Re: Skolem Again


Chris Menzel wrote:

> But it's a philosophical problem having to do with a *technical* result
> in mathematical logic. The ontological status of photons is a
> philosophical problem as well; would you expect to be taken seriously if
> you were to take up that problem with no technical competence in quantum
> mechanics, and were not even able to provide any more than a layperson's
> understanding of a wave? If so, well, I marvel at your hubris. If not,
> what's the difference in this case?

Three differences. 1 this is a usenet group, not a peer-reviewed
journal. Just look around. 2 I don't know about photons or waves,
but they sound like complex concepts, rather than foundational ones. 3
The foundational concepts of mathematics are (since Frege) very close
to those studied by philosophers. Namely reference, predication,
truth, generality, existence and so on.

> Well, P2 is scarcely grammatical, but let's focus on P3.

What is ungrammatical about "The same theory must have the same
meaning"? Absolutely standard English. Meaning, for those who prefer
it expressed that way, if x is a theory and y is a theory, and x=y,
then x has the same meaning as y. I didn't translate the 'must', but
you get the drift. Note I altered this later in order to remove the
word 'meaning'.

In any case, as you now recognise, the disagreement is bound to be
about P3.

> From the fact
> that a given interpretation of a theory makes all of its axioms true it
> does not follow that it can properly be considered the theory's meaning.

But if grasp of meaning is grasp of truth-conditions, this is terribly
difficult to avoid.

> Some of its objects may, for example, have properties that we clearly
> do not intend. Hence, the model can't be what we mean by our axioms.

In what sense do the objects have properties we "don't intend"? If the
objects all fit the interpretation, as they clearly all do, how on
earth can this happen? And how can a model be what we 'mean' by our
axioms?

Do you mean, a given interpretation may result in "x is uncountable"
being true of some set of things of which it is 'really' false? But
what does 'really' mean here? Suppose jake says he has a dolphin. He
says it has four legs, a shaggy coat, and it barks. Ah, you say, he
'really' means a dog.

But did say that it was a dolphin? No, he said it was a dog.

It's really a dog, though he said it was a dolphin. But of course he
didn't say it was a dolphin. He said it was a dog. That's because the
meaning of 'dolphin', in his language, is a dog. It had to be,
because, the truth conditions of 'Fluffy is a dolphin', in his
language, are that Fluffy is a dog.

And in what sense can objects have properties we don't intend. Did
Jake intend that Fluffy have fins and a blowhole, and make squeaky
noises? No. And would he be surprised that Fluffy has legs, and
barks? No again.

.

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