Foundations:

- 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.

Pure mathematics:

- Number theory: properties of integers, especially primes and prime factorization.
- Algebra: numerical quantities and attempts to solve equations.
- Abstract algebra;

- Geometry: figures, objects, and their relationships to each other.
- Differential geometry: distance and curvature on surfaces and manifolds

- Topology: invariant property of objects under continuous transformations.
- Analysis: real- and complex-valued continuous functions.
- Dynamical systems: description of how a complex system changes over time.
- Discrete math: objects that can assume only distinct, separated values.
- Graph;
- Combinatorics;

Applied mathematics:

- Probability and statistics;
- Computation: tasks that are theoretically possible with computing machines; the relative difficulty and complexity of these tasks.
- Optimization and Game theory;

Table: Development of Common Mathematical Structures

Structure | Origin/motivation | New Operations | Feature | Abstraction |
---|---|---|---|---|

\( \mathbb{N}\) | Counting | \( +, * \) | ||

\( \mathbb{Z}\) | Closed inversion of \(+\) | \( - \) | ||

\( \mathbb{Q}\) | Closed inversion of \( * \) | \( / \) | Polynomial (rational function) | |

\( \mathbb{R}\) | Closure/completeness | ^ | Analysis | metric, topology |

\( \mathbb{C}\) | root of negative numbers | Trigonometrics, Fourier analysis | ||

\( \mathbb{F}^n\) | Product space | \( (\cdot,\cdot) \) | inner product, norm | |

\( \mathbb{F}^{\infty}\) | Discrete process | |||

\( L_p \) | Approximation of functions |