Mathematics (from Greek μάθημα (''máthēma'') — knowledge, study, learning) is the study of quantity, structure, space, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic. Since the pioneering work of Giuseppe Peano, David Hilbert, 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 evolved 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 continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world in AD 800, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.

The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions". David Hilbert defined mathematics as follows: ''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.'' 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."

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 and sometimes leads 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, although practical applications for what began as pure mathematics are often discovered.

Etymology

The word "mathematics" comes from the Greek μάθημα (''máthēma''), which means in ancient Greek ''what one learns'', ''what one gets to know'', hence also ''study'' and ''science'', and in modern Greek just ''lesson''.

The word ''máthēma'' comes from μανθάνω (''manthano'') in ancient Greek and from μαθαίνω (''mathaino'') in modern Greek, both of which mean ''to learn''.

The word "mathematics" came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is (''mathēmatikós''), meaning ''related to learning'' or ''studious'', which likewise further came to mean ''mathematical''. In particular, (''mathēmatikḗ tékhnē''), , meant ''the mathematical art''.

The apparent plural form in English, like the French plural form (and the less commonly used singular derivative ), goes back to the Latin neuter plural (Cicero), based on the Greek plural , used by Aristotle, 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. In English, the noun ''mathematics'' takes singular verb forms. It is often shortened to ''maths'' or, in English-speaking North America, ''math''.

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, which is shared by many animals, 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. 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. 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 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. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.

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."

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; nowadays, 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. 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". 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. 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. Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to Paul Erdős. 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. 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. 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.

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.

Fields of mathematics

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

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. 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 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.

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 rapid advance of computer hardware. A famous problem is the "P=NP?" problem, one of the Millennium Prize Problems. 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.

:{| style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="15" | $p\; \backslash Rightarrow\; q\; \backslash ,$|| |- | Mathematical logic || Set theory || Category theory || Theory of computation |}

Pure mathematics

====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, 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.

:{| style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="20" | $1,\; 2,\; 3\backslash ,...\backslash !$ || $...-2,\; -1,\; 0,\; 1,\; 2\backslash ,...\backslash !$ || $-2,\; \backslash frac\{2\}\{3\},\; 1.21\backslash ,\backslash !$ || $-e,\; \backslash sqrt\{2\},\; 3,\; \backslash pi\backslash ,\backslash !$ || $2,\; i,\; -2+3i,\; 2e^\{i\backslash frac\{4\backslash pi\}\{3\}\}\backslash ,\backslash !$ |- | Natural numbers|| Integers || Rational numbers || Real numbers || Complex numbers |}

====Structure==== 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.

:{| style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="15" | || |- | Combinatorics || Number theory || Group theory || Graph theory || Order theory |}

====Space==== 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.

:{| style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="15" | |- |Geometry || Trigonometry || Differential geometry || Topology || Fractal geometry || Measure theory |}

====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 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.

{| style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="20" | |- | Calculus || Vector calculus|| Differential equations || Dynamical systems || Chaos theory || Complex analysis |}

Applied mathematics

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

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; 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.

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, a designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. 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.

Computational mathematics

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.

{| style="border:1px solid #ddd; text-align:center; margin:0 auto" cellspacing="20" |- | |- | 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

The best-known award in mathematics is the , established in 1936 and now awarded every 4 years. It is often considered the equivalent of science's Nobel Prizes. 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 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 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

upright|thumb|left|Carl Friedrich Gauss, himself known as the "prince of mathematicians", referred to mathematics as "the Queen of the Sciences".

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 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.''"

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 Gödel's incompleteness theorems convinced many mathematicians 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." 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.

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.

See also

Definitions of mathematics

Dyscalculia

Iatromathematicians

List of mathematics competitions

Mathematical anxiety

Mathematical game

Mathematical model

Mathematics and art

Mathematics education

Pseudomathematics

Notes

References

Courant, Richard 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.

Eves, Howard, ''An Introduction to the History of Mathematics'', Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.

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.

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Peterson, Ivars, ''Mathematical Tourist, New and Updated Snapshots of Modern Mathematics'', Owl Books, 2001, ISBN 0-8050-7159-8.

Further reading

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.

Davis, Philip J. and Hersh, Reuben, ''The Mathematical Experience''. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.

Gullberg, Jan, ''Mathematics — From the Birth of Numbers''. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X.

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.

Jourdain, Philip E. B., ''The Nature of Mathematics'', in ''The World of Mathematics'', James R. Newman, editor, Dover Publications, 2003, ISBN 0-486-43268-8.

External links

Free Mathematics books Free Mathematics books collection.

Encyclopaedia of Mathematics online encyclopaedia from Springer, Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics.

HyperMath site at Georgia State University

FreeScience Library The mathematics section of FreeScience library

Rusin, Dave: ''The Mathematical Atlas''. A guided tour through the various branches of modern mathematics. (Can also be found at NIU.edu.)

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.

Cain, George: Online Mathematics Textbooks available free online.

Tricki, Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques.

Mathematical Structures, list information about classes of mathematical structures.

Mathematician Biographies. The MacTutor History of Mathematics archive Extensive history and quotes from all famous mathematicians.

''Metamath''. A site and a language, that formalize mathematics from its foundations.

Nrich, a prize-winning site for students from age five from Cambridge University

Open Problem Garden, a wiki of open problems in mathematics

''Planet Math''. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the Attribution-ShareAlike license, allowing article exchange with Wikipedia. Uses TeX markup.

Some mathematics applets, at MIT

Weisstein, Eric et al.: ''MathWorld: World of Mathematics''. An online encyclopedia of mathematics.

Patrick Jones' Video Tutorials on Mathematics

Citizendium: Theory (mathematics).

