Lipton

Key Points:

1. The Foundation of Science has changed since Newtonian Era, except Biology / Medicine and Psychology which don’t keep up:

Math => Fractal Math

Newton Physics (Matter)=> Quantum Physics (Energy)

(Organic) Chemistry => Electro-Chemistry

Energy = Field = 气 Chi / Qi

3. Environment -> Mind -> Perception -> Genetic Control

4. Consciousness (5%), Subconscious-ness (95%)

Examples:

  • Driving car: inexperienced driver (consciousness), experienced driver (subconscious).
  • Dating (conscious behavior)

Note: The successful Deeplearning AI is using the concept of Perception called “Perceptrons” by analysing the Environmental (BIG DATA) pattern with the help of Mathematics (Calculus : Gradient Descent, Statistics : Bayesian Probability, Algebraic Topology, etc).

英国留学生带你回忆被数学支配的恐惧

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

Pure Mathematicians versus Applied Mathematicians

“A pure mathematician, when stuck on the problem under study, often decides to narrow the problem further and so avoid the obstruction. An applied mathematician interprets being stuck as an indication that it is time to learn more mathematics and find better tools” 

Distinguished differential geometer Eugenio Calabi

Ref: https://www.quantamagazine.org/20151203-big-datas-mathematical-mysteries/

A Day in the Mathematician’s Life 

“Mathematician is a person who turns coffee into Theorems” – Erdös Paul , the Hungarian Mathematician who had no home, no wife, travelled around the world to visit his math collaborators proving 1000+ theorems. 数学界的”丐帮洪七公”! 他是华罗庚的好友, 和 陈省身 同年得 数学Wolf Prize.

​https://anthonybonato.com/2016/09/20/a-day-in-the-life-of-a-mathematician/?from=timeline&isappinstalled=0

Ultimate Mathematician

引用巴拿赫Banach的名言,体现举一反三的境界:

A mathematician is a person who can find analogies between theorems;

A better mathematician is one who can see analogies between proofs

and

The best mathematician can notice analogies between theories.

One can imagine that the ultimate mathematician is one who can see analogies between analogies.

Algorithm with Life, Love, Happiness

Applications of Algorithm:

1) Economic : “Stable Marriage” Algorithm (2012 Nobel Economic)
◇ Boston Medical Students Matching Hospitals
◇ Schools & Students’ Choice
◇ Racial Segregation Housing Policy

2) Evolution: Genetics “Mixability” thru the algorithm “Sex” (mix of genes past down from father and mother to their children).

“God did not create species. He designed the Algorithm for creating them.”

3) Multiplicative Weights Update Algorithm: choose the best expert (C1, C2, …Cj…)

Cj = Cj (1 + 0.00001*$)
Weight = 0.000001
$ = profit or loss made by the Financial expert

Auspicious Year 2016

2016很神奇

2016=168+168+168+168+168+168+168+168+168+168+168+168

总共12个168,祝大家2016年每个月都一路发!

2016=666+666+666+6+6+6
祝大家2016年顺心隨意、顺风顺水、六六大顺!

2016=888+888+88+88+8+8+8+8+8+8+8+8,
祝大家2016年事业兴旺发达!发发发!

2016 = 999+999+9+9
祝大家在2016年幸福的日子长长久久!
恭祝2016年新年快乐,身体健康,阖家欢乐,实现梦想

Arrow’s Theorem

The Axioms for a fair voting system (eg. Political Election) :

1. Non-dictatorship:
◇ Outcome decided not by one ‘dictator’, but more than one person.

2. Unanimity
◇ If everyone votes that A is better than B, then A will be ranked higher than B in the final result.

3. Independence of irrelevant alternatives
◇ The ranking of A relative to B should not be affected by someone changing their mind about C.

Arrow’s Theorem says that if there are more than 2 people (parties) to vote for, then there is no fair voting system.

Note:
Most democratic voting systems violate the 3rd axiom (independence of irrelevant alternatives).

Truth = Knowledge + Understanding + Believe

image

K = Knowledge (comes from Proving)
U = Understanding
B = Believe

U is the mediator between K & B

Truth = KUB
Eg. Mathematics:
1 + 1 = 2

KB = Know & believe but do not understand.
Eg. Diseases like cancer, Chinese Accupuncture, …

B: Believe but do not understand or know.
Eg. Axioms, Religion

K : Know but do not understand or believe.
Eg.

KU : Know, Understand but do not believe.
Eg.

U: Understand but do not know or believe.
Eg. ?

UB: Understand, believe but do not know.
Eg. [Impossible]

张益唐:我的数学人生

[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。

Q&A:
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。

Life Algebra

image

How to solve this ‘Life’ Algebra ?

The simultaneous inequality equations with 3 unknowns {e, t, m}, resp. for energy, time, money.

It has no solution but we can get the BEST approximation :
Retire after 55 before 60, then you get optimized {e, t, m} — still have good energy (e) with plenty of time (t) and sufficient pension money (m) in CPF & investment saving.

Beyond 60 if continuing to work, the solution of {e, t, m} -> {0, 0, 0}.

Note: After death (e = t = 0), money is not yours (m = 0).

