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# Mathematics

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.

Although mathematics itself is not usually considered a natural science, the specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics. However, 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).

 Table of contents 1 Overview and history of mathematics 2 Topics in mathematics 3 Mathematical tools 4 Quotes 5 Mathematics is not... 6 Bibliography 7 External links

## 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 fieldss, structures that generalize the properties possessed by the familiar numbers. The physically important concept of vectorss, 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 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 numberss -- 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

### Discrete mathematics

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 -- Graph theory -- Game theory

### Applied mathematics

Fields in
applied mathematics use 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

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## Quotes

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 sho

AMCW2001 - Applied Mathematics in Our Changing World
Joint EMS/SIAM meeting. On-line registration. Berlin, 2--6 September 2001.
http://www.zib.de/amcw01/

GAMM 2001
Annual meeting of the Gesellschaft fuer angewandte Mathematik und Mechanik. ETH Zurich, Switzerland; 12--15 February 2001.
http://www.GAMM2001.ethz.ch/

Mathematics and its Applications
Kuwait; 10--12 March 2003.
http://www.sci.kuniv.edu.kw/ICMA03.html

BAMC2003
British Applied Mathematics Colloquium. University of Southampton; 7--10 April 2003.
http://www.maths.soton.ac.uk/bamc/

Gordon Research Conference on Nonlinear Science
Applications of methods and concepts from nonlinear dynamics to all areas of science. Mount Holyoke College, South Hadley, Mass, USA; 17--22 June 2001.
http://heracles.chem.wvu.edu/new/Grc.htm

Gordon Research Conference on Nonlinear Science
Mt. Holyoke College; June 17-22, 2001; South Hadley, MA.
http://www.grc.uri.edu/programs/2001/nonlsci.htm

HERCMA 2001
The 5th Hellenic European Conference on Computer Mathematics and its Applications. Athens, Greece; 20--22 September 2001.
http://www.aueb.gr/conferences/hercma2001/

Networks: Structure, Dynamics and Function
23rd Annual Conference. Hotel LaFonda, Santa Fe, New Mexico, USA; 12--16 May 2003.
http://cnls.lanl.gov/networks/

ICMCA 2002
Third International Conference on Mathematical and Computational Applications. Konya, Turkey; 4--6 September 2002.
http://asp.selcuk.edu.tr/asp/diger/icmca2002/default.asp

Charles-FranÃ§ois Sturm
Colloquium on the occasion of the 200th anniversary of Charles-FranÃ§ois Sturm and Workshop on Sturm-Liouville theory. Geneva, Switzerland; 15--19 September 2003.
http://theory.physics.unige.ch/~fiteo/sturm/colloquium.html

HERCMA 2003
6th Hellenic European Research Conference on Computer Mathematics and its Applications. Athens, Greece; 25--27 September 2003.
http://www.aueb.gr/conferences/hercma2003/

ICIAM2003
International Congress on Industrial and Applied Mathematics. Sydney, Australia; 7--11 July 2003.
http://www.austms.org.au/iciam2003/

SCI 2003
Seventh Multi-Conference on Systemics, Cybernetics and Informatics. Orlando, Florida, USA; 27--30 July 2003.
http://www.iiisci.org/sci2003/

New Frontiers in Computational Mathematics
University of Manchester, UK; 10--11 January 2004.
http://www.maths.man.ac.uk/MCCM/frontiers.html

Applied Mathematics
5th WSEAS International Conference. Miami, Florida, USA; 21--23 April 2004.
http://www.worldses.org/conferences/2004/florida/math/

BAMC 2004
British Applied Mathematics Colloquium. University of East Anglia, Norwich, UK; 19--22 April 2004.
http://www.mth.uea.ac.uk/bamc/

CHT-04
International symposium on Advances in Computational Heat Transfer. MS Midnatsol, between Kirkenes and Bergen, Norway; 19--24 April 2004.
http://cht04.mech.unsw.edu.au/

SoCAMS 04
4th Southern California Applied Mathematics Symposium. Claremont Colleges, CA, USA; 24 April 2004.
http://www.math.hmc.edu/socams04/

ERMAC-2004
Regional Meeting of Applied and Computational Mathematics. Rio de Janeiro, Brazil; 26--27 April 2004.
http://www.sobrapo.org.br/ermacindex.htm

International Conference on Numerical Combustion (NC04)
Focus on the integration of theory, modeling, and numerical implementation in the study of basic combustion physics and technological applications. Sedona, AZ, USA; 9--12 May 2004.
http://www.siam.org/meetings/nc04/

Student Conference on Applied Mathematics
Kaunas, Lithuania; 8 May 2004.
http://fumsa.ktusa.lt/tmp/Matematika

ECMI 2004
The 13th Conference on Mathematics for Industry. Eindhoven, The Netherlands; 21--25 June 2004.
http://www.ecmi2004.tue.nl/

