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, 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.

  Subjects

Quantity

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

  Structure

• Abstract algebra
• Linear algebra
• Number theory
• Order theory
• Function

  Space

• Geometry
• Algebraic geometry
• Trigonometry
• Differential geometry
• Topology
• Fractal geometry

  Change

• Calculus
• Vector calculus
• Differential equations
• Dynamical systems
• Chaos theory
• Analysis

  Foundations and philosophy

• Philosophy of mathematics
• Category theory
• Set theory
• Type theory

  Mathematical logic

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

  Discrete mathematics

• Combinatorics
• Cryptography
• Graph theory

  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
• 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

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