The Mathematics Ontology Bible

Preface: What This Document Is

This is a complete ontological map of mathematics — a reference that answers, for every major domain: what objects exist here, what structures govern them, what can be proven, what is left open, and how this domain depends on or gives rise to others.

The word "bible" is used deliberately. A bible is not a textbook, a proof anthology, or a research monograph. It is a canonical reference — a document that names everything, positions everything, and makes the structure of the whole visible in one place. Every entry here is a node; the dependency relations between entries are the edges; the result is a directed acyclic graph of mathematical knowledge with a unique topological sort.

Three commitments run through every section:

1. Ontological honesty. Each domain carries a note on the ontological status of its objects — are they discovered or invented, abstract or physical, set-theoretic or category-theoretic primitives? The debate is live and this document takes no side; it records the main positions.

2. Checkability. Every claim is either provable from stated axioms, undecidable from them, or empirical. The document names which is which, following the governance-layer discipline that treats unverifiability as a first-class output, not a failure.

3. Dependency. No domain is introduced before its prerequisites. The ordering is itself a theorem: it asserts that the dependency graph is acyclic, which is nontrivial and occasionally contested.

This document is part of a larger program — the LLM Governance Toolkit's epistemic infrastructure — and Part XII connects mathematics explicitly to that program.

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