Wittgenstein, Turing, and the Intelligence of Games

Abstract

One of Wittgenstein’s most quoted passages from his Remarks on the Philosophy of Psychology concerns Turing’s “machines” and says verbatim: “These machines are humans who calculate. And one might express what he [Turing] says also in the form of games.” This passage not only captures the kernel of Turing’s conceptual argument for the adequacy of his definition of “computability”, as presented in his article On Computable Numbers (1936), but also helps clarify Turing’s idea of “mechanical intelligence.” Indeed, the notion of game provides an ideal means to focus on similarities and differences between Turing and Wittgenstein’s views of mechanical procedures, mathematical understanding, and thinking activity. The live encounter between Ludwig Wittgenstein and Alan Turing took place in Cambridge in 1939, when Wittgenstein’s Lectures on the Foundations of Mathematics were regularly attended by Turing. Interestingly, during the conversations between the two, Turing seems to play the role of the Wittgenstein of the Tractatus, to allow the present Wittgenstein to reassess what he deplores as mistaken or misleading in his early work. As for Turing himself, his reflection on thinking machines from the late 1940s demonstrates the significance of his dialogue with Wittgenstein.

1. Introduction

Wittgenstein’s philosophical path—from a “picture theory” of language, which sets logic as a mirror-image of the world, to a “language game” theory—matures through a reflection on the nature of mathematics not so much with reference to logical foundations as to effective procedures, hence to the notion of proof. What makes a mathematical proof reliable? How is mathematical knowledge enhanced? Turing’s interests also converge on these issues. Although much of the opinion on a Turing-Wittgenstein comparison is inclined to emphasize their divergences, this article is mainly about possible resonances between them. The next two sections are devoted to Wittgenstein’s thought. Section 4 draws attention to specific aspects of his Cambridge lectures that may help frame the mutual perspectives of Turing and Wittgenstein. Section 5 deals with Turing’s work on “thinking machines” and his proposal of the imitation game to spell out the ineffable faculty of “intelligence.” Some final remarks encourage a parallel between Wittgenstein’s theory of meaning and Turing’s understanding of intelligence.

2. The Formula Game

Am Anfang ist das Zeichen

Hilbert 1922

As well known, Wittgenstein closed his Tractatus Logico-Philosophicus with the sentence “What we cannot speak about we must pass over in silence” [1]. Right before, we read that the only strictly correct method in philosophy would really be “to say nothing except what can be said, i.e., propositions of natural science—i.e., something that has nothing to do with philosophy.” But then how can one speak about philosophy? What is philosophy about?

The Tractatus is built on the idea that philosophical problems originate from misunderstandings about the logic of language. Thus, its major task is to provoke “a change in the way of conceiving the limits within which a speaker of any language whatsoever can put forward claims” [2] (pp. 6–7). The so called “picture theory”, which shapes the architecture of the Tractatus, presents logic as “a mirror-image of the world” (T 6.13). The logical image of facts is the thought, and the thought is the significant proposition (T 3, 4). Logic is transcendental, so it provides the conditions of possibility for experience: “Logic precedes every experience—that something is so. It is before the How, not before the What” (T 5.552). Logic is not a science concerning some category of objects, and logical truths are of a completely different kind from the truth of ordinary propositions.1 In line with the “picture theory” of language, meaning is to be seen as a correlation function between linguistic expressions and “states of affairs” in the world. An isomorphism applies between the states of affairs and the propositions expressing them: the truth conditions of propositions correspond to the existence possibilities of facts. A proposition shows a possible state of affairs and, if true, says that that state of affairs exists. Only what can be said—i.e., meaningful (sinnvolle) propositions—can be verified, compared with reality. Wittgenstein clearly understands that a decision method for logical truth allows logical propositions, which are “senseless” (sinnlos), to be distinguished from significant propositions.

Analyzed through the logical lens provided by the Tractatus, language—just like mathematics—must follow strictly determined rules, “as a calculation”. Mathematics is a logical method, and all the intuition that is needed to prove a mathematical proposition is contained in the rules of calculation. From the Tractatus:

6.233 To the question whether we need intuition for the solution of mathematical problems it must be answered that language itself here supplies the necessary intuition.

6.2331. The process of calculation brings about just this intuition.

Calculation is not an experiment.

As for logical laws, they cannot themselves be subject to logical laws, nor can they depend on experience of any other kind. If logic depended on some experience—empirical or ideal—logical truth would lose its character of necessity. Metalanguages are not allowed. The form of reality is mirrored in the form of language. The conditions of signification of propositions reside in the constituent elements immanent to the proposition itself. As a picture of a situation, a proposition cannot show how it fulfils its function. The relation between propositions and facts is not mediated by any interpretation. This would fall within the domain of the unsayable, or the mystical, a domain that the Tractatus does not mean to violate. Following Brian McGuinness, the Wittgenstein of the Tractatus is not interested in our actual language, but in the principles of symbolism as essentially a vehicle for truth and falsity [3] (p. 59). Therefore, a mechanical decision procedure, such as the method of “truth tables” that the book ushers in,2 is a device for showing that logical truth is something implicit in any method of symbolizing. “That precedent to which one would always appeal must be present in the symbol itself” (T 5.525). Thus, it is logic that provides the instrument for the clarification of thoughts. And the purpose of philosophy is but the logical clarification of thoughts. “Philosophy is not a theory but an activity” (T 4.112).

Wittgenstein’s claim of the primacy of the symbol appears very much in tune with Hilbert’s motto at the beginning of this section. Indeed, in Wittgenstein’s work at this stage and in his enduring concern with the concept of proof, we can discern relevant Hilbertian motives.3 There is a striking correspondence between the following two excerpts, the first from Hilbert’s lecture “The Foundations of Mathematics” (Hamburg 1927), the second from Wittgenstein’s Blue Book (1933–34).

The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed. [...] The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds. [6] (p. 475)

We may say that thinking is essentially the activity of operating with signs. This activity is performed by the hand, when we think by writing; by the mouth and larynx, when we think by speaking; and if we think by imagining signs or pictures, I can give you no agent that thinks. If then you say that in such cases the mind thinks, I would only draw your attention to the fact that you are using a metaphor, that here the mind is an agent in a different sense from that in which the hand can be said to be the agent in writing. [7] (BB: 6–7)

It seems very plausible that pondering the nature of mathematical thinking, Wittgenstein realized the inadequacy of his picture theory of language. Its inherent weakness comes to light in mathematical language itself, which is to say, that part of language that would seem less problematic as it follows definite rules. But how does mathematical thinking (i.e., language) operate in a proof? Within the frame of the Tractatus, insofar as it is an expression of a logical necessity, a proof simply shows which proposition must follow from the correct application of the rules of inference (T 2.9). To repeat a proof is not only to reproduce the conditions under which a certain result has been obtained (as in an experiment), but also the result itself. Unlike an experiment, a proof does not require scrutiny, and we can use it, without further ado, as a standard of validity [8]. We can think of it as an image of an experiment, in the sense that it is a model of the application of a decision procedure. But what is the characteristic use of a proof as a calculation—as opposed to its use as an experiment? What does the peculiar inexorability of mathematics consist in? [9] (RFM I.99, I.4).

The search for an answer to these questions focuses on an essential point, underestimated by logicians: the mastery of the procedures of inference, of algorithms. Understanding attained by means of a proof involves an agreement not only about definitions, but also about the correctness of proving. As Wittgenstein remarks, “understanding a mathematical proposition” is a very vague concept. And yet, “if you say ‘The point isn’t understanding at all. Mathematical propositions are only positions in a game’ that too is nonsense! ‘Mathematics’ is not a sharply delimited concept” (RFM IV-46). The crystal clarity of pure logical form cannot be a “requirement” for mathematical understanding, nor for actual language.

The more closely we examine actual language, the greater becomes the conflict between it and our requirement. (For the crystalline purity of logic was, of course, not something I had discovered: it was a requirement.) The conflict becomes intolerable; the requirement is now in danger of becoming vacuous.—We have got on to slippery ice where there is no friction, and so, in a certain sense, the conditions are ideal; but also, just because of that, we are unable to walk. We want to walk: so we need friction. Back to the rough ground! [10] (PI §-107)

3. Language Games

Im Anfang war die Tat

Goethe, Faust

The problem of mathematical understanding solicits an investigation into the conditions of demonstrability, rather than of truth, pouring over the methods of proof. Is it possible to grasp a concept without having a clear idea of its applications? Does understanding that one thing follows from another mean that one has mechanically followed the steps of derivation? As is observed in the Philosophical Investigations, there is a gulf between an order and its execution that must be closed by the process of understanding (PI §-431). All meaningful propositions, however, must be chained to the rules of a game.

1. It is of course clear that the mathematician, insofar as he really is “playing a game” does not infer. For here “playing” must mean acting in accordance with certain rules. And it would already be something outside the mere game for him to infer that he could act in this way according to the general rule.

2. Does a calculating machine calculate? [...]

I want to say: it is essential to mathematics that its signs are also employed in mufti.

It is the use outside mathematics, and so the meaning of the signs, that makes the sign-game into mathematics. (RFM IV)

The crucial question of the gap between an order and its execution, which is to say, of the relationship between rule-following and understanding, pervades Wittgenstein’s philosophical reflection. In the Blue Book (1933–1934), one of his main concerns is the “action of language” in operating with signs. He questions whether two parts should be distinguished in the action of language: “an inorganic part, the handling of signs, and an organic part, which we may call understanding these signs, meaning them, interpreting them, thinking” (BB: 3). If the activities of the first type can be located on a paper and somehow delegated to a mechanism, where do the others take place? The latter activities, Wittgenstein notes, “seem to take place in a queer kind of medium, the mind; and the mechanism of the mind, the nature of which, it seems, we don’t quite understand, can bring about effects which no material mechanism could.” Wittgenstein appropriately traces this line of thought back to Frege’s critique of the formalist conception of mathematics. According to Frege, the formalists would have confused the unimportant thing, the sign, with the important thing, the meaning. In short, Frege’s idea was that if the propositions of mathematics were only constellations of signs, they would be dead and devoid of interest, whereas they obviously have a kind of life. Thus, the conclusion would be that to give life to a proposition from dead signs, something immaterial must be added. “But if we had to name anything which is the life of the sign,” Wittgenstein claims, “we should have to say that it was its use” (BB: 4).4 As we read in his Philosophical Investigations:

Every sign by itself seems dead. What gives it life?—In use it lives. Is it there that it has living breath within it?—Or is the use its breath? (PI §-432)

For Wittgenstein, questioning what kind of activity thinking is, one can be driven to question where thinking takes place. The answer could be on paper, in the head, in the mind, but none provides the location of thinking. We must be careful not to be led astray by the similarity of the linguistic form. “It is correct to say that thinking is an activity of our writing hand, of our larynx, of our head, and of our mind, so long as we understand the grammar of these statements” (BB: 16). Hence, it is extremely important to realize how, by misunderstanding the grammar of expressions, we may consider that one of these statements gives the real seat of the activity of thinking. All this, as Wittgenstein remarks, can be also put into the question: “Can a machine think?” And the trouble is not that we don’t yet know a machine which could do the job: “The question is not analogous to that which someone might have asked a hundred years ago: ‘Can a machine liquefy a gas?’ The trouble is rather that the sentence, ‘A machine thinks (perceives, wishes)’ seems somehow nonsensical” (BB: 47). It would be like asking whether the number 3 has a color!

Going back to the claim that “thinking is essentially operating with signs,” a basic question is: “What are signs?” Now, instead of providing a general answer, Wittgenstein encourages drawing attention on specific cases of operating with signs, or ways of using signs, to grasp what he calls language games.

Language games are the forms of language with which a child begins to make use of words. The study of language games is the study of primitive forms of language or primitive languages. [...] When we look at such simple forms of language the mental mist which seems to enshroud our ordinary use of language disappears. We see activities, reactions, which are clear-cut and transparent. On the other hand, we recognize in these simple processes forms of language not separated by a break from our more complicated ones. We see that we can build up the complicated forms from the primitive ones by gradually adding new forms. (BB: 17)

Language games make it possible to see more clearly the variety of rules that guide the use of linguistic signs. When a child begins to talk, the teaching of language does not involve explanation, but training. If one studies the phenomena of language in primitive forms, in which the purpose and functioning of words can be clearly surveyed, the mist gradually disappears. For Wittgenstein, playing a language game without being engaged by philosophical problems could be interpreted as clarifying the grammar of language with a sort of “morphological investigation.” His previous concern for the general form of the proposition and language vanishes. Instead of looking for something common to all that we call language, one must regard the language phenomena as having not one thing in common, but many kinds of affinities. And it is because of all these affinities that they constitute “languages.” As an example, Wittgenstein points out the games themselves:

Consider, for example, the activities that we call “games”. I mean board-games, card-games, ball-games, athletic games, and so on. What is common to them all?—Don’t say: “They must have something in common, or they would not be called ‘games’”—but look and see whether there is anything common to all.—For if you look at them, you won’t see something that is common to all, but similarities, affinities, and a whole series of them at that. To repeat: don’t think, but look! (PI §-66)

Now, if one of the values of the Tractatus was to show the little significance of having found the final solution to the problems of philosophy through a clarification of the logic of our language, the implicit message was that those problems could “dis-solve” thanks to a new way of seeing the world, namely the language. Interestingly, in Wittgenstein’s path from Hilbert’s “formula game” to “language games,” Goethe’s morphology plays a remarkable role. The crystalline purity of the Tractatus’ logic gives way to the dynamic geometry of language games, to the “morphology of the use of an expression” [12] (p. 43). It was a mistake to look for an explanation where one must instead regard the facts as “proto-phenomena”; namely, where one must say: this is the language game that is being played (PI §-654). The poietic character of the language games is then instrumental to the essential vitality of language.

The word “language-game” is used here to emphasize the fact that the speaking of language is part of an activity, or of a form of life. (PI §-23)

The origin and the primitive form of the language game is a reaction; only from this can more complicated forms develop. Language—I want to say—is a refinement, “in the beginning was the deed” [13] (p. 67).5

4. Cambridge 1939

As already mentioned, in 1939, Wittgenstein held a lecture course on the foundations of mathematics which was regularly attended by Alan Turing. From the start, in the first lecture, Wittgenstein makes it clear that his intention is not interfering with mathematicians or entering into their calculating questions: “I must not make a calculation and say, ‘That’s the result; not what Turing says it is’” [4] (p. 14). As a philosopher, his interest turns to that kind of puzzles which arise from the use in mathematics of words from our everyday language, such as proof, number, series, order, and so on. If he can talk about them, it is because they are puzzles involving very basic mathematics, as in the case of Cantor’s argument. This allusion to Cantor’s argument offers the opportunity to make a brief remark on Turing’s computability, a topic never explicitly treated during the Cambridge Lectures and yet somehow implicated.

4.1. On Turing Machines

Throughout the Cambridge Lectures, the “Turing machine” is never named. Nevertheless, two short paragraphs from Wittgenstein’s Remarks on the Philosophy of Phycology encapsulate the core conceptual argument of Turing’s seminal article On Computable Numbers with an Application to the Entscheidungsproblem (1936). The first goes as follows:

Turing’s ‘machines’. These machines are humans who calculate. And one might express what he says also in the form of games. And the interesting games would be such as brought one via certain rules to nonsensical instructions. I am thinking of games like the “racing game”. One has received the order “Go on in the same way” when this makes no sense, say because one has got into a circle [15] (§ 1096).

The second (§ 1097) sketches the unsolvability of the Entscheidungsproblem in the terms of a game with self-referential rules. Using a variant of Cantor’s diagonal argument, it is shown how one gets into a circle. Cantor’s well-known argument is called “diagonal” because it constructs a sequence of digits from the digits on the diagonal of a list of sequences $x_1, x_2, \dots , x_n, \dots$; in fact, it constructs an “anti-diagonal” sequence $x_S$ by systematically changing each digit along the diagonal. The argument shows not only the existence of a sequence $x_S$ not on the given list, which therefore proved to be uncountable, but also that $x_S$ is as constructable as the sequences $x_1, x_2,\dots , x_n, \dots$ themselves.6 Now, Wittgenstein’s version considers that the sequences in the list are generated according to defined rules (or instructions).7 Which rule governs the formation of the “diagonal” sequence $x_D$? The “diagonal rule” should follow the rule of the first sequence $x_1$to calculate the first digit of $x_D$, the rule of the second sequence $x_2$ for the second digit, and so on. Can the diagonal sequence $x_D$ be included in the list? Where? Imagine that it is in place n = 10. Now the rule for the formation of the 10th digit only says that this digit should be equal to itself; hence, for n = 10 it is not a rule. In Wittgenstein’s words, “the rule of the game runs ‘Do the same as...’—and in the special case it becomes ‘Do the same as you are doing’!” But this makes no sense, the game jams.

