Look up Portal, portal, or portals in Wiktionary, the free dictionary.

Portal may refer to:

• Portal (architecture), a gate, door, or the extremeties (ends) of a tunnel
• Portal (fiction), a magical or technological doorway that connects two different locations.

Computing[link]

• Enterprise portal, a framework to provide a single point of access to a variety of information and tools
• Intranet portal, a gateway that unifies access to all enterprise information and applications
• Portal rendering, an optimization technique in 3D computer graphics
• Web portal, a site that functions as a point of access to information on the Internet
• Portals network programming api, a high-performance networking programming interface for massively parallel supercomputers

Entertainment[link]

• Portal (interactive novel), a 1986 video game by Activision
• Portal (video game), a 2007 video game by Valve Corporation
• Portal 2, the sequel to Valve's 2007 video game Portal
• Portal (TV series), a series about MMORPGs

Geography[link]

• Portal, California
• Portal, Georgia
• Portal, North Dakota

Music[link]

• Portall, a Thrash-Death Metal band from the Netherlands
• Portal (Australian band), a metal band from Australia
• Portal (UK band), a band from the UK
• Portals (album), a 2009 album by Arsonists Get All The Girls

Other uses[link]

• Portal (comics), a Marvel comic character
• Portal (Magic: The Gathering), a set in the Magic: The Gathering card game
• Portal Bridge, over the Hackensack River in New Jersey
• Portal Software, a company based in Cupertino, California
• Clube Atlético Portal, a football club based in Uberlândia, Brazil
• Portals Athletic F.C., a defunct football club, based in Overton, England
• Portal (magic trick), an illusion performed by David Copperfield

People with the surname[link]

• Antoine Portal (1742–1832), French anatomist, doctor and medical historian
• Charles Portal, 1st Viscount Portal of Hungerford (1893–1971), British Marshal of the RAF and Chief of the Air Staff in World War II
• Wyndham Portal, 1st Viscount Portal (1885–1949), British businessman and public servant
• Michel Portal (born 1935), French composer, saxist, and clarinetist
• Nicolas Portal (born 1979), French professional road bicycle racer
• Gerald Portal (1858–1894), British colonial diplomat

See also[link]

Portal portal

Portal Directory

• El Portal, Florida
• Hepatic portal vein, a vein that drains blood from the digestive system
• Portal venous system, an occurrence where one capillary bed drains into another through veins
• Portal frame, a construction method
• Portal stones, a type of stone monument
• Porthole

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.^[1]

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of quantity, structure, space, and change.^[2] Mathematicians seek out patterns^[3][4] and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.^[5]

Galileo Galilei (1564–1642) said, 'The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth'.^[6] Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences".^[7] Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions".^[8] David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise."^[9] Albert Einstein (1879–1955) 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".^[10]

Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.^[11]

Etymology[link]

The word "mathematics" comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means what one learns, what one gets to know, hence also study and science, and in modern Greek just lesson. The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean to learn. In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study", even in Classical times.^[12] Its adjective is μαθηματικός (mathēmatikós), meaning related to learning or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant the mathematical art.

In Latin, and in English until around 1700, the term "mathematics" more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of "mathematici" meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.

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 (384-322BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.^[13] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.

History[link]

Greek mathematician Pythagoras (c.570-c.495 BC), commonly credited with discovering the Pythagorean theorem.

The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,^[14] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years.^[15] Elementary arithmetic (addition, subtraction, multiplication and division) naturally followed.

Since numeracy pre-dated writing, further steps were needed for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.^{[citation needed]} 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.^{[citation needed]}

Mayan numerals

The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.^[16] The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.^[17]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries 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."^[18]

Inspiration, pure and applied mathematics, and aesthetics[link]

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

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.^[19] 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. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This 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".^[20] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.^[21] 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. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.^[22] Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to Paul Erdős.^[23][24] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor[link]

Leonhard Euler, who created and popularized much of the mathematical notation used today

Most of the mathematical notation in use today was not invented until the 16th century.^[25] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.^[26] Euler (1707–1783) 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 (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.

Mathematical language can be difficult to understand for beginners. Words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open and field have been given specialized mathematical meanings. Technical terms such as homeomorphism and integrable have precise meanings in mathematics. Additionally, shorthand phrases such as "iff" for "if and only if" belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof is fundamentally a matter of rigor. 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.^[27] 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. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. 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.^[28]

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.^[29]

Fields of mathematics[link]

An abacus, a simple calculating tool used since ancient times.

