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Mathematics

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Euclid, Greek mathematician, 3rd century BC, known today as the father
   of geometry; shown here in a detail of The School of Athens by Raphael.
   Enlarge
   Euclid, Greek mathematician, 3rd century BC, known today as the father
   of geometry; shown here in a detail of The School of Athens by Raphael.

   Mathematics ( colloquially, maths, or math in American English) is the
   body of knowledge centered on concepts such as quantity, structure,
   space, and change, and the academic discipline which studies them;
   Benjamin Peirce called it "the science that draws necessary
   conclusions". It evolved, through the use of abstraction and logical
   reasoning, from counting, calculation, measurement, and the study of
   the shapes and motions of physical objects. Mathematicians explore such
   concepts, aiming to formulate new conjectures and establish their truth
   by rigorous deduction from appropriately chosen axioms and definitions.

   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 ancient mathematical texts originating in ancient
   Egypt, Mesopotamia, Ancient India, and Ancient China, with increased
   rigour later introduced by the ancient Greeks. From this point on, the
   development continued in short bursts until the Renaissance period of
   the 16th century where mathematical innovations interacted with new
   scientific discoveries leading to an acceleration in understanding that
   continues to the present day.

   Today, mathematics is used throughout the world in many fields,
   including science, engineering, medicine and economics. The application
   of mathematics to such fields, often dubbed applied mathematics,
   inspires and makes use of new mathematical discoveries and has
   sometimes led to the development of entirely new disciplines.
   Mathematicians also engage in pure mathematics for its own sake without
   having any practical application in mind, although applications for
   what begins as pure mathematics are often discovered later.

Etymology

   The word "mathematics" (Greek: μαθηματικά) 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, μαθηματικὴ τέχνη (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".

   Despite the form and etymology, the word mathematics, like the names of
   arts and sciences in general, is used as a singular mass noun in
   English today. The colloquial English-language shortened forms
   perpetuate this singular/plural idiosyncrasy, as the word is shortened
   to math in North American English, while it is maths elsewhere
   (including Britain, Ireland, Australia and other Commonwealth
   countries).

History

   A quipu, a counting device used by the Inca.
   Enlarge
   A quipu, a counting device used by the Inca.

   The evolution of mathematics might be seen to be 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. Monolithic monuments testify to knowledge of geometry.

   Further steps need writing or some other system for recording numbers
   such as tallies or the knotted strings called quipu used by the Inca
   empire to store numerical data. Numeral systems have been many and
   diverse.

   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

   Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.
   Enlarge
   Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.

   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. Newton was one of the infinitesimal calculus
   inventors, Feynman invented the Feynman path integral using a
   combination of reasoning and physical insight, and today's string
   theory also inspires 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.

   Many mathematicians talk about the elegance of mathematics, its
   intrinsic aesthetics and inner beauty. Simplicity and generality are
   valued. There is beauty also in a clever proof, such as Euclid's proof
   that there are infinitely many prime numbers, and in a 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.

Notation, language, and rigor

   In modern notation, simple expressions can describe complex concepts.
   This image is generated by a single equation.
   Enlarge
   In modern notation, simple expressions can describe complex concepts.
   This image is generated by a single equation.

   Most of the mathematical notation we use today was not invented until
   the 16th century. Before that, mathematics was written out in words, a
   painstaking process that limited mathematical discovery. 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. It was said that Henri Poincaré
   was only elected to the Académie française so that he could tell them
   how to define automorphe in their dictionary. 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, while known as the "prince of mathematicians",
   did not believe that mathematics was worthy of study in its own
   right[citation needed].
   Enlarge
   Carl Friedrich Gauss, while known as the "prince of mathematicians",
   did not believe that mathematics was worthy of study in its own right.

   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. While some in
   applied mathematics feel that they are scientists, those in pure
   mathematics often feel that they are working in an area more akin to
   logic and that they are, hence, fundamentally philosophers. 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
   usually considered the equivalent of science's Nobel prize. Another
   major international award, the Abel Prize, was introduced in 2003. Both
   of these are awarded for a particular body of work, either innovation
   in a new area of mathematics 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

   Early mathematics was entirely concerned with the need to perform
   practical calculations, as reflected in this Chinese abacus.
   Enlarge
   Early mathematics was entirely concerned with the need to perform
   practical calculations, as reflected in this Chinese abacus.

   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 recognised
   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 generalised 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 formalise the
   concept of counting to infinite. 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.

   1, 2, 3\,\! -2, -1, 0, 1, 2\,\! -2, \frac{2}{3}, 1.21\,\! -e, \sqrt{2},
   3, \pi\,\! 2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!
   Natural numbers Integers Rational numbers Real numbers Complex numbers

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.

          Number theory Abstract algebra Group theory Order theory

Space

   The study of space originates with geometry - in particular, Euclidean
   geometry. Trigonometry combines space and number, 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 fibre 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 colour
   theorem, whose only proof, by computer, has never been verified by a
   human.

          Geometry Trigonometry Differential geometry Topology Fractal geometry

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 behaviour.
   Calculus Vector calculus Differential equations Dynamical systems Chaos
                                                                     theory

Foundations and philosophy

   In order to clarify the foundations of mathematics, the fields of
   mathematical logic and set theory were developed.

   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
   there are always true theorems which cannot be proven. Modern logic is
   divided into recursion theory, model theory, and proof theory, and is
   closely linked to theoretical computer science.

          P \Rightarrow Q \,
          Mathematical logic Set theory Category theory

Discrete mathematics

   Discrete mathematics is the common name for the fields of mathematics
   most generally useful in theoretical computer science. This includes
   computability theory, computational complexity theory, and information
   theory. Computability theory examines the limitations of various
   theoretical models of the computer, including the most powerful known
   model - the Turing machine. Complexity theory is the study of
   tractability by computer; some problems, although theoretically soluble
   by computer, are so expensive in terms of time or space that solving
   them is likely to remain practically unfeasible, even with rapid
   advance of computer hardware. Finally, information theory is concerned
   with the amount of data that can be stored on a given medium, and hence
   concepts such as compression and entropy.

   As a relatively new field, discrete mathematics has a number of
   fundamental open problems. The most famous of these is the " P=NP?"
   problem, one of the Millennium Prize Problems. It is widely believed
   that the answer to this problem is no.

   \begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) &
   (3,2,1) \end{matrix}
   Combinatorics Theory of computation Cryptography Graph theory

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.

          Mathematical physics • Analytical mechanics • Mathematical fluid
          dynamics • Numerical analysis • Optimization • Probability •
          Statistics • Mathematical economics • Financial mathematics •
          Game theory • Mathematical biology • Cryptography • Operations
          research

Common misconceptions

   Mathematics is not a closed intellectual system, in which everything
   has already been worked out. There is no shortage of open problems.

   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.

Relationship between 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 most systems of axioms are derived from our perceptions and
   experiments, they are not dependent on them. Nevertheless, mathematics
   remains extremely useful for solving real-world problems. This fact led
   Eugene Wigner to write an essay, The Unreasonable Effectiveness of
   Mathematics in the Natural Sciences.

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