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]
  •  

    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]

    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
    – reading and debugging proofs.

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

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    Challenge the proof: Why ?
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    Induction intuitively by:
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    Example 2: Proof by Logic
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    [Hint:]
    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

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

    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” !

    Elementary Math from an Advanced Standpoint

    Guesstimate Question  (no calculator allowed) –

    [Hint]: use Abstract Math “Homomorphism”  {f * +}

    What is 35823 x 23412 ?

    A) 845203402
    B) 838688076
    C) 812343296

    The mapping f defined by
    \forall  a_1, a_2, a_3 \in Z
    f(a_1a_2a_3) = a_1+ a_2+ a_3
    such that,

    \forall a, b \in Z,
    f (a * b) = f(a) * f(b)

    => f is an homomorphism.

    Examples:

    f (35823) = 3+5+8+2+3 = 21
    f (21)= 2+1= 3

    f (23412) = 2+3+4+1+2 =12
    f (12) =1+2 = 3

    f (35823 x 23412) = 3 x 3 = 9

    Verification  (B):
    f (838688076)= 8+3+8+6+8+8+0+7+6=54 ..5+4 =9 ☆

    Wrong :
    f(845203402)= 28…=10 = 1

    f (812343296) = 38… =11 =2

    Les maths ne sont qu’une histoire de groupes

    Clay Mathematics Seminar 2010:

    “Math is nothing but a history of Group

    Director of Institute Henri Poincaré : Cédric Villani (Fields Medalist, 2010)

    Speaker: Prof Etienne Ghys
    École Normale Supérieure de Lyon

    The Math teaching from primary schools to secondary / high schools should begin from the journey of Symmetry.

    After all, the Universe is about Symmetry, from flowers to butterflies to our body, and the celestial body of planets. Mathematics is the language of the Universe, hence
    Math = Symmetry

    It was discovered by the 19th century French tragic genius Evariste Galois who, until the eve of his fatal death at 21, wrote about his Mathematical study of ambiguities.

    Another French genius of the 20th century, Henri Poincaré, re-discovered this ambiguity which is Symmetry : Group, Differential Equation, etc.

    Only in university we study the Group Theory to explore the Symmetry.


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    Klein’s Geometry in Group

    This is the “New Geometry” introduced by Klein 200 years ago in his Erlangan Program (his PhD Thesis).
    Rigid Motion is defined by 3 components: Translation, Rotation and Reflection.
    If fixed at one point (origin), there is no translation, only Rotation ρ(θ) and Reflection r(θ) are possible around that fixed point.
    We can prove r(θ) and ρ(θ) form a Group O2, namely Orthogonal Group with this property:
    A^{T}. A = A. A^{T} = I
    where A can be any of the 2 matrices represented by ρ(θ)
    or r(θ),
    A^{T} is the transpose of A (columns => rows, rows => columns).
    1. Rotation
    ρ(θ)=
    (cos θ  -sin θ)
    (sin θ   cos θ)
    2. Reflection
    r(θ) =
    (cos θ   sin θ)
    (sin θ   -cos θ)
    when θ =0,
    r0 =
    (1 0)
    (0 -1)
    => r(θ) = ρ(θ).r0
    Change of Reference Axis:
     Make a shift from fixed origin A to another fixed original A’ by a translation t(α), the first Orthogonal Group O at A and the second Orthogonal Group O’ at A’ are related by:

    O’ = t(α).O.t^-1(α)
    ρ'(θ) = t(α).ρ(θ).t^-1(α)
    r'(θ) = t(α).r(θ).t^-1(α)

    Note: this looks analogous to Conjugate groups (Normal Subgroups).
    Einstein Relativity interpreted by Rigid Motion (M4).
    If first origin A is the Earth, second origin A’ is the spaceship traveling at speed of light, ie t(α) = c

    O’ = t(α).O.t^-1(α) ; O & O’ ∈ O4

    <=> O’.t(α) = t(α).O