Editing Alan Turing (section)

==Early life and education==

===Family===
[[File:Alan Turing (5025990183).jpg|thumb|right|upright|[[English Heritage]] [[blue plaque]] in [[Maida Vale]], London marking Turing's birthplace in 1912]]
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|British Raj government]] at [[Chatrapur]], then in the [[Madras Presidency]] and presently in [[Odisha]] state, in [[India]].<ref name = "Hodges1983P5">{{Harvnb|Hodges|1983|p=5}}</ref><ref>{{cite web |url=http://www.turing.org.uk/turing/scrapbook/early.html |title=The Alan Turing Internet Scrapbook |publisher=[[Alan Turing: The Enigma]] |access-date=2 January 2012 |archive-url=https://web.archive.org/web/20120614051614/http://www.turing.org.uk/turing/scrapbook/early.html |archive-date=14 June 2012 |url-status=live }}</ref> Turing's father was the son of a clergyman, the Rev.&nbsp;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 ({{nee|Stoney}}), daughter of Edward Waller Stoney, chief engineer of the [[Madras and Southern Mahratta Railway|Madras Railways]]. The Stoneys were a [[Protestantism in Ireland|Protestant]] [[Anglo-Irish people|Anglo-Irish]] [[gentry]] family from both [[County Tipperary]] and [[County Longford]], while Ethel herself had spent much of her childhood in [[County Clare]].<ref>{{Cite news |first=Phil |last=Maguire |title=An Irishman's Diary |page=5 |newspaper=[[The Irish Times]] |date=23 June 2012}}</ref> Julius and Ethel married on 1 October 1907 at the Church of Ireland [[St Bartholomew's Church, Dublin|St. Bartholomew's Church]] on [[Clyde Road]] in [[Ballsbridge]], [[Dublin]].<ref>Irish Marriages 1845–1958 / Dublin South, Dublin, Ireland / Group Registration ID 1990366, SR District/Reg Area, Dublin South</ref>

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]],<ref name="englishheritaget">{{cite web|url=http://www.english-heritage.org.uk/server/show/nav.001002006005/chooseLetter/T |archive-url=https://web.archive.org/web/20090903150218/http://www.english-heritage.org.uk/server/show/nav.001002006005/chooseLetter/T |archive-date=3 September 2009 |title=London Blue Plaques |access-date=10 February 2007 |work=English Heritage |url-status=live}}</ref> 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,<ref>{{Cite web |url=http://blogs.nature.com/london/2011/03/16/the-scientific-tourist-in-london-17-alan-turings-birth-place |title=The Scientific Tourist In London: #17 Alan Turing's Birth Place |archive-url=https://web.archive.org/web/20130921054045/http://blogs.nature.com/london/2011/03/16/the-scientific-tourist-in-london-17-alan-turings-birth-place |archive-date=21 September 2013 |website=Nature London Blog}},</ref><ref>{{openplaque|381}}</ref> later the [[Colonnade Hotel (London)|Colonnade Hotel]].<ref name="Hodges1983P5"/><ref name="turingorguk">{{cite web | url=http://www.turing.org.uk/turing/scrapbook/memorial.html | title=The Alan Turing Internet Scrapbook | access-date=26 September 2006 | archive-url=https://wayback.archive-it.org/all/20110720214124/http://www.turing.org.uk/turing/scrapbook/memorial.html | archive-date=20 July 2011 | url-status=live }}</ref> Turing had an elder brother, John Ferrier Turing, father of [[Dermot Turing]], 12th Baronet of the [[Turing baronets]].<ref>{{Cite web |url=https://bletchleypark.org.uk/about-us/bletchley-park-trustees/sir-john-dermot-turing |title=Sir John Dermot Turing |archive-url=https://web.archive.org/web/20171018191443/https://bletchleypark.org.uk/about-us/bletchley-park-trustees/sir-john-dermot-turing |archive-date=18 October 2017 |website=Bletchley Park}}</ref>

