The End of an Era - That of The Natural Numbers
in [Math]
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From: William Hughes on 5 May 2010 23:55 On May 5, 11:38 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Alan Smaill wrote: > > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > >> William Hughes wrote: > >>> On May 4, 9:30 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> William Hughes wrote: > >>>>> On May 4, 2:21 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>> <snip> > >>>>>>> It is interesting to note that while you have been presented with > >>>>>>> many putative "intuitions" given which the truth or falsehood > >>>>>>> of (1) is knowable (you have accepted none and explicitly rejected > >>>>>>> one), you have not presented a single "intuition" under which the > >>>>>>> truth or falsehood of (1) is not knowable. > >>>>>> That's correct: I have not - yet. That doesn't mean I'm not going to. > >>>>> I'm not holding my breath. > >>>> If you don't have a good faith on that then that's your issue and > >>>> isn't my concern. [Btw, the post about imprecision in reasoning and > >>>> the recent T post are part of the explanation. So in effect I've been > >>>> doing the explanation, whether or not you're listening to.] > >>> Can give an intuition without using multiple > >>> long posts? Forget about Observation 1. > >>> Just give an example of an intuition > >>> Any intuition will do. > >> OK. If you just want a short description intuition about (1) then here > >> it is. > > >> Intuitively, to see _either_ cGC or ~cGC as true or false, you have to > >> do the same impossible thing: transverse the entire set of natural numbers > >> to figure it out, hence (again intuitively) it's impossible to know the > >> truth value of cGC, hence of (1). > > >> [In contrast, intuitively it's not impossible to see ~GC as true since > >> a counter example is still a distinct possibility. So in principle, > >> we can't say it's impossible to know the truth value of GC, though > >> *IF* GC is genuinely true then intuitively it's impossible to know so.] > > >> And that is as short as I could put it. > > > Do you also have the intuition that it is impossible to see that > > the associativity of addition holds for natural numbers, > > where addition is defined as usual for the recursive definition > > (0 case and successor case)? > > I don't happen to have that intuition. But what would you mean by the > "natural numbers"? Those in which cGC is true? Or ~cGC is true?
From: Nam Nguyen on 6 May 2010 00:38 William Hughes wrote: > On May 5, 2:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > >> what's the difference between >> your "if GC is true we can show that it is true" and >> my "can show GC true if it's true" in my question to you? > > Nothing. However note, I am not claiming that > > A: we can show GC true if it's true > > > A is not yet known and may never be known. > I am claiming that A is my *guess*. > (In detail my guess is that T is sound > and therefore something provable in T is > true (although something true may not be > provable in T) and that GC is provable in T) > The question is not whether my guess is right > or wrong, the question is whether my guess > qualifies as an intuition. I'm probably not much interested in fine distinction between the semantics of "guess" and "intuition". Both sound the same to me in this context. My opinion is that guess and intuition _should be backed up_ by _some_ reasoning, and not some kind of whatsoever-intuition. But intuition is intuition and could be virtually any guessing, anything at all.
From: William Hughes on 6 May 2010 01:07 On May 6, 1:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 5, 2:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > >> what's the difference between > >> your "if GC is true we can show that it is true" and > >> my "can show GC true if it's true" in my question to you? > > > Nothing. However note, I am not claiming that > > > A: we can show GC true if it's true > > > A is not yet known and may never be known. > > I am claiming that A is my *guess*. > > (In detail my guess is that T is sound > > and therefore something provable in T is > > true (although something true may not be > > provable in T) and that GC is provable in T) > > The question is not whether my guess is right > > or wrong, the question is whether my guess > > qualifies as an intuition. > > I'm probably not much interested in fine distinction > between the semantics of "guess" and "intuition". > Both sound the same to me in this context. My opinion > is that guess and intuition _should be backed up_ by > _some_ reasoning, and not some kind of whatsoever-intuition. > But intuition is intuition and could be virtually any guessing, > anything at all. So the above intuition does qualify as "any intuition" and your First Observation is false as written. Presumably, by "any intuition" you did not mean just any intuition, but an intuition that is "backed up by _some_ reasoning". However, this is too vague to allow us to determine if an intuition is acceptable. (How much is _some_, who decides if the reasoning does in fact back up the intuition). Next time, define your terms precisely, _before_ you proclaim an end of an era. - William Hughes
From: Nam Nguyen on 6 May 2010 01:22 William Hughes wrote: > On May 6, 1:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> On May 5, 2:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> what's the difference between >>>> your "if GC is true we can show that it is true" and >>>> my "can show GC true if it's true" in my question to you? >>> Nothing. However note, I am not claiming that >>> A: we can show GC true if it's true >>> A is not yet known and may never be known. >>> I am claiming that A is my *guess*. >>> (In detail my guess is that T is sound >>> and therefore something provable in T is >>> true (although something true may not be >>> provable in T) and that GC is provable in T) >>> The question is not whether my guess is right >>> or wrong, the question is whether my guess >>> qualifies as an intuition. >> I'm probably not much interested in fine distinction >> between the semantics of "guess" and "intuition". >> Both sound the same to me in this context. My opinion >> is that guess and intuition _should be backed up_ by >> _some_ reasoning, and not some kind of whatsoever-intuition. >> But intuition is intuition and could be virtually any guessing, >> anything at all. > > So the above intuition does qualify as "any > intuition" and your First Observation is false > as written. > > Presumably, by "any intuition" you did not mean just > any intuition, but an intuition that is "backed up by > _some_ reasoning". However, this is too vague to allow > us to determine if an intuition is acceptable. (How much > is _some_, who decides if the reasoning does in fact back > up the intuition). Next time, define your terms precisely, > _before_ you proclaim an end of an era. So everything boils down to define term such as "intuition" in a technical debate? Great!
From: Nam Nguyen on 6 May 2010 01:24
Nam Nguyen wrote: > William Hughes wrote: >> On May 6, 1:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> William Hughes wrote: >>>> On May 5, 2:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>> what's the difference between >>>>> your "if GC is true we can show that it is true" and >>>>> my "can show GC true if it's true" in my question to you? >>>> Nothing. However note, I am not claiming that >>>> A: we can show GC true if it's true >>>> A is not yet known and may never be known. >>>> I am claiming that A is my *guess*. >>>> (In detail my guess is that T is sound >>>> and therefore something provable in T is >>>> true (although something true may not be >>>> provable in T) and that GC is provable in T) >>>> The question is not whether my guess is right >>>> or wrong, the question is whether my guess >>>> qualifies as an intuition. >>> I'm probably not much interested in fine distinction >>> between the semantics of "guess" and "intuition". >>> Both sound the same to me in this context. My opinion >>> is that guess and intuition _should be backed up_ by >>> _some_ reasoning, and not some kind of whatsoever-intuition. >>> But intuition is intuition and could be virtually any guessing, >>> anything at all. >> >> So the above intuition does qualify as "any >> intuition" and your First Observation is false >> as written. >> >> Presumably, by "any intuition" you did not mean just >> any intuition, but an intuition that is "backed up by >> _some_ reasoning". However, this is too vague to allow >> us to determine if an intuition is acceptable. (How much >> is _some_, who decides if the reasoning does in fact back >> up the intuition). Next time, define your terms precisely, >> _before_ you proclaim an end of an era. > > So everything boils down to define term such as "intuition" in > a technical debate? Great! I forgot people must also define "some" too! |