'When Are Relations Neither True Nor False? [Logic]

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From: Nam Nguyen on 11 Mar 2010 21:49

Daryl McCullough wrote:
> Nam Nguyen says...
>
>> Daryl McCullough wrote:
>
>>> I know exactly what the naturals are.
>> Don't say that out loud! Only super-natural being could have a
>> chance to know "exactly" what the naturals are!
>
> That's not true. We know exactly what the naturals are.
> Something is a natural if it is either 0 or is obtained
> from 0 by a finite number of applications of the "successor"
> operator.

Given the axiom:

Ax[Sx=0]

does *not* contradict your statement. So, we know the set of
the naturals then!

>
>>> I don't know how to answer
>>> every question about the naturals, that's not the same thing.
>> To know something, logically speaking, is to know everything,
>> while not to know something, not knowing only part of it is
>> sufficient.
>
> You can't know something unless you know everything? That's
> a pretty bizarre belief of yours. We know what the naturals are,
> we don't know every true fact about the naturals.

"know everything" about that something? Yes. You seemed to be surprised?

From: Nam Nguyen on 12 Mar 2010 00:12

Marshall wrote:
> On Mar 11, 6:39 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>> Marshall wrote:
>>
>> Why? Do you have specific technical reasons to back you up?
>
> I do, but they involve logic and reasoning,

What are those specific technical reasons that you _actually_
used to back you up?

From: MoeBlee on 15 Mar 2010 17:11

On Mar 11, 12:00 am, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Mar 9, 8:37 am, MoeBlee <jazzm...(a)hotmail.com> wrote:
>
> > On Mar 9, 12:21 am, Transfer Principle <lwal...(a)lausd.net> wrote:
> > > if a known so-called "crank," let's say
> > > JSH, were to state that the sky is blue, the _standard theorists_
> > > would be the ones to start coming up with obscure counterexamples
> > > such as the Doppler effect at velocities approaching c, alien
> > > languages in which "blue" means "red," and so forth.
> > "The standard theorists" would do that? How do you know? WHICH
> > "standard theorists"? And would you please say exactly what you mean
> > by "a standard theorist"?
>
> Fine, the non-"cranks" or anti-"cranks," then. Or perhaps the word
> often used by galathaea is more appropriate here -- "bullies."

I like 'crankbusters'.

Anyway, so you've grossly overgeneralized again. I assume you count me
an anti-crank. But, contrary to your claim, I am not inclined to
captiously dspute cranks when they happen to be correct.

> > > Case in point -- in a thread in which the standard theorists
> > > demanded that a "crank" accept Cantor's theorem as beyond dispute,
> > > I mentioned that there are some statements, such as 2+2=4, which,
> > > unlike Cantor's theorem, I do accept as unequivocally true. Then
> > > a standard theorist immediately brought up 2+2 == 1 (mod 3).
> > (1) I'd like to see the full context of that.
>
> OK, I found a couple of old posts with examples of this behavior.

> One year ago almost to the day, the morning of St. Patrick's Day of
>
> last year, I wrote:
> > > Let's start by considering the following three statements:
> > > 1) 2 + 2 = 4 (mentioned by Ullrich in another thread).
> > > 2) There are exactly five Platonic solids (Chandler).
> > > 3) 0.999... cannot equal any number other than 1.
> > > Obviously, all three statements are true in standard mathematics.
>
> Then the standard anti-"crank" Brian Chandler responded (the same
> day, approximately 5AM Greenwich time) with:
>
> "No. All three statements are true if we mean by them 'the usual
> thing'. Nothing to do with 'standard mathematics', within which all
> three statements can also be false, if we are talking about "unusual
> things": arithmetic modulo 3, 4D space, some properly formalised
> system that hasn't been presented yet, respectively. "
>
> The original context was that the "crank" MR was trying to discuss
> the possibility of 0.999... being unity minus a nonzero infinitesimal.
> So
> I argued that MR was being called a "crank" because he believes that
> "0.999... = 1," so there were some statements, like "0.999... = 1" and
> "2+2=4," which standard theorists will defend against "cranks." Then
> Chandler made the above reference to "arithmetic modulo 3," implying
> the equation 2+2=1 rather than 2+2=4.
>
> It's hard to tell what Chandler's point was.

It's not hard for me. And I don't see it as an example of someone
merely captiously disputing a correct claim happened to have been made
by a crank.

> Perhaps he was trying to
> make an argument similar to the one MoeBlee makes below:
>
> > People are not (ordinarily) called 'cranks' merely for proposing
> > alternative theories.
>
> so that MR wasn't being called a "crank" just for arguing 0.999... <
> 1, in
> the same way that someone who contradicts "2+2=4" isn't called a
> "crank" since there's a ring Z/3Z in which "2+2=1."

