***************************************************************** * * * File: 08-1-95.TXT Dateilänge: 35 KB * * * * Autor: Ray Monk, Southampton, Southampton - Great Britain * * * * Titel: Full-blooded Bolshevism: * * Wittgenstein's Philosophy of Mathematics * * * * Erschienen in: WITTGENSTEIN STUDIES, Diskette 1/1995 * * * ***************************************************************** * * * (c) 1995 Deutsche Ludwig Wittgenstein Gesellschaft e.V. * * Alle Rechte vorbehalten / All Rights Reserved * * * * Kein Bestandteil dieser Datei darf ganz oder teilweise * * vervielfältigt, in einem Abfragesystem gespeichert, * * gesendet oder in irgendeine Sprache übersetzt werden in * * irgendeiner Form, sei es auf elektronische, mechanische, * * magnetische, optische, handschriftliche oder andere Art * * und Weise, ohne vorhergehende schriftliche Zustimmung * * der DEUTSCHEN LUDWIG WITTGENSTEIN GESELLSCHAFT e.V. * * Dateien und Auszüge, die der Benutzer für seine privaten * * wissenschaftlichen Zwecke benutzt, sind von dieser * * Regelung ausgenommen. * * * * No part of this file may be reproduced, stored * * in a retrieval system, transmitted or translated into * * any other language in whole or in part, in any form or * * by any means, whether it be in electronical, mechanical, * * magnetic, optical, manual or otherwise, without prior * * written consent of the DEUTSCHE LUDWIG WITTGENSTEIN * * GESELLSCHAFT e.V. Those articles and excerpts from * * articles which the subscriber wishes to use for his own * * private academic purposes are excluded from this * * restrictions. * * * ***************************************************************** * * * Monk, Ray (1995) Full-blooded Bolshevism: Wittgenstein's * * Philosophy of Mathematics; in: * * Wittgenstein Studies 1/95, File: 08-1-95; hrsg. von * * K.-O. Apel, F. Börncke, N. Garver, B. McGuinness, P. Hacker, * * R. Haller, W. Lütterfelds, G. Meggle, C. Nyíri, K. Puhl, * * Th. Rentsch, A. Roser, J.G.F. Rothhaupt, J. Schulte, * * U. Steinvorth, P. Stekeler-Weithofer, W. Vossenkuhl * * (3 1/2'' Diskette) ISSN 0943-5727 * * * ***************************************************************** It is often maintained that Wittgenstein's later philosophy is fundamentally conservative, a claim apparently supported by one of the most frequently quoted passages from Philosophical Investigations: paragraph 124. Philosophy, Wittgenstein says there, 'may in no way interfere with the actual use of language; it can in the end only describe it. For it cannot give it any foundation either. It leaves everything as it is'. He goes on: It also leaves mathematics as it is, and no mathematical discovery can advance it. A 'leading problem of mathematical logic is for us a problem of mathematics like any other. In the face of such direct and unequivocable statements of non-revisionism, it is not surprising that commentators have taken it that it is no part of Wittgenstein's purpose to change the way we study and teach mathematics. I want to argue that this is a misunderstanding. The changes Wittgenstein wished to see are, on the contrary, I believe, so radical that the name 'full-blooded Bolshevism' suggests itself as a natural way to describe the militant tendency of his remarks. It should, however, be admitted at once that Wittgenstein's philosophy of mathematics is not revisionist in the sense that the Intuitionism of Brouwer and Weyl, for example, is revisionist. Wittgenstein does not, that is, see it as a philosophical concern to 'reconstruct' mathematics, nor to legislate on what logical principles may or may not be used in mathematical proofs, nor again to argue for or against the existence of infinitesimals or transfinite numbers. All this, as he repeatedly told his students, he regarded as 'bosh'. But it cannot be denied that his investigations into the 'interpretation' and the 'interest' of the work of mathematicians challenge attitudes and assumptions that are fundamental to the profession of pure mathematics. Indeed, it sometimes seems that the aim of his description of mathematics is precisely to strip the subject of the interest that pure mathematicians take in it. Take, for example, his strident reaction to Littlewood's suggestion that the beauty of mathematics lies not in its calculations - which are simple - but in the meaning they have. 'The only meaning they have in mathematics', Wittgenstein retorts, 'is what the calculation gives them. And if it's simple, it's simple'. He goes on: One might say, 'boys up to twenty learn complicated calculi, but you need an educated brain for this simple one, for these highly abstract notions'. As though here we had to see through the calculations to a depth beyond - This I want to say is most misleading. The calculus (system of calculations) is what it is. It has a use or it hasn't. But its use consists either in the mathematical use - (a) in the calculus which Littlewood gives, or (b) in other calculi to which it may be applied - or in a use outside mathematics. It is as pedestrian as any calculus, as pedestrian as the four dimensional cube. If you think you're seeing into unknown depths - that comes from a wrong imagery. The metaphor is only exciting as long as it's fishy. (Lectures, p254) Would the widespread adoption of this attitude really