Mathematics is a field of study that investigates topics such as number, space, structure, and change.

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Philosophy

Nature

• 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.
• Language of mathematics is the system used by mathematicians to communicate mathematical ideas among themselves, and is distinct from natural languages in that it aims to communicate abstract, logical ideas with precision and unambiguity.^[1]
• 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.

• Classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory.
• Constructive mathematics asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption.
• Predicative mathematics

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.

Concepts

• Mathematical object — an abstract concept in mathematics; an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs. Each branch of mathematics has its own objects.^[a][b]
• Mathematical structure — a set endowed with some additional features on the set (e.g., operation, relation, metric, topology). A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.

• Equivalent definitions of mathematical structures

• Abstraction — the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

Branches and subjects

Quantity

• Number theory is a branch of mathematics devoted primarily to the study of the integers and integer-valued functions.
• Arithmetic is a branch of mathematics that consists of the study of numbers and the properties of the traditional mathematical operations on them.

• Elementary arithmetic is the part of arithmetic which deals with basic operations of addition, subtraction, multiplication, and division.
• Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.
• Second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets.
• Peano axioms also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.
• Floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision.

• Numbers — a mathematical object used to count, measure, and label.

• List of types of numbers

• Natural number, Integer, Rational number, Real number, Irrational number, Transcendental number, Imaginary number, Complex number, Hypercomplex number, p-adic number
• Negative number, Positive number, Parity (mathematics)
• Prime number, Composite number
• Non-standard numbers, including: Infinity, transfinite number, ordinal number, cardinal number, hyperreal number, surreal number, infinitesimal

• List of numbers in various languages
• Numeral system, Unary numeral system, Numeral prefix, List of numeral systems, List of numeral system topics
• Counting, Number line, Numerical digit, Zero

• Radix, Radix economy, Base (exponentiation), Table of bases

• Mathematical notation, Infix notation, Scientific notation, Positional notation, Notation in probability and statistics, History of mathematical notation, List of mathematical notation systems
• Fraction, Decimal, Decimal separator

• Operation (mathematics) — an operation is a mathematical function which takes zero or more input values called operands, to a well-defined output value. The number of operands is the arity of the operation.

• Calculation, Computation, Expression (mathematics), Order of operations, Algorithm
• Types of Operations: Binary operation, Unary operation, Nullary operation
• Operands: Order of operations, Addition, Subtraction, Multiplication, Division, Exponentiation, Logarithm, Root

• Function (mathematics), Inverse function
• Commutative property, Anticommutative property, Associative property, Additive identity, Distributive property
• Summation, Product (mathematics), Divisor, Quotient, Greatest common divisor, Quotition and partition, Remainder, Fractional part
• Subtraction without borrowing, Long division, Short division, Modulo operation, Chunking (division), Multiplication and repeated addition, Euclidean division, Division by zero

• Plus and minus signs, Multiplication sign, Division sign, Equals sign
• Equality (mathematics), Inequality (mathematics), Logical equivalence
• Ratio

• Variable (mathematics), Constant (mathematics)
• Measurement

Structure

• Algebra
• Abstract algebra
• Linear algebra (Outline)
• Number theory
• Order theory
• Function

Space

• Geometry
• Algebraic geometry

• List of algebraic geometry topics
• List of algebras

• 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
• List of logic symbols
• Second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets.
• Peano axioms also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.

Discrete mathematics

• Combinatorics (outline)
• Cryptography
• Graph theory
• Number 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

Regional history

Subject history

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 algebras
• List of axioms
• List of equations
• List of mathematical functions
• List of types of functions
• List of mathematical jargon
• List of mathematical abbreviations
• List of mathematical proofs
• List of long mathematical proofs
• List of mathematical symbols
• List of mathematical symbols by subject
• List of rules of inference
• List of theorems
• List of theorems called fundamental
• List of unsolved problems in mathematics
• Table of mathematical symbols by introduction date
• Notation in probability and statistics
• List 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)

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.

Journals and 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.