Announcement of New Book Publication “What Kind of Mathematical Worldview Is AI Built Upon? — From Finite Rules to Generative Models —”

I am pleased to announce the publication of my new book:

“What Kind of Mathematical Worldview Is AI Built Upon? — From Finite Rules to Generative Models —”

The book will be available for free from today at 17:00 until tomorrow at 17:00.

If you have a chance to read it, I would greatly appreciate it if you could leave an Amazon review.

Thank you very much for your support.

This book begins with a fundamental question:

“Why is AI capable of generation?”

The Rise of Generative AI

In recent years, generative AI has rapidly begun permeating society.

Conversational AI systems such as ChatGPT can now:

• write text
• summarize information
• translate languages
• generate programs
• even behave like conversational partners

Furthermore,

• image generation AI creates artwork
• video generation AI produces moving images
• music generation AI composes songs

We are entering an era in which AI is no longer merely a “calculation tool,” but is becoming a participant in creative expression itself.

---

These behaviors appear fundamentally different from those of conventional software systems.

Early computer systems and traditional AI typically operated through relatively clear structures:

Input
↓
Rule
↓
Output

For example:

• “If this condition is met, perform this process”
• “If this word appears, assign this meaning”
• “If this input occurs, produce this output”

Humans explicitly defined the rules, and machines processed information according to those rules.

In such systems, it was comparatively possible to trace:

“Why did this result occur?”

We could generally understand:

• which rules were applied
• which branches were taken
• which processes were executed

The internal logic was, at least in principle, understandable by humans.

---

Generative AI, however, is fundamentally different.

Modern generative AI learns through enormous neural networks containing vast numbers of parameters.

In these systems, we cannot simply say:

“This answer emerged because this rule was applied.”

The reason a particular answer appeared cannot be fully explained.

Why a specific expression was generated is often unclear.

Why certain words were selected, why a sentence unfolded in a certain direction, or what internal factors became decisive are extraordinarily difficult for humans to fully comprehend.

---

Moreover, generative AI sometimes demonstrates behaviors that even humans did not anticipate.

For example:

• suddenly exhibiting advanced reasoning capabilities
• acquiring complex contextual understanding beyond certain scales
• responding to tasks it seemingly was never explicitly trained for
• producing unexpectedly natural creative expressions

These are not merely programmatic branches.

Rather, from:

• massive parameter spaces
• enormous datasets
• high-dimensional vector spaces
• probabilistic generation
• complex interactions

new global properties appear to emerge.

---

Such phenomena are often referred to as:

Emergence

Emergence refers to situations in which properties appear at the level of the whole that do not exist at the level of individual components.

For example:

• individual neurons are simple, yet intelligence emerges from the brain as a whole
• individual water molecules contain no “wave,” yet waves emerge collectively
• individual ants are simple, yet colonies display sophisticated behavior

In other words:

new forms of behavior arise from the interaction of local rules.

Similarly, generative AI was not explicitly programmed with advanced language ability or reasoning capability, and yet such capabilities appear once systems surpass certain scales.

---

This suggests that the very nature of AI itself is beginning to change.

Traditional software was:

“A designed machine.”

Generative AI, however, is becoming closer to:

“A massive informational space that learns, forms relationships, and behaves probabilistically.”

AI is no longer merely a rule-execution device.

It increasingly operates within:

• context
• relationships
• semantic spaces
• probability distributions
• emergence

---

And this transformation is not merely technological.

It is also transforming humanity’s understanding of:

• What is intelligence?
• What is understanding?
• What is creativity?

What Kind of Mathematical Worldview Is AI Built Upon?

What this book truly seeks to ask is not:

“Generative AI is mysterious.”

Rather, the more important question is:

Why has modern AI begun to develop structures fundamentally different from traditional AI?

And behind this shift lies:

“A transformation in the mathematics underlying AI itself.”

---

Earlier AI systems were primarily constructed around:

• logic
• symbols
• inference
• if-then rules
• finite states

In that era, intelligence was understood as:

“The ability to perform logically correct reasoning.”

For example:

• “Birds can fly”
• “If A, then B”
• “If condition X is satisfied, perform action Y”

The world was described through explicit rules, and intelligence was thought to emerge through manipulation of those rules.

