Suppose I say to Turing, “This is the Greek letter sigma,” pointing to the sign σ. Then when I say, “Show me a Greek sigma in this book,” he cuts out the sign I showed him and puts it in the book.—Actually these things don’t happen. These misunderstandings only immensely rarely arise—although my words might have been taken either way. This is because we have all been trained from childhood to use such phrases as “This is the letter so-and-so” in one way rather than another.
When I said to Turing, “This is the Greek sigma,” did he get the wrong picture? No, he got the right picture. But he didn’t understand the application.
One of Wittgenstein’s ambitions was to compel his students to recognize the importance of common sense even in philosophical inquiry. (“Don’t treat your common sense like an umbrella,” he told them. “When you come into a room to philosophize, don’t leave it outside but bring it in with you.”) Nor was it accidental that of all the participants in the seminar, it was Turing he singled out, time and again, to serve as the representative of what might be called the logicist position; Wittgen-stein was, in his own words, always trying to “tempt” Turing toward making claims that favored logic over common sense (though not always with success). As a practicing mathematician, Turing could be counted on to reiterate the traditional postulates of his discipline and in so doing give Wittgenstein the opportunity to pull the rug out from under them. Church, or someone like him, would have made a more convenient whipping boy, and had Wittgenstein known more about the unorthodoxy of some of Turing’s ideas, he might have taken a different tack. Instead, he assigned Turing to the role of the naysayer, as this bit of dialogue from the sixth lecture shows:
Turing [asked whether he understood]: I understand but I don’t agree that it is simply a question of giving new meanings to words.
Wittgenstein: Turing doesn’t object to anything I say. He agrees with every word. 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. But mathematicians make most important discoveries.
This was just one of many instances, during the lectures, when Wittgenstein took on, dismantled, and in a certain sense recontextualized the arguments underlying pure mathematics, including the old debate as to whether mathematics was invention or discovery.* Though he was far from being an antirealist or an intuitionist of the Brouwer school, he insisted that his students question even the most fundamental axioms of arithmetic. (“We say of a proof that it convinces us of a logical law.—But of course a proof starts somewhere. And the point is: What convinces us of the primitive propositions on which the proof is based? Here there is no proof.”) In Wittgenstein’s rooms, the most basic assumptions were subject to scrupulous analysis:
What is counting? Pointing to things and saying “1, 2, 3, 4”? But I need not say the numbers: I could point and say “Mary had a little lamb” or I might whistle “God Save the King” or anything.—But normally the process of counting is used in a different way, whereas “Mary had . . .” is not used in this way at all. If you came from Mars you wouldn’t know.
By the same token, for someone from Mars, an axiom of mathematics might be “that whenever numerals of more than five figures cropped up . . . they were thrown away and disregarded.” Or that 5(6)4 is the same as (56)4. The point was to interrogate the relationship between the mathematical meaning of words and their “ordinary” meaning, and to expose the consequences of divorcing one from the other. For example, in the third lecture, he imagined a wallpaper factory in which the pattern on the paper consisted of the proof that 21 × 36 = 756 repeated over and over. “You might call this figure the proof that 21 × 36 = 756,” Wittgenstein told his students,
and you might refuse to recognize any other proof. Why do we call this figure a proof?
Suppose I train the apprentices of wallpaper manufacturers so that they can produce perfect proofs of the most complicated theorems in higher mathematics, in fact so that if I say to one of them “Prove so-and-so” (where so-and-so is a mathematical proposition), he can always do it. And suppose that they are so unintelligent that they cannot make the simplest practical calculations. They can’t figure out if one plum costs so-and-so, how much do six plums cost, or what change you should get from a shilling for a twopenny bar of chocolate.—Would you say that they had learnt mathematics or not?
They know all the calculations but not their applications. So one might say, “They have been taught pure mathematics.”
In short, though the apprentices “would use the words ‘proof,’ ‘equals,’ ‘more,’ etc., in connexion with their wallpaper designs, . . . it would never be clear why they used them.” On the other hand, “if it were said, ‘The proof of Lewy’s guilt is that he was at the scene of the crime with a pistol in his hand’—what is the connexion between this and calling the [wallpaper] pattern a proof? They wouldn’t know why it was called a proof.”
Wallpaper hangers, soldiers and generals, white lions, collapsing bridges: Wittgenstein was always presenting analogies in his lectures, asking his students to “suppose” one thing or another. While Wittgenstein could jump easily from one analogy to the next, however, Turing tended to cling tenaciously to the examples as if, by virtue of their simple iteration, they had taken on a kind of physical reality for him. In the transcripts of the course, no other student responds to Wittgen-stein so frequently, or so readily, as Turing does, and in many cases his responses amount to proposing extensions of the analogy with which Wittgenstein has begun. On the example of the wallpaper hangers, for instance, Turing says, “The ordinary meanings of words like ‘three’ will come out to some extent if they are able to do simple things like counting the numbers of symbols in a line.” Likewise the examples that Turing himself gives suggest the degree to which he was becoming more and more preoccupied with the relationship of logic to events in the real world:
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 a way, and see which way the lever swings,” and in the other case you say, “Take these figures, look up in such-and-such tables, 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.
