The history of mathematics is a field generally characterized by careful scholarship and close attention to evidence. Sometimes, though, a story with little or no evidence in its favor takes root and propagates. We discuss a case in point. It is widely maintained in the literature that Kurt Gödel’s 1931 work on incompleteness was the foundation for Alan Turing’s work on computability and undecidability, set out in his famous 1936 paper “On Computable Numbers” (OCN) [45]. (See Figures 1 and 2 for photographs of our two lead characters.) In fact, there is very little evidence for this contention. The aim of our discussion is to arouse a skeptical attitude toward this often-repeated claim, and we hope in future to see the claim treated with greater caution in the literature.

Figure 1.
figure 1

Alan Turing, 1912–1954. (Call mark AMT/K/7/11.) (Reproduced by permission of King’s College Library, Cambridge.)

Figure 2.
figure 2

Kurt Gödel, 1906–1978. (Call mark 11/110215, Kurt Gödel papers, Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton (on deposit at Princeton University).) (Photo by Marcel Natkin; reproduced by permission of Stephane and Laurent Natkin.)

OCN is widely regarded as a founding document—perhaps the founding document—of computer science. This “seminal paper,” says Turing Award laureate Vinton Cerf, introduced “key concepts … that continue to play a central role in our industry today.”Footnote 1 Furthermore, the undecidability results that Turing proved in OCN are, along with Gödel’s 1931 incompleteness theorems, major milestones in twentieth-century mathematical logic.

Gödel repeatedly praised OCN and made fundamental use of Turing’s ideas.Footnote 2 He emphasized that “due to A. M. Turing’s work, a precise and unquestionably adequate definition of the general concept of formal system can now be given,” with the consequence, he said, that incompleteness can “be proved rigorously for every consistent formal system containing a certain amount of finitary number theory” [21, p. 71]. But what about influences in the opposite direction? Did Turing, in OCN, make use of concepts or techniques from Gödel’s 1931 incompleteness paper [19]?

At first blush, one might suspect not, since famously, Turing was a lone worker par excellence. He once told his Bletchley Park colleague Patrick Mahon that he took on Naval Enigma in 1939 “because no one else was doing anything about it and I could have it to myself” [37, p. 279]. Turing’s friend and colleague Peter Hilton described him as “so often going back to first principles for his inspiration” [30, p. 23]. Making a study of what others had written about a topic was not at all Turing’s habit. Yet there is a widespread assumption that Turing did indeed make extensive use of Gödel’s prior work. Judson Webb, for example, said in his influential book on mechanism that Turing (and Church) “only succeed because of having the basic discoveries of Gödel to guide them” [49, p. 26].

Prima facie, this assumption is a very reasonable one. After all, by 1935, Gödel’s incompleteness paper was well known to mathematical logicians—including Turing’s mentor, Max Newman—and Turing cited and discussed the paper in OCN. Moreover, Turing was in effect picking up where Gödel had left off. David Hilbert, Gödel’s target, had emphasized the need for consistency, completeness, and decidability, and Gödel’s paper dealt explicitly with the first two of these, but not the third. (As is well known, Gödel showed that the system he called PM, for Principia Mathematica, was incomplete and could not prove its own consistency.) Turing took up the third issue, the decision problem. His treatment—like Gödel’s treatment of the first two issues—made central use of the technique of diagonalization and the idea of denoting syntactical items by means of numbers. Furthermore, Turing’s sequel to OCN [46], completed in the spring of 1938, took Gödel’s 1931 work as its point of departure. The first line of that paper refers to Gödel’s “well known theorem.” Turing explained, in his opening paragraph, that he was going to explore the idea that from “a system L of logic a more complete system L′ may be obtained”—by Gödelian methods—and then moved on immediately to a discussion of “Gödel representations” [46, pp. 161–162].

Our aim, however, is to inject a note of caution into the literature, since despite the prima facie reasonableness of this assumption, there is in fact little evidence in its favor. Indeed, Turing may have encountered Gödel’s incompleteness paper only after formulating the central ideas and proofs of OCN.

First, let’s take a closer look at what can be called the Standard Story, according to which central ideas in OCN were imported from Gödel’s work. Like all good stories, the Standard Story comes in a number of different versions. Jürgen Schmidhuber, for example, claims that in OCN, Turing “merely reformulated Gödel’s work in an elegant way” [43, p. 1638]. Others present less severe versions of the story, whereby one or another—or several—of the central techniques in OCN are said to have derived from Gödel’s 1931 paper.

