Mathematics is a field of study that investigates topics such as number, space, structure, and change.
- 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.
- 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
- 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.
- 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.[lower-alpha 1][lower-alpha 2]
- 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.
- 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
- Elementary arithmetic is the part of arithmetic which deals with basic operations of addition, subtraction, multiplication, and division.
- Modular arithmetic
- 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.
- Natural number, Integer, Rational number, Real number, Irrational number, Imaginary number, Complex number, Hypercomplex number, p-adic number
- Negative number, Positive number, Parity (mathematics)
- Prime number, Composite number
- 0, Zero, Infinitesimals
- 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
- 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
Foundations and philosophy
See Lists of mathematicians.
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.