'When Are Relations Neither True Nor False?
in [Math]
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From: Marshall on 13 Apr 2010 09:43 On Apr 12, 11:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > > Yes you are. That's what's so funny. > > Are you taking a cheap shot at my typo "taliking"? No. > If not, what's > so funny about "the relativity of mathematical truth"? Buffoons are funny. Marshall
From: Marshall on 13 Apr 2010 10:03 On Apr 12, 9:18 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Apr 11, 7:51 pm, Nam Nguyen <namducngu...(a)shaw.ca> 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. > > > Not a "formal system" in this sense: > > >http://en.wikipedia.org/wiki/Formal_system > > > The naturals and their operators are a model, not a theory. > > Not only you're clueless as to exactly what the natural numbers be, > you also don't use the technical terms correctly. One of the things that makes you so amusing is the way you thrash around any time anyone speaks of something with which you are unfamiliar. It's hilarious how huffy you get whenever anyone uses a term in a way you don't like, yet at the same time you always feel free to change the meaning of your own symbols in capricious ways. > First of all, if the naturals are collectively a model of a language > then you should have not resoundingly said "yes" that they'd be "a finite > syntactical notion", which refer to syntactical symbols, formulas, axioms, > axiom-system, rules of inference: but not to interpretation, or truth values > as when we talk about model truths. Theories are syntactical. That does not mean that theories are the only things that are syntactical. > Secondly model would *require* intuition: if there's no intuition there's no > interpretation, no truth values, hence no model! _You've contradicted yourself_ > by first rejecting intuition "plays any part in our thinking about the naturals" > and now stating "The naturals and their operators are a model". Hilarious phrases in this paragraph: "model would require intuition" and "you've contradicted yourself." Oooohhhhhh you caught me. Not. In another post you used the word "intuition" to mean "vague." I'm sure even you don't know what you're using it to mean here. Something to score debating points with, I guess. > >> How would you know though such a formal system is syntactical > >> consistent? > > > The question of consistency is asked of theories, not of > > models. I guess you are not really clear about the distinction. > > I wasn't asking you about the general consistency. I was asking you > about *your contradictory claims* about the natural numbers. SURE you were. > My hope > here was that my question would help you to realize how wrong you > were in rejecting intuition "plays any part in our thinking about > the naturals". Why don't you enlighten us all, Einstein, about what you mean by "intuition". Go on. It's a simple term, and clearly very important to you. Define it exactly. > > Since the naturals and their operators are a model, > > it makes no sense to ask about their consistency. > > Again it was about one of your contradictory claims that the naturals > are of syntactical notion, which is something that we'd *only* refer > to when talking about formal systems. Your statements flip-flop back > and forth between the naturals being model theoretical and syntactical. > Being model theoretical and being syntactical are 2 different (and opposing) > notions. Hilarious! > Can you now make up your mind and tell us once for all, which > ways the natural numbers are: model theoretical, or syntactical? The naturals are a model, which happens to be a model that can be entirely captured by syntactic mechanisms. Deal with it. > >> and the 10-year old > >> would most likely never claim he could use such an intuition > >> to prove the consistency of PA, right? > > > Sure. But we were talking about the natural numbers, not PA. > > You're wrong. My conversation here started when I responded to > AK's statement: Oh, ok. Because your conversation started with talking with someone else about PA, every time you've mentioned the natural numbers when talking to me it's meant PA. Got it. > So, Marshall. Let's not forget the conversation here is about > the proof consistency of PA, as well as about intuition-or-not- > intuition of the naturals. I await your exciting answer to what "intuition" means. Of course, knowing you, the meaning of the term will change every time you use it to suit your claim-of-the-moment. > > I am a poor judge of what AK would say. > > Then perhaps you should have stayed silent instead of trying > to defend someone whose statement-validities you only have a > "poor" judgment - at best! ROFL > > For myself, I am > > unclear enough about what you mean when you say "intuition" > > to be able to answer you. > > Since you don't understand a very simple notion such as intuition > [that a 10-year old student would understand], let me explain that > to you. > > You yourself claimed that the natural numbers [collectively] > is a model [I'm assuming of the language L(PA)]. But how would you > know that for sure? The answer is you CAN NOT know that for sure, > because you yourself will NEVER be able to list out all the necessary > n-ary relations (sets) required by the definition of a language model. > The only reason left for you to have a feeling that you know the > naturals is a model at all is your INTUITION. Do you understand now > that you were wrong before when you resoundingly claimed: Buffoon. > >> I reject your claim that intuition, of whatever kind, plays > >> any part in our thinking about the naturals. > > > My best guess would be: would > > we consider the hunch of a ten year old to be a proof that > > PA is consistent? Speaking for myself, no. If you meant > > something else, please clarify. > > Above I already explained to you the simple notion of intuition that a > 10-year old would understand! You have written nothing whatsoever that says what intuition is. Your linguistic incompetence betrays you. No, I am not talking about English here. I am confident that you are just as incompetent in your native tongue about things like semantics. > > If you do decide to clarify, > > please recall that PA is not the natural numbers, that PA is > > a theory and the naturals+their operators are a model, > > and that consistency is a question that only applies to > > theories. > > Where did I claim otherwise? I could go on but I'm getting bored. Say more funny things. Dance, buffoon. Marshall
