In mathematics, the '''' (, German for 'decision problem') is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false. Such an algorithm would be able to decide, for example, whether statements such as Goldbach's conjecture or the Riemann hypothesis are true, even though no proof or disproof of these statements is known. The has often been identified in particular with the decision problem for first-order logic (that is, the problem of algorithmically determining whether a first-order statement is universally valid).

In 1936 and 1937 Alonzo Church and Alan Turing respectively, published independent papers showing that it is impossible to decide algorithmically whether statements in arithmetic are true or false, and thus a general solution to the Entscheidungsproblem is impossible. This result is now known as Church's Theorem or the Church–Turing Theorem (not to be confused with the Church–Turing thesis).

History of the problem

The origin of the goes back to Gottfried Leibniz, who in the seventeenth century, after having constructed a successful mechanical calculating machine, dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements (Davis 2000: pp. 3–20). He realized that the first step would have to be a clean formal language, and much of his subsequent work was directed towards that goal. In 1928, David Hilbert and Wilhelm Ackermann posed the question in the form outlined above.

In continuation of his "program" with which he challenged the mathematics community in 1900, at a 1928 international conference David Hilbert asked three questions, the third of which became known as "Hilbert's " (Hodges p. 91). As late as 1930 he believed that there would be no such thing as an unsolvable problem (Hodges p. 92, quoting from Hilbert).

Negative answer

Before the question could be answered, the notion of "algorithm" had to be formally defined. This was done by Alonzo Church in 1936 with the concept of "effective calculability" based on his λ calculus and by Alan Turing in the same year with his concept of Turing machines. It was later recognized that these are equivalent models of computation.

The negative answer to the '''' was then given by Alonzo Church in 1935–36 and independently shortly thereafter by Alan Turing in 1936–37. Church proved that there is no computable function which decides for two given λ calculus expressions whether they are equivalent or not. He relied heavily on earlier work by Stephen Kleene. Turing reduced the halting problem for Turing machines to the . The work of both authors was heavily influenced by Kurt Gödel's earlier work on his incompleteness theorem, especially by the method of assigning numbers (a Gödel numbering) to logical formulas in order to reduce logic to arithmetic.

Turing's argument is as follows. Suppose we had a general decision algorithm for statements in a first-order language. The question whether a given Turing machine halts or not can be formulated as a first-order statement, which would then be susceptible to the decision algorithm. But Turing had proven earlier that no general algorithm can decide whether a given Turing machine halts.

The is related to Hilbert's tenth problem, which asks for an algorithm to decide whether Diophantine equations have a solution. The non-existence of such an algorithm, established by Yuri Matiyasevich in 1970, also implies a negative answer to the Entscheidungsproblem.

Some first-order theories are algorithmically decidable; examples of this include Presburger arithmetic, real closed fields and static type systems of (most) programming languages. The general first-order theory of the natural numbers expressed in Peano's axioms cannot be decided with such an algorithm, however.

See also

Hilbert's second problem

Oracle machine

Notes

References

Alonzo Church, "An unsolvable problem of elementary number theory", American Journal of Mathematics, 58 (1936), pp 345–363

Alonzo Church, "A note on the Entscheidungsproblem", Journal of Symbolic Logic, 1 (1936), pp 40–41.

Martin Davis, 2000, ''Engines of Logic'', W.W. Norton & Company, London, ISBN 0-393-32229-7 pbk.

Alan Turing, "On computable numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, Series 2, 42 (1937), pp 230–265. Online versions: from journal website, from Turing Digital Archive, from abelard.org. Errata appeared in Series 2, 43 (1937), pp 544–546.

Martin Davis, "The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions", Raven Press, New York, 1965. Turing's paper is #3 in this volume. Papers include those by Godel, Church, Rosser, Kleene, and Post.

Andrew Hodges, Alan Turing: The Enigma, Simon and Schuster, New York, 1983. Alan M. Turing's biography. Cf Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.

Toulmin, Stephen, "Fall of a Genius", a book review of "Alan Turing: The Enigma by Andrew Hodges", in The New York Review of Books, January 19, 1984, p. 3ff.

Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge at the University Press, 1962. Re: the problem of paradoxes, the authors discuss the problem of a set not be an object in any of its "determining functions", in particular "Introduction, Chap. 1 p. 24 "...difficulties which arise in formal logic", and Chap. 2.I. "The Vicious-Circle Principle" p. 37ff, and Chap. 2.VIII. "The Contradictions" p. 60 ff.

Category:Theory of computation Category:Computability theory Category:Mathematical logic Category:Metatheorems Category:German words and phrases

ar:مشكلة القرار (رياضيات) ca:Entscheidungsproblem cs:Entscheidungsproblem de:Entscheidbar es:Entscheidungsproblem fr:Problème de la décision hr:Entscheidungsproblem it:Entscheidungsproblem pt:Entscheidungsproblem ru:Проблема разрешения simple:Entscheidungsproblem uk:Задача розв'язності zh:可判定性

Coordinates	30°19′10″N81°39′36″N
name	Alan Mathison Turing
birth date	June 23, 1912
birth place	Maida Vale, London, England, United Kingdom
death date	June 07, 1954
death place	Wilmslow, Cheshire, England, United Kingdom
residence	United Kingdom
nationality	British
field	Mathematics, Cryptanalysis, Computer science
work institutions	University of CambridgeGovernment Code and Cypher SchoolNational Physical LaboratoryUniversity of Manchester
alma mater	King's College, CambridgePrinceton University
doctoral advisor	Alonzo Church
doctoral students	Robin Gandy
known for	Halting problemTuring machineCryptanalysis of the EnigmaAutomatic Computing EngineTuring AwardTuring TestTuring patterns
prizes	Officer of the Order of the British EmpireFellow of the Royal Society }}

Alan Mathison Turing, OBE, FRS ( ; 23 June 1912 – 7 June 1954), was an English mathematician, logician, cryptanalyst and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a significant role in the creation of the modern computer. Turing is widely considered to be the father of computer science and artificial intelligence.

