'When Are Relations Neither True Nor False?
in [Math]
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From: MoeBlee on 9 Mar 2010 12:09 On Mar 9, 1:06 am, Transfer Principle <lwal...(a)lausd.net> wrote: > On Mar 7, 3:37 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > I'd like to add another Principle: > > (3) Principle of Symmetry (of Non-Logicality): > > Other than concepts that are tautologous or contradictory (in > > the underlying reasoning framework), a concept and its negation > > are totally _symmetrical_ with respect to methods of reasoning, > > in the sense that if there's a context that the concept is so-and-so > > then there's another context of the same category of contexts that > > its negation is also so-and-so. > > For instance, Let a concept be semantically reflected by a formula > > F, thenif F is provable in a system then there's another system > > that ~F is provable; if there's a language model that F is true, > > then the same could be said of ~F; if there's a "standard" model > > that F is true, then there's another model that ~F is true and > > this model could be equally considered as a "standard" model; etc... > > The symmetry of F and ~F is very much "absolute" as the symmetry > > of the elements in a 2-element set: there can't be _no preference_ > > between the 2 elements in term of set membership; if one of them > > could be used as the first element of an ordered pair, then so > > could be the other one, e.g. > > I strongly agree with what Nguyen is saying in this post. > > Maybe Nguyen's formula F refers to Cantor's theorem, or the > existence of a bijection between N and Q, or some other common > statement that standard theorists like to defend against the > so-called "cranks" here because ZFC proves F. Then I'd like to > believe that there's a theory that's every bit as good as ZFC in > ways that matter most to standard theorists (including power and > ease of use), yet proves ~F. This is how I interpret Nguyen's > Principle of Symmetry. (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. (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. 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. 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. (3) 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. > > This Principle basically would absolve any "standard-ness" in reasoning > > and would pay the way for relativity in mathematical reasoning, imho. > > One reason that mathematicians declare one theory to be "standard" > is suppose every mathematician had his or her own theory. Then no > two mathematicians would agree on what is provable, and so > communication among mathematicians would be nearly impossible. WRONG. Complete misunderstanding of BASIC concepts. 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. Of course, this may make the discourse awkward (at least at first), but in principle there is nothing stopping it. Moreover, "standard theorists" (those who study ZFC?) OFTEN study other models rather than the standard model for the language of PA. Indeed, exercise after exercise after exercise is given in model theory to find different models. And non-standard analysis may itself be formulated in ZFC and is often studied within ZFC. > And so ZFC is usually taken as the standard theory. I don't mind > having ZFC be the standard theory -- I only mind when standard > theorists fail to consider the possibility of other theories and > act as if ZFC is the only theory that matters. Would you please name JUST ONE "standard theorist" who does that? Preferably, a professional "standard theorist" or even just a graduate student who is familiar with basic mathematical logic and set theory. Damn, I've been telling you for YEARS that is simply not true that "we" only recognize ZFC. You can SEE FOR YOURSELF that many people here who you would call "standard theorists" talk, not too rarely, about various theories other than ZFC. > I believe that it's > an accident of mathematical history that ZFC is the standard > theory, rather than one which proves ~F. In a sense, of course it is. Who do you think doesn't understand that? (On the other hand, there are also good mathematical and philosophical reasons why ZFC emerged on top.) > Going back to Nguyen's red/blue alien example from earlier, just as > there may be aliens for which the word "blue" means "red," there > may be a planet on which the theory which proves ~F is the standard > theory, How do you "prove" something is a "standard theory"? A theory is more or less standard by convention, not by proof. > possibly because mathematics developed differently on that > planet from how it did here. And on that planet, those who try to > use theories which prove F (possibly a theory similar to ZFC) are > called "cranks," For god's sake, we've addressed this a THOUSAND times. People are not (ordinarily) called 'cranks' merely for proposing alternative theories. Rather, they're called 'crank' for their irrational argumentation, for their unwillingness to define their terminology, for their circular reasoning, for their poetic/imagistic rather than rigorous use of mathematical terminology, for their stubbornness to admit points that have been clearly made to them, for their continual shifting of the terms of the argument, for their blatantly false and ignorant claims about the mathematics they don't approve, for their presumption that they can pontificate about mathematics that they haven't even studied page one, etc., or some combination of these. MoeBlee
