Foundations

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

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;

Miscellaneous

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

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