During the Second World War, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre. For a time he was head of Hut 8, the section responsible for German naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine that could find settings for the Enigma machine. After the war he worked at the National Physical Laboratory, where he created one of the first designs for a stored-program computer, the ACE.

Towards the end of his life Turing became interested in mathematical biology. He wrote a paper on the chemical basis of morphogenesis, and he predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, which were first observed in the 1960s.

Turing's homosexuality resulted in a criminal prosecution in 1952, when homosexual acts were still illegal in the United Kingdom. He accepted treatment with female hormones (chemical castration) as an alternative to prison. He died in 1954, just over two weeks before his 42nd birthday, from cyanide poisoning. An inquest determined it was suicide; his mother and some others believed his death was accidental. On 10 September 2009, following an Internet campaign, British Prime Minister Gordon Brown made an official public apology on behalf of the British government for the way in which Turing was treated after the war.

Childhood and youth

Alan Turing was conceived in India. His father, Julius Mathison Turing, was a member of the Indian Civil Service. Julius and wife Ethel Sara Stoney (1881–1976, daughter of Edward Waller Stoney, chief engineer of the Madras Railways) wanted Alan to be brought up in England, so they returned to Maida Vale, London, where Alan Turing was born on 23 June 1912, as recorded by a blue plaque on the outside of the building, later the Colonnade Hotel. He had an elder brother, John. His father's civil service commission was still active, and during Turing's childhood years his parents travelled between Hastings, England and India, leaving their two sons to stay with a retired Army couple. Very early in life, Turing showed signs of the genius he was to later prominently display.

His parents enrolled him at St Michael's, a day school at 20 Charles Road, St Leonards on Sea, at the age of six. The headmistress recognised his talent early on, as did many of his subsequent educators. In 1926, at the age of 14, he went on to Sherborne School, a famous independent school in the market town of Sherborne in Dorset. His first day of term coincided with the General Strike in Britain, but so determined was he to attend his first day that he rode his bicycle unaccompanied more than from Southampton to school, stopping overnight at an inn.

Turing's natural inclination toward mathematics and science did not earn him respect with some of the teachers at Sherborne, whose definition of education placed more emphasis on the classics. His headmaster wrote to his parents: "I hope he will not fall between two stools. If he is to stay at Public School, he must aim at becoming ''educated''. If he is to be solely a ''Scientific Specialist'', he is wasting his time at a Public School". Despite this, Turing continued to show remarkable ability in the studies he loved, solving advanced problems in 1927 without having even studied elementary calculus. In 1928, aged 16, Turing encountered Albert Einstein's work; not only did he grasp it, but he extrapolated Einstein's questioning of Newton's laws of motion from a text in which this was never made explicit.

Turing's hopes and ambitions at school were raised by the close friendship he developed with a slightly older fellow student, Christopher Morcom, who was Turing's first love interest. Morcom died suddenly only a few weeks into their last term at Sherborne, from complications of bovine tuberculosis, contracted after drinking infected cow's milk as a boy. Turing's religious faith was shattered and he became an atheist. He adopted the conviction that all phenomena, including the workings of the human brain, must be materialistic, but he still believed in the survival of the spirit after death.

University and work on computability

After Sherborne, Turing went to study at King's College, Cambridge. He was an undergraduate there from 1931 to 1934, graduating with first-class honours in Mathematics, and in 1935 was elected a fellow at King's on the strength of a dissertation on the central limit theorem.

In 1928, German mathematician David Hilbert had called attention to the ''Entscheidungsproblem'' (decision problem). In his momentous paper "On Computable Numbers, with an Application to the ''Entscheidungsproblem''" (submitted on 28 May 1936 and delivered 12 November), Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with what became known as Turing machines, formal and simple devices. He proved that some such machine would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. He went on to prove that there was no solution to the ''Entscheidungsproblem'' by first showing that the halting problem for Turing machines is undecidable: it is not possible to decide, in general, algorithmically whether a given Turing machine will ever halt. While his proof was published subsequent to Alonzo Church's equivalent proof in respect to his lambda calculus, Turing was unaware of Church's work at the time.

Turing's approach is considerably more accessible and intuitive. It was also novel in its notion of a 'Universal (Turing) Machine', the idea that such a machine could perform the tasks of any other machine, or in other words, is provably capable of computing anything that is computable. Turing machines are to this day a central object of study in theory of computation.

In his memoirs Turing wrote that he was disappointed about the reception of this 1936 paper and that only two people had reacted – these being Heinrich Scholz and Richard Bevan Braithwaite.

The paper also introduces the notion of definable numbers.

From September 1936 to July 1938 he spent most of his time at the Institute for Advanced Study, Princeton, New Jersey, studying under Alonzo Church. In addition to his purely mathematical work, he studied cryptology and also built three of four stages of an electro-mechanical binary multiplier. In June 1938 he obtained his PhD from Princeton; his dissertation (''Systems of Logic Based on Ordinals'') introduced the concept of ordinal logic and the notion of relative computing, where Turing machines are augmented with so-called oracles, allowing a study of problems that cannot be solved by a Turing machine.

Back in Cambridge, he attended lectures by Ludwig Wittgenstein about the foundations of mathematics. The two argued and disagreed, with Turing defending formalism and Wittgenstein arguing that mathematics does not discover any absolute truths but rather invents them. He also started to work part-time with the Government Code and Cypher School (GCCS).

Cryptanalysis

During the Second World War, Turing was a main participant in the efforts at Bletchley Park to break German ciphers. Building on cryptanalysis work carried out in Poland by Marian Rejewski, Jerzy Różycki and Henryk Zygalski from Cipher Bureau before the war, he contributed several insights into breaking both the Enigma machine and the Lorenz SZ 40/42 (a Teleprinter (Teletype) cipher attachment codenamed ''Tunny'' by the British), and was, for a time, head of Hut 8, the section responsible for reading German naval signals.

