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

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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][2]
 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 nonexistence 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.[loweralpha 1][loweralpha 2]
 Mathematical structure — a set endowed with some additional features on the set (e.g., operation, relation, metric, topology).[3] 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.
 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 pure mathematics devoted primarily to the study of the integers and integervalued functions.
 Arithmetic — (from the Greek ἀριθμός arithmos, 'number' and τική [τέχνη], tiké [téchne], 'art') 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
 Secondorder 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.
 Floatingpoint arithmetic is arithmetic using formulaic representation of real numbers as an approximation to support a tradeoff between range and precision.
 Numbers — a mathematical object used to count, measure, and label.

 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
 Mathematical notation, Infix notation, Scientific notation, Positional notation, Notation in probability and statistics, History of mathematical notation, List of mathematical notation systems
 Infinity, Hyperreal numbers, Surreal numbers
 Fractions, Decimal, Decimal separator
 Operation (mathematics) — an operation is a mathematical function which takes zero or more input values called operands, to a welldefined output value. The number of operands is the arity of the operation.[4]
 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
Structure
Space
Change
Foundations and philosophy
Mathematical logic
 Model theory
 Proof theory
 Set theory
 Type theory
 Recursion theory
 Theory of Computation
 List of logic symbols
 Secondorder 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
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
 History of combinatorics
 History of arithmetic
 History of algebra
 History of geometry
 History of calculus
 History of logic
 History of mathematical notation
 History of trigonometry
 History of writing numbers
 History of statistics
 History of probability
 History of group theory
 History of the function concept
 History of logarithms
 History of the Theory of Numbers
 History of Grandi's series
 History of manifolds and varieties
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
 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 3111 (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.
See also
References
Bibliography
Citations
 "The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 20190801. Retrieved 20200808.
 Bogomolny, Alexander. "Mathematics Is a Language". www.cuttheknot.org. Retrieved 20170519.
 "The Definitive Glossary of Higher Mathematical Jargon — Mathematical Structure". Math Vault. 20190801. Retrieved 20191209.
 "The Definitive Glossary of Higher Mathematical Jargon — Operation". Math Vault. 20190801. Retrieved 20191210.
Notes
 For a partial list of objects, see Mathematical object.
 See Object and Abstract and concrete for further information on the philosophical foundations of objects.