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His answer was that, some day in the not-too-distant future, in a certain sense they might: “I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.”²¹⁸

Then Turing works over a number of possible objections to his prediction. “Lady Lovelace’s Objection: … she states, ‘The Analytical Engine has no pretensions to originate anything. It can do whatever we know how to order it to perform’.”^§MS1.3.2a But perhaps, says Turing, the Analytical Engine had the potential to “think for itself” to the extent that it could come up with surprising answers to mathematical problems.²¹⁹

Turing proposes an “imitation game” where a computer gives answers that seem as smart as a person.^§MS1.3.2c Such a game might require, say, 10⁹ binary digits. “At my present rate of working I produce about a thousand digits of programme a day, so that about sixty workers, working steadily through the fifty years might accomplish the job, if nothing went into the waste-paper basket.” Meanwhile, “[p]arts of modern machines which can be regarded as analogues of nerve cells work about a thousand times faster than the latter … Machines take me by surprise with great frequency …, largely because I do not do sufficient calculation.”²²⁰

Perhaps, Turing says in a footnote, in this sense Lovelace was not so wrong. Her statement about machines doing what we tell them, he says, does not include the word “only.” “There was no obligation on [Lovelace or Babbage] to claim all that could be claimed.”²²¹ Nor, on the other hand, was Turing’s claim ever so grandiose as to suggest what is now popularly called “artificial intelligence.”^§AS1.4.6c His claim was not to anything more than “mechanical intelligence,” that computers could produce surprising mathematical results with fast calculation. Lovelace surely would have agreed.

Turing first brought these ideas to print in his 1936 paper, “On Computable Numbers.” He had been appointed a fellow of Kings College Cambridge University just two years before, at the remarkably early age of 22.

“On Computable Numbers” is in some respects a strange paper, not just for its mathematical tangle written in obscure German Gothic type, but for talking about what seem to be two quite tangential things that add up to an apparently contradictory proposition. The rest of its title was “ … with an Application to the Entscheidungsproblem,” or the problem of decidability in mathematics.²²² German mathematician David Hilbert had famously argued in 1928 that mathematics was complete, that it was consistent, and that every problem it posed was decidable in the sense that every one of its propositions could be proved or disproved. Kurt Gödel had by 1931 shown that mathematics was neither complete nor consistent.^§MS1a Turing was now able to show that not every problem was decidable, so bringing down the last pillar of Hilbert’s mathematical edifice.

But, here is the apparent contradiction: incidental to making this case, Turing imagined a machine that would be able to deal with things in mathematics that were still absolutely decidable, though not feasibly decidable by humans. “According to my definition, a number is computable if its decimal can be written down by a machine.”²²³ This leaves a lot of scope for definitive decidability, including calculations that are well beyond what is practically possible for humans and without the errors to which they are prone.

Turing’s imaginary machine worked like this. It would run by reading a paper tape with a sequence of squares each capable of carrying a symbol. As the machine is fed the tape, it scans each symbol and is able to remember some of the symbols it has already scanned. It can print calculations onto blank squares, so remembering the steps it is taking. It can, in other words, write down the interim results of its calculations in a kind of working memory, then act on these. In this way, the machine emulates a sequence of minimal states of mind, breaking a sequence of mathematical operations into their most elementary steps.²²⁴

“Let us imagine the operations performed by [a person computing] to be split up into ‘simple operations’ which are so elementary that it is not easy to imagine them further divided,” said Turing. Now, “the two-dimensional character of paper is no [longer] essential of computation. I assume then that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares.” By this means, and far from being undecidable, “[i]t is my contention that these operations include all those which are used in the computation of a number.”²²⁵

Now, we find that we have a machine capable of making calculations beyond the practical realm of human decidability. “[A]n an Arabic numeral such as 17 or 999999999999999 is normally treated as a single symbol … The differences from our point of view between the single and compound symbol is that the compound symbols, if they are too lengthy, cannot be observed at one glance … We cannot tell at a glance whether 9999999999999999 and 999999999999999 are the same.” But by laborious emulation of elementary “states of mind,” a machine of the hypothetical kind he was describing would be able to deal quickly and with precision with calculations that a human could only do with such difficulty as to make them impracticable. With such a machine a whole lot more could become decidable, and definitively. “It is possible to invent a single machine which can be used to compute any computable sequence,” he concluded.²²⁶

Turing offered his first lecture course at Cambridge in 1939, “Foundations of Mathematics,” in which the final examination question was to come up with a solution to the Entscheidungsproblem. In that year, the famed philosopher of language, Ludwig Wittgenstein,^§MS2.1.1a offered a course with the same name. Turing was one of fifteen people who attended Wittgenstein’s course.

