'When Are Relations Neither True Nor False?
in [Math]
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From: Daryl McCullough on 10 Mar 2010 11:16 Nam Nguyen says... > >Daryl McCullough wrote: >> Give an example of a nontrivial theorem in such a system. I don't >> think anyone would be interested in it, not even you. > >How about ExAy[~(Sy=x)], in Q (in that edifice)? It's an arithmetic >theorem, got to be interesting, isn't it? > >"Interesting" is subjective and *not* logical/reasoning. Proving is >logical/reasoning. If you're eager to claim some truths and in the >process sacrificing the rigidity of reasoning via syntactical proofs, >what's the point? Syntactic reasoning is not a goal in itself. It's only interesting to the extent that one can argue that it is truth-preserving, and it answers questions that we care about. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 10 Mar 2010 11:18 Nam Nguyen says... > >Daryl McCullough wrote: >> Nam Nguyen says... >>> Daryl McCullough wrote: >>>> The beauty of mathematical proof is that you can be certain >>>> of the truth of a universal statement without checking every >>>> instance. >>> You meant as "certain" as the truth of GC or "There are infinitely many >>> counter examples of GC"? >> >> Neither. I mean certain as the truth of "every consistent theory has a >> countable model". > >How certain is that while you don't know exactly what the naturals >collectively is? I know exactly what the naturals are. I don't know how to answer every question about the naturals, that's not the same thing. If I am holding a locked box, I may be unable to answer the question: "What's inside the box?", but that doesn't mean that there is any ambiguity about which box we're talking about. -- Daryl McCullough Ithaca, NY
From: MoeBlee on 10 Mar 2010 12:35 On Mar 9, 10:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > What I mean is: What mathematical theorems can you prove? Come on, if > > I don't have to prove any mathematical theorems, then I wouldn't need > > to propose any principles AT ALL! > > You still could prove a lot of theorems under an edifice that complies > to these principles. Whatever you mean by an 'edifice', you're welcome to go ahead and specify your system and show the derivations of your theorems. > In fact if you take the current FOL and strip off > anything that are (or _depends on_) the naturals or models from the > role of _inference_ then you'd obtain an edifice that would conform > to these 4 principles (at least from a first glance at the matter). Ordinarily, we articulate the syntac of first order logic itself in terms of natural numbers (for example, that there are denumerably many individual variables and that predicate and operation symbols have arities, etc.). But you're welcome to propose a notion of a formal system that does not use natural numbers in its formulation. But I don't know what you mean by "models from the role of_inference". Nor would I know what you would mean by "models from the rule of inference". > Such edifice/system would be very much an Hilbert-rule-of-inference > syntactical system, where the concepts of models would play a back-seat Models have no seat, back or otherwise, in MERELY the syntax of first order logic. > role of _intuition_ but has _no final say on inference_. Models have no "say" in the pure syntax of first order derivability. > And in that > systems, all mathematical FOL theorems by rules of inference are still > mathematical FOL theorems by rules of inference: a lot of them; infinitely > many of them. Of course. Maybe you'll show us your system? > >> The knowledge of the naturals (which Godel's work and PRA is based > >> on, via recursion and truth of the relations of numbers) is uncertain, > >> not conservative, and too intuitive to be rigorous for reasoning > >> sake, which is why my 4 principles don't suggest or support it. > >> In fact, the 4 principles basically "dismantle" any foundation based > >> on knowledge of the natural numbers. The naturals then is just one > >> of infinitely models (if at all), nothing more nothing less, nothing > >> special, _nothing preferential_. > > > If you say primitive recursive arithmetic is "uncertain" then you'd > > have to show me that your principles are any more certain. Moreover, > > that your priniciples have any potency to prove very much of > > interest. > > We're talking 2 different matters here. One is a set of principles > and guidelines that you don't have to follow, but these principles > _don't assert_ "facts" or knowledge. Arithmetic of the naturals on > the other hand is a set of assertions, asserting what are supposed > to be knowledge or facts in the realm of abstraction, hence they (the > assertions) can be questioned as to whether or not they are certain, > and in this case there demonstrable facts that some arithmetic assertions > are very much questionable, hence they aren't certain. Then your principles, being merely normative (as far as I can tell from what you've said), are even more questionable than those of PRA. > Those who argue for arithmetic certainties would be at the disadvantage: > all it takes is a single reasonable doubt. And what reasonable doubt do you refer to? > As for your "[the 4] principles have any potency to prove very much > of interest", again, principles