Mathematics is the investigation of axiomatically defined abstract structures using symbolic logic and mathematical
notation. It is commonly defined as the study of patterns of structure, change, and space; even more informally, one might say
it is the study of "figures and numbers". Because it is not empirical, it is not a science.
Mathematical knowledge is constantly growing, through research and application. Mathematics is usually regarded as a tool for
science, even though the development of mathematics is not necessarily done with science in mind.
The specific structures that are investigated by mathematicians sometimes do have their origin in natural and social
sciences, including physics and economics. Some contemporary mathematics also has its origins in computer science and communication
In the formalist view, widely accepted as a description by professionals in
the field, the definition used is the one given at the beginning of this article. Mathematics might accordingly be seen as an
extension of spoken and written natural languages, with an
extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual
relationships. There are other views, and some are described in the article on the philosophy of mathematics.
Mathematics itself is usually considered absolute, without any reference.
Mathematicians define and investigate some structures for reasons purely
internal to mathematics; they may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Finally, many mathematicians work for purely aesthetic reasons, viewing mathematics more as an art ('pure mathematics')
rather than for its practical application ('applied mathematics'); this is the same kind of motivation as poets and philosophers may experience, and no more
explicable. Albert Einstein referred to the subject as the Queen
of the Sciences in his book Ideas and Opinions.
Mathematics is often abbreviated as math (American English) or maths (British
Overview and history of mathematics
See the article on the history of mathematics for
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, first 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 vectors, 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 (also applying to both more and less dimensions), 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 functions, fiber bundles,
derivatives, 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 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
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
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.
Topics in mathematics
An alphabetical and subclassified list of
mathematical topics is available. The following list of subfields and topics reflects one organizational view of mathematics.
For a fuller treatment, see Areas of mathematics
In general, these topics and ideas present explicit measurements of sizes of numbers or sets, or ways to find such
- Number – Natural
number – 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 –
These topics give ways to measure change in mathematical functions, and changes between numbers.
- Arithmetic – Calculus
– Vector calculus – Analysis – Differential equations – Dynamical systems and chaos theory – List of functions
These branches of mathematics measure size and symmetry of numbers, and various constructs.
- Abstract algebra – Number theory – Algebraic geometry
– Group theory – Monoids – Analysis –
Topology – Linear
algebra – Graph theory – Universal algebra – Category
theory – Order theory
These topics tend to quantify a more visual approach to mathematics than others.
- Topology – Geometry –
Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic
topology – Linear algebra – Fractal geometry
Topics in discrete mathematics deal with branches of
mathematics with objects that can only take on specific, separated values.
- Combinatorics – Naive set theory – Probability – Theory of computation – Finite mathematics – Cryptography –
Fields in applied mathematics use knowledge of mathematics
to solve real world problems.
- Mechanics – Numerical analysis – Optimization – Probability –
Statistics – Financial mathematics – Game theory –
Mathematical biology – Cryptography – Information
theory – Fluid dynamics
Famous theorems and conjectures
These theorems have interested mathematicians and non-mathematicians alike.
- Pythagorean theorem – Fermat's last theorem – Goldbach's conjecture – Twin Prime Conjecture – Gödel's incompleteness theorems –
Poincaré conjecture – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity –
Scholz Conjecture – Church-Turing thesis
Important theorems and conjectures
These are theorems and conjectures that have changed the face of mathematics throughout history.
- Riemann hypothesis – Continuum hypothesis – P=NP – Pythagorean theorem – Central limit theorem – Fundamental theorem of calculus – Fundamental theorem of algebra – Fundamental theorem of arithmetic –
of projective geometry – classification theorems of surfaces – Gauss-Bonnet theorem
Foundations and methods
Such topics are approaches to mathematics, and influence the way mathematicians study their subject.
- Philosophy of mathematics – Mathematical intuitionism – Mathematical constructivism – Foundations of mathematics – Set theory – Symbolic
logic – Model theory – Category theory – Logic – Reverse Mathematics – Table of mathematical symbols
History and the world of mathematicians
- History of mathematics – Timeline of mathematics – Mathematicians – Fields medal –
Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union –
Mathematics competitions – Lateral thinking – Mathematical abilities and
Mathematics and other fields
- Mathematics and architecture –
and education – Mathematics of
- List of mathematical
Mathematics is not...
Although Einstein called it "the Queen of the Sciences", mathematics itself is not a science, because it is not empirical.
Mathematics is not physics, as physics is a science and mathematics is not.
Mathematics is not numerology. Although numerology uses modular arithmetic
to reduce names and dates down to numbers, numerology assigns emotions or traits to these numbers without proving the assignments
in a logical manner, or providing exact definitions for the emotions or traits. The interactions between the assigned emotions of
the numbers are established by intuitive estimation rather than rigorous calculation.
Mathematics is not accountancy. Although arithmetic computation is crucial
to the work of accountants, they are mainly concerned with proving that the computations are true and correct through a system of
doublechecks. The proving or disproving of hypotheses is very important to mathematicians, but not so much to accountants.
Advances in abstract mathematics are irrelevant to accountancy if the discoveries can't be applied to improving the efficiency of
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition
1998 available, 1st edition 1968 . A concise
history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996.
An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (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.
- Kline, M., Mathematical Thought from Ancient to Modern
- Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles (http://www.cut-the-knot.org/). A collection of articles on various math topics, with
interactive Java illustrations
- Rusin, Dave: The Mathematical
Atlas (http://www.math-atlas.org/). A guided tour through the various
branches of modern mathematics.
- Stefanov, Alexandre: Textbooks in Mathematics (http://us.geocities.com/alex_stef/mylist.html). A list of free online textbooks and lecture
notes in mathematics.
- Weisstein, Eric et al.: MathWorld:
World of Mathematics (http://www.mathworld.com/). An online encyclopedia of
mathematics, focusing on classical mathematics.
- A mathematical thesaurus maintained by the NRICH (http://nrich.maths.org/) project at the University of Cambridge (UK), Connecting Mathematics (http://thesaurus.maths.org/)
- Planet Math (http://planetmath.org/). An online math encyclopedia under construction, focusing on modern
mathematics. Uses the GFDL, allowing
article exchange with Wikipedia. Uses TeX markup.
- Mathforge (http://www.mathforge.net/). A news-blog with topics ranging from popular mathematics to
popular physics to computer science and education.
Mathematicians Network (YMN) (http://www.youngmath.net/concerns). A
math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search,
Career, Work & Family, Teaching, Research, Misc...
- Metamath (http://metamath.org/). A site and a language, that formalize math from its foundations.
- Math in the Movies (http://world.std.com/~reinhold/dir/mathmovies.html). A guide to major motion pictures with scenes of real mathematics
- Mathematics in fiction (http://math.cofc.edu/faculty/kasman/MATHFICT/default.html). Links to works of fiction that refer to mathematics or mathematicians.