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

poopdeville_at_gmail.com
Date: 01/28/05 

Date: 27 Jan 2005 18:01:01 -0800

tchow@lsa.umich.edu wrote:
> In article <1106823984.872929.137690@z14g2000cwz.googlegroups.com>,
><snip>
> >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.
>
> What I find curious about your account here is that I largely agree
with it,
> but draw different conclusions than you do from it. You don't think
the
> regress is infinite because in practice it bottoms out somewhere. I
agree,
> and describe the situation by saying that at the bottoming-out point,
I
> grasp your meaning *without* needing to ask for truth-in-a-model. I
grasp
> your meaning simpliciter. I don't see why you don't describe the
situation
> the same way.
>
> So for example, I assert AC. You ask, "AC in which model?" I take
this to
> be a symptom of the fact that you are skeptical about sets, so you
can't
> grasp AC simpliciter. I've bottomed out at AC; you're bottoming out
> somewhere else. But either way, at some point we know what's meant
without
> having to say "In what model?" So there's some other kind of notion
of
> truth/meaning coming into play there.

Now that we have a common language to work with, many of my claims are
uncontroversial. However, we've both brought some philosophical
baggage to the table. In my understanding of your position, you're
committed to the existence of sets.

>
> >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.
>
> But don't you at least need to know what symbols are, and what
syntactic
> rules are? Are you skeptical about rules? I suppose if you like
> Wittgenstein, maybe you are. But then how can you do any mathematics
> if you don't know what symbols, strings, and rules are?

Yes! We need to know what symbols are, and what (the relevant)
syntactic rules are to do number theory. But the syntactic rules
demonstrably don't pick out a unique interpretation, which is exactly
why we need to be careful when talking about truth.

<snip>
> >Whereas what can be
> >proven (and is thus true in all relevant models) is meaningful.
>
> Is it? You have to be able to communicate your proof. What if you
can't
> do that? USENET provides spectacularly good examples of how even the
most
> perspicuously transparent proofs fail to be accepted by everyone.
>

This is an interesting point. I'll have to think about it some.

> >But what makes you so sure *you* know what a set is if you
> >can't even communicate it?
>
> If indeed I couldn't communicate it, I would be worried. But I don't
have
> any trouble in practice communicating it. I experience no
communication
> difficulties when I talk with professional set theorists, logicians,
and
> mathematicians. It's only when talking with students (and on
USENET!)
> that communication difficulties arise, and there the difficulties
arise
> not because of lack of clarity of the notion of a set, but for other
> reasons.
>

I was perhaps dishonestly attributing to you an inability to
communicate what a set is. If you look in just about any book on set
theory, it'll say that sets are collections of objects or something
just as unsatisfactory. Pinter's "Set Theory" even states:

"Every axiomatic system, as we have seen, must start with a certain
number of undefined notions.... While we are free in our own minds to
attach a "meaning," in the form of a mental picture, to each of these
notions, mathematically we must proceed "as if" we did not know what
they meant. Now an "undefined" notion has no properties except those
which are explicitly assigned to it; therefore, we must state as axioms
all the elementary properties we expect our undefined notions to have."

If you can tell me what a set is, outside of a "collection" (the
more-or-less personal "mental picture") or "an element of a collection
which satisfies the membership relation" (the public notion that
doesn't pick out one of these mental pictures), that'd be great. :-)

> >I still feel that this division is artificial. A set of axioms for
FOL
> >mimics the discourse of a particular field.
>
> The set of axioms for a group does not mimic the *discourse* of group
> theorists. Group theorists will assert things like, "Every odd order
> group is solvable." Or "Every finite simple group is either a cyclic
> group of prime order, an alternating group, a group of Lie type, or
one
> of 26 sporadic exceptions." Or "There exist infinite groups that are
> finitely generated with finite exponent." None of these statements
can
> be expressed in the first-order language of group theory. Only
certain
> properties of groups---first-order properties---can be expressed in
> this language. The function of the group axioms is to define what a
> group *is*, and not to mimic *discourse* about groups.
I had not thought of this distinction.

'cid 'ooh