don't prove, but the system that would > conform to them that I've just mentioned above would still prove (by > rules of inference) a lot of "ordinary mathematics", like theorems > about group, e.g. Okay, I'll just sit tight waiting for your system. > >>> Moreover, for each > >>> principle that Nam announces (including those I may agree with), I > >>> don't see that he's provided a basis for THEM that is any less based > >>> on "intution" than is the basis for adopting the logical and very > >>> modest mathematical principles employed in the limited version of PRA > >>> mentioned by Koskensilta. > >> As I've mentioned above, any principles based on the knowledge of the > >> naturals would *not* be "modest" at all. For one thing, this knowledge > >> is rooted in intuition It is surely RELATIVELY modest. I mean, it's not much of a mathematical leap to consider that such things as "1+1 = 2" are true, especially in some operational sense as "If we have one stroke symbol followed by another stroke symbol then we have two stroke symbols". > > And your knowledge of your principles is rooted in what? > > One might be surprised, but these principles are rooted in facts: > facts that we don't know certain things. Oops. Now you're claiming knowledge of facts, even as you claim such facts as those expressed by PRA are dubious. Sorry, I'll choose for myself what is more dubious. I find "1+1 =2" not too dubious. Moreover, in another post I mentioned something about why I find certain of your Namian-normative principles to be ill conceived. > For example, it's a fact > that we *now* don't know the truth of the 1st order formula "There > are infinitely many counter-examples of GC". Truth of that formula in WHAT model? Aside from that, applying Namian ultra-skepticism, actually by all kinds of ultra-skeptical tacts, I could doubt your mentioned fact (in WHATEVER sense you mean 'truth of' that is not model specific). > On the other hand, if > you claim you know the naturals then it's your _burden to assert_ > one way or the other about the truth of that formula. Do you now > see why I said the knowledge of the naturals isn't "modest"? Hey, if you have sincere doubts as to "1+1=2" but think you've found other non-logical facts that are less dubious, then I wish you luck and I'll not try to convince you that "1+1=2". But meanwhile, you have not shown me any non-logical facts that I see as less dubious than, e.g., "1+1=2" and such that I can use them to formulate the syntax of theories and then also to derive certain theorems about theories. > >> and however appealing as a "suggestion" force, > >> intuition should never be a foundation of reasoning. The whole reason > >> why today we have (Hilbert style) rules of inference is because we've > >> never been able to completely 100% trust intuition. Intuition and > >> reasoning as as different as the heart and the mind: one should not > >> replace the other as far as "role" is concerned. > > > You completely skipped the part I wrote about starting premises. > > By now I think it should be clear to anyone that the 4 principles > are *not* > > > embodying premises and principles of reasoning as modest as the > > limited version of PRA mentioned by Koskensilta. > > and I've explained the reasons why. Fine. Then my original point stands. > >> For another thing, there are very strong indications that there are > >> formulas in L(PA) that we can't assign truth value to them, hence a > >> reasoning foundation based on the natural numbers would most likely > >> incomplete. > > > (1) A truth assignment for sentences of the langauge assigns truth or > > falsehood to EVERY sentence in the language. We've been over this a > > thousand times already. > > So? What _specific_ "truth assignment" are _you_ talking about? No SPECIFIC truth assignment. Rather ANY truth assignment. > What I talked about is specifically some formulas we can't assign > _arithmetic/natural-number_ truths, simply because we have an > incomplete definition of the naturals. So, we might have been over > this 1000 times but you don't seem to be able to recognize the > difference. I have no idea where you find an actual mathematical fault in the claim that the domain of a truth assignment for the set of sentences of the language is indeed (redundantly!) the set of sentences of the language. > > (2) For the purposes I've mentioned, it is not > > required that a theory be negation complete. Lack of negation > > completeness doesn't at all impugn PRA for the purpose of deriving > > certain theorems about systems of finite strings of symbols. > > Sorry, I don't really know what all this "negation complete" and > "doesn't at all impugn PRA" is about. Could you elaborate a bit > more? You mentioned completeness. In this context I would think you're referring to negation completeness (i.e., that for every sentence of the language either it or its negation is a theorem of the theory). And "impugn PRA" in this context refers to your arguments that PRA is not modest and epistemologically reliable enough to accept for the purpose of proving the incompleteness theorem. MoeBlee