The idea of a contradiction “jamming” is a salient subject addressed in his Cambridge Lectures. But before getting into these lectures, it is worth adding a few comments on Wittgenstein and Turing machines. I borrow the first from Wilfried Sieg: “Wittgenstein’s terse remark about Turing machines, ‘These machines are humans who calculate,’ captures the very feature of Turing’s analysis of calculability that makes it epistemologically relevant” [18] (p. 529). In his thorough analysis of computability, Sieg recognizes the distinctive character of Turing’s work in the move from arithmetically motivated calculations to general symbolic processes that underlie them and must be carried out by human beings. To understand why Turing’s work proved to be of greater mathematical and philosophical importance than that of other authors, Robin Gandy draws attention to the different approaches or strategies adopted. Turing begins his analysis by asking “What are the possible processes which can be carried out in computing a number?” [19] (p. 74), which is significantly different from the question “What is a computable function?”, which other authors asked. And yet, as Gandy emphasises, “Turing’s analysis makes no reference whatsoever to calculating machines. Turing machines appear as a result, as a codification, of his analysis of calculation by humans” [20] (p. 74).

What is of particular interest in the present context is the crucial role of an agent in Turing’s analysis of effective calculability. Considering what a human computer does, one can determine which simple mechanical operations on symbolic configurations are carried out in (human) computing. If a computation is performed by printing symbols on a potentially infinite tape divided into squares, then there is a lower bound on the size of symbols, to distinguish one symbol from the other, and an upper bound on the number of symbols, to ensure that all symbols (which take part in a computational step) can be observed at a single glance. The states relevant to the computation must be immediately recognizable. To know the state of a computation means to know “the sequence of symbols on the tape, which of these are observed by the computer (possibly with a special order), and the state of mind of the computer” [19] (p. 76). Insofar as human computing activity can be simulated by a finite number of local operations, a Turing machine can be regarded as “computationally equivalent” to a human computer. Therefore, the finiteness conditions that must be imposed on computability can be grounded in limitations of the relevant capabilities of the agent involved. Ultimately, the Church thesis gives Turing credit for two fundamental steps: (1) the interpretation of “effectively calculable” as calculable by a computer satisfying precise boundedness conditions; (2) the thesis that a human computer is “computationally equivalent” to a Turing machine. In short, Turing machines calculate as humans. All this implies that the abstract theory of computability is anchored in the capabilities of the agent available for its execution. Here is a turning point of critical importance for subsequent technology as well as for theoretical understanding. That Wittgenstein was not acquainted with Turing’s work in 1939 seems rather implausible; that he fully grasped its significance is shown by his remarks quoted above.8

4.2. On Meaning and Use of a Contradiction

Two main topics of discussion in the Cambridge Lectures are of particular interest here: how to deal with contradiction (Lectures XVIII-XXIII), how to relate calculation and experiment (Lectures X-XV). In the Lecture XVIII, Wittgenstein recalls that he once wrote the law of contradiction and other propositions of logic in a certain symbolism to be considered as a kind of explanation. “If you write ‘p.¬p’ in this symbolism, you get a proposition which has only F’s. Then: ‘¬(p.¬p)’—we get a proposition, the law of contradiction, which has only T’s” (LFM: 177). How might this symbolism explain why a contradiction does not work, and a proposition of logic may be said to be true but not verifiable by experience? “A contradiction jams and a tautology does nothing”—How can this be an explanation? What would it go wrong if we did not recognize the law of contradiction?

As a matter of fact, the “double negation” can be worth either an affirmation or a negation (reaffirmed). Has it “different meanings”? Is the meaning that makes a certain use correct or does the meaning consist in use? Posing these questions, Wittgenstein’s aim is to show that the truths of logic are not determined by a consensus of opinion but of action:

There is no opinion at all; it is not a question of opinion. They are determined by a consensus of action: a consensus of doing the same thing, reacting in the same way. We all act the same way, count the same way. (LFM: 183–184)

Turing and Wittgenstein agree that a contradictory order does not work. Turing observes, “the most natural thing to do when ordered “p and not-p” is to be dissatisfied with anything which is done.” And Wittgenstein, “We most naturally compare a contradiction to something which jams” (LFM: 186–187).9

A detailed discussion is about the so-called “hidden contradiction”. How might it be harmful? Wittgenstein poses the case in which, after using the rules, one day we come to a contradiction. We could realize that we have not used the rules correctly, or we may want to change the rules. His main concern, however, is to show that “a contradiction is not a germ which shows general illness.” As an example, if the rules for moving chessmen are given without mentioning the hedge of the chessboard, one can interpret the rules in such a way as to jump off the board or not. Is there a hidden contradiction? Here Turing notes that the teacher would not jump off the board, but the pupil might. And to Wittgenstein’s question as to how any harm comes in, he replies that the real harm with a contradiction comes in with an application; for instance, a bridge may fall. But if a bridge falls, Wittgenstein replies, it is because of a wrong natural law or a mistake in calculation. Where is the contradiction hidden? How does it undermine calculation? In his opinion, when it is hidden it has no course, once it is manifest, it can do no harm.

Wittgenstein: You might get p.¬ p by means of Frege’s system. If you can draw any conclusion you like from it, then that, as far as I can see, is all the trouble you can get into. And I would say, “Well then, just don’t draw any conclusions from a contradiction.”