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<ref>{{Harvnb|Hodges|1983|p=6}}</ref> and India, leaving their two sons to stay with a retired [[British Army|Army]] couple. At Hastings, Turing stayed at [[Baston Lodge]], Upper Maze Hill, [[St Leonards-on-Sea]], now marked with a blue plaque.<ref name="Hastings & St. Leonards Observer - 29 June 2012 - Plaque unveiled at Turing's home in St Leonards">{{cite news|url=http://www.hastingsobserver.co.uk/news/plaque-unveiled-at-turing-s-home-in-st-leonards-1-4003535|title=Plaque unveiled at Turing's home in St Leonards|date=29 June 2012|work=[[Hastings & St. Leonards Observer]]|access-date=3 July 2017|archive-url=https://web.archive.org/web/20170912121655/http://www.hastingsobserver.co.uk/news/plaque-unveiled-at-turing-s-home-in-st-leonards-1-4003535|archive-date=12 September 2017|url-status=dead}}</ref> The plaque was unveiled on 23 June 2012, the centenary of Turing's birth.<ref name="BBC News - 25 June 2012 - St Leonards plaque marks Alan Turing's early years">{{cite news|url=https://www.bbc.com/news/uk-england-sussex-18580826|title=St Leonards plaque marks Alan Turing's early years|date=25 June 2012|work=[[BBC News]]|access-date=3 July 2017|archive-url=https://web.archive.org/web/20171203074933/http://www.bbc.com/news/uk-england-sussex-18580826|archive-date=3 December 2017|url-status=live}}</ref>

Very early in life, Turing's parents purchased a house in [[Guildford]] in 1927, and Turing lived there during school holidays. The location is also marked with a blue plaque.<ref>{{cite news |url=http://www.guildford-dragon.com/2012/11/29/founder-of-computer-science-alan-turings-guildford-stargazing/ |title=Guildford Dragon NEWS |newspaper=The Guildford Dragon |date=29 November 2012 |access-date=31 October 2013 |archive-url=https://web.archive.org/web/20131019062927/http://www.guildford-dragon.com/2012/11/29/founder-of-computer-science-alan-turings-guildford-stargazing/ |archive-date=19 October 2013 |url-status=live }}</ref>

===School===
[[File:Alan Turing Aged 16.jpg|thumb|Turing at age 16, {{Circa|1928|1929}}]]
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".<ref>{{Cite book|last=Cawthorne|first=Nigel|url=https://www.worldcat.org/oclc/890938716|title=Alan Turing : the enigma man|date=2014|isbn=978-1-78404-535-7|location=London|publisher=Arcturus Publishing|pages=18|oclc=890938716|access-date=26 August 2024|archive-date=17 November 2021|archive-url=https://web.archive.org/web/20211117213134/https://www.worldcat.org/title/alan-turing-the-enigma-man/oclc/890938716|url-status=live}}</ref>

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]]).<ref>{{cite web|url=http://oldshirburnian.org.uk/wp-content/uploads/2016/04/TURING-Alan-Mathison.pdf|title=Alan Turing Archive – Sherborne School (ARCHON CODE: GB1949)|author=Alan Mathison|work=Sherborne School, Dorset|date=April 2016|access-date=5 February 2017|archive-url=https://web.archive.org/web/20161226093015/http://oldshirburnian.org.uk/wp-content/uploads/2016/04/TURING-Alan-Mathison.pdf|archive-date=26 December 2016|url-status=live}}</ref> In 1926, at the age of 13, he went on to [[Sherborne School]],<ref>{{Cite web|date=1 September 2016|title=Alan Turing OBE, PhD, FRS (1912–1954)|url=https://oldshirburnian.org.uk/alan-turing/|access-date=10 October 2020|website=The Old Shirburnian Society|language=en-GB|archive-date=4 November 2020|archive-url=https://web.archive.org/web/20201104133822/https://oldshirburnian.org.uk/alan-turing/|url-status=live}}</ref> 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 United Kingdom general strike|1926 General Strike]], in Britain, but Turing was so determined to attend that he rode his bicycle unaccompanied {{convert|60|mi|km}} from [[Southampton]] to Sherborne, stopping overnight at an inn.<ref name=metamagical>{{Cite book|title=Metamagical Themas: Questing for the Essence of Mind and Pattern |page=[https://books.google.com/books?id=o8jzWF7rD6oC&pg=PA484 484]|first=Douglas R. |last=Hofstadter |year=1985 |publisher=Basic Books |isbn=978-0-465-04566-2 |oclc=230812136}}</ref>