From: MoeBlee on 10 Mar 2010 12:53 On Mar 10, 12:37 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > (1) Anyone who has studied mathematical logic already recognizes that > > a sentence that is neither logically true nor logically false has > > models in which it is true and models in which it is false. > > Did you yourself quantifying-ly inspect uncountably many models > of, say, PA to know that? What does "quantifying-ly inspect" mean? Anyway, I don't claim to have individually inspected uncountably many models. So what? > > (2) As far > > as calling one model 'the standard model', if you object to lack of > > neutrality, then we could just as easily refer to it as 'the blandard > > model' or whatever. > > I suppose so. > > > But what would be ridiculous would be to require > > every mathematician to be as interested in EVERY possible model as > > much as he or she is interested in certain particular models. > > You've missed the key point and that's why you don't seem to understand > the objection to lack of neutrality: each model has its own "peculiarity", > hence _each equally deserves_ the name "standard". We're *not* talking > requiring anybody anything! At least the 4 principles don't require so. It seems to me that you're requing that we recognize what certain things "deserve" or don't "deserve". Anyway, as to what "deserves" to be called 'standard', please just refer to my point above (to which you wrote 'I suppose so' and to the rest of my post. > > People > > focus on certain mathematical objects, questions, etc. for a variety > > of reasons. It is not, and should not be required that mathematicians > > promise to do what is not even humanly, not even finitely, possible to > > do, such as study EVERY SINGLE model with as much interest and > > attention as every other single model. > > But the concept of the naturals number is "not even humanly, not even > finitely, possible", don't you think? Why do people then force themselves > to study it? Of course the concept is humanly, finitely possible. In human language, I can give you finite specification as to what a natural number is. > > (3) Such matters as this kind > > of neutrality are not even themselves formal mathematical questions > > but rather heuristic and ... ethical questions. > > Iirc, I introduced the word "symmetry", not "neutrality". So I'm > not sure why the word "ethical" has to be here. I'm responding to TransferPrinciple there, not necessarily to your formulation. And DAMN, look what you just did - you BLATANTLY cut my remark to distort it: What I actually wrote: "Such matters as this kind of neutrality are not even themselves formal mathematical questions but rather heuristic and (I guess, to certain people, perhaps you) ethical questions." First, clearly I was speaking to TransferPrinciple, not you. Second, I only GUESSED and said PERHAPS that TransferPrinciple takes this matter in an ethical sense. That you wrote "..." there doesn't excuse that what goes in place of that ellipsis is CRITICALLY different from the way you represented my remark. Also, if, given the context and intent of TransferPrinciple's remarks, the word 'neutrality' is not apropos, then fine, I would easily withdraw it for merely 'symmetry'. But then, mutatis mutandis, still my remarks to him stand. > > In any given context, all two mathematicians have to do is agree as to > > which particular terminology, axioms, and rules are in play in that > > context. If some other mathematician wishes to call some other model > > the 'standard model' then all he has to do is make it clear that the > > terminology 'the standard model' now refers to such and such. > > So then you'd agree the following meta theorem: > > For any formal system T capable to carry basic notions of arithmetic > there's a formula that's false but not provable in T. > > is true? No, I'd never say that. What makes you think I'd say such a silly thing? > > Of > > course, this may make the discourse awkward (at least at first), but > > in principle there is nothing stopping it. > > Agree! Then from now on, when someone mentions 'the standard model for the language of PA' just take that as 'the blandard model of PA', which is to say <w S 0 + *>. Now, poof, all your problems with this are gone. MoeBlee
From: MoeBlee on 10 Mar 2010 12:54
On Mar 10, 1:40 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Nam Nguyen wrote: > > MoeBlee wrote: > > >> (3) Such matters as this kind > >> of neutrality are not even themselves formal mathematical questions > >> but rather heuristic and ... ethical questions. > > > Iirc, I introduced the word "symmetry", not "neutrality". So I'm > > not sure why the word "ethical" has to be here. > > My mistake: I did use the word "_neutral_" in a post. But the sense > I used the word had nothing to do with "ethical", so I don't know > the relevance of "ethical" here. Good, then I'll repeat what I just posted: And DAMN, look what you just did - you BLATANTLY cut my remark to distort it: What I actually wrote: "Such matters as this kind of neutrality are not even themselves formal mathematical questions but rather heuristic and (I guess, to certain people, perhaps you) ethical questions." First, clearly I was speaking to TransferPrinciple, not you. Second, I only GUESSED and said PERHAPS that TransferPrinciple takes this matter in an ethical sense. That you wrote "..." there doesn't excuse that what goes in place of that ellipsis is CRITICALLY different from the way you represented my remark. MoeBlee |