Intuition, Decision, Compulsion

Severin SCHROEDER Intuition, Decision, Compulsion 1 Intuition or Decision? “What you are saying, then, comes to this: a new insight—intuition—is needed at every step to carry out the order ‘+n’ correctly.”—… It would almost be more correct to say, not that an intuition was needed at every stage, but that a new decision was needed at every stage.1 Is Wittgenstein saying in this passage that every application of a rule requires a new decision? No, he is not. For one thing, ‘almost F’ doesn’t mean ‘F’; on the contrary, what is almost F is not (strictly speaking) F. For another thing, ‘more correct’ doesn’t mean ‘correct’, just as ‘closer to the truth’ doesn’t mean ‘true’. It is a common rhetorical device to reject a spectacularly false statement by the claim that even its contrary opposite, although easily recognized as false too, would be less absurd than the statement in question. And yet, the view that according to Wittgenstein rule-following requires a new decision at every stage is remarkably widespread. Michael Dummett describes how he thinks Wittgenstein applies ‘the considerations about rules presented in the Investigations and elsewhere’ to mathematical proofs: at each step we are free to choose to accept or reject the proof … If we accept the proof, we confer necessity on the theorem proved … In doing this we are making a new decision, and not merely making explicit a decision we had already made implicitly.2 And again: [Wittgenstein] appears to hold that it is up to us to decide to regard any statement we happen to pick on as holding necessarily, if we choose to do so.3 || 1 Wittgenstein, PI, § 186. 2 Dummett 1959, pp. 495–6. 3 Dummett 1959, p. 500. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM 26 | Severin Schroeder More recently such a decisionistic reading of the rule-following considerations has been propounded by Pasquale Frascolla who (in a generally excellent book) attributes to Wittgenstein the astonishing view that: what makes [1000 + 2 = 1002] is only our spontaneous decision, which is agreed upon by all those who have had a certain training and possess certain linguistic inclinations, that establishes it. What one is dealing with, in fact, is the creation ex novo of a conceptual connection, which could not be derived in any way from the concept of the rule and from the concept of the number 1,000 such as we know them before the decision is taken.4 In the first part of this paper I shall take a closer look at Wittgenstein’s thoughts on intuition and decision in following a rule, focussing in particular on §3 of Remarks on the Foundations of Mathematics ‘Part 1’ (TS 222). In that remark (RFM I-3), Wittgenstein presents a version of the question that underlies the considerations of the preceding two sections, a variant of the puzzle raised by the case of the deviant pupil (of PI § 185), namely: ‘How do I know that in working out the series + 2 I must write “20004, 20006” and not “20004, 20008”?’. Before, the key term was ‘determine’: How can a formula determine certain steps? The epistemological paraphrase ‘How do I know given a formula what steps to take?’ makes it clearer what kind of determination the question is looking for: The underlying idea is that a formula determines certain steps by telling us what steps to take. The formula is to contain the information, to give us the knowledge as to what steps to take. Elsewhere Wittgenstein makes the idea of determination through knowledge explicit: What must I know in order to be able to carry out the order? Is there some knowledge that makes the rule followable only in this way?5 But then, as the scenario of the deviant pupil illustrates, the formula doesn’t appear to contain the information required. And indeed it is hard to see how it could. After all, the series + 2 contains infinitely many steps; yet how could a laconic order or formula contain an infinite amount of knowledge? The second paragraph of RFM I-3 attempts an answer. It is only one piece of information that the formula needs to convey, namely the principle of adding 2. Once you have understood that, you know that it involves only a small number of modifications from one number to the next in the series: In the units you || 4 Frascolla, 1994, p. 135. In this and other passages Frascolla appears to set aside his own earlier observation that Wittgenstein’s use of the word ‘decision’ is ‘always accompanied by certain warnings’ or qualifications. Frascolla, 1994, p. 117. 5 Wittgenstein, RFM, 341. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Intuition, Decision, Compulsion | 27 change 2 to 4, 4 to 6, 6 to 8, or 8 to 0, and whenever you make a change to 0 you also move one up in the next digit to the left. Wittgenstein’s reply is that this doesn’t help, as the philosophical problem applies even to the simplest algorithm, even to the instruction always to repeat the same number: 2, 2, 2, 2… ‘For how do I know that I am to write “2” after the five hundredth “2”?’ We may want to reply that, surely, the instruction is clear enough: we just keep writing the same figure. But Wittgenstein objects that the problem is to know what at any given point counts as ‘the same figure’. In a parenthesis at the end of the first paragraph he compares that question to the question ‘How do I know that this colour is “red”?’. Again, one could imagine the reply that ‘red’ is called whatever is of the same colour as a given sample of red, e.g. a ripe tomato. But Wittgenstein’s concern would be that we’d still need to know what counts as ‘of the same colour’ in a given situation. And we can imagine our criteria for colour identity to be less than straightforward. They might, for example, involve a reference to the time of day, so that what counts as ‘red’ in the morning, say, would not be called ‘red’ at night. Or again, we might find it natural to apply certain concepts alternatingly in different ways: ‘that when we make this transition one time, the next time, “just for that reason”, we make a different one, and therefore (say) the next time the first one again’.6 This may sound far-fetched, but it highlights the point that, logically speaking, there always remains a gap between a general instruction (applicable to an indefinite number of instances) and its execution. That is to say that although, of course, in some sense an instruction can be said to determine some answers as correct and others as incorrect, this determination is not foolproof. Wittgenstein sets his face against a certain philosophical picture of such a determination according to which the correct answer is somehow already contained in the instruction. This was a view he found in Frege and Russell: In his fundamental law Russell seems to be saying of a proposition: “It already follows— all I still have to do is, to infer it”. Thus Frege somewhere says that the straight line which connects any two points is really already there before we draw it; and it is the same when we say that the transitions, say in the series + 2, have already been made before we make them orally or in writing—as it were tracing them.7 || 6 Wittgenstein, RFM I, § 155. 7 Wittgenstein, RFM I, § 21. Cf. G. Frege, (1884) 1953, p. 24: ‘Each [axiom of geometry] would contain concentrated within it a whole series of deductions for future use’. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM 28 | Severin Schroeder Wittgenstein insists that that is only a picture, a metaphor.8 Admittedly, giving someone a formula by which to calculate a series of 50 numbers can be just as reliable as asking him to copy an existing list of those numbers; nevertheless the former case is not one of copying or tracing what is already there. What can be produced does not, for that matter, exist already as a shadow in some Platonic underworld; not really.9 And even if it did —: ‘how would it help?’10 Even if the case were one of copying, we could still imagine a pupil doing it in an aberrant way—for instance, leaving out every tenth number11—, thinking that was the correct method. The correct answers do no follow mechanically from the rule or formula. Whenever we take the step from the general rule to an individual application, we have to take the rule in a certain way. Hence one could say that following a rule ‘always involves interpretation’.12 Not, to be sure, in the substantive sense of verbally giving it an interpretation: ‘substituting one expression of the rule for another’,13 but in the minimal sense of applying the rule in one way rather than another. At one point, in 1929, Wittgenstein was inclined to think that whenever we proceed to apply a general rule in an individual case—as the latter is not actually already contained in the former, but requires us to go beyond it—we need a new insight or intuition: Supposing there to be a certain general rule (therefore one containing a variable), I must recognize each time afresh that this rule may be applied here. No act of foresight can absolve me from this act of insight. Since the form to which the rule is applied is in fact different at every step.14 This Wittgenstein took to be an implication of the intuitionist view of mathematics, as he explained in a 1939 lecture: Intuitionism … requires that we have an intuition at each step in calculation, at each application of a rule; for how can we tell how a rule which has been used for fourteen steps applies at the fifteenth?—And [the intuitionists] go on to say that the series of cardinal || 8 Wittgenstein, RFM I, § 22. 9 Cf. Wittgenstein, PG, 281d. 10 Wittgenstein, PI, § 219. 11 Cf. Wittgenstein, PI, § 86. 12 Wittgenstein, RFM I, § 114. 13 Wittgenstein, PI, § 201. 14 Wittgenstein, PR, 171. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Intuition, Decision, Compulsion | 29 numbers is known to us by a ground-intuition—that is, we know at each step what the operation of adding 1 will give.15 Wittgenstein seems to have encountered intuitionist ideas in Hermann Weyl’s Philosophy of Mathematics and Natural Science (1927) and ‘Die heutige Erkenntnislage in der Mathematik’ (1925), which he mentions in conversation with Schlick and Waismann.16 And on 10th March 1928 he attended a lecture given by L.E.J. Brouwer, the founder of intuitionism, in Vienna on ‘Mathematics, Science and Language’. Herbert Feigl remembered that while Wittgenstein had been very reluctant to discuss philosophy with members of the Vienna Circle before, after the lecture he felt inspired to do so very volubly and at great length. It was as if Brouwer’s intuitionist ideas had given Wittgenstein some impulse to return to philosophy.17 According to L.E.J. Brouwer, ‘the falling apart of moments of life into qualitatively different parts’, the intuition of two-oneness, or two-ity, is ‘the basal intuition of mathematics’: It creates not only the numbers one and two, but also all finite ordinal numbers, inasmuch as one of the elements of the two-oneness may be thought of as a new two-oneness, which process may be repeated indefinitely.18 On this account, every step in a series is an introspective construction that we experience as a self-evident truth.19 That means that such mathematical constructions, while being human artefacts, are nonetheless true, the result of a series of insights. And the series of natural numbers, as the most straightforward development of the basic intuition of two-ity, occupies a privileged and fundamental position.—This is one aspect of Brouwer’s idea of the ‘basic intuition’ that Wittgenstein disputed, arguing that our series of natural numbers was not more correct than the alternative number series ‘1, 2, 3, 4, 5, many’,20 or even a system of cardinal numbers lacking the 5.21 More important, however, in this context is the idea that the basic intuition of two-ity is to be iterated for every new step of the number series. Every act of following the rule + 1 is to be based on a new insight of how it applies to a given || 15 Wittgenstein, LFM, p. 237; cf. Wittgenstein, WA I, p. 101. 