'When Are Relations Neither True Nor False?
in [Math]
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From: Marshall on 11 Apr 2010 20:46 On Apr 11, 5:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>> I reject your claim that intuition, of whatever kind, plays > >>> any part in our thinking about the naturals. > >> So is the naturals collectively a finite syntactical notion *to you*, > >> since you'd reject the idea they're an intuition notion? > > > I reject the idea that the naturals are anything particular *to me* > > that they are not to anyone else. > > The question was simply asking you to explain from your mathematical > knowledge what the natural number be. That's all it was asked of you. > Whether your concept of the naturals is the same as mine, or TF's, > or anybody else' isn't relevant to the question. Exactly. The "*to you*" part of your question was irrelevant, so I rejected it. Fortunately we both agree it is irrelevant. > Can you share with us your resounding, firm, answer to that simple > question? You mean, again? You want me to say "yes" again? Okay: Yes. > > Other than that, and as much as I hesitate to accede to any > > formula of yours given how unreliably you use terminology, > > my answer is "yes." We can completely capture enough > > about the naturals to uniquely characterize them up to > > isomorphism. For example, we have many syntactic > > representations of natural numbers available to us that > > can represent any natural up to resource limits, and we > > have simple algorithms on those representations that > > can compute successor, addition, etc. of any naturals > > up to resource limits. These things are entirely > > mechanical, and free of any vague or intuitive aspects. > > I'm sorry: "up to resource limits" isn't an intuition or technical > notion of the natural numbers. So you don't seem to understand > the notion of the natural numbers, despite your having claimed > otherwise. Of course it's not an intuition. I've already made it clear that I'm not talking about that. But it certainly *is* a technical notion when discussing any "entirely mechanical" implementation of an algorithm. > > I am under no delusions that you will agree with me, > > but please spare me the umpteenth repetition of your > > GC counterargument. > > Until you demonstrate precisely what you meant by the naturals, > while asserting they're not of intuition notion, I wouldn't > have a reason not to repeat my GC counter argument. I am pretty sure a marginally competent ten year old can explain what a base ten natural number is (whether or not his school uses those exact terms) and can explain how to find the successor, how to add two naturals, multiply, etc. I could certainly supply you with my own version of that explanation, but if you can't already do it yourself, then I propose you ought to first go learn that. If you can, then there is no point in my retyping it. Marshall
From: Nam Nguyen on 11 Apr 2010 22:51 Marshall wrote: > On Apr 11, 5:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>> I reject your claim that intuition, of whatever kind, plays >>>>> any part in our thinking about the naturals. >>>> So is the naturals collectively a finite syntactical notion *to you*, >>>> since you'd reject the idea they're an intuition notion? > >> Can you share with us your resounding, firm, answer to that simple >> question? > > You mean, again? You want me to say "yes" again? Okay: > > Yes. So your resounding answer is "yes" that the notion of the naturals is based on an intuition but is a notion of syntactical formal system. How would you know though such a formal system is syntactical consistent? > > >>> Other than that, and as much as I hesitate to accede to any >>> formula of yours given how unreliably you use terminology, >>> my answer is "yes." We can completely capture enough >>> about the naturals to uniquely characterize them up to >>> isomorphism. For example, we have many syntactic >>> representations of natural numbers available to us that >>> can represent any natural up to resource limits, and we >>> have simple algorithms on those representations that >>> can compute successor, addition, etc. of any naturals >>> up to resource limits. These things are entirely >>> mechanical, and free of any vague or intuitive aspects. >> I'm sorry: "up to resource limits" isn't an intuition or technical >> notion of the natural numbers. So you don't seem to understand >> the notion of the natural numbers, despite your having claimed >> otherwise. > > Of course it's not an intuition. I've already made it clear that > I'm not talking about that. But it certainly *is* a technical > notion when discussing any "entirely mechanical" implementation > of an algorithm. "Entirely mechanical" is not a technical definition, just in case you're not aware. If the naturals are of syntactical notion then they must be described by a formal system. So far you keep saying it's syntactical but giving no hint what such natural-number formal system be! > > I am pretty sure a marginally competent ten year old > can explain what a base ten natural number is (whether > or not his school uses those exact terms) and can > explain how to find the successor, how to add two > naturals, multiply, etc. But that's what we'd call _intuition_ and the 10-year old would most likely never claim he could use such an intuition to prove the consistency of PA, right? But AK would do so and you'd do the same or at least support such a claim. > I could certainly supply you > with my own version of that explanation, but if you > can't already do it yourself, then I propose you > ought to first go learn that. If you can, then there > is no point in my retyping it. Oh I learnt the naturals numbers the same way mathematicians would learn: it's an intuitive notion! It's you who'd go against the mainstream of mathematicians and made the following odd notion: >>>>> I reject your claim that intuition, of whatever kind, plays >>>>> any part in our thinking about the naturals. [What textbook's author would even hint such a bogus an idea?]
From: Nam Nguyen on 11 Apr 2010 23:01 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> To claim some formula being >> absolutely true (or false) is to destroy the notion of such symmetry. > > Do you agree that symmetry is only > broken if the cases "P" and "not P" are treated differently? Of course I'd not agree. Even in relativity, given an appropriate context, P and ~P must necessarily be treated differently because of LEM. The Principle addresses a different issue: P or ~P can't be uniformly treated as true or false in _all_ contexts. That's all the Principle of Symmetry would stipulate, and all I've really said (as above).
From: Nam Nguyen on 11 Apr 2010 23:02 Nam Nguyen wrote: > Marshall wrote: >> On Apr 11, 5:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> Marshall wrote: >>>> On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> I reject your claim that intuition, of whatever kind, plays >>>>>> any part in our thinking about the naturals. >>>>> So is the naturals collectively a finite syntactical notion *to you*, >>>>> since you'd reject the idea they're an intuition notion? >> >>> Can you share with us your resounding, firm, answer to that simple >>> question? >> >> You mean, again? You want me to say "yes" again? Okay: >> >> Yes. > > So your resounding answer is "yes" that the notion of the naturals > is based on an intuition but is a notion of syntactical formal system. > How would you know though such a formal system is syntactical > consistent? I meant "... is NOT based on an intuition but is a notion ..." > >> >> >>>> Other than that, and as much as I hesitate to accede to any >>>> formula of yours given how unreliably you use terminology, >>>> my answer is "yes." We can completely capture enough >>>> about the naturals to uniquely characterize them up to >>>> isomorphism. For example, we have many syntactic >>>> representations of natural numbers available to us that >>>> can represent any natural up to resource limits, and we >>>> have simple algorithms on those representations that >>>> can compute successor, addition, etc. of any naturals >>>> up to resource limits. These things are entirely >>>> mechanical, and free of any vague or intuitive aspects. >>> I'm sorry: "up to resource limits" isn't an intuition or technical >>> notion of the natural numbers. So you don't seem to understand >>> the notion of the natural numbers, despite your having claimed >>> otherwise. >> >> Of course it's not an intuition. I've already made it clear that >> I'm not talking about that. But it certainly *is* a technical >> notion when discussing any "entirely mechanical" implementation >> of an algorithm. > > "Entirely mechanical" is not a technical definition, just in case > you're not aware. If the naturals are of syntactical notion then > they must be described by a formal system. So far you keep saying > it's syntactical but giving no hint what such natural-number formal > system be! > >> >> I am pretty sure a marginally competent ten year old >> can explain what a base ten natural number is (whether >> or not his school uses those exact terms) and can >> explain how to find the successor, how to add two >> naturals, multiply, etc. > > But that's what we'd call _intuition_ and the 10-year old > would most likely never claim he could use such an intuition > to prove the consistency of PA, right? But AK would do > so and you'd do the same or at least support such a claim. > >> I could certainly supply you >> with my own version of that explanation, but if you >> can't already do it yourself, then I propose you >> ought to first go learn that. If you can, then there >> is no point in my retyping it. > > Oh I learnt the naturals numbers the same way mathematicians > would learn: it's an intuitive notion! > > It's you who'd go against the mainstream of mathematicians > and made the following odd notion: > > >>>>> I reject your claim that intuition, of whatever kind, plays > >>>>> any part in our thinking about the naturals. > > [What textbook's author would even hint such a bogus an idea?]
From: Nam Nguyen on 12 Apr 2010 00:55
Nam Nguyen wrote: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> To claim some formula being >>> absolutely true (or false) is to destroy the notion of such symmetry. >> >> Do you agree that symmetry is only >> broken if the cases "P" and "not P" are treated differently? > > Of course I'd not agree. Even in relativity, given an appropriate context, > P and ~P must necessarily be treated differently because of LEM. > The Principle addresses a different issue: P or ~P can't be uniformly > treated as true or false in _all_ contexts. That's all the Principle of > Symmetry would stipulate, and all I've really said (as above). To be fair the "standard theorists" and I don't seem to fight on the relativity of mathematical truth in general. They know ~(1+1=0) is true or false, relative to what kind of arithmetic we've chosen as the underlying one (e.g. arithmetic modulo 2 or arithmetic modulo 3). The battle here is they'd believe our intuition of the natural numbers (or equivalently our intuition of the standard model of PA) is an absolute notion, that everyone "must" intuit them exactly the same way, while I've been saying that they're wrong in that respect. And they're wrong by more than one account, but the first one being that the concept of the natural numbers is an intuition which is subjective (hence relative) to an individual perception. You might believe GC is true in your perception of the naturals while I may believe "there are infinitely many counter examples of GC" is true in my own perception of them, and none of us is _absolutely_ correct or incorrect. Their belief on the absolute truths of the natural numbers is very much akin to the belief before that arithmetic syntactical provability be absolute in the sense that either a formula or its negation written in the language of arithmetic must be _syntactically provable_ and there be no middle ground. Some how they've managed not to pay attention to the history that has only been fairly recent. |