Category:Mathematical sciences Category:Formal sciences Category:Greek loanwords

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name	Mos Def
background	solo_singer
birth name	Dante Terrell Smith
born	December 11, 1973Bedford-Stuyvesant, Brooklyn, New York City, U.S.
origin	Bedford-Stuyvesant, Brooklyn, New York City
genre	Hip hop
occupation	Rapper, actor, singer, activist
years active	1994–present
label	Rawkus, Priority, Geffen, Downtown, GOOD Music
associated acts	Soulquarians, Black Star, Talib Kweli, Native Tongues Posse, Kanye West, Gorillaz
notable instruments	}}

Dante Terrell Smith (born December 11, 1973) is an American actor and MC known by the stage name Mos Def (). Mos Def started his hip hop career in a group called Urban Thermo Dynamics, after which he appeared on albums by Da Bush Babees and De La Soul. With Talib Kweli, he formed the duo Black Star, which released the album ''Black Star'' in 1998. He was a major force in the late 1990s underground hip hop explosion spearheaded by Rawkus Records. As a solo artist he has released the albums ''Black on Both Sides'' in 1999, ''The New Danger'' in 2004, ''True Magic'' in 2006, and ''The Ecstatic'' in 2009.

Although he was initially recognized for his musical output, since the early 2000s, Mos Def's screen work has established him as one of only a handful of rappers who have garnered critical approval for their acting work. Mos Def has also been active in several social and political issues.

Early life

Mos Def grew up during the golden age of hip-hop and has rapped and acted since he was six. He attended Philippa Schuyler Middle School in Bushwick, Brooklyn. He majored in Musical Theater at Talent Unlimited High School of the Performing Arts in Manhattan. He studied at New York University in the Gallatin School of Individualized Study.

He has two younger brothers, Abdul Rahman (a.k.a. "Gold Medal Man"), who is Mos Def's full-time DJ, and Anwar Superstar. He also has a younger sister, Ces "Casey" Smith, and a younger half-brother, Jermone Victor Moulton, who resides in Brooklyn and shares the same mother, Sheron.

Mos Def converted to Islam. While his father was initially a member of the Nation of Islam and later an active member in the community of Imam Warith Deen Mohammed, who merged into mainstream Islam from the Nation, Mos Def was not exposed to Islam until the age of 13. At 19, he took his ''shahada'', the Muslim declaration of faith. He is friends with fellow Muslim rappers Ali Shaheed Muhammad and Q-Tip.

Career

Music career

Mos Def began his music career in 1994 in the short-lived group Urban Thermo Dynamics with his younger brother D.c.Q and younger sister Ces. Despite their contract with Payday Records, the group only had two singles, and their debut album ''Manifest Destiny'' was not released until 2004, when it was distributed by Illson Media. In 1996, he emerged as a solo artist and worked with De La Soul and Da Bush Babees, before he released his own first single, "Universal Magnetic", which was a huge underground hit.

Mos Def signed with Rawkus Records and formed the group Black Star with Talib Kweli. They released an album, ''Mos Def & Talib Kweli are Black Star'', in 1998. Mostly produced by Hi-Tek, the album featured the hit singles, "Respiration" and "Definition", which would go on to be featured in VH1's 100 Greatest Songs of Hip-Hop. Mos Def released his solo debut album ''Black on Both Sides'' in 1999, also through Rawkus. Around this time he also contributed to the Scritti Politti album ''Anomie & Bonhomie'' and Rawkus compilations ''Lyricist Lounge'' and ''Soundbombing''.

After the collapse of Rawkus, he signed to Interscope/Geffen Records, which released his second solo album ''The New Danger'' in 2004. ''The New Danger'' contained a mix of several musical genres, including soul, blues, and rock and roll, performed with his rock band Black Jack Johnson, which contained members of the bands Bad Brains and Living Colour. The singles included "Sex, Love & Money" and the B-side "Ghetto Rock"; the latter went on to receive several Grammy Award nominations in 2004.

Mos Def's final solo album for Geffen Records, ''True Magic'', was quietly released on December 29, 2006. ''True Magic'' features production from The Neptunes, Rich Harrison and Minnesota, among others. The album was released in a clear-case with no cover art. Neither Geffen nor Mos Def himself promoted the album at all, which is the main reason the album was received under the radar.

The song "Crime & Medicine" is essentially a cover of GZA's 1995 single "Liquid Swords", though it contains different verses. Also, the track "Undeniable" samples a version of the Barrett Strong/Norman Whitfield composition "Message from a Black Man". The song "Dollar Day" uses the same beat as Juvenile's "Nolia Clap".

MTV reported that this album isn't a full version, but a teaser/promotional debut. A new version of the album would be released spring 2007, with updated songs and cover art. However, on October 17, 2007, Okayplayer reported, through discussions with Mos Def's management, that these rumors were unsubstantiated. The CD was intended to be released without promotion or cover art, as per Mos Def's request. There would be no future re-release.

On November 7, 2007, Mos Def performed live in San Francisco at a venue called The Mezzanine. This performance was recorded for an upcoming "Live in Concert" DVD. During this performance Mos Def announced that he would be releasing a new album to be called ''The Ecstatic''. He sang a number of new tracks; in later shows, Def previewed tracks produced by Madlib and was rumored to be going to Kanye West for new material. Producer and fellow Def Poet Al Be Back stated that he would be producing as well. The album was released on June 9, 2009; upon its release, only Madlib's production had made the cut, along with tracks by Preservation, The Neptunes, Mr. Flash, Madlib's brother Oh No, a song by J. Dilla, and Georgia Anne Muldrow.

Mos Def appears alongside Kanye West on the track "Two Words" from The College Dropout album, the track "Drunk And Hot Girls" and the bonus track "Good Night" off West's third major album, Graduation. In 2002, he released the 12" single Fine, which was featured in the ''Brown Sugar'' Motion Picture Soundtrack.

Mos Def also appears on the debut album from fellow New Yorkers Apollo Heights on a track titled, "Concern." In October, he signed a deal with Downtown Records and appeared on a remix to the song "D.A.N.C.E." by Justice. Mos Def appeared on Stephen Marley's album ''Mind Control'' on the song "Hey Baby." In 2009, Mos Def worked together with Somali rapper K'naan to produce the track "America" for K'naan's album Troubadour.

In April 2008 he appeared on the title track for a new album by The Roots entitled ''Rising Down''. The new single, Life In Marvelous Times, was made officially available through iTunes on November 4, 2008, and is available for stream on the Roots' website Okayplayer.

