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Mathematics encompasses a growing variety and depth of subjects over its history, and comprehension of it requires a system to categorize and organize these various subjects into more general areas of mathematics or fields of mathematics.[opinion] A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve.
An ideal system of classification permits adding new areas into the organization of previous knowledge, and fitting surprising discoveries and unexpected interactions into the outline.[opinion]
For example, the Langlands program has found unexpected connections between areas previously thought unconnected, at least Galois groups, Riemann surfaces and number theory.
In the Library of Congress Classification, mathematics is assigned the many subclass QA within the class Q (Science). The LCC defines broad divisions, and individual subjects are assigned specific numerical values.
Mathematicians have always worked with logic and symbols, but for centuries the underlying laws of logic were taken for granted, and never expressed symbolically. Mathematical logic, also known as symbolic logic, was developed when people finally realized that the tools of mathematics can be used to study the structure of logic itself. Areas of research in this field have expanded rapidly, and are usually subdivided into several distinct subfields.
Proof theory and constructive mathematics: Proof theory grew out of David Hilbert's ambitious program to formalize all the proofs in mathematics. The most famous result in the field is encapsulated in Gödel's incompleteness theorems. A closely related and now quite popular concept is the idea of Turing machines. Constructivism is the outgrowth of Brouwer's unorthodox view of the nature of logic itself; constructively speaking, mathematicians cannot assert "Either a circle is round, or it is not" until they have actually exhibited a circle and measured its roundness.
Set theory: A set can be thought of as a collection of distinct things united by some common feature. Set theory is subdivided into three main areas. Naive set theory is the original set theory developed by mathematicians at the end of the 19th century. Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory. It treats sets as "whatever satisfies the axioms", and the notion of collections of things serves only as motivation for the axioms. Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal (unimaginably small) elements within the real numbers. See also List of set theory topics.
General algebraic systems: Given a set, different ways of combining or relating members of that set can be defined. If these obey certain rules, then a particular algebraic structure is formed. Universal algebra is the more formal study of these structures and systems.
Field theory and Galois theory: Field theory studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined. A polynomial is an expression in which constants and variables are combined using only addition, subtraction, and multiplication.
Number theory is the study of numbers and the properties of operations between them. Number theory is traditionally concerned with the properties of integers, but more recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers.
Arithmetic: An elementary part of number theory that primarily focuses upon the study of natural numbers, integers, fractions, and decimals, as well as the properties of the traditional operations on them: addition, subtraction, multiplication and division. Up until the 19th century, arithmetic and number theory were synonyms, but the evolution and growth of the field has resulted in arithmetic referring only to the elementary branch of number theory.
Elementary number theory: The study of integers at a higher level than arithmetic, where the term 'elementary' here refers to the fact that no techniques from other mathematical fields are used.
Fractal geometry: The study of fractals, geometric objects exhibiting self-similar behavior. Common examples such as the Mandelbrot set and the Cantor set motivate the study of fractal geometry, and the subject enjoys applications toward many other disciplines both within mathematics and the sciences.
Deals with the properties of a figure that do not change when the figure is continuously deformed. The main areas are point set topology (or general topology), algebraic topology, and the topology of manifolds, defined below.
Statistics: The science of making effective use of numerical data from experiments or from populations of individuals. Statistics includes not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments. See also the list of statistical topics.
Computer algebra: This area is also called symbolic computation or algebraic computation. It deals with exact computation, for example with integers of arbitrary size, polynomials or elements of finite fields. It includes also the computation with non numeric mathematical objects like polynomial ideals or series.
Classical Mechanics: Addresses and describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
Mechanics of structures: Mechanics of structures is a field of study within applied mechanics that investigates the behavior of structures under mechanical loads, such as bending of a beam, buckling of a column, torsion of a shaft, deflection of a thin shell, and vibration of a bridge.
Particle mechanics: In mathematics, a particle is a point-like, perfectly rigid, solid object. Particle mechanics deals with the results of subjecting particles to forces. It includes celestial mechanics—the study of the motion of celestial objects.
Mathematical programming: Mathematical programming (or mathematical optimization) minimizes (or maximizes) a real-valued function over a domain that is often specified by constraints on the variables. Mathematical programming studies these problems and develops iterative methods and algorithms for their solution.
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives.
The history of mathematics is inextricably intertwined with the subject itself. This is perfectly natural: mathematics has an internal organic structure, deriving new theorems from those that have come before. As each new generation of mathematicians builds upon the achievements of their ancestors, the subject itself expands and grows new layers.
From magic squares to the Mandelbrot set, numbers have been a source of amusement and delight for millions of people throughout the ages. Many important branches of "serious" mathematics have their roots in what was once a mere puzzle and/or game.