From: Nam Nguyen on 14 Apr 2010 23:26 David Bernier wrote: > > Somebody referred to Strawson, the logician/philosopher. > Would you know if Strawson thought of a "Principle of Symmetry" > similar to yours? I read briefly the first few sections of his writing from the link you gave but didn't see anything about any "Principle of Symmetry". Iirc, what I read is in PS form so I couldn't search for the phrase. If anyone knows the page number (if he actually wrote something about it), much appreciated if you could let me know. > > Presumably, this "Principle of Symmetry" you mention has been > discussed in some book or article. Would you know > of some reference for this? I actuallly came up with that in an impromptu way, though I might have unconsciously remembered a similar phrase "Symmetry Breaking" in some kind of physics book (perhaps QM) I had read years ago. I've googled "Principle of Symmetry" but found nothing directly for mathematical logic, in the way I stipulated it. I don't think anyone has thought of the term in conjunction with mathematical reasoning, since the phrase implies a high degree of "relativity" but our mathematical reasoning has been (rather wrongly) based in absolutism (Platonism).
From: Nam Nguyen on 14 Apr 2010 23:43 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Alan Smaill wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> 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. >>> Now you've really lost me. >>> LEM *is* symmetric itself, isn't it? >> I have no idea what you meant here by "LEM *is* symmetric itself". >> You have to explain what you meant by that. >> >> The word "symmetry" I've used here is in the context of the Principle >> of Symmetry I articulated before with some clear examples and are about >> concepts, formulas - but not about LEM]. > > LEM can be expressed by all formulae of a certain syntactic shape > (P v not P) in the language to hand. > Whether you take P as true or P as false makes no difference to > LEM. In that sense, swapping P with not P makes no difference, > LEM will apply in either case. > > Perhaps you can explain why "P and ~P must necessarily be treated > differently because of LEM"? (Are you maybe thinking of what > is called the law of non-contradiction, rather than the excluded middle?) I think you misunderstood the context in which I've been talking. The point I've been making is that a formula (other that those that are tautologous or contradictory) would never have an absolute truth. For example Ax[~(Sx=0)] cann't be absolutely true or false: there must always be a context when we say it's true, or false. So without a context the formula is neither true nor false, seemingly violating LEM. But once we choose a context, say a model of a language, then the formula must conform to LEM: it must either true or false and its negation must be opposite (hence different); and there's no violation of LEM whatsoever. And that's what I meant in "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). >>> So how do you know it is relevant to the particular context at hand, >>> that of assuming "not P" hypothetically (ie for sake of argument), deriving >>> a contradiction, and concluding "P". And also in the context where "P" >>> and "not P" are switched systematically in such an argument? >> I'm not sure what you're trying to argue here. My argument only concerns >> the alleged "absolute" truths that DB tried to suggest, and in which I >> counter claimed any such absolute truth would break the Principle of >> Symmetry. That's all I've really argued in this conversation! > > Still not convinced on my side. > I think we've gone as far as we can productively go, however. >
From: Nam Nguyen on 15 Apr 2010 00:34
Marshall wrote: > On Apr 12, 9:18 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Apr 11, 7:51 pm, Nam Nguyen <namducngu...(a)shaw.ca> 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. >>> Not a "formal system" in this sense: >>> http://en.wikipedia.org/wiki/Formal_system >>> The naturals and their operators are a model, not a theory. >> Not only you're clueless as to exactly what the natural numbers be, >> you also don't use the technical terms correctly. > > One of the things that makes you so amusing is the way > you thrash around any time anyone speaks of something > with which you are unfamiliar. It's hilarious how huffy you > get whenever anyone uses a term in a way you don't > like, yet at the same time you always feel free to change > the meaning of your own symbols in capricious ways. There's no "like" or "dislike" on my part, here. You clearly claimed the "naturals and their operators are a model" but you haven't proved they satisfy the technical definition of model, have you? And since you didn't prove the naturals are model theoretical and yet wrongly assert rejected they are of an intuitive notion, I just pointed out whatever you said about them including your rejection are incorrect. That's all. So again, how did you arrive at the conclusion that the naturals meet the definition of language model? Did you, as I mentioned before, define all the n-ary relations (sets) required by model- definition? Also, did you now realize that the naturals are of intuitive notion? These are technical matters and questions. Either you have some clues what they're or you don't but there's no "like" or "dislike" in these matters. |