The Eve Before Calculus : Fermat


“这个人”并不存在,但他却是20世纪最伟大的数学家之 – Bourbaki

Dieudoné 1930



Audio book ]: It all adds up – the story of people & Mathematics

数论的世界 Number Theory

【数学之美 | 数论的世界】
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The Math we learnt in High school is pre-19th Century up to Newtonian Calculus of 17CE.

The birth of Modern Math since 19CE till WW2 is the “Abstract Algebra” from French Revolution Galois “Group Theory” 群论.

After WW2 till now, Mathematics faces the crisis of “Truth”: whether its Foundation is correct.

3 schools of fight on the Fondation of Mathematics:

1. Russell (Logic with Types to fix the “Russell Paradox” in Set Theory Crisis)

2. Hilbert (Axiomatization of all Mathematics)

3. Brouwer (Intuitionism) against "排中律" (Law of the Excluded Middle)

Winner: Godel “The Incomplete Theorem” (不完备定律)

However, the by-product of these 3 school fights give rise to new Math discovery in Machine Proofing (2010s):

Homotopy Type Theory (HoTT) = Logic (Proof) + Type (Intuitionist) + Topology (Homotopy).

ie. Math Proof = Computer Program



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).

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张益唐 天才的野心: 《朗道-西格尔零点》猜想

【张益唐 天才的野心】

Dr. Zhang YiTang (1955 -) the “Subway Sandwich” mathematician who, before 50 was still a temporary hourly worker… until 2013 at 58 proved the “70-million Twin Prime Gap“.

He becomes overnight famous worldwide, now a tenure professor.

His next ambition is the “Landau-Siegel Zero Conjecture”, a weaker form of the Millenium Problem “Riemann Conjecture”.


Landau–Siegel zeros and zeros of the derivative of the Riemann zeta function




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数学最高奖沃尔夫数学奖的历史 Wolf Prize

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Fields Medal : why Age limit atv40

【因为他的私心,“数学界的诺奖”菲尔兹奖才有了 40 岁限制】
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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}}}}

13 classic mathematics books for lifelong learners

View at

Out of 13 must-read Popular Math books, our Singapore National Library NLB has 6 which I borrowed and read: eg. “Prime Obsession” , “What is Mathematics” , etc.

Popular math books are better than the boring textbooks (Axiom – Theorem – Proof – Exercise). They are motivational, more concrete instead of abstract, philosophical analogy with the Nature (afterall, the math ideas derived from the universe, eg. Pi, e, golden ratio, infinity, limit, …), plus the historical background in which these math ideas were first discovered, and the beauty of these inter-connected ideas such as the Euler’s Identity:

\boxed{e^{i. \pi} + 1 = 0}

Proof from the Book” – the name “Book” (God’s Theorem Proof Book) is coined by Paul Erdos the Hungarian ‘vagabond’ (homeless, single, no nationality) mathematician, who had proven 1000+ theorems (some co-operated with his students). He said he had peeped into God’s “Book” to discover these theorems.

My favorite Popular Math book which inspired me in 2005 to re-pick up the fearsome Abstract (aka Modern) Algebra is : “Unknown Quantity” by Prof John Derbyshire, avail at NLB.

Math before university is the “What and How“, whereas the University Math is the “Why” – after WW2 the French Bourbaki Reform in Math Education worldwide based on Set Theory, the post-war Philosophy Trend “The Structurism 结构主义” shaking the basic foundation of Math: Algebraic Structures. eg Group (群) , Ring (环) , Field (域) , Vector Space (向量空间) , Cateogy Theory (范畴论) .

The 70 years of WW1 & 2 taught the world Anarchism (无政府主义) was chaotic & disastrous to society, hence the more orderly “Structurism” Philosophy was born, influencing all Sciences: Chomsky Linguistics, Sociology, IT Structured Programming ‘Pascal’ , Anthropology 人类学, Abstract Algebraic Structure Math…

Mathematics (Mathematikos) = “Fond of Learning”

Μαθηματικος (Mathematikos) = Mathematics = Fond of Learning

Three simple A-level to undergraduate Mathematics a MUST for Data Science and AI:

  1. Linear Algebra (aka Matrix)
  2. Probability (Baeysian)
  3. Statistics
  4. etc


eFinancialCareers: The mathematics you must learn for a job in data science and AI.

How you pronounce German Mathematician’s Names ( and Physicists )

Germany produced many Mathematicians and Physicists who succeeded the 18 CE Newtonian England, and the 19CE Napoleonic France before WW2, after which the Americans (mostly the Jewish German immigrants) take over till now.

German names are difficult to pronounce for foreigners.










For due respect, please learn to pronounce their German name correctly.

My favorite Fermat Little Theorem with Pascal Triangle

Fermat Little Theorem: For any prime integer p, any integer m

\boxed {m^{p} \equiv m \mod p}

When m = 2,

\boxed{2^{p} \equiv 2 \mod p}

Note: 九章算数 Fermat Little Theorem (m=2)

Pascal Triangle (1653 AD France )= (杨辉三角 1238 AD – 1298 AD)

1 \: 1 \implies sum = 2 = 2^1 \equiv 2 \mod 1

1\: 2 \:1\implies sum = 4 = 2^2 \equiv 2 \mod 2 \;(\equiv 0 \mod 2)

1 \:3 \:3 \:1 \implies sum = 8= 2^3 \equiv 2 \mod 3

1 4 6 4 1 => sum = 16= 2^4 (4 is non-prime)

1 \:5 \:10\: 10\: 5\: 1 \implies sum = 32= 2^5 \equiv 2 \mod 5


Celebrates Mathematician Gottfried Wilhelm Leibniz’s 372nd Birthday

My favorite mathematician is German Leibniz, who co-invented Calculus with Newton.

Today we thank Leibniz for his elegant Calculus symbols:

\boxed {\frac{dy}{dx}}


Leibniz also invented 01 binary algebra, which he later found it was already in the 3,000-year-old Chinese “Yin-Yang” (阴阳 八卦), so impressed that he recommended to the most powerful western (French) king Louis XIV (14th) to use Chinese as the Universal Language of the world.

The rich Newton sued Leibniz for plagiarism of Calculus, until Leibniz died poor in bankruptcy, buried in a common unknown grave.

The war between Newton & Leibniz extended & lasted 100 years between UK Math Community and Continental Europe Math Community. As a result UK lost its math leadership after Newton, France (Lagrange, Fourier, Cauchy, Galois… ) followed by Germany (Felix Klein, Gauss, Hilbert, Riemann …) took over as the world center of math. After WW2 many German mathematicians (mostly Jewish eg. Noether, Gödel, Artin, …) fled to the USA which is now the Kingdom of Advanced Math.

How Mathematicians Think

Hadamard estimated that :

About 90% of mathematicians think visually, 10% think formally.

Usually, they think in steps:

  1. Get the right idea, often think vaguely about structural issues, leading to some kind of strategic vision;
  2. Tactics to implement it;
  3. Rewrite everything in formal terms to present a clean, logical story. (Gauss’s removal of ‘scaffolding’ – middle working steps)

Source: [NLB #510.922]

Seven Fields Medalists

The 7 Fields Medalists are:

2014 – Maryam Mirzakhani (1977-2017) – 1st lady Fields medalist

2010 – Cédric Villani (1973- )

2006 – Grigori Perelman (1966- ) – 1st declined the award

1998 – Andrew Wiles (1953- ) [silver plaque] – Fermat’s Last Theorem

1990 – Edward Witten (1951- ) – Physicist won Fields medal

1982 – Alain Connes (1947- ) – Quantum Theory

1966 – Alexander Grothendieck (1928-2014) – Hermit mathematician

Joseph Fourier is Still Transforming Science

Key Words: 250 years anniversary

  • Yesterdays: Fourier discovered Heat is a wave , Fourier Series, Fourier Transformation, Signal processing…
  • Today: IT imaging JPEG compression, Wavelets, 3G/4G Telecommunications, Gravitational waves …
  • Friends / bosses: Napoleon, Monge… Egypt Expedition with Napoleon Army.
  • Taught at the newly established Military Engineering University “Ecole Polytechnique”.
  • Scientific Research: Short period but intense.
  • Before Fourier died (he wrapped himself with thick blanket in hot summer), he was reviewing another young Math genius Evariste Galois’s paper on “Group Theory”.

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.