Outline of Mathematics

Mathematics is a field of study that investigates topics including number, space, structure, and change. For more on the relationship between mathematics and science, refer to the article on science.

Nature of mathematics

Definitions of mathematics – Mathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions, all of which are controversial.
Philosophy of mathematics – its aim is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in people's lives.

Mathematics is

an academic discipline – branch of knowledge that is taught at all levels of education and researched typically at the college or university level. Disciplines are defined (in part), and recognized by the academic journals in which research is published, and the learned societies and academic departments or faculties to which their practitioners belong.
a formal science – branch of knowledge concerned with the properties of formal systems based on definitions and rules of inference. Unlike other sciences, the formal sciences are not concerned with the validity of theories based on observations in the physical world.

General reference
Classification systems

Mathematics in the Dewey Decimal Classification system
Mathematics Subject Classification – alphanumerical classification scheme collaboratively produced by staff of and based on the coverage of the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH.

Reference databases

Mathematical Reviews – journal and online database published by the American Mathematical Society (AMS) that contains brief synopses (and occasionally evaluations) of many articles in mathematics, statistics and theoretical computer science.
Zentralblatt MATH – service providing reviews and abstracts for articles in pure and applied mathematics, published by Springer Science+Business Media. It is a major international reviewing service which covers the entire field of mathematics. It uses the Mathematics Subject Classification codes for organizing their reviews by topic.

Subjects
Quantity

Quantity

Arithmetic
Natural numbers
Integers
Rational numbers
Real numbers
Complex numbers
Hypercomplex numbers
Infinity

Structure

Algebra
Abstract algebra
Linear algebra
Number theory
Order theory
Function (mathematics)

Space

Geometry
Algebraic geometry
Trigonometry
Differential geometry
Topology
Fractal geometry

Change

Calculus
Vector calculus
Differential equations
Dynamical systems
Chaos theory
Analysis

Foundations and philosophy

Foundations of mathematics

Philosophy of mathematics
Category theory
Set theory
Type theory

Mathematical logic
See also: Outline of mathematical logic

Mathematical logic

Model theory
Proof theory
Set theory
Type theory
Recursion theory
Theory of Computation

Discrete mathematics

Combinatorics (outline)
Cryptography
Graph theory

Applied mathematics

Mathematical chemistry
Mathematical physics
Analytical mechanics
Mathematical fluid dynamics
Numerical analysis
Control theory
Dynamical systems
Mathematical optimization
Operations research
Probability
Statistics
Game theory
Engineering mathematics
Mathematical economics
Financial mathematics
Information theory
Cryptography
Mathematical biology

History

History of mathematics

Babylonian mathematics
Egyptian mathematics
Indian mathematics
Greek mathematics
Chinese mathematics
History of the Hindu–Arabic numeral system
Islamic mathematics
Japanese mathematics
History of algebra
History of geometry
History of mathematical notation
History of trigonometry
History of writing numbers

Psychology

Mathematics education
Numeracy
Numerical Cognition
Subitizing
Mathematical anxiety
Dyscalculia
Acalculia
Ageometresia
Number sense
Numerosity adaptation effect
Approximate number system
Mathematical maturity

Influential mathematicians

See Lists of mathematicians.
Mathematical notation

Mathematical notation

List of mathematical abbreviations
List of mathematical symbols
List of mathematical symbols by subject
Table of mathematical symbols by introduction date
Notation in probability and statistics
Table of logic symbols
Physical constants
Greek letters used in mathematics, science, and engineering
Latin letters used in mathematics
Mathematical alphanumeric symbols
Mathematical operators and symbols in Unicode
ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology)