# Alan Turing in America

*“Beyond the way they speak, there is only one (no two!) features of American life which I find really tiresome. The impossibility of getting a bath in the ordinary sense and their ideas on room temperature”* — Alan Turing (1936)

Pioneering British computer scientist and mathematician Alan Turing first arrived in America on the 28th of September 1936. Born in 1912, Turing was twenty-four years old when he was invited attend graduate school at Princeton University, studying under Alonzo Church. A Master’s student of mathematics at the University of Cambridge, Turing was at the time working on the theory of computability and would in the coming months be publishing his now renowned paper “On Computable Numbers, with an Application to the *Entscheidungsproblem*” which re-proved his eventual Ph.D. supervisors’ result known as Church’s theorem.

Turing would travel to America twice, first in ’36 and next in ’42. His first visit was for graduate studies in mathematics, his second as a liaison for the UK’s team of cryptoanalysts at Bletchley Park. This essay aims to recount some of his experiences while there.

**Background (1912–1936)**

Alan Mathison Turing was born in London on June 23rd, 1912. The son of Julius Mathison and Ethel Sara Turing, he was the youngest of two children, the brother of the elder son John. Civil servants, his parents lived in India and Alan and his brother spent their early years living with another family near Hastings. He was educated first at Hazelhurst (1922–26), then Sherborne school (1926–31).

**At King’s College, Cambridge (1931–36)**

Following his graduation from secondary school at Sherborne, Turing in 1931 won a scholarship to the renowned King’s College, Cambridge where he in 1934 graduated with first-class honors in mathematics. The following year, in 1935, he was elected the youngest ever fellow of King’s College at the age of 22, on the strength of his undergraduate thesis proving the central limit theorem.

While still at Cambridge, Turing in the spring of 1935 attended a course by the mathematician Max Newman (1897–1984) entitled “Foundations of Mathematics”, which covered Hilbert’s *Entscheidungsproblem*(“decision problem”) of whether it is possible to provide a decision procedure that *“allows one to decide the validity of a sentence”* (Soare, 2013). Turing soon showed that no such algorithm can exist, but despite Newman’ encouragements, was hesitant to write his result up, waiting until the spring of 1936 to do so (Zitarelli, 2015). When he finally did, soon thereafter his future Ph.D. supervisor Alonzo Church published a paper showing the same result, now known as “Church’s theorem” that Peano arithmetic is undecidable.

Rather than defeat, Turing reportedly felt a zeal from the confirmation of his discovery. As the story goes, Newman wrote directly to Church the same day, seeking support to have Turing travel to America for graduate studies:

After a full year of work, Turing gave Newman a draft of his paper in April of 1936. "Max's first sight of Alan's masterpiece must have been a breathtaking experience, and from this day forth Alan became on of Max's principle protégés". [...] Max lobbied for the publication of "On Computable Numbers, with an Application to the Entscheidungsproblem" in the Proceedings of the London Mathematical Society, and arranged for Turing to go to Princeton to work with Alonzo Church. "This makes it all the more important that he should come into contact as soon as possible with the leading workers on this line, so that he should not develop into a confirmed solitary", Newman wrote to Church."- Excerpt,Turing's Cathedralby George Dyson (2012)

The leading “*workers on the line*” of the study of computation and computability in 1936 essentially consisted of a handful of mathematicians several of whom had settled in America in fear of a second World War. Most notably, as Dyson (2012) writes, was perhaps logician Alonzo Church (1903–1995) at Princeton University, who had earned his Ph.D. under Oswald Veblen (1880–1960) in the 1920s and would eventually assume the role as Turing’s Ph.D. thesis advisor. Also worth mentioning is logician Kurt Gödel (1906–1978) who in 1931 had proved the breathtaking incompleteness theorems, ending Hilbert’s program to establish a formal foundation for mathematics. By 1936 Gödel was spending extended periods of time as a Visiting Fellow at the Institute for Advanced Study (IAS) in Princeton, unsettled by the political events taking place in his native Austria. Also there at that time, on the IAS’ permanent faculty, was mathematician John von Neumann (1903–1957). Ten years prior to Turing’s arrival in Princeton, von Neumann had studied Hilbert’s theory of consistency with German mathematician Hermann Weyl (1885–1955), before traveling to Göttingen to work with the great man himself.

**At Princeton University (1936–38)**

“Americans can be the most insufferable and insensitive creatures you could wish”

Turing arrived in Princeton on the 29th of September 1936 after a five day journey by sea onboard the Star Liner *Berengaria*. While onboard, he wrote his mother Sara a letter complaining about his impressions of the Americans on board:

Excerpt, letter from Alan to Sara Turing (28th of September, 1936)

It strikes me that Americans can be the most insufferable and insensitive creatures you could wish. One of them has just been talking to me and telling me all the worst aspects of America with evident Pride.

