今月不曾照古人——Classical Mathematics in Modern Light

Why Revisit Classical Mathematics through Modern Perspectives Historical Context and LimitationsSignificance of Modern ReinterpretationObjectivesCalculusA Higher Perspective on Basic Problems in CalculusModule over a Noncommutative Algebra and Taylor seriesA New Proof of King’s Rule Using InvolutionBeyond Sequences: A Topological Approach to Density ArgumentsNormal Distribution as Orbit of Standard Normal Distribution under Affine Group ActionDifferential AlgebraFrom Commutative Algebra to Linear Constant Coefficient Ordinary Differential EquationLinear AlgebraTopics in Tensor and Categorical Linear AlgebraComplex Analysis A natural proof for Cauchy-Riemann ConditionThe Residue Theorem via Fundamental Groups and Homology Group: Integration as a Group Homomorphism

The title inspired by and contrasting with Li Bai's 把酒问月·故人贾淳令予问

Why Revisit Classical Mathematics through Modern Perspectives

Historical Context and Limitations

Mathematical development has often been constrained by its historical context. Many classical theories are presented in ways that reflect historical contingencies rather than mathematical necessity. emerged primarily to address 17th-century physics problems of motion and change, at a time when mathematicians had yet to develop modern concepts like algebraic and topological structures.

These historical constraints manifest in several ways:

1. Artificial Introduction of Concepts

   • Many fundamental concepts appear through seemingly arbitrary constructions

   • Traditional derivation of Taylor series lacks clear motivation

   • Complex analysis concepts often rely on computational tricks rather than structural understanding

2. Hidden Mathematical Unity

   • Deep connections between different branches are obscured by computational details

   • Unified mathematical principles are fragmented into isolated theorems

   • For instance, the intrinsic connection between homotopy theory and complex functions is rarely emphasized

3. Pedagogical Challenges

   • Learners must memorize apparently disconnected results

   • Core mathematical insights are buried under technical details

   • The natural development of ideas is obscured by historical presentation

Significance of Modern Reinterpretation

Reexamining classical theories through modern mathematics serves multiple purposes:

1. Revealing Mathematical Essence

   • Modern frameworks demonstrate the natural emergence of theories

   • Technical derivations give way to conceptual understanding

   • Example: Taylor series emerges naturally from operator algebra

2. Unifying Mathematical Perspective

   • Illuminates profound connections between different mathematical branches

   • Establishes more coherent theoretical frameworks

   • Reveals common patterns underlying seemingly distinct concepts

3. Enhancing Mathematical Education

   • Provides clearer conceptual development

   • Helps learners understand the evolution of mathematical thought

   • Reduces reliance on memorization in favor of deeper understanding

Objectives

This column aims to:

1. Reexamine Classical Theories

   • Apply modern mathematical tools to classical results

   • Reveal the inherent structure behind historical presentations

   • Demonstrate how modern perspectives simplify complex ideas

2. Bridge Mathematical Disciplines

   • Show how different areas of mathematics naturally connect

   • Demonstrate the unity of mathematical thought

   • Reveal structural patterns across different theories

3. Advance Mathematical Understanding

   • Provide new insights into familiar concepts

   • Develop more natural approaches to classical results

   • Offer fresh perspectives for mathematical education

Through this approach, we are not merely revisiting classical mathematics but exploring its essential structure and development patterns. The title "今月不曾照古人" (The Moon Shines Differently Than in Ancient Times) suggests this novel perspective on classical theories, emphasizing how modern viewpoints can illuminate mathematical concepts in ways unavailable to their original discoverers.

This endeavor is both a mathematical and philosophical journey, seeking to understand not just the what of classical mathematics, but the why and how of its development and inner structure.

Calculus

A Higher Perspective on Basic Problems in Calculus

Module over a Noncommutative Algebra and Taylor series

A New Proof of King’s Rule Using Involution

Beyond Sequences: A Topological Approach to Density Arguments

Normal Distribution as Orbit of Standard Normal Distribution under Affine Group Action

Differential Algebra

Math Essays: Differential Ring (1): Some general result and interesting application (marco-yuze-zheng.blogspot.com)

Math Essays: Differential Ring (2): Arithmetic Function Ring (marco-yuze-zheng.blogspot.com)

Math Essays: Sheaf of differential ring over manifold (marco-yuze-zheng.blogspot.com)

Math Essays: Differential Ring (3): An example from a monoid ring (marco-yuze-zheng.blogspot.com)

Math Essays: Derivations from Rings to Modules: A Categorical Perspective Exploration (marco-yuze-zheng.blogspot.com)

Math Essays: Differential Ring (4): a way to constrcut differential ring. (marco-yuze-zheng.blogspot.com)

From Commutative Algebra to Linear Constant Coefficient Ordinary Differential Equation

Linear Algebra

Topics in Tensor and Categorical Linear Algebra

Complex Analysis

A natural proof for Cauchy-Riemann Condition

Remark. Cauchy-Riemann Condition is equivalent to $({\partial }_x+i{\partial }_y)f=0$$(\partial_x+i\partial_y)f=0$. Notice that $\mathrm{\Delta }={\partial }_x^2+{\partial }_y^2$$\Delta=\partial_x^2+\partial_y^2$, we get that

$$\begin{array}{}\text{(1)}& \mathrm{\Delta }=({\partial }_x-i{\partial }_y)({\partial }_x+i{\partial }_y)\end{array}$$

Hence holomorphic implies harmonic. By the way, if you want to prove that if $f$$f$ is holomorphic, then $p({\partial }_x,{\partial }_y)f$$p(\partial_x,\partial_y) f$ is holomorphic as well, just observe that $\mathbb{C}[{\partial }_x,{\partial }_y]\subset {\mathrm{End}}_{\mathrm{Ab}}{C}^{\mathrm{\infty }}({\mathbb{R}}^2,\mathbb{C})$$\mathbb C[\partial_x,\partial_y]\subset \mathrm{End}_{\mathrm{Ab}}C^{\infty}(\mathbb R^2,\mathbb C)$​.

Hence we could view $C^{\mathrm{\infty }}({\mathbb{R}}^2,\mathbb{C})$$C^{\infty}(\mathbb R^2,\mathbb C)$ as $\mathbb{C}[{\partial }_x,{\partial }_y]$$\mathbb C[\partial_x,\partial_y]$ module and notice that ${\partial }_x+i{\partial }_y$$\partial_x+i\partial_y$ is an endomorphism of this module, hence its kernel is a submodule, which is also closed under scalar multiplication.

The Residue Theorem via Fundamental Groups and Homology Group: Integration as a Group Homomorphism