April 2009 saw him traveling to South Africa for the first time where he performed accompanied by The Robert Glasper Experiment at the renowned Cape Town International Jazz Festival. He enticed his bemused African following with an encore introduced by his own rendition of John Coltrane's "Love Supreme" followed by a sneak preview of the track "M.D. (Doctor)", much to the delight of the fans.

Mos Def also designed two pairs of limited edition Converse shoes. The shoes were released to Foot Locker stores on August 1, 2009 in very limited amounts.

In late 2009, Mos Def created a brand of clothing line with UNDRCRWN called the "Mos Def Cut & Sew Collection." All clothing items will be sold in select stores located around the U.S. and almost exclusively on the UNDRCRWN website. 2009 also found Mos Def among the MCs collaborating with the Black Keys on the first Blakroc album, a project headed by the Black Keys and Damon Dash. Mos Def appeared with Jim Jones and the Black Keys on the Late Show with David Letterman to perform the Blakroc track "Ain't Nothing Like You (Hoochie Coo)".

In March 2010, Mos Def's song Quiet Dog Bite Hard was featured in Palm's "Life moves fast. Don't miss a thing." campaign.

Mos Def features on the first single, "Stylo", from the third Gorillaz album, ''Plastic Beach'', alongside soul legend Bobby Womack. He also appears on the track titled "Sweepstakes".

In September 2010, after appearing on Kanye West's G.O.O.D. Friday track "Lord Lord Lord", Mos Def confirmed his signing with GOOD Music.

Mos Def has been an active contributor to the recovery of the oil spill in the Gulf, performing concerts and raising money towards the repair of the damages. In June 2010, he recorded a cover of the classic New Orleans song originally by Smokey Johnson, "It Ain't My Fault" with the Preservation Hall Jazz Band, Lenny Kravitz and Trombone Shorty.

Acting career

He began his professional acting career at the age of fourteen, appearing in the TV movie ''God Bless the Child'', starring Mare Winningham. He then played the oldest child in the short-lived family sitcom, ''You Take the Kids'', starring Nell Carter and Roger E. Mosley. His most notable acting role before his music career was that of Bill Cosby's sidekick on the short-lived detective show, ''The Cosby Mysteries''. He also starred in a 1996 Visa check card commercial featuring Deion Sanders. In 1997 he had a small role alongside Michael Jackson in his short film and music video "Ghosts".

After brief appearances in ''Bamboozled'' and ''Monster's Ball'', Mos re-invigorated his acting career with his performance as a talented rapper who is reluctant to sign to a major label in ''Brown Sugar''. He was nominated for an Image Award and a Teen Choice Award.

In 2001, he took a supporting role to Beyoncé Knowles and Mehki Phifer in the MTV movie Carmen: A Hip Hopera as Lt. Miller, a crooked cop.

In 2002, he played the role of Booth in Suzan-Lori Parks' ''Topdog/Underdog'', a Tony-nominated and Pulitzer-winning Broadway play. He and co-star Jeffrey Wright won a Special Award from the Outer Critics Circle Award for their joint performance. He also received positive notices as the quirky Left Ear in the blockbuster hit, ''The Italian Job'' in 2003. He also appeared in 2003 in the music video ''You Don't Know My Name'' of the song by Alicia Keys.

In television, Mos Def has appeared on Comedy Central's ''Chappelle's Show'', and has hosted the award-winning HBO spoken word show, ''Def Poetry'' since its inception. The show's sixth season aired in 2007. He also appeared on the sitcom ''My Wife And Kids'' as the disabled friend of Michael Kyle (Damon Wayans).

Mos Def won Best Actor, Independent Movie at the 2005 Black Reel Awards for his portrayal of Detective Sgt. Lucas in ''The Woodsman''. For his portrayal of Vivien Thomas in HBO's film ''Something the Lord Made'', he was nominated for an Emmy Award and a Golden Globe, and won the Image Award. He also played a bandleader in HBO's ''Lackawanna Blues''. He then landed the role of Ford Prefect in the 2005 movie adaptation of ''The Hitchhiker's Guide to the Galaxy''.

In 2006, Mos Def appeared in ''Dave Chappelle's Block Party'' alongside fellow Black Star companion Talib Kweli, while also contributing to the film's soundtrack. Also, Mos Def was featured as the black banjo player in the infamous "Pixie Sketch" from ''Chappelle's Show: The Lost Episodes''. He was later edited out of it on the DVD. Additionally, Mos Def starred in the action film ''16 Blocks'' alongside Bruce Willis and David Morse. He has a recurring guest role on ''Boondocks'', starring as "Gangstalicious". He is also set to be in ''Toussaint'', a film about Haitian revolutionary Toussaint Louverture, opposite Don Cheadle and Wesley Snipes. He made a cameo appearance — playing himself — in the movie ''Talladega Nights: The Ballad of Ricky Bobby''.

In 2008, Mos Def starred in the Michel Gondry movie ''Be Kind Rewind'', playing a video rental store employee whose best friend is played by co-star Jack Black. He also portrayed Chuck Berry in the film ''Cadillac Records'', for which he was nominated for a Black Reel Award and an Image Award.

In 2009, he appeared in the ''House'' episode entitled "Locked In" as a patient suffering from locked-in syndrome. His performance was well-received, with E! saying that Mos Def "delivers an Emmy-worthy performance." He was also in the 2009 film ''Next Day Air''.

In 2010, he appeared on the children's show Yo Gabba Gabba! as Super Mr. Superhero. He also appeared in ''A Free Man of Color'', John Guare's play at the Vivian Beaumont Theatre.

In 2011, it was announced he would appear on the Showtime television series ''Dexter''. He will play Brother Sam, an ex-con who has supposedly found religion despite finding himself in violent situations.

Social and political views

By the early 1990s, a brand of socially conscious hip hop that had been pioneered and popularized by Public Enemy, KRS-One, and De La Soul, among others, had been eclipsed in popularity by gangsta rap. Mos Def, as well as Talib Kweli, The Roots, Common, and others, helped socially aware rap music experience something of a comeback in the late 1990s and early 2000s. Mos Def's collaboration with Talib Kweli, ''Black Star'', was released during the aftermath of the deaths of 2Pac and The Notorious B.I.G. and focused on violence and negativity in hip-hop, in collaboration with other acts that did the same. His music has also made reference to his Islamic faith, and his contention that black artists receive little credit for their role in the birth of rock and roll.

On Mos Def's 2004 album ''The New Danger'', he took his penchant for experimentation to a new level. Most of the songs were more hip-hop flavored stylings of blues and rock, with fewer raps thrown in. This threw off fans who were expecting another full-blown rap album. ''The New Danger'' also featured the controversial song, "The Rape Over," a parody of Jay-Z's ''The Blueprint'' hit "Takeover". His label made him take the song off releases of the album, citing clearance issues with Jay-Z and The Doors, a band which the song samples. The song has garnered controversy over its veiled reference to Israeli-American record executive Lyor Cohen (the "tall Israeli" who then was head of The Island Def Jam Music Group).

Mos Def and Immortal Technique released a similarly controversial song, "Bin Laden" in 2004, which blamed the Reagan Doctrine and President George W. Bush for the September 11, 2001 attacks. A club remix song, featuring Eminem, was released the following year, in 2005.

In September 2005, Mos Def released the single "Katrina Clap," renamed "Dollar Day" for ''True Magic'', (utilizing the instrumental for New Orleans rappers UTP's "Nolia Clap"). The song is a criticism of the Bush administration's response to Hurricane Katrina. On the night of the MTV Video Music Awards, Mos Def pulled up in front of Radio City Music Hall on a flatbed truck and began performing the "Katrina Clap" single in front of a crowd that quickly gathered around him. He was subsequently arrested despite having a public performance permit in his possession.

On September 7, 2007, Mos Def appeared on ''Real Time with Bill Maher'' where he spoke about racism against African Americans, citing the government response to Hurricane Katrina, the Jena Six and the murder conviction of Mumia Abu-Jamal. Mos Def also claimed that Al-Qaeda was not responsible for 9/11, and that Al-Qaeda is not responsible for as much terrorism as they are portrayed to be. He appeared on ''Real Time'' again on March 27, 2009, and spoke about the risk of nuclear weapons. Mos Def said that he did not listen to any of Osama Bin Laden's messages because he did not trust the translations.

In 2000, Mos Def performed a benefit concert for death row inmate Mumia Abu-Jamal.

Personal life

In 1996, Def married Maria Yepes. After having two daughters, Chandani and Jauhara Smith, he filed for divorce in 2005. The divorce became final in 2006. Two years later, Def's divorce lawyers Blank Rome sued Def for more than $60,000 in unpaid legal bills.

In October, 2006 Mos Def appeared on ''4Real'', a documentary television series. Appearing in the episode "City of God," he and the 4Real crew traveled to City of God, a slum in Rio de Janeiro, Brazil, to meet Brazilian MC MV Bill and discover the crime and social problems of the community.

He has recently taken up skateboarding and said he's looking to host a skateboarding event in the United Arab Emirates.

Nominations

Black Movie Awards

* 2006 Source Awards

Black Reel Awards

* 2008, Best Supporting Actor: ''Cadillac Records''

* 2003, Best Actor- Independent: ''Civil Brand''

* 2004, Best Supporting Actor: ''The Italian Job''

* 2005, Best Actor TV Movie/Mini-Series: ''Something the Lord Made''

* 2005, Best Indie Actor: ''The Woodsman'' (won)

Emmy Award

* 2004, Best Actor in a Television Movie or Mini-Series: ''Something the Lord Made''

Golden Globes

* 2005, Best Actor in a Television Movie or Mini-Series: ''Something the Lord Made''

Grammy Awards

* 2005, Best Urban/Alternative Performance: "Sex, Love & Money"

* 2006, Best Urban/Alternative Performance: "Ghetto Rock"

* 2007, Best Rap Solo Performance: "Undeniable"

* 2010, Best Rap Solo Performance: "Casa Bey"

* 2010, Best Rap Album: "The Ecstatic"

* 2011, Best Short Form Music Video: "Stylo" (with Bobby Womack and Gorillaz)

* 2009, Outstanding Supporting Actor in a Motion Picture: ''Cadillac Records''

* 2003, Outstanding Supporting Actor in a Motion Picture: ''Brown Sugar''

* 2005, Outstanding Actor in a Television Movie or Mini-Series: ''Something the Lord Made''

Discography

''Black on Both Sides'' (1999)

''The New Danger'' (2004)

''True Magic'' (2006)

''The Ecstatic'' (2009)

Filmography

''Ghosts (Michael Jackson film) ''Where's Marlowe?'' ''Freestyle: The Art of Rhyme'' ''Carmen: A Hip Hopera'' ''Showtime (film) ''The Italian Job (2003 film) ''The Woodsman'' Vivien Thomas ''Lackawanna Blues'' ''Dave Chappelle's Block Party'' ''Prince Among Slaves (film) ''Be Kind Rewind'' ''Next Day Air'' ''I'm Still Here (film) ''Yo Gabba Gabba

