Abstract “Nonsenses” in Abstract Math make “Sense”

After 40 years of learning Abstract Algebra (aka Modern Math yet it is a 200-year-old Math since 19CE Galois invented Group Theory), through the axioms and theorems in math textbooks and lectures, then there is an Eureka “AHA!” revelation when one studies later the “Category Theory” (aka “Abstract Nonsense”) invented only in 1950s by 2 Harvard professors.

A good Abstract Math teacher is best to be a “non-mathematician” , who would be able to use ordinary common-sense concrete examples to explain the abstract concepts: …

Let me explain my points with the 4 Pillars of Abstract Algebra :

\boxed {\text {(1) Field (2) Ring (3) Group (4) Vector Space}}

Note: the above “1-2-3 & 4″ sequence is a natural intuitive learning sequence, but the didactical / pedagogical sequence is “3-2-1 & 4″, that explains why most students could not grasp the philosophical essence of Abstract Algebra, other than the “technical” axioms & theorems.

If a number system (Calculator arithmetic) has 4 operations (+ – * ÷ ), then it is a “Field” (域) – Eg. Real, Complex, Z/pZ (Integer mod Prime)…

If a number system with +, – and * (but no ÷), then it is a “Ring” (环).
eg1. Clock arithmetic {1,2, 3,…,12} = Z/12 (note: 12 is non-prime). [Note: the clock shape is like a ring, hence the German called this Clock number a “Ring”.]
eg2: Matrix (can’t ÷ matrices)
eg3. Polynomial is a ring (can’t ÷ 0 which is also a polynomial).

If a system (G) with 1 operation (○) and a set of elements {x y z w …} that is “closed” (kaki-lang 自己人, any 2 elements x ○ y = z still stay inside G ) , associative (ie bracket orderless) :(xy)z = x(yz), a neutral element (e) s.t. x+e = x = e+x, and inversible (x^{-1}, y^{-1} … still inside G), then G is a Group.
eg. {Integer, +}: 2’s inverse (-2), neutral 0, (2+3)+4=2+(3+4)
eg2. Triangle rotation 120 degree & flip about 3 inner axes.

If a non-empty system V ={v u w z …} that is “closed”if any of its 2 elements (called vectors v, u) v + u = w still in V,
AND if any vector multiply it by a scalar “λ” s.t. “λv” still in V, then V is a Vector Space (向量空间)。
eg1. Matrix (Vector) Space
eg2. Function (Vector) Space
eg3. Polynomial (Vector) Space

Summarise the above 4 or more systems into 1 Big System called “Category” (C) 范畴, then study relation (arrow or morphism) between f: C1 -> C2, this is “Category Theory“.

In any number system (aka algebraic structure), you can find the “Yin / Yang” (阴阳) duality : eg. “Algebra” [#] / “Co-Algebra”, Homology (同调) / Co-Homology (上同调)… if we find it difficult to solve a problem in the “Yang”-aspect. eg. In “Algebraic Topology”: “Homology” (ie “Holes”) with only “+” operation, then we could study its “Yin”-aspect Co-Homology in Ring structure, ie with the more powerful “*” multiplicative operation.

Note [#]: “Algebra” (an American invented structure) is a “Vector Space” plus multiplication between vectors. (Analogy in Physics : Cross Product of vectors).

Reference:https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6

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Pre-requisites For Abstract Algebra

The 2 important pre-requisites for Abstract Algebra are “abstract thinking”, namely :

  1. You must not think of “concrete” math objects (geometrical shapes, Integer, Real, complex numbers, polynomials, matrices…), but rather their “generalised” math objects (Group, Ring, Field, Vector Space…).
  2. Rigourous Proof-oriented rather than computation-oriented.

The foundation of Abstract Algebra is “Set Theory”, make effort to master the basic concepts : eg.

  • Sub-Set,
  • Equivalence Relation (reflexive, symmetric, transitive),
  • Partitioning, Quotient Set, Co-Set
  • Necessary condition (“=>”), Sufficient condition (“<=”). Both conditions (“<=>”, aka “if and only if”)
  • Proof : “A = B” if and only if a A, a B => A B, and, b B, b A =>B A [我中有你, 你中有我 <=>你我合一]
  • Check if a function is “well-defined”. (定义良好)
  • These concepts / techniques repeat in every branch of Abstract Algebra which deals with all kinds of “Algebraic Structures”, from Group Theory to Ring Theory to Field Theory … to (Advanced PhD Math) Category Theory – aka “The Abstract Nonsense”.

Integral Domains 整环 (Abstract Algebra)

Remember when you cancel a common factor at both sides of an equation, you must check if the factor is non-zero, otherwise you would miss some answers.

This is about Cancellation Law, related to few Number Theory Properties : 

  • Zero Divisors,
  • Integral Domain.

