Torkel Franzen on truth
in [Logic]
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From: Aatu Koskensilta on 19 Nov 2007 14:08 On 2007-11-15, in sci.logic, abo wrote: > 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. Right. When saying that humans and computers can do multiplication we have in mind a perfectly clear idealised picture, and can explain that we mean by this ability that we've an explicit algorithm for carrying our the task, and obviously only limitation of time and space prevent actual humans and computers carrying out multiplications with arbitrary large figures. So what idealised picture could be involved in talking about truths that a human would ever claim to be "unassailably true"? Unless that question is answered it is utterly obscure what counts as an "unassailable truth". > This strikes me as perhaps not what people have in mind. Probably not. It's up to them to explain what it is they have in mind if anything is to be made of claims about what humans would -- "in principle" -- find acceptable, unassailable, and so on. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Daryl McCullough on 19 Nov 2007 14:34 Newberry says... > >On Nov 17, 5:01 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Now, if you want to go for the whole ball of wax and >> come up with a theory T_ultimate with the following >> property: >> >> T_ultimate proves >> Ex (P_ultimate(x,#F)) -> F >> >> There is no such theory T_ultimate except for an >> inconsistent theory. > >Do you mean that there is no such extension of PA (classical logic >with Peano axioms) or do you mean there is no such extension of ANY >theory capable of arithmetic. I mean the latter, but I'm not sure about the distinction you are making. It's hard to see how something could be "capable of arithmetic" without being an extension of PA (or at least an extension of Robinson arithmetic). >If you mean the later then I think it is >not true. Yes, it is. That's what Godel's second incompleteness theorem proves: If T_ultimate is consistent, then it cannot prove its own consistency. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 19 Nov 2007 14:38 Newberry says... > >On Nov 17, 4:47 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> The problem here is that you haven't defined what >> the heck you are talking about. What is T(F)? >> How about defining your terms before using them? >> >> T(F) --> F >> >> isn't compelling to me, because I don't even >> know what it means. >P(x,y) is a provability predicate For what theory? >F is any wff >T(F) is true How are you defining T(F)? Tarski showed that no consistent language can have a truth predicate for that very language. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 19 Nov 2007 14:48 Aatu Koskensilta says... >Right. When saying that humans and computers can do multiplication we have >in mind a perfectly clear idealised picture, and can explain that we mean by >this ability that we've an explicit algorithm for carrying our the task, and >obviously only limitation of time and space prevent actual humans and >computers carrying out multiplications with arbitrary large figures. So what >idealised picture could be involved in talking about truths that a human >would ever claim to be "unassailably true"? Unless that question is answered >it is utterly obscure what counts as an "unassailable truth". Well, to be fair, while Roger Penrose didn't define what "unassailably true" means, he did characterize it. Basically, it includes the following: 1. Any proof using standard mathematics (say, ZFC) is unassailably true. 2. If it is unassailably true that all the axioms of theory T are unassailably true statements, then any theorem of T is unassailably true. 3. No contradiction is unassailably true. 4. If any statement is unassailably true, then the fact that it is unassailably true is unassailably true. -- Daryl McCullough Ithaca, NY
From: abo on 19 Nov 2007 14:50
On Nov 19, 8:08 pm, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > On 2007-11-15, in sci.logic, abo wrote: > > > 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. > > Right. When saying that humans and computers can do multiplication we have > in mind a perfectly clear idealised picture, and can explain that we mean by > this ability that we've an explicit algorithm for carrying our the task, and > obviously only limitation of time and space prevent actual humans and > computers carrying out multiplications with arbitrary large figures. So what > idealised picture could be involved in talking about truths that a human > would ever claim to be "unassailably true"? I'm not sure there is an idealised picture, or whether there is any connection between an idealized picture and unassailable truth. I have no money on this particular horse in this thread. What I was replying to was Daryl's argument that "no single human can do any mathematical reasoning that is noncomputable." The fact (if not a fact, anyway I think it's true) that single humans are limited to a finite number of beliefs does not really have a bearing on whether their mathematical reasoning is non-computable. Sure in some sense it's computable because any of their actual reasonings are finite; but that's not I think really what's under discussion when someone suggests that the reasoning is non-computable. > it is utterly obscure what counts as an "unassailable truth". > > > This strikes me as perhaps not what people have in mind. > > Probably not. It's up to them to explain what it is they have in mind if > anything is to be made of claims about what humans would -- "in principle" > -- find acceptable, unassailable, and so on. Again, I have no money on the unassailable horse in this thread. I wanted to reply to the claim that mathematical reasoning by any particular human is computable; I don't think this is really follows from Daryl's argument. > > -- > Aatu Koskensilta (aatu.koskensi...(a)xortec.fi) > > "Wovon man nicht sprechen kann, daruber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |