Human
Understand
Intuition, examples, pictures, conjectures and clear human proofs. Not just how — but why it’s true.
Chloe, the mathematician: “Why is this true?”
A free, open-source school of mathematics
Mathematics, from counting to categories — for everyone, everywhere.
The Lumenary is a growing curriculum where every idea is explored three ways: understood by people, proved precisely, and tested by computation. Learn at your own pace. No accounts, no ads, no streaks.
One mathematics. Many languages. Every nation.
Master the present. Glimpse the future. Never require the future to understand the present.
Three ways of knowing, woven together
Each concept is taught through three complementary modes that reinforce one another rather than living in separate books.
Formal
Prove
Precise definitions and machine-checked proofs in Lean 4. Assumptions made explicit; nothing claimed that didn’t compile.
Gideon, the formalist: “Can we prove this precisely?”
Computational
Compute
Python experiments you can run in your browser. Learn the difference between evidence, approximation and proof.
Margaret, the computationalist: “What happens if we calculate it?”
The first path is open
Five connected lessons, from sorting a basket of fruit to composing functions — the doorway to category theory.
- 1K–2CollectionsSorting, belonging, and the surprising empty box — the very first ideas of sets.Ages 5–8
- 2Grades 3–5SetsMembership, subsets, unions and intersections — the language almost all of mathematics is written in.Ages 8–12
- 3Grades 6–8IntegersNumbers below zero, the opposites that cancel them, and a proof of why negative times negative is positive.Ages 11–14
- 4Grades 6–8FunctionsRules that give exactly one output for every input — drawn as arrows, built from ordered pairs, and sorted into injective, surjective and bijective.Ages 12–15
- 5Grades 9–12CompositionDoing one function and then another — why order matters, why brackets don't, and how this simple idea opens the door to category theory.Ages 14–18
Optional · never graded
Wonder Trails
Optional side-paths for the curious. They never count against you, and they always lead back to where you were.
K–2Boxes inside boxesWhat is inside a box that contains only an empty box? Is it empty? And can every description you can think of make a collection?Follow the trail Grades 6–8How many infinities?There are infinitely many counting numbers and infinitely many infinite 0–1 sequences. Are these two infinities the same size?Follow the trail Grades 6–8Clock arithmeticIt's 9 o'clock. What time will it be in 5 hours — and in 29 hours? Can a clock teach us a whole new kind of arithmetic?Follow the trail Grades 6–8Inverses and structureEvery integer can be added to another integer to get . So why can almost no integer be multiplied by another integer to get ?Follow the trail Grades 6–8Remainder familiesSort every integer — positive, negative and zero — into families by its remainder after dividing by 3. If you add a member of one family to a member of another, does the family of the answer depend on which members you chose?Follow the trail Grades 9–12The product, by its mapsCan you describe the set of pairs without ever looking inside it — only by how functions go into it?Follow the trail
Find your starting point
Every learner begins somewhere. Pick the door that fits — you can always wander.
Young learnersAges 5–8Sorting, counting and collections.ExplorersAges 8–14Sets, integers and patterns.SecondaryAges 14–18Functions, composition, proof.UniversityUndergraduateAnalysis, algebra, categories.Graduate & researchFirst-year graduate +Pathways toward research.Adult beginnersAny ageA respectful fresh start.EducatorsTeachers & parentsPrintables, maps, activities.
What exists today — honestly
The Lumenary is at the very beginning. Here is exactly what has been built and checked so far.
No content has yet been reviewed by a human mathematician or educator. That review is the next essential step.
See the verification report- lessons
- 5
- Wonder Trails
- 7
- topics mapped
- 169
- Lean declarations compiled
- 88
- experiments tested
- 12
- demo languages
- 6
Inspired by old virtues
Truthfulness, humility, generosity and wonder shape how The Lumenary is built — and every learner is welcome, whatever they believe.
Why we build this way