Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as ''speculative mathematics'', and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering, and so on. Another insightful view put forth is that ''pure mathematics is not necessarily applied mathematics''.

History

Ancient Greece

Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called number theory, and "logistic", now called arithmetic. Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn the art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being." Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must needs make gain of what he learns." The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book IV of ''Conics'' to which he proudly asserted,

They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.

And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of ''Conics'' that the subject is one of those that "...seem worthy of study for their own sake."

19th century

The term itself is enshrined in the full title of the Sadleirian Chair, founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of ''pure'' mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between ''pure'' and ''applied''. In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent.

20th century

At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of ''rigorous proof''.

In fact in an axiomatic setting ''rigorous'' adds nothing to the idea of ''proof''. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved. Pure mathematician became a recognized vocation, achievable through training.

Generality and abstraction

One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality.

Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures

Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.

One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently, or using results from other areas of mathematics.

Generality can facilitate connections between different branches of mathematics. Category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math.

Generality's impact on intuition is both dependent on the subject and a matter of personal preference or learning style. Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition.

As a prime example of generality, the Erlangen program involved an expansion of geometry to accommodate non-Euclidean geometries as well as the field of topology, and other forms of geometry, by viewing geometry as the study of a space together with a group of transformations. The study of numbers, called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level; and the study of functions, called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level. Each of these branches of more ''abstract'' mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. A steep rise in abstraction was seen mid 20th century.

In practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1980. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, not enough Poincaré. The point does not yet seem to be settled (unlike the foundational controversies over set theory), in that string theory pulls one way, while discrete mathematics pulls back towards proof as central.

Purism

Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the most famous (but perhaps misunderstood) modern examples of this debate can be found in G.H. Hardy's ''A Mathematician's Apology''.

It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that Hardy preferred pure mathematics, which he often compared to painting and poetry, Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express ''physical'' truth in a mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. Hardy made a separate distinction in mathematics between what he called "real" mathematics, "which has permanent aesthetic value", and "the dull and elementary parts of mathematics" that have practical use.

Hardy considered some physicists, such as Einstein and Dirac, to be among the "real" mathematicians, but at the time that he was writing the ''Apology'' he also considered general relativity and quantum mechanics to be "useless", which allowed him to hold the opinion that only "dull" mathematics was useful. Moreover, Hardy briefly admitted that—just as the application of matrix theory and group theory to physics had come unexpectedly—the time may come where some kinds of beautiful, "real" mathematics may be useful as well.

Another insightful view is offered by Magid:

"I've always thought that a good model here could be drawn from ring theory. In that subject, one has the subareas of commutative ring theory and noncommutative ring theory. An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the former: a noncommutative ring is a not necessarily commutative ring. If we use similar conventions, then we could refer to applied mathematics and nonapplied mathematics, where by the latter we ''mean not necessarily applied mathematics''… [emphasis added]"

Subfields

Analysis is concerned with the properties of functions. It deals with concepts such as continuity, limits, differentiation and integration, thus providing a rigorous foundation for the calculus of infinitesimals introduced by Newton and Leibniz in the 17th century. Real analysis studies functions of real numbers, while complex analysis extends the aforementioned concepts to functions of complex numbers. Functional analysis is a branch of analysis that studies infinite-dimensional vector spaces and views functions as points in these spaces.

Abstract algebra is not to be confused with the manipulation of formulae that is covered in secondary education. It studies sets together with binary operations defined on them. Sets and their binary operations may be classified according to their properties: for instance, if an operation is associative on a set that contains an identity element and inverses for each member of the set, the set and operation is considered to be a group. Other structures include rings, fields and vector spaces.

Geometry is the study of shapes and space, in particular, groups of transformations that act on spaces. For example, projective geometry is about the group of projective transformations that act on the real projective plane, whereas inversive geometry is concerned with the group of inversive transformations acting on the extended complex plane. Geometry has been extended to topology, which deals with objects known as topological spaces and continuous maps between them. Topology is concerned with the way in which a space is connected and ignores precise measurements of distance or angle.