Mathematics can, broadly speaking, be subdivided 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.

Foundations and philosophy[link]

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.^[30] Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer-Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective 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). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a complete 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^{[citation needed]}, as well as to Category Theory.

Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware. A famous problem is the "P=NP?" problem, one of the Millennium Prize Problems.^[31] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.

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Mathematical logic	Set theory	Category theory	Theory of computation

Pure mathematics[link]

Quantity[link]

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, from which come such popular results as Fermat's Last Theorem. The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory.

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 "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.

Failed to parse (Missing texvc executable; please see math/README to configure.): 1, 2, 3\,...\!	Failed to parse (Missing texvc executable; please see math/README to configure.): ...-2, -1, 0, 1, 2\,...\!	Failed to parse (Missing texvc executable; please see math/README to configure.): -2, \frac{2}{3}, 1.21\,\!	Failed to parse (Missing texvc executable; please see math/README to configure.): -e, \sqrt{2}, 3, \pi\,\!	Failed to parse (Missing texvc executable; please see math/README to configure.): 2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!
Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers

Structure[link]

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix}
Combinatorics	Number theory	Group theory	Graph theory	Order theory

Space[link]

The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it 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. Convex and discrete geometry was developed to solve problems in number theory and functional analysis but now is pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. 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; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with the help of computers.

Geometry	Trigonometry	Differential geometry	Topology	Fractal geometry	Measure theory

Change[link]

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 functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. 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.

Calculus	Vector calculus	Differential equations	Dynamical systems	Chaos theory	Complex analysis

Applied mathematics[link]

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation, study, and use of mathematical models in science, engineering, and other areas of mathematical practice.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Statistics and other decision sciences[link]

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments;^[32] the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data.^[33]

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.^[34] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.^[35]

Computational mathematics[link]

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic matrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

Mathematical physics	Fluid dynamics	Numerical analysis	Optimization	Probability theory	Statistics	Cryptography
Mathematical finance	Game theory	Mathematical biology	Mathematical chemistry	Mathematical economics	Control theory

Mathematics as profession[link]

Arguably the most prestigious award in mathematics is the Fields Medal,^[36][37] established in 1936 and now awarded every 4 years. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize.

The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field.

A famous list of 23 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.

Mathematics as science[link]

Carl Friedrich Gauss, known as the "prince of mathematicians".^[38]

Gauss referred to mathematics as "the Queen of the Sciences".^[7] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means a "field of knowledge", and this was the original meaning of "science" in English, also. Of course, mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to natural science follows the rise of Baconian science, which contrasted "natural science" to scholasticism, the Aristotelean method of inquiring from first principles. Of course, the role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as psychology, biology, or physics. Albert Einstein 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."^[10] More recently, Marcus du Sautoy has called mathematics 'the Queen of Science...the main driving force behind scientific discovery'.^[39]

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.^[40] However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians^[who?] that mathematics cannot be reduced to logic alone, 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."^[41] 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.^[42] 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.^{[citation needed]}

The opinions of mathematicians on this matter are varied. Many mathematicians^[who?] 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^[who?] 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.^{[citation needed]}

See also[link]

Mathematics portal

	Book: Mathematics
Wikipedia books are collections of articles that can be downloaded or ordered in print.

• Definitions of mathematics
• Mathematics and art
• Mathematics education

Notes[link]

References[link]

Further reading[link]

External links[link]

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The School of Mathematics

Method Man
December 21, 2010
Background information
Birth name	Clifford Smith
Also known as	Meth, Tical, Johnny Blaze, Methtical, Shakwon, Iron Lung, Hot Nikkels, MZA, John J
Born	(1971-04-01) April 1, 1971 (age 41)[1][2] Staten Island, New York City, United States
Genres	Hip hop
Occupations	Rapper, actor
Years active	1990–present (rapping) 1995–present (acting)
Labels	Def Jam
Associated acts	Wu-Tang Clan, Redman, Erick Sermon, LL Cool J, The Notorious B.I.G., DMX, Xzibit, Shyheim
Website	method-man.com