The Standard Story

The idea that Turing’s 1936 work was inspired and informed by Gödel’s is common among authors with a detailed knowledge of the relevant technical developments, including computer scientists and historians of computing. The Standard Story is simply taken for granted in two monographs devoted to OCN, Charles Petzold’s 2008 book [40] and Christopher Bernhardt’s 2016 book [4]. Petzold writes, “Turing was undoubtedly inspired by the approach Gödel took in his Incompleteness Theorem in converting every mathematical expression into a unique number” [40, p. 138]. Bernhardt writes, “The idea of converting algorithms into binary strings might seem fairly natural to us today, but in Turing’s day it was not. Turing got the idea of encoding his machines from the work of Kurt Gödel who, for his work on the incompleteness theorems, had encoded statements in mathematics into strings of numbers … Turing borrowed the concept and applied it to algorithms” [4, p. 88]. Biographer Andrew Hodges writes, “In 1936, Turing did follow Gödel’s 1931 revolution in logic. His first citation was to Gödel, and he described his mathematical argument as similar to Gödel’s” [33, p. 1639].

The Standard Story is also endorsed by historians of computing. Liesbeth De Mol asserts—without providing any evidence—that “Turing relied on Gödel’s insight that these kind of problems can be encoded as a problem about numbers” [16]. Edgar Daylight states that “Turing used Gödel numbering to uniquely identify each of his machines with integers, a key element in the construction of his universal machine” [15, p. 16]. To substantiate this claim, Daylight rests content with a reference to the lines by Petzold that were quoted above; but Petzold offers no evidence for his claim that Turing’s approach was “inspired” by Gödel’s. Furthermore, Daylight’s claim is in fact false. Recursion theory pioneer Stephen Kleene pointed out that “Turing uses a totally different method of numbering linguistic objects than Gödel—one perhaps more natural to a person who is steeped in machines” [35, p. 492]. Gödel numbering involves coding via exponentiation of primes [19, pp. 178–179], whereas Turing encoded a machine’s instruction table by means of a simple substitution code, employing a key assigning a numeral to each component of his “instruction tables” (e.g., the letter A is coded by 1, the semicolon by 7, and so forth).

Other parts of the Standard Story concern Gödel’s influence on further central concepts in OCN. Mark Priestley claims that “Turing applied and generalized existing work in logic, particularly that of Gödel,” and he gives several alleged examples in which Gödel’s work was “transcribed and used,” including the use of substitution and recursion in what Turing called skeleton tables and m-functions, as well as “Gödel’s technique of arithmetization … in the definition of the universal machine” [41, pp. 87, 96, 98]. Teresa Numerico, discussing the origins of the stored-program concept, says the idea that “instructions can be stored in memory coded as numbers,” essential to Turing’s universal machine, “came directly from the arithmetization technique used in logic by Gödel” [39, p. 183]. William Aspray says that “without question,” Gödel’s incompleteness theorem “led” Turing to the idea of “encoding instructions as numbers” [2, p. 192]. Again no evidence is presented. Diagonalization, central to Turing’s proof of undecidability, is another illustration. Bernhardt says Turing “borrowed another argument from Gödel who, in turn, had borrowed it from Cantor” [4, p. 12]. Yet when discussing the diagonal argument in OCN [45, pp. 246–248]. Turing in fact referred not to Gödel but to Ernest Hobson’s 1921 book [31, The Theory of Functions of a Real Variable.Footnote 3

As we shall explain, we do not see any evidence for the Standard Story, either in the historical record, in OCN itself, or in Turing’s other writings from the time. We believe the Standard Story should therefore be regarded with skepticism. Worse yet, the historical record presents some serious challenges to the Standard Story. We explain these challenges in what follows. But first, we briefly discuss Turing’s library borrowing records. These have only recently come to light, and they contain some relevant information.

Turing’s Library Records

In March 1935, Turing wrote to his mother Sara (Figure 3) that he had “just joined the Cambridge Philosophical Society” [52]. Always hopeful of unearthing unknown documents, we recently contacted the Society—and struck gold. Detailed records of Turing’s borrowings from the Society’s library emerged into the light of day.Footnote 4 These provide fresh information about the timeframes of Turing’s various research projects and are a fascinating window on his mathematical interests, from the mid-1930s up to his final borrowing from the library, in July 1950.

Figure 3.
figure 3

Sara Turing, 1881–1976. (Call mark AMT/K/7/24.) (Courtesy of King’s College Library, Cambridge.)

The Philosophical Society, founded in 1819, embraced not only natural philosophy narrowly construed (i.e., physics) but also “Chemistry, Mineralogy, Geology, Botany, Zoology, and other branches of Natural Science”Footnote 5 and had many notable members over the years (including Charles Babbage, Niels Bohr, George Boole, Charles Darwin, Paul Dirac, Arthur Eddington, Godfrey Harold Hardy, and Ernest Rutherford). The Society elected Turing a Fellow at its meeting of February 25, 1935 (the great Rutherford delivered a lecture on radioactivity at the same meeting).Footnote 6 Turing said he joined “principally because they have an excellent Mathematical & Scientific library” [52]. Situated along Pembroke Street, the library was close to King’s Parade, and a few minutes’ walk would take Turing from his rooms in King’s to one of the best collections of mathematical journals in Europe. The Society’s library held “the widest range of journals in Cambridge,” explains historian Susannah Gibson [18, p. 62]. By the 1920s, the library was taking approximately 300 scientific journals not found in the University Library [24, p. 68]. “[I]t was to the Society that people went when they wanted to read the most up-to-date research … The colleges and University could not compete [24, p. 68].

Turing was able to borrow from the Society’s library from November 1934, probably on the strength of his fellowship proposal (Figure 4).Footnote 7 His first recorded borrowing, on November 29, 1934—a month or so before the start of the Foundations of Mathematics lectures by Newman that are discussed below—was the July 1934 issue of the Transactions of the American Mathematical Society. This contained John von Neumann’s article “Almost Periodic Functions in a Group. I” [48], and Turing borrowed the 1934 Transactions again in January 1935.Footnote 8 He saw a way of improving on von Neumann’s argument, and soon, toward the end of April, the manuscript of his first publication was ready for submission. It comprised a short proof that von Neumann’s twin concepts of left almost periodicity and right almost periodicity, which von Neumann had treated as being independent, were equivalent [44].

Figure 4.
figure 4

Turing’s fellowship proposal certificate, Cambridge Philosophical Society. (Call mark CPS/6/2/3/22.) (Reproduced by permission of the Cambridge Philosophical Society.)

The library records show that on February 4, 1935, Turing borrowed the December 1930 volume of Mathematische Annalen, not returning it until March (having made one trip to the library to renew it, at the end of February). This volume contains David Hilbert’s “Probleme der Grundlegung der Mathematik” (Problems Concerning the Foundation of Mathematics), the text of an address he had delivered in Bologna in 1928 [27]. Here Hilbert famously stated that “in mathematics there is no ignorabimus, rather we are always able to answer meaningful questions,” and he described his “new foundation for mathematics, which can appropriately be called proof theory” [27, pp. 9, 3]. Hilbert mentioned the Entscheidungsproblem (decision problem), reporting [27, p. 8) that the decidability of the monadic predicate calculus—a special case of the Entscheidungsproblem—had been established in 1915 by Leopold Löwenheim [36] and “later in final form” by Hilbert’s protégé Heinrich Behmann [3].Footnote 9 Turing, of course, went on to show that “the Hilbert Entscheidungsproblem can have no solution” in the case of the full predicate calculus and all mathematical systems containing it [45, p. 259].

On April 18, 1936, Turing borrowed the December 1935 volume of Monatshefte für Mathematik und Physik, returning to the library to renew it on May 5 and then again on May 22. Strikingly, this volume contains a paper by Rudolf Carnap titled “Ein Gültigkeitskriterium für die Sätze der klassischen Mathematik” (A Criterion of Validity for the Propositions of Classical Mathematics). Carnap wrote:

One can pose the problem of finding a definite criterion of validity, i.e., one whose satisfaction or non-satisfaction in any individual case could be decided in a finite number of steps by a fixed procedure … Were such a criterion found, we would have a decision procedure [Entscheidungsverfahren] for mathematics; we could then—so to speak—calculate the truth or falsity of every given proposition [6, p. 163, translated by Copeland and Sommaruga].

Assuming that Turing read these words, they must have held considerable interest. For Carnap was anticipating Turing’s hard-won, and at that stage unpublished, negative result—and, worse still, attributing it to Gödel. “Based on Gödel’s latest results,” Carnap said, “it appears hopeless to search for a definite criterion of validity for the entire system of mathematics,” adding, “However, the question of solving this so-called Entscheidungsproblem for certain classes of propositions remains important and fertile” [6, pp. 163–164].

On the day he first borrowed the December 1935 Monatshefte (April 18, 1936), Turing also borrowed the 1931 volume of the journal, volume 38. This contained Gödel’s incompleteness paper, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I.” Figure 5 shows the library borrowing record with Turing’s signature.Footnote 10 No previous borrowings of this volume are recorded against Turing’s name. We will evaluate the significance of this information later in the discussion. Might Turing not have read Gödel’s incompleteness paper until mid-April 1936? But first, let’s examine Turing’s earliest known presentation of his undecidability results.

Figure 5.
figure 5

Turing borrowed the volume of Monatshefte für Mathematik und Physik containing Gödel’s 1931 paper on April 18, 1936. He renewed it on May 5 and 22, 1936. (Call mark CPS 9/3/1/6.) (Reproduced by permission of the Cambridge Philosophical Society.)

Turing’s Précis of “On Computable Numbers”

Turing’s précis is an important source of evidence concerning the development of OCN. Just over a page and a half in length, it is Turing’s earliest surviving account of the core ideas and arguments of OCN. Written in French, it is little known; advocates of the Standard Story never mention it. Small fragments have been translated in [11] and [12], but a full translation has not so far been published.