leave mathematics untouched? Yes, it would, in a sense: the calculi (and, for Wittgenstein, 'mathematics consists entirely of calculations' - PG p468) would not be interfered with at all. Some of them would, however, be left unused. They would, one might say, be left completely untouched. Cantor's paradise would not be destroyed, it would be left completely alone - and empty. (see Lectures, p103). What is perhaps most startling about Wittgenstein's allegedly non-revisionist 'description' of mathematics is that pure mathematics seems to be left out, or at best marginalised. On Wittgenstein's view, it seems, every mathematician is an applied mathematician: the only distinction to be made is between interesting and uninteresting applications; between applying mathematics to the physical world and applying it to a lot of mumbo-jumbo. Physical applications of calculi are interesting because of the seriousness of the predictions that may be based upon them. Applications on which no physical predictions are made are a relatively trivial matter; they may puzzle, mystify, amuse, charm, or infuriate us, depending on the imagery we use to accompany the calculations. And, as Wittgenstein insisted in his discussions with Waismann and Schlick, you cannot tell whether a calculus has an interesting use or not simply by looking at it (see, for example, WWK, p170). The published secondary work has in my opinion taken too little account of the attitudes that inform Wittgenstein's philosophy. For whether we find them fascinating or infuriating, it cannot be denied that they are the most striking thing about his remarks on mathematics. We can hardly read a single page without being aware that Wittgenstein is expressing first and foremost an attitude, and that he is trying to influence us, not in the sense of persuading us of the truth of this or that proposition or set of propositions, but in the sense of persuading us to take a certain view, a certain attitude towards mathematics and formal contradiction. Moreover, the attitudes he is propagating are the exact reversal of the ones which predominate in the debate to which his remarks are addressed: the debate about the logical foundations of mathematics. Anyone capable of understanding the issues on which his remarks are intended to shed light must, it seems, be predisposed to take the alternative view. If we did not find it deeply perplexing that, though it works in practice, the reasoning on which the differential calculus is based is not logically sound; if we did not think Russell's Paradox a serious matter; if we could not see any serious motivation behind Hilbert's formalisation programme; if we did not care about logical consistency; if we did not think the liar paradox worthy of our attention, is it likely that we would ever have shown any interest in the problem of providing logical foundations for mathematics? And if we have shown an interest in these things, is it likely that we should be impressed by being told that the fear of contradiction is a superstition?, that the reasoning behind Russell's Paradox is all bosh? that the liar paradox is a useless game? that set theory might just as naturally be interpreted as a joke as a serious piece of mathematics, and that the problem of the infinitesimal was, all along, simply a matter of succumbing to the picture of 'very tiny things'? Wittgenstein's philosophy of mathematics is, then, doubly paradoxical: not only is he attempting to contribute to an area of discussion in which attitudes are held which he does not hold, and in which techniques of argument are expected which he does not use, his contribution is precisely to attack those attitudes and techniques. Nothing more Quixotic could be imagined. It is simply impossible, I believe, to interpret Wittgenstein's remarks as constituting a serious philosophy of mathematics in the sense that logicism, formalism, intuitionism and strict-finitism are serious philosophies of mathematics - i.e., serious attempts to make sense of the subject studied by professional mathematicians. And this is for the strong and simple reason that Wittgenstein does not take that subject seriously. Indeed, as I have said, he hardly seems to believe it exists. The only activity that might deserve the name 'pure mathematics' that emerges from his 'description' is the construction of calculi for either use or amusement; that is, an activity that is either indistinguishable from applied mathematics or else is a frivolous pastime that has nothing to do with science. He cannot, however, altogether rule out the possibilty that set theory, transfinite number theory and four-dimensional geometry - his three particular bugbears - might have a 'reasonable' use in the future. His attempts to conceive such possible uses, however, are not very encouraging for any attempt to take these theories seriously. Imagine, he suggests (RFM, V 5), the geometry of four-dimensional space done with a view to learning about the living conditions of spirits. Or imagine (RFM, V 7) infinite numbers used in a fairy tale: The dwarves have piled up as many gold pieces as there are cardinal numbers - etc. And again (RFM, V 7): Imagine set theory's having been invented by a satirist as a kind of parody on mathematics - Later a reasonable meaning was seen in it and it was incorporated into mathematics. (For if one person can see it as a paradise of mathematicians, why should not another see it as a joke?) Wittgenstein's remark: 'Later a reasonable meaning was seen in it and it was incorporated into mathematics' is very telling; it indicates what it is that makes his philosophy of mathematics unique. And that is that to him it is essential to mathematics that it have a non-mathematical use and, therefore, a non-mathematical meaning. The non-mathematical use is the keystone to his whole conception of mathematics. 