Turing: But that would not be enough. For if one made that rule, one could get round it and get any conclusion which one liked without going through the contradiction. (LFM: 220)

Although it might seem that Wittgenstein overlooks the logical issue of having contradictions involved in formal systems,10 his main point is to shift the focus from abstract meaning to use: “the point of avoiding a contradiction is not to avoid a peculiar untruth about logical matters but to avoid the ambiguity that results—to avoid getting to that place from which you can go in every direction” (LFM: 224). Pondering the case in which we have a calculus and later find a contradiction in it, the question is how. If we think of a particular way in which a contradiction may arise, we can examine the rules and check them in this respect. But a vague search for contradictions in a system only warns that a contradiction can hide anywhere. And we might also say that once a contradiction comes up, a calculus is no longer the same. Indeed, any rule may be reinterpreted (naturally or unnaturally) and then, if it were again interpreted in some new way, a contradiction may arise. “But in practice—Turing remarks—the question of rules being reinterpreted does not come in seriously.” And this is very important, Wittgenstein stresses: “We do not imagine a case of interpreting a rule!” (LFM: 226) As with the law of contradiction, the fact is not that we are convinced of a particular truth. But rather that we want to do so-and-so (LFM: 230). The relevant distinction is between having a use and not having a use, not between having a meaning and not having a meaning.

This is very important when, for example, the question arises of whether mathematics is just a game with symbols or whether it depends on the meanings of its signs. This question vanishes when one ceases to think of meaning as being something in the mind. If you say, “The sign ‘2’ has no meaning”, do you want to say we don’t count chairs? Or do you want to distinguish between mathematics and its application? There is no question of giving it a meaning apart from an application. (LFM: 223)

4.3. Calculation vs. Experiment

Engaging mathematical meaning with application, the problem of “logical truths” goes to the background whereas the view of mathematics as an activity comes into the limelight. The question then arises as to how mathematical discoveries or inventions come about. Is there a way through experiment? Considering Turing’s suggestion to see a multiplication as an experiment, Wittgenstein queries how the result of the calculation can be regarded as the result of the experiment. In the case of the calculation, what is called “obeying the rules” of the calculation has already been fixed; therefore, the result of the calculation is right or wrong. As for the result of the experiment, it is not what you get if you follow the rules, but whether you follow them or not. Turing refines his idea.

Turing: One could make this comparison between an experiment in physics and a mathematical calculation: in the one case you say to a man, “Put these weights in the scale pan in such-and-such way, and see which way the lever swings”, and in the other case you say, “Take these figures, look up in such-and-such table, etc., and see what the result is”.

Wittgenstein: Yes, the two do seem very similar. But what is this similarity?

Turing: In both cases one wants to see what will happen in the end. (LFM: 96)

Wittgenstein warns against confusion. First, interest in the result does not make an activity an experiment; it can be an experiment. Second, if something is considered a calculation, then it is not considered an experiment. Perhaps a calculation can be regarded as the image of an experiment. Can a calculation always be put into the archive of measurements?

Turing: The difficulty is that there is not a finite number of multiplications... and when I do a multiplication which is not in your archives, what then?

Wittgenstein: Well, what then?—This is like counting to a number which has not been counted to. (LFM: 105)

To address Turing’s difficulty about the number of multiplications, Wittgenstein asks whether one “might say every new multiplication made is a new rule made.” But then what is the point of making multiplications?

Turing: If we were only concerned with multiplications up to 10, we could put them all in the archives; but as it is, the case is quite different.

Wittgenstein: Yes, and it is important to see that the two cases are different. (LFM: 106)

Wittgenstein submits the case in which, after putting into the archives a general rule and a few examples, a new example occurs. It might be a new rule, to be put in the archives or not? “The fact is that we recognize it.” The number of examples (whether finite or not) is entirely irrelevant.

The discussion extends from the rule that makes us get the result of calculation, to the proof that makes us see the truth. How do we recognize a mathematical proof? Is the proof “a kind of telescope” (LFM: 132) that makes us see a truth that has always been there? What is the criterion for the mathematical proof to be true if not the proof itself? For Wittgenstein, the idea that God sees the truth of the proof while the mathematician can come to see it in different ways, more or less stringent, does not hold. “We know as much as God does in mathematics” (LFM: 104). As for the not uncommon view that mathematics is a game, to be compared, for instance, with chess, it is worth examining how it can be both useful and misleading. An argument against the idea of comparing mathematics to chess highlights the distinction between games played by different people and the theory of the game. Then, Wittgenstein notes, “if you say the rules of chess are arbitrary, your opponents will say the theory of chess is not arbitrary” (LFM: 143). To what extent is it correct to say that the mathematical theory of chess expresses truths? Could we not say that the mathematical theory of chess is another game?

Turing: From the mathematical theory one can make predictions.

Wittgenstein: Yes, one can. But what sort of predictions? What is the relation between the mathematics of the chess and the predictions? (LFM: 150)

In a sense, Wittgenstein remarks, the pure mathematics of chess makes no predictions, as the calculus makes no predictions, but you can make predictions “by means of it”. Indeed, chess playing is not used for making predictions, but it can be—whereas the theory is used for making predictions; and that is the difference. Nonetheless, they could both be used for making predictions. They can both be called mathematics and both be called games. “The theory of chess is not arbitrary” only in the sense that has an obvious application. “Whereas chess hasn’t got an obvious application in that way. That’s why it is a game.”

4.4. Wittgenstein’s Further Remarks

Wittgenstein returns on the relationship between proof as a calculation vs. experiment and the idea of mathematics as a game, in his Remarks on the Foundations of Mathematics.11 One question he poses is whether the application of calculation suggests that it is the calculation itself, rather than the mathematician, that performs the operations. Why should mathematics always be considered under the aspect of discovery and not of doing? When we say that “a proof is a picture”, Wittgenstein stresses, this picture stands in need of ratification, which we give it when we work over it. It does not seem implausible to detect here a resonance with Hilbert’s proof theory, though with a different concern. According to Hilbert, “a proof is a figure, which we must be able to view as such” [27] (p. 1127), we must be able to make a proof as such something concrete and displayable, hence, the object of our investigation. Just as the physicists investigate their apparatus and the philosophers practice the critique of reason, so the mathematicians should practice a critique of their proofs. Up to this point, Hilbert’s view is shared by both Wittgenstein and Turing. Regarding their respective motivations, if Hilbert’s is to secure mathematical procedures, Wittgenstein’s seems to be more oriented towards enhancing ways of doing mathematics, i.e., proving, calculating, and experimenting. In some ways, as we shall see, Turing’s work acts as a hinge between the two.

