# 20世纪初数学界的三国混战时代，最终被他一统天下】

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

# Logic & Math (Set Theory)

Cambridge Prof Peter Smith:

https://boingboing.net/2018/12/26/beyond-a-equals-a.html/amp?from=singlemessage

Philosophy using Math – that is cool!

Set Theory is Philosophy, see this “Set Proofing Technique” taught in French Baccalaureate high school but Cambridge GCE A level ignores :

Prove : A = B
You need to prove 2 ways:
A ⊂ B
and
B ⊂ A
=> A = B

In the Bible 《John 14:11》
Jesus said to his disciples:
“Believe me when I say that I am in the Father and the Father is in me”.

Proof: Father (God) = Jesus

“Father in Me” :
<=>
Father ⊂ Jesus
and
“I am in the Father” :
<=>
Jesus ⊂ Father
Hence,
Jesus = Father (God)
[QED. ]

Merry Xmas! 圣诞快乐！

# Cours Raisonnements (Logics) , Ensembles ( Sets), Applications (Mappings)

This is an excellent quick revision of the French Baccalaureat Math during the first month of French university. (Unfortunately common A-level Math syllabus lacks such rigourous Math foundation.)

Most non-rigourous high-school students / teachers abuse the use of :

“=> ” , “<=>” .

Prove by “Reductio ad Absurdum” 反证法 (Par l’absurde / By contradiction) is a clever mathematical logic :

$\boxed {(A => B) <=> (non B => non A)}$

Famous Examples: 1) Prove $\sqrt 2$ is irrational ; 2) There are infinite prime numbers (by Greek mathematician Euclid 3,000 years ago).

Example: Prove $\forall n \in {\mathbb{N}}^{*}, \frac {2n+1}{2 \sqrt {n(n+1)} } \geq 1$ … (*)

Assume the opposite of (*) is true:
$\forall n \in {\mathbb{N}}^{*}, \frac {2n+1}{2 \sqrt {n(n+1)} } < 1$
$\iff {2n+1} < 2 \sqrt {n(n+1)}$
$\iff (2n+1) ^{2} < 4.n(n+1)$
[Rigor: Square both sides, “<“ relation still kept since both sides are positive and Square is a strictly monotonous (increasing) function]

$\iff 1 < 0 , \text {(False!) }$
Hence, (*) is True : $\boxed {\forall n \in {\mathbb{N}}^{*},\frac {2n+1}{2 \sqrt {n(n+1)} } \geq 1}$

The young teacher showed the techniques of proving Functional mapping:
$E \to F$
$x \mapsto y = f (x)$

Caution: A Function from E to F always has ONE and ONLY ONE image in F.

I.) Surjective (On-to) – best understood in Chinese 满射 (Full Mapping).
By definition:
$\boxed { \forall y \in F, \exists x \in E, f (x) = y}$

To prove Surjective:
$\text {Let } y \in F$

$\text {find } \exists x \in E, f (x) = y$

He used an analogy of (the Set of) red Indians shooting (the Set of) bisons 野牛: ALL bisons are shot by arrows from 1 or more Indians. (Surjective shoot)

II.) Injective (1-to-1) 单射

By definition: $\boxed { \forall (x,x') \in E^{2}, x \neq x' \implies f (x) \neq f (x') }$

To prove Injective, more practical to prove by contradiction:

$\forall (x,x') \in E^{2}, \text { Suppose: } f (x) = f (x')$

prove: x = x’

III.) Bijective (On-to & 1-to-1) 双射
Definition:
$\boxed{\forall y \in F, \exists ! x\in E, f (x) = y }$

To prove Bijective,
$\text {Let } y\in F, \text {let } x\in E, f(x) = y$
$\iff \text {...}$
$\iff \text {...}$
$\iff x = g (y)$
$\iff \boxed { g = f^{-1}}$

My example: Membership cards are issued to ALL club members (Surjective or On-to), and every member has one unique membership card identity number (1-to-1 or Injective), thus

“Cards – Members” mapping is Bijective.

