Re: Moore on Skolem's Paradox

On 29 Sep 2005 11:26:04 -0700, "William of Ockham"
<d3uckner@xxxxxxxxxxxxxx> wrote:

>(I sent this twice as I'm not sure what "reply to author" actually
>does)

Right. If you're not sure what some button in some program
does the best thing is to just push the button and find out.
Aargh.

Evidently what it does is send email to the author. Please
don't do that.

>Ockham
>>A consequence of the theorem as I understand it, is that, if there is a
>>countable model for set theory as expressed in some language *English,
>>the reference of "the real numbers" as used in *English is different
>>from its reference in actual English.
>
>> Hmm, wait ... maybe you meant "model for
>> [set theory as expressed in some language]".
>
>Hurrah - I'll ignore your earlier comments. But you say
>
>> To which someone will probably reply that an
>> _interpretation_ of a formal language supplies
>> referents for the terms in the language. That's
>> correct, given a pair consisting of a formal
>> language plus an interpretation of it the terms
>> do have referents, those referents being elements
>> of the "universe" of. the interpretation But that
>> does not mean that terms in a formal language
>> _per se_ have referents, which is what we need
>> to make sense of your assertion.
>
>Of course, no terms of any language have referents per se. The
>expressions themselves are just noises. What we call "learning a
>language" is assigning a reference to all of the terms that we learn.
>So we learn that "hamburgers" mean hamburgers, "Socrates" means
>Socrates, "two"means the number two, and so on. An interpretation (as
>I understand) is the information which we must have about a sentence in
>order to understand it.

You really haven't been paying attention. I _know_ that that's
what you understand an interpretation to be. That's one of
the reasons I say you simply don't understand what you're talking
about - when we say things like "an interpretation of set theory"
that is simply _not_ what the word "interpretation" means.

>So by "set theory as expressed in some language *English" I mean the
>language+ the interpretation given to it as a part of learning the
>language itself.

If we regard set theory as an axiomatic system, as I do, then
there simply is no such interpretation of the terms.

When someone says you're wrong about something you can't
necessarily conclude that he didn't understand what you
meant. Another possibility is that he did understand what
you meant, and he's asserting that what you meant is false.

> In that case, the reference for "the real numbers",

[Speaking of English, this locution of your has been
bugging me, in a fingers-on-blackboard sort of way. I've
tried to hint at a correction by just expressing things
correctly in my replies. To be explicit: If A refers
to B then A is a reference to B. B is not "the reference
of A", B is the referent of A.]

>given the expression with its meaning (reference) in that language, has
>to be different from the reference of "the real numbers" as used in
>actual English (assuming "English" includes the meaning of all the
>English words as normally learnt and used, as is the convention in
>philosophy).

As far as I can see from various things you've said I should
read (and other things I've read over the years), a convention
in philosophy is to take a term used in a technical field like
mathematics and assume that what it means in mathematics is
the same as what it means in ordinary English, ignoring the
fact that the term has a specific technical meaning in mathematics.
This is a necessary step in showing that various true facts
are false.

>Is that a problem? Well, I don't see that L-S rules out the
>possibility of a world where mathematics was learned in exactly the
>same way as this world, but where the interpretation involved a
>countable number of objects only (perhaps because only countably many
>objects were available).

This is just silly. When Keith posted a lengthy reply to something
you said you replied saying you agreed with most of his points,
and then said something like this, indicating that you'd totally
missed what seemed to me to be the main point to his post. Let
me try again:

In that imaginary world you're speaking of, where everything
is countable, mathematics is simply nothing whatever like
mathematics in our world. If you want to imagine a world
where the sun is green and apples fall up, fine. But when
you imagine a world where the proof of the existence of an
uncountable set is invalid you're just not making any
sense.

We can imagine a world where there does not exist a language
in which the proof can be expressed. That has no bearing
on anything, any more than the fact that we can't teach
a chimpanzee the proof shows that it might be wrong.

Maybe what you have in mind is a world where the proof
is valid but nonetheless everything is "really"
countable. There is no such world. Elsewhere you say
that mathematical truth should be the same in any
imaginary world. That's correct. Then you say "but
what about a world where..." - your suggesting that
we imagine a world that cannot exist does not show
that mathematical truth could be different, the fact
that mathematical truth cannot be different shows
that there simply is no such world.

But what if the world was a L-S countable model
of set theory? Uh, no, that makes no sense. A
model of set theory is not a "world", imaginary
or otherwise. It's at most part of a world
(please don't ask me exactly what that means -
in my view any talk of imaginary worlds where
things are other than as they are is incoherent.)

I mean it really seems a little too easy. No matter
_what_ anyone says about _anything_, you're free to say
"but what about an imaginary world where this is false!"
The fact that you can _say_ that does not mean that
you can in fact refute anything that anyone can say
about anything - we're a little suspicious about the
validity of such an awesomely powerful method of proof.

>Nothing about the learning process seems to rule this possibility out.
>In what sense are the referents of mathematically referring expressions
>like "2" accessible to us? How do we learn what "2" refers to? Godel
>thought we have a mental faculty of direct intuition for numbers. But
>this (as Graham Priest has argued) seems to fall foul of the Private
>Language Argument in one way or another. If the fixing of reference is
>performed solely by subjective mental acts, then there is nothing to
>prevent each of us fixing reference in a quite different way, which is
>another way of saying that (in respect of our public language),
>reference is not fixed at all.

This seems extremely silly, at least in the present context. The
fact that we can never be certain that what we mean by a term
is the same as what someone else means by the same term has
nothing to do specifically with the existence of uncountable
sets or Skolem's "paradox" - again, it's the sort of thing
you could use to refute anything anyone said about anything.

In particular, I have not asserted that anyone actually does
know what I mean by anything I say. It seems awesomely clear
that you don't know what a lot of us mean by a lot of things,
for example. If you actually learned the math that might
help.

Yes, the supposed existence of a non-ignorant logician,
who of course turned out not to be a logician at all,
who feels that the LS theorem does have some significance
regarding the existence of infinite sets does refute the
notion that if you learned what you were talking about
you wouldn't speak so much nonsense. But you might note
that a statement is not the same as its converse: even
if such a logician exists, that has no bearing on the
fact that someone who _is_ ignorant of the meaning of
the basic terms in a field is not likely to have the
ability to make sensible statements about that field.

>The language of set theory is, after
>all, a public and shared language. So the the criteria for
>reference-fixing must equally be public.


************************

David C. Ullrich
.

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