The End of an Era - That of The Natural Numbers
in [Math]
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From: Jesse F. Hughes on 19 Jun 2010 09:00 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > You got to read people's caveat more carefully, before jumping to > conclusion and .... > >> >> Lordy, but you're incapable of reading a relatively clear and simple >> text. > > and before committing yourself at kind of pathetic dialog, as above. > > As far as I'm concerned, you're in such attack mode whenever you want > to create a smokescreen to hide your technical mistakes. Let me just ask you: isn't it remarkable that a single self-taught logician like you is the *only one* who understands Shoenfield's text and also the *only one* who really gets FOL? That no one else present here can read that text and understand it correctly? Strange, isn't it, that those of us who have been formally trained and evaluated all misunderstand these remarkably basic things, but that a self-taught individual like yourself grasps this text (a very standard text) perfectly. No point, I guess. I just wonder how it is that something like this happens. And no, I'm not offering this to rebut your brilliant "technical" analysis. I see now that this is impossible. I just am curious as to your reaction to your singular brilliance in a newsgroup of well-trained morons. -- "Rob Enderle [predicts] that Longhorn will provide 'vast improvements in security.' We can cheer this happy prospect, but at the same time we must ignore the snide laughs of Macintosh users who have yet to encounter a virus..." -- New York Times
From: Aatu Koskensilta on 19 Jun 2010 10:56 Marshall <marshall.spight(a)gmail.com> writes: > This banana on the counter over here is a model of the > language of basic arithmetic. It doesn't have a member > of the domain that models the constant zero, but it *is* > high in potassium. Don't you rather mean "high on potassium"? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 21 Jun 2010 02:36 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> You got to read people's caveat more carefully, before jumping to >> conclusion and .... >> >>> Lordy, but you're incapable of reading a relatively clear and simple >>> text. >> and before committing yourself at kind of pathetic dialog, as above. >> >> As far as I'm concerned, you're in such attack mode whenever you want >> to create a smokescreen to hide your technical mistakes. > > Let me just ask you: isn't it remarkable that a single self-taught > logician like you is the *only one* who understands Shoenfield's text > and also the *only one* who really gets FOL? > > That no one else present here can read that text and understand it > correctly? > > Strange, isn't it, that those of us who have been formally trained and > evaluated all misunderstand these remarkably basic things, but that a > self-taught individual like yourself grasps this text (a very standard > text) perfectly. On June 6, you wrote: > I said that p.19 from Shoenfield defines truth in a structure only for > closed formulas. (Someone else -- Daryl? -- says that according to > Shoenfield, x=x is in fact *not* true or false, but rather valid, > because true/false applies only to closed formulas. I assume he's > correct, but I haven't checked.) > > Again: p. 19 does not define truth for open formulas. Which is wrong because you failed to recognize that a) Shoenfiel's "valid" on pages 18, 19, and 22 are actually "true" [and I had already mentioned this on May 16th, 8:31 am], and b) his sense of "valid" is also different from _another_ sense of valid on page 20. Now, I had this request to Aatu (June 16): > So what's in you mind the definition of x=x being true be, model > theoretically speaking? Can you _explicitly_ state the definition? to which he had this 1-sentence terseness, which really sounded like a joke than any seriousness: > Just read some more Shoenfield. I don't know for fact, but I never doubt you both are much more trained in the matter than I'm (who's just an amateur at best). But you both should have _NOT_ taken it as _a blank check_ that your reasoning would be "perfect" , never have any problem, and could stay above all scrutinies that might be required as part of reasoning _discipline_! If you failed to recognize Shoenfield's definition of model truth, and if Aatu failed the "obligation" to articulate the very definition he'd need to defend x=x is true in all contexts of FOL, then you both _didn't give me much of a choice_ to trust what you were saying. And that's not an insult or anything personal: that's just how mathematical arguments would go. *** Now, earlier this year, you corrected a mistake of mine on some definition. In my note, I wrote following: > x Lnext y <-> (x < y) & Az[(z < y) -> (y=z)] // Incorrect! > x Lnext y <-> x < y & (Az)(x < z -> y <= z) // Correct from Jesse! (If you request I'll email to you a zip file containing the file [containing this note] that has the last