Torkel Franzen on truth
in [Logic]
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From: Daryl McCullough on 7 Dec 2007 11:17 Newberry says... >There are three possibilities: >1) We do not have the foggiest idea if PA is consistent and we will >never know. Hence we do not know if the Goedel sentence is true. >2) The human mind surpasses any computer >3) There exists a formalization of arithmetic that can prove its own >consistency. I believe that all three of those are false, so I reject your claim that those are the only three possibilities. Your number 1 assumes that: To know X means that X is provable in some system T0 such that T0 is provably consistent in some system T1 such that T1 is provably consistent in some system T2 such that ... By that definition, we never know anything. I suppose that might be correct, in some sense, but the way most people use the word "know" is something short of proof. Or I should say *different* from proof. Proof is neither necessary nor sufficient for knowledge. In any case, whether or not we can be said to *know* with certainty that PA is consistent, it is certainly false to say that "we do not have the foggiest idea if PA is consistent". That's completely wrong. The consistency of PA is as certain as *anything*. There is no reason to believe otherwise, and plenty of reason to believe it. >But it is not possible to reject 1 and 3 and at the same time claim >that the human mind does NOT surpass any machine. Sure, it's possible. I reject 1 and 3 and at the same time calim that the human mind does not surpass any machine. -- Daryl McCullough Ithaca, NY
From: Peter_Smith on 7 Dec 2007 13:56 On 7 Dec, 16:17, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >There are three possibilities: > >1) We do not have the foggiest idea if PA is consistent and we will > >never know. Hence we do not know if the Goedel sentence is true. > >2) The human mind surpasses any computer > >3) There exists a formalization of arithmetic that can prove its own > >consistency. > > I believe that all three of those are false, so I reject your > claim that those are the only three possibilities. Your number 1 > assumes that: > > To know X means that X is provable in some system T0 > such that T0 is provably consistent in some system T1 > such that T1 is provably consistent in some system T2 > such that ... > > By that definition, we never know anything. I suppose that > might be correct, in some sense, but the way most people > use the word "know" is something short of proof. Or I should > say *different* from proof. Proof is neither necessary nor > sufficient for knowledge. > > In any case, whether or not we can be said to *know* with > certainty that PA is consistent, it is certainly false to > say that "we do not have the foggiest idea if PA is consistent". > That's completely wrong. The consistency of PA is as certain > as *anything*. There is no reason to believe otherwise, and > plenty of reason to believe it. > > >But it is not possible to reject 1 and 3 and at the same time claim > >that the human mind does NOT surpass any machine. > > Sure, it's possible. I reject 1 and 3 and at the same time calim > that the human mind does not surpass any machine. > > -- > Daryl McCullough > Ithaca, NY That seems a pretty good diagnosis. It does seem, indeed, that some of the suspect claims people make e.g. about our supposedly not be able know that PA is consistent have nothing much to do the philosophy of mathematics itself, but rather arise from importing the sort of dubious assumptions about what is required for knowledge generally that leads to rampant across-the-board scepticism.
From: pboparis on 7 Dec 2007 14:07 On Dec 7, 5:17 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > In any case, whether or not we can be said to *know* with > certainty that PA is consistent, it is certainly false to > say that "we do not have the foggiest idea if PA is consistent". > That's completely wrong. The consistency of PA is as certain > as *anything*. Well, it seems to me that this last assertion is completely wrong. "2 + 2 = 4" seems more certain to me than the consistency of PA, and I imagine the same is true for you, as well. After all, the consistency of PA - that a particular logico-mathematical system does not ever produce among its deductions "not 0 = 0" - presumably depends on an argument for you to believe it. "2 + 2 = 4" does not. > There is no reason to believe otherwise, and > plenty of reason to believe it. There is plenty of reason to believe that water boils at 100 degrees centigrade, and no reason to believe otherwise. That hardly means that it is "as certain as anything."
From: pboparis on 7 Dec 2007 14:08 On Dec 7, 7:56 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 7 Dec, 16:17, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > > Newberry says... > > > >There are three possibilities: > > >1) We do not have the foggiest idea if PA is consistent and we will > > >never know. Hence we do not know if the Goedel sentence is true. > > >2) The human mind surpasses any computer > > >3) There exists a formalization of arithmetic that can prove its own > > >consistency. > > > I believe that all three of those are false, so I reject your > > claim that those are the only three possibilities. Your number 1 > > assumes that: > > > To know X means that X is provable in some system T0 > > such that T0 is provably consistent in some system T1 > > such that T1 is provably consistent in some system T2 > > such that ... > > > By that definition, we never know anything. I suppose that > > might be correct, in some sense, but the way most people > > use the word "know" is something short of proof. Or I should > > say *different* from proof. Proof is neither necessary nor > > sufficient for knowledge. > > > In any case, whether or not we can be said to *know* with > > certainty that PA is consistent, it is certainly false to > > say that "we do not have the foggiest idea if PA is consistent". > > That's completely wrong. The consistency of PA is as certain > > as *anything*. There is no reason to believe otherwise, and > > plenty of reason to believe it. > > > >But it is not possible to reject 1 and 3 and at the same time claim > > >that the human mind does NOT surpass any machine. > > > Sure, it's possible. I reject 1 and 3 and at the same time calim > > that the human mind does not surpass any machine. > > > -- > > Daryl McCullough > > Ithaca, NY > > That seems a pretty good diagnosis. It does seem, indeed, that some of > the suspect claims people make e.g. about our supposedly not be able > know that PA is consistent have nothing much to do the philosophy of > mathematics itself, but rather arise from importing the sort of > dubious assumptions about what is required for knowledge generally > that leads to rampant across-the-board scepticism. Sigh.
From: Daryl McCullough on 7 Dec 2007 16:44
pboparis(a)gmail.com says... > >On Dec 7, 5:17 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> In any case, whether or not we can be said to *know* with >> certainty that PA is consistent, it is certainly false to >> say that "we do not have the foggiest idea if PA is consistent". >> That's completely wrong. The consistency of PA is as certain >> as *anything*. > >Well, it seems to me that this last assertion is completely wrong. "2 >+ 2 = 4" seems more certain to me than the consistency of PA, and I >imagine the same is true for you, as well. Well, I should probably say as certain as any nontrivial mathematics. I don't think it is possible to do anything nontrivial without PA (or something equivalent). >After all, the consistency of PA - that a particular >logico-mathematical system does not ever produce among >its deductions "not 0 = 0" - presumably depends on an >argument for you to believe it. "2 + 2 = 4" does not. Sure it does. >> There is no reason to believe otherwise, and >> plenty of reason to believe it. > >There is plenty of reason to believe that water boils at 100 degrees >centigrade, and no reason to believe otherwise. That hardly means >that it is "as certain as anything." Well, I'm mainly taking issue with the claim that "we do not have the foggiest idea if PA is consistent". That's not true. -- Daryl McCullough Ithaca, NY |