From September 1938, Turing had been working part-time (notionally for the British Foreign Office) with the Government Code and Cypher School (GCCS), the British code breaking organisation. He worked on the problem of the German Enigma machine, and collaborated with Dilly Knox, a senior GCCS codebreaker. On 4 September 1939, the day after the UK declared war on Germany, Turing reported to Bletchley Park, the wartime station of GCCS.

In 1945, Turing was awarded the OBE for his wartime services, but his work remained secret for many years.

Turing had something of a reputation for eccentricity at Bletchley Park. Jack Good, a cryptanalyst who worked with him, is quoted by Ronald Lewin as having said of Turing:

in the first week of June each year he would get a bad attack of hay fever, and he would cycle to the office wearing a service gas mask to keep the pollen off. His bicycle had a fault: the chain would come off at regular intervals. Instead of having it mended he would count the number of times the pedals went round and would get off the bicycle in time to adjust the chain by hand. Another of his eccentricities is that he chained his mug to the radiator pipes to prevent it being stolen.

While working at Bletchley, Turing, a talented long-distance runner, occasionally ran the to London when he was needed for high-level meetings, and he was capable of world-class marathon standards.

Turing–Welchman bombe

Within weeks of arriving at Bletchley Park, Turing had specified an electromechanical machine which could help break Enigma faster than bomba from 1938, the bombe, named after and building upon the original Polish-designed bomba. The bombe, with an enhancement suggested by mathematician Gordon Welchman, became one of the primary tools, and the major automated one, used to attack Enigma-protected message traffic.

Jack Good opined:

Turing's most important contribution, I ''think'', was of part of the design of the bombe, the cryptanalytic machine. He had the idea that you could use, in effect, a theorem in logic which sounds to the untrained ear rather absurd; namely that from a contradiction, you can deduce ''everything.''

The bombe searched for possibly correct settings used for an Enigma message (i.e., rotor order, rotor settings, etc.), and used a suitable ''crib'': a fragment of probable plaintext. For each possible setting of the rotors (which had of the order of 10¹⁹ states, or 10²² for the four-rotor U-boat variant), the bombe performed a chain of logical deductions based on the crib, implemented electrically. The bombe detected when a contradiction had occurred, and ruled out that setting, moving onto the next. Most of the possible settings would cause contradictions and be discarded, leaving only a few to be investigated in detail. Turing's bombe was first installed on 18 March 1940. More than two hundred bombes were in operation by the end of the war.

Hut 8 and Naval Enigma

Turing decided to tackle the particularly difficult problem of German naval Enigma "because no one else was doing anything about it and I could have it to myself". In December 1939, Turing solved the essential part of the naval indicator system, which was more complex than the indicator systems used by the other services. The same night that he solved the naval indicator system, he conceived the idea of ''Banburismus'', a sequential statistical technique (what Abraham Wald later called sequential analysis) to assist in breaking naval Enigma, "though I was not sure that it would work in practice, and was not in fact sure until some days had actually broken". For this he invented a measure of weight of evidence that he called the ''Ban''. Banburismus could rule out certain orders of the Enigma rotors, substantially reducing the time needed to test settings on the bombes.

In 1941, Turing proposed marriage to Hut 8 co-worker Joan Clarke, a fellow mathematician, but their engagement was short-lived. After admitting his homosexuality to his fiancée, who was reportedly "unfazed" by the revelation, Turing decided that he could not go through with the marriage.

In July 1942, Turing devised a technique termed ''Turingery'' (or jokingly ''Turingismus'') for use against the Lorenz cipher messages produced by the Germans' new Geheimschreiber machine (''secret writer''). This was codenamed ''Tunny '' at Bletchley Park. He also introduced the Tunny team to Tommy Flowers who, under the guidance of Max Newman, went on to build the Colossus computer, the world's first programmable digital electronic computer, which replaced a simpler prior machine (the Heath Robinson) and whose superior speed allowed the brute-force decryption techniques to be applied usefully to the daily changing cyphers. A frequent misconception is that Turing was a key figure in the design of Colossus; this was not the case.

Turing travelled to the United States in November 1942 and worked with U.S. Navy cryptanalysts on Naval Enigma and bombe construction in Washington, and assisted at Bell Labs with the development of secure speech devices. He returned to Bletchley Park in March 1943. During his absence, Hugh Alexander had officially assumed the position of head of Hut 8, although Alexander had been ''de facto'' head for some time—Turing having little interest in the day-to-day running of the section. Turing became a general consultant for cryptanalysis at Bletchley Park.

Alexander wrote as follows about his contribution:

There should be no question in anyone's mind that Turing's work was the biggest factor in Hut 8's success. In the early days he was the only cryptographer who thought the problem worth tackling and not only was he primarily responsible for the main theoretical work within the Hut but he also shared with Welchman and Keen the chief credit for the invention of the Bombe. It is always difficult to say that anyone is absolutely indispensable but if anyone was indispensable to Hut 8 it was Turing. The pioneer's work always tends to be forgotten when experience and routine later make everything seem easy and many of us in Hut 8 felt that the magnitude of Turing's contribution was never fully realised by the outside world.

In the latter part of the war he moved to work at Hanslope Park, where he further developed his knowledge of electronics with the assistance of engineer Donald Bayley. Together they undertook the design and construction of a portable secure voice communications machine codenamed ''Delilah''. It was intended for different applications, lacking capability for use with long-distance radio transmissions, and in any case, Delilah was completed too late to be used during the war. Though Turing demonstrated it to officials by encrypting/decrypting a recording of a Winston Churchill speech, Delilah was not adopted for use. Turing also consulted with Bell Labs on the development of SIGSALY, a secure voice system that was used in the later years of the war.

Early computers and the Turing test

From 1945 to 1947 Turing lived in Church Street, Hampton while he worked on the design of the ACE (Automatic Computing Engine) at the National Physical Laboratory. He presented a paper on 19 February 1946, which was the first detailed design of a stored-program computer. Although ACE was a feasible design, the secrecy surrounding the wartime work at Bletchley Park led to delays in starting the project and he became disillusioned. In late 1947 he returned to Cambridge for a sabbatical year. While he was at Cambridge, the Pilot ACE was built in his absence. It executed its first program on 10 May 1950.

