Maths

Math notes from foundations outward.

Orientation

• PREAMBLE.md: why the project starts with foundations.
• INDEX.md: complete entry index.
• maps/buildup-path.md: a route through the prerequisites.
• maps/progression-map.md: dependency overview.
• formal/README.md: Lean formalizations.

Subjects

• 00-foundations/: type theory, set theory, foundational viewpoints.
• 01-logic/: propositions, quantifiers, proof principles.
• 02-sets-functions-relations/: sets, functions, relations, quotients, orders.
• 03-number-systems/: natural numbers, divisibility, primes, real and complex numbers.
• 04-algebra/: operations, monoids, groups, rings, fields, modules.
• 05-linear-algebra/: vector spaces, bases, linear maps, matrices.
• 06-analysis/: sequences, limits, continuity, differentiation, integration.
• 07-topology/: open sets, metric spaces, compactness, homeomorphisms.
• 08-probability/: probability spaces, random variables, expectation.
• 09-discrete-math/: graphs and finite structures.
• 10-geometry/: Euclidean space and manifolds.
• 11-number-theory/: congruences, modular arithmetic, Diophantine equations.
• 12-combinatorics/: counting, arrangements, recurrences, generating functions.
• 13-statistics/: statistics, estimation, hypothesis tests, regression, Bayesian inference.
• 14-differential-equations/: ODEs, PDEs, initial value problems, dynamical systems.
• 15-numerical-analysis/: approximation, error, root finding, interpolation, numerical linear algebra.
• 16-optimization/: objectives, convexity, linear programming, gradient descent.
• 17-complex-analysis/: complex functions, holomorphicity, contour integrals, residues.
• 18-functional-analysis/: normed spaces, Banach spaces, Hilbert spaces, operators.
• 19-category-theory/: categories, functors, natural transformations, universal properties.
• 20-computability-complexity/: algorithms, Turing machines, decidability, complexity classes.
• 21-mathematical-physics/: state spaces, symmetries, conservation laws, field equations.
• 22-frontier/: conjectures, research programs, sources, and experiments.

First Route

Start with maps/buildup-path.md for the full route. The shortest route is:

1. 00-foundations/type-theory.md
2. 00-foundations/set-theory.md
3. 01-logic/proposition.md
4. 01-logic/quantifier.md
5. 02-sets-functions-relations/set.md
6. 02-sets-functions-relations/function.md
7. 03-number-systems/natural-number.md
8. 04-algebra/group.md
9. 05-linear-algebra/vector-space.md
10. 06-analysis/limit.md
11. 07-topology/topology.md
12. 08-probability/probability-space.md
13. 10-geometry/manifold.md

Lean Setup

The formal layer is a Lean 4 Lake project:

lake exe cache get
lake build