Outline of mathematics

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

Philosophy edit

Nature edit

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 edit

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 edit

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 edit

Quantity edit

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued 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 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

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

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

Structure edit

Space edit

Change edit

Foundations and philosophy edit

Mathematical logic edit

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 edit

Applied mathematics edit

History edit

Psychology edit

Influential mathematicians edit

See Lists of mathematicians.

Mathematical notation edit

Classification systems edit

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 edit

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.

References edit

Bibliography edit

Citations edit

^ Bogomolny, Alexander. "Mathematics Is a Language". www.cut-the-knot.org. Retrieved 2017-05-19.

Notes edit

^ For a partial list of objects, see Mathematical object.
^ See Object and Abstract and concrete for further information on the philosophical foundations of objects.

External links edit

[1] Bogomolny, Alexander. "Mathematics Is a Language". www.cut-the-knot.org. Retrieved 2017-05-19.

[2] For a partial list of objects, see Mathematical object.

[3] See Object and Abstract and concrete for further information on the philosophical foundations of objects.

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Outline of mathematics

Contents

Philosophy edit

Nature edit

Mathematics is edit

Concepts edit

Branches and subjects edit

Quantity edit

Structure edit

Space edit

Change edit

Foundations and philosophy edit

Mathematical logic edit

Discrete mathematics edit

Applied mathematics edit

History edit

Regional history edit

Subject history edit

Psychology edit

Influential mathematicians edit

Mathematical notation edit

Classification systems edit

Journals and databases edit

See also edit

References edit

Bibliography edit

Citations edit

Notes edit

External links edit