In 1948, he was appointed Reader in the Mathematics Department at Manchester (now part of The University of Manchester). In 1949, he became Deputy Director of the computing laboratory at the University of Manchester, and worked on software for one of the earliest stored-program computers—the Manchester Mark 1. During this time he continued to do more abstract work, and in "Computing machinery and intelligence" (Mind, October 1950), Turing addressed the problem of artificial intelligence, and proposed an experiment which became known as the Turing test, an attempt to define a standard for a machine to be called "intelligent". The idea was that a computer could be said to "think" if a human interrogator could not tell it apart, through conversation, from a human being. In the paper, Turing suggested that rather than building a program to simulate the adult mind, it would be better rather to produce a simpler one to simulate a child's mind and then to subject it to a course of education. A reversed form of the Turing test is widely used on the Internet; the CAPTCHA test is intended to determine whether the user is a human or a computer.

In 1948, Turing, working with his former undergraduate colleague, D. G. Champernowne, began writing a chess program for a computer that did not yet exist. In 1952, lacking a computer powerful enough to execute the program, Turing played a game in which he simulated the computer, taking about half an hour per move. The game was recorded. The program lost to Turing's colleague Alick Glennie, although it is said that it won a game against Champernowne's wife.

His Turing test was a significant and characteristically provocative and lasting contribution to the debate regarding artificial intelligence, which continues after more than half a century.

He also invented the LU decomposition method in 1948, used today for solving an equations matrix.

Pattern formation and mathematical biology

Turing worked from 1952 until his death in 1954 on mathematical biology, specifically morphogenesis. He published one paper on the subject called ''The Chemical Basis of Morphogenesis'' in 1952, putting forth the Turing hypothesis of pattern formation. His central interest in the field was understanding Fibonacci phyllotaxis, the existence of Fibonacci numbers in plant structures. He used reaction–diffusion equations which are central to the field of pattern formation. Later papers went unpublished until 1992 when ''Collected Works of A.M. Turing'' was published. His contribution is considered a seminal piece of work in this field.

Conviction for indecency

In January 1952, Turing met Arnold Murray outside a cinema in Manchester. After a lunch date, Turing invited Murray to spend the weekend with him at his house, an invitation which Murray accepted although he did not show up. The pair met again in Manchester the following Monday, when Murray agreed to accompany Turing to the latter's house. A few weeks later Murray visited Turing's house again, and apparently spent the night there.

After Murray helped an accomplice to break into his house, Turing reported the crime to the police. During the investigation, Turing acknowledged a sexual relationship with Murray. Homosexual acts were illegal in the United Kingdom at that time, and so both were charged with gross indecency under Section 11 of the Criminal Law Amendment Act 1885, the same crime for which Oscar Wilde had been convicted more than fifty years earlier.

Turing was given a choice between imprisonment or probation conditional on his agreement to undergo hormonal treatment designed to reduce libido. He accepted chemical castration via oestrogen hormone injections.

Turing's conviction led to the removal of his security clearance, and barred him from continuing with his cryptographic consultancy for GCHQ. His British passport was not revoked, though he was denied entry to the United States after his conviction. At the time, there was acute public anxiety about spies and homosexual entrapment by Soviet agents, because of the recent exposure of the first two members of the Cambridge Five, Guy Burgess and Donald Maclean, as KGB double agents. Turing was never accused of espionage but, as with all who had worked at Bletchley Park, was prevented from discussing his war work.

Death

On 8 June 1954, Turing's cleaner found him dead; he had died the previous day. A post-mortem examination established that the cause of death was cyanide poisoning. When his body was discovered an apple lay half-eaten beside his bed, and although the apple was not tested for cyanide, it is speculated that this was the means by which a fatal dose was delivered. An inquest determined that he had committed suicide, and he was cremated at Woking Crematorium on 12 June 1954. Turing's mother argued strenuously that the ingestion was accidental, caused by her son's careless storage of laboratory chemicals. Biographer Andrew Hodges suggests that Turing may have killed himself in an ambiguous way quite deliberately, to give his mother some plausible deniability. David Leavitt has suggested that Turing was re-enacting a scene from the 1937 film ''Snow White'', his favourite fairy tale, pointing out that he took "an especially keen pleasure in the scene where the Wicked Witch immerses her apple in the poisonous brew."

Epitaph

Recognition and tributes

A biography published by the Royal Society shortly after Turing's death (and while his wartime work was still subject to the Official Secrets Act) recorded:

Since 1966, the Turing Award has been given annually by the Association for Computing Machinery to a person for technical contributions to the computing community. It is widely considered to be the computing world's highest honour, equivalent to the Nobel Prize.

''Breaking the Code'' is a 1986 play by Hugh Whitemore about Alan Turing. The play ran in London's West End beginning in November 1986 and on Broadway from 15 November 1987 to 10 April 1988. There was also a 1996 BBC television production. In all cases, Derek Jacobi played Turing. The Broadway production was nominated for three Tony Awards including Best Actor in a Play, Best Featured Actor in a Play, and Best Direction of a Play, and for two Drama Desk Awards, for Best Actor and Best Featured Actor.

On 23 June 1998, on what would have been Turing's 86th birthday, Andrew Hodges, his biographer, unveiled an official English Heritage Blue Plaque at his birthplace and childhood home in Warrington Crescent, London, later the Colonnade hotel. To mark the 50th anniversary of his death, a memorial plaque was unveiled on 7 June 2004 at his former residence, Hollymeade, in Wilmslow, Cheshire.

On 13 March 2000, Saint Vincent and the Grenadines issued a set of stamps to celebrate the greatest achievements of the twentieth century, one of which carries a recognisable portrait of Turing against a background of repeated 0s and 1s, and is captioned: "1937: Alan Turing's theory of digital computing".

On 28 October 2004, a bronze statue of Alan Turing sculpted by John W Mills was unveiled at the University of Surrey in Guildford, marking the 50th anniversary of Turing's death; it portrays him carrying his books across the campus.

