mathematics

the discipline that studies math — structure, proof, and abstraction. uniquely among disciplines, mathematics maps to a single crystal domain because its phenomena are internally coherent: everything proved in math follows from axioms without empirical input

this does not mean mathematics is simple. it contains vast sub-fields that took centuries to develop:

branches

algebra — structures with operations: groups, rings, fields

geometry — shape, space, curvature, Euclid to Riemann

topology — properties preserved under continuous deformation

calculus — change, limits, differential equations, fourier transform

number theory — integers, primes, Diophantine equations

combinatorics — counting, arrangements, graph enumeration

set theory — foundations, cardinality, the axiom of choice

category theory — structure-preserving maps between structures

linear algebra — vectors, matrices, eigenvalues, spectral gap

probability — uncertainty, distributions, stochastic processes

statistics — inference from data, estimation, hypothesis testing

logic — formal reasoning, propositional logic, predicate logic, modal logic

key figures

Euclid, Archimedes, Leonhard Euler, Carl Friedrich Gauss, Emmy Noether, Kurt Goedel, Stefan Banach, Gottfried Leibniz