The foundational assumptions of that period were:

• the world can be represented symbolically
• knowledge can be organized into rules
• reasoning can be formalized logically

In other words:

intelligence was understood as the correct manipulation of symbols.

This was an extremely modern — and in some sense mechanistic — view of intelligence.

---

Modern AI, however, has shifted dramatically away from this perspective.

In neural networks and deep learning systems, AI no longer operates through explicitly written rules.

What matters instead are:

• continuous spaces
• vector representations
• differentiability
• gradients
• optimization

For example, an image-recognition AI does not contain a rule like:

“Cats have two ears.”

Instead, through enormous datasets, it repeatedly performs calculations such as:

“Adjusting weights slightly in this direction reduces error.”

Millions upon millions of optimization steps gradually form “features” inside high-dimensional spaces.

Knowledge no longer exists as explicit rules.

Instead, it exists as:

distributions of weights.

---

In generative AI, this transformation becomes even more fundamental.

Large Language Models (LLMs) do not generate text through simple logical rules.

What becomes central are:

• context
• relationships
• embedding spaces
• high-dimensional vectors
• probabilistic generation
• emergence

Words and concepts are no longer treated as fixed symbolic entities.

Instead, they are positioned within spaces according to:

their relationships to other concepts.

For example:

• the relationship between “king” and “queen”
• the relationship between “Tokyo” and “Japan”
• the relationship between “doctor” and “hospital”

are represented not through logical rules, but through directions and distances inside vector spaces.

Modern AI is therefore beginning to treat:

meaning itself

as:

a geometry of relationships.

---

Furthermore, generative AI is not deterministic.

It generates the next word from probability distributions.

As a result, modern AI is shifting from:

“A machine that logically derives the single correct answer”

to:

“A machine that generates semantically natural outputs from enormous possibility spaces.”

---

This is also a transformation in mathematical worldview itself.

Traditional AI was built upon:

• discrete mathematics
• symbolic logic
• formal systems
• automata theory
• determinism

Modern AI, however, is increasingly built upon:

• linear algebra
• calculus
• probability
• geometry
• information spaces
• optimization
• high-dimensional spaces

AI is shifting from:

“A machine of logic”

to:

“A machine of continuity and relationships.”

And this shift is not merely technological evolution.

It is also transforming humanity’s understanding of:

• What is intelligence?
• What is meaning?
• What is understanding?

What This Book Explores

This book traverses topics such as:

• symbolic AI
• logic
• Answer Set Programming
• possible worlds
• calculus
• neural networks
• Transformers
• nonlinearity
• chaos
• emergence
• AGI
• relational intelligence
• Multi-Agent systems
• Decision Trace Model

while exploring:

“What kind of mathematical worldview modern AI is built upon.”

This is not merely a technical explanation of AI.

Rather, it is also an attempt to reconsider:

• Can the world truly be closed through finite rules?
• Is intelligence merely reasoning?
• Can meaning be fixed?
• Does AI actually “understand”?
• Can society operate through a single correct answer?

In other words, it is an inquiry into:

the worldview of the AI era itself.

---

This book may also be read as a philosophical and theoretical background volume connected to the broader “Relational Intelligence (Chinōba)” series:

• AI Is Not Prediction. It Is Decision. — A Practical Guide to Decision Trace Model —
• Intelligence as Relationship — Chinōba
• Intelligence Field Economics — Trust, Knowledge and Coordination in the AI Era

Themes explored in Decision Trace Model (DTM), such as:

• the distinction between Signal and Decision
• Multi-Agent Coordination
• Runtime structures
• governance
• boundary conditions
• relational intelligence

are deeply connected to the mathematical and philosophical worldview discussed in this book.

---

If modern AI is no longer merely:

“A prediction machine,”

but is instead becoming:

“An entity that generates meaning, forms structures, and behaves emergently within relationships,”

then perhaps we must fundamentally reconsider the concept of:

intelligence itself.

I hope this book becomes one small contribution toward thinking more deeply about the age of AI.

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Chinoba — Runtime Society and Coordination Systems:
chinoba.org