# Extensive Definition

Mathematics is the body of knowledge centered on
such concepts as quantity, structure, space, and change, and also the academic
discipline that studies them. Benjamin
Peirce called it "the science that draws necessary
conclusions". Other practitioners of mathematics maintain that
mathematics is the science of pattern, and that mathematicians seek out
patterns whether found in numbers, space, science, computers,
imaginary abstractions, or elsewhere. Mathematicians explore such
concepts, aiming to formulate new conjectures and establish
their truth by rigorous
deduction
from appropriately chosen axioms and definitions. Through the use
of abstraction
and logical reasoning, mathematics evolved
from counting, calculation, measurement, and the
systematic study of the shapes and motions
of physical objects. Knowledge and use of basic mathematics have
always been an inherent and integral part of individual and group
life. Refinements of the basic ideas are visible in mathematical
texts originating in the ancient
Egyptian, Mesopotamian,
Indian,
Chinese,
Greek
and Islamic
worlds. Rigorous arguments
first appeared in Greek
mathematics, most notably in Euclid's Elements.
The development continued in fitful bursts until the Renaissance
period of the 16th
century, when mathematical innovations interacted with new
scientific
discoveries, leading to an acceleration in research that
continues to the present day.

Today, mathematics is used throughout the world
in many fields, including natural
science, engineering, medicine, and the social
sciences such as economics. Applied
mathematics, the application of mathematics to such fields,
inspires and makes use of new mathematical discoveries and
sometimes leads to the development of entirely new disciplines.
Mathematicians also engage in pure
mathematics, or mathematics for its own sake, without having
any application in mind, although applications for what began as
pure mathematics are often discovered later.

## Etymology

The word "mathematics" (Greek: μαθηματικά or
mathēmatiká) comes from the Greek
μάθημα (máthēma), which means learning, study, science, and
additionally came to have the narrower and more technical meaning
"mathematical study", even in Classical times. Its adjective is
μαθηματικός (mathēmatikós), related to learning, or studious, which
likewise further came to mean mathematical. In particular,
(mathēmatikḗ tékhnē), in Latin ars
mathematica, meant the mathematical art.

The apparent plural form in English,
like the French
plural form les mathématiques (and the less commonly used singular
derivative la mathématique), goes back to the Latin neuter plural
mathematica (Cicero), based on
the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and
meaning roughly "all things mathematical". In English, however, the
noun mathematics takes singular verb forms. It is often shortened
to math in English-speaking North America and maths
elsewhere.

## History

The evolution of mathematics might be seen as an
ever-increasing series of abstractions, or
alternatively an expansion of subject matter. The first abstraction
was probably that of numbers. The realization that two
apples and two oranges have something in common was a breakthrough
in human thought. In addition to recognizing how to count physical
objects, prehistoric
peoples also recognized how to count abstract quantities, like
time — days, seasons, years. Arithmetic
(addition, subtraction, multiplication and
division),
naturally followed.

Further steps need writing or some other system for
recording numbers such as tallies or
the knotted strings called quipu used by the Inca to store
numerical data. Numeral
systems have been many and diverse, with the first known
written numerals created by Egyptians in Middle
Kingdom texts such as the Rhind
Mathematical Papyrus. The Indus
Valley civilization developed the modern decimal
system, including the concept of zero.

From the beginnings of recorded history, the
major disciplines within mathematics arose out of the need to do
calculations relating to taxation and commerce, to understand the
relationships among numbers, to measure
land, and to predict astronomical events. These
needs can be roughly related to the broad subdivision of
mathematics into the studies of quantity, structure, space, and
change.

Mathematics has since been greatly extended, and
there has been a fruitful interaction between mathematics and
science, to the benefit of both. Mathematical discoveries have been
made throughout history and continue to be made today. According to
Mikhail B. Sevryuk, in the January 2006 issue of the
Bulletin of the American Mathematical Society, "The number of
papers and books included in the Mathematical
Reviews database since 1940 (the first year of operation of MR)
is now more than 1.9 million, and more than 75 thousand items are
added to the database each year. The overwhelming majority of works
in this ocean contain new mathematical theorems and their proofs."

## Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, Richard Feynman invented the Feynman path integral using a combination of mathematical reasoning and physical insight, and today's string theory continues to inspire new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. The remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics."As in most areas of study, the explosion of
knowledge in the scientific age has led to specialization in
mathematics. One major distinction is between pure
mathematics and applied
mathematics. Several areas of applied mathematics have merged
with related traditions outside of mathematics and become
disciplines in their own right, including statistics, operations
research, and computer
science.