modification date of Feb., many months before this thread was even created!) So you see, even if I argue with people I don't disrespect their knowledge and I always try to better my limited knowledge by recognizing when I err and by learning what they know and I don't. That doesn't mean though I won't point out to them what I believe as their mistakes and neither would that mean they'd have a black check to say whatever they'd feel pleased and disregard the wisdom, something to the effect that, "to be human is to err". I think in mathematical reasoning we _all_ should have the same amount of, say, "discipline" and expectation to argue in the correct manner: adhering to agreed definitions, to agreed methods of reasoning, and being impartial in making logical judgment. I think one of the mistake you and a couple of others on your side have made is to have taken your opponent for granted: you don't know my motivation and you don't recognize my discipline adhering to definition and reasoning process, and consequently you've badly impaired your judgment, impartiality, and straightforwardness. > > No point, I guess. I just wonder how it is that something like this > happens. I think I know why it happened: (a) a few people on your side have a tendency to "blast" me first (as crank, troll, having bizarre ideas), before even "half-heartedly" considering what I was presenting; and (b) you and a few others already made up your mind about the absoluteness of arithmetic truths and would slam the door shut when I tried to explain "relativity in reasoning", lest that if you even hear that phrase somehow we all would become cranks! > And no, I'm not offering this to rebut your brilliant > "technical" analysis. I see now that this is impossible. I'm sorry: if you already made up your mind there's no relativity in reasoning, then it's impossible. (But I think you know there's a saying something to the effect that smashing an atom is easier than smashing an opinion) > I just am > curious as to your reaction to your singular brilliance in a newsgroup > of well-trained morons. But I don't have such reaction because that's you accusation, not mine. The question I have for you, Aatu, and Marshall though is why you three think you're infallible and above the rigorousness of mathematical reasoning?
From: Marshall on 21 Jun 2010 11:02 On Jun 20, 11:36 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Jesse F. Hughes wrote: > > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > >> You got to read people's caveat more carefully, before jumping to > >> conclusion and .... > > >>> Lordy, but you're incapable of reading a relatively clear and simple > >>> text. > >> and before committing yourself at kind of pathetic dialog, as above. > > >> As far as I'm concerned, you're in such attack mode whenever you want > >> to create a smokescreen to hide your technical mistakes. > > > Let me just ask you: isn't it remarkable that a single self-taught > > logician like you is the *only one* who understands Shoenfield's text > > and also the *only one* who really gets FOL? > > > That no one else present here can read that text and understand it > > correctly? > > > Strange, isn't it, that those of us who have been formally trained and > > evaluated all misunderstand these remarkably basic things, but that a > > self-taught individual like yourself grasps this text (a very standard > > text) perfectly. > > On June 6, you wrote: > > > I said that p.19 from Shoenfield defines truth in a structure only for > > closed formulas. (Someone else -- Daryl? -- says that according to > > Shoenfield, x=x is in fact *not* true or false, but rather valid, > > because true/false applies only to closed formulas. I assume he's > > correct, but I haven't checked.) > > > > Again: p. 19 does not define truth for open formulas. > > Which is wrong because you failed to recognize that a) Shoenfiel's > "valid" on pages 18, 19, and 22 are actually "true" [and I had > already mentioned this on May 16th, 8:31 am], and b) his sense > of "valid" is also different from _another_ sense of valid on page > 20. > > Now, I had this request to Aatu (June 16): > > > So what's in you mind the definition of x=x being true be, model > > theoretically speaking? Can you _explicitly_ state the definition? > > to which he had this 1-sentence terseness, which really sounded > like a joke than any seriousness: > > > Just read some more Shoenfield. > > I don't know for fact, but I never doubt you both are much more > trained in the matter than I'm (who's just an amateur at best). > But you both should have _NOT_ taken it as _a blank check_ that > your reasoning would be "perfect" , never have any problem, and > could stay above all scrutinies that might be required as part of > reasoning _discipline_! > > If you failed to recognize Shoenfield's definition of model truth, > and if Aatu failed the "obligation" to articulate the very definition > he'd need to defend x=x is true in all contexts