- 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

- 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;

- Probability and Statistics;
- Optimization and Game theory;
- Computation theory: tasks that are theoretically possible with computing machines, and their relative difficulty and complexity.
- Numerical methods;

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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