Mathematics





Mathematics (often abbreviated to maths or, in American English, math) is
commonly defined as the study of patterns of structure, change, and space.
In the modern formalist view, it is the investigation of axiomatically
defined abstract structures using logic and mathematical notation.

These specific structures investigated often have their origin in the
natural sciences, most commonly in physics, but mathematicians also define
and investigate structures for reasons purely internal to mathematics,
because the structures may provide, for instance, a unifying generalization
for several subfields, or a helpful tool for common calculations. Finally,
many mathematicians study the areas they do for purely aesthetic reasons,
viewing mathematics as an art form rather than as a practical or applied science.

Historically, the major disciplines within mathematics arose out of the need
to do calculations in commerce, to measure land and to predict astronomical
events. These three needs can be roughly related to the broad subdivision of
mathematics into the study of structure, space and change.

The study of structure starts with numbers, firstly the familiar natural
numbers and integers and their arithmetical operations, which are recorded
in elementary algebra. The deeper properties of whole numbers are studied in
number theory. The investigation of methods to solve equations leads to the
field of abstract algebra, which, among other things, studies rings and
fields, structures that generalize the properties possessed by the familiar
numbers. The physically important concept of vector, generalized to vector
spaces and studied in linear algebra, belongs to the two branches of
structure and space.

The study of space originates with geometry, first the Euclidean geometry
and trigonometry of familiar three-dimensional space, but later also
generalized to non-Euclidean geometries which play a central role in general
relativity. Several long standing questions about ruler and compass
constructions were finally settled by Galois theory. The modern fields of
differential geometry and algebraic geometry generalize geometry in
different directions: differential geometry emphasizes the concepts of
coordinate system, smoothness and direction, while in algebraic geometry
geometrical objects are described as solution sets of polynomial equations.
Group theory investigates the concept of symmetry abstractly and provides a
link between the studies of space and structure. Topology connects the study
of space and the study of change by focusing on the concept of continuity.

Understanding and describing change in measurable quantities is the common
theme of the natural sciences, and calculus was developed as a most useful
tool for doing just that. The central concept used to describe a changing
variable is that of a function. Many problems lead quite naturally to
relations between a quantity and its rate of change, and the methods to
solve these are studied in the field of differential equations. The numbers
used to represent continuous quantities are the real numbers, and the
detailed study of their properties and the properties of real-valued
functions is known as real analysis. For several reasons, it is convenient
to generalise to the complex numbers which are studied in complex analysis.
Functional analysis focuses attention on (typically infinite-dimensional)
spaces of functions, laying the groundwork for quantum mechanics among many
other things. Many phenomena in nature can be described by dynamical systems
and chaos theory deals with the fact that many of these systems exhibit
unpredictable yet deterministic behavior.

In order to clarify and investigate the foundations of mathematics, the
fields of set theory, mathematical logic and model theory were developed.

When computers were first conceived, several essential theoretical concepts
were shaped by mathematicians, leading to the fields of computability
theory, computational complexity theory, information theory and algorithmic
information theory. Many of these questions are now investigated in
theoretical computer science. Discrete mathematics is the common name for
those fields of mathematics useful in computer science.

An important field in applied mathematics is statistics, which uses
probability theory as a tool and allows the description, analysis and
prediction of phenomena and is used in all sciences. Numerical analysis
investigates the methods of efficiently solving various mathematical
problems numerically on computers and takes rounding errors into account.

     Mathematics may be defined as the subject in which we never know what
     we are talking about, nor whether what we are saying is true.
          -Bertrand Russell