'It is the use outside mathematics', he says (RFM, V 2), 'and so the meaning of the signs, that makes the sign-game into mathematics'. If a calculus has no non-mathematical use then it simply isn't mathematics. In such a case, it is necessary - as above - to invent a quite fantastic, imaginary use to even make sense of the calculus. And then: 'The question is what use the image is to us' (RFM, V 7). When Wittgenstein says: I should like, so to speak, to show that we can get away from logical proofs in mathematics. (RFM, III 44) we can see how seriously to take the idea that his philosophy of mathematics would leave mathematics as it is. For 'logical proofs in mathematics' have been the norm in pure mathematics ever since the first attempts were made to resolve the logical inconsistencies of the differential calculus. What was disturbing about the notion of the infinitesimal was not that it lacked meaning, an intelligible use - as Wittgenstein repeatedly emphasised, the phrase 'infinitely small' could have and does have (among, e.g. engineers) a perfectly intelligible use - what it lacked, rather, was a consistent definition. And that, though it did not disturb the users of the calculus in non-mathematical contexts, disturbed mathematicians greatly. For they had inherited from the use of 'logical proofs in mathematics', from, say, Euclid, the notion of a valid proof. The reason that the contradictions in the notion of an infinitesimal were disturbing was that they made the proofs of the differential calculus formally invalid. On Wittgenstein's view, however, there is simply no such thing as a formally invalid proof. A proof, for him, is a picture; you can find it compelling, or you can take no interest in it; you can use it as a model to make physical predictions, or you can refuse to use it. But you cannot refute a picture. But on Wittgenstein's account there is no room for the notions of a formally valid proof, a formally invalid proof and a refutation. And if we conceive of a theorem as a true proposition that forms the conclusion to a formally valid proof, then there are no theorems either. But for centuries we have had an academic subject, which many consider to be a science, which consists in the attempt to construct formally valid proofs, to gain the acceptance of theorems or to give refutations of other proofs. What, one might ask, has been going on in this subject? Well, says, Wittgenstein: 'it is an interesting fact that people set up rules for the fun of it, and then keep to them'. For this strange phenomenon of a search for the removal of logical contradictions in mathematical proofs is, for him, of only psychological and anthropological interest, a product of the 'exasperation' (RFM, III 88) of mathematicians. The fear of contradiction it exemplifies is of philosophical interest only in so far as it shows how tormenting problems can grow out of the misuse of language and also 'what kind of things can torment us' (RFM, III 13). The target of Wittgenstein's philosophy of mathematics, then, is not Platonism - or any other specific philosophical doctrine -, but rather the 'game' of pure mathematics as a supposedly scientific activity. To quote from Philosophical Grammar: Confusions in these matters are entirely the result of treating mathematics as a kind of natural science. And this is connected with the fact that mathematics has detached itself from natural science; for as long as it is done in immediate connection with physics, it is clear that it isn't a natural science. (Similarly, you can't mistake a broom for part of the furnishing of a room as long as you use it to clean the furniture. (PG, p375) Wittgenstein's work is an attempt to provide a completely fresh look at the subject, to look at it, so to speak, with the eyes of a child, or at least with the eyes of someone unencumbered by a mathematical or philosophical training. 'We shall see contradiction in a quite different light', he writes (RFM, III 87) 'if we look at its consequences as it were anthropologically - and when we look at it with a mathematician's exasperation. That is to say, we shall look at it differently, if we try merely to describe how the contradiction influences language-games, and if we look at it from the point of view of the mathematical law-giver'. Wittgenstein's philosophy of mathematics is not an attempt to understand the subject studied by pure mathematicians. If it has any connection with that subject it is as an attempt to undermine attitudes and assumptions that are basic to it. The debate about 'revisionism' was thus of no interest to him. He was not interested either in reconstructing mathematics on intuitionistic principles ('Intuitionism', he said, 