Number theory is the theory of the positive integers. It is based on ideas such as divisibility and congruence. Its fundamental theorem states that each positive integer has a unique prime factorization. In some ways it is the most accessible discipline in pure mathematics for the general public: for instance the Goldbach conjecture is easily stated (but is yet to be proved or disproved). In other ways it is the least accessible discipline; for example, Wiles' proof that Fermat's equation has no nontrivial solutions requires understanding automorphic forms, which though intrinsic to nature have not found a place in physics or the general public discourse.

Quotations

Notes

See also

Applied mathematics

Logic

Metalogic

Metamathematics

External links

''What is Pure Mathematics?'' Department of Pure Mathematics, University of Waterloo

'' What is Pure Mathematics?'' by Professor P.J. Giblin The University of Liverpool

'' The Principles of Mathematics '' by Bertrand Russell

How to Become a Pure Mathematician (or Statistician) - a List of Undergraduate and Basic Graduate Textbooks and Lecture Notes, with several comments and links to solution, companion site, data set, errata page, etc.

Category:Mathematical terminology Category:Mathematical science occupations

ar:رياضيات بحتة ca:Matemàtica pura cs:Čistá matematika cy:Mathemateg bur et:Puhas matemaatika es:Matemáticas puras eo:Pura matematiko fa:ریاضیات محض fr:Mathématiques pures gl:Matemática pura hi:शुद्ध गणित io:Pura matematiko id:Matematika murni ia:Mathematica pur it:matematica pura ka:წმინდა მათემატიკა ms:Matematik tulen nl:Zuivere wiskunde ja:純粋数学 no:Ren matematikk nov:Puri matematike pl:Czysta matematyka pt:Matemática pura ru:Чистая математика simple:Pure mathematics uk:Чиста математика vo:Matemat klinöfik zh:純粹數學

Name	Gregory Chaitin
Birth place	Chicago
Residence	United States
Nationality	American
Ethnicity	Argentinian-American
Fields	BiologyMathematicsComputer science
Workplaces	Federal University of Rio de JaneiroIBM Thomas J. Watson Research Center
Alma mater
Doctoral advisor
Academic advisors
Known for	Chaitin-Kolmogorov complexityChaitin's constantChaitin's algorithm
Signature
Footnotes	}}

Gregory John Chaitin (born 1947) is an Argentine-American mathematician and computer scientist.

Mathematics and computer science

Beginning in 2009 Chaitin has worked on metabiology, a field parallel to biology dealing with the evolution of artificial software (computer programs) instead of natural software (DNA).

Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a new incompleteness theorem in reaction to Gödel's incompleteness theorem. He attended the Bronx High School of Science and City College of New York, where he (still in his teens) developed the theories that led to his independent discovery of Kolmogorov complexity.

Chaitin has defined Chaitin's constant Ω, a real number whose digits are equidistributed and which is sometimes informally described as an expression of the probability that a random program will halt. Ω has the mathematical property that it is definable but not computable.

Chaitin's early work on algorithmic information theory paralleled the earlier work of Kolmogorov.

Chaitin is also the originator of using graph coloring to do register allocation in compiling, a process known as Chaitin's algorithm.

Other scholarly contributions

Chaitin also writes about philosophy, especially metaphysics and philosophy of mathematics (particularly about epistemological matters in mathematics). In metaphysics, Chaitin claims that algorithmic information theory is the key to solving problems in the field of biology (obtaining a formal definition of ‘life’, its origin and evolution) and neuroscience (the problem of consciousness and the study of the mind).

Indeed, in recent writings, he defends a position known as digital philosophy. In the epistemology of mathematics, he claims that his findings in mathematical logic and algorithmic information theory show there are ''“mathematical facts that are true for no reason, they're true by accident. They are random mathematical facts”''. Chaitin proposes that mathematicians must abandon any hope of proving those mathematical facts and adopt a quasi-empirical methodology.

