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.
- 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.
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;
Miscellaneous
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 |
|
|
|
MathWorld Classroom
🏷 Category=Topics