Wittgenstein: Does one want to see that? In the mathematical case, does one want to see what chalk mark the man makes? Surely there is something queer about this.—Does one want to see what he will get if he multiplies, or what he will get if he multiplies correctly—what the right result is?
Turing: One can never know that one has not made a mistake.
It was as characteristic of Wittgenstein to seize on the chalk mark as it was of Turing to lay emphasis on the correlation between figures looked up “in such-and-such tables” (perhaps by a Turing machine) and the direction in which a lever swung. The odd thing was, for all the head butting that they did, both were fundamental pragmatists in a way that Alonzo Church could never have been. Turing might play the role of the defender of logic as a “clean and gentle” discipline, remote from the ugliness of human endeavor and human conflict, yet his own imagination was taking him as far from Hardy’s idealism as Wittgenstein’s. At the same time, he was not prepared to accept Wittgenstein’s dismissal (for example) of the liar’s paradox, which lay at the root of Turing’s investigations into the Entscheidungsproblem, as nothing but a “useless language-game”: “If a man says ‘I am lying’ we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you were black in the face. Why not? It doesn’t matter.” For Turing, it did matter—not in some abstract or ideal sense but because he believe
d that hidden contradictions could result in things “going wrong.” Their argument extended over the course of the entire year and reached its culmination in a long discussion about the role of the law of contradiction in logic and (again) “ordinary” life:
I may give you the rules for moving chessmen without saying that you have to stop at the edge of the chessboard. If the case arises that a man wishes to make a piece jump off the chessboard, we can then say, “No, that is not allowed.” But this does not mean that the rules were either false or incomplete.—Remember what was said about counting. Just as one has freedom to continue counting as one likes, so one can interpret the rule in such a way that one may jump off the board or in such a way that one may not.
But it is vitally important to see that a contradiction is not a germ which shows general illness.
Turing: There is a difference between the chess case and the counting case. For in the chess case, the teacher would not jump off the board but the pupil might, whereas in the counting case we should all agree.
Wittgenstein: Yes, but where will the harm come?
Turing: The real harm will not come in unless there is an application, in which case a bridge may fall down or something of that sort.
The bridge came up (and fell down) again and again. Just as Turing was adamant that Wittgenstein should admit the possibility of a bridge disaster brought on by the misapplication of “a logical system, a system of calculations,” Wittgen-stein was adamant that Turing should draw a distinction between the world of logic and the world of bridge building. Thus to Turing’s repeated assertion that “practical things may go wrong if you have not seen the contradiction,” Wittgenstein replied,
The question is: Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc., outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics? Turing says, “Because something may go wrong with the application.” But nothing need go wrong. And if something does go wrong—if the bridge breaks down—then your mistake was of the kind of using a wrong natural law.
Yet Turing would not forget the bridge. It was as if, sitting in Wittgenstein’s rooms, he could see it collapsing, hear the cries of the pedestrians as they fell into the river. His point was simple, and he would not let it go: Turing, in his own words, objected “to the bridge falling down.”
Wittgenstein: But how do you know that it will fall down? Isn’t that a question of physics? It may be that if one throws dice in order to calculate the construction of the bridge it will never fall down.
Turing: If one takes Frege’s symbolism and gives someone the technique of multiplying in it, then by using a Russell paradox he could get a wrong multiplication.
Wittgenstein: This would come to doing something which we would not call multiplying. You give him a rule for multiplying; and when he gets to a certain point he can go in either of two ways, one of which leads him all wrong.
Suppose I convince Rhees of the paradox of the Liar, and he says, “I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2 × 2 = 369.” Well, we should not call this “multiplication”; that is all.
It is as if you give him rules for multiplying which lead to different results—say, in which a × b ≠ b × a. That is quite possible. You have given him this rule. Well, what of it? Are we to say that you have given him the wrong calculus?
Turing: Although you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere.
Wittgenstein: But nothing has ever gone wrong that way yet. And why has it not?
Wittgenstein appears to have been not unsympathetic to Turing. Indeed, he went to great lengths to make sure that Turing felt he had his say. Yet his impatience was visceral, and obviously exacerbated by Turing’s stubborn refusal to distinguish between the collapse of logic and the collapse of a bridge:
[To Turing] Before we stop, could you say whether you really think that it is the contradiction which gets you into trouble—the contradiction in logic? Or do you see that it is something quite different?—I don’t say that a contradiction may not get you into trouble. Of course it may.
Turing: I think that with the ordinary kind of rules which one uses in logic, if one can get into contradictions, then one can get into trouble.
Wittgenstein: But does this mean that with contradictions one must get into trouble?
Or do you mean the contradiction may tempt one into trouble? As a matter of fact it doesn’t. No one has ever yet got into trouble from a contradiction in logic. [It is] not like saying “I am sure that that child will be run over; it never looks before it crosses the road.”
If a contradiction may lead you into trouble, so may anything. It is no more likely to do so than anything else.
Turing: You seem to be saying that if one uses a little common sense, one will not get into trouble.
Wittgenstein: No, that is NOT what I mean at all.—The trouble described is something you get into if you apply the calculation in a way that leads to something breaking. This you can do with any calculation, contradiction or no contradiction.
What is the criterion for a contradiction leading you into trouble? Is it specially liable to lead you into trouble?
It cannot be a question of common sense; unless physics is a question of common sense. If you do the right thing by physics, physics will not let you down and the bridge will not collapse.
At one point in the course of the lectures, citing Hilbert, Wittgenstein insisted that his purpose was not to drive his students “out of the paradise which Cantor has created”; rather, it was to compel them to question whether that paradise was really worth staying in in the first place:
I would say, “I wouldn’t dream of trying to drive anyone out of this paradise.” I would try to do something quite different: I would try to show you that it is not a paradise—so that you’ll leave of your own accord. I would say, “You’re welcome to this; just look about you.”
Turing, however, had long since abandoned that paradise—which made Wittgenstein’s insistence that he play the role of its defender doubly ironic: once again, he was compelled to dress up as someone he was not, and to wear the mantle of a thesis in the validity of which he did not ultimately believe.
3.
Wittgenstein was fond of battle metaphors. “Suppose I am a general and I receive reports from reconnaissance parties,” he asked in the twenty-first lecture. “One officer comes and says, ‘There are 30,000 enemy,’ and then another comes and says, ‘There are 40,000 enemy.’ Now what happens, or what might happen?”
The imminence of war was clearly on Wittgenstein’s mind. “Suppose I am a general and I give orders to two people,” he proposed in the next lecture.
Suppose I tell Rhees to be at Trumpington at 3:00 and at Grantchester at 3:30, and I tell Turing to be at Grantchester at 3:00 and to be at Grantchester at the same time as Rhees. Then these two compare their orders and they find “That’s quite impossible: we can’t be there at the same time.” They might say the general has given contradictory orders.
Turing never received orders to go to Grantchester. Instead, on September 4, 1939, he reported to Bletchley Park, a stately pile in Buckinghamshire, about fifty miles northwest of London. Bletchley Park—or B.P., as it came to be known—had begun its life around the time of the Battle of Hastings, and was a fairly modest red-brick farmhouse until Sir Herbert Leon, a London financier, purchased the estate in 1883. Wanting a mansion grand enough to suit his wealth, Sir Herbert made numerous additions to the building, including an ice house, an entrance hall, a library, and a ballroom. Less judiciously, he added a brick-and-stone façade in what the historian Stephen Budiansky calls “a sort of Victorian mock-Tudor” style, replete with “arches, pillars, gables, domes, and parapets. . . . The interior was equally overdecorated in a
n unsettling combination of carved oak and red plush.” So hideous was Bletchley Park that in an essay entitled “Architecture and the Architect,” David Russo gives it as an example of what not to do when designing a house, noting that “even to the untrained eye the structure seems to consist of a variety of forms as though it was not built as a whole but rather built in parts that were later juxtaposed by whim. . . . The resultant building . . . appears to be part castle, part turreted Indian gazebo overlaid with a variety of styles ranging from Romanesque gates to neo-Norman pediments.”*
Architectural integrity, however, was not on the mind of Admiral Quex Sinclair (better known as “C”) when he bought the house to serve as the base for the continuing activities of the General Code and Cipher School during the Second World War. What attracted him to the property was its spaciousness, its accessibility to London, and its situation exactly midway between Oxford and Cambridge on the railway line then connecting the two universities. Already it was obvious to C that on the cryptanalysis front, at least, the war was going to be fought by intellectuals.
Turing arrived at Bletchley as part of a corps of scientists and mathematicians traveling under the guise of “Captain Ridley’s Shooting Party.” Among the other members of the group were two fellow Cambridge mathematicians, Gordon Welchman (who later wrote the first memoir of Bletchley Park) and John Jeffreys. Along with a fourth mathematician, Peter Twinn, they were given space in a low building not far from the main house called the Cottage, where they settled down to the task of breaking Germany’s Enigma code. Turing probably imagined that his stay at Bletchley would last for a few months; in fact the Crown Inn in the village of Shenley Brook End, where he was billeted and from whence he bicycled each day to Bletchley, was to become his home for the duration of the war.
The Man Who Knew Too Much: Alan Turing and the Invention of the Computer (Great Discoveries) Page 13