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".<ref>{{Harvnb|Hodges|1983|p=26}}</ref> 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.<ref>{{Harvnb|Hodges|1983|p=34}}</ref>

===Christopher Morcom===
At Sherborne, Turing formed a significant friendship with fellow pupil Christopher Collan Morcom (13 July 1911 – 13 February 1930),<ref>{{Cite web|url=https://oldshirburnian.org.uk/wp-content/uploads/2015/01/Obituary-for-Christopher-Morcom-The-Shirburnian-March-1930.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://oldshirburnian.org.uk/wp-content/uploads/2015/01/Obituary-for-Christopher-Morcom-The-Shirburnian-March-1930.pdf |archive-date=9 October 2022 |url-status=live|title=''The Shirburnian''}}</ref> who has been described as Turing's first love.<ref>{{cite book |title=The Alyson Almanac: A Treasury of Information for the Gay and Lesbian Community |date=1989 |publisher=Alyson Publications |isbn=978-0-932870-19-3 |page=192 |url=https://books.google.com/books?id=v-8gAQAAMAAJ |language=en |quote=After his first love, Christopher Morcom, died of tuberculosis ... |access-date=26 August 2024 |archive-date=7 November 2023 |archive-url=https://web.archive.org/web/20231107005431/https://books.google.com/books?id=v-8gAQAAMAAJ |url-status=live }}</ref><ref>{{cite book |last1=Hodges |first1=Andrew |title=Alan Turing: The Enigma |date=1992 |publisher=Vintage |isbn=978-0-09-911641-7 |page=35 |url=https://books.google.com/books?id=VWvPIWm75XIC |quote=This was first love, which Alan would himself come to regard as the first of many for others of his own sex. |access-date=26 August 2024 |archive-date=8 September 2023 |archive-url=https://web.archive.org/web/20230908030210/https://books.google.com/books?id=VWvPIWm75XIC |url-status=live }}</ref><ref>{{cite book |title=Tekhnema: Journal of Philosophy and Technology |date=1995 |publisher=American University of Paris |page=57 |url=https://books.google.com/books?id=wkvGAAAAIAAJ |quote=... Turing's first platonic love, Christopher Morcom ...}}</ref> Their relationship provided inspiration in Turing's future endeavours, but it was cut short by Morcom's death, in February 1930, from complications of [[bovine tuberculosis]], contracted after drinking infected cow's milk some years previously.<ref name=NYReviewBooks>{{cite web|author=Caryl, Christian|title=Poor Imitation of Alan Turing|url=http://www.nybooks.com/blogs/nyrblog/2014/dec/19/poor-imitation-alan-turing/|newspaper=[[New York Review of Books]]|date=19 December 2014|access-date=9 January 2015|archive-url=https://web.archive.org/web/20150107010418/http://www.nybooks.com/blogs/nyrblog/2014/dec/19/poor-imitation-alan-turing/|archive-date=7 January 2015|url-status=live}}</ref><ref>{{cite web |first=Rachel |last=Hassall |url=http://oldshirburnian.org.uk/wp-content/uploads/2014/03/The-Sherborne-Formula-Vivat-2012-2013-optimised.pdf |title=The Sherborne Formula: The Making of Alan Turing |archive-url=https://web.archive.org/web/20140415082353/http://oldshirburnian.org.uk/wp-content/uploads/2014/03/The-Sherborne-Formula-Vivat-2012-2013-optimised.pdf |archive-date=15 April 2014 |work=Vivat! |date=2012–2013}}</ref><ref name=teuscher>{{Cite book|editor-last=Teuscher |editor-first=Christof|editor-link=Christof Teuscher |title=Alan Turing: Life and Legacy of a Great Thinker |year=2004 |publisher=[[Springer Science+Business Media|Springer-Verlag]] |isbn=978-3-540-20020-8 |oclc=53434737 }}</ref>

