Six books that have shaped my mathematical worldview

First book: The 17th CE French Mathematician-cum-Judge Fermat is the name of my alma-mater French Engineering (undergraduate) University.《birth-of-a-theorem-a-mathematical-adventure》/

费马大定理 Fermat’s Last Theorem

费马大定理 Fermat’s Last Theorem (FLT): 17世纪业余数学家法国大法官费马开的一个”玩笑”, 推动350年来近代数学(Modern Mathematics)的突飞猛进。

1977秋 ~1979秋 笔者在法国-图卢斯(Toulouse, Southern France, Airbus 产地)费马学院 (College Fermat, aka Lycée Pierre de Fermat: Classe Préparatoire, 178th Batch)读两年的大学近代数学 (Mathématiques Supérieures et Spéciales), 尝过一生读书的”地狱”生活, 严谨(Mathematical Rigor)的思考训练, 像地鼠般(法国人戏称taupe)不见天日, 废寝忘食的煎熬。 当年对数学的恐惧, 终生牢牢铭记在心; 30年后”由惧转爱”, 数学竟然成为半退休后的业余嗜好, 享受数学的美 — 也是造物者宇宙天地的美!


FLT 350年数学长征英雄人物:
1. Fermat (费马 1601@ Toulouse, France)
2. Galois (伽罗瓦): Group Theory (群论)
3. Gauss (高斯)
4. Cauchy (柯西) Lamé (拉梅) Kummer (库马)
5. Solphie Germain
6. Euler (欧拉)
7. Taniyama (谷山丰), Shimura (志村五郎)

“数风流人物, 还看今朝”集大成者 :
8. Andrew Wiles (怀尔斯) 证明 (1994 -1995)”盒外思路” (Think Out of The Box): The Great Moment of 1994 Proof (YouTube)

\boxed {(1) = (2) = (3) }
(1). Elliptic Curve (椭圆曲线)
(2). Modular Form (模形式)
(3). Fermat’s Last Theorem (费马大定理)

费马大法官品尚清高, 讨厌政界官僚逢场作戏的应酬, 工余爱躲在家里玩数学, 然后写信和好友(巴斯卡 Pascal, 笛卡儿 Descartes,…)讨论, 无心中发明了物理(Optics)定律, 或然率 (Probability – 和Pascal合作), 解析几何 (Analytical Geometry – 和Descartes合作)…尤其他是近代数论(Number Theory)的开山鼻祖 (他的另一个Fermat’s Little Theorem今天用在电脑密码RSA Encryption)。
他偶然读到3,000年前希腊数学家Diaophantine的书 (10世纪阿拉伯人保存, 16世纪拉丁文翻译自阿拉伯文)。他心血来潮, 在书眉写道: “我找到一个漂亮的证明这题Diaophantine Equation, 但此书旁地方太小, 不能写下”。 他死后, 儿子整理遗作而发现此书, 就成为350年来的数学疑案。


Great Math Popular Books


日本数学科普作家写的好书。深入浅出, 适合中学生读最高深数学。

結城 浩 Hiroshi Yuki (1963 -) is a Japanese Math Popular Book Writer for Secondary and High School students. In the “Galois Theory” (Chapter 10) he boldly attempted to explain to them such complicated concepts: Quotient Group, Field Extension, Group Order, Normal Sub-Group, Solvable Group …

Evariste Galois‘s genius is he built a “bridge” between Field (域/体) and Group (群) – both new concepts invented by him. The “bridge” is called Galois Group, or by Emile Artin the “Group Automorphism”. He transferred the difficult problem of solving complicated 4 -ops (+-×/) Field (coefficients) to the single-op (permutation of roots) Group.

Galois Group is the ultimate TRUTH of all Math — Fermat’s Last Theorem, and any advanced Math, will use Galois Group or Field, to solve. Prof C.N. Yang 杨振宁 Nobel-Prize Physics discovery was based on Group Theory.

Evariste Galois was a French Math genius, died at 21 in a duel during French Revolution. He is the ‘Father’ of Modern Algebra. Failed 2 years in Ecole Polytechnique CONCOURS Entrance Exams, then kicked out by Ecole Normale Supérieure, his Math was not understood by all the 19th century World’s greatest Math Masters : Gauss, Fourier, Poisson, Cauchy…

“Galois Theory” — the ultimate Math “葵花宝典” (a.k.a. “Kongfu Bible“) — is only taught in the Math Honors Undergraduate or Masters degree Course.

