Deductive reasoning didn't start with Geometry. It began with the Law Courts & Administrative Tribunals & guys arguing in political or military councils for one rather than another course of collective action.
Math adopted the notion that laws existed and judgments (theorems) must be formulated after a process of scrutiny or cross-examination of a categorical type.
A paradox was something against expectations or received opinion. But it could arise if a witness affirmed two contradictory things- e.g. 'I did not have sex with that woman' and 'I jizzed copiously on her dress'. The prosecutor would use deductive reasoning to show that a guy who wasn't sexually involved with a particular woman would not think his jizz would be found on her dress.
In this context, the following exchange between Witless & Turning, remarked upon by Ash Jogalekar, is revealing...of the former's stupidity.
I've always found this conversation between Wittgenstein and Turing (reprinted in Andrew Hodges's Turing biography) endlessly fascinating: "WITTGENSTEIN: … Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone – much more extraordinary than you might think. … Because the thing works like this: if a man says ‘I am lying’
He is admitting to be a person whose testimony can't be relied on. Does this mean it is 'inadmissible'? Not necessarily. Still, if you have been relying on his previous statements, you now no longer expect the same outcome. That is the 'paradox'.
we say that it follows that he is not lying,
Nothing necessarily follows from any statement whatsoever. Still, for a particular purpose, there may be protocols such that there is a buckstopped assignment of truth values to statements.
from which it follows that he is lying and so on. Well, so what?
To get the liar to tell the truth, make sure he is under oath & understands the penalty for perjury
You can go on like that until you are black in the face. Why not? It doesn’t matter. … it is just a useless language-game, and why should anybody be excited?
Set theory is useful. It isn't a 'language game'. Scrabble is.
TURING: What puzzles one is that one usually uses a contradiction as a criterion for having done something wrong.
If you do the same sum twice and your answers contradict each other- you have indeed done something wrong.
But in this case one cannot find anything done wrong.
I may say I'm a liar but that doesn't mean I'd lie under oath and risk going to jail for perjury. Still, I would be well-advised not to say 'I'm a liar' or 'I'm a pedophile. Could I baby-sit for you?'
WITTGENSTEIN: Yes – and more: nothing has been done wrong. … where will the harm come?
You say you are a liar and you get your head kicked in coz the thug next door no longer believes you are his wife's gay Yoga-instructor and Yoga is best performed nude and in a comfortable double bed.
TURING: The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort.
People who say they are liars aren't trusted. This may entail getting your head kicked in.
WITTGENSTEIN: … The question is: Why are people afraid of contradictions?
Coz if you say x or believe x and someone contradicts you or the evidence is that x is definitely not the case, then you might be in big trouble. It's like when you ring your banker to confirm that the
It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc., outside mathematics.
Math is protocol bound- just like the law. True, you could have mathematicians or lawyers who don't mind contradicting themselves in the course of their professional activities. But they aren't going to be pulling down the big bucks.
The question is: Why should they be afraid of contradictions inside mathematics?
It will turn into stuff done by drunken hobos if contradictions are ubiquitous within it.
Turing says, ‘Because something may go wrong with the application.’ But nothing need go wrong. And if something does go wrong – if the bridge breaks down – then your mistake was of the kind of using a wrong natural law.
Which is fine if you are a drunken hobo for whom dumpster diving provides adequate nutrition.
… TURING: You cannot be confident about applying your calculus until you know that there is no hidden contradiction in it.
If Bletchley Park had been staffed by drunken hobos, the military would have placed no confidence in its Enigma decrypts.
WITTGENSTEIN: There seems to me to be an enormous mistake there. … Suppose I convince Rhees of the paradox of the Liar, and he says, ‘I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2 × 2 = 369.’
This is ex falso quodlibet- the fact that any nonsense can be deduced from a lie.
Well, we should not call this ‘multiplication’, that is all.
No. We call it multiplication of an erroneous sort. If you are paid to do arithmetic, you get sacked if you keep making mistakes in multiplication.
… TURING: Although you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere. WITTGENSTEIN: But nothing has ever gone wrong that way yet.
Lots of bridges would have collapsed, as, indeed, would civilizations, if there weren't a contiuous process of weeding out engineers & economists who get their sums wrong.
Turing's work contributed to the creation of the digital computer. That was very fucking useful. Witless did nothing useful.
Self-reference (as in the Liar's paradox) & Impredicativity do create problems for computing. Maybe that is no longer true. But, the fact is, by Wells thorem, full impredicative type inference (where type applications are left implicit and automatically inferred by the compiler) is undecidable. Because of this, mainstream languages traditionally restricted themselves to predicative inference or required explicit type annotations for higher-rank/polymorphic use. I don't know what the state of play is now- maybe Vovevodksy resizing has been implemented without Girard paradox- but, the fact is, the liar's paradox suddenly became important for commercial and military reasons at precisely the time when Turing & Witless had this discussion
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