Origin of “Integral” => Integers

Definition of Integral Domian:

Property: Cancellation Law  

Reading:

THEOREMS(PROOF Here)

  1. Every Field is an Integral Domain.
  2. Every finite Integral Domain is a Field

北大 高等代数 (1) Beijing University Advanced Algebra

辛弃疾的《青玉案·元夕》:“…众里寻他千百度;蓦然回首,那人却在灯火阑珊处。” –表达出了我的一种 (网上)意外相逢的喜悦,又表现出对心中(名师)的追求。

2011 年 北京大学教授 丘维声教授 被邀给清华大学 物理系(大学一年级) 讲一学期课 : (Advanced Algebra) 高等代数, aka 抽象代数 (Abstract Algebra)。

丘维声(1945年2月-)生于福建省龙岩市,中国数学家、教育家。16岁时以全国高考状元的成绩考入北京大学,1978年3月至今担任北京大学数学科学学院教授,多年坚持讲授数学专业基础课程。截至2013年,共著有包括《高等代数(上册、下册)》、《简明线性代数》两本国家级规划教材在内的40部著述。于1993-97年的一系列文章中逐步解决了n=3pr情形的乘子猜想,并取得了一系列进展。[Ref: Wekipedia ]

———————

72岁的丘教授学问渊博, 善于启发, 尤其有别于欧美的”因抽象而抽象”教法, 他独特地提倡用”直觉” (Intuition) – 几何概念, 日常生活例子 (数学本来就是源于生活)- 来吸收高深数学的概念 (见: 数学思维法), 谆谆教导, 像古代无私倾囊相授的名师。

全部 151 (小时) 讲课。如果没时间, 建议看第1&第2课 Overview 。

http://www.bilibili.com/mobile/video/av7336544.html?from=groupmessage

第一课: 导言 : n 维 方程组 – 矩阵 (Matrix)- n 维 向量空间 (Vector Space) – 线性空间 (Linear Space)

第二课

上表 (左右对称): 

左。双线性函数 (Bi-linear functions) – eg.內积 (Inner Product x.y = |x|.|y|.cos A)

右:  线性映射 (Linear Map) – 保存 线性 (linearity  ie + 法 , 数乘法 scalar multiplication)

线性 : 一切 平面上的, 或球体表面的点平面 (如: 地球的某点是平地)

线性空间 + 度量 norm =>

  • Euclidean Space (R) => (正交 orthogonal , 对称 symmetric) 变换
  • 酉空间 Unitary Space (C)…  => 变换, Hermite变换

近代代数 (Modern Math since 19CE  Galois) : 从 研究 结构 (环域群) 开始: Polynomial Ring, Algebraic structures (Ring, Field, Group).

第三课: 简化行阶梯形矩阵 Reduced Row Echelon Matrix

第四课: 例子 (无解)

第五课: 证明 无解/唯一解/无穷解, 行列式 (Determinant, Det)
[几何直觉]: 任何2线 1) 向交(唯一解) ; 2) 平行 (无解) ; 3) 重叠 (无穷解)。

n次方程組的解也只有3个情况:

无解
: O = d Det = 0
有解:

  • Rank r < n : 无穷解Det = 0
  • Rank r  = n : 唯一解 Det \neq 0

继续: n阶 行列式 

In Search for Radical Roots of Polynomial Equations of degree n > 1

Take note: Find roots (根) to solve polynomial (多项式方程式) equations, but find solutions (解) to solve simultaneous equations (联式方程式).

Radical : (Latin Radix = root): \sqrt [n]{x}

Quadratic equation (二次方程式) [最早发现者 : Babylon  和 三国时期的吴国 数学家 赵爽]

{a.x^{2} + b.x + c = 0}

\boxed{x= \frac{-b \pm \sqrt{b^{2}-4ac} }{2a}}

Cubic Equation: 16 CE Italians del Ferro,  Tartaglia & Cardano
{a.x^{3} = p.x + q }

Cardano Formula (1545 《Ars Magna》):
\boxed {x = \sqrt [3]{\frac {q}{2} + \sqrt{{ (\frac {q}{2})}^{2} - { (\frac {p}{3})}^{3}}} + \sqrt [3]{\frac {q}{2} -\sqrt{ { (\frac {q}{2})}^{2} - { (\frac {p}{3})}^{3}}}}

Example:
{x^{3} = 15x + 4}
By obvious guess,  x = 4
Using Cardano formula,
x = \sqrt[3]{2+ 11 \sqrt{-1}} + \sqrt [3]{2 - 11 \sqrt{-1}}

They discovered the first time in history the “Imaginary” number (aka Complex number):
\boxed {i = \sqrt{-1}}
then
(2 + i)^{3} =2+11i
(2 - i)^{3} =2-11i
x = (2 + i) + (2 - i) = 4

Quartic Equation: by Cardano’s student Ferrari
{a.x^{4} + b.x^{3} + c.x^{2} + d.x + e = 0}

Quintic Equation:
{a.x^{5} + b.x^{4} + c.x^{3} + d.x^{2} + e.x + f = 0}

No radical solution (Unsolvability) was suspected by Ruffini (1799), proved by Norwegian Abel (1826), but explained by French 19-year-old boy Évariste Galois (discovered in 1831, published only after his death in 1846) with his new invention : Abstract Algebra “Group“(群) & “Field” (域)。

Notes:

Group Theory is Advanced Math.
Field Theory is Elementary Math.

Field is the Algebraic structure which has 4 operations on calculator (+ – × ÷). Examples : Rational number (\mathbb{Q}), Real (\mathbb{R}), Complex (\mathbb{C}), \mathbb{Z}_{p}  (Integer modulo prime, eg.Z2 = {0, 1}) , etc.

If \mathbb{Q}   (“a”, “b”) is adjoined with irrational (eg. \sqrt {2})  to become a larger Field (extension) \mathbb{Q} (\sqrt {2}) = a +b\sqrt {2}
it has a beautiful “Symmetry” aka Conjugate
(a - b\sqrt {2}) 

Field Extension of \mathbb{Q} (\sqrt {2}) = a +b\sqrt {2} :

Any equation P(x) = 0
with root in \mathbb{Q} (\sqrt {2}) = a +b\sqrt {2} will have
another conjugate root (a - b\sqrt {2})

Galois exploited such root symmetry in his Group structure to explain the unsolvability for polynomial equations of quintic degree and above.

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