[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 
> value.

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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