A lay-readable book on the mathematics of optimal intelligence, the specification problem, and what they imply about modern AI.
by Alex Towell
We taught sand to think. We did it before we learned to say what we wanted it to do.
There is a real, settled, and genuinely beautiful theory of what intelligence is: Bayes' theorem, Solomonoff's universal prior, and expected-utility maximization, synthesized as AIXI, the optimal agent. And there are the messy, bounded systems we actually built, the large language models everyone now uses. Between the clean theory and the crude practice lies a gap, and that gap is where the safety of all of it is decided.
This book builds both halves from first principles, for the determined lay reader, and points at the gap honestly. The hardest problem turns out not to be making machines capable. It is saying what we want them to do. The specification problem is the whole game.
Status: Complete. 17 chapters, 4 parts, 214 pages. Both editions published via Amazon KDP (2026): paperback and reflowable Kindle eBook.
The thesis runs through every chapter. There is a clean theoretical limit, AIXI, which captures what optimal agency means under standard assumptions about computation and rationality. There are bounded real systems, the LLMs of 2026, which approximate that limit. The distance between them is real, structural, and consequential. Most of what people argue about under the heading of AI safety, or alignment, is, in different vocabulary, an argument about that gap.
It is not a textbook and not a polemic. Math is used where it has to be, explained conceptually before it is formalized, with diagrams carrying as much of the work as the prose. The reader is treated as intelligent and willing to work. Closer to a Penguin science book than to a survey. No formal background is required.
Seventeen chapters across four parts. Parts I to III develop the theory in clean settings; Part IV moves to messy practice.
| Part | Chapters | Focus |
|---|---|---|
| I. Prediction | 1 to 4 | Bayes, the prior problem, description and probability, Solomonoff induction |
| II. Decision | 5 to 8 | The agent, reinforcement learning, generalization, AIXI |
| III. The Specification Problem | 9 to 12 | Reward modeling, inner alignment, why optimization is dangerous, mitigations and their limits |
| IV. Reality | 13 to 17 | Large language models, reward and reasoning, the gap, what's ahead, and the finale (Teaching Sand to Think) |
The closing chapter takes a committed stand the mathematics earns: prediction carried far enough is intelligence; these are real minds built of sand; the prize is enormous and the danger is the same capability; the burden of proof on the trajectory has shifted.
Both editions are live on Amazon:
- Kindle eBook: amazon.com/dp/B0H4DMBHND (reflowable).
- Paperback: amazon.com/dp/B0H4CCNVRQ (6 x 9, 214 pages, ISBN 979-8-1803407-5-7).
Or read it straight from this repository (the build outputs are tracked):
on-intelligence.pdf, the print edition.on-intelligence-kindle.epub, the reflowable Kindle edition (math as MathML, the TikZ diagrams as images).
The book is written in LaTeX. The print PDF builds with latexmk (it reruns until cross-references and the table of contents converge). The Kindle eBook builds with pandoc: since pandoc cannot render TikZ, scripts/render_tikz.py first pre-renders each diagram to a transparent PNG and rewrites the chapters into build/, then pandoc emits the math as reflowing MathML and carries the diagrams as images. This is the toolchain Amazon KDP's converter accepts.
make # build the print PDF (default)
make epub # build the Kindle eBook (on-intelligence-kindle.epub)
make wordcount # word count
make clean # remove build intermediates (outputs preserved)
make help # list all targetsRequirements: a TeX Live distribution (with latexmk and pdflatex) for the PDF and the diagram rendering, plus pandoc and poppler's pdftocairo for the eBook.
on-intelligence/
├── on-intelligence.tex # main LaTeX file
├── chapters/ # one .tex per chapter (+ archive/ of the prior framing)
├── lore/ # the editorial bible: outline, themes, direction, math reference
├── figures/ # TikZ diagrams (most are inline in the chapters)
├── kdp/ # cover assets (front + print-ready full-wrap PDF)
├── docs/ # editorial review and integration records
├── scripts/ # render_tikz.py (TikZ -> PNG for the pandoc eBook build)
├── Makefile # build system (PDF via latexmk, EPUB via pandoc)
├── CITATION.cff # citation metadata ("Cite this repository")
└── .zenodo.json # Zenodo DOI metadata
The book was restructured once. It began as a philosophical companion to the author's Worldlines; the technical core was the strongest part, and the new direction takes that material as its spine and points it at a concrete destination, AI safety understood mathematically. The prior framing is preserved under chapters/archive/ and lore/archive/.
The book sits alongside Brian Christian's The Alignment Problem, Stuart Russell's Human Compatible, and Max Tegmark's Our Mathematical Universe, and it shares a structural move with the author's own Worldlines: The Indifference of Geometry: settled mathematics at the bottom, lived implications at the top, no hand-waving in between.
If you reference this book, please cite it using the metadata in CITATION.cff, or via the "Cite this repository" button on GitHub. Author: Alexander Towell, ORCID 0000-0001-6443-9897, Southern Illinois University Edwardsville.
Alex Towell. lex@metafunctor.com | metafunctor.com | @queelius
CC-BY-NC-SA-4.0 (Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International). Pedagogical adaptation is welcome with attribution.