No, that's no my specific point. I've made my point ALL-CAPS GIANT
BILLBOARD SIZE clear over and over already. I can't help that you
ignore it over and over again.

> Of course, I don't
> agree with this, since almost every time a sci.math poster writes the
> inequality 0.999... < 1, out come the "bullies." So the standard
> theorists
> may claim that people aren't called "cranks" just for writing 0.999...
> < 1,
> but the evidence in sci.math posts doesn't support that claim at all.

Okay, I'll try ONE MORE TIME: What is crank is not the particular
mathematical claims, but rather the WAY in which they argued along
with certain other behaviours reflecting irrationality, ignorance, and
intellectual dishonesty.

> An older post in which 2+2=4 leads to a modular reponse is occurred
> back on the morning of 19th November 2008. I had written:
>
> > > I don't believe that having a
> > > set theory in which, say, Cantor's theorem
> > > fails is equivalent to having a set theory in
> > > which 2+2 is anything other than 4.
>
> And at 7:32 AM Greenwich time that morning, the standard anti-"crank"
> Denis Feldmann gave the following response:
>
> "Your beliefs are of no concern to me. You are wrong, period. If you
> believe you are right, please exhibit such a theory, or, better,
> solve
> the halting problem (this is equivalent to Cantor thoerem [sic],
> actually)
> Learn to read. And, BTW, in Z/4Z, 2+2=0"
>
> In this case, the so-called "crank" tommy1729 was trying to come up
> with an alternate theory with a set V of all sets, which obviously
> can't
> adhere to Cantor's Theorem.

You sure? You're sure that "~(Cantor's theorem & ExAy yex) is a
logical truth?

> Once again, I compared Cantor's Theorem
> to 2+2=4 in that standard theorists would defend them. Here Feldmann
> argued that Cantor's Theorem is even _ more_ solid than 2+2=4, since
> in the ring Z/4Z, 2+2=0, but Cantor's Theorem is equivalent to the
> _halting problem_, and of course I can't solve the halting problem (by
> producing a program/Turing machine that takes another program as input
> and determines whether it halts or not). So in a way, Feldmann implies
> that Cantor's Theorem has even _more_ empirical evidence supporting it
> than 2+2=4 does.

> At the time, I hadn't learned about NFU yet, or otherwise I would have
> told him that NFU is an example of a theory in which Cantor's Theorem
> actually fails. But then, what about the halting problem? One would
> think that simply switching from ZFC to NFU would make the halting
> problem suddenly solvable.

I don't wish to opine on his view or your interpretation of what he
wrote, but so far I don't see an example of someone merely captiously
disputing a correct claim happened to have been made by a crank.

> It could be that when Feldmann wrote:
>
> "the halting problem (this is equivalent to Cantor thoerem [sic],
> actually)"
>
> he meant that _ZFC_ proves that they are equivalent

They're both theorems of Z set theory, so, of course, quite trivially
they are equivalent in Z set theory.

> -- but they aren't
> necessary equivalent in another theory such as NFU. So it possible
> that
> Cantor's Theorem fails in NFU without solving the halting problem.

> > (2) So because one
> > "standard theorist" said such and such in one instance, then you
> > conclude that "standard theorists" (whatever you mean by that)
> > generally say such and such?
>
> Counting the Z/4Z example as well as the Z/3Z example, that's _two_
> standard theorists (anti-"cranks"/"bullies") who said that. In one
> case,
> the poster was trying to argue that merely proposing an alternate
> theory
> doesn't make one a "crank," while the other was trying to argue that
> contradicting Cantor's Theorem _does_ make one a "crank," even while
> contradicting simple arithmetic doesn't.

First, I don't take either as examples of what you were first talking
about. Second, even if they were, from TWO examples (or even a whole
bunch more out of the many crankbusters) you generalize as to
crankbusters in general. Personally worse for me, you include me in
those generalizations.

> But what I've never seen is one standard theorist declaring "2+2=4"
> and
> a fellow non-"crank" mentioning Z/3Z or Z/4Z as counterexamples.

The examples you cited were about possibilities of non-logical
statements being theorems in some theories and not theorems in other
theories. That's not akin to "2+2=4" being true in the standard model
of the language of PA or even as finitistically true statement given
the ordinary understanding of the '2', '4'. '+' and '='.

> That's
> why I gave the generalization (which might be viewed as a "lie") that
> standard anti-"cranks" only mention counterexamples like Z/3Z and Z/4Z
> when an opponent is making claims like 2+2=4.

