Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

poopdeville_at_gmail.com
Date: 01/27/05 

Date: 27 Jan 2005 03:06:24 -0800

Our posts are quickly becoming very long. I apologize for the lack of
snipping.

tchow@lsa.umich.edu wrote:
> In article <1106723962.638685.9760@f14g2000cwb.googlegroups.com>,
> <poopdeville@gmail.com> wrote:
> >Well, given that V is a model for ZFC -- Zermelo-Fraenkel with
Choice
> >-- V |= AC is the fact that V satisfies AC.
>
> But how do we know that V is a model for ZFC? V is just the class of
> all sets. People only say that V is a model of ZFC because they
> believe that the cartesian product of a nonempty family of nonempty
> sets is nonempty. If you didn't believe that, you also wouldn't
> believe that V is a model of ZFC.

Well, you claimed that V was a model for ZFC when you wrote:

"When I ask whether AC is true, I'm asking whether it's
true in V, the class of all sets, which is a [proper class] model of
ZFC."

I just assumed you had some technical knowledge I didn't, since you've
obviously studied this much more than I have.

<snip>

> First of all, the way mathematicians *in fact* use the word "true"
> violates your norms. They may be guilty of a philosophical
transgression,
> but if you want to understand what they're saying, you have to get
used
> to this usage whether you like it or not. I think this is how the
whole
> (sub)thread went flailing off in the first place: You just weren't
used
> to normal mathematical talk.
>

Of course. I addressed this in the paragraph you snipped. However,
normal mathematical talk is still quite confused. Here I refer to your
(and Jeffrey Ketland's, but mostly his) talk of disquotational schemes,
with emphasis of Jeffrey's use of models and the real world and your
group theory example at the end of your message.

> That aside, the more interesting question is whether mathematicians
are
> guilty of abusing language, or of implicitly adhering to a
"discredited"
> theory of truth, or of some other philosophical sin.
>

I think mathematicians make outrageous ontological claims when doing
philosophy. But in most cases the result of their mathematical work is
going to be independent of their philosophical biases.

> The major problem I see with your insistence that truth makes sense
only
> relative to a model is that the infinite regress that we briefly
discussed
> earlier becomes vicious. I assert a sentence S. You claim not to
> understand what S means unless I say which model it's true *in*. So
I
> assert the sentence S' = "S is true in model M." But there's no
reason
> to stop here: Why not challenge S' as meaningless unless you say
which
> model it's true in? Then when I try to assert S'' = "S' is true in
M'"
> ...well, it's obvious where this leads.

It's not vicious if you use new models to provide a foundation for the
old ones. The way I see it (philosophically, anyways) is that a model
is something like a linguistic explanation of meaning.

Consider the following example -- it's relevance will hopefully become
clear: You're walking on a forest trail, but it suddenly ends, right
in the middle of the forest. However, you see a sign with an arrow on
it: <-. Presumably, if you go in the correct direction, you'll find
the trail again. How are you justified in going left? Of course,
convention dictates that left is the correct direction. But suppose
you need to explain what the glyph "<-" means. Would you say "That
means left" or point in that direction? "<-" is just a symbol and must
be interpreted. And your explanation is too. If a symbol is
misinterpreted, and if one still wishes to explain what is meant, one
must provide a new symbol. (By induction, kaboom) Explanation piles
upon explanation until (hopefully) the listener understands. But he
might never understand. Does that mean that explanations are useless?

Clearly not. Just because there *might* be an infinite regress doesn't
mean that there *is* one. Explanations are very often successful in
communicating what we intend. The analogy here is that an explanation
provides a foundation for what it attempts to explain, just as each
term in my (possibly transfinite) sequence of models provides a
foundation for the previous one. Once you get what I'm driving at,
there's no need for any more.

