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Mathematics is commonly defined as the study of patterns of
structure, change, and space; more informally,
one might say it is the study of 'figures and numbers'. In the formalist view, it
is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a
simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of
describing and exploring physical and conceptual relationships.
The specific structures that are investigated by mathematicians 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. Some mathematicians like to refer to their subject as
"the Queen of Sciences".
Mathematics is often abbreviated to math (in American English) or maths (in British English).
Overview and history of mathematics
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, 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, 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 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 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.
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.
Quantity
In general, these topics and ideas present explicit measurements of sizes of numbers or sets, or ways to find such
measurements.
- Number -- Natural number
-- Pi -- 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 -- Base
Change
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
Structure
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
Space
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
Such topics 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 -- Graph theory -- Game
theory
These fields apply knowledge of mathematics to real world problems.
- Mechanics -- Numerical analysis -- Optimization -- Probability -- Statistics -- Financial mathematics
Famous theorems and conjectures
These theorems have interested mathematicians and non-mathematicians alike.
- 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
These are theorems 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
--Fundamental theorem 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 -- Theorem-proving -- 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
Mathematics and other fields
- Mathematics and architecture --
Mathematics
and education -- Mathematics of musical
scales
Mathematical coincidences
- List of mathematical
coincidences
Mathematical tools
Old:
New:
Quotes
Referring to the axiomatic method, where certain properties of an (otherwise unknown) structure are assumed and consequences
thereof are then logically derived, Bertrand Russell said:
- 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.
This may explain why John Von Neumann once said:
- In mathematics you don't understand things. You just get used to them.
About the beauty of Mathematics, Bertrand Russell said in
Study of Mathematics:
- Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of
sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely
pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the
sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as
poetry.
Elucidating the symmetry between the creative and logical aspects of mathematics, W.S. Anglin observed, in Mathematics and
History:
- Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the
explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have
gone elsewhere.
Mathematics is not...
Bibliography
- 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.
- 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 Times (1973);
External links
- Rusin, Dave: The Mathematical
Atlas . A guided tour through the various branches
of modern mathematics.
- Planet Math . An online math encyclopedia under construction, focusing on modern
mathematics. Uses the GFDL license, allowing article exchange with Wikipedia. Uses TeX markup.
- Weisstein, Eric et al.: World of
Mathematics . An online encyclopedia of mathematics,
focusing on classical mathematics.
- Stefanov, Alexandre: Textbooks in Mathematics . A list of free online textbooks and lecture notes
in mathematics.
- A mathematical thesaurus maintained by the NRICH
project at the University of Cambridge (UK), Connecting Mathematics
- Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles . A huge collection of articles on various math topics with more than
400 illustrated with Java applets.
- Mathforge . A news-blog with topics ranging from popular mathematics to popular
physics to computer science and education.
- Metamath . A site and a language, that formalize math from its foundations.
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