'When Are Relations Neither True Nor False?
in [Math]
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From: Aatu Koskensilta on 11 Apr 2010 11:13 Transfer Principle <lwalke3(a)lausd.net> writes: > To repeat, what may be intuitive to one poster may be counterintuitive > to another. And I see no reason to favor one poster's intution over > another's, no matter what the standard theorists try to say. I intuit it for you that in general there's no reason to pay much any notice to our intuitions. I don't recall anyone claiming we should favour any poster's intuition over that of others. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 11 Apr 2010 11:18 Newberry <newberryxy(a)gmail.com> writes: > If it absolutely certain that PA is consistent why don't we formalize > the reasoning? Absolute certainty is irrelevant. Consistency proofs are every bit as formalizable as other proofs. We can formalize the trivial consistency proof for PA in the subtheory ACA of second-order arithmetic, in formal set theory, in Per-Martin L�f's constructive type theory (by a detour through a double-negation interpretation), and so on. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 11 Apr 2010 11:18 Newberry <newberryxy(a)gmail.com> writes: > Such a proof in ZF that PA is consistent is obviously wothless. Why? Are all proofs in ZF worthless? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 11 Apr 2010 11:25 Marshall <marshall.spight(a)gmail.com> writes: > I don't see any reason to pay much attention to anyone's > intuition, my own included. This is a sober attitude. > "Intuition" is just a fancy word for "hunch." "Intuition" can mean pretty much anything, from a vague hunch to something very specific, as in e.g. Kant's thought. For example, when people say that, say, the principle of mathematical induction is intuitively evident it would be odd to take them to be telling us they've a hunch it might hold. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 11 Apr 2010 11:28
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > If you don't remember or have any doubt about my position, please > allow me to clearly reiterate my position: > > It's impossible to have logically acceptable methods for proving a > syntactical consistency. But there's nothing clear about this reiteration. What is a "logically acceptable method"? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |