• Set theory: mathematical theories of sets, in varying consistency strengths.
  • Category theory: algebraic properties of collections of transformations between mathematical objects of the same type.
  • Mathematical logic

Pure Mathematics

  • Number theory: properties of integers, especially primes and prime factorization.
  • Algebra: numerical quantities, equations, and structures.
  • Geometry: shapes of and spatial relationships (e.g. distance) between objects.
    • Axiomatic geometry: Euclidean geometry, non-Euclidean geometries (e.g. elliptic, hyperbolic).
    • Differential Geometry: distance and curvature on manifolds.
  • Topology: invariant property of objects under continuous transformations.
  • Analysis: real- and complex-valued continuous functions; operators.
  • Dynamical system: description of how a complex system changes over time.
  • Discrete math: objects that can assume only distinct, separated values.
    • Graph;
    • Combinatorics;

Applied Mathematics


Table: Development of Common Mathematical Structures

Set Origin/Motivation New Operator Feature; Abstraction
$\mathbb{N}$ counting $+$, $*$
$\mathbb{Z}$ closed inversion of $+$ $-$
$\mathbb{Q}$ closed inversion of $*$ $/$ rational function
$\mathbb{R}$ complete metric ^ analysis; metric, topology
$\mathbb{C}$ root of negative numbers harmonic analysis
$\mathbb{F}^n$ product space $\cdot$ inner product, norm
$\mathbb{F}^\infty$ discrete process/function
$L^p$ approximation of functions


🏷 Category=Topics