'is all bosh - entirely'), or in defending classical mathematics on formalist principles. What he was seeking to defend - against the 'mathematical law-giver' - was the mathematics of the engineer. He was not seeking to resolve logical problems, but to encourage an attitude of indifference to them. He hoped that the solution of the problem would be seen in the vanishing of it. His analysis of the problem is so fundamental and the change of attitude he advocates so radical, that if the 'enormous consequences' he was hoping for were to be mathematical consequences, the implications for the profession of mathematics would be quite staggering. So staggering that it is absurd to expect mathematicians to do anything about it but shudder with horror. As Wittgenstein put it (PG, p381): A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop. What, exactly, are these thoughts and doubts? And why must they be horrifying to any trained mathematician? To see this let us take the example of the differential calculus, for that is, I think, Wittgenstein's paradigmatic example of a calculus in which contradiction is harmless, and his prime case in which the exasperation of the 'mathematical lawgiver' has done more harm than good. For a mathematician to agree with Wittgenstein in his analysis of this case would be for him to regard the history of his science, not as progress, but as a series of confusions extending all the way from the definition of a derivative as the limit of a function to the construction of the Russellian calculus. For Wittgenstein's diagnosis locates the source of the problem at the point when the calculus began to be analysed rather than used; at the point, that is, of the creation of pure mathematics. The reason, according to Wittgenstein, that the contradictions of the differential calculus were harmless is that the real proofs never were the logically connected series of propositions that justified the calculations and which were illustrated by the geometrical drawing. What really enabled engineers, for example, to understand and to have confidence in the calculus were those calculations and the geometrical illustrations themselves. It was they that were used as a model on which to base physical predictions. The 'proofs' that mathematicians gave to justify the use of these models were never used by engineers. What was used was the technique of calculation that the mathematicians had invented. And for them the proof that this type of calculation had to work was the geometrical illustration. If one has a continuous function relating the time and distance of, say, a falling object, and one represents this function on a graph as a continuous curve, then one can see that a point on the curve can be approached from either side. And one can see that each 'little bit' of the curve can be represented by a differential ratio dx/dy and that as the 'little bits' get smaller one may as well, at some point, regard the ratio dx/dy as representing a point on the curve rather than a 'little bit'. At this point one has found the derivative, and calculated the instantaneous velocity of the falling object at a particular moment. An engineer will not mind that he or she hasn't quite calculated the velocity of the object at an instant in time, but only the average velocity of the object during a very short period of time. For practical purposes, to have calculated the average velocity of a falling object between the first minute after it has fallen and the millionth of a second after that first minute is to have calculated its velocity at the first minute. That this technique of calculating a derivative has to work if the function one starts with can be represented by a continuous curve is something you will more easily persuade someone of by showing them a picture and teaching them the technique of calculation than by giving them a logically connected series of definitions and propositions - especially if this latter series has as its first step Dedekind's definition of a real number. Is there, one might ask, really anybody who wanted to use the differential calculus (and not, say, someone who was interested in mathematical logic independently of its usefulness in providing a foundation for mathematics) that would feel more secure in using it by being persuaded of the Platonic objectivity of real numbers, and persuaded that a 'Dedekind cut' is a real number, that continuity is not a spatial or temporal notion but an arithmetic one, that the existence of the derivative is guaranteed by its existing as a unique point on the real number continuum - is there anyone who would use the calculus with more confidence after being persuaded of the truth of all that than he would if he had simply learned to see the derivative as a 'little bit' on a continuous curve? And of not the users of the calculus, then who, exactly, are the foundations of the calculus supposed to reassure? And of what? These are the 'thoughts and doubts' that Wittgenstein developed and sought to develop in others. They are not mathematical questions. Their form is not: 'Is such-and-such true?' or 'Is such-and-such valid?', but rather: 'What is the point of all this?' Wittgenstein was not interested in doing mathematical logic, but he was certainly interested in