直观 数学 Intuition in Abstract Math

Can Abstract Math be intuitive, ie understood with concrete examples from daily life objects and phenomena ?

Yes! and Abstract Math should be taught by intuitive way!

1. 直观 线性空间 : Intuition in Linear Space

(Part I & II) 矩阵 (Matrix), 线性变换 (Linear Transformation)

http://m.blog.csdn.net/myan/article/details/647511

(Part III)

http://m.blog.csdn.net/myan/article/details/1865397

Animation: English (Chinese subtitles)

http://m.bilibili.com/video/av6731067.html

2. 直观 群论 (Intuition in Group Theory)

https://www.zhihu.com/question/23091609

https://www.zhihu.com/question/23091609/answer/127659716

Advertisements

What is Motif (Motive 目的)

Below is an excellent intuitive explanation (in Chinese) of the abstract concept Motif by Grothendieck:

Brief SummaryMotif is the source of all “beautiful things” expressed in different forms.

Example : God created Natural Numbers (N), we express N in different forms: Binary (0, 1), Decimal (0, 1, 2 …9), Hexadecimal (0,1, 2…9, a, b, c, …f), etc. However, the “Motif” behind these forms is they all follow for (+, *) operations the same TWO Laws : 1) Commutative; 2) Distributive.

Similarly, in Algebraic Geometry applying the cohomology from Algebraic Topology: étale cohomology, crystalline cohomology, de Rham cohomology are the different forms (~ Binary, Decimal, Hexadecimal), factored through the common “Motif” of the Universal cohomology (~N).

[My Analogy in IT Language]:
Motif is like Interface or Generic, it spells out only the specification, leaving out the implementation (method) in actual classes / functions.

[怎麼理解代數幾何概念 motive?]

https://www.zhihu.com/question/20518518/answer/105460719

Ref: Alain Connes [Paragraph : Motif in a Nutshell]

https://link.zhihu.com/?target=http%3A//www.alainconnes.org/docs/bookwebfinal.pdf

Univalent Foundation – Computer Proof of All Maths

The scary complex field of Math worried the mathematicians who would prove a theorem relying on the previous theorems assumed proven correct by other mathematicians.

A sad example was Zhang YiTang (1955 – ) who prepared his PhD Thesis based on a previous “flawed” Theorem proved by none other than his PhD Advisor Prof Mok in Purdue University. Unfortunately his Thesis was found wrong, and the tragic happened to Zhang as he had revealed the mistake of his PhD advisor who insisted his (Mok’s) Theorem was correct. As a result Zhang failed the 7-year PhD course without any teaching job recommendation letter from his angry advisor. He ended with a Subway Sandwich Kitchen job offered by his Chinese friend, sleeping in another Chinese music conductor’s house on a sofa. It was there he spent another 7 years thinking on Math, finally an Eureka breakthrough one 2013 morning in the backyard wild forest – the proof of the famous “70 million Prime Gap Conjecture”!

Univalent Foundation was born out of the same requirement by the (late) mathematician Vladimir Voevodsky (1966 – 2017) [#] – Use computer to prove Mathematics !

https://www.quantamagazine.org/univalent-foundations-redefines-mathematics-20150519/

Note : [#] Vladimir Voevodsky

Visionary mathematician Vladimir Voevodsky

A self-study Russian mathematician, kicked out 3 times in high schools, expelled from Moscow University, all because he did not attend classes, preferred to self-study in a broader scope for his curiosity, at his own faster speed than the rigid curriculum and boring test-and-exams regime in classrooms.

He did the PhD in Harvard by invitation even he did not have a Bachelor degree, and he barely passed the Harvard’s QE (Qualifying Exams) in Algebraic Geometry, a field in which he made a revolutionary discovery few years later, and for which he was awarded the highest honor : Fields Medal.

This is the typical trait of the geniuses like Evariste Galois, Albert Einstein, Ramanujian, Hua Luogeng (华罗庚), Zhang YiTang (张益唐 – proved “70m Twin Prime Gap”) [#] , Chen Jingrun (陈景润, proved Goldbach Conjecture “1+2”) [##]… with self-motivated curiosity in their field of passion, with reading from the Masters’ works by themselves, PLUS thinking on a problem over many years with perseverance, they finally made great discoveries !

Keywords:

  1. Topology 拓扑
  2. Algebraic Geometry 代数几何: Geometry and Algebraic Equations
  3. Motif: Grothendieck
  4. Homotopy Theory 同伦

[Full Article] :
https://www.quantamagazine.org/visionary-mathematician-vladimir-voevodsky-dies-at-51-20171011/

Notes:

[#] : https://tomcircle.wordpress.com/2014/03/01/gaps-between-primes-zhang-yitang/

[##] : https://tomcircle.wordpress.com/2015/07/24/goldbach-conjecture-theorem-chen/

Category Theory – Purest of pure mathematical disciplines may also be a cornerstone of applied solutions in computational science

There exists in almost all Universities a clear division between pure and applied mathematics. A friendly (and sometimes not so friendly) rivalry exists between both sides of the divide, with separate conferences, separate journals and in many cases even a whole separate language. Category Theory was seen as such an abstract area of research that even pure mathematicians started to refer to it as “abstract nonsense“, and until the mid 1980’s almost all category theorists occupied a place hidden somewhere up above the ‘cloud level’ in the highest reaches of the peaks that defined “pure” maths.

By the mid 1990’s and then by the turn of the millenium, a whole world of computer programmers were learning basic category theory as part of their induction into functional programming. The best known product of these efforts is the Haskell language, but even in the past 7 or 8 yrs, workshops on category theory for computer programmers of all types have flourished and proliferated. It is almost as if there are two separate communities masquerading as one – mathematical category theory and computer programming category theory – and never the twain would meet. Or so it seemed, until now.

https://www.linkedin.com/pulse/category-theory-classic-dichotomy-purest-pure-may-also-khan-ksg/