Mathematics: The Next Generation

Historical Backgroud:

Math evolves since antiquity, from Babylon, Egypt 5,000 years ago, through Greek, China, India 3,000 years ago, then the Arabs in the 10th century taught the Renaissance Europeans the Hindu-Arabic numerals and Algebra, Math progressed at a condensed rapid pace ever since: complex numbers to solve cubic equations in 16th century Italy, followed by the 17 CE French Cartersian Analytical Geometry, Fermat’s Number Theory,…, finally by the 19 CE to solve quintic equations of degree 5 and above, a new type of Abstract Math was created by a French genius 19-year-old Evariste Galois in “Group Theory”. The “Modern Math” was born since, it quickly develops into over 4,000 sub-branches of Math, but the origin of Math is still the same eternal truth.

Math Education Flaw: 本末倒置 Put the cart before the horse.

Math has been taught wrongly since young, either is boring, or scary, or mechanically (calculating).

This lecture by Queen Mary College (U. London) Prof Cameron is one of the rare Mathematician changing that pedagogy. Math is a “Universal Language of Truths” with unambiguous, logical syntax which transcends over eternity.

I like the brilliant idea of making the rigorous Math foundation compulsory for all S.T.E.M. (Science, Technology, Engineering, Math) undergraduate students. Prof S.S. Chern 陈省身 (Wolf Prize) after retirement in Nankai University (南开大学, 天津, China) also made basic “Abstract Algebra” course compulsory for all Chinese S.T.E.M. undergraduates in 2000s.

The foundations Prof Cameron teaches are centered around 4 Math Objects:

1. SET 集合
– Set is the founding block of the 20th century Modern Math, hitherto introduced into the world’s university textbooks by the French “Bourbaki” school (André Weil et al) after WW1.

Note: The last “Bourbaki” grand master Grothendieck proposed to replace Set by Category. That will be the next century Math for future Artificial Intelligence Era, aka “The 4th Human Revolution”.

2. FUNCTION 函数
– A vision first proposed by the German Gottingen School’s greatest Math Educator Felix Klein, who said Functions can be visualised in graphs, so it is the best tool to learn mathematical abstractness.

3. NUMBERS
– The German mathematician Leopold Kronecker, who once wrote that “God made the integers; all else is the work of man.”

– The universe is composed of numbers in “NZQRC” (ie Natural numbers, Integers, Rationals, Reals, Complex numbers). After C (Complex), no more further split of new numbers. Why?

4. Proofs

Example 1: Proof by Contradiction, aka Reductio ad Absurdum (Euclid’s Proof on Infinitely Many Prime Numbers)

Challenge the proof: Why ?

Induction intuitively by:

Example 2: Proof by Logic

[Hint:]
By Reasoning (which is unconscious), most would get “2 & A” (wrong answer)

By Logic (using consciousness), then you can proof …
Test on all 3 Truth cases below in Truth Table:
p = front side
q = back side

Matteo Ricci – The 16th-Century Jesuit who brought Euclid to China

Matteo Ricci (利马窦) the Italian Jesuit Priest who came to China during the 16th Century  Ming Dynasty of WanLi 万历 Emperor, tried to evangelise the Imperial Court Mandarins with his impressive western Science knowledge and Mathematics. In1607 Ricci translated Euclid’s Masterpiece “The Elements” in Chinese with the Prime Minister Xu GuangQi (徐光啓), the later named Geometry as “几何”, a Chinese term still being used today. Paul Xu and few Scientist Officers were baptised by Ricci as the first Catholics in Chinese history. Ricci stopped the Euclid translation at Book 6 of 15 (Euclid’s 13 books + 2 add-on books from Ricci’s Jesuit professor Clavius), albeit the earnest plea from Xu to translate the remaining 9 books (likely because Ricci himself was not taught in Rome before he left for China), which were later translated by a British Alexander Wylie and a Chinese mathematician 李善兰 in 1857 of Qing Dynasty.

Ricci spent his remaining life in China because the Catholic Pope in Rome, influenced by the Macau-based Franciscans (then the opponent of Jesuits), prosecuted him for being ‘compromised’ by Chinese rituals of Ancestor Worship.

Now the current Pope Francis wants to befriend China, which broke diplomatic tie with Vatican in 1950s, intents to beatify Matteo Ricci : The first Saint ‘Mathematician’ !

Prime Numbers * +

Prime Numbers

God multiplies Primes: Fundamental Law of Arithmetic (Euclid):

Any number (n) can be UNIQUELY Prime Factorized: n = P1 x P2 x …Pn

e.g. 12= 2 x 2 x 3,
28=2 x 2 x7