The Math Inside the Machine: How Intelligence Emerges from Eleven Simple Operations
bookA pop-science book explaining that the math inside ChatGPT is the same math you learned between kindergarten and calculus
You already know the math inside ChatGPT. You just don’t know you know it.
Available on Amazon: Paperback ($19.99) | Kindle ($9.99)
Overview
A Napoleonic revelation in the land of Ra. An escape from The Plague. A beachside think tank, gaming the apocalypse. A dying genius, guards at his door, and one unfinished question. These events don’t seem connected to each other, let alone your last chatbot query, but they are.
Inside every AI system that appears to think is a machine built from eleven operations, the same ones you learned between kindergarten and calculus. This book explains not just what the math means and how it works, but how the course of human history wandered its way to this moment.
The Eleven Operations
| Operation | Grade Level | Transformer Role |
|---|---|---|
| Counting | K-2 | Tokenization |
| Addition | 1-3 | Residual connections |
| Multiplication | 3-5 | Embeddings, attention scores |
| Number Representation | 6-8 | Precision, quantization |
| Exponentiation | 6-8 | Softmax |
| Averaging | 7-9 | Attention output |
| Logarithms | 9-11 | Cross-entropy loss |
| Comparison | 2-4 / Advanced | Reward modeling, RLHF |
| Trigonometry | 10-11 | Positional encoding |
| Derivatives | 11-12+ | Backpropagation |
| Iteration | Throughout | Training, generation |
Stories from History
Each chapter weaves mathematical explanation with historical narrative:
- Fourier before a skeptical Lagrange — presenting his radical claim that any pattern can be built from waves
- Shannon’s statue and shadow — the birth of information theory at Bell Labs
- Newton and Leibniz’s bitter priority war — a dispute whose modern echo has spawned its own memes
- The Y2K bug — how counting constraints cost hundreds of billions globally
- The Pentium FDIV bug — when Intel’s chip lied about division
- Galton’s ox — the 1906 country fair that demonstrated the wisdom of crowds
Two Ways to Read
Not in the mood for math? This book is built for two kinds of reading:
- Follow every equation — find rigor enough to tackle the original papers afterward
- Let the stories carry you — each chapter is structured so you’ll know when it’s safe to skim
The Payoff
By the final chapter, you’ll trace a word through the transformer itself. You’ll see the system as engineers do. The black box becomes glass, even if the wonder of emergence remains.
You’ll be left with one last question: which is more unbelievable — the superhuman technology we are building, or the fragile history that brought us here?
Purchase
- Paperback ($19.99)
- Kindle ($9.99)
- ISBN: 979-8-9943457-8-8 (paperback), 979-8-9943457-9-5 (ebook)
- Book Website: themathinside.com