> The only way I see of breaking out of this regress is to simply
assert
> by fiat that we know what certain sentences mean. We know what
symbols
> are, what rules are, what integers are, and so forth, so we know what
> we mean by "ZFC is consistent" simpliciter. We might not know
whether
> ZFC *is* in fact consistent, but we know what it would *mean* for it
to
> be consistent.
>
> In a sense this "assertion by fiat" is arbitrary. I'm arbitrarily
saying
> that I know, for example, that I know what it means for a proof to
have
> "finite" length, and that I know what the standard integers are and
that
> they're not the nonstandard integers. However, this is the only way
that
> I see to get off the ground, and symbols, rules, integers, etc., are
about
> the simplest mathematical objects around; if we can't assume that we
know
> what *they* are, then we're basically taking the defeatist position
that
> doing mathematics is impossible. Therefore I am not bothered by the
> arbitrariness.

You don't need to know what numbers are to do number theory. Or sets
for set theory. Frege struggled for years trying to prove that Julius
Caesar wasn't a number. But this wasn't for any mathematical reason,
just a philosophical one. I'm skeptical of your (and my) knowledge of
what a number is, but I don't claim we can't reason about them.

> The advantage of this point of view is that we can then develop
mathematics
> as usual. Once in a while we might get into factional arguments;
some
> people might feel that infinite sets are perfectly clear while others
> might not, and some might like the law of the excluded middle and
others
> might not. The general outlines, however, are the same. We have
some
> statements that we make, and we know what they mean. *Then* you can
go
> about developing mathematical logic, and analyzing mathematical
discourse
> by mimicking it with formal languages.
>

My view is also that mathematicians drive mathematics. However, the
existence of these factional differences implies that not all
mathematicians share the same intuitions about the sorts of objects
they're dealing with. In order to communicate their results with other
mathematicians, one has to give them enough information to evaluate the
truth of their claims. In intuitionist analysis, every function is
continuous. This is spectacularly false in classical analysis. But a
classical mathematician must accept the proof that intuitionist
analysis implies that every function is continuous since every step in
the proof is valid in classical logic. They're just playing different
games. If they want to play with one another, they need to communicate
their rules.

> One then discovers that if I mimic mathematical discourse with a
formal
> language, then that formal language admits different interpretations.
> This should not be a surprise; *of course* I can redefine words and
> sentences to mean anything I want them to.

Agreed.

> In particular, I don't see
> why I should suddenly lose faith in my original "arbitrary" meaning
of
> "ZFC is consistent" and doubt that I knew what I meant in the first
> place, just because someone else can make "ZFC is consistent" mean
> something totally different.

Because if you can't communicate your arbitrary meaning of "ZFC is
consistent," the phrase really is meaningless. Whereas what can be
proven (and is thus true in all relevant models) is meaningful. As is
a statement if evaluated with respect to a particular interpretation.

>
> What I've just said is typically convincing to many people, but those
> same people often balk at, say, AC or the continuum hypothesis. Then
> the independence results seem to make people want to say that AC or
CH
> is "meaningless" (or some such) unless you specify a model. However,
> I maintain that this is just because these folks didn't feel they had
> a clear idea of what sets were in the first place. Independence
results
> by themselves don't necessarily force this upon you, as the case of
> "ZFC is consistent" should make clear.

Perhaps. I have no issue *using* AC, but I certainly fall into the
categories you describe (assuming that "(or some such) is a modulo
relation) But what makes you so sure *you* know what a set is if you
can't even communicate it?

<snip>

> The case of group theory is different. We can use the same formal
> apparatus, but now we're not mimicking general mathematical
discourse.
> In general mathematical discourse we don't say "For all a, b, and c,
> (a*b)*c = a*(b*c)" and feel that we know exactly what we're saying
> without any explanation of what a, b, and c are or what * is. So
> then it doesn't make sense to ask whether the statement is "true"
> without further explication. But this doesn't undermine the previous
> discussion, because this is a different ball game.

I still feel that this division is artificial. A set of axioms for FOL
mimics the discourse of a particular field. But Intuitionism
demonstrates that even FOL + arbitrary axioms isn't enough to mimic the
discourse of an arbitrary field. In short, there is no "general
mathematical discourse," only instances of mathematical discourse.
Also note that by this standard it doesn't make sense to say that "AC
is true" without explication of what is meant by "true" -- truth
relative to V.

'cid 'ooh

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