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”.

– 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.

– 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
– reading and debugging 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

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

By Logic (using consciousness), then you can proof …
Correct Answer: 2 and B
Test on all 3 Truth cases below in Truth Table:
p = front side
q = back side


Recommended The Best Book on Abstract Algebra by Prof. Peter J. Cameron

Math Foundations

MathFoundations (all videos):

This is the Felix Klein’s Vision of Elementary Math from an Advanced Standpoint“.

All the Math we learn are taught as such by teachers and professors, but why so? what are the foundations ? These 200 videos answer them !

Good for students to appreciate Math and, hopefully, they will love the Math subject after viewing most of these 200 great videos.

Video 1: Natural Number This should be taught in kindergartens to 3-year-old kids.

◇ What is number ? (strings of 1s),
◇ Equal, bigger, smaller concepts are “pairing up” (1-to-1 mapping) two strings of 1s.
◇ Don’t teach the kids how to write first 12345…, without prior building these mathematical foundational concepts.

Video 2: Arithemetic
Distributive Law:
k x (m+n) = (k x m) + (k x n) = k x m + k x n

Convention: ‘x’ (or its reverse ‘÷’) takes precedent over ‘+’ (or its reverse ‘-‘).
(先 x ÷ 后 + -)

Video 31: Affine Geometry (仿射几何)

Video 32: Geometry Education in Primary Schools
◇ Use Grid paper to visualize concrete objects (line, square, triangle, circle…)
◇ Arithmetic is closely linked to Geometry which has been ignored in schools in the last 3 decades since the introduction of Modern Math (vector algebra).

Video 34: Signed Area of Polygon

Video 106: What exactly is a Limit