Wittgenstein’s remark on the need of ratification for a proof continues as follows: “But if it got ratification from one person, but not from another, and they could not come to any understanding—would what we had here be calculation?” Then, he claims, “it is not the ratification by itself that makes it calculation but the agreement of ratifications” (RFM: V-6). And this agreement is the “pre-condition of our language-game”. Therefore, “if a calculation is an experiment and the conditions are fulfilled, then we must accept whatever comes, as the result;” the proposition that produces a certain result is the proposition that under such conditions this kind of sign comes out. Accordingly, when, under given conditions, a result appears at one moment and another result at another moment, we must not say, “there is something wrong here,” we should rather say “this calculation does not always yield the same result.” For Wittgenstein, to come to an understanding about the difference in the result of a calculation is to get a calculation free of discrepancies. And if this is not the case, then we cannot say that the one and the other are calculating, only with different results. Although the procedure here involved might be even more interesting, “what we have here now is no longer a calculation” (RFM V-7 (164e)). On the other hand, if a calculation is not seen as an experiment per se, but as a part of the technique of an experiment, one overlooks that a particular application is an integral part of the making of an experiment, and the calculation is an instrument of the application. Finally, it might not seem inappropriate to place in Wittgenstein’s next passage the gist of his discussion with Turing a few years earlier.

If calculation is to be practical, then it must uncover facts. And only experiment can do that. [...] Mathematics—I want to say—teaches you, not just the answer to question, but a whole language-game with questions and answers. (RFM V-14)

5. Imitation Game

After 1939, the destinies of Wittgenstein and Turing took different paths through the Second World War. As early as September 1939, when war broke out, Turing moved to Bletchley Park,12 where his work would prove decisive for the future of Europe. He played a crucial role in breaking the German Naval Enigma cipher. Wittgenstein, by contrast, worked as a dispensary porter at Guy’s Hospital. Once the war was over, Turing begins to focus on the design of a digital electronic computer, as a concrete form of the “universal Turing machine”. In 1947, in a lecture to the London Mathematical Society, he presents his project for an Automatic Computing Engine (ACE) as a work explicitly connected with his early investigations on the theoretical possibilities and limitations of computing machines.

It has been said that computing machines can only carry out the processes that they are instructed to do. This is certainly true in the sense that if they do something other than what they were instructed then they have just made some mistake. It is also true that the intention in constructing these machines in the first instance is to treat them as slaves... Up till the present machines have only been used in this way. But is it necessary that they should always be used in such a manner? [29] (pp. 392–393)

In 1948, the possibility for machinery to show intelligent behaviour is further explored in his Intelligent Machinery.13 Then, in his Computing Machines and Intelligence (1950), Turing proposes to consider the question “Can machines think?” in terms of what he calls “imitation game.” The game involves three players, a person, a machine, and an interrogator. The interrogator is required to determine, only based on the answers written by the other two players, which of them is the person and which is the machine [30] (p. 443). Now, if human and mechanical responses are “indistinguishable”, does this prove that the machine thinks? In what sense does Turing regard the ability to play the imitation game successfully as a “criterion for thinking” [28] (pp. 433–440)?

5.1. Where Is the Mind Hidden?

The idea that a machine can think or participate in that peculiar language-game that is the imitation game originates from the development of research into what digital computers can do. As Turing explains:

The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer. The human computer is supposed to be following fixed rules; he has no authority to deviate from them in any detail. We may suppose that these rules are supplied in a book, which is altered whenever he is put on to a new job. He has also an unlimited supply of paper on which he does his calculations. [30] (p. 435)

If the logic model of the Turing machine was drawn from human calculability, digital computers are now designed as “practical versions” of the universal Turing machine, a machine that can simulate the behavior of any (single-purpose) Turing machine. But then, if Turing machines are “humans who calculate”, are digital computers “humans who think”?

Among those who seem concerned by this conclusion is also Kurt Gödel. In his view, if Turing’s work were to be criticized from a philosophical point of view, the “error” would have to be traced to his 1936 essay. Indeed, while giving Turing credit for providing the “most satisfactory” definition of the concept of finite procedure [31] (pp. 304–305), Gödel points out an error in Turing’s conceptual analysis: that of having reduced mental procedures to mechanical procedures. Turing’s argument, however, seems not only insightful, but also correct. As mentioned above, since his aim was to identify the essential ingredients of a computation performed by a human agent and convert them into a logical model, his analysis relied on a preliminary exam of the limits of human capabilities involved in computing. These are observability limits: the number of symbols that a (human) calculator can observe at each stage of calculation is finite; and memory limits: the number of “states of mind”, which can be taken into account at each stage, is finite. If one admitted an infinity of symbols or states of mind, some of them will be “arbitrarily close” and will be confused, whereas symbols and states relevant to computation must be “immediately recognisable”. For Gödel, these limits apply to the “mind” only at a given stage of the procedure. Thus, Turing’s error would have been to consider the mind as a finite-state system.

