[FOM] Possible worlds
Timothy Y. Chow
tchow at alum.mit.edu
Wed Dec 20 11:47:34 EST 2006
Thanks to everyone for all the responses.
Stephen Pollard <spollard at truman.edu> wrote:
> If FLT were false, then no argument for it would be sound.
> If FLT were false, then Andrew Wiles would deserve eternal fame.
> Folks who feel that these statements have different truth values are,
> naturally, dissatisfied with semantics that assign them the same truth
This response comes closest to addressing my concern. I'm having
temporary trouble getting to the projecteuclid.org website, but I'll check
it out later.
Several others introduced the word "epistemic" into the discussion as a
proposed solution to my quandary, but I'm not convinced. I agree that on
a platonistic view, either FLT is true or it is false, and so "FLT might
be false" can only be meaningfully parsed as, "There is currently no known
proof of FLT." That is, "FLT might be false" is analogous to
"There might be life on Mars"
---a statement that is presumably either true or false but we just don't
know which. But isn't there another point of view (an intuitionist view?)
under which "FLT might be false" is more analogous to
"The international community might send a manned mission to Mars in
the next 50 years"
---which seems to involve a different sense of possibility, one that
doesn't seem to be "just" a matter of epistemology?
Another, somewhat fuzzier, argument that there "ought" to be a way to fit
"FLT might be false" into a possible-worlds framework is that (in my
opinion) the whole point of possible worlds is to create a microcosm in
which some hypothesis is true, so that you have a clear framework in which
to derive consequences of the hypothesis. This is really not that
different from what people are trying to achieve when they hypothesize
this or that mathematical assertion. Therefore it seems to me that if a
possible-worlds formalism fails to deal naturally and correctly with
mathematical statements, but has to handle them separately, it is a sign
that something may be wrong with the formalism.
The suggestion that some have made, of imposing some kind of limit on the
length or intricacy of the deductive consequences allowed (where the limit
depends on context, I suppose), also sounds promising.
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