Clifford Smith (born April 1, 1971),^[3] better known by his stage name Method Man, is an American hip hop artist, record producer, actor and member of the hip hop collective Wu-Tang Clan. He took his stage name from the 1979 film The Fearless Young Boxer, also known as Method Man. He is one half of the rap duo Method Man & Redman. He won a Grammy Award for Best Rap Performance by a Duo or Group for "I'll Be There for You/You're All I Need to Get By" with Mary J. Blige. In 2007, the writers of About.com placed him on their list of the Top 50 MCs of Our Time (1987–2007).^[4]

Method Man appeared in the motion pictures Belly, How High, Garden State as well as a minor role in The Wackness. More recently, he appeared as a crewman in George Lucas' film Red Tails. On television, he and hip-hop collaborator Redman co-starred on the short-lived Fox sitcom Method & Red, and he had a recurring role on HBO's Oz as Tug Daniels and as Calvin "Cheese" Wagstaff on the acclaimed HBO drama series The Wire.

Early life[link]

Born in Staten Island, New York, on April 1 , 1971, Clifford Smith divided his childhood between his father's Long Island residence and his mother's home in the Park Hill section of Staten Island aka Killa Hill.^[5] He has two sisters, Terri and Missy.^[5]

Rapping career[link]

[edit] 1992–96: Enter the Wu-Tang (36 Chambers) and Tical

As Wu-Tang Clan ascended to hip hop stardom, Method Man was always one of the most visible members of the collective. He was one of only two members to get a solo song on the group's debut album Enter the Wu-Tang: 36 Chambers and he was the first to release a solo album under the Clan's unusual contract which allowed its members to release albums under any record label (Method chose to sign with rap label Def Jam). Method Man's solo debut, Tical (1994) was critically acclaimed and well received, entering the American charts at #4 and eventually selling in excess of one million copies. That album featured the hit single "All I Need" which was later remixed featuring Mary J Blige, which won a Grammy "I'll Be There for You/You're All I Need,"). during this time Method Man also became close friends with The Notorious B.I.G., and was the only guest rapper featured on his debut album Ready to Die. He was also featured on Spice 1's album "AmeriKKKa's Nightmare" on the track "Hard 2 Kill". In 1995, he was also featured on "Got the Flava" off Showbiz and A.G.'s album Goodfellas. In 1996, Method Man appeared on Tupac Shakur's album All Eyez on Me, on the song "Got My Mind Made Up" alongside his rhyme partner Redman, The Dogg Pound (Daz and Kurupt) and Inspectah Deck (Rebel INS), whose verse did not make the released album version (although his nickname "Rebel INS" can be heard as the song fades).

[edit] 1997–98: Wu-Tang Forever and Tical 2000: Judgement Day

On June 3, 1997 the Wu-Tang Clan released their Grammy-nominated multiplatinum double CD Wu-Tang Forever, the long-awaited follow up to 36 Chambers. The album has sold over 8.3 million copies to date worldwide.

His second solo album was Tical 2000: Judgement Day, released in 1998, which was heavily influenced by the apocalypse theories surrounding the forthcoming end of the millennium, and which featured myriad guest appearances from his fellow Wu-Tang MCs, proving for the world that Method Man is in fact America's most blunted(with the help of Red Man). The album was certified double platinum. Other guest appearances include Lisa "Left Eye" Lopes, D'Angelo, Chris Rock, Mobb Deep, Redman, and brief cameos from Russell Simmons, Bishop Don "Magic" Juan, Janet Jackson, and Donald Trump. The album sold better than his first fueled by the party track "Judgment Day" and the D'Angelo collaboration "Break Ups 2 Make Ups", earning Platinum and Gold certifications in the U.S. and Canada respectively. Reviews for the album were mixed and its long runtime and abundance of intermittent comedy skits were widely criticized. Producers on this album included True Master, 4th Disciple and the RZA.

[edit] 1999–2000: Blackout! (with Redman)

Method Man was part of the very successful Hard Knock Life Tour with Jay-Z, Redman, Ja Rule, and DMX. During this tour, Method Man & Redman recorded Blackout!, a light-hearted, bass-heavy, profanity-laced, party record with an EPMD-evoking emphasis on funky beats and the mischievous wit and cool flows and good rhythm of the two MCs. The album reached platinum status quickly, both in the U.S. and Canada, fueled by "Da Rockwilder", "Cereal Killa", "1, 2, 1, 2", "Tear It Off" and "Y.O.U.". This album also featured three previously released tracks on which the two collaborated.