The typescript (Figure 6) survived among Sara Turing’s papers and is marked on the first page “Copy of first rough draft of précis of ‘Computable Numbers’ made for ‘Comptes Rendues’” [53]. Turing intended to publish it in Comptes rendus hebdomadaires des séances de l’Académie des Sciences. Turing and Sara evidently worked together on the draft; Sara wrote, “we embarked together on a précis in French of ‘Computable Numbers’ ” [47, p. 49]. It seems the précis was never published. Turing wrote to Sara on May 29, 1936, saying:

The situation with regard to the note for Comptes Rendus was not so good. It appears that the man who I wrote to, and whom I asked to communicate the paper for me had gone to China, and moreover the letter seems to have been lost in the post … [55].

Figure 6.
figure 6

Sara Turing’s typescript of the précis. (Call mark AMT/K/4/1.) (Reproduced by permission of King’s College Library, Cambridge.)

Sara later summed matters up: “the war supervened and Alan, as far as I know, heard no more” [47, p. 49].

The typescript is undated, but its approximate date is clear from a letter Turing wrote to Sara on May 4. He told her he had had the précis “vetted by a French expert” and had “sent it off” [54]. Probably the draft was written in the Cambridge Easter vacation (March 26–April 15) while Turing was staying at the family home in Guildford [54].

In the same letter (May 4), Turing mentioned having sought Max Newman’s (Figure 7) opinion on the précis, saying that Newman “examined my note for C.R. and approved it after some alterations.” Newman, a Fellow of St John’s, had lectured the previous year on the Foundations of Mathematics, in the Cambridge Lent term of 1935. He presented Hilbert’s Entscheidungsproblem to his audience, and said 40 years later (in a tape-recorded interview) that Turing’s journey leading to OCN had “started because he attended a lecture of mine.”Footnote 11

Figure 7.
figure 7

Max Newman, 1897–1984. (Reproduced by permission of Edward Newman.)

Since the précis bears importantly on our topic, we translate it in full.Footnote 12

One can call the numbers whose decimals can be written down by a machine “computable.” Such a machine has a tape, in some ways analogous to paper, running through it. The tape is divided into sections called “squares.” Each square can bear a symbol, but this is not necessary. The squares that bear no symbol are called “blank squares.” The machine is capable of several m-configurations, q1.........qn; i.e. levers, wheels, et cetera can be arranged in several ways, called “m-configurations.” At any moment only one square appears in the machine. This square is called “the viewed square,” the symbolFootnote 13 on it is called “the viewed symbol.” “The viewed symbol” and the m-configuration together are called simply “the configuration.” The configuration determines the next motion of the machine which can go to the left or to the right, or write a new symbol on “the viewed square,” if it is empty, or erase “the viewed symbol.” Then it can change the m-configuration.

The symbols written by the machine include the figures of the number that it is computing and other symbols. The machine must never erase a figure.

A veritable “computing machine” must write as many figures as one wants. Thus one calls a machine M “méchante” [literally “nasty” or “naughty”] if there is a number N such that M never writes N figures. A sequence of figures computed by a “non-méchante” machine is called a “computable sequence.” A number whose decimal expression is a computable sequence is called a “computable number.”

We show that the computable numbers comprise a quite broad yet enumerableFootnote 14 class. It seems, on the contrary, that the application of Cantor’s diagonal process would demonstrate that this class cannot be enumerable. But the reasoning is fallacious. The correct application of the diagonal process shows there is no general process for deciding whether a machine is “méchante” or not.Footnote 15In other words, there is no machine which supplied with a description of any machine M decides whether M is “méchante.”

This theorem has many applications. This comes about as follows. It can be shown that there is no general process for deciding whether a machine M never writes the symbol 0. If “Machine M sometimes writes the symbol 0” is expressed in Hilbert’s calculus, a formula F(M) is obtained. If one had a solution of the Entscheidungsproblem one would have a general process for deciding whether F(M) is provable. That is to say a general process for deciding whether M ever writes 0. It has been shown that this is not possible. Therefore the Entscheidungsproblem has no solution.

There is much of interest in the précis. First, there are small but intriguing variations in terminology between the précis and OCN. The delightfully mechanical “levers” and “wheels” of the précis do not occur in OCN, whereas the perfectly machine-like “scanned square” and “scanned symbol” of OCN do not occur in the précis. The awkward terms “circular machine” and “circle-free machine” of OCN had also apparently not yet been thought of, and “méchante” is used in the précis instead of “circular.” It is difficult to guess what English term the playful “méchante” translated, but perhaps “badly behaved” or even “wretched.”

Second, it is clear from the précis that the main ideas and fundamental structure of OCN were well established at the time of writing, including the definition of a computing machine and computable number, the diagonal argument, and the unsolvability of the Entscheidungsproblem. Of course, much of importance in OCN is necessarily omitted from the brief précis—including the reasons, amply spelled out in the published paper, for thinking that (what is now called) the Church–Turing thesis is true, the thesis that “the ‘computable’ numbers include all numbers which would naturally be regarded as computable” [45, p. 249].Footnote 16 Turing’s insouciant phrase “in other words” in the penultimate paragraph of the précis conceals some of the most important argumentation of OCN.