In 2006, Boston Pride named Turing their Honorary Grand Marshal.

Turing was one of four mathematicians examined in the 2008 BBC documentary entitled "Dangerous Knowledge".

The Princeton Alumni Weekly named Turing the second most significant alumnus in the history of Princeton University, second only to President James Madison.

A 1.5-ton, life-size statue of Turing was unveiled on 19 June 2007 at Bletchley Park. Built from approximately half a million pieces of Welsh slate, it was sculpted by Stephen Kettle, having been commissioned by the late American billionaire Sidney Frank.

Turing has been honoured in various ways in Manchester, the city where he worked towards the end of his life. In 1994, a stretch of the A6010 road (the Manchester city intermediate ring road) was named Alan Turing Way. A bridge carrying this road was widened, and carries the name Alan Turing Bridge. A statue of Turing was unveiled in Manchester on 23 June 2001. It is in Sackville Park, between the University of Manchester building on Whitworth Street and the Canal Street gay village. The memorial statue, depicts the "father of Computer Science" sitting on a bench at a central position in the park. The statue was unveiled on Turing's birthday.

Turing is shown holding an apple—a symbol classically used to represent forbidden love, the object that inspired Isaac Newton's theory of gravitation, and the means of Turing's own death. The cast bronze bench carries in relief the text 'Alan Mathison Turing 1912–1954', and the motto 'Founder of Computer Science' as it would appear if encoded by an Enigma machine: 'IEKYF ROMSI ADXUO KVKZC GUBJ'.

A plinth at the statue's feet says 'Father of computer science, mathematician, logician, wartime codebreaker, victim of prejudice'. There is also a Bertrand Russell quotation saying 'Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture.' The sculptor buried his old Amstrad computer, which was an early popular home computer, under the plinth, as a tribute to "the godfather of all modern computers".

In 1999, ''Time Magazine'' named Turing as one of the 100 Most Important People of the 20th Century for his role in the creation of the modern computer, and stated: "The fact remains that everyone who taps at a keyboard, opening a spreadsheet or a word-processing program, is working on an incarnation of a Turing machine." Turing is featured in the 1999 Neal Stephenson novel "Cryptonomicon."

In 2002, Turing was ranked twenty-first on the ''BBC'' nationwide poll of the 100 Greatest Britons.

The logo of Apple computer is often erroneously referred to as a tribute to Alan Turing, with the bite mark a reference to his method of suicide. Both the designer of the logo and the company deny that there is any homage to Turing in the design of the logo.

In 2010, actor/playwright Jade Esteban Estrada portrayed Turing in the solo musical, "ICONS: The Lesbian and Gay History of the World, Vol. 4."

Turing is mentioned several times in the DLC for the 2K Games "Bioshock 2" He is mentioned several times by the voice actor who portrays C.M. Porter. There is also a telegram in Porter's office requesting he come to London and work with Turing. He is also mentioned in Assassin's Creed: Brotherhood, when an employee of Abstergo Industries orders him to be killed and commands the would-be executioner to "make it look biblical".

In February 2011, Turing's papers from the Second World War were bought for the nation with an 11th-hour bid by the National Heritage Memorial Fund, allowing them to stay at Bletchley Park.

Government apology

In August 2009, John Graham-Cumming started a petition urging the British Government to posthumously apologise to Alan Turing for prosecuting him as a homosexual. The petition received thousands of signatures. Prime Minister Gordon Brown acknowledged the petition, releasing a statement on 10 September 2009 apologising and describing Turing's treatment as "appalling":

Thousands of people have come together to demand justice for Alan Turing and recognition of the appalling way he was treated. While Turing was dealt with under the law of the time and we can't put the clock back, his treatment was of course utterly unfair and I am pleased to have the chance to say how deeply sorry I and we all are for what happened to him ... So on behalf of the British government, and all those who live freely thanks to Alan's work I am very proud to say: we're sorry, you deserved so much better.

Tributes by universities

A celebration of Turing's life and achievements arranged by the British Logic Colloquium and the British Society for the History of Mathematics was held on 5 June 2004.

The University of Surrey has a statue of Turing on their main piazza.

Istanbul Bilgi University organises an annual conference on the theory of computation called "Turing Days". The University of Texas at Austin has an honours computer science programme named the Turing Scholars. In the early 1960s Stanford University named the sole lecture room of the Polya Hall Mathematics building "Alan Turing Auditorium". One of the amphitheatres of the Computer Science department (LIFL) at the University of Lille in northern France is named in honour of Alan M. Turing (the other amphitheatre is named after Kurt Gödel).

The Department of Computer Science at Pontifical Catholic University of Chile, the Polytechnic University of Puerto Rico, Los Andes University in Bogotá, Colombia, King's College, Cambridge and Bangor University in Wales have computer laboratories named after Turing.

The University of Manchester, The Open University, Oxford Brookes University and Aarhus University (in Århus, Denmark) all have buildings named after Turing.

Alan Turing Road in the Surrey Research Park is named for Alan Turing.

Carnegie Mellon University has a granite bench, situated in The Hornbostel Mall, with the name "A. M. Turing" carved across the top, "Read" down the left leg, and "Write" down the other.

The École Internationale des Sciences du Traitement de l'Information has named its recently acquired third building "Turing".

The University of Ghent has one of its main computer rooms (in a building used mostly by mathematicians and computing scientists) named for Alan Turing.

The University of Oregon has a bust of Turing on the side of the Deschutes Hall, the computer science building.

Centenary commemoration

180px|border|rightTo mark the 100th anniversary of Turing's birth, the Turing Centenary Advisory Committee (TCAC) is coordinating the Alan Turing Year, a year-long programme of events around the world honouring Turing's life and achievements. The TCAC working with The University of Manchester faculty members and a broad spectrum of people from Cambridge University and Bletchley Park, is chaired by S. Barry Cooper, with Alan Turing's nephew Sir John Dermot Turing acting as TCAC Honorary President.

Events are scheduled in many countries around the world including the USA, Brazil, China, Czech Republic, the Philippines, New Zealand, Israel, Spain, Norway, Italy, Portugal and Germany. The keystone events will be a three-day conference in Manchester, UK in June examining Turing's mathematical and code-breaking achievements, and a Turing Centenary Conference in Cambridge organised by King's College, Cambridge and the association Computability in Europe.

See also

Good–Turing frequency estimation

Turing degree

Turing machine examples

Turing switch

Unorganized machine

Notes

References

Petzold, Charles (2008). "The Annotated Turing: A Guided Tour through Alan Turing's Historic Paper on Computability and the Turing Machine". Indianapolis: Wiley Publishing. ISBN 978-0-470-22905-7

Smith, Roger (1997). ''Fontana History of the Human Sciences''. London: Fontana.

Weizenbaum, Joseph (1976). ''Computer Power and Human Reason''. London: W.H. Freeman. ISBN 0-7167-0463-3

Turing's mother, who survived him by many years, wrote this 157-page biography of her son, glorifying his life. It was published in 1959, and so could not cover his war work. Scarcely 300 copies were sold (Sara Turing to Lyn Newman, 1967, Library of St John's College, Cambridge). The six-page foreword by Lyn Irvine includes reminiscences and is more frequently quoted. This 1986 Hugh Whitemore play tells the story of Turing's life and death. In the original West End and Broadway runs, Derek Jacobi played Turing and he recreated the role in a 1997 television film based on the play made jointly by the BBC and WGBH, Boston. The play is published by Amber Lane Press, Oxford, ASIN: B000B7TM0Q

Williams, Michael R. (1985) ''A History of Computing Technology'', Englewood Cliffs, New Jersey: Prentice-Hall, ISBN 0-8186-7739-2

Further reading

Gleick, James, ''The Information: A History, A Theory, A Flood'', New York: Pantheon, 2011, ISBN 9780375423727

Leavitt, David, ''The Man Who Knew Too Much: Alan Turing and the Invention of the Computer'', W. W. Norton, 2006

External links

Alan Turing RKBExplorer

Alan Turing Year

CiE 2012: Turing Centenary Conference

Visual Turing

Turing Machine calculators at Wolfram Alpha

Alan Turing site maintained by Andrew Hodges including a short biography

AlanTuring.net – Turing Archive for the History of Computing by Jack Copeland

The Turing Archive – contains scans of some unpublished documents and material from the Kings College, Cambridge archive

Papers

An extensive list of Turing's papers, reports and lectures, plus translated versions and collections BibNetWiki

Oral history interview with Donald W. Davies, Charles Babbage Institute, University of Minnesota; Davies describes computer projects at the U.K. National Physical Laboratory, from the 1947 design work of Alan Turing to the development of the two ACE computers

Oral history interview with Nicholas C. Metropolis, Charles Babbage Institute, University of Minnesota. Metropolis was the first director of computing services at Los Alamos National Laboratory; topics include the relationship between Alan Turing and John von Neumann

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Coordinates	30°19′10″N81°39′36″N
name	Kurt Gödel
birth date	April 28, 1906
birth place	Brno, Moravia, Austria–Hungary
death date	January 14, 1978
death place	Princeton, New Jersey, United States
field	Mathematics, Mathematical logic
work institutions	Institute for Advanced Study
alma mater	University of Vienna
doctoral advisor	Hans Hahn
known for	Gödel's incompleteness theorems, Gödel's completeness theorem, the consistency of the Continuum hypothesis with ZFC, Gödel's ontological proof
prizes	Albert Einstein Award (1951); National Medal of Science (USA) in Mathematical, Statistical, and Computational Sciences (1974)
religion	Theist
signature	Kurt Gödel signature.svg
footnotes	}}

Kurt Friedrich Gödel (; April 28, 1906 – January 14, 1978) was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for "working mathematicians" to assume the axiom of choice in their proofs. There are many important results whose only known proofs use the axiom of choice, which is equivalent to Zorn's lemma. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

Life

Childhood

Gödel was born April 28, 1906, in Brno, Austria-Hungary into the ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh). At the time of his birth the town had a slight German-speaking majority, and this was the language of his parents. The ancestors of Kurt Gödel were often active in the cultural life of the Brno city. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the "Brünner Männergesangverein".

Although he spoke very little Czech himself, Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian empire broke up at the end of World War I. According to his classmate Klepetař, "Gödel considered himself always Austrian and an exile in Czechoslovakia" ("''ein Österreicher im Exil in der Tschechoslowakei''") during this time. He chose to become an Austrian citizen at age 23. When Nazi Germany annexed Austria, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became an American citizen.

In his family, young Kurt was known as ''Herr Warum'' ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage.

Gödel attended the ''Evangelische Volksschule'', a Lutheran school in Brno from 1912 to 1916, and was enrolled in the ''Deutsches Staats-Realgymnasium'' from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's ''Theory of Colours'' and criticisms of Isaac Newton, and the writings of Immanuel Kant.

Studying in Vienna

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics. Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism. He read Kant's ''Metaphysische Anfangsgründe der Naturwissenschaft'', and participated in the Vienna Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book ''Introduction to Mathematical Philosophy'', he became interested in mathematical logic. According to Gödel mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."

Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published ''Grundzüge der theoretischen Logik'' (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: ''Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?''

This was the topic chosen by Gödel for his doctorate work. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established the completeness of the first-order predicate calculus (Gödel's completeness theorem). He was awarded his doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.

The Incompleteness Theorem

In 1931 and while still in Vienna, Gödel published his incompleteness theorems in "Über formal unentscheidbare Sätze der ''Principia Mathematica'' und verwandter Systeme" (called in English "On Formally Undecidable Propositions of ''Principia Mathematica'' and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that: # If the system is consistent, it cannot be complete. # The consistency of the axioms cannot be proven within the system. These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in ''Principia Mathematica'' and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to produce a method to encode statements, proofs, and the concept of provability as natural numbers. He did this using a process known as Gödel numbering.

In his two-page paper "Zum intuitionistischen Aussagenkalkül" (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).

