'When Are Relations Neither True Nor False?
in [Math]
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From: Marshall on 12 Apr 2010 01:36 On Apr 11, 9:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > 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). In other words, what a sentence means is "relative" to what meaning we choose for the symbols it is composed of. Of course, calling this "the relativity of mathematical truth in general" is inappropriately grandiose. The only thing this says about anything at all is: The meaning of a symbol depends on context. I'll alert the media. Marshall
From: Nam Nguyen on 12 Apr 2010 01:43 Marshall wrote: > On Apr 11, 9:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> 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). > > In other words, what a sentence means is "relative" to > what meaning we choose for the symbols it is composed of. NO. Semantics and truth are 2 _different notions_! And I'm taliking about "the relativity of mathematical truth". If you read people's writing carefully.
From: Marshall on 12 Apr 2010 02:08 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. > 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. Since the naturals and their operators are a model, it makes no sense to ask about their consistency. > >>> 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. It's a term; I never claimed it was the definition of anything. I guess this is another area where you don't know very much. I will merely note your lack of familiarity with the technical terminology around resource limits and around the mechanical execution of algorithms and move on. > If the naturals are of syntactical notion then > they must be described by a formal system. Is that your intuition? > So far you keep saying > it's syntactical but giving no hint what such natural-number formal > system be! You haven't been asking about "formal systems". I'm just describing the natural numbers, not any formal system. I've said on other occasions that you would really do well to learn the difference, but oh well. > > 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_ You are of course free to call it whatever you want. I do not call it that however. Has my repeated rejection of the term "intuition" really gone unnoticed by you, or are you just teasing? > 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. > But AK would do > so and you'd do the same or at least support such a claim. I am a poor judge of what AK would say. For myself, I am unclear enough about what you mean when you say "intuition" to be able to answer you. 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. 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. Also recall that the fact that all the axioms of PA are true for the naturals is a model-theoretic proof that PA is consistent. > Oh I learnt the naturals numbers the same way mathematicians > would learn: it's an intuitive notion! The above sentence does not signify anything clearly enough for me either to agree nor to disagree. Every time I engage with you in any least substantive way, I end up feeling like I have wasted my time. Not sure why I do it. I really would be better served just to laugh at your silly ideas and be done with it. Marshall
From: Marshall on 12 Apr 2010 02:13 On Apr 11, 10:43 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Apr 11, 9:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> 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). > > > In other words, what a sentence means is "relative" to > > what meaning we choose for the symbols it is composed of. > > NO. No? You don't think the meaning of a sentence depends on the meaning of the symbols it contains? Really? > Semantics and truth are 2 _different notions_! Is there any relationship between them at all? > And I'm taliking about "the relativity of mathematical truth". Yes you are. That's what's so funny. Marshall
From: Alan Smaill on 12 Apr 2010 07:12
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? > 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? -- Alan Smaill |