of FOL, then you both > _didn't give me much of a choice_ to trust what you were saying. And > that's not an insult or anything personal: that's just how mathematical > arguments would go. > > *** > > Now, earlier this year, you corrected a mistake of mine on some definition. > In my note, I wrote following: > > > x Lnext y <-> (x < y) & Az[(z < y) -> (y=z)] // Incorrect! > > x Lnext y <-> x < y & (Az)(x < z -> y <= z) // Correct from Jesse! > > (If you request I'll email to you a zip file containing the file > [containing this note] that has the last modification date of Feb., > many months before this thread was even created!) > > So you see, even if I argue with people I don't disrespect their > knowledge and I always try to better my limited knowledge by recognizing > when I err and by learning what they know and I don't. > > That doesn't mean though I won't point out to them what I believe as their > mistakes and neither would that mean they'd have a black check to say > whatever they'd feel pleased and disregard the wisdom, something to the > effect that, "to be human is to err". I think in mathematical reasoning > we _all_ should have the same amount of, say, "discipline" and expectation > to argue in the correct manner: adhering to agreed definitions, to agreed > methods of reasoning, and being impartial in making logical judgment. > > I think one of the mistake you and a couple of others on your side > have made is to have taken your opponent for granted: you don't know > my motivation and you don't recognize my discipline adhering to definition > and reasoning process, and consequently you've badly impaired your judgment, > impartiality, and straightforwardness. > > > > > No point, I guess. I just wonder how it is that something like this > > happens. > > I think I know why it happened: (a) a few people on your side have a > tendency to "blast" me first (as crank, troll, having bizarre ideas), > before even "half-heartedly" considering what I was presenting; and (b) > you and a few others already made up your mind about the absoluteness > of arithmetic truths and would slam the door shut when I tried to > explain "relativity in reasoning", lest that if you even hear that phrase > somehow we all would become cranks! > > > And no, I'm not offering this to rebut your brilliant > > "technical" analysis. I see now that this is impossible. > > I'm sorry: if you already made up your mind there's no relativity > in reasoning, then it's impossible. (But I think you know there's > a saying something to the effect that smashing an atom is easier than > smashing an opinion) > > > I just am > > curious as to your reaction to your singular brilliance in a newsgroup > > of well-trained morons. > > But I don't have such reaction because that's you accusation, not mine. > The question I have for you, Aatu, and Marshall though is why you > three think you're infallible and above the rigorousness of mathematical > reasoning? tldnr
From: Nam Nguyen on 22 Jun 2010 00:48
Nam Nguyen wrote: > > Any rate, consider the context of an inconsistent theory T, is x=x > true in in such a T? Or is it that one could assign the formula > x=x to true or false as one wishes, relatively speaking? Indeed, one is free to assign x=x to either true or false as one wishes in such case. FOL formal systems (with or without =) are always syntactical objects in the first place, before any semantics or truth values can be defined for any formulas of the systems. In FOL- (without =) the truth value of x=x would be relative to which system it's even a formula. In FOL=, x=x must be true in a model where U isn't empty, and x=x is either false or could be assigned to either value when U is empty. So in the final analysis, x=x is true is relative not just between FOL- and FOL=, but within FOL= itself. Assuming we're talking about FOL=, the above observation means that given a particular language L, together with the collection K of its models, and given a formula A that's neither tautological nor contradictory, A does _NOT_ have an intrinsic truth or falsehood. If it's true, it's true _relative_ to the models in which it's interpreted as true, and likewise for false. Note though once we completely construct (define) a model M, the relativity of A's truth or falsehood would end in that model M, and M would be called a terminal model. The important consequence of the above observation is if we have an _incompletely defined model_, we'd have a collection K' of incompletely defined models in which a certain formula A would be completely relative over K', in the sense that _it's impossible to determine a terminal model_ in K' in which A would be true/false. And this case, we're _free to assume_ A is true or false, _with no adverse consequence_ in reasoning_. We're not quite there yet, but the above paragraph has brought us closer to the title of the thread, since it marks the beginning of an end of the absoluteness of the truths of the natural numbers. |