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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

Intuitionistic Logic
Intuitionistic logic encompasses the principles of logical reasoning which were used by L. E. J. Brouwer in developing his intuitionistic mathematics, beginning in [1907]. Because these principles also underly Russian recursive analysis and the constructive analysis of E. Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics. From the Stanford Encyclopedia.
http://plato.stanford.edu/entries/logic-intuitionistic/

Indispensability Arguments in the Philosophy of Mathematics
From the fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine and Putnam have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities. From the Stanford Encyclopedia.
http://plato.stanford.edu/entries/mathphil-indis/

Constructive Mathematics
Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase `there exists' as `we can construct'. In order to work constructively, we need to re-interpret not only the existential quantifier but all the logical connectives and quantifiers as instructions on how to construct a proof of the statement involving these logical expressions. From the Stanford Encyclopedia.
http://plato.stanford.edu/entries/mathematics-constructive/

Inconsistent Mathematics
Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem. By Chris Mortensen, from the Stanford Encyclopedia.
http://plato.stanford.edu/entries/mathematics-inconsistent/

Philosophy of Mathematics Class Notes
Notes to a class by Carl Posy at Duke University, Fall 1992.
http://www.cs.washington.edu/homes/gjb/doc/philmath.htm

Nineteenth Century Geometry
Philosophical-historical survey of the development of geometry in the 19th century. From the Stanford Encyclopedia, by Roberto Toretti.
http://plato.stanford.edu/entries/geometry-19th/

19th Century Logic between Philosophy and Mathematics
Online article by Volker Peckhaus.
http://www.phil.uni-erlangen.de/~p1phil/personen/peckhaus/texte/logic_phil_math.html

Social Constructivism as a Philosophy of Mathematics
Article by Paul Ernest.
http://www.ex.ac.uk/~PErnest/soccon.htm

Paul Ernest's Page
Based at School of Education, University of Exeter, United Kingdom, includes the text of back issues of the Philosophy of Mathematics Education Journal, and other papers on the philosophy of mathematics and related subjects.
http://www.ex.ac.uk/~PErnest/

PHILTAR - Philosophy of Mathematics
Links to pages on individual philosophers.
http://philtar.ucsm.ac.uk/philosophy_of_mathematics/individual_philosophers/

The Logical and Metaphysical Foundations of Classical Mathematics
ArchÃ© Research Project at the University of St Andrews. Description of the project, sponsors, researchers and publications.

Mathematical Structures Group
Research topics include mathematical models and theories in the empirical sciences, models and theories in mathematics, category theory, and the use of mathematical structures in theoretical computer science. Bibliographic data.
http://www.mmsysgrp.com/mathstrc.htm

On GÃ¶del's Philosophy of Mathematics
A paper by Harold Ravitch, Los Angeles Valley College.
http://www.friesian.com/goedel/

Foundations: Philosophy of Mathematics
A study guide on the Philosophy of Mathematics provided by The Objectivist Center, including a study guide on the subject.
http://ios.org/articles/foundations_phil-of-mathematics.asp

Canadian Society for History and Philosophy of Mathematics
Bulletin, members' pages, meetings.

Structuralism, Category Theory and Philosophy of Mathematics
By Richard Stefanik (Washington: MSG Press,1994).
http://www.mmsysgrp.com/strctcat.htm

Holistic Math
An enlarged paradigm of mathematical reality that includes psychology as an integral component.
http://www.iol.ie/~peter/

Hilbert's Program
In 1921, David Hilbert made a proposal for a formalist foundation of mathematics, for which a finitary consistency proof should establish the security of mathematics. From the Stanford Encyclopedia, by Richard Zach.
http://plato.stanford.edu/entries/hilbert-program/

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