Third, the existence of the universal machine is not noted in the précis. This is perhaps unsurprising, since despite Turing’s detailed exposition of the universal machine in OCN, the machine figures only in a supplementary argument (in fact, one of the arguments hidden beneath that glib “in other words”).

Fourth, and most importantly, Turing does not so much as mention Gödel. The précis contains no indication whatsoever that Turing was familiar with Gödel’s work. We will say more about this later.

We next examine some important remarks by Church’s colleague Stephen Kleene, remarks that those who repeat the Standard Story seem to overlook.

Kleene’s Remarks

According to the Standard Story, Turing’s work was founded on, and was a development of, Gödel’s: Turing, it is said, “recast” Gödel’s findings “in the guise of the Halting Problem” [14, p. 133]. Kleene’s view of the matter was very different, however. Noting that “One sometimes encounters statements asserting that Gödel’s work laid the foundation for Church’s and Turing’s results,” Kleene went on: “It seems to me that the truth is that Church’s approach through λ-definability and Turing’s through his machine concept had quite independent roots (motivations), and would have led them to their main results even if Gödel’s paper [19] had not already appeared” [35, p. 491]. Then he said:

Whether or not one judges that Church would have proceeded from his thesis to these [undecidability] results without his having been exposed to Gödel numbering, it seems clear that Turing in [“On Computable Numbers”] had his own train of thought, quite unalloyed by any input from Gödel. One is impressed by this in reading Turing [“On Computable Numbers”] in detail [35, pp. 491–492].

These remarks—from someone who worked closely with Church and played a leading part in the development of computability theory in the 1930s—are not to be taken lightly. Moreover, Kleene’s claim that Turing’s thought was “unalloyed by any input from Gödel” certainly fits with what Turing said, and didn’t say, in his précis. At the time he wrote this summary of his ideas, Gödel’s work—if he even knew about it—did not merit a mention. In OCN, on the other hand, he gave a short, careful statement of the difference between his results and Gödel’s. That his findings were significantly different from Gödel’s could easily have been stated in the précis as well, for example using much the same words as in OCN: “It should perhaps be remarked that what I shall prove is quite different from the well-known results of Gödel” [45, p. 259]. But he did not do so.

We will take a close look at the text of OCN below. But first we introduce another relevant document, a letter by Cambridge don Richard Braithwaite (Figure 8) describing Turing’s knowledge of Gödel’s work at the time in question.

Figure 8.
figure 8

Richard Braithwaite, 1900–1990. (Call mark KCAS/39/4/1/78.) (Reproduced by permission of King’s College Library, Cambridge.)

Braithwaite on “Turing’s Complete Ignorance of Gödel’s Work”

According to the Standard Story, Turing learned about Gödel’s work on incompleteness during Max Newman’s 1935 lectures on the Foundations of Mathematics. Newman delivered the lectures on Tuesdays, Thursdays, and Saturdays during Lent term.Footnote 17 Andrew Hodges says that “Newman’s lectures finished with the proof of Gödel’s theorem, and thus brought Alan up to the frontiers of knowledge” [32, p. 93]. Ivor Grattan-Guinness summarizes: “Turing, newly graduated from the tripos, sat in on [Newman’s] course in 1935 and learnt about decision problems and Gödel numbering from one of the few Britons who was familiar with them” [23, p. 439]. However, there is disagreement in the literature about this tidy picture of the transmission of ideas.

Margaret Boden raised a difficulty for the idea that Turing’s knowledge of Gödel’s incompleteness theorems stemmed from Newman’s 1935 lectures, and indeed for the Standard Story in toto:

One might expect that Turing’s first thoughts about computability would have been informed by Gödel’s theorem. But they weren’t. It’s not clear that he even knew about it when he wrote his paper in 1935. Certainly, Newman’s final lecture had mentioned it—but perhaps Turing hadn’t been there? There’s good evidence that he was introduced to Gödel’s work not by Newman, but by the philosopher Richard Braithwaite [5, p. 174].

Although Boden seems to indicate otherwise, the exact content of the 1935 lectures is in fact somewhat uncertain. But detailed notes were taken during Newman’s 1934 Foundations of Mathematics lectures, by his student Frank Smithies, later a distinguished mathematician, and these provide much valuable information about the lecture content in 1934 [51].

There were nineteen lectures in total in 1934. Lecture 16 began with a discussion of the problem of proving the consistency of the Peano axioms and moved on to a presentation of the Entscheidungsproblem. Newman explained that the Entscheidungsproblem had been settled for the monadic predicate calculus. Then at the end of lecture 16 came a wide-ranging list of references: Russell and Whitehead’s Principia Mathematica, Russell’s Introduction to Mathematical Philosophy, Ramsey’s Foundations of Mathematics, Wittgenstein’s Tractatus Logico-Philosophicus, Hilbert and Bernays’s Grundlagen der Mathematik, papers by Hilbert, Brouwer, and Church, and also Gödel’s 1931 paper in the Monatshefte, which was to be the topic of the next three lectures (lectures 17–19).