自己学习《Galois Theory》(Page 365):
— “高中会教这种困难的数学吗 ?”
— “…我觉得比起給高中老师教, 不如自己好好学吧。”
— ” 重要的是自己学习。”

The Gap of Today’s Math Education: Rigor 严谨

This professor criticized the lack of rigor in today’s math education, in particular, there exists universally a prevalent ‘ambiguous’ gap between high school and undergraduate math education.

I admire his great insight which is obvious to those postwar baby boomer generation.

I remember I was the last Singapore batch or so (early 1970s) taking the full Euclidean Geometry course at 15 years old, and strangely in that year of Secondary 3 Math (equivalent to 3ème in Baccalaureate) my (Chinese) school had 2 separate math teachers for Geometry and Elementary/Additional (E./A.) Math.

Guess what ? the Geometry teacher was an Art teacher. It turned out it was a blessing in disguise, as my class of average Math students who hated E./A. Maths all scored 90% distinctions in Geometry. We did not treat Geometry like the other boring maths. The lady Art teacher started on the first day from Euclid’s 5 axioms, one by one as the days went by through the year, she derived each theorem with rigourous proof based on axioms and previous proven theorems. When a tough problem seemed too difficult, we were amazed by her astute ‘dotted line’ (补助线) technique, and voilà, we suddenly saw the relationship of the angles or triangles.

(This ‘dotted line’ once was used by my university French professor in Physics, he was very proud of finding a similar triangle relationship in the trajectory of a satellite revolving the earth – he was ‘awed’ by the French classmates who skipped the traditional Euclidean Geometry in lieu of the modern abstract Affine Geometry).

It may be a wise decision that Euclidean Geometry, now excluded from the main stream Math syllabus, be embedded in Art Class. Afterall, the great artists like Leonardo da Vinci applied Perspective Geometry in their art.

The professor challenged the development of these Math coursewares from the “First Principles”, a la Euclid’s Rigor: Group Theory, Combinatorics, etc.


Don’t forget history tells us that Mathematics originated from the Greek Geometry (Plato Math School “Lycée” : “Let no one who ignores Geometry enters this (school) gate. “), Euclid’s rigor influenced us for 3,000 years until 1900. The Modern Math conjectures like The Fermat’s Last Theorem from Number Theory was finally solved after 380 years by Geometry (Elliptic Curve from Shimura-Taniyama-Weil Theorem) in 1994.

The most difficult Math Challenge of today is the Langrands Program in unifying all Mathematics, with the hope of using the André Weil’s “Rosetta Stone” to ‘translate’ 3 distinct Math languages: Algebra, Analysis via the bridge of Geometry (Alexander Grothendieck’s Sheaf).

From the past IMO (International Math Olympiad) champion countries, notice that the Russian and Chinese school of Math do better than the western countries, because out of 6 IMO questions there are often 2 in Euclidean Geometry – a weakness in western school math syllabus (including Singapore which models on UK) for dropping Euclidean Geometry in schools, but the Russian and Chinese wisely keep it.

Shimura-Taniyama-Weil Conjecture (Modularity Theorem)

Shimura and Taniyama are two Japanese mathematicians first put up the conjecture in 1955, later the French mathematician André Weil re-discovered it in 1967.

The British Andrew Wiles proved the conjecture and used this theorem to prove the 380-year-old Fermat’s Last Theorem (FLT) in 1994.

It is concerning the study of these strange curves called Elliptic Curve with 2 variables cubic equation:

\boxed {y^{2} + y = x^{3} - x^{2} }  (I)

There are many solutions in integers N, real R or complex C numbers, but solutions in modulo N hide the most beautiful gem in Mathematics.

For modulo 5, the above equation has 4 solutions:
(x, y) = (0, 0)
(x, y) = (0, 4)
(x, y) = (1, 0)
(x, y) = (1, 4)

Note 1: the last solution when y=4,
Left-side = 16 + 4 = 20 = 4×5 = 0 (mod 5).
Right-side = 1-1= 0 (mod 5).

Note 2: We call the equation (I) a “Curve over a finite field” since {0, 1,2,.. p-1} is a Field with finite p elements.

Mathematicians for some time have known that if N is a prime number (p), there will be roughly p solutions.

However, the most interesting number is a_{p} = the difference between p and the actual number of solutions.

For N = p = 5, the above equation has actually 4 solutions,
\boxed {a_{5} = 5 - 4 = 1}

Note: a_{p} can be positive or negative.

There is a ‘general rule’ (generating function) to predict a_{p}, and it is inspired from the ubiquitous Fibonacci numbers.