For those who are mathematically inclined, there
is often a definite aesthetic aspect to much of mathematics. Many
mathematicians talk about the elegance of mathematics, its
intrinsic aesthetics
and inner beauty.
Simplicity
and generality are
valued. There is beauty in a simple and elegant proof, such as
Euclid's
proof that there are infinitely many prime
numbers, and in an elegant numerical method that speeds
calculation, such as the fast
Fourier transform. G. H. Hardy
in A
Mathematician's Apology expressed the belief that these
aesthetic considerations are, in themselves, sufficient to justify
the study of pure mathematics. Mathematicians often strive to find
proofs of theorems that are particularly elegant, a quest Paul
Erdős often referred to as finding proofs from "The Book" in
which God had written down his favorite proofs. The popularity of
recreational
mathematics is another sign of the pleasure many find in
solving mathematical questions.

## Notation, language, and rigor

Most of the mathematical notation in use today
was not invented until the 16th
century. Before that, mathematics was written out in words, a
painstaking process that limited mathematical discovery. In the
18th
century, Euler was
responsible for many of the notations in use today. Modern notation
makes mathematics much easier for the professional, but beginners
often find it daunting. It is extremely compressed: a few symbols
contain a great deal of information. Like musical notation, modern
mathematical notation has a strict syntax and encodes information
that would be difficult to write in any other way.

Mathematical language also is hard for
beginners. Words such as or and only have more precise meanings
than in everyday speech. Also confusing to beginners, words such as
open and
field
have been given specialized mathematical meanings. Mathematical
jargon includes technical terms such as homeomorphism and integrable. But there is a
reason for special notation and technical jargon: mathematics
requires more precision than everyday speech. Mathematicians refer
to this precision of language and logic as "rigor".

Rigor is
fundamentally a matter of mathematical
proof. Mathematicians want their theorems to follow from axioms
by means of systematic reasoning. This is to avoid mistaken
"theorems", based on
fallible intuitions, of which many instances have occurred in the
history of the subject. The level of rigor expected in mathematics
has varied over time: the Greeks expected detailed arguments, but
at the time of Isaac Newton
the methods employed were less rigorous. Problems inherent in the
definitions used by Newton would lead to a resurgence of careful
analysis and formal proof in the 19th century. Today,
mathematicians continue to argue among themselves about computer-assisted
proofs. Since large computations are hard to verify, such
proofs may not be sufficiently rigorous. Axioms in traditional
thought were "self-evident truths", but that conception is
problematic. At a formal level, an axiom is just a string of
symbols,
which has an intrinsic meaning only in the context of all derivable
formulas of an axiomatic
system. It was the goal of Hilbert's
program to put all of mathematics on a firm axiomatic basis,
but according to
Gödel's incompleteness theorem every (sufficiently powerful)
axiomatic system has
undecidable formulas; and so a final axiomatization of
mathematics is impossible. Nonetheless mathematics is often
imagined to be (as far as its formal content) nothing but set theory in
some axiomatization, in the sense that every mathematical statement
or proof could be cast into formulas within set theory.

## Mathematics as science

Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein has stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."Many philosophers believe that mathematics is not
experimentally falsifiable,
and thus not a science according to the definition of Karl Popper.
However, in the 1930s important work in mathematical logic showed
that mathematics cannot be reduced to logic, and Karl Popper
concluded that "most mathematical theories are, like those of
physics and biology, hypothetico-deductive: pure mathematics
therefore turns out to be much closer to the natural sciences whose
hypotheses are conjectures, than it seemed even recently." Other
thinkers, notably Imre
Lakatos, have applied a version of falsificationism to
mathematics itself.

An alternative view is that certain scientific
fields (such as theoretical
physics) are mathematics with axioms that are intended to
correspond to reality. In fact, the theoretical physicist, J. M.
Ziman, proposed that science is public knowledge and thus
includes mathematics. In any case, mathematics shares much in
common with many fields in the physical sciences, notably the
exploration of the logical consequences of assumptions. Intuition
and experimentation
also play a role in the formulation of conjectures in both
mathematics and the (other) sciences. Experimental
mathematics continues to grow in importance within mathematics,
and computation and simulation are playing an increasing role in
both the sciences and mathematics, weakening the objection that
mathematics does not use the scientific
method. In his 2002 book A
New Kind of Science, Stephen
Wolfram argues that computational mathematics deserves to be
explored empirically as a scientific field in its own right.