To judge from scattered references elsewhere in Smithies’s notes, Newman seems to have been using the 1934 Hilbert and Bernays Grundlagen der Mathematik [29] as a textbook, along with Hilbert and Ackermann’s 1928 Grundzüge der Theoretischen Logik [28]. If so, Newman’s lectures 17–19 ran well ahead of his 1934 textbook. Hilbert and Bernays did refer to Gödel’s 1931 paper (mentioning in particular his Proposition VII, that every primitive recursive relation is arithmetical), but only in connection with their sustained argument that primitive recursive definitions, and some more complicated recursive definitions, can be reduced to explicit definitions [29, p. 415, Footnote 1]. There was no mention of Gödel’s incompleteness theorems. The closest the book came to discussing these was the following remark by Hilbert, in his brief foreword to the book:

I want to emphasize that the opinion, which has arisen from time to time, that the unfeasibility of my proof theory follows from certain recent results of Gödel’s, has been established to be erroneous. In point of fact, that result shows only that for more far-reaching consistency proofs, one must exploit the finitist standpoint in a sharper way than is necessary in the consideration of elementary formalisms [29, p. v; translated by Copeland and Sommaruga].Footnote 18

Smithies’s notes are our best (and only) evidence regarding the content of the 1935 lectures. In his lectures 17–19, during the final week of the 1934 course, Newman gave a tour of the central ideas of Gödel’s 1931 paper. He covered Gödel numbering, the construction of Gödel’s xBy (x is a proof of y) and Bew(x) (x is provable), Gödel’s Proposition V (every recursive relation is definable), and the concept of ω-consistency, followed by deliciously brief proofs of the first and second incompleteness theorems. Did Newman end his lectures in the same way in 1935? It is conjecture to say so, since no syllabus or notes seem to have survived, and there was no examination question relating to Gödel’s theorems in the 1935 Tripos.Footnote 19 Nevertheless, in the absence of any evidence to the contrary, it seems not unreasonable to assume that Newman would have covered much the same material in the 1934 and 1935 versions of the course.

Boden questions whether Turing attended the final lectures of the course. Perhaps that week found him fully engrossed in completing what King’s mathematician Philip Hall (quoted in [47, p. 45) described as a “very pretty little proof,” submitted a few weeks later as “Equivalence of Left and Right Almost Periodicity.” Boden is right that whether Turing attended all the lectures is a matter for conjecture.Footnote 20 (Newman reported only that Turing “attended a lecture of mine.”) On the other hand, Turing might indeed have attended every lecture and heard Newman give much the same introduction to Gödel’s results as he gave in 1934. To be clear, we believe there is no evidence either way, and the facts of the matter may by this time be unknowable.

Turning to Boden’s claim that it was not Newman but Braithwaite who introduced Turing to Gödel’s work, her “good evidence” is a letter Braithwaite wrote to her in 1982. A Fellow of King’s from 1924, Braithwaite was a mathematically and scientifically oriented philosopher, working in the broadly logical positivist tradition, and he was involved in Turing’s election to a King’s junior research fellowship in March 1935. Braithwaite authored the 32-page introduction to the English translation of Gödel’s 1931 incompleteness paper [20], and he was intimately familiar with the pioneering work done on the Entscheidungsproblem by his close friend Frank Ramsey, also a Fellow of King’s. In 1928, in a paper Braithwaite edited for his 1931 posthumous collection of Ramsey’s papers, Ramsey described the Entscheidungsproblem as “one of the leading problems of mathematical logic,” and he solved it for important special cases [42].

In his letter to Boden, Braithwaite remarked on “Turing’s complete ignorance of Gödel’s work when he wrote his ‘Computable Numbers’ paper.”Footnote 21 Braithwaite continued: “I consider I played some part in drawing Turing’s attention to the relation of his work to Gödel’s.” Braithwaite’s recollection (if accurate) clearly undermines the Standard Story. However, contrary to Boden’s view (quoted above), it may not have been Braithwaite alone who was responsible for drawing Turing’s attention to the relation of his work to Gödel’s. Newman may have played a key role too (although more than a year later than the Standard Story relates), as we will explain.

The Alternative Picture

Not only is the précis the earliest surviving exposition of the central ideas of OCN, it seems it was also Newman’s entrée to Turing’s work on the topic. Turing wrote to Sara on May 4, 1936:

I saw Mr Newman four or five days after I came up. He is very busy with other things just at present and says he will not be able to give his whole attention to my theory for some week or so yet. However he examined my note for C.R and approved it after some alterations [54].

Concerning OCN itself, Turing said in the same letter, “I don’t think the full text will be ready for a fortnight or more yet.”