Year !! Film !! Role !! Notes
1991	The Hard Way (1991 film)>The Hard Way''	Dead Romeos Gang Member
1997	Ghosts'' ||	Townsperson
1998	|	Wilt Crawley
rowspan="3"	2000	|	Himself
''Bamboozled''	Big Blak Afrika
''Island of the Dead (2000 film)	Island of the Dead''	Robbie J
rowspan="2"	2001	|	Lieutenant Miller
''Monster's Ball''	Ryrus Cooper
rowspan="4"	2002	Showtime'' ||	Lazy Boy
''Civil Brand''	Michael Meadows
''Brown Sugar (2002 film)	Brown Sugar''	Chris 'Cav' Anton Vichon
''My Wife and Kids''
2003	The Italian Job'' ||	Left Ear
rowspan="2"	2004	|	Detective Lucas
''Something the Lord Made''	| Nominated - Emmy for Outstanding Lead Actor in a Miniseries or a Movie Nominated - Golden Globe for Best Performance by an Actor in a Mini-Series or a Motion Picture Made for TelevisionNominated - Image Awards for Outstanding Actor in a Mini-Series or Television Movie
rowspan="3"	2005	|	The Bandleader
''The Boondocks (TV series)	The Boondocks (2005-2008)	Voice Of Gangstalicious
''The Hitchhiker's Guide to the Galaxy (film)	The Hitchhiker's Guide to the Galaxy''	Ford Prefect (character)>Ford Prefect
rowspan="4"	2006	|	Himself
''16 Blocks''	Eddie Bunker
''Talladega Nights: The Ballad of Ricky Bobby''	Himself
''Journey to the End of the Night (2006 film)	Journey to the End of the Night''	Wemba
2007	Prince Among Slaves'' ||	Narrator
rowspan="2"	2008	|	Mike
''Cadillac Records''	Chuck Berry
rowspan="2"	2009	|	Eric
''House (TV series)	House''	Lee
2010	I'm Still Here'' ||	Himself
2010	(TV series)	Yo Gabba Gabba!''	Super Mr. Superhero

References

External links

Mos Def / Dante Smith discographies at Discogs.

Mos Def at Geffen Records.

Mos Def Fan Message Board

Category:1973 births Category:Actors from New York City Category:African American actors Category:African American Muslims Category:Converts to Islam Category:African American Muslims Category:African American rappers Category:American vegetarians Category:Living people Category:People from Bedford–Stuyvesant, Brooklyn Category:Rappers from New York City Category:Slam poets

ar:موس ديف bg:Мос Деф cs:Mos Def da:Mos Def de:Mos Def es:Mos Def fr:Mos Def fy:Mos Def ko:모스 데프 id:Mos Def it:Mos Def he:מוס דף nl:Mos Def ja:モス・デフ no:Mos Def pl:Mos Def pt:Mos Def sq:Mos Def simple:Mos Def fi:Mos Def sv:Mos Def uk:Мос Деф

name	Georg Cantor
birth name	Georg Ferdinand Ludwig Philipp Cantor
birth date	March 03, 1845
birth place	Saint Petersburg, Russian Empire
death date	January 06, 1918
death place	Halle, Province of Saxony, German Empire
residence	Russian Empire (1845–1856),German Empire (1856–1918)
field	Mathematics
work institutions	University of Halle
alma mater	ETH Zurich, University of Berlin
doctoral advisor	Ernst KummerKarl Weierstrass
doctoral students	Alfred Barneck
known for	Set theory
religion	Lutheran
prizes	}}

Georg Ferdinand Ludwig Philipp Cantor ( ; ; – January 6, 1918) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God, on one occasion equating the theory of transfinite numbers with pantheism. The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics, and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth." Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries, but these episodes can now be seen as probable manifestations of a bipolar disorder.

The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. Cantor believed his theory of transfinite numbers had been communicated to him by God. David Hilbert defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created."

Life

Youth and studies

Cantor was born in 1845 in the western merchant colony in Saint Petersburg, Russia, and brought up in the city until he was eleven. Georg, the oldest of six children, was an outstanding violinist, having inherited his parents' considerable musical and artistic talents. Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In 1862, Cantor entered the Federal Polytechnic Institute in Zürich, today the ETH Zurich. After receiving a substantial inheritance upon his father's death in 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a very important center for mathematical research. In 1867, Berlin granted him the PhD for a thesis on number theory, ''De aequationibus secundi gradus indeterminatis.

Teacher and researcher

In 1867, Cantor completed his dissertation, on number theory, at the University of Berlin. After teaching briefly in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle.

In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday.

Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible. Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians. Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Cantor came to believe that Kronecker's stance would make it impossible for Cantor ever to leave Halle.

In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.

In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle. Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal ''Acta Mathematica''. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to ''Acta''. He asked Cantor to withdraw the paper from ''Acta'' while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but wrote to a third party:

Cantor then sharply curtailed his relationship and correspondence with Mittag-Leffler, displaying a tendency to interpret well-intentioned criticism as a deeply personal affront.

Cantor suffered his first known bout of depression in 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 attacked Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:

This emotional crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study of Elizabethan literature in an attempt to prove that Francis Bacon wrote the plays attributed to Shakespeare (see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.

Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–1884. He eventually sought a reconciliation with Kronecker, which Kronecker graciously accepted. Nevertheless, the philosophical disagreements and difficulties dividing them persisted. It was once thought that Cantor's recurring bouts of depression were triggered by the opposition his work met at the hands of Kronecker. While Cantor's mathematical worries and his difficulties dealing with certain people were greatly magnified by his depression, it is doubtful that they were its cause. Rather, his posthumous diagnosis of bipolarity has been accepted as the root cause of his erratic mood.

In 1890, Cantor was instrumental in founding the ''Deutsche Mathematiker-Vereinigung'' and chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity he felt towards Kronecker, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time.

Late years

After Cantor's 1884 hospitalization, there is no record that he was in any sanatorium again until 1899. Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly (while Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics. Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since it had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, even momentarily questioning God. Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the ''Deutsche Mathematiker–Vereinigung'' in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newly published ''Principia Mathematica'' repeatedly cited Cantor's work, but this did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.

Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life.

Mathematical work

Cantor's work between 1874 and 1884 is the origin of set theory. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of Aristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics.

In one of his earliest papers, Cantor proved that the set of real numbers is "more numerous" than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes, some infinite sets being larger than others. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and uncountable sets (nondenumerable infinite sets).

Cantor introduced fundamental constructions in set theory, such as the power set of a set ''A'', which is the set of all possible subsets of ''A''. He later proved that the size of the power set of ''A'' is strictly larger than the size of ''A'', even when ''A'' is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter $\backslash aleph$ (aleph) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega). This notation is still in use today.

The ''Continuum hypothesis'', introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his famous address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher Charles Sanders Peirce praised Cantor's set theory, and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Cantor's religious ideas. This was later published, as were several of his expository works.