Their success would lead the duo on to star in movies and TV shows, become product spokespersons and household names, but also associated them with marijuana use in the media. The most immediate results of their success was their co-starring roles in the major motion picture film How High, their endorsement deal for Right Guard, Redman's starring role in Seed of Chucky and a short-lived sitcom on Fox Television entitled Method & Red.

[edit] 2000–04: The W, Iron Flag, and Tihkal 0: The Prequel

The Wu-Tang Clan released The W on November 21, 2000 and Iron Flag on December 18, 2001. The W received both critical and commercial success for the group, while Iron Flag did receive some but not to the effect of The W. The efforts earned two more platinum plaques for the Wu-Tang Clan.

In 2004, Meth released his third solo album Tical 0: The Prequel, which featured the hit party single "What's Happenin'" with Busta Rhymes. Hip hop critics voiced their displeasure with the album, many agreeing that Tical 0 felt like generic party rap and featured too many mainstream guests, detracting from his own performances. Regardless, this album sold reasonably well and was certified gold record by the RIAA relatively quickly, but would not see the platinum success of his previous solo releases. There was trouble even before the album's release when Method apparently complained to the press about excessive interference from Def Jam over the album's beats (Meth supposedly desired more input from Wu-Tang leader RZA). On its release, many fans and critics were taken aback by its strong "mainstream" or "commercial" sound, highlighted by the guest appearances of pop-rap stars like Missy Elliott and P. Diddy, two artists that are involved with much different facets of rap music.

P. Diddy was one of the executive producers for the album, although Meth later voiced his displeasure with the final product. "On the third LP, it was suggested (by Def Jam) to bring in Harve Pierre and P. Diddy. Who am I to argue? Puff knows how to sell some records. But that wasn't the direction to go in, and I know that."^[6]

[edit] 2006–07: 4:21... The Day After

Method Man at the 2007 Eurockéennes.

Method Man's fourth album, entitled 4:21: The Day After was released in August 2006 with a star lineup of producers featuring Havoc, Erick Sermon, Scott Storch, Allah Mathematics, Mr. Porter, and, most importantly to Meth, RZA. This time around, a more focused Method Man went back to his hip hop roots and both hip hop fans and the media took notice. He did an interview on the ItsHipHop.Tv.^[7] Despite this being one of Meth's strongest solo efforts to date, the album failed to do well commercially due to it having no single or video, which Method Man has held discontent towards his own label for. However, he has been touring strongly all over the world to promote the album, and has appeared onstage with fellow Wu-Tang member Inspectah Deck, as well as New York up and comers Saigon, and Gat Murdah. Meth contributed various reasons for the problems between him and his label, Def Jam. While he puts most of the blame on personal agendas in the Def Jam offices, Meth did take some blame, himself, for giving into his record label.

In early May 2007, Method Man's camp leaked the street single "New York New York" which became a popular track on the internet.^[8]

[edit] 2007–10: Blackout! 2

On March 27, 2007, Redman confirmed on BET's Rap City: Tha Bassment that the sequel to How High, How High 2, is currently being written.

In an April 10, 2007 Onion AV Club interview,^[9] Redman hinted that there would be a second collaborative album with Method Man, with work beginning in midsummer or early September.

In early 2008, a remake of the Smoothe da Hustler and Trigger tha Gambler classic Broken Language was released to the internet by the duo entitled Broken Language 2008, fueling rumors of a Blackout! sequel coming soon. This rumor was further fueled by the duo while performing in Gainesville, Florida at the University of Florida. A Blackout! 2 LP was scheduled for a December 9, 2008 release but was recently pushed back to the first quarter of 2009, with a new release date on May 19. Bun B has confirmed that he will guest star on Blackout! 2 – in April 2009, a single has been released named City Lights, produced by Nasty Kutt ^[10] Also producers such as Erick Sermon, Rockwilder and Pete Rock have announced their presence on the album.

The duo has finished their Still High tour with Termanalogy, the Alchemist, and Evidence of Dilated Peoples.

[edit] 2011–present: Crystal Meth

Crystal Meth is Method Man's upcoming fifth studio album.^[11][12][13] The album has no confirmed release date. The Crystal Meth was first announced before Method Man began working on his and Redman's sophomore collaboration album, Blackout! 2. In the liner notes of that album, it was given a scheduled release date for 2009. The album, however, was further postponed until it was later mentioned in an MTV interview it would be released in early 2010.^[11] no update on the album's progress or release date has been made. At a concert on the 23 of December 2010 Method Man told the crowd to look for his album in March 2011. However the album was pushed back to an unknown release date.