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 (née Swan), Turing wrote:

{{blockquote|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&nbsp;... 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.<ref>{{Harvnb|Hodges|1983|p=61}}</ref>}}

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.<ref name=":1">{{cite book |title=Alan Turing: The Enigma |publisher=Princeton University Press |author-link=Andrew Hodges |last=Hodges |first=Andrew |page=[https://archive.org/details/alanturingenigma0000hodg/page/87 87] |year=2012 |isbn=978-0-691-15564-7 |url=https://archive.org/details/alanturingenigma0000hodg|url-access=registration }}</ref> A day before the third anniversary of Morcom's death (13 February 1933), he wrote to Mrs. Morcom:

{{blockquote|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.<ref>{{cite book |title=Alan Turing: The Enigma |publisher=Princeton University Press |author-link=Andrew Hodges |last=Hodges |first=Andrew |page=[https://archive.org/details/alanturingenigma0000hodg/page/90 90] |year=2012 |isbn=978-0-691-15564-7 |url=https://archive.org/details/alanturingenigma0000hodg|url-access=registration }}</ref>}}

Some have speculated that Morcom's death was the cause of Turing's [[atheism]] and [[materialism]].<ref>{{Cite news |first=Paul |last=Gray |url=http://www.time.com/time/magazine/article/0,9171,990624,00.html |title=Alan Turing |archive-url=https://web.archive.org/web/20110119181237/http://www.time.com/time/magazine/article/0,9171,990624,00.html |archive-date=19 January 2011 |work=Time Magazine's Most Important People of the Century |page=2}}</ref> 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:

{{blockquote|Personally, I believe that spirit is really eternally connected with matter but certainly not by the same kind of body&nbsp;... 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.<ref>{{Harvnb|Hodges|1983|pp=82–83}}</ref><ref>{{Cite web|url=https://oldshirburnian.org.uk/alan-turing-and-the-nature-of-spirit/|title=Alan Turing and the 'Nature of Spirit'|website=oldshirburnian.org.uk|date=15 August 2020|access-date=26 August 2024|archive-date=20 December 2017|archive-url=https://web.archive.org/web/20171220185715/http://oldshirburnian.org.uk/alan-turing-and-the-nature-of-spirit/|url-status=live}}</ref>}}