challenging it. When he asked himself: 'Why do I want to take the trouble to work out what mathematics is?' he answered as follows: Because we have a mathematics, and a special conception of it, as it were an ideal of its position and function - and this needs to be clearly worked out. It is my task, not to attack Russell's logic from within, but from without. That is to say: not to attack it mathematically, otherwise I should be doing mathematics - but its position, its office. (RFM, VII 19) What he wanted to show was that we were free to accept or reject any law that the 'mathematical lawgiver' were to give us, that it was up to us whether we wanted to use the calculi he had invented or not. I think when we see it in this light it becomes clear why it seems so artificial to discuss the role of decision in Wittgenstein's philosophy of mathematics as though the purpose was to argue for the truth of a theory about logical necessity: 'full-blooded conventionalism'. Apart from the general point that, on Wittgenstein's terms, the programme of arguing for such a theory would be an attempt to do the impossible, this particular theory - foisted on Wittgenstein by Professor Michael Dummett - was one that he explicitly rejected. The theory is, in Dummett's words, that 'there is nothing which forces us to accept [a mathematical] 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'. This is a position that Wittgenstein considers and discounts - on more than one occasion - in his 1939 lectures. 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. (p237) Suppose that I tell you to multiply 418 by 563. Do you decide how to apply the rule for multiplication? No: you just multiply....It is not a decision. (p238) If we try to foist any theory upon Wittgenstein's remarks on mathematics, if we concentrate on what he says rather than on why he is saying it, we will do him an injustice. How can you construct a logically consistent argument for the irrelevance of logical consistency? The distinction between saying and showing was one Wittgenstein used as well as mentioned. The point of emphasising the role of decision in the acceptance or rejection of mathematical calculi was to challenge the position, the office, of the mathematical law-giver. Part of the purpose behind investigating the 'ordinary' use of mathematical terms like 'proof', 'equals', 'inference', 'contradiction' was to raise the question: Is this the way they are used by mathematicians? The most striking example of this is Wittgenstein's investigation into the use of the word 'contradiction', and the role it plays in our language-games, i.e. the role it plays in our life. The outcome he wanted from the investigation is 1that of seeing that, though the use of 'contradiction' by logicians is analogous to the ordinary use, there is something unnatural about it. Russell's Paradox, he says, is disquieting, not because it is a contradiction, but because 'the whole growth culminating in it is a cancerous growth, seeming to have grown out of the normal body aimlessly and senselessly' (RFM, VII 11) That growth, of course, is the attempt to provide the calculus with logical foundations. In discussing the role that contradiction plays in our 'ordinary', our 'natural' language, Wittgenstein mentions that we do sometimes allow contradictions like the following: It might for example be said of an object in motion that it existed and did not exist in this place; change might be expressed by means of contradiction. (RFM, VII 11) Now if this is a perfectly natural way to express change; if this is a paradigm case of the kind of harmless contradiction we allow in our ordinary language, then the contradiction at the heart of the differential calculus was founded is perfectly natural. For it should be noticed that Wittgenstein's paradigmatic example of a harmless contradiction is precisely the one that gave centuries of mathematicians so much trouble. The differential calculus is the attempt to express change, and the technique of calculating the derivative of a function to find an instantaneous velocity was so successful precisely because the people who used it chose to ignore the contradiction of supposing an object in motion to exist and not exist at the point at which the derivative is found. In suggesting that this contradiction is perfectly natural, Wittgenstein is, by implication, asking us to look at the whole history of pure mathematics - and not just that of mathematical logic - as an unnatural aberration, going right back to the very beginnings of calculus. For the contradiction at the heart of the differential calculus, expressed in kinematic terms, is exactly the one to which Bishop Berkeley drew attention so effectively in his attack on Newton's characterisation of a differential as a 'Fluxion': The great Author of the Method of Fluxions [...] gave into these nice Abstractions and Geometrical Metaphysics, without which he saw nothing could be done one the received Principles; and what in the way of Demonstration he hath done with them the Reader will judge. It must, indeed, be acknowledged, that he used Fluxions, like the scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities, nor Quantities infinitely small not yet nothing. May we not call them the Ghosts of departed