16 Wittgenstein, WVC 37, pp. 81ff. 17 Feigl 1986, p. 64; Marion, 2003, pp. 104–5; cf. Hacker, 1986, pp. 120–8. 18 Brouwer, 1912, p. 69. 19 Cf. Brouwer, 1940. 20 Wittgenstein, VW, pp. 66–9. 21 Wittgenstein, BT, pp. 570-1; cf. Wittgenstein, LFM, pp. 82–3. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM 30 | Severin Schroeder number. Although initially Wittgenstein was inclined to think so himself, he soon came to reject this view. ‘If intuition is needed to continue the series + 1, then it is also needed to continue the series + 0’;22 but it isn’t. Returning to his earlier remark quoted above: ‘No act of foresight can absolve me from this act of insight [for every new step]’, he wrote in the margin: ‘Act of decision, not of insight’.23 The same point is elaborated in The Brown Book: It is no act of insight, intuition, which makes us use the rule as we do at the particular point of the series. It would be less confusing to call it an act of decision, though this too is misleading, for nothing like an act of decision must take place, but possibly just an act of writing or speaking. … We need to have no reason to follow the rule as we do. The chain of reasons has an end.24 To know something ‘by intuition’ means: to know it immediately, without reasoning, as if by an act of direct perception. You just see that something is the case, though not literally, but with your mind’s eye. (Brouwer also speaks of ‘introspection’.) However, you can only have a (true) intuition that something is the case, if indeed it is the case. The object of an intuition must be an objective fact and hence ascertainable independently of someone’s intuition; just as the object of one’s visual experience—if it is not a hallucination—must have an independent existence that can be ascertained in other ways (e.g. by touch). Normally, when we say that somebody had an intuition that something is the case, we know of another, more pedestrian way of recognizing the matter. Thus, you can be said to know by intuition that 27 × 177 = 4779 if you can immediately produce the answer that others derive by calculation.25 In short, what is known by intuition—without reasoning (or evidence, or sense perception)—must also be ascertainable by reasoning (or on the basis of evidence or through sense perception). In the case in hand, however, it is not only that an answer is given immediately and without reasoning; the important point is that ultimately no reason can be given. That is to say, the reason why you write 2002 after 2000 is of course: that you are applying the rule + 2; but when you are questioned further: why when applying the rule + 2 you write 2002 after 2000, you can only say that that’s what adding 2 requires at this point. Your reasons soon give out; and then you act without reasons.26 It is not only an immediate apprehension of || 22 Wittgenstein, RFM I, § 3c; cf. Wittgenstein, PI, § 214. 23 Wittgenstein, PR, 171. 24 Wittgenstein, BB, 143; cf. Wittgenstein,PI, § 186. 25 Wittgenstein, LFM, 30. 26 Wittgenstein, PI, § 211. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Intuition, Decision, Compulsion | 31 something that can also be established or justified by reasons; in the end there just are no more reasons. And that means that the concept of an intuition (although tempting to invoke) is out of place. In another passage Wittgenstein suggests that if from the notion of an intuition we remove the metaphor of an inner perception, what it boils down to is: ‘guessing correctly’.27 That is perhaps not entirely accurate, as the experience of an intuition would more likely be one of confidence and understanding, rather than that of a mere guess. But the crucial point remains that correctness can never be guaranteed by an experience of being right, however confident. Getting it right requires an independent criterion of correctness. It is an illusion to think that the lack of such a criterion can be compensated by an intense experience of assurance and clarity. Wittgenstein rejects the talk of intuitions of how to continue a series as just another version of the Frege-Russell idea that the steps to be taken are, in some shadowy sense, already taken. In order for something to be seen by intuition it must already be there. In this respect the word ‘intuition’ is akin to ‘discovery’.28 As an antidote to the idea of an intuition guiding our steps in following a rule, Wittgenstein suggests that it would be closer to the mark to speak of a decision. To say that I have to decide to write 2002 after 2000 (in following the rule + 2) makes it clear that the number cannot be read off anywhere: that the application of the rule does not already exist in some Platonic realm. Rather, I have to take full responsibility myself for producing it. Already in a 1936 lecture, Wittgenstein indicated that his use of the term ‘decision’ was somewhat hyperbolical: a poignant way of bringing out that in some sense it is like taking a decision: ‘I can’t give reasons ad infinitum’.29 In 1939 he explained more clearly that it was also not correct to speak of a decision: We might as well say that we need, not an intuition at each step, but a decision.—Actually there is neither. You don’t make a decision: you simply do a certain thing. It is a question of a certain practice.30 In MS 164 (1943/44), once again Wittgenstein feels inclined to say that ultimately in applying a rule we take a ‘spontaneous decision’; but again, he ex- || 27 Wittgenstein, RFM, pp. 235f. 28 Wittgenstein, LFM, p. 82. 29 Wittgenstein, PO, p. 354. 30 Wittgenstein, LFM, p. 237. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM 32 | Severin Schroeder plains that to mean simply: ‘that’s how I act; ask for no reason!’.31 And he adds the disclaimer: When I say “I decide spontaneously”, naturally that does not mean: I consider which number would really be the best one here and then plump for...32 That is why to speak of a decision is ultimately misleading: for it wrongly suggests (and it did suggest to Dummett) that one chooses freely, to write, say, 2002. In fact, in following the rule I feel compelled to write this number.33 ‘When I obey a rule, I do not choose’.34 Where I choose, I do not follow a rule.35 There is no denying that I ‘must’ write this number; it’s only that, when pressed beyond the first explanation, I cannot say what compels me: ‘I can give no reason’.36 There is a danger when reading Wittgenstein to mistake his rejection of a natural philosophical explanation of a given phenomenon for a rejection of the phenomenon itself. It is tempting to think of following a rule as tracing steps that, in some way, have already been taken—be it objectively, in some Platonic realm, or subjectively, in some mysterious act of meaning. This way of thinking of rules can be so deeply rooted in one’s mind that Wittgenstein’s objections to it sound like an attack on the very possibility of following a rule. Thus it can appear that if in following a rule the step to be taken at a given point cannot be intuited, cannot be perceived—it doesn’t exist. It would appear that there is no correct answer to the question of how to apply the rule (which would of course mean that there was no such thing as following a rule.37 And this radically destructive reading would seem to be supported by Wittgenstein’s use of the word ‘decision’ in this context, especially if one doesn’t pay attention to the exact wording, which presents it as right in only one respect, but wrong in another. The following paragraph (d) of RFM I–3 makes it clear that Wittgenstein had no such absurdly radical consequences in mind. It is perhaps the most elegant account of (the core of) Wittgenstein’s solution to the rule-following puzzle: || 31 Wittgenstein, RFM, p. 326. 32 Wittgenstein, RFM, p. 326. 33 Wittgenstein, PI, § 231. 34 Wittgenstein, PI, § 219. 35 Wittgenstein, RFM, p. 413d. 36 Wittgenstein, RFM, p. 326. 37 Cf. Schroeder, 2006, pp. 187–201. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Intuition, Decision, Compulsion | 33 “But do you mean to say that the expression ‘+ 2’ leaves you in doubt what you are to do e.g. after 2004?”—No; I answer “2006” without hesitation. But for that very reason it is superfluous to suppose that that this was determined earlier on. My having no doubt in face of the question does not mean that it has been answered in advance.38 Our initial inclination was to think that in a straightforward case of rulefollowing, such as + 2, where there is never any doubt about the next step, it must somehow have been determined in advance: (1) Transition is not doubtful  transition has been determined in advance. For it seemed that in order to be certain of a transition at a given point we must somehow be able to ascertain that that transition has been laid down as correct. When more careful consideration shows that we don’t in fact find the correct answer laid down in advance, one may be inclined to argue from (1) by modus tollens that we cannot be certain of it. That is the response given by the first sentence of RFM I–3d. Yet Wittgenstein insists on the truth of the antecedent (which it would be absurd to deny), but rejects his interlocutor’s conditional, suggesting a different one instead: (2) Transition is not doubtful  transition need not be determined in advance. That of course raises the question of how we can be so certain of the application of a rule at a given point if that application has not been laid down anywhere in advance. The answer was already given in RFM I–1a and is repeated more explicitly in RFM I–22: There is an indirect way of determining the development of a series, without specifying each individual step in advance, namely through training. Thus, for instance, children get training ‘in the multiplication tables and in multiplying, so that all who are so trained do random multiplications (not previously done in the course of being taught) in the same way and with results that agree’; and the same holds for the series + 2.39 As we were reminded in the first section, a formula determines a series of transitions only for those who have been trained to use it in a certain way. And this training provides a determination that is not mediated by any theoretical knowledge. Having mastered the technique of adding 2, I know at each point of the series what to write next. But the question ‘How do you know?’ (from the beginning of RFM I–3) cannot be answered. My knowledge is a practical certainty, based on training, || 38 Wittgenstein, RFM, I, § 3d. 39 Wittgenstein, RFM, I, § 22. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM 34 | Severin Schroeder not based on reasons. In the same way, our mastery of colour words is not mediated by theoretical knowledge.40 Why do I call that colour ‘red’? I cannot give a reason, I can only cite a cause: ‘I have learned English’.41 2 Logical Compulsion Wittgenstein does not regard following a rule as freely deciding what to do— which would amount to denying the very possibility of being guided by a rule. And yet, does he not reject the view that when following a rule we are logically compelled to go one way rather than another? The deviant pupil who systematically misunderstands our explanations is a recurring theme in Wittgenstein. If he insists on a perverse application of the formula, what can we do? It would appear that we cannot force him to agree with us. That makes it sound as if Wittgenstein wanted to deny logical compulsion: “But am I not compelled, then, to go the way I do in a chain of inferences?”—Compelled? After all I can presumably go as I choose!—“But if you want to remain in accord with the rules you must go this way.”— Not at all, I call this ‘accord’.—“Then you have changed the meaning of the word ‘accord’, or the meaning of the rule.”—No;—who says what ‘change’ and ‘remaining the same’ mean here?42 However many rules you give me—I give a rule which justifies my employment of your rules. The inverted commas in this dialogue seem to suggest that an imagined interlocutor tries to defend the idea of logical compulsion, whereas Wittgenstein himself champions a radical idea of freedom. (Such was Dummett’s impression when he asked uncomprehendingly: ‘whence does a human being gain a freedom of choice in this matter’?43 Yet in fact, what is at issue here is not logical compulsion, but only a wrong idea of it: namely that a formula, or a sequence of words, can compel me to understand and apply it in a certain way. Of course it cannot. There is no causal necessity by which rules or laws of inference could compel someone ‘to say or to write such and such like rails compelling a loco- || 40 Wittgenstein, RFM, I , § 3a. 41 Wittgenstein, PI, § 381. 42 Wittgenstein, RFM, I, § 113. 43 Dummett, 1959, p. 496. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Intuition, Decision, Compulsion | 35 motive’.44 And even if I want to remain in accord with a given formula I can do all sorts of things as long as I interpret the formula accordingly. In short, what this section45 rejects is not logical compulsion, but what one could call hermeneutic compulsion: by which one would be coerced to interpret a formula in a certain way. What then is logical compulsion? Shall we say: given a certain understanding of an expression, we are forced to accept that it has certain implications? For example: “From ‘all’, if it is meant like this, this must surely follow!”. Wittgenstein objects: ‘No, it is not true that it must—but it does follow: we perform this transition’.46 That is to say, he objects to giving logic a modal underpinning. Logical relations can of course be expressed with modal verbs. We say: ‘This must be true’ to indicate that something follows from a trusted premise, but it is a mistake to use both indicators of a logical inference, ‘must’ and ‘it follows’, together. The expression ‘it must follow’ is a pleonasm, which may be philosophically misleading: suggesting that logic itself is logically necessary. Elsewhere Wittgenstein characterises logical inferences as ‘the steps which are not brought in question’.47 A related manuscript remark elaborates: “The necessary inference [denknotwendige Folge].” That is the inference that is not brought in question. (I don’t say: “cannot be”.)48 Logical necessity is due not to some eternal metaphysical structures,49 but to contingent linguistic conventions. If Jones is a bachelor, it follows that he is unmarried. Because that is the way we use the term ‘bachelor’. To say that it must follow would presumably mean that we must use the word ‘bachelor’ to apply to unmarried men, which is obviously not true: we could just as well agree to use the word to designate any young man. “” Furthermore, logic does not force us to do what we might have preferred not to do, or prevent us from doing something we would have liked to do. There is, so to speak, no substance to logical compulsion. For, unlike physical compulsion, it cannot go against my will. When I am physically compelled to leave, I || 44 Wittgenstein, RFM, I, § 116b. 45 Wittgenstein, RFM, I, § 113. 46 Wittgenstein, RFM, I, § 12. 47 Wittgenstein, RFM, I, § 156. 48 “Die denknotwendige Folge.” Das ist die Folge, die nicht in Frage [90r] gezogen wird. (Ich sage nicht: “werden kann”.) Wittgenstein, MS 118, 89v–90r. 49 Cf. Frege: logical laws as ‘boundary-stones fixed in an eternal ground’ [Grenzsteine in einem ewigen Grunde befestigt] Frege, 1893, XVI. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM 36 | Severin Schroeder may have preferred to stay. When I am logically compelled to accept that being a bachelor Jones must be unmarried, I am not debarred from any alternative. Logic does not compel me to go one way rather than another, for there is no other way. What is ruled out by logic just does not make sense, it is nothing meaningful—so nothing is ruled out by logic. Wittgenstein criticises Frege for misrepresenting the laws of logic as substantive norms, telling us how we ought to think. Frege, in his Grundgesetze, argues against a psychological account of logic as describing the regularities of human thinking. Thus, from a psychological point of view we may say that it is ‘impossible for human beings to recognize an object as different from itself’.50 This may be so, Frege concedes, but it is merely a contingent matter. Future generations or creatures in far away places might think differently, illogically. A psychological logician would have to accept that such creatures just follow different laws of thought, whereas Frege insists that by the timeless norms of logic such creatures’ thinking must be judged flawed, indeed a kind of insanity.51—Like the psychologistic view he criticises, Frege’s own position takes logic to consist of substantive norms: which one may accept or reject. An example he gives is: ‘Every object is identical with itself’.52 But, Wittgenstein queries, how could one possibly fail to accept this logical law? Could one believe an object not to be identical with itself? Indeed, the psychological version of this law (which Frege dismisses as being concerned not with truth, as befits logic, but merely with what people take to be true) states that it is ‘impossible for human beings to recognize an object as different from itself’. But is there really something that it is impossible to do? ‘Well’, says Wittgenstein ironically, ‘if only I had an inkling how it is done,—I should try at once!’53 And further: Frege calls it ‘a law about what men take for true’ that ‘It is impossible for human beings... to recognize an object as different from itself’.