Showing his upper-middle-class British prejudices, he however adds that *“they may not all be like that.” *(Hodges, 2014). His impression did not warm as the ship arrived in the New World the following morning:

We were practically in New York at 11.00 a.m. on Tuesday but what with going through quarantine and passing immigration officers we were not off the boat until 5.30 p.m. Passing the immigration officers involved waiting in a queue for over two hours with screaming children around me. The, after getting through the customs I had to go through the ceremony of initiation to the U.S.A., consisting of being swindled by a taxi-driver. I considered his charge perfectly preposterous, but as I had already been charged more than double English prices for sending my luggage, I thought it was possibly right.

On October 3rd, the page proofs for his new paper arrived. The paper *On Computable Numbers, with an Application to the Entscheidungsproblem*was eventually published on the 12th of November 1936 in Proceedings of the London Mathematical Society. Despite having been published after Church’s paper, Turing’s approach to the problem would be the solution that has endured, likely due to Church’s solution’s reliance on his so-called λ**-**calculus, which Gödel reportedly rejected outright as *“thoroughly unsatisfactory”* (Soare, 2013), writing (Gödel, 1995):

Steps taken in a calculus must be of a restricted character and they are [here] assumed without argument to be recursive. [...] This is precisely where we encounter the major stumbling block for Church's analysis"

Unlike Church’s, Turing’s definition did not rely on the recursive functions of Gödel (1931) or Herbrand-Gödel (1934). Rather, Turing instead narrated an analysis of how humans would go about carrying out a calculation, showing step by step that the same procedure could also be conducted by a machine. In terms of the rigor of this analysis, Gandy (1988) later wrote *“Turing’s analysis does much more than provide an argument, it proves a theorem*”, without *“making no references whatsoever to calculating machines”*. Instead, *“Turing machines appear as a result, a codification, of his analysis of calculations by humans” *(Soare, 2013). Turing’s definition hence aligned much more closely with Gödel’s view of computation (Soare, 2013):

"But Turing has shown more. He has shown that the computable functions defined in this way are exactly those for which you can construct a machine with finite number of parts which will do the following thing. If you write down any number n₁, n₂, ..., nᵣ on a slip of paper and put the slip into the machine and turn the crank then after a finite number of turns the machine will stop and the value of the function for the argument n₁, n₂, ..., nᵣ will be printed on the paper."

Alan’s first report home to his mother Sara occurred three days after his proof pages arrived. As Hodges (2014) writes, his report home *“betrayed no lack of self-confidence*”:

Letter from Alan to Sara Turing (October 6th 1936)

The mathematics department here comes fully up to expectations. There is a great number of the most distinguished mathematicians here. John von Neumann, Weyl, Courant, Hardy, Einstein, Lefschetz, as well as hosts of smaller fry. Unfortunately there are not nearly as many logic people here as last year. Church is here of course, but Gödel, Kleene, Rosser and Bernays who were here last year have left. I don't think I mind very much missing any of these except Gödel.

Despite their mutual admiration, Turing and Gödel’s paths never met (Welch, 2010).

Following his first year of study at Princeton, Turing was by mid-1937 ready to undertake his Ph.D. thesis. Although initially intending to stay for a year (Copeland, 2004), the offer of a Procter Visiting Fellowship in 1937 persuaded him to stay a second year. The Dean of the Graduate School at Princeton was sufficiently impressed by Turing that he wrote the Procter Foundation directly on his behalf:

Mr. Alan Mathison Turing informs me that he is making application for the Procter Visiting Fellowship to Princeton for next year. No doubt he will have members of our staff who have been in close touch with him write to you and give you an estimate of the quality of his work. I have seen him a number of times. From this experience and what I have learned from those in residence in the Graduate College I think that his personal relations here have been highly satisfactory so that, if after considering the other applications you decide to nominate him, we will be very glad to receive him. However, I do not wish you to interpret this letter as in anyway way suggesting that he be appointed in preference to any other candidate. Sincerely yours, Luther P. Eisenhardt Dean of the Graduate SchoolPrinceton University

The letter of recommendation helped, and alongside a glowing letter from the Vice-Chancellor of his alma mater Cambridge, Turing was awarded the fellowship, which would cover his costs of free tuition, a suite and meals in the graduate college, as well as an additional $1,300.

John von Neumann reportedly supported Turing’s application for the Procter fellowship as well, gossiping to Oswald Veblen in a letter in the summer of 1935 that he is “*quite promising*”:

Letter from John von Neumann to Oswald Veblen (July 6th 1935)

Dear Oswald,

[...]

By the way, there seems to be quite a traffic-jam on the road to Princeton. There are 4 or more advanced students or PhD.'s who will come next year to Cambridge: A Commonwealth fellow M. Price, who is very able, he worked in quantum theory until now, but he wants to change to group theory; a National Research Council fellow, whose name is, I think, Lewinsohn, who is a pupil of Norbert Wiener (!) and spent his first year in Cambridge (Hardy thinks that he is very good), Touring, whom you mentioned and who seems to be strongly supported by the Cambridge mathematicians, for the Proctor fellowship (I think that he is quite promising); and one or two more, whose names I forgot.

[...]

