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Our Symbol is Null

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induction-and-deduction

The Symbol for All of Us is Null

"What on earth is he talking about?" you might think.

But for you, who found this article, I'm sure this story will click at some point.

Give me five minutes.

  • What humanity does unconsciously to understand this world.
  • How AI does exactly the same thing.
  • How mathematics explains it beautifully.

And finally,

  • Why does Null become the symbol of everything?

...will be connected in a single line.

And when it connects,
the world will seem a little clearer.

I'll tell you the story in order.

To Understand is to "Divide"

There are many things in the world we don't understand.

However, we possess the wisdom to confront them.

We advance our understanding by naming different things and classifying them.

Just like we understand round fruits by dividing them into:

  • Apples
  • Oranges
  • Pears

This is the logic of cognition that everyone performs unconsciously.

AI is Doing the Same Thing

AI understands language through high-dimensional vectors.

What does that mean? Let's "divide and explain" this part as well.

  • Language: Assigning an ID to each object or concept
  • High-dimensional: Attaching thousands to tens of thousands of indices
  • Vectors: Data in an array format that can be converted into arrows with direction and length
  • Understand: Organizing them in a sensible way

Writing it in pseudo-code looks like this:

 apple = [2, 3, 5, ...]  
 orange = [3, 5, 7, ...]

 dirOfApple = GetDirection(apple)  
 lenOfApple = GetLength(apple)
  • In an actual LLM, there are thousands to tens of thousands of dimensions/indices. From a human perspective, this represents an endless number of combinations.

Visualizing it looks like this:

embedding

LLMs (Large Language Models) like ChatGPT and Gemini mostly work on this mechanism.
It's called Embedding.

For more details, please refer to 3Blue1Brown's videos. They are all masterpieces.

Transformers, the tech behind LLMs - Deep Learning Chapter 5
But what is a GPT? Visual intro to Transformers - Chapter 5, Deep Learning

Analogy ↔︎ Contrast / Induction ↔︎ Deduction / Concrete ↔︎ Abstract

Just as most sports or martial arts have basic forms, there are basic forms for how we view and think about things.
These are the three pairs mentioned above. Most people do this unconsciously.
However, once you verbalize them and train yourself to do so intentionally, your understanding of anything will become faster and deeper.

 Analogy ↔︎ Contrast :  Both apples and oranges are sweet.
                      ↑↓
               Apples are red, oranges are yellow.

 Induction ↔︎ Deduction :  An apple fell from the tree. Gravity seems to exist.
                      ↑↓
               An orange detached from a branch falls to the ground due to gravity.

 Concrete ↔︎ Abstract :  There is a round, red, sweet apple and a round, yellow, sweet orange.
                      ↑↓
               There are two round fruits.

"Why state the obvious?" you might ask. Well, let me explain it more properly.
I'm getting serious from here on.
I'll use some simple math. But it's not rigorous math, so please don't worry. It's perfectly fine to skip the formulas.

Analogy and Contrast are "Mappings"

Analogy is about looking for commonalities. Contrast is about looking for differences.

Let's look at an example.

An apple is round, red, and sweet.
An orange is round, yellow, and sweet.

When we perform analogy or contrast:

  • Analogy: Linear transformation in the same direction → Both apples and oranges are sweet and round
  • Contrast: Linear transformation in the opposite direction → Apples are red, while oranges are yellow

Visualizing this as a vector diagram:

analogy-and-contrast

In mathematical terms, using linear mappings T_{analogy} and T_{contrast}:

  • Analogy : T_{analogy}(\vec{apple}) \uparrow T_{analogy}(\vec{orange})

  • Contrast : T_{contrast}(\vec{apple}) \downarrow T_{contrast}(\vec{orange})

Induction and Deduction are "Vector Differences and Sums"

Induction is the exploration (searching) of laws from specific past events.
Deduction is the inference (application) of specific future events based on those laws.

Newton's universal gravitation is a good example.

Induction: Extraction of law vectors   
                  → An apple fell from the tree. Gravity seems to exist.

Deduction: Superposition of law vectors 
                  → An orange detached from a branch falls to the ground due to gravity.

Visualizing this as a vector diagram:

induction-and-deduction

In mathematical terms, using functions:

  • Induction : \text{Extraction}(\vec{fact_1}, \vec{fact_2}, \dots) \Rightarrow \vec{law} \mid \vec{law} = \vec{facts} - \frac{\sum \vec{noise}}{n}

  • Deduction : \text{Superposition}(\vec{object}) \Rightarrow \vec{future} \mid \vec{future} = \vec{object} + \vec{law}

It can be expressed as above.