(My Remark): If the mappings f and g are both surjective (满射), then

the composed mapping f(g) is also 满 (满) 射 = 满射 surjective ! (Trivial). [#]

Note [#]: “Abstract” Math concepts expressed in rich Chinese characters are more intuitive than the esoteric “anglo-franco-greco-germanic” terminologies. Some good examples are: homo-/endo-/iso-/auto-/homeo-morphism (同态/自同态/同构/自同构/同胚), homology (同调), homotopy (同伦), matrix (矩阵), determinant (行列式), eigen-value/vector (特征 值 /向量), manifold (流形), simplicial (单纯) complex (复形), ideal (理想), topology (拓扑), monad (单子), monoid (么半群)…

No wonder André Weil (WW2 Modern Math French/USA “Bourbaki School” Founder) had remarked:

“One day the westerners will have to learn Math in Chinese.”

# Proof: “Any Statement Implies Itself”

Let p = any statement

“Any Statement Implies Itself”
written in logic:
(p => p)

Example: “I am Me.”

Prove: (p => p)
Proof :
(p => ((p => p) => p)) =>
((p => (p => p)) => (p => p))

p => ((p =>p) => p)

(p => (p => p)) => (p => p)

p => (p => p)

p => p [QED]

# 白马非马

Let 马 = H = {w, b, r, y …}
w : 白马
b : 黑马
r ：红马
y：黄马

Let 白马 = W = {w}

To prove:
H = W
We must prove:
(1)H ⊂ W and
(2)H ⊃ W

From definition we know:
(2) fulfilled because
$w \in H \supset W$
But, (1) not true since
$H \nsubseteq W$

$\implies H \neq W$

[QED]

# Solution Birthday

Be careful of the sequence of dialogues which lead to correct sequence of eliminations.
Dialogue 1: (by red color)
Ben says: I do not  know which month (M), also sure that  Mark does not know which date (N)
=> they are both confused by duplicate M & N
=> eliminate unique dates N: 7/6, 1/12
Ben infers that not in M = 6, 12
=> eliminate 4/6 , 1/12, 8/12 (duplicate Month M)
Dialogue 2: (by Blue color)
Mark says: Initially I don’t know, now I know (N)
=> eliminate duplicate dates N : 5/3, 5/9
Dialogue 3: (by yellow)
Ben says: now I also know (which month M)
=> eliminate duplicate month M: 4/3, 8/3
[QED]

# Logic: Monkey’s Dilemma

On the way to the West, the team came to a Y-junction, either go Left or Right.
The Monkey King (孙悟空 ) went out reconnaissance, only came back few days later.
The Pig (猪八戒) suspects this fellow is a monster in disguise of the monkey, may not tell the truth.
The smart Monk (三藏) asks only 2 questions to detect his identity :

Q1. Is ‘Left’ direction correct ?
Q2. If I ask you Q1, you would say “Yes”, would-you?

The true Monkey will answer “Yes/Yes” or “No/No”, regardless the correct direction.
Why ?

Case 1. If the Correct direction =Left

Monkey: (Q1, Q2) = (Y,Y)
Monster (tell lie) = (N,Y)

Case 2. If the Correct direction =Right

Monkey : =(N,N)
Monster (tell lie) =(Y, N)

# Logic: Pascal Wager

Pascal Wager:

1. We can choose to believe God exists, or we can choose not to so believe.
2. If we reject God and act accordingly, we risk everlasting agony and torment if He does exist (Type I error in Statistics lingo) but enjoy fleeting earthly delights if He doesn’t exist.
3. If we accept God and act accordingly, we risk little if He doesn’t exist (Type II error) but enjoy endless heavenly bliss if He does exist.
4. It’s in our self-interest to accept God’s existence.
5. Therefore God exists!

Mathematical Proof:
Pascal assumed
Probability of God exists = p
Probability God doesn’t exist = 1-p

You lead 2 lives, either Worldly (世俗) or Piously （虔,诚) , you get rewards X, Y, infinity or Z, as shown in table below.
In Worldly Life, the Expectation in probability is
Ew = p.X + (1-p).Y
In pious life, the Expectation is
Ep = p.∞  + (1-p).Z
Regardless how big the rewards in X (negative, punishment by God) ,Y, Z (self-sacrifice, negative reward)
Ep is infinitely bigger than Ew.
(Ew is good only if p= 0)
Conclusion: no matter what, pious life is better.