The opinions of mathematicians on this matter are
varied. Many mathematicians feel that to call their area a science
is to downplay the importance of its aesthetic side, and its
history in the traditional seven liberal
arts; others feel that to ignore its connection to the sciences
is to turn a blind eye to the fact that the interface between
mathematics and its applications in science and engineering has driven much
development in mathematics. One way this difference of viewpoint
plays out is in the philosophical debate as to whether mathematics
is created (as in art) or discovered (as in science). It is common
to see universities
divided into sections that include a division of Science and
Mathematics, indicating that the fields are seen as being allied
but that they do not coincide. In practice, mathematicians are
typically grouped with scientists at the gross level but separated
at finer levels. This is one of many issues considered in the
philosophy
of mathematics.

Mathematical awards are generally kept separate
from their equivalents in science. The most prestigious award in
mathematics is the Fields Medal,
established in 1936 and now awarded every 4 years. It is often
considered, misleadingly, the equivalent of science's Nobel Prizes.
The
Wolf Prize in Mathematics, instituted in 1979, recognizes
lifetime achievement, and another major international award, the
Abel
Prize, was introduced in 2003. These are awarded for a
particular body of work, which may be innovation, or resolution of
an outstanding problem in an established field. A famous list of 23
such open problems, called "Hilbert's
problems", was compiled in 1900 by German mathematician
David
Hilbert. This list achieved great celebrity among
mathematicians, and at least nine of the problems have now been
solved. A new list of seven important problems, titled the
"Millennium
Prize Problems", was published in 2000. Solution of each of
these problems carries a $1 million reward, and only one (the
Riemann
hypothesis) is duplicated in Hilbert's problems.

## Fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.### Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, whence such popular results as Fermat's last theorem. Number theory also holds two widely-considered unsolved problems: the twin prime conjecture and Goldbach's conjecture.As the number system is further developed, the
integers are recognized as a subset of the rational
numbers ("fractions").
These, in turn, are contained within the real numbers,
which are used to represent continuous quantities. Real numbers are
generalized to complex
numbers. These are the first steps of a hierarchy of numbers
that goes on to include quarternions and octonions. Consideration of the
natural numbers also leads to the transfinite
numbers, which formalize the concept of counting to infinity.
Another area of study is size, which leads to the cardinal
numbers and then to another conception of infinity: the
aleph
numbers, which allow meaningful comparison of the size of
infinitely large sets.

### Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure. The structural properties of these objects are investigated in the study of groups, rings, fields and other abstract systems, which are themselves such objects. This is the field of abstract algebra. An important concept here is that of vectors, generalized to vector spaces, and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics: quantity, structure, and space. Vector calculus expands the field into a fourth fundamental area, that of change.### Space

The study of space originates with geometry - in particular, Euclidean geometry. Trigonometry combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles and calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics, and includes the long-standing Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.### Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and real-valued functions is known as real analysis, with complex analysis the equivalent field for the complex numbers. The Riemann hypothesis, one of the most fundamental open questions in mathematics, is drawn from complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.### Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed, as well as category theory which is still in development.Mathematical logic is concerned with setting
mathematics on a rigid axiomatic framework, and studying
the results of such a framework. As such, it is home to
Gödel's second incompleteness theorem, perhaps the most widely
celebrated result in logic, which (informally) implies that any
formal
system that contains basic arithmetic, if sound (meaning that
all theorems that can be proven are true), is necessarily
incomplete (meaning that there are true theorems which cannot be
proved in that system). Gödel showed how to construct, whatever the
given collection of number-theoretical axioms, a formal statement
in the logic that is a true number-theoretical fact, but which does
not follow from those axioms. Therefore no formal system is a true
axiomatization of full number theory. Modern logic is divided into
recursion
theory, model
theory, and proof
theory, and is closely linked to
theoretical computer
science.

### Applied mathematics

Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. Most experiments, surveys and observational studies require the informed use of statistics. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group.) Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically too large for human numerical capacity; it includes the study of rounding errors or other sources of error in computation.## Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Mathematicians publish many thousands of papers embodying new discoveries in mathematics every month.Pseudomathematics
is a form of mathematics-like activity undertaken outside academia, and occasionally by
mathematicians themselves. It often consists of determined attacks
on famous questions, consisting of proof-attempts made in an
isolated way (that is, long papers not supported by previously
published theory). The relationship to generally-accepted
mathematics is similar to that between pseudoscience and real
science. The misconceptions involved are normally based on:

- misunderstanding of the implications of mathematical rigor;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, often in the belief that the journal is biased against the author;
- lack of familiarity with, and therefore underestimation of, the existing literature.

The case of Kurt
Heegner's work shows that the mathematical establishment is
neither infallible, nor unwilling to admit error in assessing
'amateur' work. And like astronomy, mathematics owes
much to amateur contributors such as Fermat
and Mersenne.

### Mathematics and physical reality

Mathematical concepts and theorems need not
correspond to anything in the physical world. Insofar as a
correspondence does exist, while mathematicians and physicists may
select axioms and postulates that seem reasonable and intuitive, it
is not necessary for the basic assumptions within an axiomatic
system to be true in an empirical or physical sense. Thus, while
many axiom
systems are derived from our perceptions and experiments, they
are not dependent on them.

For example, we could say that the physical
concept of two apples may be accurately modeled
by the natural
number 2. On the other hand, we could also say that the natural
numbers are not an accurate model because there is no standard
"unit" apple and no two apples are exactly alike. The modeling idea
is further complicated by the possibility of fractional
or partial apples. So while it may be instructive to visualize the
axiomatic definition of the natural numbers as collections of
apples, the definition itself is not dependent upon nor derived
from any actual physical entities.

Nevertheless, mathematics remains extremely
useful for solving real-world problems. This fact led physicist
Eugene
Wigner to write an article titled "The
Unreasonable Effectiveness of Mathematics in the Natural
Sciences".

## See also

## Notes

## References

- Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford University Press, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4.
- Boyer, Carl B., A History of Mathematics, Wiley; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Courant, R. and H. Robbins, What Is Mathematics? : An Elementary Approach to Ideas and Methods, Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.— A gentle introduction to the world of mathematics.
- Eves, Howard, An Introduction to the History of Mathematics, Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
- Gullberg, Jan, Mathematics—From the Birth of Numbers. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X. — 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 mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online http://eom.springer.de/default.htm.
- Jourdain, Philip E. B., The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover, 2003, ISBN 0-486-43268-8.
- Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.
- Oxford English Dictionary, second edition, ed. John Simpson and Edmund Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.
- The Oxford Dictionary of English Etymology, 1983 reprint. ISBN 0-19-861112-9.
- Pappas, Theoni, The Joy Of Mathematics, Wide World Publishing; Revised edition (June 1989). ISBN 0-933174-65-9.
- ">http://links.jstor.org/sici?sici=0002-9327%281881%294%3A1%2F4%3C97%3ALAA%3E2.0.CO%3B2-X}} JSTOR.
- Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
- A Mathematician Reads the Newspaper
- In Search of a Better World: Lectures and Essays from Thirty Years
- Gauss zum Gedächtniss ">http://www.amazon.de/Gauss-Ged%e4chtnis-Wolfgang-Sartorius-Waltershausen/dp/3253017028}}

## External links

sisterlinks Mathematics- Online Encyclopaedia of Mathematics http://eom.springer.de from Springer. Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics.
- Some mathematics applets, at MIT
- Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics. (Can also be found here.)
- Stefanov, Alexandre: Textbooks in Mathematics. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: MathWorld: World of Mathematics. An online encyclopedia of mathematics.
- Polyanin, Andrei: EqWorld: The World of Mathematical Equations. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- Planet Math. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- Metamath. A site and a language, that formalize mathematics from its foundations.
- Mathematician Biographies. The MacTutor History of Mathematics archive Extensive history and quotes from all famous mathematicians.
- Cain, George: Online Mathematics Textbooks available free online.
- Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas.
- Nrich, a prize-winning site for students from age five from Cambridge University
- 'FreeScience Library->Mathematics ' The mathematics section of FreeScience library
- Open Problem Garden, a wiki of open problems in mathematics
- Applications of High School Algebra

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mathematic in Sundanese: Matematika

mathematic in Finnish: Matematiikka

mathematic in Swedish: Matematik

mathematic in Tagalog: Matematika

mathematic in Tamil: கணிதம்

mathematic in Tetum: Matemátika

mathematic in Thai: คณิตศาสตร์

mathematic in Vietnamese: Toán học

mathematic in Tajik: Математика

mathematic in Turkish: Matematik

mathematic in Turkmen: Matematika

mathematic in Ukrainian: Математика

mathematic in Venetian: Matemàtega

mathematic in Volapük: Matemat

mathematic in Võro: Matõmaatiga

mathematic in Waray (Philippines):
Matematika

mathematic in Wolof: Xayma

mathematic in Yiddish: מאטעמאטיק

mathematic in Yoruba: Mathematiki

mathematic in Contenese: 數學

mathematic in Dimli: Matematik

mathematic in Samogitian: Matematėka

mathematic in Chinese: 数学

mathematic in Slovak:
Matematika