From: MoeBlee on 9 Mar 2010 12:14 On Mar 9, 8:17 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > If someone is speaking > informally (which is almost always) there are many possible areas in > which what he says is literally not true, but it's usually not worth > correcting, because everyone can see that either the mistake is easily > corrected, and it's clear what was meant, or else the mistake is minor, > and irrelevant to the conclusion. An example of the sort of mistake > or sloppiness that people make informally is failing to distinguish > between a natural number and the corresponding numeral. > > In accepting an informal argument, there is almost always a principle > of charity at work: you assume that the speaker means something nonstupid, > if you can figure out what that is. You only bother to point out errors > or ambiguities if it is unclear what the speaker meant, or if the > error seems like it cannot be corrected without affecting his conclusion. Or perhaps just to make a correction for the record. But aside from my added nit, what Daryl wrote is right on the money and well said! MoeBlee
From: Nam Nguyen on 9 Mar 2010 23:32 MoeBlee wrote: > On Mar 8, 10:52 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: > >>> he does discuss some general principles later in his post >>> and in later posts. However, I don't find anything in his principles >>> that supply or even suggest a system that has any potency at all while >>> embodying premises and principles of reasoning as modest as the >>> limited version of PRA mentioned by Koskensilta. >> Of course by presenting those principles I already had in my mind >> the kind of potency a reasoning system (edifice) conforming to these >> 4 principles would have. And I think I've expressed that before in >> some forms. For one thing, such a system could be conducive to >> relativity in mathematical reasoning, which is more _realistic_ >> than the "absoluteness" kind of reasoning we'd see in PRA or what >> Godel used to prove his meta proof. For another, it'd make our >> reasoning _more conservative_ in the sense that if we can't possibly >> know some "truth" then we have to acknowledge that impossibility, >> rather than acting as if we could know; this is nothing more than >> the principle of "conservation of knowledge" in mathematical >> reasoning, so to speak. > > 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. 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). Such edifice/system would be very much an Hilbert-rule-of-inference syntactical system, where the concepts of models would play a back-seat role of _intuition_ but has _no final say on inference_. 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. > >> 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. Those who argue for arithmetic certainties would be at the disadvantage: all it takes is a single reasonable doubt. 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. > >>> 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 > > 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. 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". 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"? > >> 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. > >> 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? 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. > (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?
From: Nam Nguyen on 9 Mar 2010 23:38 Transfer Principle wrote: > On Mar 7, 3:37 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> I'd like to add another Principle: >> (3) Principle of Symmetry (of Non-Logicality): >> Other than concepts that are tautologous or contradictory (in >> the underlying reasoning framework), a concept and its negation >> are totally _symmetrical_ with respect to methods of reasoning, >> in the sense that if there's a context that the concept is so-and-so >> then there's another context of the same category of contexts that >> its negation is also so-and-so. >> For instance, Let a concept be semantically reflected by a formula >> F, thenif F is provable in a system then there's another system >> that ~F is provable; if there's a language model that F is true, >> then the same could be said of ~F; if there's a "standard" model >> that F is true, then there's another model that ~F is true and >> this model could be equally considered as a "standard" model; etc... >> The symmetry of F and ~F is very much "absolute" as the symmetry >> of the elements in a 2-element set: there can't be _no preference_ >> between the 2 elements in term of set membership; if one of them >> could be used as the first element of an ordered pair, then so >> could be the other one, e.g. > And these alternate theories which Nguyen states exists by his > Principle of Symmetry are the theories I'd like to investigate. This Principle (3) actually says nothing about "alternate theories".
From: Nam Nguyen on 10 Mar 2010 00:34
Transfer Principle wrote: > The point I'm trying > to make is that if ZFC proves a formula F, there's no reason why > there can't be another _dual_ theory, one which is just as powerful > and elegant as ZFC, which proves ~F. I wouldn't call that "another _dual_ theory": just "another theory". Also, "elegant" is a subjective notion, not a mathematical (FOL) notion. Principle 3. actually is just a generalized reflection of a known fact in FOL: if a formal system is consistent, then there exists a different consistent formal system, all of which has nothing to do with "elegance". |