In a recent interview with The Come Up Show following a recent performance in Canada in April 2011, Method Man replied with this regarding his current 'weed-loving' image:"When you get older and you've got kids and your kids are going to school and you know [their] teachers...and they see how active you are and concerned [you are] with your kids' education or well being, it's hard to sit there and be taken seriously if people are always talking about he's always high...which is totally not the case," he said. "When I first came out, I was young, we were doing our thing, we smoked a lot...and we didn't care if the world knew. Now, I have to use more discretion because of my kids. This is not for me; everything I do is for them now, so I use a bit more discretion and I don't put weed as a forefront any more."

On October 5, 2011, a new single from Method Man, titled "World Gone Sour (The Lost Kids)", was released on iTunes.^[14] Recently, he confirmed that the album will come out next year and will have the usual line-up of producers including RZA, Rockwilder, Erick Sermon, 3rd Digala of Digem Tracks and some other new underground producers he's begun to work with.^[15] He also expressed a desire to work with Odd Future frontman Tyler, The Creator.^[16] It was recently announced via Facebook that Crystal Meth will be his last album & to be fully produced by RZA.

Acting career[link]

In the early 2000s Method Man began a career in acting. He has had recurring roles in critically acclaimed television shows such as HBO's Oz as Tug Daniels, HBO's The Wire in which he plays Prop Joe's nephew Cheese, The Twilight Zone and CSI: Crime Scene Investigation. He and Redman hosted a pilot on MTV called Stung. He has made numerous appearances as himself on TV shows such as Mind of Mencia and Chappelle's Show.

Method Man portrays Drops on CSI, a wealthy Las Vegas party promoter who clashes with the CSI team, specifically investigator Nick Stokes, in their investigations involving his clubs or entourage. His first appearance on CSI as Drops was in the 2006 episode "Poppin' Tags". He resumed the role in the 2007 episode "Big Shots" and again in the 2008 episode "Drops Out".

His first prominent role came in 1998 with the film Belly along with fellow rappers Nas and DMX. He has since added many credits to his name, including roles in the films Garden State, One Eight Seven, and many others, with starring roles in the feature films such as How High and Soul Plane. On March 27, 2007 Redman confirmed on BET's show Rap City that the sequel to How High was being written. The script for How High 2 is being written by Dustin Lee Abraham of CSI, who also wrote the first movie. In 2005 Method Man also had a cameo in the horror movie Venom, where he played a deputy who shortly into the movie is killed. He appears in the 2008 movies The Wackness and also, Meet the Spartans.

Method Man stars in the episode "Snitch" of Law & Order SVU as the main antagonist. The episode was first broadcast December 4, 2007.

Method Man has made an appearance in the Def Jam series of video games. In Fight for NY he voiced Blaze, one of the main characters. In Icon, he voiced Gooch, a major character in the storyline. In Underground, He voiced Meth, one of the major characters. He made a guest appearance in the music video for the 2003 "If I Ain't Got You" by Alicia Keys, where he played the role of her boyfriend. He also appeared in Beanie Sigel's music video "Feel It in the Air", where Method Man played an undercover cop leading an operation against Sigel.

Method Man has fallen back from pursuing more acting roles after the situation with his sitcom on Fox left a bad taste in his mouth,^[17] and now mostly just acts if the project is being handled by a friend of his, as was the case with CSI and The Wire. He also played an arsonist and a wealthy executive as well as main antagonist in an episode of the fox TV show The Good Guys.

Method Man appears as a hip hop business mogul in an episode of Burn Notice.

Method man had a cameo appearance in the 1997 film Cop Land as a physically violent fleeing criminal that throws Peter Berg's character off of a New York rooftop. Method Man has also appeared in the TV drama Wonderland, as a patient in a mental hospital.

Method man plays a small role in the 2011 film "The Sitter" starring Jonah Hill. In 2012 he played crewman "Sticks" in George Lucas' movie Red Tails about the Tuskegee airmen.

Other ventures[link]

Method Man appeared in the 1995 documentary entitled The Show. There is a memorable scene in which Method Man, on a train in Japan, gets into an argument with U-God and Ghostface Killah, over camera time, radio interviews, and clothing mishaps.

In 2006, Method Man appeared on the MTV reality game show Yo Momma in the first episode of Season 1.