Yours truly, John

Indeed, von Neumann appears to have been rather enchanted by the young Turing, and his work in the nascent theory of computing and computability, as recounted by pioneering computer scientist Julian Bigelow (1913–2003):

After having been here for a month, I was talking to von Neumann about various kinds of inductive processes and evolutionary processes, and just as an aside he said, "Of course that's what Turing was talking about." And I said, "Who's Turing?". And he said, "Go look up Proceedings of the London Mathematical Society, 1937". —Julian Bigelow

Turing likely first met von Neumann during the latter’s sabbatical visit to Cambridge in 1935, in which von Neumann lectured on group theory. A year before, von Neumann had published a paper entitled *Almost periodic functions in a group*whichwould serve as the inspiration for Turing’s first published paper *Equivalence of Left and Right Almost Periodicity* (1935) published while von Neumann was still in Cambridge. In the paper, Turing showed that the concepts of right and left periodicity are equivalent.

Turing obtained his Ph.D. from the Department of Mathematics at Princeton in 1938 with the dissertation *Systems of Logic Based on Ordinals*in which he introduced ordinal logic and the notion of relative computing — augmenting his previously devised Turing machines with so-called oracle machines, allowing the study of problems that lay beyond the capability of Turing machines. Although inquired about by von Neumann for a position as a postdoctoral research assistant following his Ph.D., Turing declined and instead travelled back to England.

**In Washington D.C. (1942–43)**

Turing again travelled to America in 1942, arriving by boat in New York on the Queen Elizabeth on November 12th. Reportedly, he was detained at Ellis Island by the Immigration Authorities who were

“Very snooty about me carrying no orders and no evidence to connect me with the Foreign Office”

By that point, Turing was committed to the British war effort, working for the secret British code-breaking team at Bletchley Park. He travelled to the headquarters of the US Secret Service (now the CIA) in Washington D.C. on a two-fold and rather ‘cloak-and-dagger’ mission:

Officially, Turing was meant to disclose everything he and his team at Bletchley Park knew about the workings of Enigma. In reality, he was under strict instructions from MI6 to act as its official liar and keep the Americans in the dark as much as possible.- Excerpt,The Imitation Game: How Alan Turing played dumb to fool US intelligenceby David Cox (2014) inThe Guardian

Remarkably, many of the details of Turing’s involvement with the MI6 are known because of Ian Fleming’s diaries (author of the James Bond novels) who at that point was a lieutenant commander in Britain’s naval intelligence division:

“Most of the time he’s writing about how annoyed he is with Turing. Fleming was regularly proposing plans and Turing was apparently in a position to be able to reject them. Fleming makes it clear he doesn’t like the look of him. There’s a memorable extract where he writes that Turing came in like an undertaker and shot down whatever he was saying.“ —Graham Moore (2014)

MI6 was wary of letting Britain’s American ally in on its progress in cracking the infamous German enigma code, fearing that the information would leak. Turing’s role was hence to convince his American counterparts in cryptoanalysis that Britain was struggling to match the US’ expertise, as well as document their progress in building encryption devices.

“He absolutely hated it because he had to play dumb while being grilled by these mathematicians” —Graham Moore (2014)

Turing’s own accounts of the meetings are reportedly filled with disdain. *“I am persuaded that one cannot very well trust these people where a matter of judgement in cryptography is concerned”*, ironically expressing a perceived over-reliance on technology:

“It astonished me to find that they make these elaborate calculations before they had really grasped the main principles.”

Turing would however also find inspiration during this final trip to America, in particularly from his meetings with Claude Shannon (1916–2001), the so-called “father of information theory”. The two reportedly enjoyed lengthy conversations about the idea of building an electronic brain (Cox, 2014), meeting daily over tea in the Bell Labs cafeteria (Soni & Goodman, 2017):

Turing and I had an awful lot in common, and we would talk about that kind of question [of building computers that will think and what you can do with computers and all that]. He had already written his famous paper about Turing Machines, so called, as they call them now, Turing Machines. They didn't call them that then. And we spent much time discussing the concepts of what's in the human brain. How the brain is built, how it works and what can be done with machines and whether you can do anything with machines that you can do with the human brain and so on. And that kind of thing. And I talked to him several times about my notions on Information Theory, I know, and he was interested in those.- Excerpt, A Mind at Play by Jimmy Soni & Rob Goodman (2017)

Their discussions excited Turing enough to buy a book on electronic components to read on his sail back to Britain. Shannon and John McCarthy (1927–2011) later requested that Turing contribute to their forthcoming book on Automata Studies in the Annals of Mathematics Studies 1953. Turing politely declined.

“I think Turing had a great mind, a very great mind” — Claude Shannon

By the end of the war, it appears that Turing’s view of America and Americans had sufficiently soured to where he refused to return ever again (Daley, 2017). In 2017, while cleaning out a filing cabinet in a university storage room, Professor Jim Miles discovered 148 letters written by Turing in the period 1949–1954 when he was the deputy director of the University of Manchester’s computing laboratory. In one of the letters, a short reply to an invitation to speak at a conference in the United States in 1953, Turing summarizes his view in a rather harsh yet eloquent tone, stating:

“I would not like the journey, and I detest America.”

The archive of letters is available here.

Thank you for reading and subscribing to the Privatdozent newsletter.

Have a great week!

Sincerely,

Jørgen.

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