Honors

In 1995 he was given the degree of doctor of science ''honoris causa'' by the University of Maine. In 2002 he was given the title of honorary professor by the University of Buenos Aires in Argentina, where his parents were born and where Chaitin spent part of his youth. In 2007 he was given a Leibniz Medal by Wolfram Research. In 2009 he was given the degree of doctor of philosophy ''honoris causa'' by the National University of Córdoba. He was formerly a researcher at IBM's Thomas J. Watson Research Center and is now a professor at the Federal University of Rio de Janeiro.

Criticism

Some philosophers and logicians strongly disagree with the philosophical conclusions that Chaitin has drawn from his theorems. The logician Torkel Franzén criticizes Chaitin’s interpretation of Gödel's incompleteness theorem and the alleged explanation for it that Chaitin’s work represents.

See also

Digital philosophy

Halting problem

Bibliography

''Information, Randomness & Incompleteness'' (World Scientific 1987)

''Algorithmic Information Theory'' (Cambridge University Press 1987) online, ''Information-Theoretic Incompleteness'' (World Scientific 1992) online,

''The Limits of Mathematics'' (Springer-Verlag 1998),

''The Unknowable'' (Springer-Verlag 1999),

''Exploring Randomness'' (Springer-Verlag 2001),

''Conversations with a Mathematician'' (Springer-Verlag 2002),

''From Philosophy to Program Size'' (Tallinn Cybernetics Institute 2003),

''Meta Math!: The Quest for Omega'' (Pantheon Books 2005) (reprinted in UK as ''Metamaths: The Quest for Omega'', Atlantic Books 2006),

''Teoria algoritmica della complessità'' (G. Giappichelli Editore 2006),

''Thinking about Gödel & Turing'' (World Scientific 2007),

''Mathematics, Complexity and Philosophy'' (Editorial Midas 2011),

''Gödel's Way'' (CRC Press 2012),

''Proving Darwin: Making Biology Mathematical'' (Pantheon Books 2012 (upcoming)).

Notes

References

Ugo Pagallo, ''Introduzione alla filosofia digitale. Da Leibniz a Chaitin'', (G. Giappichelli Editore, 2005) ISBN 88-348-5635-X

Cristian S. Calude (ed.), ''Randomness and Complexity. From Leibniz to Chaitin'', (World Scientific, 2007) ISBN 13 978-981-277-082-0

External links

G J Chaitin Home Page

List of publications of G J Chaitin

Video of lecture on metabiology: "Life as evolving software"

New Scientist article (March, 2001) on Chaitin, Omegas and Super-Omegas

A short version of Chaitin's proof

Category:1947 births Category:Living people Category:The Bronx High School of Science alumni Category:City College of New York alumni Category:American mathematicians Category:Argentine mathematicians Category:Argentine computer scientists Category:20th-century mathematicians Category:21st-century mathematicians Category:Information theorists Category:IBM employees

ca:Gregory Chaitin de:Gregory Chaitin es:Gregory Chaitin fr:Gregory Chaitin ht:Gregory Chaitin nl:Gregory Chaitin ja:グレゴリー・チャイティン ru:Хайтин, Грегори sl:Gregory Chaitin

Name	Add N to (X)
Background	group_or_band
Origin	London, England, United Kingdom
Instruments	EMS Synthi A
Genre	Electroclash, Electronic music
Years active	1994–2003
Label	Blow Up RecordsSatelliteMute Records
Website	Official Add N to (X) Website
Current members	Barry Smith (1994–2003)Ann Shenton (1994–2002)Andrew Aveling (1994–1996)Steven Claydon (1997–2003)
Past members	}}

Add N to (X) were a three-piece British band specializing in electronic music performed on analogue synthesizers, formed in London in 1994.

The original band members were Andrew Aveling, Barry Smith (aka Barry 7) and Ann Shenton. Steven Claydon replaced Aveling in 1997.

After several releases on small labels, they turned down offers from major labels and signed to large independent label Mute Records in 1998, and achieved a modest commercial success before splitting in 2003.

Several of their songs and video clips have been adult/sex-related; the video for "Metal Fingers in My Body" is an animated short featuring a woman having sex with a robot, and their video for "Plug Me In" is famous for featuring porn actresses playing with sex toys.

Biography

In 1994 Andrew Aveling met Justin Anderson from Freaky Realistic and together they started a band named Radix Couplment. Andrew was dating Ann Shenton at this time and got her involved on the project. Andrew then asked his friend Barry (Former Radio Prague DJ Smith) if he too would be interested in joining. They then spent some time gigging under this name before a fall out which led to Justin's departure from the band. There being only three remaining Andrew then renamed the group Add N to X. They released Vero Electronics (1996) on the Blow Up label, and after did many other recordings for the next album.

They kept the name but placed brackets around the X due to legal reasons. They then enlisted Steven Claydon who remained with the group until its dissolution.

1997 saw the band twice awarded ''Single of the Week'' by the NME (for ''The Black Regent'' and ''King Wasp'').

Second album ''On the Wires of Our Nerves'' was released in 1998, and was described as "like Stereolab/Suicide with a rocket shoved up their rectum". The album was played heavily by BBC Radio 1 DJ Mary Anne Hobbs on her show, ''The Breezeblock''.

The group performed live regularly, often augmenting their core three-piece line up with either one or two acoustic drummers, and sometimes additional musicians playing extra synths and/or electric guitar.

They often utilized distinctive artwork for the videos and record sleeves, a fetishistic collage of sexual imagery with analogue electronic equipment, based in part on the movie and book ''Demon Seed''

The band signed to Mute Records and released three more albums, ''Avant Hard'', ''Add Insult to Injury'', and ''Loud Like Nature''. They also released the single Little Black Rocks In The Sun, which was issued on 10 inch hexagonal vinyl. Shenton was reportedly overwhelmed by the pressures of the ''Loud Like Nature'' tour, and either left the group or was fired. In 2003 Smith and Claydon continued touring the United States without Shenton. The band broke up shortly thereafter. Barry Smith runs the Horseglue Records store and label with his partner Ethan Reid. Ann Shenton has formed a new group, Large Number. and record label White Label Music. Steven Claydon is now known for his artwork and in 2006 was included in a group show at Tate Modern.

Discography

Albums

''Vero Electronics'' (January 1996) Blow Up

''On the Wires of Our Nerves'' (February 1998) Satellite

''Avant Hard'' (April 1999) Mute

''Add Insult to Injury'' (October 2000) Mute

''Loud Like Nature'' (October 2002) Mute

Singles

*"The Black Regent" (1997) Satellite

*"King Wasp" (1997) Satellite

Fridge (band)>Fridge)

*"Little Black Rocks in the Sun" (1998) Mute

*"Metal Fingers in My Body" (1999) Mute

*"Revenge of the Black Regent" (1999) Mute

*"Live 1940" (1999) Slut Smalls

*"Plug Me In" (2000) Mute

*"The Poke 'Er 'Ole" (2001) Mute

*"And Another Thing" (2001) Rocket Girl (as ADD N TO FUXA)

*"Take Me To Your Leader" (2002) Mute

References

Notes

http://www.phinnweb.org/5HT/interviews/largenumber/

http://www.connexionbizarre.net/all-numbers-great-andsmall-an-interview-with-ann-shenton

http://www.kindamuzik.net/q_and_a/article.shtml?id=350

http://www.sabotagetimes.com/music/ann-shenton-from-add-n-to-x-interviewed-on-stage-it-was-like-warfare/

External links

Category:English electronic music groups Category:Mute Records artists

fr:Add N to (X) pt:Add N to (X) ru:Add N to (X)