— When I think of this as impossible for me, then I think of trying to do it. So I look at my lamp and say: “This lamp is different from itself”. (But nothing stirs.) It is not that I see it is false, I can’t do anything with it at all. (Except when the lamp shimmers in sunlight; then I can quite well use the sentence to express that.) One can even get oneself into a thinking-cramp, in which one acts as if one tried to think the impossible and did not succeed. Just as one can also act as if one tried (vainly) to draw an object to oneself from a distance by mere willing (in doing this one || 50 Frege, 1893, XVII. 51 Frege, 1893, XVI. 52 Frege, 1893, XVII. 53 Wittgenstein, RFM, 404. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Intuition, Decision, Compulsion | 37 makes e.g. certain faces, as if one were trying, by one’s expression, to give the thing to understand that it should come here.)54 It is not impossible to think of a thing not being identical with itself, however hard one tries;55 either ‘not identical with itself’ is a meaningless expression, or—if we give it a meaning (e.g. for describing a peculiar visual impression)—we can think of a thing not being identical with itself. Either way, there is nothing there for logic to disallow. To think of logic as demarcating what is licit or accessible for our thinking produces a philosophical mirage that may find expression in such charades or grimaces as Wittgenstein describes. In a related passage, written some six years later than TS 222, Wittgenstein considers another way in which one could give meaning and application to the expression ‘thinking of an object as different from itself’, and then points out that not only what logic is meant to forbid isn’t really there; what it prescribes cannot really be thought either: But, if it is impossible for us to recognize an object as different from itself, is it quite possible to recognize two objects as different from one another? I have e.g. two chairs before me and I recognize that they are two. But here I may sometimes believe that they are only one; and in that sense I can also take one for two.—But that doesn’t mean that I recognize the chair as different from itself! Very well; but then neither have I recognized the two as different from one another.56 Sometimes we mistake one object for two, which one could perhaps call: mistakenly thinking of it as ‘different from itself’. Thus, when I think that the author of David Copperfield is different from the author of Great Expectations, that could perhaps be described by saying that I think of Charles Dickens that he is different from himself. (Of course while I have that belief I myself could never express it in those words, but others may. Just as I could never truly say of myself that I believe of a cat that it is a dog, and yet in the dark I may easily make that mistake.) The natural response is that that is clearly not what the logicians’ ‘law of identity’ is meant to rule out. Similarly, acknowledgement of the grammatical truth that a bachelor is an unmarried man is not incompatible with occasionally mistaking a bachelor for a married man (—when one doesn’t know that he’s a bachelor). To be wrong about somebody’s marital status is obviously not to be || 54 Wittgenstein, RFM, I, § 132; translation changed. 55 Wittgenstein, RFM, I, § 116b. 56 Wittgenstein, RFM, pp. 404–5. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM 38 | Severin Schroeder guilty of an inconsistency. Very well, but if mistaking a bachelor for a married man is not to be regarded as a logical mistake, then correctly judging one’s bachelor friend Jones to be unmarried would also be an empirical judgement and not a confirmation of the analytic truth that a bachelor is an unmarried man. Similarly, if thinking that one has seen two chairs in a dark room when in fact there was only one is not a breach of logic, but simply an empirical error, then the corresponding correct judgement (that there are two chairs when indeed there are two) is not an acknowledgement or a confirmation of logic either: but simply a correct empirical judgement. In short, to perceive (or think) that there are two chairs is not to perceive (or think) that two chairs are two chairs— which, like its negation, is not something that can be perceived (or thought). Frege wrote that to think illogically would be a kind of madness,57 but (Wittgenstein observes) ‘he never said what this “madness” would really be like’.58 This remark might be taken as ironic: inviting the reader to try to imagine the kind of madness in question only to find that it could not be done, for Frege was quite wrong to think that people could have mistaken logical beliefs. In the sense explained logical norms have no substance: no content with which one could agree or disagree. Hence to imagine people disagreeing with, say, the law of identity would not be to imagine a tribe of lunatics, but would simply be impossible. Nothing would count as rejecting the law of identity, for anybody’s disagreement with a formulation of the law of identity would just indicate that they take that formulation in a different sense (as illustrated by Wittgenstein’s examples). However, in a 1939 lecture (given some two years after his comment in RFM I–152 on Frege’s madness remark) Wittgenstein takes a different line, suggesting that one could indeed imagine such madness, albeit not as merely an absurd theoretical conviction. He considers the example of people selling wood not by cubic measure, but by the surface area taken up by the pile regardless of its height. Or again, he imagines that people may calculate 4 × 3 = 9 and regularly apply that calculation when distributing things. We should find their behaviour pointless and utterly incomprehensible.59— It is not so clear, however, that we should be entitled to accuse those people of a breach of the laws of logic. In the case of the wood sellers, Wittgenstein himself says in their defence that ‘there is nothing wrong with giving wood away’.60 And there is nothing illogical in fixing || 57 Frege, 1893, XVI. 58 Wittgenstein, RFM, I, § 152. 59 Wittgenstein, LFM, pp. 202–3. 60 Wittgenstein, LFM, p. 202. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Intuition, Decision, Compulsion | 39 prizes in a whimsical way. Even a false equation is not obviously an example of a breach of logic, but would only lead to a contradiction when used as part of our arithmetical system; and it is not clear from Wittgenstein’s example what the rest of those people’s arithmetic looks like. Let us try to find a clearer example of madly illogical thinking. We know what it means for Jones to believe that his neighbour is at home and we know what it means for Jones to believe that his neighbour is away; what sense can we make of the idea that Jones believes that his neighbour is at home and not at home? It could mean that Jones believes his neighbour to be at home, but not wanting it to be known and refusing to communicate with anybody. It is often possible thus to make sense of a contradiction by taking it to describe some intermediary state. But suppose that is decidedly not what Jones means: he declares in no uncertain terms that by ‘at home’ he means simply being physically in one’s home, regardless of one’s communicative intentions. How then could we possibly attribute to him the belief that his neighbour is both at home and abroad?— Roughly speaking, each of those two beliefs tends to go with a certain kind of behaviour: Believing his neighbour to be at home, Jones would avoid making too much noise after 10 p.m.; needing a tape measure he might ring his neighbour’s doorbell to ask if he could borrow his; or again he might try to call on him to invite him round for a drink. Believing his neighbour to be away, he might feel no qualms playing loud music after midnight; he would not try to invite his neighbour for a drink; he might enter his house with his spare key in order to water the flowers and feed the cat, certain that he’d find nobody there. Now imagine he does all those things in close succession: not as if he first believed that his neighbour was at home and then remembered that he was away, but behaving for one moment as if he had one belief, then the next moment the other, then the first again, and so on, without any awareness of changing his mind or being inconsistent. Asked if he thought his neighbour was at home, he says yes.—But didn’t you just say he was away?—‘Yes, that’s right, too. He’s away.’— So he’s at home and away?—‘Yes, that’s right, he’s at home. And he’s away.’ On the grounds of such behaviour one could perhaps say of a person that he did not adhere to the law of contradiction. Note that in order to flesh out that idea it seemed necessary to imagine somebody with a pathologically flaccid and desultory memory. It is not that he has one patently inconsistent belief—the belief that his neighbour is simultaneously away and not away —for that doesn’t mean anything. Rather, he oscillates between two contradictory beliefs in such a way that we can ascribe both to him—after a fashion: for of course his constant wavering between those two beliefs means that in a more demanding Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM 40 | Severin Schroeder sense he can’t be said to have either, being too confused to believe anything clearly and firmly. That is the crucial point: Logic’s ban of contradictions is not at all like a ban on smoking—something feasible, but harmful. For a patent contradiction is meaningless, hence nothing that one could possibly believe (or be admonished not to believe). Therefore, to make any sense of the idea of somebody having such a belief we must either give a contradiction a meaning so that it ceases to be illogical (e.g.: ‘It’s raining and it’s not raining’, meaning: there’s a hint of a drizzle in the air), or we must settle for an extenuated version of belief, such as we may ascribe to a person ‘madly’ oscillating between the acceptance of contradictories without batting an eyelid. And yet we can be said to be compelled by logic or the meanings of words. They have no causal power: they cannot force people to do anything, e.g. to follow certain rules, but they can sanction deviant behaviour by withholding certain descriptions: “Then according to you everybody could … infer anyhow!” In that case we shan’t call it … “inference”.61 So there is what might be called ‘semantic compulsion’. The meaning of the word forces bachelors to be unmarried—: for as soon as they get married they are no longer called ‘bachelors’. Meaning forces them, as bachelors, to be unmarried; though of course it doesn’t force them, as men, to remain unmarried. A deviant pupil cannot be forced to accept that the series +2 should be continued: 1000, 1002, 1004 etc., since he cannot be forced to understand our words in the right way. But if he insists on writing 1000, 1004, 1008 instead, we shall not call that the series +2, the series of even numbers. There is ‘a connexion in grammar’ between the concept of the series of even numbers and the sequence 1000, 1002, 1004, which is indeed stricter and harder than any causal connection.62 Likewise, there is a connexion in grammar between ‘p & q’ and ‘p’ such that we call the transition from the former to the latter a logical inference. Those rigid connections are due to the fact that we insist, sometimes quite rigorously, on using words with a fixed meaning. One aspect of semantic compulsion is (as Wittgenstein observes at the very beginning of TS 222) that we characterise certain formulae as determining a certain solution, thus distinguishing them from other formulae. The expression || 61 Wittgenstein, RFM, I, § 116. 62 Wittgenstein, RFM, I, § 128. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Intuition, Decision, Compulsion | 41 ‘The steps are determined by the formula …’ indicates that the formula is of a certain kind, such that it does not allow more than one solution or correct response at a given point. Thus the formula ‘y = x² + 1’ determines one and only one number for any given value of x. The formula ‘y > x² + 1’, by contrast, is not in this way determinate: a given value of x does not determine one number y.63 And this is of course the distinction between following a rule and deciding freely what to do at each point. In this manner Wittgenstein64 gives the common sense answer to the question what it means that the transitions are determined by a formula: It characterises the formula as having only one solution. Moreover, semantic or logical compulsion tends to be reinforced by social compulsion. We insist on our concepts, especially when teaching a new generation. So in a certain sense the deviant pupil can be forced after all. Part of the insistence on a given concept is the insistence on certain logical implications. Hence: …the laws of inference can be said to compel us; in the same sense, that is to say, as other laws in human society. The clerk who infers as [as we normally do] must do it like that; he would be punished if he inferred differently. If you draw different conclusions you do indeed get into conflict, e.g. with society; and also with other practical consequences.65 And people do indeed feel compelled by arguments: In what sense is logical argument a compulsion?—“After all you grant this and this; so you must also grant this!” That is the way of compelling someone. That is to say, one can in fact compel people to admit something in this way.— Just as one can e.g. compel someone to go over there by pointing over there with a bidding gesture of the hand.66 So logical compulsion can be employed in order to exert psychological compulsion. Finally, the reason why logical compulsion is reinforced by social compulsion is of course that our concepts and their stability are important to us. In particular, concepts such as inference or think or reason are not arbitrary, but reflections of essential features of human life. They are ‘bounded for us, not by an arbitrary definition, but by natural limits corresponding to the body of what can be called the role of thinking and inferring in our life’.67 On the other hand, || 63 Wittgenstein, RFM, I, § 1. 64 Wittgenstein, RFM, I, § 1. 65 Wittgenstein, RFM, I, § 116. 66 Wittgenstein, RFM, I, § 117. 67 Wittgenstein, RFM, I, § 116. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM 42 | Severin Schroeder those concepts are not sharply defined, but vague.68 Therefore, a certain extent of derivation from our existing practices of reasoning and inference could be tolerated within the realm of what we are still prepared to call ‘logic’ or ‘thinking’. (It is, for instance, often acceptable to infer in a way that is not truthpreserving, but probabilistic, and we could perhaps imagine people accepting, and acting on, forms of argument whose inductive strength is far weaker than what we would insist on.) In conclusion: Wittgenstein’s discussion of rule-following does not give us any reason to deny the obvious. Of course it is possible to follow a rule, which means: not to decide freely what to do at each stage, but to be guided, indeed even compelled by a general rule or concept.69 His point is merely that reasons come to an end: the understanding of a rule, the ability to apply it to new cases is a practical skill that cannot, ultimately, be derived from any piece of theoretical knowledge. In order to counter a deep-seated philosophical inclination to think otherwise, to invoke intuition as a specious justification for any particular application of a rule, Wittgenstein presents the idea of a decision as a didactic exaggeration in the opposite direction. But his claim is merely that an application of a rule is comparable to a decision: in so far as it cannot ultimately be justified. That is, although such an application (e.g. writing 1002 as the result of applying +2 to 1000) is of course justified by the rule ‘+2’— hence we are not freely deciding, but following the rule—we cannot give a second order justification of why we take this to be the correct application of the rule. Similarly, Wittgenstein does not deny that there is logical compulsion— which would be, absurdly, to deny the existence of logic, and hence of language. His concern is only to clear away misconceptions of logical compulsion, especially the idea of an utterly irresistible causal power. 3 References BROUWER, L.E.J., 1912: Intuitionism and Formalism, in: P. Benacerraf and H. Putnam (eds): Philosophy of Mathematics. Selected Readings. Englewood Cliffs, New Jersey, Prentice-Hall, 1964, pp. 66–77. BROUWER, L.E.J., 1940: Consciousness, Philosophy, and Mathematics, in: P. Benacerraf and H. Putnam (eds): Philosophy of Mathematics. Selected Readings. Englewood Cliffs, New Jersey, Prentice-Hall, 1964, pp. 78–84. || 68 Wittgenstein, RFM, I, § 116b. 69 Wittgenstein, RFM, I, § 413d. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Intuition, Decision, Compulsion | 43 DUMMETT, M., 1959: Wittgenstein’s Philosophy of Mathematics, in: P. Benacerraf and H. Putnam (eds), Philosophy of Mathematics. Selected Readings. Englewood Cliffs, New Jersey, Prentice-Hall, 1964, pp. 491–509. FEIGL, H., 1968: The Wiener Kreis in America, in: Inquiries and Provocations. Selected Writings 1929-1974, ed.: R.S. Cohen. Dordrecht, Reidel, 1981, pp. 57–93. FRASCOLLA, P., 1994: Wittgenstein’s Philosophy of Mathematics. London, Routledge. FREGE, G., 1884: The Foundations of Arithmetic (1884), tr. J.L. Austin. Oxford, Blackwell. FREGE, G., 1893: Grundgesetze der Arithmetik. Jena, Pohle. HACKER, P.M.S., 1986: Insight and Illusion, rev. ed. Oxford, Clarendon Press. MARION, M., 2003: Wittgenstein and Brouwer, Synthese, 137, pp. 103–127. SCHROEDER, S., 2006: Wittgenstein: The Way Out of the Fly-Bottle. Cambridge, Polity. WEYL, H., 1925: Die heutige Erkenntnislage in der Mathematik, Symposion "Erlangen", Vol. I: 1, pp. 1–32. WEYL, H., 1927: Philosophy of Mathematics and Natural Science, tr.: O. Helmer. Princeton, Princeton University Press, 1949. Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM Brought to you by | De Gruyter / TCS Authenticated Download Date | 4/20/16 9:40 PM