Quantities? (The Analyst, para 35) What Wittgenstein is suggesting in using the expression of change as a paradigm example of a harmless contradiction is: why not call them the 'Ghosts of departed Quantities?' We can see an object in motion, we can see that the differential calculus is an effective instrument. If we cannot express that motion in language except by means of a contradiction, then that points to the limitations of language not to the insecurity of the foundations of the calculus. The solution lies in a correct understanding of those limitations - an understanding of what can be said and what has to be shown - rather than in a revision of the calculus. Wittgenstein's attitude to the problem posed by Berkeley, then, would be: if the technique is useful, but inconsistent with the 'received Principles', well, then, so much the worse for the received Principles: For the point of a new technique of calculation is to supply us with a new picture, a new form of expression and there is nothing so absurd as to try and describe this new schema, this new kind of scaffolding, by means of the old expressions. (RFM, II 46) So, firstly, the contradiction was not in the calculations but arose from the attempt to describe the calculations in another form of expression, another calculus - the English language; and, secondly, the contradiction as expressed in English is, in any case, harmless. It would seem to follow that all the attempts to provide a consistent derivation of the theorems of the calculus - all the definitions of the derivative, and of continuity, etc, and all the theoretical machinery that was constructed to render the calculus rigorous and consistent - was all a waste of time, all based on an inadequate understanding of the confusions that arise from 'the tendency assimilate to each other expressions which have very different functions in the language'. If we consider that this makes the whole history of the 'arithmetisation of analysis', from Cauchy to Russell, a huge mistake, we will realise why it is that, on Wittgenstein's view, philosophical clarity will have - and would have had - the same effect on the growth of mathematics as sunlight has on the growth of potato shoots. It is not too much to say that Wittgenstein's 'way of seeing' mathematics undermines the significance of almost everything done in pure mathematics since the eighteenth century, and interprets as 'fanciful applications' even such basic things as the accepted definitions of the derivative and continuity, the definitions given to every undergraduate learning pure mathematics. We can see, then, that, though he constructed no calculi - and, therefore, on his terms, left mathematics alone - his challenge to the authority of mathematicians could hardly have been more fundamental. This, I think, is what he had in mind in comparing himself to Frank Ramsey in the following terms: Ramsey was a bourgeois thinker. I.e. he thought with the aim of clearing up the affairs of some particular community. He did not reflect on the essence of the state - or at least did not like doing so - but on how this state might reasonably be organised. The idea that this state might not be the only possible one in part disquieted him and in part bored him. He wanted to get down as quickly as possible to reflecting on the foundations of this state. This is what really interested him; whereas real philosophical reflection disturbed him until he put its result (if it had one) to one side and declared it trivial. (CV, p17) It was Ramsey - the 'bourgeois thinker' - who spoke of the 'Bolshevik menace' of Brouwer and Weyl, a phrase Wittgenstein was no doubt consciously echoing when he tried to reassure Turing that he was not 'introducing Bolshevism into mathematics' (Lectures, p67). Unlike Ramsey, Wittgenstein was not interested in the foundations of this state. But neither was he interested in introducing Bolshevism into this state (pure mathematics). His Bolshevism took the form of reflecting on the 'essence of the state', challenging the authority of this state, the mathematics of the law-givers. His reflections on the essence of mathematics have the result that, whereas the mathematics of the engineer is essential the mathematics of the mathematician is not. Therefore, in using mathematics, there is no reason to adopt the attitude of mathematicians to logical contradiction, to look at contradiction with a 'mathematician's exasperation', and no reason not to adopt the attitude of an engineer: I should like to ask something like: 'Is it usefulness you are out for in your calculus?' - In that case you do not get any contradiction. And if you aren't out for usefulness -then it doesn't matter if you do get one. (RFM, III 80) If it is, as Wittgenstein insists again and again, the use outside mathematics and so the meaning of the signs that makes the sign-game into mathematics, then the question that arises is not whether what engineers do is mathematics, but whether what mathematicians do can rightly be called mathematics. And of that, as we have seen, he entertained grave doubts, doubts which, if taken seriously, would lead to an assault on the Russellian Palace far more radical and threatening than anything envisaged by those comparatively tame Bolsheviks, Brouwer and Weyl.