The approximate date of Turing’s meeting with Newman can be deduced. As previously mentioned, Turing had spent Easter at the family home in Guildford [54]. That year, Easter Sunday fell on April 12, and the new Cambridge term began on Thursday, April 16. If Turing “came up” on the Monday or Tuesday (and in the UK, traveling back home from Easter family gatherings on Easter Monday is often the norm), then his meeting with Newman was most likely on the 17th or 18th.

Newman may have been surprised by the précis (to say the least). He did not then know of Church’s contemporaneous proof of unsolvability, and Turing had not previously said much about his own research. A few weeks later, Newman wrote that “Turing’s work is entirely independent: he has been working without any supervision or criticism from anyone” [50]. As he read through the précis, might Newman have quizzed Turing about the conspicuous absence of any mention of Gödel’s work? At any rate, on the 18th, Turing went to the Philosophical Society Library and borrowed the volume of Monatshefte containing Gödel’s 1931 paper.Footnote 22 He kept it until June (Figure 5), while finalizing OCN for submission to the London Mathematical Society, where his typescript was received on May 28.

As we noted earlier, Turing had not previously borrowed the 1931 volume, according to the library records. Might he have obtained it on prior occasions from another source, or perhaps read it in the Philosophical Society Library without borrowing it? This certainly cannot be ruled out. Turing’s own college library did not take the Monatshefte, and there were no copies elsewhere in the Cambridge library system apart from the University Library. This was more than twice as far from King’s as the Philosophical Society Library, which Turing could walk to from his rooms in three minutes or less. He might have borrowed the 1931 Monatshefte from the University Library, though he was, as we have seen, in the habit of borrowing the journals he needed from the Philosophical Society Library. Later, once an offprint of Church’s Entscheidungsproblem paper had arrived in Cambridge—an event Newman described as “painful” for Turing [50]—some modifications to OCN were necessary. Turing added a short appendix “Computability and Effective Calculability,” dated August 28, 1936. His only reference in the appendix was to a paper by Church’s student Kleene, in the 1935 volume of the American Journal of Mathematics. Once again, Turing used the Philosophical Society Library, borrowing the 1935 volume of the journal on August 15, 1936.

Braithwaite’s letter, in conjunction with Kleene’s remarks discussed above, and the fact that the March or early April 1936 précis did not mention Gödel, make an alternative proposition worthy of consideration, namely that Turing first came to appreciate “the relation of his work to Gödel’s” (Braithwaite’s phrase) in approximately mid-April 1936, and by then, of course, OCN was virtually complete. Turing’s thinking on how best to present his results does seem to have changed markedly over a period of a few weeks. In the précis, he saw no need to refer to Gödel, whereas OCN contains brief but deep remarks on Gödel’s work (these remarks are examined in the next section).

This alternative proposition allows that Turing may already have had some prior knowledge of Gödel’s work, from attending Newman’s lectures. Braithwaite’s phrase “complete ignorance” (our italics) may have been emphatic rather than literal. Braithwaite’s essential point seems to be that, as he said, he “played some part in drawing Turing’s attention to the relation of his work to Gödel’s” (our italics).

While the considerations we have set out so far in the discussion are suggestive, we certainly do not mean to be understood as claiming that this alternative proposition is correct. We keep an open mind. The presently available evidence is visibly insufficient for such a claim. Our point is rather that given the available evidence, this alternative proposition is no less probable than the Standard Story, and indeed sits rather more comfortably with the known and reported facts (as we shall continue to argue in the next section). Our aim is simply to show that the Standard Story transcends the evidence and is highly speculative.

Let us indulge in some speculation of our own. When Turing consulted Newman about the précis in April 1936, it seems not unlikely that Newman would have advised him to familiarize himself with the relation of his work to Gödel’s. Furthermore, it is probable that soon after returning to Cambridge from the Easter vacation, Turing met not only Newman but also Braithwaite (since Braithwaite and Turing were both Fellows of King’s).Footnote 23 If Turing sought Braithwaite’s opinion on the précis—and why would he not have consulted an expert on the Entscheidungsproblem from his own college?—then to judge by Braithwaite’s letter, he would have received the same advice, to familiarize himself with the relation of his work to Gödel’s. Thus, Turing’s visit to the Philosophical Society Library on April 18 may have been precipitated by both Braithwaite and Newman. Indeed, Newman showed no great interest in Carnap, whereas Braithwaite had a keen interest (his 1930s papers contain discussions of Carnap’s ideas, and his 1953 opus major Scientific Explanation was strongly influenced by Carnap’s view of scientific theories as interpreted deductive calculi).

The Text of “On Computable Numbers”

As is well known, OCN referred to Gödel’s 1931 article. In this section, we examine each mention of Gödel in Turing’s text.

When Turing turned to modifying his draft in order to take account of Church’s 1936 publications [7, 8, he added merely two sentences to the text [45, p. 231]: “In a recent paper Alonzo Church has introduced an idea of ‘effective calculability,’ which is equivalent to my ‘computability,’ but is very differently defined. Church also reaches similar conclusions about the Entscheidungsproblem.” Turing dealt with Gödel’s 1931 article in a similar fashion. His text includes no more than three short mentions of Gödel.

