Outline of mathematics

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.

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.

Quantity

Quantity –

Arithmetic –

Natural numbers –

Integers –

Rational numbers –

Real numbers –

Complex numbers –

Hypercomplex numbers –

Infinity –

Structure

Structure –

Abstract algebra –

Linear algebra –

Number theory –

Order theory –

Function (mathematics) –

Space

Space –

Geometry –

Algebraic geometry –

Trigonometry –

Differential geometry –

Topology –

Fractal geometry –

Change

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

Mathematical logic –

Model theory –

Proof theory –

Recursion theory –

Set theory –

Type theory –

Discrete mathematics

Discrete mathematics –

Combinatorics

Theory of computation

Cryptography

Graph theory

Applied mathematics

Applied mathematics –

Mathematical physics –

Analytical mechanics –

Mathematical fluid dynamics –

Numerical analysis –

Control theory –

Dynamical systems –

Mathematical optimization –

Operations research –

Probability –

Statistics –

Game theory –

Mathematical economics –

Financial mathematics –

Information theory –

Cryptography –

Mathematical biology –

History

Babylonian mathematics

Egyptian mathematics

Indian mathematics

Greek mathematics

Chinese mathematics

Abacus

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

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)