Torkel Franzen on truth
in [Logic]
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From: Newberry on 15 Nov 2007 01:56 On Nov 14, 9:13 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Nov 14, 4:56 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> Why does it matter whether the machine can prove it, > >> if the human can't prove it, either? As I said, for > >> a computer program to be as powerful as a human in > >> recognizing truth, all that's necessary is for the > >> program to be able to *recognize* true statements. > >> It's not necessary that the program be able to > >> *prove* them. > > >How can a computer recognize that PA is consistent? > > By producing an output "true" when given the > input question "Do you believe that PA is consistent?". > > Are you asking how one would go about *programming* > a computer program that would emulate a human's > mathematical reasoning? If so, I have no idea. > The issue is not whether we *currently* know > how to make an artificial intelligent computer > program. The issue is whether Godel's theorem > implies that it is impossible. It doesn't > imply any such thing. > > There is what I think is a pretty air-tight argument > that no single human can do any mathematical reasoning > that is noncomputable: A real human has a finite > memory capacity, and so there are only finitely > many different statements of mathematics that we > can ever hold in our heads at one time. So the > collection of all statements that any *actual* > human would ever claim to be "unassailably true" > is a finite set. Every finite set of formulas > is computable. Bingo!! You got it! So we have the human mind surpasses any machine and no single human can do any mathematical reasoning that is noncomputable. A contradiction! That is what I was trying to say all along. > > Now, you could argue about what an idealized > human could do, where we idealize the human > to have an infinite memory capacity. Could > such an idealized human do something that > no Turing machine could do? Well, it depends > on the details of how the "ideal human" is > idealized. > > -- > Daryl McCullough > Ithaca, NY
From: abo on 15 Nov 2007 03:26 On Nov 15, 6:13 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > There is what I think is a pretty air-tight argument > that no single human can do any mathematical reasoning > that is noncomputable: A real human has a finite > memory capacity, and so there are only finitely > many different statements of mathematics that we > can ever hold in our heads at one time. So the > collection of all statements that any *actual* > human would ever claim to be "unassailably true" > is a finite set. Every finite set of formulas > is computable. > A finite machine can't compute the palindrome or multiplication function, so it would seem by the same token neither any actual human nor any actual computer can "do" multiplication. This strikes me as perhaps not what people have in mind.
From: Daryl McCullough on 15 Nov 2007 06:12 Newberry says... > >On Nov 14, 9:13 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> There is what I think is a pretty air-tight argument >> that no single human can do any mathematical reasoning >> that is noncomputable: A real human has a finite >> memory capacity, and so there are only finitely >> many different statements of mathematics that we >> can ever hold in our heads at one time. So the >> collection of all statements that any *actual* >> human would ever claim to be "unassailably true" >> is a finite set. Every finite set of formulas >> is computable. > >Bingo!! You got it! So we have the human mind surpasses any machine >and no single human can do any mathematical reasoning that is >noncomputable. A contradiction! That is what I was trying to say all >along. No, we *don't* have that the human mind surpasses any machine. There is no reason to believe that's true. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 15 Nov 2007 06:52 abo says... >A finite machine can't compute the palindrome or multiplication >function, so it would seem by the same token neither any actual human >nor any actual computer can "do" multiplication. I think that's all perfectly true. We can't multiple two trillion-digit numbers. We can't compute the palindrome of a trillion-letter word. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 15 Nov 2007 08:34
Daryl McCullough says... > >abo says... > >>A finite machine can't compute the palindrome or multiplication >>function, so it would seem by the same token neither any actual human >>nor any actual computer can "do" multiplication. > >I think that's all perfectly true. We can't multiple two >trillion-digit numbers. We can't compute the palindrome >of a trillion-letter word. Whoops! What I meant was that we can't decide whether a trillion-letter word is a palindrome. -- Daryl McCullough Ithaca, NY |