Wittgenstein on consistency and contradictions

p. 101, RFM:

And if they now demand a proof of consistency, because otherwise they would be in danger of falling into the bog at every step — what are they demanding? Well, they are demanding a kind of order. But was there no order before? — Well, they are asking for an order which appeases them now. — But are they like small children, that merely have to be lulled asleep?

Well, multiplication would surely become unsuable in practice because of its ambiguity — that is for the former normal purposes. Predictions based on multiplications would no longer hit the mark…

Is this kind of calculation wrong, then? — Well, it is unusable for these purposes. (Perhaps usable for other ones.) Isn't it as if I were once to divide instead of multiplying? (As can actually happen.)

p. 104, RFM:

Can we say: 'Contradiction is harmless if it can be sealed off'? But what prevents us from sealing it off? That we do not know our way about in the calculus. Then that is the harm. And this is what one means when one says: the contradiction indicates that there is something wrong about our calculus. It is merely the (local) symptom of a sickness of the whole body. But the body is only sick if we do not know our way about.
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."

pp. 104-5, RFM:

The idea of the predicate which is true of itself etc. does of course lean on examples — but these examples were stupidities, for they were not thought out at all. But that is not to say that such predicates could not be applied, and that the contradiction would not then have its application!
Is the question this: "Where did we forsake the region of usability?"? —

For might we not possibly have wanted to produce a contradiction? Have said — with pride in a mathematical discovery: "Look, this is how we produce a contradiction"? Might not e.g. a lot of people possibly have tried to produce a contradiction in the domain of logic, and then at last one person succeeded?

But why should people have tried to do this? Perhaps I cannot at present suggest the most plausible purpose. But why not e.g. in order to show that everything in this world is uncertain?

These people would then never actually employ expressions of the form f(f), but still would be glad to lead their lives in the neighbourhood of a contradiction.

"Can I see an order which prevents me from unwittingly arriving at a contradiction?" That is like saying: shew me an order in my calculus to convince me that I can never in this way arrive at a number which… Then I shew him e.g. a recursive proof.

But is it wrong to say: "Well, I shall go on. If I see a contradiction, then will be the time to do something about it."? — Is that: not really doing mathematics? Why should that not be calculating?! I travel this road untroubled; if I should come to a precipice I shall try to turn around. Is that not 'travelling'?

p. 106, RFM:

I want to ask: must a proof of consistency (or of non-ambiguity) necessarily give me greater certainty than I have without it? And, if I am really out for adventures, may I not go out for ones where this proof no longer offers me any certainty?

p. 107, RFM:

The misuse of the idea of mechanical insurance against contradiction. But what if the parts of the mechanism fuse together, break or bend?

84. "Only the proof of consistency shews me that I can rely on the calculus."

What sort of proposition is it, that only then can you rely on the calculus? But what if you do rely on it without that proof! What sort of mistake have you made?

p. 109, RFM:

At certain places the calculus led me to its own abrogation. Now I want a calculus that does not do this sand that excludes these places.—-Does this mean, however, that any calculus in which such an exclusion does not occur is an uncertain one? "Well, the discovery of these places was a warning to us." — But did you not misunderstand this 'warning'?!
The proof of consistency must give us reasons for a prediction; and that is its practical purpose.

p. 130, RFM:

It is one thing to use a mathematical technique consisting in the avoidance of contradiction, and another to philosophize against contradiction in mathematics.
Why should not a calculation made for a practical purpose, with a contradictory result, tell me: "Do as you please, I, the calculation, do not decide the matter?"
"Why should contradiction be disallowed in mathematics?" Well, why is it not allowed in our simple language-games? (There is certainly a connexion here.) Is this then a fundamental law governing all thinkable language-games?

Let us suppose that a contradiction in an order, e.g. produces astonishment and indecision — and now we say: that is just the purpose of contradiction in this language-game.

p. 131, RFM:

The pernicious ting is not, to produce a contradiction in the region in which neither the consistent nor the contradictory proposition has any kind of work to accomplish; no, what is pernicious is: not to know how one reached the place where contradiction no longer does any harm.

pp. 166-71, RFM:

Say that we quite often arrived at the results of our calculations through a hidden contradiction. Does that make them illegitimate? — But suppose that we now absolutely refuse to accept such results, but still are afraid that some might slip through. — Well then, in that case we have an idea which might serve as a model for a new calculus. As one can have the idea of a new game.