Number theory and function theory

Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the nth derived set p(n) of a set S of zeros of a trigonometric series. Thus p(1) is the set of limit points of S, p(2) is the set of limit points of p(1), and so on. By taking the intersection of p(1), p(2), p(3),... he formed p(ω), and then he noticed that p(ω) had a set of limit points p(ω + 1), and so on. Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics". Cantor also published an erroneous "proof" of the inconsistency of infinitesimals

Set theory

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"). This article was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements). Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable. His proof is more complex than the more elegant diagonal argument that he gave in 1891. Cantor's article also contains a new method of constructing transcendental numbers. Transcendental numbers were first constructed by Joseph Liouville in 1844.

Cantor established these results using two constructions. His first construction shows how to write the real algebraic numbers as a sequence ''a''₁, ''a''₂, ''a''₃, …. In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence — that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers. Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.

Between 1879 and 1884, Cantor published a series of six articles in ''Mathematische Annalen'' that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole. Cantor also introduced the Cantor set during this period.

The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883, was the most important of the six and was also published as a separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.

In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set ''A'' is strictly larger than the cardinality of ''A''. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem.

Cantor wrote on the Goldbach conjecture in 1894.

In 1895 and 1897, Cantor published a two-part paper in ''Mathematische Annalen'' under Felix Klein's editorship; these were his last significant papers on set theory. The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if ''A'' and ''B'' are sets with ''A'' equivalent to a subset of ''B'' and ''B'' equivalent to a subset of ''A'', then ''A'' and ''B'' are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schroeder theorem.

One-to-one correspondence

Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Dedekind, Cantor proved a far stronger result: for any positive integer ''n'', there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an ''n''-dimensional space. About this discovery Cantor famously wrote to Dedekind: "''Je le vois, mais je ne le crois pas''!" ("I see it, but I don't believe it!") The result that he found so astonishing has implications for geometry and the notion of dimension.

In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence, and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that ''n''-dimensional Euclidean space R^''n'' has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.

This paper displeased Kronecker, and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.

Continuum hypothesis

Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is ''exactly'' aleph-one, rather than just ''at least'' aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").

Paradoxes of set theory

Discussions of set-theoretic paradoxes began to appear around the end of the nineteenth century. Some of these implied fundamental problems with Cantor's set theory program. In an 1897 paper on an unrelated topic, Cesare Burali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.

In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set ''A'', the cardinal number of the power set of ''A'' is strictly larger than the cardinal number of ''A'' (this fact is now known as Cantor's theorem). This paradox, together with Burali-Forti's, led Cantor to formulate a concept called ''limitation of size'', according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as proper classes.

One common view among mathematicians is that these paradoxes, together with Russell's paradox, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naive set theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.

Philosophy, religion and Cantor's mathematics

The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's. He directly addressed this intersection between these disciplines in the introduction to his ''Grundlagen einer allgemeinen Mannigfaltigkeitslehre,'' where he stressed the connection between his view of the infinite and the philosophical one. To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications—he identified the Absolute Infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.

Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind. Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all." Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.

Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. In particular, Neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity". Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:

Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism—and was shocked when he realized that he was the only faculty member at Halle who did ''not'' hold to deterministic philosophical beliefs.

In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim, as well as theologians such as Cardinal Johannes Franzelin, who once replied by equating the theory of transfinite numbers with pantheism. Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.

Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his famous assertion that "the essence of mathematics is its freedom." These ideas parallel those of Edmund Husserl.

Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering:

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, Descartes, Berkeley, Leibniz, and Bolzano on infinity.

Cantor's ancestry

"Very little is known for sure about the origin and education of George Woldemar Cantor." However, Cantor was frequently described as Jewish in his lifetime. Cantor's paternal grandparents were from Copenhagen, and fled to Russia from the disruption of the Napoleonic Wars. There is very little direct information on his grandparents, consequently their level of Jewish observance is unknown. Jakob Cantor, Cantor's grandfather, gave his children Christian saints' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they, or their ancestors, converted to Orthodox Christianity. Cantor's father, Georg Waldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. His mother, Maria Anna Böhm, was an Austrian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, saying which could be read to imply that she was of Jewish ancestry.

There are documented statements that call this Jewish ancestry into question:

It is also later said in the same document: (the rest of the quote is finished by the very first quote above)

Thus Cantor has been called Jewish in his lifetime, but has also variously been called Russian, German, and Danish as well.

Historiography

Until the 1970s, the chief academic publications on Cantor were two short monographs by Schönflies (1927)—largely the correspondence with Mittag-Leffler—and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by Eric Temple Bell's ''Men of Mathematics'' (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the history of mathematics"; and as "one of the worst". Bell presents Cantor's relationship with his father as Oedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics, and fills the picture with stereotypes. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell—including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents.

See also

Cantor cube

Cantor function

Cantor medal—award by the Deutsche Mathematiker-Vereinigung in honor of Georg Cantor.

Cantor space

Cantor's back-and-forth method

Controversy over Cantor's theory

Heine–Cantor theorem

Pairing function

German inventors and discoverers

Notes

References

:''Older sources on Cantor's life should be treated with caution. See Historiography section above.'' ; Primary literature in English: . .

; Primary literature in German: . . Almost everything that Cantor wrote. .

; Secondary literature: . ISBN 0-7607-7778-0. A popular treatment of infinity, in which Cantor is frequently mentioned. . . The definitive biography to date. ISBN 978-0-691-02447-9 . Internet version published in Journal of the ACMS 2004. . . ISBN 3-7643-8349-6 Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory. . . ISBN 978-0-691-05858-0 . . ISBN 0-19-853283-0 . ISBN 3-540-90092-6 . ISBN 0-8126-9538-0 Three chapters and 18 index entries on Cantor. . . . ISBN 0-679-77631-1 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist. . ISBN 0-8176-1770-1 . ISBN 0-387-04999-1 . ISBN 0-553-25531-2 Deals with similar topics to Aczel, but in more depth. . . . ISBN 0-486-61630-4 Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics. . ISBN 0-393-00338-8 .

External links

Mainly devoted to Cantor's accomplishment.

Stanford Encyclopedia of Philosophy: Set theory by Thomas Jech.

