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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 mathēmatik�) comes from
the
Greek μάθημα (m�thēma), 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 (mathēmatik�s), related to learning, or studious,
which likewise further came to mean mathematical. In particular,
μαθηματικὴ τέχνη (mathēmatikḗ t�khnē),
in Latin ars
mathematica, meant the mathematical art.
The apparent plural form in
English, like the
French plural form les math�matiques (and the less commonly used
singular derivative la math�matique), goes back to the Latin neuter
plural mathematica (Cicero),
based on the Greek plural τα μαθηματικά (ta mathēmatik�), 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.
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