What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. ... although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge towards infinity in the course of the application of the procedure. [32] (p. 306)

In Gödel’s opinion, the process of creating increasingly powerful axioms of infinity—as in set theory—might require something of that kind. Our understanding of infinitely complicated combinatorial relations may require an increasing number of ever more precise abstract terms. While the complexity of the states of a “human” mind might increase indefinitely due to the ever-increasing precision of abstract terms, this possibility would remain out of reach for the states of a “mechanical” mind. Putting aside the fascinating questions on the nature of mind and the potential infinity of its states, one crucial issue raised by Gödel’s passage is the difficulty of reconciling the abstractness and complexity of certain mathematical structures with mechanical procedures. But Turing had not disregarded that point: the “universal Turing machine” was somehow designed to cope with that kind of difficulties. Given enough time and memory, it can produce the output of any calculation performed by a single-purpose Turing machine. Here germinates Turing’s idea of creating a machine capable not only of calculating but also of “thinking”, of imitating a brain. Turing undoubtedly agrees with Gödel that some aspects of mathematical reasoning cannot be formalized [33] (pp. 192–193), but, in contrast with Gödel, he is inclined to consider “mental” processes as ultimately consequences of physical processes. In line with Wittgenstein, he regards the effectiveness of mathematical procedures as engaged with mathematical practice.

5.2. Learning from Machines

Considering the question “Can machines think?”, once it is established that “machines” are digital computers, what definition could appropriately reflect the normal use of the word “think”? The imitation game offers a way to approach the problem without having to give such a vexed definition. Turing is well aware of the reasons for a widespread attitude against the possibility for machines to show intelligent behaviour. “To behave like a brain seems to involve free will, but the behaviour of a digital computer, when it has been programmed, is completely determined” [34] (p. 484). How to reconcile mechanical behaviour and free will?

We may say then that in so far as a man is a machine he is one that is subject to very much interference. ... He is in frequent communication with other men, and is continually receiving visual and other stimuli which themselves constitute a form of interference. It will only be when the man is “concentrating” with a view to eliminating these stimuli or “distractions” that he approximates a machine without interference. [35] (p. 421)

What about a machine “with interference”? Could it be feasible? Even the universal Turing machine is a closed and deterministic system, which cannot interfere with the outside, nor intervene to change its initial settings. And yet, a machine which is supposed to imitate a brain must behave as if it had free will, and one may wonder how this can be achieved. A way to release machines from their condition of “slavery” may be to train them to imitate the intelligence of chance. The introduction of random elements into a machine’s instructions can render its behaviour indeterminate. This is also crucial in connection with learning machines. For a machine, as for a child, “learning from experience” means being able to go wrong and interfere with the outside world.

If the untrained infant’s mind is to become an intelligent one, it must acquire both discipline and initiative. So far we have been considering only discipline. To convert a brain or machine into a universal machine is the extremest form of discipline. Without something of this kind one cannot set up proper communication. But discipline is certainly not enough in itself to produce intelligence. That which is required in addition we call initiative. ... Our task is to discover the nature of this residue as it occurs in man, and to try and copy it in machines. [35] (p. 429)

Turing elaborates first on the idea of “random machines,” then on the idea of “unorganized machines” with interference.14 In his view, the rules that lead to the partial organization of an unorganized machine and to stability are determined by the interference produced by increments of information from outside. One could then have an unlimited progression from one Turing machine to another. The choice between different machines would result in a procedure that, combining discipline and initiative, represents the evolution of the rule system as a non-deterministic process. Connecting different machines could also be effective in the search for mathematical proofs.

If a machine is used as a “proof-finding machine”, different machines provide different methods for finding proofs. By choosing more and more suitable machines, one can approximate “truth” by “probability” as much as one likes.15 Seeing a mathematician as a proof-finding machine, Turing asks whether, when mathematicians discover new methods and techniques of proof by intuition, we might not consider them as “machines that learn from experience” and become competent in a broader set of proofs. A challenge would then be to devise a machine able to mimic the activity of intuition, by taking advantage of random elements in its search through proof-finding machines [28] (pp. 465–470). Generalising, insofar as intellectual activity consists of various kinds of search, it can be translated in a programme that can be executed by a machine; insofar as an unexpected result can be read as the work of chance, it can be the output of a “non-deterministic” machine. Nonetheless, it is well known that incompleteness and undecidability results impose limitations on the powers of any discrete-state machine. Hence Gödel’s claim that “the human mind (even within the realm of pure mathematics) infinitely surpasses the power of any finite machine” [31] (p. 310). Turing is more cautious:

although it is established that there are limitations to the powers of any particular machine, it has only been stated, without any sort of proof, that no such limitations apply to the human intellect. [30] (p. 451)

According to Turing, we should not attach too much importance to that genuine feeling of superiority we get from knowing that a definite answer given by a machine must be wrong. Our superiority may only be occasional, with respect to special cases of machine fallibility, but there would be no question of triumphing simultaneously over all machines. In short, “there might be men cleverer than any given machine, but then again there might be other machines cleverer again, and so on.” Now, while those who use a “mathematical argument” to dispute about thinking machines would most likely accept the imitation game as a basis for discussion, dealing with the “consciousness argument” may be more demanding. This argument can be expressed saying that only when a machine not just writes something—an answer, a sonnet, a piece of music—but “knows” it has written it, we may regard it as equal to a brain. Undoubtedly, no mechanism could “feel” pleasure or pain. Taking this view to its limits, however, the only way to be sure that a machine thinks is to be that machine and to feel oneself think. But this might also apply to a person: the only way to know that a person thinks is to be that person. “It may be the most logical view to hold,” Turing notes, “but it makes communication of ideas difficult.” This is not to say that there is no mystery about consciousness, but to ponder how relevant the solution of these mysteries is to the issue at hand.

A further objection to the claim that a machine can think is that “it can do whatever we know how to order it to perform.” This was Lady Lovelace’s objection to Babbage’s Analytical Engine, which Turing revises as: “a machine can never ‘take us by surprise’” [30] (p. 455). And yet, if a surprise involves some “creative mental act” on our part, the appreciation of something as surprising, Turing remarks, “requires as much of a ‘creative mental act’ whether the surprising event originates from a man, a book, a machine or anything else.” Moreover, the belief that machines cannot provide any new knowledge would rest on the false assumption that as soon as a fact is presented to a mind all its consequences spring into the mind simultaneously with it. As a natural conclusion, one should assume that there is no virtue in the mere working out of consequences from data and general principles.

