Mathematics

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Mathematics (often abbreviated to math or, in British English, maths) is commonly defined as the study of patterns of structure, change, and space. It has been called the "science of measurement", measurement itself being the application of metrics) and perception.

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defined by practices, not proofs

In the modern view, mathematics is usually considered the investigation of axiomatically defined abstract structures using formal logic as the common or foundational framework. This was the most common view in the early 20th century and it remains common today.

However, through that century, many dissenters stated and tried to prove that this is not necessary or desirable - that social or cognitive factors specific to humans and their interactions are more basic than logic, sets or other abstractions - see philosophy of mathematics, foundations of mathematics, and Foundations and Methods references below.

In general the philosophy of mathematics one adopts has little effect on mathematical practice: mathematicians all over the world can rely on mathematics as a language even if there are arguments about the meaning or reliability of certain constructs or "words" or "phrases" used in any given "sentence". It is the practices, not the proofs, that define mathematics as a discipline, though the proofs remain persistent over time to a remarkable degree: Euclid's are still in use and are 2000 years old.

By contrast to science, politics or religion, the rationale for "why it works" has remained remarkably stable for mathematics, which is why the ability to do or check mathematical proofs is often considered to be the most basic human knowledge.

where mathematics comes from

The specific structures investigated in mathematics often are those found useful in the natural sciences, most commonly in physics but increasingly in biology, to construct models. Indeed this is so successful that Eugene Wigner called it the The Unreasonable Effectiveness of Mathematics in the Natural Sciences in the 1960 paper of that name.

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. It is finding a motivation for these more abstract investigations, discoveries or inventions that ofte

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 even consider mathematics created or invented, not discovered or revealed.

Another view is that mathematics is simply a human way to organize things that is not shared by other beings: recent experiments with near human relatives like chimpanzees seem to prove though that counting and even adding and substracting with symbols (arithmetic) are skills learnable by non-humans. Wigner found it "useful, in epistemological discussions, to abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species." Anthropology and primatology now often study "folk mathematics", informal and intuitive beliefs about mathematical constructs.

history / origins

The word "mathematics" comes from the Greek μάθημα (m�thema) which means "science, knowledge, or learning"; μαθηματικός (mathematik�s) means "fond of learning".

There is some study of historicism in mathematics but because of the very durable nature of mathematical practices and proofs, there is a strong bias against explaining any given mathematical idea as a product of mere "history".

Historically, 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 origin of mathematics is generally believed to be related to a certain level of cultural complexity required to manage engineering, agriculture and large scale social organization. Early accounting for instance emerged in Sumeria, and in China, alongside sophisticated devices like the abacus and the very complex systems of measuring stellar precession. In many places the emergence of mathematics was as a mystical or religious practice, no less in Greece, where Pythagoras ran a school of such adepts, even going so far as to drown a student in a barrel for revealing that irrational numbers (which violated Pythagoras' ideology) actually existed.

In general, today's mathematicians disdain such brute force methods. But there are still allegations that some beliefs in mathematics are so convenient they are accepted without adequate proof, and that social pressures define what is accepted as such proof. To the degree this is so, history may play a role in what we believe about mathematics, and so, about our larger universe.

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 coordinate system, 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.

foundations and practices

In order to clarify and investigate the foundations of mathematics in the 20th century, the fields of set theory, mathematical logic and model theory were developed, and were partially successful at explaining where mathematics "comes from". The rise of quasi-empiricism in mathematics and an increasing concern with cognition and perception has stalled these lines of inquiry however: the so-called meta-mathematics which is part of mathematics itself now is generally not believed to be adequate to explain what Wigner called The Unreasonable Effectiveness of its symbols.

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.

Computer-generated proofs have also come under scrutiny as they obscure the boundary between mathematical practices and software engineering.

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.

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.
-Bertrand Russell

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

Change

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

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 versus Quasi-empiricism in mathematics versus cognitive science of mathematics -- Symbolic logic, Set theory, Model theory andCategory theory -- Mathematical proof or theorem proving ("mathematical practice") -- mathematics as a language - folk 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 thought

Mathematics is Not...

Further Reading

  • Davis, Philip J.; Hersh, Reuben: The Mathematical Experience. Birkh�user, 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.
  • Mathematical Society of Japan: Encyclopedic Dictionary of Mathematics, 2nd ed.. MIT Press, Cambridge, Mass., 1993. Definitions, theorems and references.
  • Michiel Hazewinkel (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.

External Links

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References

References

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References

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

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 versus Quasi-empiricism in mathematics versus cognitive science of mathematics -- Symbolic logic, Set theory, Model theory andCategory theory -- Mathematical proof or theorem proving ("mathematical practice") -- mathematics as a language - folk 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 thought

Mathematics is Not...

Further Reading

  • Davis, Philip J.; Hersh, Reuben: The Mathematical Experience. Birkh�user, 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.
  • Mathematical Society of Japan: Encyclopedic Dictionary of Mathematics, 2nd ed.. MIT Press, Cambridge, Mass., 1993. Definitions, theorems and references.
  • Michiel Hazewinkel (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.

External Links

[[ar:�������]][[ca:Matem�tiques]][[es:Matem�ticas]][[fr:Math�matiques]][[pt:Matem�tica]]

References

References

[[ar:�������]][[ca:Matem�tiques]][[es:Matem�ticas]][[fr:Math�matiques]][[pt:Matem�tica]]

References

[[ar:�������]][[ca:Matem�tiques]][[es:Matem�ticas]][[fr:Math�matiques]][[pt:Matem�tica]]

References

References

[[ar:�������]][[ca:Matem�tiques]][[es:Matem�ticas]][[fr:Math�matiques]][[pt:Matem�tica]]

References

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