Grammar school Georg-Cantor Halle (Saale): Georg-Cantor-Gynmasium Halle

Category:People from Saint Petersburg Category:German mathematicians Category:German logicians Category:Set theorists Category:19th-century mathematicians Category:20th-century mathematicians Category:German philosophers Category:19th-century philosophers Category:20th-century philosophers Category:University of Halle-Wittenberg faculty Category:ETH Zurich alumni Category:German Lutherans Category:People with bipolar disorder Category:Baltic Germans Category:1845 births Category:1918 deaths

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name	Christof Koch
birth date
birth place	Kansas City, Missouri
alma mater	University of TübingenMax Planck Institute for Biological Cybernetics
field	Biophysics
doctoral advisor	Valentin BraitenbergTomaso Poggio
nationality	American
doctoral students	Virgil Griffith
website	http://www.klab.caltech.edu/~koch/
religion	"Romantic Reductionist" }}

Christof Koch (born November 13, 1956, Kansas City) is an American neuroscientist working on the neural basis of consciousness. He is the Lois and Victor Troendle Professor of Cognitive and Behavioral Biology at California Institute of Technology, where he has been since 1986. In early 2011, he also became the Chief Scientific Officer of the Allen Institute for Brain Science, leading their high through-put, large scale cortical coding project.

He is the son of German parents; his father was a diplomat. He was raised as a Roman Catholic and attended a Jesuit high school in Morocco. He received a PhD in nonlinear information processing from the Max Planck Institute in Tübingen, Germany in 1982. He then worked for four years at the Artificial Intelligence Laboratory at MIT. In 1986, he joined the newly started Computation and Neural Systems PhD program at Caltech.

He has been active since the early 1990s in the promotion of consciousness as a scientifically tractable problem, and has been particularly influential in arguing that consciousness can now be approached using the modern tools of neurobiology. His primary collaborator in the endeavour of locating the neural correlates of consciousness was the late Francis Crick.

Together with James Bower, he founded in 1988 the Methods in Computational Neuroscience summer course at the Marine Biological Laboratory in Woods Hole, which remains ongoing. In 1993, he founded, together with Rodney Douglas and Terrence Sejnowski, the Neuromorphic Engineering Summer School in Telluride, Colorado, which remains ongoing.

Koch was the executive officer of the ''Computation and Neural Systems'' program at Caltech from 2000 to 2005. In 2005 he was the local organizer of the Association for the Scientific Study of Consciousness meeting.

Miscellaneous Trivia

After his dog died he became a vegetarian, and has expressed sympathy for animal rights due to the neurological similarities between human and non-human animals.

He has an apple logo tattooed on his right deltoid and a drawing from Ramon y Cajal of a cortical microcircuit tattooed on his left deltoid.

He has an Erdős number of three.

He is allergic to strawberries.

Publications

''Biophysics of Computation: Information Processing in Single Neurons'', Oxford U. Press, (1999), ISBN 0-19-518199-9

''The Quest for Consciousness: a Neurobiological Approach'', Roberts and Co., (2004), ISBN 0-9747077-0-8

References

External links

Homepage of Christof Koch

Homepage of Koch's laboratory

His book ''The Quest for Consciousness''

Online lecture videos from an undergraduate course taught by Christof Koch at Caltech in 2003 on the neurobiological basis of consciousness.

Neural Correlates of Consciousness, Scholarpedia article by Christof Koch

Association for the Scientific Study of Consciousness

Radio interview with Christof Koch and Francis Crick, 2001

Radio interview with Christof Koch and Constanze Hofstoetter, 2005

Interview on the Brain Science Podcast, 2007

Christof Koch, a short clip

Interview with Christof Koch on the neuroscience of reading and the movie in your mind

Category:1956 births Category:Living people Category:American neuroscientists Category:Consciousness researchers and theorists Category:American vegetarians Category:California Institute of Technology faculty

bn:ক্রিস্টফ কখ de:Christof Koch fa:کریستف کخ nl:Christof Koch ja:クリストフ・コッホ

Name	Black Thought
Background	solo_singer
Birth name	Tariq Trotter
Origin	Philadelphia, Pennsylvania
Birth date	October 03, 1971
Genre	Hip hop
Years active	1987–present
Label	DGC/Geffen/MCA RecordsDef Jam Recordings
Associated acts	The Roots, Money Making Jam Boys, Black Star, Mos Def, Talib Kweli, Common
Website	www.myspace.com/blackthought }}

Tariq Trotter (born on October 3, 1971), better known as Black Thought, is an American hip-hop artist who is the lead MC of the Philadelphia-based hip hop group The Roots and occasional actor. Black Thought, who co-founded The Roots with drummer ?uestlove (Ahmir Thompson), is widely lauded for his complex and politically aware lyrical content, and his sharply honed live performances.

Early life

Black Thought was born Tariq Trotter and attended the Philadelphia High School for Creative and Performing Arts and Millersville University. In 1987, Trotter became friends with drummer Ahmir "?uestlove" Thompson and formed a drummer/MC duo performing on the streets of Philadelphia and at talent shows. Trotter would subsequently spend some time as one of two MCs in the group The Square Roots: the other one was Malik B., who Tariq met in college. In high-school, Black Thought converted to the Islamic sect of The Nation of Gods and Earths.

The Roots

The Square Roots renamed themselves the Roots and released its debut album ''Organix'' in 1993, to prepare for a concert in Germany. The Roots signed to DGC and followed up with ''Do You Want More?!!!??!'' in 1995. Recorded without any sampling, the album was more popular among alternative music fans than those of hip hop. Around the release of the album, the Roots performed at the Lollapalooza alternative music festival and Montreux Jazz Festival. ''Illadelph Halflife'', the band's 1996 album, became its first album to chart within the top 40 spots on the Billboard 200 because of the successful single "What They Do". ''Things Fall Apart'' followed in 1999, the year the band played at the Woodstock 99 concert.

In 2000, the Roots won the Grammy Award for Best Rap Performance by a Duo or Group for "You Got Me", with guest performances by Erykah Badu and Eve. The Roots' album ''Things Fall Apart'' was nominated for the Best Rap Album award. For Jay-Z's acoustic concert for the television program ''MTV Unplugged'', The Roots provided instrumentals. Succeeding albums were ''Phrenology'' (2002), ''The Tipping Point'' (2004), ''Game Theory'' (2006), ''Rising Down'' (2008), and ''How I Got Over'' (2010).

Other work

Films in which Black Thought starred in include ''Bamboozled'' (2000), ''Perfume'', ''Love Rome'', and ''Brooklyn Babylon'' (both 2001).

Thought has also made guest performances on several other records including "Pimpas Paradise" by Damian "Jr. Gong" Marley, ''Team'' by Dilated Peoples, ''One Day It'll All Make Sense'' by Common, ''Reanimation'' by Linkin Park, ''Pick a Bigger Weapon'' by The Coup, ''The Rising Tied'' by Fort Minor, Mos Def & Talib Kweli's Black Star album and ''A Ma Zone'' by Zap Mama.

Black Thought recorded a solo album to be titled ''Masterpiece Theatre'' and released in summer 2001, but the project was scrapped after learning that the album wouldn't count towards The Roots' current contract commitments. Most of the songs demoed wound up on The Roots' ''Phrenology'' album. In 2006, he began working on a collaborative project with producer Danger Mouse titled ''Dangerous Thoughts''. In a June 2008 interview with Brian Kayser of the website ''HipHopGame'', Black Thought spoke of yet another solo project, which will come out on the label Razor and Tie. He stated that there would be the possibility of Questlove working on production. In 2008, Peta2.com nominated Black Thought as among the "World's Sexiest Vegetarians".

In February 2011, Black Thought, along with 10.Deep and his side collective "Money Making Jam Boys", which includes Dice Raw, S.T.S., Truck North, & P.O.R.N., released the mixtape titled ''The Prestige''.

Discography

''The Talented Mr. Trotter'' (2011)

Appearances

1995: "Meiso", ''Meiso'', DJ Krush

1996: "Da Jawn", Kollage, Bahamadia

1997: "Get This Low", ''The Psycho-Social,'' ''Jedi Mind Tricks''

1997: "Stolen Moments, Pt. 2", ''One Day It'll All Make Sense'', Common

1997: "Listen To This", Listen To This, Walkin' Large

1998: "It's About That Time", ''Soul Survivor'', Pete Rock

1998: "Live From the Sretch Armstrong Show", ''Lyricist Lounge Volume One'', Various Artists

1998: "Super Lyrical", ''Capital Punishment'', Big Pun

1998: "Tabou (Roots Remix)", ''Princesses Nubians'', Les Nubians

1999: "Burnin' and Lootin'", ''Chant Down Babylon (Post-Humous Remix)'', Bob Marley

1999: "Respiration (Flying High Remix)", ''Respiration 12"'', Black Star

2000: "Cold Blooded", ''Like Water for Chocolate'', Common

2000: "Hurricane", ''The Hurricane Soundtrack'', Black Thought, Common, Dice Raw, Flo Brown, Jazzyfatnastees and Mos Def

2000: "Network", ''Plain Rap'', The Pharcyde

2001: "Hard Hitters", ''Expansion Team'', Dilated Peoples

2001: "Zen Approach", ''Zen'', DJ Krush

2002: "Clap!", ''Next'', Soulive

2002: "Guerilla Monsoon Rap", ''Quality'', Talib Kweli

2002: "X-Ecutioner Style", ''Reanimation'', Linkin Park

2004: "Live From The PJs", ''Revolutions'', The X-Ecutioners

2005: "Appreciate", ''Love and Life'', LaToya London

2005: "Flutlicht", ''Sinnflut'', Curse

2005: "Pimpa's Paradise", ''Welcome to Jamrock'', Damian Marley

2005: "Right Now", ''The Rising Tied'', Fort Minor

2006: "Love Movin'", ''The Shining'', J Dilla

2006: "My Favorite Mutiny", ''Pick A Bigger Weapon'', The Coup

2006: "Yes, Yes Y'all",''Timeless'', Sérgio Mendes

2007: "Clean Up", ''Deep Hearted'', Strong Arm Steady

2008: "Give It Up", Thee Adventures Of A B-Boy D-Boy, Muja Messiah

2008: "Hold Tight", The Million Dollar Backpack, Skillz

2008: "Cause I'm Black", ''Super Gangster (Extraordinary Gentleman)'', Styles P

2009: "Hot Shyt", "Back to the Feature", Wale

2009: "Live Forever", ''Velvet Ballads'', Cradle Orchestra

2009: "Philly Boy", Rádio do Canibal, BK-One

2009: "Reality TV", Jay Stay Paid, J Dilla

2009: "Slow Down", Chiddy Bang

2010: "Philadelphia Born and Raised", Meek Mill

2010: "Ill Street Blues", More Demand 2, STS

2010: "In tha Park", Apollo Kids, Ghostface Killah

2010: "Philly Sh*t Remix, Young Chris, Meek Mill & Eve

2011: "The Masters of Our Fates", Raekwon

2011: "Too Long (Prod. DJ Corbett)" (Saigon) (From Sai's Album The Greatest Story Never Told)

2011: "Million Star Motel", Pennies In A Jar, Nikki Jean

Filmography

''Bamboozled'' (2000)

''Brooklyn Babylon'' (2001)

''Perfume'' (2001)

''Brown Sugar'' (2002)

''Love Rome'' (2004)

''Explicit IIIs'' (2008)

''In NorthWood'' (2009)

''Night Catches Us'' (2010)

References

External links

TheRoots.com - Official website of The Roots

Black Thought discography

Category:Living people Category:African American actors Category:African American rappers Category:American vegetarians Category:Def Jam Recordings artists Category:Geffen Records artists Category:Millersville University of Pennsylvania alumni Category:Members of the Nation of Gods and Earths Category:Rappers from Philadelphia, Pennsylvania Category:The Roots members Category:1971 births

de:Black Thought es:Black Thought pl:Black Thought pt:Black Thought