'When Are Relations Neither True Nor False?
in [Math]
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From: Transfer Principle on 9 Mar 2010 01:30 On Mar 6, 10:13 am, David Bernier <david...(a)videotron.ca> wrote: > Nam Nguyen wrote: > > I'm less interested for example whether or not, say, PA is consistent > > but I'm interested in based from what existing and historical reasoning > > backgrounds and by what methods one would logically conclude - with no > > emotion or "belief" - PA is consistent - or not. > There's an ultrafinitist named Nelson, who rose > in professorial (USA) ranks to Associate Professor > of Mathematics or higher. That's a sign > of a definitely competent mathematician, generally > speaking. For years, PA and/or ZFC were ok > for him. Now, his belief is that PA *might* > be inconsistent. Ah yes, Ed Nelson, one of my favorite mathematicians. (Indeed, notice that my username refers to the Transfer Principle, a schema from one of Nelson's theories.) In one of his works, Nelson describes an attempt to prove that PA (and hence ZFC) is inconsistent. The proof relies on the use of large numbers -- numbers so large that one can't prove that they are numerals (of the form SSS...SSS0) without resorting to the Induction Schema. It would be freakin' _hilarious_ if Nelson's attempt to prove PA inconsistent were successful. The standard theorists would then be _forced_ to come crawling back to the ultrafinitists like RE, Y-V, and others, if they want to find a theory that would survive his inconsistency proof!
From: Transfer Principle on 9 Mar 2010 02:06 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. > 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. 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. I believe that it's an accident of mathematical history that ZFC is the standard theory, rather than one which proves ~F. 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, 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," just as those who try to use theories which prove ~F or argue in favor of ~F are called "cranks" here. The existence of such a planet is completely hypothetical, of course (yet, as I wrote earlier, the standard theorists would be the _first_ to bring up aliens who call red "blue" if a "crank" was the one who wrote "the sky is blue"). 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. And these alternate theories which Nguyen states exists by his Principle of Symmetry are the theories I'd like to investigate.
From: Jesse F. Hughes on 9 Mar 2010 07:42 Transfer Principle <lwalke3(a)lausd.net> writes: > The truth is, most people -- standard theorists and "cranks" > alike -- point out errors or obscure counterexamples only when an > opponent makes the statement, even though the same statement would > go unchallenged if an ally makes the statement. That's true in many endeavors, but this is not my experience in mathematical discussions. In typical math discussions, a sensible person is unconcerned with friends and enemies and instead corrects mistakes on both sides. > It's simply human nature -- I admit that I am guilty of the same. Oh, I'm sure you are. This is part of your own idiosyncratic view of intellectual disagreements: there are good guys and bad guys in every argument. -- "The needs of the many outweigh the needs of the few [...] I must make the same choice as those who came before me without regard to the impact today, for the sake of the children of humanity, the children of tomorrow." -- JSH channels Spock and generic politicians everywhere
From: Jesse F. Hughes on 9 Mar 2010 07:48 Transfer Principle <lwalke3(a)lausd.net> writes: > It would be freakin' _hilarious_ if Nelson's attempt to prove > PA inconsistent were successful. The standard theorists would > then be _forced_ to come crawling back to the ultrafinitists > like RE, Y-V, and others, if they want to find a theory that > would survive his inconsistency proof! You're absolutely nuts. I have no opinion on Y-V, but no, Russell Easterly is no expert on ultrafinitism. Mathematicians, if faced with an inconsistency in PA, would not turn to a mathematical incompetent like Easterly. (To be sure, compared to many other posters, Russell is reasonably lucid and coherent. Nonetheless, he has shown a clear inability to distinguish valid arguments from invalid.) -- "He isn't capable of actually defining his terms, or axiomatizing them, or deriving consequences from them. The kindest course of action is to humor him[...]Just pat him on the head and say 'Tony, aren't you the cutest little mathematician!'" -- Daryl McCullough on Tony Orlow.
From: Daryl McCullough on 9 Mar 2010 09:17
Jesse F. Hughes says... > >Transfer Principle <lwalke3(a)lausd.net> writes: > >> The truth is, most people -- standard theorists and "cranks" >> alike -- point out errors or obscure counterexamples only when an >> opponent makes the statement, even though the same statement would >> go unchallenged if an ally makes the statement. > >That's true in many endeavors, but this is not my experience in >mathematical discussions. In typical math discussions, a sensible >person is unconcerned with friends and enemies and instead corrects >mistakes on both sides. That's mostly true. But there are gray areas. 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. -- Daryl McCullough Ithaca, NY |