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* Autor: Ray Monk, Southampton, Southampton - Great Britain *
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* Titel: Full-blooded Bolshevism: *
* Wittgenstein's Philosophy of Mathematics *
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* Erschienen in: WITTGENSTEIN STUDIES, Diskette 1/1995 *
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* 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, *
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* (3 1/2'' Diskette) ISSN 0943-5727 *
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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.