The mid 1930s: further work and visits to the US

Gödel earned his habilitation at Vienna in 1932, and in 1933 he became a ''Privatdozent'' (unpaid lecturer) there. In 1933 Adolf Hitler came to power in Germany and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians. In June 1936, Moritz Schlick, whose seminar had aroused Gödel's interest in logic, was assassinated by a pro-Nazi student. This triggered "a severe nervous crisis" in Gödel. He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend. He delivered an address to the annual meeting of the American Mathematical Society. During this year, Gödel also developed the ideas of computability and recursive functions to the point where he delivered a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering.

In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled ''On undecidable propositions of formal mathematical systems''. Stephen Kleene, who had just completed his Ph.D. at Princeton, took notes of these lectures which have been subsequently published.

Gödel would visit the IAS again in the autumn of 1935. The traveling and the hard work had exhausted him, and the next year he took a break to recover from a depression. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis; he would go on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.

He married Adele Nimbursky (née Porkert, 1899–1981), whom he had known for over 10 years, on September 20, 1938. Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he.

Subsequently, he left for another visit to the USA, spending the autumn of 1938 at the IAS and the spring of 1939 at the University of Notre Dame.

Gödel and his wife Adele spent the summer of 1942 in Blue Hill, Maine, in the Blue Hill Inn at the top of the bay. Gödel was taking a vacation from the IAS.

Gödel was not merely vacationing, and had a very productive summer of work. Using Heft 15 [volume 15] of Gödel's still-unpublished Arbeitshefte [working notebooks], John W. Dawson, Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.

Relocation to Princeton, Einstein and US citizenship

After the Anschluss in 1938, Austria had become a part of Nazi Germany. Germany abolished the title of ''Privatdozent'', so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. The University of Vienna turned his application down. His predicament intensified when the German army found him fit for conscription. World War II started in September 1939. Before the year was up, Gödel and his wife left Vienna for Princeton. To avoid the difficulty of an Atlantic crossing, the Gödels took the trans-Siberian railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the U.S. by train to Princeton, where Gödel would accept a position at the Institute for Advanced Study (IAS).

Gödel very quickly resumed his mathematical work. In 1940, he published his work ''Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory'' which is a classic of modern mathematics. In that work he introduced the constructible universe, a model of set theory in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with the Zermelo–Fraenkel axioms for set theory (ZF). Paul Cohen later constructed a model of ZF in which AC and GCH are false; together these proofs mean that AC and GCH are independent of the ZF axioms for set theory.

Albert Einstein was also living at Princeton during this time. Gödel and Einstein subsequently developed a strong friendship, and were known to take long walks together to and from the Institute for Advanced Study. The nature of their conversations was a mystery to the other Institute members. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that his "own work no longer meant much, that he came to the Institute merely...to have the privilege of walking home with Gödel".

On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his chances. Fortunately, the judge turned out to be Phillip Forman. Forman knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.

Later years and death

Gödel became a permanent member of the Institute of Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.

In 1951, Gödel demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. He gave this elaboration to Einstein as a present for his 70th birthday. These "rotating universes" would allow time travel and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric.

During his many years at the Institute, Gödel's interests turned to philosophy and physics. He studied and admired the works of Gottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed. To a lesser extent he studied Immanuel Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of Anselm of Canterbury's ontological proof of God's existence. This is now known as Gödel's ontological proof. Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.

In later life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, Adele was hospitalized for six months and could no longer prepare Gödel's food. In her absence, he refused to eat, eventually starving to death. He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.

Religious views

Gödel was a convinced theist. He rejected the notion of others like his friend Albert Einstein that God was impersonal.

He believed firmly in an afterlife, stating: "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of. But I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world [Welt] is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."

In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is ''theistic'', not pantheistic, following Leibniz rather than Spinoza."

Legacy

The Kurt Gödel Society, founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of logic, philosophy, and the history of mathematics. The University of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. The Association of Symbolic Logic has invited an annual Kurt Gödel lecture each year since 1990.

Five volumes of Gödel's collected works have been published. The first two include Gödel's publications; the third includes unpublished manuscripts from Gödel's ''Nachlass'', and the final two include correspondence.

A biography of Gödel was published by John Dawson in 2005. Gödel was also one of four mathematicians examined in the 2008 BBC documentary entitled "Dangerous Knowledge".

Douglas Hofstadter wrote a popular book in 1979 called ''Gödel, Escher, Bach: An Eternal Golden Braid'' to celebrate the work and ideas of Gödel, along with those of artist M. C. Escher and composer Johann Sebastian Bach. The book partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete computational system, which may include the human brain.

Important publications

In German:

1930, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls." ''Monatshefte für Mathematik und Physik'' 37: 349–60.

1931, "Über formal unentscheidbare Sätze der ''Principia Mathematica'' und verwandter Systeme, I." ''Monatshefte für Mathematik und Physik'' 38: 173–98.

1932, "Zum intuitionistischen Aussagenkalkül", ''Anzeiger Akademie der Wissenschaften Wien'' 69: 65–66.

In English:

1940. ''The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory.'' Princeton University Press.

1947. "What is Cantor's continuum problem?" ''The American Mathematical Monthly 54'': 515–25. Revised version in Paul Benacerraf and Hilary Putnam, eds., 1984 (1964). ''Philosophy of Mathematics: Selected Readings''. Cambridge Univ. Press: 470–85.

1950, "Rotating Universes in General Relativity Theory." ''Proceedings of the international Congress of Mathematicians in Cambridge,'' 1: 175–81

In English translation:

Kurt Godel, 1992. ''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. B. Meltzer, with a comprehensive introduction by Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.

Kurt Godel, 2000. ''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. Martin Hirzel

Jean van Heijenoort, 1967. ''A Source Book in Mathematical Logic, 1879–1931''. Harvard Univ. Press.

*1930. "The completeness of the axioms of the functional calculus of logic," 582–91.

*1930. "Some metamathematical results on completeness and consistency," 595–96. Abstract to (1931).

*1931. "On formally undecidable propositions of ''Principia Mathematica'' and related systems," 596–616.