Going through Turing’s answers to the various objections to the idea that machines can think, it hardly seems inappropriate to perceive the echoes of the arguments discussed during the Cambridge Lectures, as well as a certain underlying harmony with Wittgenstein’s attitude on the issue. The imitation game seems to show that what Wittgenstein states for meaning might apply to intelligence: there is no point in trying to explain what intelligence is, it appears in use. The game was not designed for testing machines’ cognitive abilities. Turing did not regard it as a “test” to be passed by the machine but as a means to question how intelligence can be discerned through answers. And the interesting result is when the questioner is unable to identify the author of the answers. If intelligence is considered as an essential human property, the inability to recognise its “owner” by playing the imitation game should cast doubts not so much on the machine’s intelligence as on our understanding of intelligence as a property. Ultimately, the value of Turing’s and Wittgenstein’s engagement with games lies in the deeper insights they provide into the very nature of intelligence and meaning. Like meaning, intelligence is not something that can be trapped in artificial cages and used “privately”. It arises from interference, creates connections, questions limits and rules, lives in action; in Wittgenstein’s words, “the use is its breath”.

6. Conclusive Remarks

Looking back at the various forms of games that intertwine in Wittgenstein’s thought and resonate across Turing’s work, how to assess the encounter between the Austrian philosopher and the British mathematician? As a clash, a convergence of views or a productive exchange of mutual insights?

From their discussion in the sixth Cambridge lecture, one can glean a reasonable suggestion. It hinges on the concept of “analogy” and originates from the problem of finding for the heptagon (a polygon classically not constructible with ruler and compass) a construction “analogous” to that of the pentagon. Assuming a drawing of the pentagon, how could the construction of the heptagon be derived from it by “analogy”? Wittgenstein presents two ways of finding an analogy in two drawings of the pentagon: (1) to draw a picture of a pentagon from the picture; (2) to draw the construction of a heptadecagon from the construction of the pentagon. In his opinion, the latter is not drawing something in a “different projection”, as “Turing might say”. It is not by following a rule for projecting (somehow on a different scale) that the mathematician achieved the construction of the heptadecagon from the picture, but rather by inventing a new mode of projection, i.e., by discovering a new kind of analogy. Doing something in a new way involves grasping a different kind of analogy, “seeing” the picture in a different way. The picture does not provide a rule for creating an “analogous” geometrical drawing. The point at issue is how the meaning of “analogous” is itself performative.

Turing: It certainly isn’t a question of inventing what the word “analogous” means; for we all know what “analogous” means.

Wittgenstein: Yes, certainly, it’s not a question merely of inventing what it is to mean. [...] The point is indeed to give a new meaning to the word “analogous”. (LFM, p. 66)

How? As Wittgenstein clarifies, “the new meaning must be such and such that we who have had a certain training will find it useful in certain ways.” In other words, the new meaning must be shared by the community of speakers, by the people capable of participating in the “language game”. Interestingly, he makes a comparison with the case of definitions: “Definitions do not merely give new meanings to words. We do not accept some definitions; some are uninteresting, some will be entirely muddling, others very useful, etc.” In this context—it is worth noting—Turing’s definition of computability itself could provide a case in point, as it reveals a new kind of analogy in the notion of “effective calculability”, allowing us to see computability differently. The example may be significant, even though Wittgenstein does not mention that definition and, most probably, neither he nor Turing were thinking of it at that moment. Indeed, their respective points of view may appear in conflict [5] (pp. 136–138). Asked whether he understood, Turing replies: “I understand but I don’t agree that it is simply a question of giving new meanings to words” (LFM, p. 67). Then, Wittgenstein gives a rather peculiar interpretation of what Turing thinks.16

Wittgenstein: Turing doesn’t object to anything I say. ... He objects to the idea he thinks underlies it. He thinks we’re undermining mathematics, introducing Bolshevism into mathematics. But not at all. We are not despising the mathematicians; we are only drawing a most important distinction—between discovering something and inventing something.

Although in this lecture Wittgenstein traces the distinction between discovering and inventing to two different types of analogy, he would later refine his thought. In his Philosophical Investigations, he points out two uses of the word “see” [12] (p. 203). When asked “What do you see there?”, one person answers “I see this”, the other “I see a likeness in these two faces”. The one can accurately draw two faces; the other can see a resemblance in the drawing that the former did not see. Between the two “objects” of sight there is a “categorial difference”. Looking at one face, we notice its similarity to another: the face has not changed, yet we see it differently. The two “objects” of sight may correspond to two ideas of “representation”: one closer to the pictorial theory of the Tractatus, built on the idea of language as a mirror; the other built through intermediate links, centered on the concept of “perspicuous representation” (übersichtlichen Darstellung) that conveys a kind of understanding as a seeing just the connections. “Hence the importance of finding intermediate links” [36] (p. 46).

Although there is no explicit evidence of the influence of Wittgenstein’s philosophy on Turing’s work, an insightful connection, nevertheless, might be outlined between Wittgenstein’s stance on the two types of analogy, or the two uses of the word “see”, and Turing’s efforts to discover the nature of that residue called “initiative”, which, in addition to discipline, he believed was required to produce intelligence. To copy initiative in machines is not reducible to drawing a picture of it; it involves enabling machines to see similarities and elaborate on them. Thus, the challenge would be to discover what allows a person to see “structural similarities” or to find “intermediate connections” and try and copy it in machines. Mental effects are not called into play, Wittgenstein and Turing would certainly agree on this.