June 2008 Maths Paper 2

[Midas Souza]

Bornin London, Turing was raised in southern England. He graduated in maths from King's College, Cambridge, and in 1938, earned a maths PhD from Princeton University. During the Second World War, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre that produced Ultra intelligence. He led Hut 8, the section responsible for German naval cryptanalysis. Turing devised techniques for speeding the breaking of German ciphers, including improvements to the pre-war Polish bomba method, an electromechanical machine that could find settings for the Enigma machine. He played a crucial role in cracking intercepted messages that enabled the Allies to defeat the Axis powers in many crucial engagements, including the Battle of the Atlantic.[10][11]

In 1952, Turing was prosecuted for homosexual acts. He accepted hormone treatment, a procedure commonly referred to as chemical castration, as an alternative to prison. Turing died on 7 June 1954, aged 41, from cyanide poisoning. An inquest determined his death as suicide, but the evidence is also consistent with accidental poisoning.[15] Following a campaign in 2009, British prime minister Gordon Brown made an official public apology for "the appalling way [Turing] was treated". Queen Elizabeth II granted a pardon in 2013. The term "Alan Turing law" is used informally to refer to a 2017 law in the UK that retroactively pardoned men cautioned or convicted under historical legislation that outlawed homosexual acts.[16]

Turing has an extensive legacy with statues and many things named after him, including an annual award for computer science innovations. He appears on the current Bank of England 50 note, which was released on 23 June 2021 to coincide with his birthday. A 2019 BBC series, as voted by the audience, named Turing the greatest person of the 20th century.

Turing was born in Maida Vale, London, while his father, Julius Mathison Turing, was on leave from his position with the Indian Civil Service (ICS) of the British Raj government at Chatrapur, then in the Madras Presidency and presently in Odisha state, in India.[17][18] Turing's father was the son of a clergyman, the Rev. John Robert Turing, from a Scottish family of merchants that had been based in the Netherlands and included a baronet. Turing's mother, Julius's wife, was Ethel Sara Turing (ne Stoney), daughter of Edward Waller Stoney, chief engineer of the Madras Railways. The Stoneys were a Protestant Anglo-Irish gentry family from both County Tipperary and County Longford, while Ethel herself had spent much of her childhood in County Clare.[19] Julius and Ethel married on 1 October 1907 at St. Bartholomew's Church on Clyde Road in Ballsbridge, Dublin.[20]

Julius's work with the ICS brought the family to British India, where his grandfather had been a general in the Bengal Army. However, both Julius and Ethel wanted their children to be brought up in Britain, so they moved to Maida Vale,[21] London, where Alan Turing was born on 23 June 1912, as recorded by a blue plaque on the outside of the house of his birth,[22][23] later the Colonnade Hotel.[17][24] Turing had an elder brother, John Ferrier Turing, father of Sir John Dermot Turing, 12th Baronet of the Turing baronets.[25]

Turing's father's civil service commission was still active during Turing's childhood years, and his parents travelled between Hastings in the United Kingdom[26] and India, leaving their two sons to stay with a retired Army couple. At Hastings, Turing stayed at Baston Lodge, Upper Maze Hill, St Leonards-on-Sea, now marked with a blue plaque.[27] The plaque was unveiled on 23 June 2012, the centenary of Turing's birth.[28]

Turing's parents enrolled him at St Michael's, a primary school at 20 Charles Road, St Leonards-on-Sea, from the age of six to nine. The headmistress recognised his talent, noting that she "...had clever boys and hardworking boys, but Alan is a genius".[30]

Between January 1922 and 1926, Turing was educated at Hazelhurst Preparatory School, an independent school in the village of Frant in Sussex (now East Sussex).[31] In 1926, at the age of 13, he went on to Sherborne School,[32] an independent boarding school in the market town of Sherborne in Dorset, where he boarded at Westcott House. The first day of term coincided with the 1926 General Strike, in Britain, but Turing was so determined to attend that he rode his bicycle unaccompanied 60 miles (97 km) from Southampton to Sherborne, stopping overnight at an inn.[33]

