2025 Winter Omakase: A Concept Overwhelmed by Form 📂JOF

2025 Winter Omakase: A Concept Overwhelmed by Form

Introduction

The year 2025, the Year of the Blue Snake, has arrived. In this course, instead of introducing difficult and complex new content, we will present examples where the form of equations that repeatedly appear throughout the history of science has swallowed intuition.

Vectors

What is the first definition of a vector you encountered? As geometry and physics developed after Descartes’ coordinate system, scholars felt the need to handle multiple numbers simultaneously and conceived vectors arranged as follows: $\left( x , y , z \right)$ As abstract algebra advanced, the contemplation of the essence of vectors began. If a vector is merely an arrangement of numbers, nothing prevents us from calling a matrix, just a rectangular arrangement, a vector. On the other hand, if for that reason matrices can also be called vectors, there’s no reason a polynomial function, where coefficients are handled individually, cannot be called a vector.

Now, a vector is defined not simply by the arrangement of numbers but by possessing properties that we think are vector-like. Accordingly, in modern mathematics, vectors are not defined directly; instead, if the elements of a set satisfy certain conditions, the set is defined as a vector space, and its elements are called vectors.

"A vector is an arrangement of numbers"
▼
"An element of a vector space is a vector"

Angles and Trigonometric Functions

In elementary school, an angle is defined as a shape formed where two lines meet at a point, and the angle is described by its size. In middle school, trigonometric ratios are defined as the ratio of the adjacent side and opposite side to the hypotenuse from an angle in a right triangle. In high school, trigonometric functions are defined as correspondences to the trigonometric ratios of a given angle $\theta$ . Such a build-up is geometrically very intuitive and neat, but it has its limitations when extended to multidimensional spaces $\mathbb{R}^{n}$ . However, considering the dot product, the angle between two vectors $\mathbf{u}, \mathbf{v}$ can be expressed as follows: ${\frac{ \left< \mathbf{u}, \mathbf{v} \right> }{ \left| \mathbf{u} \right| \left| \mathbf{v} \right| }} = \cos \theta$ Now, an angle is defined as a value that satisfies the equation form without geometric concepts like straight lines or angles.

"Calculates trigonometric function values based on angles"
▼
"Angles are determined based on the values of trigonometric functions"

Variance

$s^{2} := {\frac{ 1 }{ n-1 }} \sum_{x} \left( x - \bar{x} \right)^{2}$ When I first encountered statistics in high school, I remember being baffled by the calculation of sample variance $s^{2}$ as follows. Dividing by $n-1$ was one thing, but I couldn’t accept the subtraction to square the differences. From the high school student’s perspective, calling $s^{2}$ variance and arbitrarily inventing terms like $s$ for standard deviation seemed formless without reason. $\begin{align*} \sigma^{2} \left( X \right) :=& \min_{a \in \mathbb{R}} E \left[ \left( X - a \right)^{2} \right] \\ \mu \left( X \right) :=& \argmin_{a \in \mathbb{R}} E \left[ \left( X - a \right)^{2} \right] \end{align*}$ However, with familiarity with concepts like the method of least squares, this definition becomes convincing. On a much broader scale, it is more reasonable to define the mean as the value that minimizes variance rather than measuring how far data deviates from the mean.

"Variance is defined as the mean of the squared deviations from the mean"
▼
"Variance is the minimum sum of squared deviations, making what minimizes variance the mean"

Derivatives

In the curriculum, derivatives are introduced as applications like instantaneous rate of change or the slope of a curve, and the derivative is defined as a limit of the continuous function. $\begin{align*} ( f + g ) ' =& f ' + g ' \\ ( f g ) ' =& f ' g + f g ' \end{align*}$ Interestingly, such forms appear repeatedly beyond analysis across mathematics. Even in number theory, derivatives are defined as in $f ' (n) := f(n) \log n$ , achieving the form above without the concept of limits. Meanwhile, automatic differentiation, researched for dual numbers, can be viewed as an outcome derived solely from the form itself.

"The derivative is the limit of the average rate of change of a function"
▼
"If it has the sum rule and product rule of differentiation, then it is a derivative"