The first mention is at the foot of the first page, immediately before Turing’s brief remark about Church. He said simply, “By the correct application of one of these arguments, conclusions are reached which are superficially similar to those of Gödel” [45, p. 230]. The second mention does not come until more than halfway through the paper and is no more than a passing remark in parentheses [45, p. 248]. To the sentence “For each of these ‘general process’ problems can be expressed as a problem concerning a general process for determining whether a given integer n has a property G(n), and this is equivalent to computing a number whose n-th figure is 1 if G(n) is true and 0 if it is false,” Turing added a parenthesis, immediately following the first occurrence of G(n):

[e.g. G(n) might mean “n is satisfactory” or “n is the Gödel representation of a provable formula”].

His inconsistent use of square brackets was probably a result of lack of care as he made the insertion; elsewhere in the paper he always used round brackets for within-sentence parentheses (including parentheses of the same length as the above).

The third and last mention of Gödel comes almost right at the end of the paper, in the final section, “Application to the Entscheidungsproblem” [45, p. 259]. Turing’s principal aim in mentioning Gödel at that point was evidently (as with the mention on the first page) to distance his work from Gödel’s. He said succinctly:

It should perhaps be remarked that what I shall prove is quite different from the well-known results of Gödel. Gödel has shown that (in the formalism of Principia Mathematica) there are propositions \(\mathfrak{A}\) such that neither \(\mathfrak{A}\) nor –\(\mathfrak{A}\) is provable. As a consequence of this, it is shown that no proof of consistency of Principia Mathematica (or of K) can be given within that formalism. On the other hand, I shall show that there is no general method which tells whether a given formula \(\mathfrak{A}\) is provable in K, or, what comes to the same, whether the system consisting of K with –\(\mathfrak{A}\) adjoined as an extra axiom is consistent [45, p. 259].

Then, in the next sentence, Turing noted:

If the negation of what Gödel has shown had been proved, i.e. if, for each \(\mathfrak{A}\), either \(\mathfrak{A}\) or –\(\mathfrak{A}\) is provable, then we should have an immediate solution of the Entscheidungsproblem. For we can invent a machine \(\mathcal{K}\) which will prove consecutively all provable formulae [45, p. 259].

Turing’s point here—plainly if modestly expressed—is that Gödel’s first incompleteness result follows immediately from his own quite different undecidability result. This was the first mention of a road gratefully followed by many a logician and computer scientist since. Gödel’s first incompleteness theorem can be regarded as an immediate corollary of (what Scott Aaronson calls) Turing’s conceptually prior undecidability theorem [1]—and with no need to mention Gödel numbers or the labyrinth of subsidiary definitions involved in defining Gödel’s Bew(x).

There is, then, no evidence for the Standard Story to be found in Turing’s references to Gödel’s work in OCN. The text of OCN contains nothing at all to indicate that Gödel’s work was the foundation for Turing’s, or that Turing used concepts or techniques drawn from Gödel’s 1931 paper. On the contrary, Turing’s brief remarks concerning Gödel served only to emphasize—firmly—that his own work was very different. The text of OCN itself provides the greatest challenge to the Standard Story, as Kleene observed.

Concluding Remarks

We have assembled an array of challenges to the Standard Story, and we have developed a plausible alternative picture that sits more comfortably with the record. Both accounts are short on decisive evidence. We conclude that no one should claim that central ideas in OCN were imported from Gödel’s work unless both the existence of a conflicting account and a lack of deciding evidence is acknowledged.

It is one of the great ironies of the history of logic that all along, incompleteness sufficed to establish undecidability. This was first noted more than thirty years after Gödel’s paper appeared, by Martin Davis [13, p. 109], [34, p. 136]. Naturally, this does not diminish Turing’s, nor Church’s, great achievement. Back in the day, Carnap and others suspected the existence of an implication from Gödel’s results to a negative solution of the Entscheidungsproblem, but no proof existed. In 1931, commenting on Gödel’s recently published incompleteness results, Herbrand said cautiously, “[A]lthough at present it seems unlikely that the decision problem can be solved, it has not yet been proved that it is impossible to do so” [26, p. 259]. Newman later summed up matters as they had appeared at the time, before Turing and Church produced their transformational proofs:

A first blow was dealt [to the Hilbert program] by Gödel’s incompleteness theorem (1931), which made it clear that truth or falsehood of A could not be equated to provability of A or not-A in any finitely based logic, chosen once for all; but there still remained in principle the possibility of finding a mechanical process for deciding whether A, or not-A, or neither, was formally provable in a given system [38, p. 256].

If as Turing developed the arguments and proofs of “On Computable Numbers” he was not even aware of the relation of Gödel’s results to his own, irony is piled on irony.