Method Man is the first of the Wu-Tang Clan to produce a series of eponymous graphic novels for Hachette Book Group USA's imprint Grand Central Publishing^[18] (to be followed by GZA and Ghostface Killah).^[19]

Personal life[link]

He became engaged to his wife in 1999 and they married in 2001.^[5] He has three children: two sons (born in 1996 and 2001), and one daughter (born in 1997). Method Man also has collabed with Rochester rapper "Dale Mahoney" on his project "Jordans&Drugs" in 2012.^[20]

Legal issues[link]

On Thursday May 17, 2007 Method Man was arrested in New York City on marijuana charges. His Mercedes-Benz convertible was pulled over at the Battery Tunnel en route to Manhattan, and when he rolled his window down the officer noticed a strong smell of marijuana. "It was like something out of Cheech & Chong. He rolls down the window and the smoke would choke a horse," a source later said.^[21] The arresting officer said he noticed two blunts and a plastic bag with more marijuana in plain view. Upon further inspection more marijuana was found under the driver's seat. The arrest was made at about 10 P.M. Thursday night near the Battery Tunnel toll booths on Hamilton Ave. in Carroll Gardens. He was charged with unlawful possession of marijuana, criminal possession of marijuana, operating a motor vehicle under the influence and driving an uninspected motor vehicle. He later made many public service appearances to try to make up for it, most notably at Abraham Lincoln High School in Brooklyn.^[22]

On October 5, 2009, Method Man was arrested at his home in Staten Island for income tax violations.^[23] He was accused of failing to file income tax returns for the state of New York between 2004 and 2007 and owes nearly $33,000 in taxes. On June 28, 2010,^[24] Method Man plead guilty to tax evasion and was sentenced to a conditional discharge^[25] and paid a $106,000.00 fine.^[24]

Controversies[link]

Wu-Tang management[link]

In 2003, Method Man criticized Oli "Power" Grant and Mitchell "Divine" Diggs, the managers of the Wu-Tang business.^[26] "Number 1 on my shit list right now is Divine from Wu-Tang management. He took something major from me that he had no intention of giving back."^[27] Divine is multiplatinum record producer RZA's brother who along with Oli "Power" Grant took over the business end of Wu-Tang around 1997.

Aside from the financial issues, Method Man was unhappy with the decision to bring Wu-Tang into the fashion world for a brief period of time with Wu-Wear, despite the brand being a major money-maker for the group. "When Wu-Wear started making shoes and sneakers and pants, it was shoddy material. I never rocked that shit."^[27]

Wendy Williams[link]

In 2006 Method Man had a highly personal and highly publicized conflict with TV talk show host Wendy Williams on The Breakdown, an internet show on ONLOQ.com. Williams talked on air about Method Man's wife having cancer, which was something he had wanted to keep very private and even her own family members had not yet known about. He said that people who lived next door to him didn't even know, but Williams had dug it up and made it public over the radio. Because of this leaked information, Method Man and his wife had many problems.^[28]

Williams also reported rumors that Method Man had even been having an affair with his wife's doctor. Method Man first heard of this while in recording sessions in Los Angeles.

Sean Combs[link]

During a concert in 2006, Method Man criticized Diddy's decisions on the posthumous The Notorious B.I.G. Duets: The Final Chapter album, saying that Biggie never would have worked with some of the subpar rappers. "They got niggas on that album Big would have never rocked with," Meth said of the album. "Musically, I ain't fuckin' with Puff Daddy".^[30] He also brought up the fact that he was the only other rapper that Biggie chose to have on his debut album Ready To Die. He was featured on the track "The What".

Previously, Diddy had been one of the executive producers for Method Man's 2004 album Tical 0: The Prequel, to which Meth later voiced his displeasure on the final product. From then on, he focused on production from within the clan and its closely tied collaborators.

Fox Television[link]

He starred in a Fox sitcom called Method & Red in late 2004, however after only a short time on the air the show was put on hiatus and never returned. Method Man later complained in the press about Fox's influence on the show's style, claiming that "there's been too much compromise on our side and not enough on their side" and bemoaning the network's decision to add a laugh track. Before the show even aired for the first time, he was telling fans not to bother watching it. He told the newspaper:

Discography[link]

Filmography[link]

Method Man filmography

References[link]

External links[link]

Wikiquote has a collection of quotations related to: Method Man

• Method Man at the Internet Movie Database
• Official website