ICMA 2004
International Conference on Mathematics and Its Applications. City University of Hong Kong; 28--31 May 2004.
http://www.cityu.edu.hk/rcms/ICMA2004/

International Conference on Nonlinear Problems in Aviation and Aerospace
ICNPAA 2004: Mathematical Problems in Engineering and Aerospace Sciences. The West University of Timisoara, Romania; 2--4 June 2004.
http://www.icnpaa.com/

GEOMED 2001
Third International Workshop on Geomedical Systems. Paris, France; 17--19 October 2001.
http://oms.u444.jussieu.fr/geomed2001/

Mathematical and Computer Techniques for Agro-food Technologies
Barcelona, Spain; 26--27 November 2001.
http://www.cimne.upc.es/congress/food/

MISG2001
Mathematics-in-Industry Study Group 2001. Centre for Industrial and Applicable Mathematics, University of South Australia; 29 January -- 2 February 2001.
http://www.unisa.edu.au/misg/Default.htm

SIAM Southwestern Regional Mathematics in Industry Workshop
Houston, Texas, USA; 27--28 April 2001.
http://www.siam.org/meetings/mii01sw/

ICIAM 99
Fourth International Congress on Industrial and Applied Mathematics. Edinburgh, Scotland; 5--9 July 1999. Newsletters, schedules, proceedings.
http://www.ma.hw.ac.uk/iciam99/

2000 SIAM Annual Meeting
Rio Grande, Puerto Rico. 10--14 July 2000.
http://www.siam.org/meetings/an00/

ICIAM 95
Proceedings of the Third International Congress on Industrial and Applied Mathematics. Table of contents and order info.
http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Mennicken/CICIAM.html

2001 SIAM Annual Meeting
Jointly with the SIAM Conference on Control and its Applications. San Diego, CA, USA; 9--13 July 2001.
http://www.siam.org/meetings/an01/

SIAM Conference Archives
By subject and date.
http://www.siam.org/meetings/archives/index.htm

ANZIAM 2002
The 38th Applied Mathematics Conference of the ANZIAM division of the Australian Mathematical Society. Rydges Eaglehawk Resort, Canberra; 2--6 February 2002.

First PIMS Industrial Problem Solving Workshop
UBC, Vancouver, Canada; 22 August 1997. Report on the meeting.
http://www.pims.math.ca/sections/whatsnew/ipswreport.html

GAMM 2002
Annual meeting of the Gesellschaft fuer Angewandte Mathematik und Mechanik. Augsburg, Germany; 25--28 March 2002.
http://gamm2002.uni-augsburg.de/

Annual Meeting of the UK and Republic of Ireland Section of SIAM
Report from the 2002 annual meeting held in Leeds on 11 January 2002.
http://www.comp.leeds.ac.uk/siam/

Inverse Problems in Engineering
4th International Conference. Angra dos Reis, Rio de Janeiro, Brazil; 26--31 May 2002.
http://www.lttc.coppe.ufrj.br/4icipe/index.htm

SIMAI 2002
6th Congress of The Italian Society for Applied and Industrial Mathematics (SIMAI). Chia Laguna, Italy; 27--31 May 2002.
http://www.iac.rm.cnr.it/simai/simai2002/engindex.htm

6th PIMS Industrial Problem Solving Workshop
University of British Columbia, Vancouver, BC, Canada; 27--31 May 2002.
http://www.pims.math.ca/industrial/2002/ipsw/

Wave Scattering in Complex Media: From Theory to Applications
NATO ASI. Institut d'Etudes Scientifiques de CargÃ¨se, Corsica, France; 10--22 June 2002.
http://lpm2c.polycnrs-gre.fr/Themes/tiggelen/cargese/

Mathematics and Computers in Sport
6th Australian Conference. Bond University, Queensland, Australia; 1--3 July 2002.
http://www.maths.uq.edu.au/6mcs/

SIAM50
SIAM 50th Anniversary Annual Meeting. Philadelphia, PA, USA; 8--12 July 2002.
http://www.siam.org/meetings/an02/

ICCAM 2002
Tenth International Congress on Computational and Applied Mathematics. Katholieke Universiteit Leuven, Belgium; 22--26 July 2002.
http://www.cs.kuleuven.ac.be/conference/iccam2002/iccam.htm

Engineering Mathematics and Applications
5th Biennial Conference. Brisbane, Australia; 29 September -- 2 October 2002.
http://www.icms.com.au/emac02/

Conference on Applied Mathematics
University of Central Oklahoma, Edmond, OK, USA; 25--27 October 2002.
http://www.math.ucok.edu/dms/CAM/cam.html

Workshop on Industrial Mathematics
King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia; 27--29 October 2002.
http://users.kfupm.edu.sa/imath/

Applied Mathematics and Applications of Mathematics
First EMS/SMAI/SMF Joint Conference. Nice, France; 10--13 February 2003.
http://acm.emath.fr/amam/

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