(1) That was not original claim. You didn't just claim that
crankbusters only cite alternative readings of mathematical terms, but
rather the more general claim (via a particular example) at the top of
this post.

(2) When I've said you "lied" about me, I wasn't even including such
generalizations about crankbusters but rather certain of your comments
about ME SPECIFICALLY. And what I mentioned as 'lies' certainly were
lies.

MoeBlee

From: MoeBlee on 15 Mar 2010 17:25

On Mar 11, 12:44 am, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Mar 9, 9:09 am, MoeBlee <jazzm...(a)hotmail.com> wrote:
>
> > On Mar 9, 1:06 am, Transfer Principle <lwal...(a)lausd.net> wrote:
> > > 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.
>
> Yet if someone who's been labeled a "crank" is the one to discuss such
> models, then he is criticized for not adhering to the standard model.

You're completely NUTS. Look in any textbook on mathematical logic or
on model theory, and you will find all kinds of discussion about non-
standard models. I mean, your claim here is just LUDICROUS. One is not
called a crank (or at least it would be rare to find) just for
mentioning non-standard models.

> > (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.
>
> Conversely, they shouldn't ridicule _every_single_ model just because
> the person who mentions it has been labeled a "crank." I'd be
> satisfied
> if standard theorists just gave _some_ constructive discussion of the
> alternate models.

Who ridicules MODELS? What an insane idea! A model is a mathematical
object. How would one ridicule a mathematical object?

And if you want to find discussion about all kinds of models
(standard, non-standard, schmandard, non-schmandered, whatever!) then
just look in a book on mathematical logic, model theory, or any of
probably THOUSANDS of journal articles about the subject.

> > 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,
>
> Actions speak louder than words. Repeating that people aren't called
> "cranks" for proposing alternate theories doesn't erase the numerous
> times I've seen people who post alternate theories called "cranks,"
> right here on sci.math, with my own eyes.

YOU DID IT AGAIN!!! You just SKIPPED the VERY point I just made.

Would you please LISTEN this time?:

The cranks weren't called 'cranks' MERELY for proposing an alternative
theory.

> > for their circular reasoning, for their poetic/imagistic rather than
> > rigorous use of mathematical terminology
>
> Does "poetic" refer to their use of English, rather than formal
> symbolic
> language, to describe concepts?

NO. It refers to virtually complete woozy ambiguity. It refers to
using suggestive CONNOTATIONS of mathematical terminology in
deductions (formal or informal deductions). It refers to using the
metaphorical suggestiveness of terms such as 'line' or whatever in
deductions as opposed to confining to the actual deductive capability
wrought from axioms and definitions. Also to ersatz terminology that
is not declared either primitive or defined but is used merely at the
whim of the creator of the terminology. ETC. with a bunch of points
I've made in numerous posts.

> To me, I find "<= is a total order"
> much easier to read and understand than:
>
> Axyz (x<=x & ((x<=y & y<=x) -> x=x) & ((x<=y & y<=z) -> x<=z))
>
> even if the latter is considered more "rigorous."

Wonderful. I have no problem with "<= is a total order" when spoken by
someone who is using that terminology in its ordinary sense or is
using that terminology in some special sense but with alternative
definitions back to some alternative or standard primitives.

MoeBlee

From: MoeBlee on 15 Mar 2010 17:33

On Mar 11, 2:18 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> MoeBlee wrote:
> > On Mar 10, 8:47 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >> 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?
>
> > Ha! (If your question is rhetorical, which it sure appears to be), you
> > just committed an obvious fallacy.
>
> Just so you know (and you should have), Daryl challenged me a straight
> forward task: "give *an* example of a nontrivial theorem in such a system"
> which he himself believed I wouldn't be interested in. I directly responded
> to him with a straight forward example and through question-style I informed
> him he was wrong since it's interesting to me, it being an arithmetic
> theorem in Q.
>
> If you interpreted that straight forward answer as an "obvious fallacy"
> then obviously you were incapable of comprehending a short conversation
> between people here.

What you did is to give an example of an uninteresting theorem in Q.
Of course, I'm taking your "got to be interesting, isn't it?" as
sarcasm, but even if not sarcasm, it's easy to counter just by saying,
"No, we don't take that theorem as particularly interesting. But
rather such theorems as as used for more profound scientific
calcululations or even of merely abstract but still fascinating
mathematical results."

> > Damn, you're not even LISTENING - as usual.
>
> Damn, your posting is as idiotic as ever!

Damn, you exceed as an unwitting ironist.

MoeBlee