Is This really a Proof that Standard Math is Inconsistent?
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From: Andrew Usher on 24 Jan 2010 21:24 On Jan 24, 7:11 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> Rules of inference are valid or not independently of the theory in > >> which they are used. The definition of validity does not refer to the > >> theory. > > >Well, not my definition. If you say so, then your logic is just > >meaningless symbol manipulation. > > Well, in a sense, that's what logic is all about; the study of arguments > that are valid because of their form. The point of a logically valid > argument is that its validity should be checkable *without* any knowledge > of the meanings of the function symbols, predicate symbols, or the domain > of discourse. You're right, of course, but the context really determines the meaning of this argument. Hughes was rebutting the OP, who said that a proof by contradiction assumes the consistency of the theory. I think that his criticism is nonsense because when we lay out a proof, we are claiming that it is true, not just logically valid. Andrew Usher
From: Marshall on 25 Jan 2010 00:24 On Jan 24, 6:24 pm, Andrew Usher <k_over_hb...(a)yahoo.com> wrote: > On Jan 24, 7:11 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: > > > >> Rules of inference are valid or not independently of the theory in > > >> which they are used. The definition of validity does not refer to the > > >> theory. > > > >Well, not my definition. If you say so, then your logic is just > > >meaningless symbol manipulation. > > > Well, in a sense, that's what logic is all about; the study of arguments > > that are valid because of their form. The point of a logically valid > > argument is that its validity should be checkable *without* any knowledge > > of the meanings of the function symbols, predicate symbols, or the domain > > of discourse. > > You're right, of course, but the context really determines the meaning > of this argument. Hughes was rebutting the OP, who said that a proof > by contradiction assumes the consistency of the theory. I think that > his criticism is nonsense because when we lay out a proof, we are > claiming that it is true, not just logically valid. I agree that the context is extremely important. In this case, the context is the standard terminology of mathematical logic, which is why you are so very very wrong. Marshall
From: David C. Ullrich on 25 Jan 2010 06:22 On Sun, 24 Jan 2010 16:26:24 -0800 (PST), Andrew Usher <k_over_hbarc(a)yahoo.com> wrote: >On Jan 21, 7:57�am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > >> >Only in a vacuous sense. Mathematicians do assume 'P xor not P' >> >because it is true, that is, true in real, informal logic. The fact >> >that Goedel's theorem shows that it is not always so in any formal >> >system >> >> For heaven's sake, where did you get the idea that Godel's >> theorem says that "P xoe not P" is not always so in any >> formal system? > >It shows that it isn't always provable, which in the context of >rigorous proof means the same thing. This is simply nonsense. Godel's theorem does not show that "P xor not P" is not provable. For any formula P, the formula "P xor not P" _is_ provable in the formal systems discussed in the theorem. At some point you may realize that you're very confused about a lot of this stuff. >Andrew Usher
From: Aatu Koskensilta on 25 Jan 2010 07:28 Don Stockbauer <don.stockbauer(a)gmail.com> writes: > Actually, Hofstadter does spend a lot of time on Godel's results in > GEB What is the relevance of this observation? As far as I can recall no one here has claimed Hofstadter doesn't consider G�del's theorems in some detail in GEB. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Jesse F. Hughes on 25 Jan 2010 11:21
Andrew Usher <k_over_hbarc(a)yahoo.com> writes: > On Jan 24, 7:11 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: > >> >> Rules of inference are valid or not independently of the theory in >> >> which they are used. The definition of validity does not refer to the >> >> theory. >> >> >Well, not my definition. If you say so, then your logic is just >> >meaningless symbol manipulation. >> >> Well, in a sense, that's what logic is all about; the study of arguments >> that are valid because of their form. The point of a logically valid >> argument is that its validity should be checkable *without* any knowledge >> of the meanings of the function symbols, predicate symbols, or the domain >> of discourse. > > You're right, of course, but the context really determines the meaning > of this argument. Hughes was rebutting the OP, who said that a proof > by contradiction assumes the consistency of the theory. I think that > his criticism is nonsense because when we lay out a proof, we are > claiming that it is true, not just logically valid. I was responding to the following: >> Joshua Cranmer <Pidgeo...(a)verizon.invalid> writes: >> > Ultimately, a proof by contradiction assumes that a contradiction cannot >> > be proved in said formal system. You assumed that a contradiction exists >> > in T, which renders invalid a proof by contradiction. Perhaps you will suggest that he really didn't mean to use "invalid" here, but surely I'm not as clever as you are. I assumed that he used the word "invalid" because he meant invalid. (If I were really pedantic, I would ask you what you mean when you say we're claiming that a proof is "true", by the way.) -- Jesse F. Hughes "Philosophy, Socrates, if pursued in moderation and at the proper age, is an elegant accomplishment, but too much philosophy is the ruin of human life." -- Callicles, in "Gorgias" |