See the article on
the
history of mathematics for details.
The word
"mathematics" comes from the
Greek μάθημα (máthema) which means "science, knowledge, or
learning"; μαθηματικός (mathematikós) means "fond of learning".
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
An alphabetical
list of mathematical topics is available; together with the
"Watch links" feature, this list is useful to track changes in
mathematics articles. The following list of subfields and topics
reflects one organizational view of mathematics.
Numbers --
Natural numbers --
Integers --
Rational numbers --
Real numbers --
Complex numbers --
Hypercomplex numbers --
Quaternions --
Octonions --
Sedenions --
Hyperreal numbers --
Surreal numbers --
Ordinal numbers --
Cardinal numbers --
p-adic numbers --
Integer sequences --
Mathematical constants --
Number names --
Infinity
Arithmetic --
Calculus --
Vector calculus --
Analysis --
Differential equations --
Dynamical systems and chaos theory --
Fractional calculus --
List of functions
Abstract algebra --
Number theory --
Algebraic geometry --
Group theory --
Monoids --
Analysis --
Topology --
Linear algebra --
Graph theory --
Universal algebra --
Category theory
Topology --
Geometry --
Trigonometry --
Algebraic geometry --
Differential geometry --
Differential topology --
Algebraic topology --
Linear algebra --
Fractal geometry
Combinatorics --
Naive set theory --
Probability --
Theory of computation --
Finite mathematics --
Cryptography --
Graph theory --
Game theory
Mechanics --
Numerical analysis --
Optimization --
Probability --
Statistics
Fermat's last theorem --
Riemann hypothesis --
Continuum hypothesis --
P=NP --
Goldbach's conjecture --
Twin Prime Conjecture --
Gödel's incompleteness theorems --
Poincaré conjecture --
Cantor's diagonal argument --
Pythagorean theorem --
Central limit theorem --
Fundamental theorem of calculus --
Fundamental theorem of algebra --
Fundamental theorem of arithmetic --
Four color theorem --
Zorn's lemma --
"The most remarkable formula in the world"
Philosophy of mathematics --
Mathematical intuitionism --
Mathematical constructivism --
Foundations of mathematics --
Set theory --
Symbolic logic --
Model theory --
Category theory --
Theorem-proving --
Logic --
Reverse Mathematics --
Table of mathematical symbols
History of mathematics --
Timeline of mathematics --
Mathematicians --
Fields medal --
Abel Prize --
Millennium Prize Problems (Clay Math Prize) --
International Mathematical Union --
Mathematics competitions --
Lateral thought
Mathematics and architecture
Numerology
Old:
New:
- Davis, Philip
J.; Hersh, Reuben: The Mathematical Experience. Birkhäuser,
Boston, Mass., 1980. A gentle introduction to the world of
mathematics.
- Gullberg, Jan:
Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An
encyclopedic overview of mathematics presented in clear, simple
language.
- Mathematical
Society of Japan: Encyclopedic Dictionary of Mathematics, 2nd
ed.. MIT Press, Cambridge, Mass., 1993. Definitions, theorems
and references.
- Michiel
Hazewinkel (ed.): Encyclopaedia of Mathematics. Kluwer Academic
Publishers 2000. A translated and expanded version of a Soviet
math encyclopedia, in ten (expensive) volumes, the most complete
and authoritative work available. Also in paperback and on
CD-ROM.