*1931a. "On completeness and consistency," 616–17.

"My philosophical viewpoint", c. 1960, unpublished.

"The modern development of the foundations of mathematics in the light of philosophy", 1961, unpublished.

''Collected Works'': Oxford University Press: New York. Editor-in-chief: Solomon Feferman.

*Volume I: Publications 1929–1936 ISBN 978-0-19-503964-1 / Paperback:ISBN 978-0-19-514720-9,

*Volume II: Publications 1938–1974 ISBN 978-0-19-503972-6 / Paperback:ISBN 978-0-19-514721-6,

*Volume III: Unpublished Essays and Lectures ISBN 978-0-19-507255-6 / Paperback:ISBN 978-0-19-514722-3,

*Volume IV: Correspondence, A–G ISBN 978-0-19-850073-5,

*Volume V: Correspondence, H–Z ISBN 978-0-19-850075-9.

See also

Notes

References

Dawson, John W., 1997. ''Logical dilemmas: The life and work of Kurt Gödel''. Wellesley MA: A K Peters.

1911 Encyclopædia Britannica/Brünn. (2007, September 19). In Wikisource, The Free Library. Retrieved 10PM EST March 13, 2008.

Rebecca Goldstein, 2005. ''Incompleteness: The Proof and Paradox of Kurt Gödel''. W. W. Norton & Company, New York. ISBN 0-393-32760-4 pbk.

Further reading

John L. Casti and Werner DePauli, 2000. ''Gödel: A Life of Logic'', Basic Books (Perseus Books Group), Cambridge, MA. ISBN 0-7382-0518-4.

John W. Dawson, Jr. ''Logical Dilemmas: The Life and Work of Kurt Gödel''. AK Peters, Ltd., 1996.

John W. Dawson, Jr, 1999. "Gödel and the Limits of Logic", ''Scientific American'', vol. 280 num. 6, pp. 76–81

Torkel Franzén, 2005. ''Gödel's Theorem: An Incomplete Guide to Its Use and Abuse''. Wellesley, MA: A K Peters.

Ivor Grattan-Guinness, 2000. ''The Search for Mathematical Roots 1870–1940''. Princeton Univ. Press.

Jaakko Hintikka, 2000. ''On Gödel''. Wadsworth.

Douglas Hofstadter, 1980. ''Gödel, Escher, Bach''. Vintage.

Stephen Kleene, 1967. ''Mathematical Logic''. Dover paperback reprint ca. 2001.

Stephen Kleene, 1980. ''Introduction to Metamathematics''. North Holland ISBN 0-7204-2103-9 (Ishi Press paperback. 2009. ISBN 978-0-923891-57-2)

J.R. Lucas, 1970. ''The Freedom of the Will''. Clarendon Press, Oxford.

Ernest Nagel and Newman, James R., 1958. ''Gödel's Proof.'' New York Univ. Press.

Procházka, Jiří, 2006, 2006, 2008, 2008, 2O1O. ''Kurt Gödel: 1906–1978: Genealogie''. ITEM, Brno. Volume I. Brno 2006, ISBN 80-902297-9-4. In Ger., Engl. Volume II. Brno 2006, ISBN 80-903476-0-6. In Germ., Engl. Volume III. Brno 2008, ISBN 80-903476-4-9. In Germ., Engl. Volume IV. Brno, Princeton 2008, ISBN 978-80-903476-5-6. In Germ., Engl.Volume V,Brno,Princeton 2O1O, ISBN 8O-9O3476-9-X.In Germ.,Engl.

Ed Regis, 1987. ''Who Got Einstein's Office?'' Addison-Wesley Publishing Company, Inc.

Raymond Smullyan, 1992. ''Godel's Incompleteness Theorems''. Oxford University Press.

Olga Taussky-Todd, 1983. Remembrances of Kurt Gödel. Engineering & Science, Winter 1988.

Hao Wang, 1987. ''Reflections on Kurt Gödel.'' MIT Press.

Hao Wang, 1996. ''A Logical Journey: From Godel to Philosophy''. MIT Press.

Yourgrau, Palle, 1999. ''Gödel Meets Einstein: Time Travel in the Gödel Universe.'' Chicago: Open Court.

Yourgrau, Palle, 2004. ''A World Without Time: The Forgotten Legacy of Gödel and Einstein.'' Basic Books. Book review by John Stachel in the Notices of the American Mathematical Society (54 (7), p 861–868):

External links

Kennedy, Juliette. "Kurt Gödel." In Stanford Encyclopedia of Philosophy.

Time Bandits – an article about the relationship between Gödel and Einstein by Jim Holt

"Gödel and the limits of logic" by John W Dawson Jr. (June 2006)

Notices of the AMS, April 2006, Volume 53, Number 4 Kurt Gödel Centenary Issue

Paul Davies and Freeman Dyson discuss Kurt Godel

"Gödel and the Nature of Mathematical Truth" Edge: A Talk with Rebecca Goldstein on Kurt Gödel.

It's Not All In The Numbers: Gregory Chaitin Explains Gödel's Mathematical Complexities.

Dangerous Knowledge Google Video of a BBC documentary made by David Malone about the life and work of Kurt Gödel and other revolutionary mathematical thinkers.

Gödel photo g.

Category:1906 births Category:1978 deaths Category:20th-century philosophers Category:20th-century mathematicians Category:American mathematicians Category:American people of Austrian descent Category:Austrian emigrants to the United States Category:Austrian mathematicians Category:Austrian logicians Category:Austrian philosophers Category:Christian philosophers Category:Ontologists Category:Platonists Category:Deaths by starvation Category:Austrian people of Moravian German descent Category:German-language philosophers Category:Institute for Advanced Study faculty Category:National Medal of Science laureates Category:Naturalized citizens of the United States Category:People from Brno Category:Princeton University faculty Category:Set theorists Category:Vienna Circle Category:University of Vienna alumni Category:Burials at Princeton Cemetery Category:Foreign Members of the Royal Society Category:American logicians Category:American philosophers Category:Austrian Lutherans Category:American Lutherans

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