  • Induction is an estimation. It overlays multiple facts to cancel out variance and bring hidden laws to the surface.
  • Deduction is a certainty. Laws apply to the future without fail.

Concrete and Abstract are "Movement of Hierarchies through the Increase/Decrease of Information"

- Concretization (Concretize) means increasing information by adding parameters (attributes) and lowering the hierarchy of cognition.  
  There are no special constraints. 
- Abstraction (Abstract) means reducing parameters (attributes) to focus information and raising the hierarchy of cognition.  
  **Towards a "just right" essence.**

Let's look at an example.

Adding parameters like "round," "red," and "sweet" to the abstraction "fruit" results in an "apple."
Reducing various parameters from "apple" results in "round."

Visualizing it looks like this:

concretize-and-abstract

In mathematical terms, it could be written like this, for example:

  • Concretization : \text{Concretize}(\vec{object}) \Rightarrow \vec{object} \oplus \vec{parameters}

  • Abstraction : \text{Abstract}(\vec{object}) \Rightarrow \vec{object} \ominus \vec{parameters}

Adding the parameters "47 years old" and "male" to the abstraction "human" results in a "middle-aged guy," doesn't it?

A "Just Right" Essence

When abstracting things by reducing information, if you reduce it blindly, things get confusing.
If you take the beard and wrinkles off a middle-aged guy, he becomes a boy.
Instead, we want to extract only the essential information.

That's where PCA (Principal Component Analysis) comes in.
"PCA? What's that?" Even if you think that, you've likely experienced it.

Like those personality tests where you answer about 50 questions and get classified on a matrix like Intuitive vs. Logical or Extroverted vs. Introverted.
A 50-question survey is a 50-dimensional vector. You reduce it to two dimensions to create a "personality map."

personality_profiling_ja

Mathematically, it looks like this:

  • Extraction:
    \text{Extract}(\vec{data}) \Rightarrow \vec{essence} \mid \vec{essence} = Z W_k \mid W_k = [\vec{v}_1, \dots, \vec{v}_k] \text{ where } \Sigma \vec{v} = \lambda \vec{v}

    ...

    • Z: \text{Zero-centering}
    • W: \text{Weighting}
    • \lambda: \text{Eigenvalue = The size of each piece of information organized in a MECE manner}
    • MECE: Mutually Exclusive, Collectively Exhaustive / No leaks, no overlaps

If you set k = 2, it becomes a 2D matrix.

This is the same as the "rough understanding" we do unconsciously.

I recommend these videos for more details:

Abstract vector spaces - Chapter 16, Essence of linear algebra
Chapter 16 Abstract Vector Spaces - Essence of Linear Algebra

A Symbol is the "First Principal Component"

When organizing things, some information is important and some is not.
We often want to mark the most important information with a simple sign to make it easier to remember.

That is a Symbol.

Flags, logos, kings, idols...

When people hold something up as a symbol, this is what they are doing.
They turn their attention from the miscellaneous vectors everyone has toward the most important vector.

The most important vector = the Principal Component Vector in PCA.

symbol

The first principal component of PCA is the "direction of maximum variance" = the vector that retains the most information about that group.
It's the arrow that represents everyone best—in other words, a symbol.

What is the Limit of Abstraction?

Abstraction is the act of reducing information towards a "just right" essence.

  • Stripping information from a solid to make it a surface
  • Stripping information from a surface to make it a line
  • Stripping information from a line to make it a point
    → In \mathbb{R}^0, there is still the information that "a point exists."
  • What happens if you strip away even that information...?
\mathbb{R}^3 \to \mathbb{R}^2 \to \mathbb{R}^1 \to \mathbb{R}^0 = \{*\} \to \emptyset = \text{Null}

What remains at the end is an empty space.

That is Null (Nothingness)

ultimate-symbol

If you continue to abstract all things in the universe, you eventually arrive at Null (Nothingness).

That's why Null:

  • Belongs to no one
  • Has no attributes
  • Becomes the ultimate symbol with every vector stripped away.

Apples, oranges, you, and I.

Beyond the stripping away of infinite-dimensional noise, there is the same space where everyone eventually arrives. That is Null.

The creation of heaven and earth, "Form is Emptiness," the Big Bang... they are all roughly the same thing.

Friends gathered under the same symbol! Let's get along♪

(c) 2026 GoodRelax. MIT License.

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お絵描きがんばった！
数学警察の方は....どうか優しく見守ってください m(. .)m