Turing's natural inclination towards mathematics and science did not earn him respect from 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".[34] Despite this, Turing continued to show remarkable ability in the studies he loved, solving advanced problems in 1927 without having studied even elementary calculus. In 1928, aged 16, Turing encountered Albert Einstein's work; not only did he grasp it, but it is possible that he managed to deduce Einstein's questioning of Newton's laws of motion from a text in which this was never made explicit.[35]

The event caused Turing great sorrow. He coped with his grief by working that much harder on the topics of science and mathematics that he had shared with Morcom. In a letter to Morcom's mother, Frances Isobel Morcom (ne Swan), Turing wrote:

I am sure I could not have found anywhere another companion so brilliant and yet so charming and unconceited. I regarded my interest in my work, and in such things as astronomy (to which he introduced me) as something to be shared with him and I think he felt a little the same about me ... I know I must put as much energy if not as much interest into my work as if he were alive, because that is what he would like me to do.[43]

Turing's relationship with Morcom's mother continued long after Morcom's death, with her sending gifts to Turing, and him sending letters, typically on Morcom's birthday.[44] A day before the third anniversary of Morcom's death (13 February 1933), he wrote to Mrs. Morcom:

I expect you will be thinking of Chris when this reaches you. I shall too, and this letter is just to tell you that I shall be thinking of Chris and of you tomorrow. I am sure that he is as happy now as he was when he was here. Your affectionate Alan.[45]

Some have speculated that Morcom's death was the cause of Turing's atheism and materialism.[46] Apparently, at this point in his life he still believed in such concepts as a spirit, independent of the body and surviving death. In a later letter, also written to Morcom's mother, Turing wrote:

Personally, I believe that spirit is really eternally connected with matter but certainly not by the same kind of body ... as regards the actual connection between spirit and body I consider that the body can hold on to a 'spirit', whilst the body is alive and awake the two are firmly connected. When the body is asleep I cannot guess what happens but when the body dies, the 'mechanism' of the body, holding the spirit is gone and the spirit finds a new body sooner or later, perhaps immediately.[47][48]

Between the springs of 1935 and 1936, at the same time as Church, Turing worked on the decidability of problems, starting from Gdel's incompleteness theorems. In mid-April 1936, Turing sent Max Newman the first draft typescript of his investigations. That same month, Alonzo Church published his An Unsolvable Problem of Elementary Number Theory, with similar conclusions to Turing's then-yet unpublished work. Finally, on 28 May of that year, he finished and delivered his 36-page paper for publication called "On Computable Numbers, with an Application to the Entscheidungsproblem".[57] It was published in the Proceedings of the London Mathematical Society journal in two parts, the first on 30 November and the second on 23 December.[58] In this paper, Turing reformulated Kurt Gdel's 1931 results on the limits of proof and computation, replacing Gdel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as Turing machines. The Entscheidungsproblem (decision problem) was originally posed by German mathematician David Hilbert in 1928. Turing proved that his "universal computing 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 decision problem by first showing that the halting problem for Turing machines is undecidable: it is not possible to decide algorithmically whether a Turing machine will ever halt. This paper has been called "easily the most influential math paper in history".[59]

From September 1936 to July 1938, Turing spent most of his time studying under Church at Princeton University,[4] in the second year as a Jane Eliza Procter Visiting Fellow. In addition to his purely mathematical work, he studied cryptology and also built three of four stages of an electro-mechanical binary multiplier.[64] In June 1938, he obtained his PhD from the Department of Mathematics at Princeton;[65] his dissertation, Systems of Logic Based on Ordinals,[66][67] introduced the concept of ordinal logic and the notion of relative computing, in which Turing machines are augmented with so-called oracles, allowing the study of problems that cannot be solved by Turing machines. John von Neumann wanted to hire him as his postdoctoral assistant, but he went back to the United Kingdom.[68]

When Turing returned to Cambridge, he attended lectures given in 1939 by Ludwig Wittgenstein about the foundations of mathematics.[69] The lectures have been reconstructed verbatim, including interjections from Turing and other students, from students' notes.[70] Turing and Wittgenstein argued and disagreed, with Turing defending formalism and Wittgenstein propounding his view that mathematics does not discover any absolute truths, but rather invents them.[71]

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