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 Cathedral by 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 (1903–1955) 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.

A photo of Alan Turing believed to be taken during his time at Princeton University. Photo: Convergence Portrait Gallery

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).