===University and work on computability===
[[File:Alan Turing az 1930-as években.jpg|thumb|Turing in the 1930s]]
After graduating from Sherborne, Turing applied for several Cambridge colleges scholarships, including [[Trinity College, Cambridge|Trinity]] and [[King's College, Cambridge|King's]], eventually earning an £80 per annum scholarship (equivalent to about £4,300 as of 2023) to study at the latter.<ref>{{cite book |last1=Hodges |first1=Andrew |title=Alan Turing: The Enigma |date=10 November 2014 |publisher=Princeton University Press |isbn=978-0-691-16472-4 |pages=74–5 |edition=2014}}</ref><ref>{{cite web | url=https://www.bankofengland.co.uk/monetary-policy/inflation/inflation-calculator | title=Inflation calculator | access-date=26 August 2024 | archive-date=5 October 2018 | archive-url=https://web.archive.org/web/20181005211045/https://www.bankofengland.co.uk/monetary-policy/inflation/inflation-calculator | url-status=live }}</ref> There, Turing studied the undergraduate course in Schedule B <ref>(Schedule B was a three-year scheme consisting of Parts I and II, of the [[Mathematical Tripos]], with extra courses at the end of the third year, as Part III only emerged as a separate degree in 1934</ref> from February 1931 to November 1934 at [[King's College, Cambridge]], where he was awarded first-class honours in mathematics. His dissertation, ''On the Gaussian error function'', written during his senior year and delivered in November 1934 (with a deadline date of 6 December) proved a version of the [[central limit theorem]]. It was finally accepted on 16 March 1935. By spring of that same year, Turing started his master's course ([[Part III of the Mathematical Tripos|Part III]])—which he completed in 1937—and, at the same time, he published his first paper, a one-page article called ''Equivalence of left and right almost periodicity'' (sent on 23 April), featured in the tenth volume of the ''[[London Mathematical Society|Journal of the London Mathematical Society]]''.<ref>{{cite web | url=https://turingarchive.kings.cam.ac.uk/publications-lectures-and-talks-amtb/amt-b-10 | title=AMT-B-10 &#124; the Turing Digital Archive }}</ref> Later that year, Turing was elected a [[Fellow]] of King's College on the strength of his dissertation<ref>{{Cite journal |last=Aldrich |first=John |date=December 2009 |title=England and Continental Probability in the Inter-War Years |journal=Electronic Journ@l for History of Probability and Statistics |volume=5 |number=2 |url=https://www.jehps.net/Decembre2009/Aldrich.pdf |pages=7–11}}</ref> where he served as a [[lecturer]].<ref name=lecturer>{{cite web|first=Alan|last=Turing|year=1939|url=https://archivesearch.lib.cam.ac.uk/repositories/7/archival_objects/272543|website=cam.ac.uk|title=Letter From Alan Turing to his mother, Sara Turing, 1939-01-23|quote=“My lectures are going off rather well. There are 14 people coming to them at present. No doubt the attendance will drop off as the term advances.”}}</ref> However, and, unknown to Turing, this version of the theorem he proved in his paper, had already been proven, in 1922, by [[Jarl Waldemar Lindeberg]]. Despite this, the committee found Turing's methods original and so regarded the work worthy of consideration for the fellowship. [[Abram Besicovitch]]'s report for the committee went so far as to say that if Turing's work had been published before Lindeberg's, it would have been "an important event in the mathematical literature of that year".<ref>{{cite book |last=Turing |first=Dermot |author-link=Dermot Turing |title=[[Prof: Alan Turing Decoded]] |year=2015 |publisher=[[The History Press]] |isbn=9781841656434 |page= 69}}</ref>{{sfn|Hodges|1983|p=113}}<ref>{{cite journal|title=Alan Turing and the Central Limit Theorem |first=S. L. |last=Zabell |pages=483–494 |journal=The American Mathematical Monthly |volume=102 |year=1995 |number=6 |doi=10.1080/00029890.1995.12004608}}</ref>

Between the springs of 1935 and 1936, at the same time as [[Alonzo Church]], Turing worked on the decidability of problems, starting from [[Gödel's incompleteness theorems]]. In mid-April 1936, Turing sent Max Newman the first draft typescript of his investigations. That same month, 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]]".<ref name=computable>{{Cite journal | last = Turing | first = A. M. | publication-date = 1937 | title = On Computable Numbers, with an Application to the Entscheidungsproblem: A correction | periodical = Proceedings of the London Mathematical Society | series = 2 | volume = 43 | pages = 544–46 | doi = 10.1112/plms/s2-43.6.544 | year = 1938 | issue = 1 }}</ref><ref>{{Harvnb|Turing|1937}}</ref> 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.<ref>{{cite book |url=https://books.google.com/books?id=MlsJuSj2OkEC&pg=PA211 |page=211 |title=Computability: Turing, Gödel, Church, and Beyond |author1=B. Jack Copeland |author2=Carl J. Posy |author3=Oron Shagrir |publisher=MIT Press |year=2013|isbn=978-0-262-01899-9 }}</ref> In this paper, Turing reformulated [[Kurt Gödel]]'s 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as [[Turing machine]]s. 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 [[Decision problem|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".<ref>{{cite book |page=15 |title=Mathematics and Computation |author=Avi Wigderson |publisher=Princeton University Press |year=2019|isbn=978-0-691-18913-0 }}</ref>