The Russellian contradiction 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.
But you can't allow a contradiction to stand! — Why not? We do sometimes use this form in our talk, of course not often — but one could imagine a technique of language in which it was a regular instrument.

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.
'We take a number of steps, all legitimate — i.e. allowed by the rules — and suddenly a contradiction results. So the list of rules, as it is, is of no use, for the contradiction wrecks the whole game!' Why do you have it wreck the game?

But what I want is that one should be able to go on inferring mechanically according to the rule without reaching any contradictory results. Now, what kind of provision do you want? One that your present calculus does not allow? Well, that does not make that calculus a bad piece of mathematics, — or not mathematics in the fullest sense. The meaning of the word "mechanical" misleads you.

9. When, for some practical purpose, you want to avoid a contradiction mechanically, as your calculus so far cannot do, this is e.g. like looking for a construction of the …-gon, which you have up to now only been able to draw by trial and error; or for a solution of a third degree equation, to which you have so far only approximated.

What is done here is not to improve bad mathematics, but to create a new bit of mathematics.
‘So long as freedom from contradiction has not been proved I can never be quite certain that someone who calculates without thinking, but according to the rules, won’t work out something wrong.' Thus so long as this provision has not been obtained the calculus is untrustworthy.—But suppose that I were to ask: "How untrustworthy?" — If we spoke of degrees of untrustworthiness mightn't this help us to extract the metaphysical thorn?

Were the first rules of the calculus not good? Well, we gave them only because they were good. — If a contradiction results later, — have they failed in their office? No, they were not given for this application.

I may want to supply my calculus with a particular kind of provision. This does not make it into a proper piece of mathematics, but e.g. into one that is more useful for a certain purpose.

The idea of the mechanization of mathematics. The fashion of the axiomatic system.
I at one time inferred via a contradiction without realizing it. Is my result then wrong, or at any rate wrongly got?

If the contradiction is so well hidden that no one notices it, why shouldn't we call what we do now proper calculation?

We say that contradiction would nullify the calculus. But suppose it only occurred in tiny doses in lightning flashes as it were, not as a constant instrument of calculation, would it nullify the calculus?
Let us suppose that the Russellian contradiction had never been found. Now — is it quite clear that in that case we should have possessed a false calculus? For aren't there various possibilities here?
And suppose that the contradiction had been discovered but we were not excited about it, and had settled e.g. that no conclusions were to be drawn from it. (As no one does draw conclusions from the 'Liar'.) Would this have been an obvious mistake?

"But in that case it isn't a proper calculus! It loses all strictness!" Well, no all. And it is only lacking in full strictness, if one has a particular ideal of strictness, wants a particular style in mathematics.

'But a contradiction in mathematics is incompatible with its application.'

'If it is consistently applied, i.e. applied to produce arbitrary results, it makes the application of mathematics into a farce, or some kind of superfluous ceremony. Its effect is e.g. that of non-rigid rulers which permit various results of measuring by being expanded and contracted.' But was measuring by packing not measuring at all? And if people worked with rulers made of dough, would that of itself have to be called wrong?

Couldn't reasons be easily imagined, on account of which a certain elasticity in rulers might be desirable?

"But isn't it right to manufacture rules out of ever harder, more unalterable material?" Certainly it is right; if that is what one wants!

'Then you are in favour of contradiction?' Not at all; any more than of soft rulers.

There is one mistake to avoid: one thinks that a contradiction must be senseless: that is to say, if e.g. we use the signs 'p', '~', '.' consistently, then 'p.~p' cannot say anything. — But think: what does it mean to continue such and such a use 'consistently'? ('A consistent continuation of this bit of a curve.')
One could even imagine a savage's having been given Frege's logic as an instrument with which to derive arithmetical propositions. He derived the contradiction unawares, and now he derives arbitrary true and false propositions form it.

'Up to now a good angel has preserved us from going this way.' Well, what more do you want? One might say, I believe: a good angel will always be necessary, whatever you do.

p. 181, RFM:

if a contradiction were now actually found in arithmetic — that would only prove that an arithmetic with such a contradiction in it would render very good service; and it will be better for us to modify our concept of the certainty required, than to say that it would really not yet have been a proper arithmetic.

"But surely this isn't ideal certainty!" — Ideal for what purpose?

The rules of logical inference are rules of the language-game.

p. 186, RFM:

Something surprising, a paradox, is a paradox only in a particular, as it were defective, surrounding. ONe needs to complete this surrounding in such a way that what looked like a paradox no longer seems one.