The Map of Mathematics

Show to your schooling children why they need to study Maths – the Queen of all Sciences – which pushes the frontier of human evolution in last 3,000 years. Maths is always invented few centuries or decades before it becomes useful. For examples:  Complex numbers invented accidentally by the 16th century Italian Mathematicians for solving polynomial equation of 3rd degree, became useful in Physics Electrical and Magnetic Fields (19 CE) ; Invention of Analytic Geometry (17 CE) allowed Newton to trace the earth-sun orbit; Calculus propelled Physics and Physical Chemistry; Leibniz’s Binary Math (18 CE) discovery applied in Computing (20 CE)…

Latest Examples

1. Topology was invented in 1900 by French PolyMath Henri Poincaré, today applied in Big Data, AI…

2. His PhD student invented “Derivatives” Partial Differentiation, today applied in Commodity Trading, Stock Trading, Financial Derivatives… with Black-Sholes formula. 1998 USA Sub-Prime Crisis due to the misuse and lack of understanding of its limitation (“fat tail” ).

3. Mathematician SS Chern 陈省身and Nobel Physicist Yang Zhen-Ning 杨振宁were working independently in the USA for 40 years, Chern on Differential Geometry, Yang on Yang-Mill Equation (one the 7 unsolved Math Problems in 21st century). Through a common friend the hedge fund billionaire James Simons – Chern’s former PhD Math student and university colleague of Yang – they realised that the Math “Fiber Bundles” (纤维丛) invented by Chern 30 years earlier could apply in Yang’s Physics (Gauge Theory).

Does Abstract Math belong to Elementary Math ? 

The answer is : “Yes” but with some exceptions.

Most pedagogy mistake made in Abstract Algebra teaching is in the wrong order (by historical chronological sequence of discovery):

[X] Group -> Ring -> Field 

It would be better, conceptual wise, to reverse the teaching order as:

Field -> Ring -> Group

or better still as (the author thinks):

Ring -> Field -> Group

  • Reason 1: Ring is the Integers, most familiar to 8~ 10-year-old kids in primary school arithmetic class involving only 3 operations: ” + – x”.
  • Reason 2: Field is the Real numbers familiar in calculators involving 4 operations: ” + – × ÷”, 1 extra division operation than Ring.
  • Reason 3: Group is “Symmetry”, although mistakenly viewed as ONLY 1 operation, but not as easily understandable like Ring and Field, because group operation can be non-numeric such as “rotation” of triangles, “permutation” of roots of equation, “composition” of functions, etc. The only familiar Group is (Z,+), ie Integers under ” +” operation.

Some features which separate Advanced Math from Elementary Math are:

  • Proof [1]
  • Infinity [2]
  • Abstract [3]
  • Non Visual [4]
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    Note [1]: “Proof” is, unfortunately, postponed from high-school Math to university level. This does not include the Euclidean Geometry axiomatic proof or Trigonometry Identity proof, which are still in Secondary school Elementary Math but less emphasized since the 1990s (unfortunately).

    Note [2]: However, some “potential” infinity still in Elementary math, such as 1/3 = 0.3333…only the “Cantor” Infinity of Real number, {\aleph_{0}, \aleph_{1}} etc are excluded.

    Note [3]: Some abstract Algebra like the axioms in Ring and Field  (but not Group) can be in Elementary Math to “prove” (as in [1]): eg. By distributive law
    (a + b).(a - b) = a.(a - b) + b.(a - b)
    (a + b).(a - b) = a^{2}- ab + ba - b^{2}
    By commutative law
    (a + b).(a - b) = a^{2}- ab + ab- b^{2}
    (a + b). (a - b) = a^{2} - b^{2}

    Note [4]: Geometry was a “Visual” Math in Euclidean Geometry since ancient Greek. By 17 CE, Fermat and Descartes introduced Algebra into Geometry as the Analytical Geometry, still visual in (x, y) coordinate graphs. 

    20 CE Klein proposed treating Geometry as Group Transformation of Symmetry.

    Abstract Algebra concept “Vector Space” with inner (aka dot) product is introduced into High School (Baccalaureate) Elementary Math – a fancy name in  “AFFINE GEOMETRY” (仿射几何 , see Video 31). 

    eg. Let vectors
    u = (x,y), v = (a, b)
     
    Translation:
    \boxed {u + v = (x,y) + (a, b) = (x+a, y+b)}
     
    Stretching by a factor { \lambda} (“scalar”):
    \boxed {\lambda.u = \lambda. (x,y) = (\lambda{x},  \lambda{y})}

    Distance (x,y) from origin: |(x,y)|
    \boxed {(x,y).(x,y) =x^{2}+ y^{2} = { |(x,y)|}^{2}}
     

    Angle { \theta} between 2 vectors {(x_{1},y_{1}), (x_{2},y_{2})} :

    \boxed { (x_{1},y_{1}).(x_{2},y_{2}) =| (x_{1},y_{1})|.| (x_{2},y_{2})| \cos \theta}

    Ref: 《Elements of Mathematics – From Euclid to Gödel》by John Stillwell (Princeton University Press, 2016) [NLB # 510.711]

    Trump’s Speaking Math Formula

    The lower in the score the better : Trump (4.1) beats Hillary (7.7) who beats Sanders (10.1)

    Trump defied most expectation from the world to win the 2017 President of the USA. His victory over the much highly educated Ivy-league Yale lawyer-trained Hilary Clinton who speaks sophisticated English is “SIMPLE English“: seldom more than 2-syllable words.

    1-syllable words mostly: eg.dead, die, point, harm,…

    2-syllable words to emphasize: eg. pro-blem, ser-vice, bed-lam (疯人院), root cause, …

    3-syllable words to repeat (seldom): eg. tre-men-dous

    His speech is of Grade-4 level, reaching out to most lower-class blue-collar workers who can resonate with him. That is a powerful political skill of reaching to the mass. Hilary Clinton’s strength of posh English is her ‘fatal’ weakness vis-a-vis connecting to the mass.

    In election time, it is common to see candidates who win the heart of voters by using the local dialects of the mass, never mind they are discouraged in schools or TV: Hokkien, Teochew, etc.

    Math Foundations

    MathFoundations (all videos): http://www.youtube.com/playlist?list=PL5A714C94D40392AB

    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 + -)


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    Video 31: Affine Geometry (仿射几何)
    image

    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


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    Video 106: What exactly is a Limit

    Math Education Evolution: From Function to Set to Category

    Interesting Math education evolves since 19th century.

    “Elementary Math from An Advanced Standpoint” (3 volumes) was proposed by German Göttingen School Felix Klein (19th century) :
    1)  Math teaching based on Function (graph) which is visible to students. This has influenced  all Secondary school Math worldwide.

    2) Geometry = Group

    After WW1, French felt being  behind the German school, the “Bourbaki” Ecole Normale Supérieure students rewrote all Math teachings – aka “Abstract Math” – based on the structure “Set” as the foundation to build further algebraic structures (group, ring, field, vector space…) and all Math.

    After WW2, the American prof MacLane & Eilenburg summarised all these Bourbaki structures into one super-structure: “Category” (范畴) with a “morphism” (aka ‘relation’) between them.

    Grothendieck proposed rewriting the Bourbaki Abstract Math from ‘Set’ to ‘Category’, but was rejected by the jealous Bourbaki founder Andre Weil.

    Category is still a graduate syllabus Math,  also called “Abstract Nonsense”! It is very useful in IT  Functional Programming for “Artificial Intelligence” – the next revolution in “Our Human Brain” !