Definition of the Fibonacci sequence as a recurrence relation:
\boxed{ F_{n}= \begin{cases} 0, & \text{for }n=0\\ 1, & \text{for }n=1\\ F_{n-2} + F_{n-1} , & \text{for } n \geq { 2} \end{cases} }

Alternatively there is also a generating function for Fibonacci numbers:
q + q(q+q^{2})+  q(q+q^{2})^{2} + q(q+q^{2})^{3} + q(q+q^{2})^{4} + ...

Let’s expand it we get the infinite series:
q + q^{2} + 2q^{3} + 3q^{4 } + 5q^{5} +  8q^{6} + 13q^{7} +...

The above coefficients coincide with
Fibonacci sequence: {0, 1, 1, 2, 3, 5, 8, 13…}

In 1954, the genius German mathematician Martin Eichler took the cue from the above, discovered another generating function:

\boxed {q(1-q^{1})^{2} (1-q^{11})^{2} (1-q^{2})^{2} (1-q^{22})^{2}(1-q^{3})^{2} (1-q^{33})^{2}(1-q^{4})^{2} (1-q^{44})^{2} ... }  — (II)

Let’s expand it, we get:
q-2q^{2} -q^{3}+  2q^{4} + q^{5}+2 q^{6}-2q^{7} -2q^{9} -2q^{10}+ q^{11} -2q^{12}+ 4q^{13}
Let b_{m} denotes the coefficient of the term q^{m}:
b_{1} = 1, b_{2} = -2, b_{3} = -1, b_{4} = 2, b_{5} = 1, ...

Eichler discovered that for any prime p,
\boxed { b_{p} = a_{p}}

Check: b_{5} = 1 = a_{5}

The random numbers of solutions in the elliptic curve equation (I) lies on the generating function (II).

If we view q as a point inside a unit disc on the complex plane, there is a group of symmetries and the function (II) is invariant under this group. The function (II) is called a modular form.


The advanced generalisation of the Shimura-Taniyama-Weil Conjecture : we replace each cubic equation by a Representation of the Galois Group; and the modular form generalised by the generating function the “automorphic” function.:


1. The Shimura-Taniyama-Weil Conjecture is a special case of Langlands Program.

2. Weil’s “Rosetta stone”:
Number Theory -> Curves over Finite Fields -> Riemann Surfaces


Love and Math by Edward Frenkel


Click below for more free loan at the Singapore National Library Branches:

Probability by 2 Great Friends

Today Probability is a “money” Math, used in Actuarial Science, Derivatives (Options) in Black-Scholes Formula.

In the beginning it was “A Priori” Probability by Pascal (1623-1662), then Fermat (1601-1665) invented today’s “A Posteriori” Probability.

“A Priori” assumes every thing is naturally “like that”: eg. Each coin has 1/2 chance for head, 1/2 for tail. Each dice has 1/6 equal chance for each face (1-6).

“A Posteriori” by Fermat, then later the exile Protestant French mathematician De Moivre (who discovered Normal Distribution), is based on observation of “already happened” statistic data.

Cardano (1501-1576) born 150 years earlier than Pascal and Fermat, himself a weird genius in Medicine, Math and an addictive gambler, found the rule of + and x for chances (he did not know the name ‘Probability’ then ):

Addition + Rule: throw a dice, chance to get a “1 and 2” faces:
1/6 +1/6 = 2/6 = 1/3
(Correct: 1 & 2 out of other six faces)

Multiplication x Rule: throw two dices, chance to get a “1 followed by a 2 ” faces : 1/6 x 1/6 = 1/36
(Assume 1st throw does not affect or influence the 2nd throw: independent events)

Pascal never met Fermat personally, only through correspondences (like emails today), but Pascal regarded with respect Fermat as superior in Math than himself.

Fermat and Descartes were not so. Descartes openly criticized Fermat as a second class mathematician. Both independently discovered Analytical Geometry, but Descartes scored the credit in ‘Cartesian’ coordinates.

Fermat never published any books in Math. As a successful judge in Toulous, he spent his free time as an amateur mathematician, especially in Number Theory. He showed his Math discoveries to friends in letters but never provided proofs. Hence the Fermat’s Last Theorem made the world mathematicians after him (Gauss, Euler, Kummer, Sophie Germaine, Andrew Wiles…) busy for 380 years until 1994.

Fermat died 3 years after Pascal. Another Modern Mathematics was being born – Calculus – in UK by Newton and Germany by Leibniz. Probability was put in the back seat over-taken by Calculus.