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.

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.

Mathematics is often abbreviated to math in North America and maths in other English-speaking countries.

Table of contents

1 Overview and history of mathematics 2 Topics in mathematics 2.1 Quantity 2.2 Change 2.3 Structure 2.4 Space 2.5 Discrete mathematics 2.6 Applied Mathematics 2.7 Famous Theorems and Conjectures 2.8 Foundations and Methods 2.9 History and the World of Mathematicians 2.10 Mathematics and other fields 3 Mathematical tools 4 Quotes 5 Mathematics is Not... 6 Bibliography 6.11 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, 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 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 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.

Quantity

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 numberss -- Integer sequences -- Mathematical constants -- Number names -- Infinity

Change

Arithmetic -- Calculus -- Vector calculus -- Analysis -- Differential equations -- Dynamical systems and chaos theory -- Fractional calculus -- List of functions

Structure

Abstract algebra -- Number theory -- Algebraic geometry -- Group theory -- Monoids -- Analysis -- Topology -- Linear algebra -- Graph theory -- Universal algebra -- Category theory

Space

Topology -- Geometry -- Trigonometry -- Algebraic geometry -- Differential geometry -- Differential topology -- Algebraic topology -- Linear algebra -- Fractal geometry

Discrete mathematics

Combinatorics -- Naive set theory -- Probability -- Theory of computation -- Finite mathematics -- Cryptography -- Graph theory -- Game theory

Applied Mathematics

Mechanics -- Numerical analysis -- Optimization -- Probability -- Statistics -- Financial mathematics

Famous Theorems and Conjectures

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

Foundations and Methods

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

Mathematical tools

Old:

• Abacus
• Napier's bones, Slide Rule
• Ruler and Compass

New:

• Calculators and Computers
• Programming languages
• Computer algebra systems:
  • Maple
  • Mathematica
  • Maxima
• Internet shorthand notation

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.

Mathematics is Not...

• Numerology

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

• A mathematical thesaurus maintained by the University of Cambridge (UK), http://thesaurus.maths.org/
• Rusin, Dave: The Mathematical Atlas, http://www.math-atlas.org/. A tour through the various branches of modern mathematics.
• Weisstein, Eric: World of Mathematics, http://www.mathworld.com/. An online encyclopedia of mathematics.
• Planet Math, http://planetmath.org/. An online math encyclopedia under construction. Uses the GFDL license, 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.
• Metamath, http://metamath.org/. A site, and a language, that formalize math from its foundations.

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