What is a Measure? (Measure Theory)

A person can have ‘Measure’ in Internet’s Big Data sense:
A man’s Self = Measure = ∪ {body, psychic, clothes, house, wife, children, ancestors, friends, reputation, job, car, bank-account, websites visited, online buying pattern, hobbies, interests, lifestyle, …}

You become your algorithmic self: identity and identification shifted to an entirely digital (therefore measurable) plane.

Google, Amazon, etc … collect your quantified “Measure”. Not only they amass large amount of data about you, but also use algorithms to make sense of these data.

Your ‘Measure’ is a big business in the age of Big Data, or ‘DT’ (Data Technology) Age — as coined by Jack Ma of Alibaba.com.

image

Singapore Maths Tuition

In layman’s terms, “measures” are functions that are intended to represent ideas of length, area, mass, etc. The inputs for the measure functions would be sets, and the output would be a real value, possibly including infinity.

It would be desirable to attach the value 0 to the empty set $latex emptyset$ and measures should be additive over disjoint sets in X.

Definition (from Bartle): A measure is an extended real-valued function $latex mu$ defined on a $latex sigma$-algebra X of subsets of X such that
(i) $latex mu (emptyset)=0$
(ii) $latex mu (E) geq 0$ for all $latex Ein mathbf{X}$
(iii) $latex mu$ is countably additive in the sense that if $latex (E_n)$ is any disjoint sequence ($latex E_n cap E_m =emptyset text{if }nneq m$) of sets in X, then

$latex displaystyle mu(bigcup_{n=1}^infty E_n )=sum_{n=1}^infty mu (E_n)$.

If a measure does not take on $latex +infty$, we say…

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Crows Are Problem Solvers

Are crows the ultimate problem solvers?

Wild crows inhabiting the city use it to their advantage:

Scientists have also discovered crows can count up to 3. Weren’t human primitives so too ? In Chinese, English, French, etc, anything up to 3 is equivalent to many.

Crows are ugly black birds, found commonly in the hawker centers here scrounging left over food on the tables or rubbish dumps. They are good children with strong filial piety, bringing food back to feed old parents. Below is a Tang dynasty poem by Bai Juyi 白居易 (800AD) praising the crows:

image

Music and Mathematics are Apolitical

This “Butterfly Lovers Violin Concerto” (梁山伯与祝英台: 梁祝 小提琴协奏曲 ) composed 50 years ago by 2 Chinese music students, now played so lovely by a Japanese lady violinist: 诹访内晶子 (Akiko Suwanai), who is the current user of the violin ”Dolphin”, one of the top 3 violins in the world made by Antonio Stradivarius

Only in the kingdom of Music (the other one is Mathematics) where human political hatred does not exist between countries due to past wars: Japan and China, Germany and the Allied Nations, … Just only yesterday China President Xi and Japan PM Abe both showed awkward “poker face” hand-shake at the APEC Beijing meeting; contrast to the 20th century’s greatest mathematician David Hilbert from Nazi Germany was welcome  in America to chair  the inauguration of the International Congress of Mathematics.


If more students love Math and Music, the world of tomorrow will be more peaceful.

Watch 諏訪內晶子 -《梁祝小提琴協奏曲》   Butterfly Lovers Violin Concerto. Notice her 1714 ‘magical’ violin is the treasure gift “Dolphin Stradivarius”, once owned and played by the virtuoso Jascha Heifetz (1901–1987).

Notes:
The legendary love story is the Chinese version of “Romeo & Juliet“:

http://en.m.wikipedia.org/wiki/Butterfly_Lovers’_Violin_Concerto

Interview with Akiko Suwanai

The history of the 2 composers (1958)

Math and Music

image

If you want your kids to grow up with Math talent, start young in Music, be it playing a simple drum or flute, later then at age 4 or 5 progressing to piano or violin, along the way pick up musical theory…

Notice that great mathematicians (or Physicists the close cousins of Math) are often music talented, but the converse not true! Einstein performed violin with an orchestra formed by a group of Nobel Prize Physicists; never heard Mozart or Langlang (朗朗) or any great musicians proved any Math Conjectures.

Music trains the abstract thinking mind, good for mastering the most abstract subjects like Abstract Algebra or ‘Abstract’ Physics (Quantum Physics).

image

http://www.thecrimson.com/article/1988/11/30/music-math-a-common-equation/?page=1

Calculus and Novels

Tolstoy’s “War and Peace”:

“Only by taking an infinitesimal small unit for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.”

If you understand Calculus, you will understand Tolstoy’s masterpiece.

太极 Taiji = 1 = 2^0

0 = 无极 (Wuji)
1 = 太极 (Taiji)

“无极生太极, 太极生两仪, 两仪生四象, 四象生八卦”

2^0 = 1
2^1 = 2 (两仪)
2^2 = 4 (四象)

2^3 = 8 (八卦)
2^4 = 64 (重卦/六十四卦)

注: 《周礼》称八卦为”经卦”, 六十四卦为”别卦”。

Leibniz 在1703年4月收到清朝康熙皇帝的数学老师法国传教士白晉 (Joachim Bouvet) 的信, 第一次看到“伏羲64卦主观方位图”, 发现阴爻(- -)是0, 阳爻 (一)是1。