The Core Course

This course is separate from the reference tracks on this site. The reference tracks are an encyclopaedia: alphabetical, cross-linked, no required order. This course is the opposite – a linear sequence where page N assumes page N−1, designed so that no concept you meet depends on a concept you have not yet learned.

The course assumes one thing: you can read, and you can recognise a pattern. It does not assume that you can add. Counting is built from axioms, addition and multiplication are defined recursively, their laws are proved as theorems, the integers and the rationals are constructed as equivalence classes, and the real line is built as Dedekind cuts. Everything after that is earned in the same way.

The order is dependency, not the school curriculum. Nothing appears before everything it needs, and the map below is machine-checked: a forward dependency, a cycle, or an unpaid debt fails the build.

Three rules govern every chapter:

graph TD
    A["A: Reasoning (5)"]
    B["B: Counting (4)"]
    C["C: Expanding the line (4)"]
    D["D: The continuum (5)"]
    E["E: Completing algebra (3)"]
    F["F: Structure and linear algebra (9)"]
    G["G: Euclidean analysis (8)"]
    H["H: The canvas (2)"]
    I["I: Measure (7)"]
    J["J: Calculus as linearisation (8)"]
    K["K: Complex analysis (3)"]
    L["L: Algebra and symmetry (3)"]
    A --> B
    A --> C
    A --> D
    A --> F
    A --> H
    A --> I
    B --> C
    B --> D
    B --> E
    B --> F
    B --> G
    B --> H
    C --> D
    C --> F
    C --> L
    D --> E
    D --> F
    D --> G
    D --> H
    D --> I
    E --> F
    E --> G
    E --> K
    E --> L
    F --> G
    F --> J
    F --> L
    G --> J
    G --> K
    H --> J
    I --> J
    I --> K
    J --> K

Each node is a stage, with its chapter count; each arrow is a dependency that crosses a stage boundary. The arrows only ever point forward, which is the property the Lean artifact checks chapter by chapter.

The reader arrives knowing how to read. The course builds the rest in dependency order: logic before counting, counting before the number line, the number line before algebra, algebra before structure, structure before the continuum, and the continuum before everything that measures it. The full course is 61 chapters across 12 dependency stages, from logic to Galois, with 14 written. The graph is machine-checked: docs/course-graph.json is validated for acyclicity and order-consistency on every push, and the same data generates a Lean 4 proof, checked in CI, that every dependency of every chapter descends the course order.

Two ledgers, not one. A debt is a declared input with a named chapter that pays it; the machine check refuses a debt without a payee. An axiom is a foundation the course deliberately adopts and never proves, listed in the graph so that no chapter can quietly assume one. Zermelo’s axiom of choice enters at D5 and is what the basis theorem, the maximal ideal theorem, and Tychonoff’s theorem actually consume.

The flagship debt is the fundamental theorem of algebra, declared at E3, consumed at F7 and F8, and paid at K3, roughly forty chapters later. Dependencies that run upstream (a stage-7 chapter declaring something a stage-6 chapter proves) are legal and visible: the ledger checks the payee, not the direction of travel.

The reference tracks in the sidebar remain the encyclopaedia; this course is the path through it.

This course integrates five learning protocols designed for self-directed study without the artificial forcing function of exams.

Never read a proof straight through. Read the theorem statement. Close the book. Spend fifteen minutes attempting the proof or identifying the exact technical roadblock. If stuck, open the book and read only the first sentence of the author’s proof – that sentence almost always introduces the trick or auxiliary object. Close the book again and finish the proof yourself. This turns passive consumption into active discovery.

Write your own unified exposition targeting an intelligent version of yourself from six months ago. For every core concept, include:

1. The Defect: what broke that made this definition necessary?
2. The Universal Property or Diagram: the commutative diagram or the precise statement.
3. The Extremes: the trivial example, the canonical workhorse, and the counterexample that breaks when you drop one axiom.
4. The Pivot: the single line in any long proof where the crucial conceptual shift happens.

Skip computational drills. Do classification problems (“classify all groups of order 8”), counterexample challenges (“show the theorem fails if axiom X is omitted”), and omitted lemmas (when an author defers a verification to the reader, treat that as mandatory).

Mathematics is learned by doing, not by reading. Every numbered section of every page ends with a Quick check: five short questions with terse answers, testing exactly what the section proved. Do them before moving on; if one fails, reread only that section. Every page then ends with Problems: multi-step chains that require written arguments, with hints where the jump is large. The One-Line Reveal tells you what to attempt before reading; the quick checks keep you honest while reading; the problems are where you find out whether you can do it.

Every week, spend forty-five minutes on a blank-page retrieval drill from two levels back. Write the definition, the theorem hypotheses, the mental picture, and the proof architecture from memory. If you cannot, you have found a crack before it causes structural collapse.

When stuck: drop one dimension (solve on a 2-sphere or in Set instead), or apply the 48-hour incubation rule – write the obstacle in one plain English question, close the book, and walk away. Subconscious processing in mathematics is a documented reality.

Levels 4–5 are under construction. The dependency graph above shows the planned structure.