TURING: You cannot be confident about your calculus until you know there is no hidden contradiction in it …

WITTGENSTEIN: … Why should [people] be afraid of contradictions inside mathematics? …

TURING: The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort.²²⁷

Or perhaps, to imagine a further question and its answer, why even bother with the Entscheidungsproblem? Because it tells of a whole lot more that is definitively decidable by calculation, even if much of what is possible cannot be done without the help of a machine. We calculate, machines help us to calculate, and bridges mostly stay up.

As it happened, a professor at the Institute of Advanced Study at Princeton University, Alonzo Church, had at the same time come to the same conclusion as Turing about the Entscheidungsproblem, though his method of proof was different and his version of the idea he called “effective calculability.”²²⁸ Turing acknowledged Church in his paper, and for his part Church recognized the ingenuity of Turing’s hypothetical apparatus by calling it a “Turing Machine.”²²⁹ The principles of calculability have come to be known as the “Church–Turing thesis.”

The Institute of Advanced Study was where the intellectual action was – not only were Church and Gödel there, but at various times, other key figures in the creation of computing, including John von Neumann and Claude Shannon. Turing spent time there in 1936–38, writing his PhD on ordinal numbers.

In 1938 Shannon come up with the idea that, instead of paper tape, relay circuits or on/off switches could represent mathematical symbols as zeros and ones, and do the work of calculation electrically. Applying the elementary logic of nineteenth-century mathematical philosopher George Boole, he suggested that when the circuit is closed a proposition could be considered false, and when it is open, it could be considered true: “any circuit is represented by a set of equations, the terms of the equations corresponding to the various relays and switches in the circuit. A calculus is developed for manipulating these equations by simple mathematical processes.”²³⁰ Such an electronic machine would work sequentially through a series of yes/no binaries: no decision problems or contradictions here.

By the mid-2010s, a smart phone would have over three billion such switches, performing exactly the kinds of humanly impractical calculations that Lovelace and Turing had both anticipated, though on a scale unimaginably larger again. Perhaps these things should be called Lovelace-Turing Machines.

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²¹⁸ A.M. Turing, 1950, “Computing Machinery and Intelligence,” Mind 59:433–60, p. 442.

²¹⁹ Ibid., p. 450.

²²⁰ Ibid., pp. 455, 450.

²²¹ Ibid., pp. 459, 450.

²²² A.M. Turing, 1936, “On Computable Numbers, with an Application to the Entscheidungsproblem,” Proceedings of the London Mathematical Society 42(2):230–65.

²²³ Ibid., p. 230.

²²⁴ Ibid., p. 230.

²²⁵ Ibid., pp. 250, 249, 232.

²²⁶ Ibid., pp. 250, 241.

²²⁷ Andrew Hodges, 1983, Alan Turing: The Enigma, New York, NY: Simon and Schuster, p. 254.

²²⁸ Alonzo Church, 1936, “A Note on the Entscheidungsproblem,” The Journal of Symbolic Logic 1(1):40–41.

²²⁹ Alonzo Church, 1937, “Review of ‘on Computable Numbers, with an Application to the Entscheidungsproblem’ by A.M. Turing,” The Journal of Symbolic Logic 2(1):42–43.

²³⁰ Claude E. Shannon, 1938, “A Symbolic Analysis of Relay and Switching Circuits,” Transactions of the American Institute of Electrical Engineers 57:471–95, p. 471; Claude E. Shannon and Warren Weaver, 1949, The Mathematical Theory of Communication, Urbana, IL: University of Illinois Press; Jimmy Soni and Rob Goodman, 2017, A Mind at Play: How Claude Shannon Invented the Information Age, New York, NY: Simon and Schuster.

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Cope, Bill, and Mary Kalantzis. 2020. Making Sense: Reference, Agency and Structure in a Grammar of Multimodal Meaning. Cambridge UK: Cambridge University Press, pp. 159-163. https://doi.org/10.1017/9781316459645

Alan Turing’s “Mechanical Intelligence” §MS1.3.2b