[[File:20130808 Kings College Front Court Fountain Crop 03.jpg|thumb|right|[[King's College, Cambridge]], where Turing was an undergraduate in 1931 and became a Fellow in 1935. The computer room is named after him.]]
Although [[Turing's proof]] was published shortly after Church's equivalent proof using his [[lambda calculus]],<ref>{{Harvnb|Church|1936}}</ref> Turing's approach is considerably more accessible and intuitive than Church's.<ref>{{cite web|last1=Grime|first1=James|title=What Did Turing Do for Us?|url=https://nrich.maths.org/8050|website=[[NRICH]]|publisher=[[University of Cambridge]]|access-date=28 February 2016|date=February 2012|archive-url=https://web.archive.org/web/20160304175703/http://nrich.maths.org/8050|archive-date=4 March 2016|url-status=live}}</ref> It also included a notion of a 'Universal Machine' (now known as a [[universal Turing machine]]), with the idea that such a machine could perform the tasks of any other computation machine (as indeed could Church's lambda calculus). According to the [[Church–Turing thesis]], Turing machines and the lambda calculus are capable of computing anything that is computable. [[John von Neumann]] acknowledged that the central concept of the modern computer was due to Turing's paper.<ref>"von Neumann&nbsp;... firmly emphasised to me, and to others I am sure, that the fundamental conception is owing to Turing—insofar as not anticipated by Babbage, Lovelace and others." Letter by [[Stanley Frankel]] to [[Brian Randell]], 1972, quoted in [[Jack Copeland]] (2004) ''The Essential Turing'', p.&nbsp;22.</ref> To this day, Turing machines are a central object of study in [[theory of computation]].<ref>{{Citation |last=De Mol |first=Liesbeth |title=Turing Machines |date=2021 |url=https://plato.stanford.edu/archives/win2021/entries/turing-machine/ |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |access-date=12 July 2023 |edition=Winter 2021 |publisher=Metaphysics Research Lab, Stanford University |archive-url= https://web.archive.org/web/20221016224306/https://plato.stanford.edu/archives/win2021/entries/turing-machine/ |archive-date= 16 October 2022 |url-status=live }}</ref>

From September 1936 to July 1938, Turing spent most of his time studying under Church at [[Princeton University]],<ref name="bowen19" /> in the second year as a [[Jane Eliza Procter Fellowship|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]].<ref>{{Harvnb|Hodges|1983|p=138}}</ref> In June 1938, he obtained his PhD from the [[Princeton University Department of Mathematics|Department of Mathematics]] at Princeton;<ref>{{Cite journal
| last1 = Turing | first1 = A.M.
| author-link = Alan Turing
| title = Systems of Logic Based on Ordinals
| doi = 10.1112/plms/s2-45.1.161
| journal = Proceedings of the London Mathematical Society
| pages = 161–228
| year = 1939
| volume=s2-45
| issue = 1
| hdl = 21.11116/0000-0001-91CE-3
| hdl-access = free
}}</ref> his dissertation, ''[[Systems of Logic Based on Ordinals]]'',<ref name="turingphd">{{cite thesis |degree=PhD |first=Alan|last=Turing |title=Systems of Logic Based on Ordinals|journal=Proceedings of the London Mathematical Society |publisher=Princeton University |year=1938 |volume=s2-45 |issue=1 |doi=10.1112/plms/s2-45.1.161|author-link=Alan Turing|id={{ProQuest|301792588}}|hdl=21.11116/0000-0001-91CE-3|hdl-access=free}}</ref><ref>{{cite web | last = Turing | first = A.M. | author-link = Alan Turing | title = Systems of Logic Based on Ordinals | year = 1938 | url = https://webspace.princeton.edu/users/jedwards/Turing%20Centennial%202012/Mudd%20Archive%20files/12285_AC100_Turing_1938.pdf | access-date = 4 February 2012 | archive-url = https://web.archive.org/web/20121023103503/https://webspace.princeton.edu/users/jedwards/Turing%20Centennial%202012/Mudd%20Archive%20files/12285_AC100_Turing_1938.pdf | archive-date = 23 October 2012 | url-status = dead }}</ref> introduced the concept of [[ordinal logic]] and the notion of [[Turing reduction|relative computing]], in which Turing machines are augmented with so-called [[oracle machine|oracles]], allowing the study of problems that cannot be solved by Turing machines. John von Neumann wanted to hire him as his [[Postdoctoral researcher|postdoctoral assistant]], but he went back to the United Kingdom.<ref>''John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More'', Norman MacRae, 1999, American Mathematical Society, Chapter 8</ref>