French Math is unique in treating these 3 conic curves: (ellipse, parabola, hyperbola), always starts from the first principle – a la the Cartesian Spirit “I think therefore I am” (我思故我在).
“Catersian” Analytical Geometry was co-invented by two 17CE French mathematicians René Déscartes and Pierre de Fermat.
Note: The “elliptic curve” is a powerful geometry tool used in Number Theory (proved the 350-year-old Fermat’s Last Theorem in 1994 by Andrew Wiles), also in the most advanced Encryption algorithm.
Cédric Villani (Field Medal 2010) the French mathematician becomes a deputé (equivalent to Member of Parliament) in President Emmanuel Macron’s new party “En Marche” consisting of 90% non politicians.
His new revolution in French Primary School Math Education is introducing “Singapore Math” : the 1960s Chinese Secondary School One Math (算术 Arithmetic) modified by the ex-Nantah (南大, now Nanyang Technological University) Prof Lee Peng Yee (李秉彝) with the Polya Problem Solving Method aided by visual Model diagrams.
France excels in Abstract (aka New / Modern / Bourbaki) Math but poor in Applied Math, while Asian countries (China, Singapore, Korea, Japan etc) are opposite. This demonstrates clearly in the Asian 15-year-old students scoring Top PISA Math Tests & Math Olympiad Championships (even in the USA teams), while the French young Mathematicians (< 40 years old) take 1/3 of the world’s Field Medals – “the Nobel Prize of Math”.
It is ideal to combine the Asian (applied) Math pedagogy for Primary school (from 7 to 13 years old) math and the French (abstract) Math (eg. Set, Group, Ring, Field, Vector Space, epsilon-delta Analysis…) in high-school math (from 15 to 18 years old).
The wise Cédric Villani is just doing this Ideal Combination of “East and West” for French Primary School Math Education .
Note: In the video (12:12 mins) Villani described his “Eureka discovery” inspiration at 4 am in Princeton Institute of Advanced Math. The same experience found by many mathematicians eg. Poincaré, Zhang Yitang, etc. Read Villani’s book where he called this “Strange Head Voice” (definitely not hallucination!!) experience as “The direct phone call from God “.
- Category and Functor are above the underlying algebraic structures (Set, Group, Ring, Vector Spaces, etc), study the relations between these structures.
- Early 19 CE mathematicians before “Category Theory” already knew there is 1:1 mapping between the Field Extension and Galois Group.
- Treat Structures and Relation between them (Functors) on equal footing.
Kotlin is a functional programming (FP) language which has the Category concept of co-Variance, contra-Variance. Java is object-oriented not FP, it has only Invariance.
Covariance and contravariance are terms that refer to the ability to use a less derived (less specific = CoVariance eg. Apple instead of Fruit) or more derived type (more specific = ContraVariance, eg. Gala instead of Apple) than originally specified.
Generic type parameters support covariance and contravariance to provide greater flexibility in assigning and using generic types.
Means that you can use only the type originally specified; so an invariant generic type parameter is neither covariant nor contravariant. (Java is always invariance.)
This is French “Abstract Algebra” in Math Superieure (Baccalauréat + 1 year) : rigorous and unique French way!
Others: Injective, Injective