FOM: The meaning of truth
charles silver
silver_1 at mindspring.com
Mon Nov 6 09:52:37 EST 2000
J. Shipman:
>> > IF a mathematician accepts {0,1,2,3,...} (along with the
>> > operations + and *) as a *well-defined* and *determinate* object
(which is
>> > all I mean by ontological commitment), THEN he has no trouble
making sense out
>> > of "GC is true and GC is not provable" because "true" means
true-in-the-model
>> > [{0,1,2,3,...},+,*].
V. Kanovei:
>> This thesis is absolutely similar to the following:
>>
>> (a')
>> IF a zoologist accepts unicorn as a *well-defined* and
*determinate* object
>> THEN he has no trouble to admit that unicorns should habituate yet
unknown
>> continent in the Earth,
I'm a tiny bit sympathetic with what I take to be Kanovei's view,
but unfortunately he seems unable to develop arguments for his
positions, preferring instead to joke about unicorns and the like. I
believe in this context he may be objecting to the claim that there's
a *model*, the so-called "standard model", which has certain
properties that can be ascertained. I suppose he's saying that this
(or possibly any) model is really just a fictional entity, somewhat
like a unicorn.
It seems to me there's a point here, but I don't know exactly
where to locate it (and unfortunately, Kanovei doesn't seem very
helpful). Suppose someone unaware of Gödel's results wanted to prove
that (first-order) PA has a model. We certainly would not accept the
following "proof":
"Just take the 'standard model,' or, if the term 'model' in this
context is balked at, call it the 'standard structure.' It is
easily seen that every sentence of PA is true in this
structure. Therefore, PA is consistent. QED."
Of course, the above "proof" is faulty. But, in establishing
that there's at least one sentence, G ("This sentence is unprovable"),
that is true but unprovable, this same model is alluded to. The
model is singled out in order to establish what it is that the
sentence G is true of. I am imagining that Kanovei objects
to this reference to the "standard model" (as being similar
to referring to unicorns), yet this reference is needed to
establish the truth of G.
I would like Kanovei to address whether the above loose argument
characterizes one aspect of his objection to the notion of Truth, as I
would like to better understand his view. I would also be interested
in refutations of this loosely stated argument, in order to better
understand precisely why it fails--supposing that it does.
Charlie Silver
More information about the FOM
mailing list